Méthodologie de réalisation de modèles anatomiques maillés ...theses.insa-lyon.fr/publication/2008ISAL0034/these.pdfCNRS UMR 5558 Université Claude Bernard Lyon 1 Bât G. Mendel

Numéro d’ordre : 2008-ISAL-0034 Année 2008

Thèse

Méthodologie de Réalisation de Modèles Anatomiques

Maillés : Application à l’Imagerie du Petit Animal

Présentée devant

L’Institut National des Sciences Appliquées de Lyon

Pour obtenir

LE GRADE DE DOCTEUR

ÉCOLE DOCTORALE : ÉLECTRONIQUE, ÉLECTROTECHNIQUE, AUTOMATIQUE

FORMATION DOCTORALE : IMAGES ET SYSTÈMES

Par

Yasser KHADRA

Ingénieur Biomédical

Soutenance le 26 Juin 2008 devant la Commission d’examen

Jury

Rapporteur Fabrice MÉRIAUDEAU Professeur Université de Bourgogne

Rapporteur Jean-Charles PINOLI Professeur ENSM de Saint-Étienne

Examinateur Marc JANIER Professeur Université Claude Bernard Lyon 1

Examinateur Hugues BENOIT- CATTIN Professeur, INSA de Lyon

Directeur de thèse Christophe ODET Professeur, INSA de Lyon

Centre de Recherche et d’Applications en Traitement de l’Image et du Signal (CREATIS)

CREATIS-LRMN - CNRS UMR 5220 - INSERM U630 - Lyon, France

INSA DE LYON, DEPARTEMENT DES ETUDES DOCTORALE, Ecoles Doctorales 2008

SIGLE ECOLE DOCTORALE NOM ET COORDONNEES DU RESPONSABLE

CHIMIE

CHIMIE DE LYON http://sakura.cpe.fr/ED206 Insa : R. GOURDON

M. Jean Marc LANCELIN Université Claude Bernard Lyon 1 Bât CPE 43, bd du 11 novembre 1918 69622 VILLEURBANNE Cedex Tél : 04 72 43 13 95 [email protected]

E.E.A.

ELECTRONIQUE, ELECTROTECHNIQUE, AUTOMATIQUE http://www.insa-lyon.fr/eea Insa : D. BARBIER

M. Alain NICOLAS Ecole Centrale de Lyon Bâtiment H9 36, avenue Guy de Collongue 69134 ECULLY Tél : 04 72 18 60 97 - Fax : 04 78 43 37 17 Secrétariat : M.C. HAVGOUDOUKIAN [email protected] Secrétariat : M. LABOUNE Tél. (AM) 04 72 43 64 43 Fax : 04 72 43 64 54 [email protected]

E2M2

EVOLUTION, ECOSYSTEME, MICROBIOLOGIE, MODELISATION http://biomserv.univ-lyon1.fr/ E2M2 Insa : H. CHARLES

M. Jean-Pierre FLANDROIS CNRS UMR 5558 Université Claude Bernard Lyon 1 Bât G. Mendel 43, bd du 11 novembre 1918 69622 VILLEURBANNE Cedex Tél : 04 26 23 59 50 - Fax 04 26 23 59 49- Portable : 06 07 53 89 13 [email protected]

EDIIS

INFORMATIQUE ET INFORMATION POUR LA SOCIETE http://ediis.univ-lyon1.fr

M. Alain MILLE Université Claude Bernard Lyon 1 LIRIS - EDIIS Bâtiment Nautibus 43, bd du 11 novembre 1918 69622 VILLEURBANNE Cedex Tél : 04.72. 44 82 94 - Fax 04 72 44 80 53 [email protected] - [email protected] Secrétariat : I. BUISSON

EDISS

INTERDISCIPLINAIRE SCIENCES-SANTE Insa : M. LAGARDE

M. Didier REVEL Hôpital Cardiologique de Lyon Bâtiment Central 28, Avenue Doyen Lépine - 69500 BRON Tél : 04 72 68 49 09 - Fax :04 72 35 49 16 [email protected] Secrétariat : Safia Boudjema

Matériaux

MATERIAUX DE LYON M. Jean Marc PELLETIER INSA de Lyon - MATEIS Bâtiment Blaise Pascal 7, avenue Jean Capelle - 69621 VILLEURBANNE Cedex Tél : 04 72 43 83 18 - ax 04 72 43 85 28 [email protected] Secrétariat : C. BERNAVON Tél : 04 72 43 83 85

Math IF

MATHEMATIQUES ET INFORMATIQUE FONDAMENTALE Insa : G. BAYADA

M. Pascal KOIRAN Ecole Normale Supérieure de Lyon 46, allée d’Italie - 69364 LYON Cedex 07 Tél : 04 72 72 84 81 - Fax : 04 72 72 89 69 [email protected] Secrétariat : Fatine Latif - [email protected]

MEGA

MECANIQUE, ENERGETIQUE, GENIE CIVIL, ACOUSTIQUE

M. Jean Louis GUYADER INSA de Lyon - Laboratoire de Vibrations et Acoustique Bâtiment Antoine de Saint Exupéry 25, bis avenue Jean Capelle - 69621 VILLEURBANNE Cedex Tél : 04 72 18 71 70- Fax : 04 72 18 87 12 [email protected] Secrétariat : M. LABOUNE Tél. (PM) : 04 72 43 71 70 - Fax : 04 72 43 87 12

ScSo ScSo* Insa : J.Y. TOUSSAINT

M. BRAVARD Jean Paul Université Lyon 2 86, rue Pasteur 69365 LYON Cedex 07 Tél : 04 78 69 72 76 - Fax : 04 37 28 04 48 [email protected]

* ScSo : Histoire, Geographie, Aménagement, Urbanisme, Archéologie, Science politique, Sociologie, Anthropologie

1

Remerciements

J'exprime mes profonds remerciements à mon directeur de thèse, le professeur Christophe ODET,

directeur de formation de l’INSA de Lyon, pour l'aide compétente qu'il m'a apportée, pour sa

patience et son encouragement à finir un travail commencé il y a bien longtemps. Son oeil

critique m'a été très précieux pour structurer ce travail et pour améliorer la qualité des différentes

sections. J’ai apprécié également son aide précieuse pour les problèmes administratifs pendant

mon séjour en France.

Je remercie tous particulièrement Monsieur Jean-Charles PINOLI, directeur adjoint

chargé de la recherche à l’Ecole Nationale Supérieure des Mines (Saint-Étienne), ainsi que

Monsieur Fabrice MERIADEAU, directeur du Centre Universitaire de Condorcet, qui ont

accepté de juger ce travail et d'en être les rapporteurs.

Je tiens à remercier Hugues BENOIT-CATTIN, professeur à l’INSA de Lyon, directeur

du département télécommunications, d’avoir accepté d’être examinateur et président du jury. Je

tiens également à remercier Marc JANIER, professeur à l’université Claude Bernard Lyon-1,

directeur de la plate-forme ANIMAGE, d’avoir été examinateur.

Je voudrais également remercier tous mes collègues du laboratoire CREATIS-LRMN,

avec qui j’ai partagé ses années de travail, pour leurs soutiens, leurs disponibilités et leurs

sympathies. Je tiens également à exprimer ma sympathie à tous les membres du laboratoire,

scientifiques, techniques et administratif, qui, soit par leur aide, soit par leurs encouragements

ou tout simplement par leur amitié ont rendu mon travail plus facile et plus agréable. Je souhaite

remercier tout particulièrement l’équipe du 401: Chantal, Delphine, Sorina, Jérôme, thomas et

Jean-loïc pour leurs gentillesses, les bons moments, les conseils et l’aide qu’ils m’ont apportés.

Les moments passés ensemble resteront inoubliables.

A travers ce travail, je souhaite exprimer à mes parents, à l’ensemble de ma famille mon

affectueuse reconnaissance pour leurs encouragements et leur soutien.

Je remercie enfin celle qui est la plus proche de moi, pour son amour et sa compréhension,

son soutien morale de tous les instants, ma femme Abir. Je dédie cette thèse à mes filles Shahed

et Hala sans qui rien à ce jour n’aurait la même signification.

Yasser

2

Table des matières

3

Table des matières

REMERCIEMENTS....................................................................................................................................... 1

TABLE DES MATIERES .............................................................................................................................. 3

LISTE DES FIGURES.................................................................................................................................... 7

LISTE DES TABLES.................................................................................................................................... 13

LISTE DES ABREVIATIONS..................................................................................................................... 14

RESUME........................................................................................................................................................ 15

SUMMARY.................................................................................................................................................... 16

INTRODUCTION GENERALE.................................................................................................................. 17

I. CONTEXTE ET ETAT DE L’ART......................................................................................................... 19

INTRODUCTION A LA PREMIERE PARTIE ........................................................................................ 20

CH.1. ATLAS ANATOMIQUE............................................................................................................. 21

1.1 INTRODUCTION ............................................................................................................................... 22 1.2 L’HISTOIRE DE L’ATLAS ANATOMIQUE ........................................................................................... 22 1.3 ATLAS ANATOMIQUE NUMERIQUE ET LES METHODOLOGIES DE RECONSTRUCTION......................... 24 1.4 APPLICATIONS DES ATLAS ANATOMIQUES NUMERIQUES ................................................................ 26

1.4.1 La base de données.................................................................................................................... 26 1.4.2 Vers l’automatisation de la segmentation des images médicales .............................................. 27 1.4.3 L’assistance passive pour la thérapie........................................................................................ 29 1.4.4 L’analyse de la forme et des déformations ................................................................................ 29 1.4.5 Vers l’automatisation du diagnostic et de la thérapie ............................................................... 29

1.5 CREATION D’ATLAS ANATOMIQUE.................................................................................................. 30 1.5.1 Atlas anatomique de l’homme ................................................................................................... 30 1.5.2 Atlas anatomique du petit animal .............................................................................................. 31

1.6 CONCLUSION .................................................................................................................................. 35

CH.2. CONSTRUCTION DU MODELE MOYEN ENRICHI........................................................... 37

2.1 INTRODUCTION ............................................................................................................................... 38 2.2 PROBLEME DE RECONSTRUCTION.................................................................................................... 38 2.3 METHODE DISCRETE DE RECONSTRUCTION..................................................................................... 39

2.3.1 Segmentation et classification des données ............................................................................... 39 2.3.2 Extraction directe de modèle surfacique ................................................................................... 40

2.4 SEGMENTATION/RECONSTRUCTION PAR MODELE DEFORMABLE ..................................................... 42

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2.5 ATLAS NUMERIQUE MOYEN ............................................................................................................ 43 2.6 ENRICHISSEMENT DU MODELE MAILLE ........................................................................................... 47 2.7 CONCLUSION .................................................................................................................................. 49

CONCLUSION DE LA PREMIERE PARTIE .......................................................................................... 51

II. ANATOMICAL AVERAGE MODEL CONSTRUCTION METHODOLOGY................................ 55

INTRODUCTION TO THE SECOND PART............................................................................................ 56

CH.3. OVERVIEW OF THE PROPOSED METHODOLOGY ........................................................ 57

3.1 INTRODUCTION ............................................................................................................................... 58 3.2 GENERAL DESCRIPTION .................................................................................................................. 58 3.3 AVERAGE MODEL CONSTRUCTION .................................................................................................. 59 3.4 MODEL ENRICHMENTS .................................................................................................................... 60 3.5 CONCLUSION .................................................................................................................................. 61

CH.4. SURFACE REGISTRATION..................................................................................................... 63

4.1 INTRODUCTION ............................................................................................................................... 64 4.2 BACKGROUND ................................................................................................................................ 64

4.2.1 Definition................................................................................................................................... 64 4.2.2 Classification of registration ..................................................................................................... 66 4.2.3 Iterative closest point ICP ......................................................................................................... 67 4.2.4 Discussion.................................................................................................................................. 69

4.3 FEATURE POINTS-BASED SURFACE REGISTRATION .......................................................................... 69 4.3.1 Overview of the proposed method ............................................................................................. 70 4.3.2 Surface normalization................................................................................................................ 71 4.3.3 Feature Points Extraction ......................................................................................................... 73 4.3.4 Feature points matching............................................................................................................ 76 4.3.5 Final Registration...................................................................................................................... 78 4.3.6 Qualitative and quantitative evaluation of the registration results ........................................... 78 4.3.7 Results ....................................................................................................................................... 80

4.4 ELASTIC REGISTRATION ................................................................................................................. 82 4.5 CONCLUSION .................................................................................................................................. 85

CH.5. ANATOMICAL AVERAGE MODEL CONSTRUCTION ..................................................... 87

5.1 INTRODUCTION ............................................................................................................................... 88 5.2 GENERALIZED PROCRUSTES ANALYSIS GPA.................................................................................. 88

5.2.1 Affine average model construction (AAM) ................................................................................ 89 5.2.2 Elastic average model construction EAM ................................................................................. 91 5.2.3 Results and Discussions............................................................................................................. 92

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5.3 MORPHOLOGICAL-BASED AVERAGE MODEL CONSTRUCTION .......................................................... 96 5.3.1 Overview of the proposed method ............................................................................................. 96 5.3.2 Background of the voxelization ................................................................................................. 97 5.3.3 Background of 3D Morphology ................................................................................................. 99 5.3.4 Average volume extraction using morphological operators.................................................... 101 5.3.5 Model creation......................................................................................................................... 108 5.3.6 Results and Discussions........................................................................................................... 109

5.4 COMBINING GPA AND MORPHOLOGICAL-BASED AVERAGE MODEL EXTRACTION TECHNIQUES .... 115 5.5 CONCLUSION ................................................................................................................................ 116

CH.6. MODEL ENRICHMENTS ....................................................................................................... 119

6.1 INTRODUCTION ............................................................................................................................. 120 6.2 FEATURE LINES-BASED MODEL ENRICHMENT ............................................................................... 121

6.2.1 Introduction ............................................................................................................................. 121 6.2.2 Basic background of surface geometry.................................................................................... 122 6.2.3 Normal estimation ................................................................................................................... 125 6.2.4 Curvature estimation ............................................................................................................... 126 6.2.5 Feature lines on surface .......................................................................................................... 128 6.2.6 Feature lines extraction........................................................................................................... 129 6.2.7 Feature lines thresholding....................................................................................................... 130

6.3 DISTANCE MAP-BASED MODEL ENRICHMENTS .............................................................................. 134 6.3.1 Distance maps-based average surface enrichments ................................................................ 134 6.3.2 Distance maps-based average volume enrichments ................................................................ 137 6.3.3 Probability map ....................................................................................................................... 140

6.4 ADDITIONAL MEASUREMENTS ...................................................................................................... 141 6.5 CONCLUSION ................................................................................................................................ 141

CONCLUSION OF THE SECOND PART............................................................................................... 142

III. APPLICATIONS .................................................................................................................................. 143

INTRODUCTION OF THE THIRD PART ............................................................................................. 144

CH.7. AVERAGE MODEL IN MOUSE PHENOTYPING PROCESS........................................... 145

7.1 INTRODUCTION ............................................................................................................................. 146 7.2 MATERIAL AND DATA ACQUISITION METHODS.............................................................................. 147

7.2.1 Animals.................................................................................................................................... 147 7.2.2 Imaging and segmentation....................................................................................................... 148

7.3 INITIAL COMPREHENSIVE ANALYSIS OF DATA ............................................................................... 149 7.3.1 Data analysis ........................................................................................................................... 149 7.3.2 Results ..................................................................................................................................... 152

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7.4 AVERAGE MODEL-BASED PHENOTYPING ANALYSIS ...................................................................... 153 7.4.1. Enriched normal average model construction .................................................................... 153 7.4.2. Phenotyping process ........................................................................................................... 156 7.4.3. Results ................................................................................................................................. 158

7.5 CONCLUSION AND DISCUSSION ..................................................................................................... 164

CH.8. ENRICHED AVERAGE MODEL-BASED SEGMENTATION........................................... 165

8.1 INTRODUCTION ............................................................................................................................. 166 8.2 MODEL-BASED SEGMENTATION.................................................................................................... 166 8.3 REGION GROWING METHOD INTEGRATING SHAPE PRIOR (RGISP) [ROS' 07]................................ 167 8.4 SEGMENTATION PROCESS FRAMEWORK........................................................................................ 168 8.5 EXAMPLES .................................................................................................................................... 169 8.6 CONCLUSION ................................................................................................................................ 171

CONCLUSION OF THE THIRD PART .................................................................................................. 172

CONCLUSION AND FUTURE WORKS................................................................................................. 173

BIBLIOGRAPHY ....................................................................................................................................... 175

Liste des figures

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Liste des figures

FIGURE 1 - 1. (A) PLANCHE D’ANATOMIE DE LEONARD DE VINCI, (B) PLANCHE ANATOMIQUE MONTRANT LE

SYSTEME ALIMENTAIRE ET LES ARTERES TRAITE D'ANATOMIE DE MANSUR IBN MUHAMMAD IBN

AMHAD AL-KASHMIRI AL-BALKHI (1396). ................................................................................. 23 FIGURE 1 - 2. VUES 3D DE « CAVEMAN » ATLAS ( WWW.VISUALGENOMICS.CA ) ......................................... 23 FIGURE 1 - 3. UN EXEMPLE DES POSSIBILITES DE L’ATLAS ANATOMIQUE VOXEL-MAN COMMERCIALISE PAR

L’EDITEUR ALLEMAND SPRINGER-VERLAG : VISUALISATION TRIDIMENSIONNELLE ET BASE DE

DONNEES [HOH' 92]. .................................................................................................................. 26 FIGURE 1 - 4. QUAND LE PROJET « VOXEL-MAN » RENCONTRE LE PROJET « VISIBLE HUMAN »..................... 27 FIGURE 1 - 5. (A) UNE COUPE EXTRAITE DE LA BASE DE DONNEES A SEGMENTER. (B) INFORMATION D’ATLAS.

RESULTATS DE SEGMENTATION (C) SANS ET (D)AVEC L’INFORMATION D’ATLAS [PAR' 03]. ...... 31 FIGURE 1 - 6. UN EXEMPLE DE SECTION DU CERVEAU DE LA SOURIS, DANS LE CADRE DU PROJET MAP

(HTTP://WWW.LONI.UCLA.EDU/MAP) ......................................................................................... 32 FIGURE 1 - 7. LE MODELE DE L’EMBRYON DE LA SOURIS DEVELOPPE DANS LE CADRE DU PROJET « CALTECH

µMRI ATLAS OF MOUSE DEVELOPMENT (HTTP://MOUSEATLAS.CALTECH.EDU) » ; (A) UN MODELE

3D DE L’EMBRYON (B) DIFFERENTES ETAPES DU DEVELOPPEMENT DE L’EMBRYON.................... 33 FIGURE 1 - 8. VUES 3D DE L’EMBRYON DE LA SOURIS AU 9EME JOURS OU L’AMNION A ETE ENLEVE [BRU' 99].

(A) LA SURFACE DE L’ EMBRYON (B) TISSU NEURAL (JAUNE), VESICULES OPTIQUES (ORANGE),

TROU OPTIQUE (ORANGE PALE), METAMERE (VERT FONCE), ARTERES (ROUGE ORANGE), VEINES

(BLEU FONCE), COEUR (LILAS), VENTRICULE PRIMITIF (MAGENTA), CORDIS DE BULBUS (ROUGE),

ET INTESTIN (BLEU CLAIR). ......................................................................................................... 34 FIGURE 1 - 9. LES MODELES 3D DE L’EMBRYON DE LA SOURIS [DHE' 01] (A) TOUTES LES ORGANES, (B)

SYSTEME NERVEUX CENTRAL, (C) SQUELETTE, (D) TOUTES LES COMBINAISONS DES SYSTEMES

(DANS CE CAS-CI, ENTERIQUE, PULMONAIRE, CIRCULATOIRE, ETC.)............................................ 34

FIGURE 2 - 1. EXTRACTION DE FORMES COMPOSANT UNE IMAGE VOLUMETRIQUE PAR : (A) CLASSIFICATION DE

SES VOXELS PUIS RECONSTRUCTION DISCRETE, (B) MODELE DEFORMABLE. ............................... 39 FIGURE 2 - 2. CREATION D’ENSEMBLE D’APPRENTISSAGE VOLUMIQUE [BAI' 03]............................................ 46 FIGURE 2 - 3. UN EXEMPLE DE LIGNES DE CRETES ET DE VALLEES SUR LE CERVEAU [SUB' 95], LIGNES EN BLEU

CORRESPONDANTS AUX LIGNES DE CRETES ET EN ROUGE AUX LIGNES DE VALLEES.................... 48 FIGURE 2 - 4. UN EXEMPLE DES POSSIBILITES DE L’ATLAS ANATOMIQUE VOXEL-MAN COMMERCIALISE PAR

L’EDITEUR ALLEMANDE SPRINGER-VERLAG : VISUALISATION TRIDIMENSIONNELLE AVEC

COLORISATION ARTIFICIELLE DE CHAQUE COMPOSANT DU CRANE.............................................. 48

FIGURE 3 - 1. SCHEMATIC DIAGRAM OF THE PROPOSED METHODOLOGY. ........................................................ 58 FIGURE 3 - 2. THE PIPELINE OF THE PROPOSED METHOD TO CONSTRUCT AN AVERAGE MODEL. ....................... 59

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FIGURE 3 - 3. AVERAGE MODEL ENRICHMENTS ............................................................................................... 60 FIGURE 4 -1. SIGNED DISTANCE EVALUATION; DISTANCE IS POSITIVE IN P2 AND NEGATIVE IN P1..................... 66 FIGURE 4 - 2. DIAGRAM OF THE PROPOSED INITIAL REGISTRATION STAGE....................................................... 70 FIGURE 4 - 3. DIAGRAM OF THE PROPOSED FINAL REGISTRATION STAGE......................................................... 71 FIGURE 4 - 4. A VISUALIZATION OF AN ELLIPSOID MODEL AFTER ONE, TWO AND THREE ITERATIONS OF

ANISOTROPIC SCALING SHOWN AT THE RIGHT. NOTE THAT, THE TRANSFORMED ELLIPSOID IS VERY

ISOTROPIC AFTER THE THIRD ITERATIONS OF ANISOTROPIC SCALING (CONVERGING TO SPHERE).73 FIGURE 4 - 5. A VISUALIZATION OF A TOOTH MODEL TRANSFORMED USING AN ITERATIVE ANISOTROPIC

SCALING TRANSFORMATION........................................................................................................ 73 FIGURE 4 - 6. VERTEBRA AND FOOT MODELS WITH SOME OF ITS FEATURE POINTS (GREEN BALLS PRESENT SOME

LOCAL MAXIMUM FEATURE POINTS, BLUE BALL PRESENTS A LOCAL MINIMUM FEATURE POINT) 75 FIGURE 4 - 7. FEATURE POINTS ON A HAND MODEL (A) WITH DIFFERENT RESOLUTION (B, C, D), GREEN BALLS

PRESENT SOME LOCAL MAXIMUM FEATURE POINTS, BLUE BALL PRESENTS A LOCAL MINIMUM

FEATURE POINT. .......................................................................................................................... 76 FIGURE 4 - 8. THE THREE SELECTED POINTS (P1, P2, P3) AND THEIR CORRESPONDING POINTS......................... 77 FIGURE 4 - 9. RIGID REGISTRATION RESULTS FOR A PAIR OF VERTEBRA MODELS: (A) INITIAL MODELS

POSITIONS WITHOUT ANY PRE-ALIGNMENT; (B) THE ICP REGISTRATION RESULT STARTING FROM

THE MODELS POSITIONED IN (A); (C) THE INITIAL REGISTRATION RESULT USING THE PROPOSED

FEATURE POINT MATCHING. ........................................................................................................ 78 FIGURE 4- 10. SCREENSHOT OF MESH MEASURING TOOL.............................................................................. 80 FIGURE 4 - 11. INITIAL AND FINAL REGISTRATION FOR TWO FEMUR MODELS. (A) SOURCE, (B) TARGET , (C)

SOURCE AND TARGET IN THE SAME RENDERING WINDOWS, (D) INITIAL REGISTERED SOURCE, (E)

SOURCE AND TARGET AFTER INITIAL REGISTRATION, (F) DISTRIBUTED HAUSDORFF DISTANCES ON

THE TARGET AFTER INITIAL REGISTRATION, (G) FINAL REGISTERED SOURCE, (H) SOURCE AND

TARGET AFTER FINAL REGISTRATION, (I) DISTRIBUTED HAUSDORFF DISTANCES ON THE TARGET

AFTER FINAL REGISTRATION, (J) THE SCALAR BARS CORRESPONDING TO THE HAUSDORFF

DISTANCE DISTRIBUTION ON THE TARGET MODEL IN (E), (F). ...................................................... 81 FIGURE 4 - 12. AFFINE AND ELASTIC REGISTRATION FOR SPHERE AND CUBE. .................................................. 84 FIGURE 5- 1. PSEUDO CODE OF THE ITERATIVE ALGORITHM TO COMPUTE THE AVERAGE AFFINE MODEL AAM.

................................................................................................................................................... 89 FIGURE 5- 2. THE AFFINE AVERAGE MODEL AAM CONSTRUCTION TEST. THE BASE MODEL COMPOSED OF 5650

POINTS, THE NUMBER OF ITERATIONS OF GPA ALGORITHM IS 2.................................................. 91 FIGURE 5- 3. THE ELASTIC AVERAGE MODEL OF A TRAINING SET COMPOSED OF 3 MODELS (SPHERE, ELLIPSOID

AND CUBE), WHERE THE REFERENCE IS THE SPHERE MODEL. THE NUMBER ITERATIONS IN GPA IS

3 ITERATIONS, THE NUMBER OF SELECTED LANDMARKS IN EACH ITERATION OF THE GENERAL

ELASTIC REGISTRATION IS 100 LANDMARKS. .............................................................................. 92

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FIGURE 5- 4. EFFECTS OF CHANGING THE REFERENCE MODEL AND THE NUMBER OF ITERATIONS ON THE

AVERAGE MODEL CONSTRUCTION METHOD BASED ON AFFINE GPA ALGORITHM. ...................... 93 FIGURE 5- 5. THE AFFINE AVERAGE MODELS (AAM0, AAM1 AND AAM2) AND THE ELASTIC AVERAGE MODELS

(EAM0, EAM1 AND EAM2) CONSTRUCTED FORM THE TRAINING SET MODELS TS (M0, M1 AND

M2). ............................................................................................................................................ 94 FIGURE 5- 6. THE AFFINE AVERAGE MODELS (AAM0 AND AAM1) AND THE ELASTIC AVERAGE MODELS (EAM0

AND EAM1) CONSTRUCTED FORM THE TRAINING SET MODELS TS (M0 AND M1). THE AVERAGE

MODELS ARE CONSTRUCTED USING THE SAME NUMBER OF ITERATIONS (5 ITERATIONS) AND THE

SAME NUMBER OF LANDMARKS (150 LANDMARKS). ................................................................... 95 FIGURE 5- 7. MORPHOLOGICAL-BASED AVERAGE MODEL CONSTRUCTION PIPELINE. ...................................... 97 FIGURE 5- 8. SLICES THROUGH A SPHERE: (A) SURFACE REPRESENTATION, (B) SOLID REPRESENTATION......... 98 FIGURE 5- 9. VOLUMETRIC REPRESENTATION OF A MOUSE BRAIN MODEL USING DIFFERENT 3D GRIDS

RESOLUTION. .............................................................................................................................. 99 FIGURE 5- 10. SOME LOGIC OPERATIONS BETWEEN BINARY IMAGES; BLACK REPRESENTS A BINARY 0 AND

WHITE A BINARY 1. ................................................................................................................... 100 FIGURE 5- 11. THE CONDITIONAL DILATION RESULTS OF A BASE (A) USING A MASK (M).(A) AFTER 5

ITERATION, (B) AFTER 20 ITERATIONS, (C) AFTER 30 ITERATIONS, (D) AFTER 40 ITERATIONS. .. 101 FIGURE 5- 12. NETWORK FOR THE CONSTRUCTING OF AN AVERAGE VOLUME FOR TRAINING SET OF 8 VOLUMES

USING AN AVERAGING OPERATOR (AO) IN EACH NODE OF THE NETWORK. ............................... 102 FIGURE 5- 13. EXTREME VOLUME EXTRACTION. ........................................................................................... 103 FIGURE 5- 14. CONDITIONAL DILATION AND EROSION DIAGRAM................................................................... 103 FIGURE 5- 15. THE AVERAGE IMAGE AI OF TWO BINARY IMAGES I0, I1 EXTRACTED USING AO. .................... 105 FIGURE 5- 16. THE TWO LAYERS AVERAGE OF A SET OF FOUR 256 X 256 BINARY IMAGES (I0, I1, I2, I3). ........ 106 FIGURE 5- 17. USING THE MORPHOLOGICAL-BASED AVERAGE MODEL EXTRACTION IN 3D CASE. ................. 107 FIGURE 5- 18. MODEL CREATION STEPS. ....................................................................................................... 108 FIGURE 5- 19. THE REGIONS A AND B SUPERIMPOSED [GOU' 03]. ................................................................ 109 FIGURE 5 - 20. AVERAGE IMAGES OF THE THREE POSSIBLE SEQUENCES OF THE FOUR INPUT BINARY IMAGES (I0,

I1, I2, I3)..................................................................................................................................... 111 FIGURE 5- 21. DIFFERENCE BETWEEN THE EXTRACTED AVERAGE IMAGES. ................................................... 112 FIGURE 5 - 22. AVERAGE MODELS OF THE THREE POSSIBLE ORDERING OF THE FOUR INPUT MODELS (M0, M1,

M2, M3) USING THE PROPOSED AVERAGE MODEL CONSTRUCTION MODEL................................. 113 FIGURE 5 - 23. VISUAL COMPARISON OF THE EXTRACTED AVERAGE MODELS IN FIGURE 5 - 22. (A) MAPPING OF

M0123 AND M0321, (B) MAPPING OF M0213 AND M0321. DISTRIBUTED ABSOLUTE RELATIVE

EUCLIDEAN DISTANCES BETWEEN M0321 AND (C) M0123, (D) M0213. ........................................... 114 FIGURE 5- 24. COMBINED AVERAGE METHOD CONSTRUCTION PIPELINE........................................................ 116 FIGURE 6 - 1. AVERAGE MODEL ENRICHMENTS. ............................................................................................ 120 FIGURE 6 - 2. TOPOGRAPHIC FEATURES IN TRIANGULAR MESHES. ................................................................. 122

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FIGURE 6 - 3. RED CREST AND BLUE VALLEY LINES DETECTED ON MICHELANGELO’S DAVID HEAD MODEL

[OHT' 04]. ................................................................................................................................ 122 FIGURE 6 - 4. NORMAL AND TANGENTS VECTORS ON PARAMETRIC SMOOTH SURFACE.................................. 123 FIGURE 6 - 5. DEFINITIONS OF GEODESIC AND NORMAL CURVATURES, NORMAL SECTION CURVE. ................ 124 FIGURE 6 - 6. ONE-RING NEIGHBORHOOD OF A VERTEX VI. ............................................................................ 126 FIGURE 6 - 7. CURVATURE MEASURES FOR A BRAIN MODEL USING DONG’S CURVATURE ESTIMATOR [DON'

05]: (A) MINIMUM CURVATURE VALUES , (B) MAXIMUM CURVATURE. ...................................... 128 FIGURE 6 - 8. CREST-VALLEY POINTS DETECTED ON VARIOUS MODELS. (A) CREST POINTS,(B) VALLEY POINTS,

DETECTED ON STANDARD BUNNY MODEL, (C) CREST POINTS DETECTED ON PYRAMID MODEL, (A1,

B1, C1) CREST, VALLEY POINTS ALONE ARE SUFFICIENT FOR RECOGNIZING THE MODELS, (D)

CREST POINTS ON CUBE MODEL, (E) VALLEY POINTS ON THE MAX-PLANCK MODEL, (F) VALLEY

POINTS DETECTED ON FELINE MODEL. ...................................................................................... 131 FIGURE 6 - 9. CREST AND VALLEY POINTS ON FANDISK MODEL (A) (LEFT) CREST POINTS BEFORE THE

THRESHOLDING (5356 POINTS) (TOP) AND AFTER THE THRESHOLDING (T=1.5) (BOTTOM), (RIGHT)

FORCE OF CREST LINES, (B) (LEFT) VALLEY POINTS BEFORE THE THRESHOLDING (5454 POINTS)

(TOP) AND AFTER THE THRESHOLDING (T=2.0) (BOTTOM), (RIGHT) FORCE OF VALLEY LINES... 132 FIGURE 6 - 10. CREST AND VALLEY POINTS ON HORSE MODEL (A) (LEFT) CREST POINTS BEFORE THE

THRESHOLDING (5356 POINTS) (TOP) AND AFTER THE THRESHOLDING (T=2.0) (BOTTOM), (RIGHT)

FORCE OF CREST LINES, (B) (LEFT) VALLEY POINTS BEFORE THE THRESHOLDING (5454 POINTS)

(TOP) AND AFTER THE THRESHOLDING (T=1.0) (BOTTOM), (RIGHT) FORCE OF VALLEY LINES... 133 FIGURE 6 - 11. SCHEMATIC DIAGRAM INDICATING TRAINING MODELS (M0, M1…, MN) AND THE CONSTRUCTED

MODEL. ..................................................................................................................................... 135 FIGURE 6 - 12. ENRICHED AVERAGE SURFACE COMPUTED FROM FOUR TRAINING SURFACES (M0, M1, M2, M3).

................................................................................................................................................. 136 FIGURE 6 - 13. EXAMPLES OF SOME EUCLIDEAN DISTANCE MAPS (2D) COMPUTED FROM TWO BINARY IMAGES

(I0, I1) AND ITS AVERAGE IMAGE. .............................................................................................. 138 FIGURE 6 - 14. THE COMPUTED DISTANCE MAPS FROM FOUR TRAINING MODELS (M0, M1, M2, M3) AND ITS

AVERAGE MODEL AM............................................................................................................... 139 FIGURE 6 - 15. PROBABILITY MAP OF THE TRAINING SET (M0, M1, M2, M3) USED IN FIGURE 6 - 14. .............. 140 FIGURE 7- 1. IMAGING / SEGMENTATION STAGE ............................................................................................ 148 FIGURE 7- 2. EXAMPLES OF INTRACRANIAL MODEL OF MASTOMYS NATALENSIS MICE ................................. 149 FIGURE 7- 3. MALE MICE INTRACRANIAL MODELS CHARACTERISTICS........................................................... 150 FIGURE 7- 4. FEMALE MICE INTRACRANIAL MODELS CHARACTERISTICS........................................................ 151 FIGURE 7- 5. MALE-FEMALE NORMAL MICE CHARACTERISTICS..................................................................... 152 FIGURE 7- 6. AVERAGE MODEL-BASED PHENOTYPING ANALYSIS PIPELINE.................................................... 153 FIGURE 7- 7. ENRICHED AVERAGE MODEL OF THE NORMAL 3RD AGE CLASS (MALE MICE).............................. 154

Liste des figures

11

FIGURE 7- 8. THE PERCENTAGE OF AVERAGE MODEL FACES, FOR DIFFERENT NORMAL MODELS, (A) NORMAL

MODELS USED TO CONSTRUCT THE AVERAGE MODEL, (B) NEW NORMAL MODELS NOT USED IN THE

AVERAGE MODEL CONSTRUCTION STAGE.................................................................................. 155 FIGURE 7- 9. (A) THE PERCENTAGE OF AVERAGE MODEL FACES, FOR DIFFERENT ABNORMAL NORMAL MODELS,

(B) TABLE OF THE COMPUTED P-VALUES TO EVALUATE THE STOCHASTIC SIGNIFICANCE BETWEEN

NORMAL AND ABNORMAL MODELS. .......................................................................................... 156 FIGURE 7- 10. PHENOTYPING PROCESS .......................................................................................................... 157 FIGURE 7- 11. THE PERCENTAGE OF THE AVERAGE MODEL FACES CLASSIFIED AS NORMAL FACES FOR THE

ABNORMAL MODELS. ................................................................................................................ 159 FIGURE 7- 12. PHENOTYPING PROCESS CLASSIFICATION RESULTS FOR THE ABNORMAL MODELS. FOR EACH

MODEL: (LEFT) THE Z-SCORES DISTANCE MAPS, (RIGHT) THE DETECTED ABNORMAL REGIONS

(RIGHT). RED, BLUE REGIONS INDICATE THE REGIONS WHERE ( )1 20.3, 3%thr thr= = ........... 163 FIGURE 8- 1. FLOW DIAGRAM OF THE SEGMENTATION PROCESS USING RGISP ............................................. 169 FIGURE 8- 2. EXAMPLE OF SUING ENRICHED AVERAGE MODEL AS A REFERENCE MODEL IN A SEGMENTATION

PROCESS BASED ON RGIPS METHOD [ROS' 07]. (I1, I2, I3, I4) ARE THE TRAINING SET FROM WHICH

THE AVERAGE IMAGE I1324 WILL BE CONSTRUCTED, TO IS THE TARGET OBJECT, I IS THE NOISED

INPUT IMAGE (GAUSSIAN NOISE 20Nσ = ). R1, R2 ARE THE DETECTED REGIONS USING THE

CLASSICAL REGION GROWING METHOD (WITHOUT SHAPE PRIOR), RGISP METHOD ( 20, 0.5λ = ∂ = ),

RESPECTIVELY. ......................................................................................................................... 170 FIGURE 8- 3. SEGMENTATION RESULTS ON MICRO-CT IMAGES OF MOUSE SKULL.......................................... 171

Liste des figures

12

Liste des tables

13

Liste des tables

TABLE 4 - 1. THE RELATIVE DISTANCES BETWEEN THE REGISTERED SOURCE AND TARGET MODELS IN THE TWO

PHASES OF THE PROPOSED REGISTRATION METHOD, ILLUSTRATED IN FIGURE 4 - 11................... 82 TABLE 5- 1. RELATIVE MEASURED DISTANCES BETWEEN AAM AND BASE MODEL SURFACES......................... 90 TABLE 5- 2. AND, OR AND XOR OPERATIONS TABLE................................................................................... 100 TABLE 5 - 3. DISCREPANCY MEASURES COMPUTED ON THE AVERAGE IMAGES (I0213, I0123) OF FIGURE 5- 16

COMPARED TO I0321. .................................................................................................................. 112 TABLE 5 - 4. DISCREPANCY MEASURES COMPUTED ON THE AVERAGE VOLUMES OF FIGURE 5 - 22, USING THE

FIRST AVERAGE VOLUME V0321 AS A REFERENCE VOLUME......................................................... 114 TABLE 5- 5. RELATIVE MEASURED DISTANCES BETWEEN RESULTS AVERAGE MODELS AND THE FINAL AVERAGE

MODEL M0321 (FIGURE 5 - 22).................................................................................................... 115 TABLE 7 - 1. COLLECTION OF MASTOMYS MICE INTRACRANIAL MODELS. ..................................................... 148 TABLE 7 - 2. TABLE SUMMARIZES THE P-VALUE USING THE WORDS IN THE MIDDLE COLUMN........................ 150 TABLE 7 - 3. STATISTICAL COMPARISON BETWEEN THE MEASURED FEATURES OF THE INTRACRANIAL MODELS

OF MALE MICE (3RD AGE CLASS) (MEAN ± SEM (STANDARD ERROR OF MEAN)), THE

CORRESPONDING P-VALUES. ..................................................................................................... 151 TABLE 7 - 4. THE RESULTS OF THE PROPOSED PHENOTYPING PROCESS CLASSIFICATION OF 24 MODELS (16

NORMAL (LASSA -) AND 8 ABNORMAL (LASSA +)) USING DIFFERENT VALUES OF THRESHOLDS.160 TABLE 7 - 5. THE MEDICAL STATISTICS OF THE CLASSIFICATION RESULTS IN TABLE 7 - 4 ............................. 162

Liste des abréviations

14

Liste des abréviations

AAM Affine Average Model

AO Averaging Operator

B-rep Boundary Representation

CD Conditional Dilation

CE Conditional Erosion

CT Computed Tomography

d.p.c. Day Post Conception

DM Distance Map

EAM Elastic Average Model

EDD Euclidean Distance Distribution

EDR External Distortion Rate

emax, emin Maximum Extremality, Minimum Extremality

FN, FP False Negative, False Positive

GPA Generalized Procrustes Analysis

ICP Iterative Closest Points

IDR Internal Distortion Rate

IRM Imagerie par Résonance Magnétique

kmax, tmax Maximum Curvature, Maximum Curvature Direction

kmin, tmin Minimum Curvature, Minimum Curvature Direction

MC Marching Cubes

MMC Modified Marching Cubes

MSE Mean squared Error

NPV, PPV Negative Predective Value, Positive Predective Value

OBB Oriented Bounding Box

ORL Oto-Rhino-Laryngologiste

PCA Principal Component Analysis

RM Reference Model

SD Standard Deviation

TN True Negative

TP True Positive

TPS Thin Plat Spline

TS Training Set

Résumé

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Résumé

En imagerie médicale, la robustesse de l’analyse et de la segmentation d’image est améliorée grâce à l’utilisation de connaissances à priori. Les atlas anatomiques constituent une base de connaissances à priori utiles pour localiser certains organes, ainsi que certaines structures très difficiles à distinguer sur des images complexes. Les atlas construits à partir d’un seul jeu de données ne permettent pas de prendre en compte les variations morphologiques et pathologiques inter-individus contrairement aux atlas moyens (construits à partir de plusieurs jeux de données). Dans cette thèse, nous nous intéressons à la construction d’un modèle moyen enrichi à partir des jeux de données surfaciques (modèles maillés). Ce modèle enrichi consiste en un modèle moyen (élément de base pour un atlas moyen) complété par des informations de variations géométriques de la structure anatomique étudiée (enrichissement). Pour atteindre cet objectif, nous construisons d’abord un modèle moyen déduit d’un ensemble d’apprentissage. Ensuite, nous procédons à l’enrichissement du modèle par des informations quantitatives, statistiques et géométriques extraites à partir de modèle moyen lui-même et de tous les modèles utilisés pour le construire. Les informations d’enrichissement du modèle moyen permettent ainsi la caractérisation de la variabilité d’une structure anatomique pathologique. La simplicité ou la complexité de l’enrichissement du modèle dépendront de l’application envisagée. Dans le cadre de cette thèse, nous proposons deux applications basées sur l’utilisation de ce modèle moyen enrichi :

Modèle de comparaison pour un processus de phénotypage anatomique du petit animal. Modèle de référence pour la segmentation des images médicales in vivo intégrant des a

priori sur la forme de la structure anatomique à segmenter. Mots-clés: atlas moyen, modèle maillé, recalage de surface, modèle moyen, modèle de référence, enrichissement de modèle, géométrie différentielle, lignes de crête, carte de distance, analyse statistique, phénotypage.

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Summary

The medical image analysis and segmentation are improved thanks to the use of a priori knowledge. The anatomical atlases provide an a priori knowledge base useful to locate certain anatomical structures very difficult to distinguish on complex images. The atlases built from only one subject don’t take into account the morphological variations and pathological inter-individuals, contrary to the average atlases (built from several subjects). In this thesis, we are interested in the construction of an enriched average model from a set of training meshed models. The constructed model consists of the average model (basic element for an average atlas) and the geometrical variations of the anatomical structure under consideration (enrichments). To achieve this objective, first we construct an average model from a set of training models representing the same anatomical structure. Next, we enrich the constructed average model with quantitative, statistical and geometrical information extracted from the average model itself and all the models from which it was constructed. The enrichments of the average model characterize the variability of a pathological anatomical structure. The simplicity and the complexity of the enrichments depend on the envisaged applications using the constructed model. In the framework of this thesis, we propose two applications based on the use of the enriched average model as:

Comparison model for an anatomical phenotyping process of small animal models. Reference model for in-vivo model-based image segmentation.

Key-words: average atlas, meshed model, surface registration, average model, reference model, model enrichments, differential geometry, crest lines, distance maps, statistical analysis, phenotyping.

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Introduction générale

Voir à l'intérieur du corps humain sans nuire, tel était le rêve d'Hippocrate il y a plus de 23 siècles. Ce rêve a réellement commencé à se réaliser en 1895 grâce à la découverte des rayons X par Röntgen. Depuis, l'imagerie médicale a sans cesse évoluée pour parvenir maintenant à décrire de manière non invasive la totalité du corps humain par des images numériques. De nombreuses modalités d’imagerie médicale permettent une exploration volumique (3D) de plus en plus précise des structures anatomiques. Les images volumiques fournissent aux spécialistes en médecine un accès direct à l’intérieur du corps d’un patient ce qui réduit le recours à des explorations invasives. L’analyse des images médicales est très complexe car ces images sont représentées par de grandes quantités de données et elles présentent parfois des effets non désirables comme le bruit. Cependant, des techniques de prétraitement, de segmentation et de reconstruction sont progressivement mises au point pour construire des représentations surfaciques des structures anatomiques cartographiées dans ces données. L’utilisation d’information a priori pendant le traitement d’images peut faciliter beaucoup leur analyse. Normalement, cette information a priori est représentée par les images dites de référence ou atlas, qui déterminent un espace commun où l’anatomie humaine peut être précisément représentée, comparée et corrélée. Un atlas, construit à partir d’une base de données, doit prendre en compte aussi bien les ressemblances que les diversités des exemplaires afin de construire un ensemble de caractéristiques qui vont servir de repères. Dans cette thèse, nous proposons une méthodologie pour construire un modèle maillé, moyen (modèle de référence) et enrichi représentant une structure anatomique. Le modèle moyen enrichi constitue une information exploitable dans beaucoup d'autres applications. La méthodologie proposée est composée de deux étapes :

La construction d’un modèle surfacique moyen (modèle de référence) à partir d’un ensemble d’apprentissage composé de N modèles maillés représentant la même structure anatomique.

L’enrichissement du modèle moyen par des informations quantitatives, statistiques, géométriques ou fournies par un expert en biologie.

Le mémoire de thèse est organisé en trois parties. La première partie porte sur le contexte général des recherches réalisées dans le cadre de cette thèse. Cette partie est composée de deux chapitres. Dans le premier, nous commençons par présenter les origines et l’état de l’art dans le domaine de la construction d’atlas anatomique et ses applications dans différents domaines médicaux. Ensuite, quelques travaux récents concernant la construction d’atlas anatomique numérique pour l’homme et l’embryon de souris seront présentés. Les étapes nécessaires afin de réaliser ce type d’atlas seront analysées à la fin de ce chapitre. Le deuxième chapitre est consacré à la construction du modèle surfacique à partir de données volumiques issues des différentes modalités de l’imagerie médicale. Ensuite, nous expliquerons la nécessité d’enrichir le modèle

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surfacique simple par d’autre information pour une application de haut niveau. Enfin, nous analyserons quelques travaux existants concernant l’extraction d’un modèle moyen à partir d’ensembles de modèles qui représentent la même structure anatomique. Dans la deuxième partie, nous détaillons la méthodologie proposée pour la construction d’un modèle moyen enrichi en quatre chapitres. Une vue globale de la méthodologie proposée est présentée dans le troisième chapitre. Le recalage de surfaces qui est une étape nécessaire pour toutes les phases de la méthodologie est exposé dans le quatrième chapitre. Dans le chapitre suivant, la méthode de construction du modèle moyen maillé à partir d’un ensemble d’apprentissage composé de N modèles sera détaillée. Enfin, dans le sixième chapitre nous abordons l’enrichissement du modèle moyen. La troisième partie présente deux applications du modèle moyen sur des données réelles. Dans la première application, un modèle moyen est construit à partir d’une base de données composée de 16 modèles normaux de crânes de souris. Le modèle moyen normal est enrichi par des cartes de distance obtenues à partir de tous les modèles de l’ensemble d’apprentissage. Ensuite, un processus de phénotypage sera présenté afin d’analyser l’aptitude du modèle moyen, d’une part, de décrire la population à partir de laquelle il a été construit (modèles normaux), d’autre part de différencier des modèles qui appartiennent à une population différente (modèles anormaux). Dans une deuxième application, le modèle moyen enrichi servira comme modèle de référence pour la segmentation des images médicales du petit animal, en intégrant des informations a priori. Enfin, nous terminons ce manuscrit par quelques perspectives pour les travaux à venir.

Partie 1 : Contexte et état de l’art

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I. Contexte et état de l’art


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Introduction à la première partie

Dans cette partie, nous présentons le contexte général des recherches réalisées dans le cadre de cette thèse, en deux chapitres. Dans le premier, nous commençons par présenter un aperçu de l’histoire de l’atlas anatomique et la comparaison entre l’atlas anatomique numérique et sur papier. Puis nous présentons les deux types de reconstruction d’une structure anatomique dans un atlas numérique. Ensuite, nous exposons quelques applications de l’atlas numérique dans le domaine médical. Enfin, quelques travaux récents concernant la construction d’atlas anatomiques numériques pour l’homme et l’embryon de souris sont présentés, les étapes nécessaires afin de réaliser ce type d’atlas sont analysées à la fin de ce chapitre. Nous nous concentrons dans le deuxième chapitre sur la construction d’atlas numérique moyen enrichi (modèle moyen enrichi). Tout d’abord, une présentation non exhaustive des méthodes existantes pour extraire des représentations surfacique des données volumique (images 3D) est exposée. Ensuite, l’atlas anatomique moyen (modèle moyen) est présenté à travers la deuxième partie de ce chapitre. Quelques exemples des méthodes existantes afin de créer ce type de modèle volumique et surfacique sont présentées et analysées. Nous présentons aussi un résumé de quelques exemples pour enrichir un modèle surfacique afin d’utiliser ce modèle enrichi dans une application de haut niveau. Enfin, une analyse des méthodes existantes pour créer un atlas anatomique moyen est présentée à la fin de ce chapitre.

Ch 1 : Atlas anatomique

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Ch.1. Atlas Anatomique


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1.1 Introduction

Le modèle anatomique moyen enrichi est un atlas anatomique surfacique particulier qui décrit la forme d’une structure anatomique donnée et ses caractéristiques. Dans ce contexte, nous présenterons dans ce chapitre les origines et l’état de l’art dans le domaine de la construction d’atlas anatomique, ainsi que ses applications dans différents domaines. Quelques travaux concernant la construction d’atlas anatomiques numériques pour l’homme et l’embryon de souris seront présentés. Les étapes nécessaires afin de réaliser un atlas anatomique numérique seront présentées à la fin de ce chapitre suite à l’analyse de ces travaux.

1.2 L’histoire de l’atlas anatomique

Explorer le corps humain sans opérer, ou alors avec une précision extrême, est possible, grâce en partie à l’imagerie médicale. Des cryptes sombres à l’homme transparent 4D [TUR' 08], les atlas anatomiques font partie de la vie de tous les jours pour les futurs médecins et surtout pour les futurs chirurgiens. L’atlas anatomique permet d’acquérir la connaissance technique et détaillée de chaque partie du corps humain. L’atlas anatomique a évolué des premières planches grossières du XVéme siècle, à l’homme transparent en quatre dimensions « CaveMan » [TUR' 08] du XXIéme siècle. Au Moyen Âge, les médecins connaissaient l’existence des organes, et ils savaient que le corps humain était constitué de liquides divers, appelés communément humeurs. Ces humeurs variaient en quantité et qualité, selon l’état de santé. Grâce aux animaux avec lesquels ils vivaient en relation très étroite, les premiers médecins savaient, que comme les bêtes, les humains étaient constitués d’os, de nerfs et de tendons. Mais dans l’ensemble la connaissance anatomique était très limitée. Dès le 17ème siècle, les connaissances se raffinent. Léonard de Vinci lui-même s'est penché sur le sujet, les esprits curieux voulaient en savoir davantage. Le problème est qu'à l'époque, il était fort mal vu d'ouvrir un corps humain, car c'eût été profaner le réceptacle de l'âme. Ceux qui étaient passionnés par le corps humain devaient parfois prendre des risques. A la tombée de la nuit, ils envoyaient leurs apprentis à la morgue. Si celle-ci était vide, ils allaient marauder du côté des fosses communes pour dérober l'un des corps que l'on venait tout juste d'y déposer. C'est dans des souterrains humides et faiblement éclairés, que les premiers anatomistes se firent à la main. Grâce à ce travail ingrat, les premières planches anatomiques voient le jour (Figure 1 - 1). Enfin, la médecine quitta le domaine des pratiques secrètes pour acquérir le droit de cité dans les écoles et dans les universités. Depuis 1850, les planches utilisent la couleur et gagnent de plus en plus en précision. Il existe des pages spécialisées sur chaque organe et chaque composante du corps humain. C'est une recherche continuelle.


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(a) (b)

Figure 1 - 1. (a) planche d’anatomie de Léonard de Vinci, (b) Planche anatomique montrant le système

alimentaire et les artères Traité d'anatomie de Mansur ibn Muhammad ibn Amhad al-Kashmiri al-Balkhi

(1396).

Figure 1 - 2. Vues 3D de « CaveMan » atlas ( www.visualgenomics.ca )

Le corps humain ne livre pas ses trésors facilement. Mais grâce à l'informatique moderne, les chercheurs font un pas de géant : l'homme transparent en quatre dimensions « CaveMan »


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[TUR' 08] (www.visualgenomics.ca) (Figure 1 - 2). Ce mannequin représentant le corps humain fournit les outils permettant d’étudier les fonctions métaboliques et de reproduire les expressions génétiques dans leur totalité, ce qui était auparavant impossible. Il est également possible de déplacer une caméra à l’intérieur des organes comme si on réalisait une endoscopie. Il montre également comment diverses concentrations de nutriments et de protéines dans les organes internes changent en raison de facteurs externes. La quatrième dimension simule des activités en temps réel notamment les battements de coeur, la circulation du sang et la respiration.

1.3 Atlas anatomique numérique et les méthodologies de

reconstruction

L'imagerie tridimensionnelle regroupe l'acquisition, l'exploitation et l'interprétation de données volumique. C'est un domaine qui connaît un plein essor depuis une vingtaine d'années, notamment grâce aux progrès techniques et à ses nombreux débouchés dans les domaines médicaux et biologiques. Les médecins et les biologistes doivent comparer les images médicales entre elles pour établir leur diagnostic, préparer le traitement ou planifier les gestes chirurgicaux et suivre l'évolution post-opératoire. Nous pouvons distinguer trois types de comparaison :

la comparaison entre les images du même patient pour suivre l'évolution d'une maladie ou les conséquences d'une opération chirurgicale ;

la comparaison entre les images de deux patients différents (l'un sain et l'autre malade) afin de mettre en évidence les structures pathologiques ;

la mise en correspondance avec un atlas anatomique numérique. Cette dernière application est certainement la plus riche car elle permet un repérage absolu des structures anatomiques, ce qui ouvre des perspectives dans le cadre d'automatisation du diagnostic et de la chirurgie. Tout d’abord, définissons un atlas anatomique, en supposant que nous ayons plusieurs exemplaires d’une structure anatomique provenant de patients différents. Un atlas, construit à partir de ces bases de données, doit prendre en compte aussi bien les ressemblances que les diversités des exemplaires afin de construire un ensemble de caractéristiques qui vont servir de repères [SUB' 95]. Pour cela, les caractéristiques doivent être :

génériques : c’est-à-dire, présentes dans tous les exemplaires de la base de données ; invariantes : dans chaque exemplaire, les caractéristiques doivent être à la même position ; anatomiquement significatives : les caractéristiques devront mettre en évidence les zones

qui ont un intérêt anatomique, par exemple, les zones pathologiques. Notons que l’intérêt dépend fortement de l’application médicale envisagée.

Il existe de nombreuses catégories d'atlas, selon le type et le nombre d'examens utilisés pour leur réalisation, les structures anatomiques répertoriées, les espaces de représentation et les domaines d'utilisation. On distingue d'abord les atlas papier, et les atlas numériques. De plus, à la


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représentation numérique de l’anatomie peuvent être associés des outils informatiques permettant de l'exploiter. Les atlas anatomiques sur papier sont couramment utilisés et avec beaucoup de succès par les médecins. Néanmoins, si nous consultons un de ces atlas, nous nous apercevons qu'il reste difficiles à appréhender pour trois raisons : une visualisation malaisée, l'absence de données quantitatives et une précision limitée [SUB' 95].

Une visualisation malaisée : Les atlas anatomiques sur papier sont essentiellement descriptifs et très peu synthétiques. Ils sont composés de grandes planches qui sont surchargées par les descriptions de caractéristiques de types différents. L'absence de troisième dimension nécessite de multiplier les couleurs et les points de vue ou d'introduire une multitude de coupes. Ainsi, consulter un atlas exige une certaine expérience et un débutant est rapidement submergé par la quantité et la diversité des informations.

L'absence de données quantitatives : Les atlas anatomiques sur papier sont en général qualitatifs. Nous ne trouvons ni les coordonnées des caractéristiques, ni des études sur leurs variabilités. Cela empêche toute automatisation dans la localisation des caractéristiques.

Une précision limitée : Les difficultés de visualisation et l'absence de valeurs quantitatives limitent la précision dans la localisation des caractéristiques.

Tout ceci est d'autant plus regrettable que les systèmes modernes d'imagerie médicale permettent d'obtenir des images tridimensionnelles très précises. Ces problèmes peuvent être résolus en combinant les outils performants de visualisation tridimensionnelle disponibles sur les stations de travail graphiques, par des algorithmes de traitement d'images ou par la précision des systèmes d'acquisition d'images médicales. Les méthodologies de reconstruction d’une structure anatomique se divisent en deux approches duales : les représentations surfaciques et les représentations volumiques. Globalement, il existe la même dualité entre volumes et surfaces dans les volumes numériques (images 3D) que dans les images numériques 2D entre régions et contours. Les forces et faiblesses de ces deux approches peuvent se résumer de la façon suivante :

les approches volumiques bénéficient naturellement des potentialités visuelles considérables, issues de leur nature même de données échantillonnées. Les outils d'analyse que l'on peut y appliquer sont directement issus des outils "classiques" du traitement d'image dont elles sont les correspondants 3D (filtrage, segmentation, morphologie mathématique, …etc.). Il est par contre évident que cette puissance se paye, en coût de stockage et temps de traitement. Il est assez difficile de travailler en temps réel et les visualisations doivent être adaptées.

les approches surfaciques agissent à un niveau en fait plus élevé : celui de la structure même des données tridimensionnelles que l'on souhaite visualiser ou représenter. Dans tous les cas, une approche reconstructrice est nécessaire (reconstruction ou modélisation). Le nombre de données est alors considérablement réduit et on peut alors s'orienter vers des outils de travail en temps réel (mesure, simulation, réalité virtuelle). Comparativement aux approches


26

volumiques ( voxel-based ), les approches surfaciques s'appuient sur un ensemble bien plus réduit de points : ceux pouvant être considérés comme appartenant à la surface des organes à modéliser. Ces points résultent d'une analyse préalable des images destinée à détecter les contours des formes en présence et à extraire des points pertinents qui constitueront l'ensemble de données de la méthode de reconstruction.

1.4 Applications des Atlas anatomiques numériques

1.4.1 La base de données

Les praticiens utilisent l’atlas informatique comme instrument de consultation et d’apprentissage. Cette application nécessite de puissantes techniques de visualisation volumique pour afficher les structures anatomiques sous n’importe quel point de vue ainsi que des fonctionnalités comme la section ou la transparence. Les atlas les plus développés, tels que « VOXEL-MAN » [HOH' 03, HOH' 96, HOH' 92, POM' 07, QAT' 06] ( Figure 1 - 3 ), utilisent l’ordinateur pour la construction d’un modèle tridimensionnelle du corps humain et pour sa manipulation interactive. Ces systèmes anatomiques d’atlas peuvent fournir une reconstruction spatiale relativement précise du corps. De plus, certains proposent de visualiser les organes en fonctionnement par le biais de vidéo ou d’animation.

Figure 1 - 3. Un exemple des possibilités de l’atlas anatomique Voxel-Man commercialisé par l’éditeur

allemand Springer-Verlag : visualisation tridimensionnelle et base de données [HOH' 92].

La NLM « National Library of Medicine » s’est engagée à fournir un ensemble d’images digitalisées du corps humain pour un usage dans l’éducation et la recherche. Le « Visible Human


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Project » [ACK' 95, ACK' 98] a été crée des images numériques de cadavres humains homme et femme avec des photographies anatomiques numériques, des données d’IRM et de tomographie computérisée (CT). Le « Visible Man » est un ensemble d’images numériques du corps d’un homme âgé de 39 ans, qui a donné son corps à la science après avoir été condamné à mort pour meurtre. Il a été exécuté par injection mortelle au Texas en 1993. Les données ont été rendues disponibles en 1994. Les données de « Visible Women » proviennent d’une femme âgée de 59 ans qui est décédée de mort naturelle. Ces données ont été rendues disponibles en décembre 1995. Pour l’homme et la femme, des images du corps ont été obtenues en utilisant l’IRM et scanner avant leur décès. Leur corps a été ensuite trempé dans la gélatine, gelé, et découpé tranche par tranche en coup transversales, de 1 millimètre d’épaisseur pour l’homme et 1/3 mm pour la femme, la surface du corps étant photographiée après chaque tranche et digitalisée. La taille des données de cette base représente environ 14 giga-octets pour l’homme et 39 giga octets pour la femme. Il ne fallait pas grand-chose pour que les deux projets « Visible Human » et « Voxel-Man » se croisent. Les données de « Visible Human » ont donc été traitées par le logiciel de « Voxel-Man » pour obtenir, entre autres, les rendus, où les couleurs ne sont plus cette fois issues de procédés artificiels mais d’une réalité photographique (Figure 1 - 4).

Figure 1 - 4. Quand le projet « Voxel-Man » rencontre le projet « Visible Human »

1.4.2 Vers l’automatisation de la segmentation des images médicales

L’utilisation d’atlas anatomiques dans un processus de segmentation des images médicales est très fréquente. Dans ce contexte, la première approche possible est la segmentation uniquement fondée sur le recalage à partir d'un atlas. En général, la correspondance entre un volume de référence et le volume traité est établie en deux étapes, d'abord avec un recalage rigide ou affine, puis à l'aide d'un


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algorithme de recalage élastique. L'atlas associé au volume de référence peut alors être projeté sur le volume traité au moyen de cette transformation. Le résultat de cette projection fournit la segmentation des structures délimitées dans l'atlas. Dans le domaine de la neuro-imagerie, une référence classique est l'atlas de Talairach [TAL' 88]. Il permet de replacer le cerveau dans un référentiel défini à partir des points caractéristiques standard et peu variables d’un sujet à l'autre, en l'orientant et en appliquant des facteurs de proportions. L'utilisation de l'atlas lui-même est plutôt dédiée au repérage des noyaux internes pour la chirurgie, mais le référentiel associé est considéré comme standard et utilisé dans de nombreuses méthodes de segmentation automatiques. Collins et al. [COL' 97] proposent de segmenter simultanément diverses structures cérébrales en calculant un champ de déformation dense entre deux sujets. Dans le même ordre d'idée, Dawant et al. [DAW' 99] appliquent une transformation globale puis locale de façon à segmenter le cerveau, le cervelet et le noyau caudé sur une série de neuf volumes. La méthode est ensuite utilisée pour quantifier des atrophies cérébrales [BON' 05, DAW' 99]. Une autre possibilité consiste à utiliser un atlas pour initialiser ou guider un processus de segmentation. Warfield et al. [WAR' 95] proposent de segmenter le cortex à partir d'un recalage combiné à un algorithme de croissance de région qui prend en compte la distribution d'intensité de l'image. Ce procédé sert ensuite à améliorer la segmentation des lésions de la matière blanche, dont les intensités sont proches de celles du cortex. Dans [SEG' 04], un atlas est utilisé pour guider un snake (contour actif) afin d'affiner l'extraction du cerveau en corrigeant de petites erreurs locales. D'autre part, Bach, Pollo et al. [BAC' 04, POL' 05] proposent de segmenter des structures cérébrales sur des sujets présentant une lésion de grande taille. Les auteurs commencent par effectuer le recalage d'un atlas vers le sujet traité, puis injectent un germe de lésion dans l'atlas et le déforment avec un algorithme modélisant le grossissement de la lésion. De manière un peu plus spécifique, on peut considérer un sujet sur lequel la segmentation a bien fonctionné comme volume de référence, et calculer la transformation vers le sujet traité. La structure segmentée obtenue sur le volume de référence peut alors être projetée sur le sujet traité pour être utilisée comme initialisation de la segmentation [BAI' 01]. Enfin, les informations fournies par un atlas peuvent être combinées à certaines connaissances anatomiques sur la courbure, l’épaisseur ou le positionnement relatif des structures cibles afin d’affiner petit à petit une image étiquetée pour conduire à une segmentation simultanée de plusieurs structures [BOS' 03].


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1.4.3 L’assistance passive pour la thérapie

Il s’agit d’assistance passive dans le sens où l’ordinateur aide le médecin en affichant des informations utiles mais n’intervient pas dans le diagnostique ou la thérapie elle-même. Ainsi dans le cas du simulateur de chirurgie craniofaciale présenté dans [DEl' 94], où l’utilisateur découpe « virtuellement » des fragments du crâne pour les réorganiser, la mise en correspondance des données du patient avec l’atlas permettrait d’étiqueter automatiquement certaines zones, de mettre en évidence les difformités et par la même de planifier la procédure chirurgicale. Olivier Commowick [COM' 07] a utilisé un atlas anatomique numérique pour la radiothérapie des tumeurs cérébrales et des tumeurs de la sphère ORL. Le but principal était de proposer une méthode automatique permettant de segmenter les structures d'intérêt dans deux régions principales : le cerveau et la sphère ORL. Cette segmentation automatique permettra d'aider les thérapeutes à effectuer la planification en leur proposant directement une segmentation des structures d'intérêt de la région souhaitée.

1.4.4 L’analyse de la forme et des déformations

Le but de cette application est d’obtenir des paramètres morphométriques quantitatifs (la morphométrie est « l’étude quantitative des formes biologiques ») en particulier, les coordonnées moyennes des structures anatomiques et leurs variances. Ces paramètres permettent l’analyse de la forme et des déformations des structures biologiques. La morphométrie permet aussi de comparer deux structures biologiques, soit l’une saine et l’autre pathologique (syndrome de Down [WEI' 91] , maladie d’Alzheimer [MAR' 94]) , soit deux structures qui ont varié [DEA' 96]. Il devient alors aussi possible de créer un patient virtuel, c’est-à-dire, une modélisation dynamique d’un atlas du patient. Le patient virtuel peut être déformé vers des cas pathologiques en fonction des paramètres morphométriques caractérisant une maladie donnée [TUR' 08]. Une des applications du patient virtuel est la simulation chirurgicale [HEN' 95, MES' 00, MES' 97].

1.4.5 Vers l’automatisation du diagnostic et de la thérapie

Les résultats quantitatifs permettent d’envisager des applications avec diagnostic automatique. Le recalage entre l’atlas morphométrique et les données du patient permet d’analyser statistiquement la localisation des structures anatomiques. Un bon exemple se trouve dans [SUZ' 95] où une base de connaissance permet d’examiner les données du patient et de trouver les parties pathologiques du système nerveux. Un autre exemple se trouve dans [GLE' 03] pour une application du diagnostic des maladies du foie. En combinant un tel diagnostic automatique et la planification de trajectoire étudiée par les roboticiens, il devient possible d’automatiser certaines procédures d’opérations chirurgicales ou de radiothérapie [COM' 07, DAV' 06].


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1.5 Création d’atlas anatomique

De nombreuses méthodes ont été explorées dans la littérature afin de créer des atlas anatomiques numériques. Dans cette section, quelques travaux choisis concernant la construction d’atlas anatomiques de l’homme et du petit animal (embryons de souris) seront présentés.

1.5.1 Atlas anatomique de l’homme

Les travaux présentés ci-après utilisent, d’une part les données de l’imagerie médicale, et d’autre part les techniques de traitement d’image (filtrage, segmentation, recalage, reconstruction surfacique..) pour construire un atlas anatomique humain. Nous pouvons classer les étapes principales de la construction de chaque atlas en fonction de leur degré d’automaticité : entièrement manuelle, semi-automatique, entièrement automatique. U. Tiede et al [TIE' 93] ont présenté une méthode pour construire un atlas anatomique tridimensionnel du crâne et du cerveau humain à partir de séries d’images scanographiques (CT) et IRM bidimensionnelles. L’objectif était de développer un logiciel pour visualiser la tête et ses composantes sur différents axes de vue, avec la possibilité de visualiser les coupes quelle que soit la direction choisie. La segmentation des organes était faite manuellement par des experts radiologues sur un seul individu. G. Subsol [SUB' 95] a utilisé dans sa thèse une méthode entièrement automatique et générique pour créer aussi bien un atlas du crâne (Images CT) que du cerveau (Images IRM). Sa méthode était basée sur l’extraction et la mise en correspondance des caractéristiques représentatives du crâne de du cerveau (lignes de crêtes § 6.2.5). Les données de départ étaient des images médicales tridimensionnelles préalablement segmentées. La segmentation des images du crâne a été réalisée par seuillage simple, tandis que pour segmenter les images du cerveau il a utilisé le seuillage avec des outils de morphologie mathématiques (ouverture). En 2000, J. M. Sulivan et al. [SUL' 00] ont proposé une méthode pour générer un modèle 3D des organes humains, à partir d’images scanographiques et MRI prises directement de la base de données du projet « Visible Human ». La segmentation était manuelle pour quelques structures et semi-automatique pour d’autres. La reconstruction 3D a été faite en utilisant la technique du « Marching cube [HAR' 98] ». Dans [NIK' 00], nous trouvons une procédure de construction d’un modèle du cerveau tridimensionnel statistique déformable à partir de 50 volumes MRI de patients correctement choisis par les experts. La segmentation était automatique, en utilisant une méthode de modèle déformable. Ensuite ils ont utilisé un recalage rigide entre les modèles. Ce modèle donne des paramètres servant pour la segmentation et le recalage d’un nouveau volume MR d'un autre patient n'existant pas dans la base de données. M. Ferrant et ses collègues [FER' 99, FER' 02] ont construit un atlas anatomique tridimensionnel du cerveau afin d’utiliser celui-ci pour la localisation et l'identification automatique


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des structures du cerveau dans les images MR. La méthode proposée est basée sur un recalage entre les structures d’atlas et les images MR d’un nouveau patient. La segmentation a été automatisée en utilisant la technique « watershed [VIN' 91]». Dans le travail de H. Park et ses collègues [PAR' 03], nous trouvons une méthode de construction d’un atlas anatomique abdominal pour quatre organes (le foie, les deux reins et la moelle épinière). 32 images MRI ont été prises puis segmentées manuellement par des experts. Ensuite, toutes les images ont été recalées entre elles afin de donner un modèle moyen de chaque structure anatomique. L’atlas a été utilisé comme une base de données pour aider à la segmentation d’une nouvelle image qui n’existe pas dans la base de données (Figure 1 - 5 ).

(a) (b)

(c) (d)

Figure 1 - 5. (a) une coupe extraite de la base de données à segmenter. (b) Information d’atlas.

Résultats de segmentation (c) sans et (d)avec l’information d’atlas [PAR' 03].

1.5.2 Atlas anatomique du petit animal

Les chercheurs se sont également intéressés à la construction d'atlas anatomique du petit animal. Notamment la création des modèles tridimensionnels des différentes structures de l'embryon de la souris. L'objectif est d'étudier l'évolution de l'embryon, le développement des structures


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anatomiques et les déformations liées à des modifications génétiques. Depuis une dizaine d’années quelques projets sont réalisés afin de créer des atlas plus précis et surtout plus détaillés. Le projet Emap « Edinburgh Mouse Atlas Project » de l’université d’Edinburgh est basé sur le livre de développement embryonnaire de la souris de Theiler [THE' 89] et Kaufman [KAU' 92]. L’objectif est de créer une série de modèles anatomiques tridimensionnels interactifs des embryons de souris aux étapes successives du développement (http://genex.hgu.mrc.ac.uk/). MAP « Mouse Atlas Project 2003» du laboratoire LONI ( Laboratory of Neuro Imaging ) de l’université à California de Los Angeles est un autre projet pour construire un atlas du cerveau de la souris ( http://www.loni.ucla.edu/MAP ). Le but de ce projet est de produire des bases de données et de rassembler l'architecture du cerveau, l'expression des gènes, et l'information de la formation de l’image dans une interface simple, pour mesurer les changements de l'expression des gènes, de l'anatomie et des taux de croissance dans le cerveau de la souris C57BL/6J. Dans ce projet les chercheurs ont utilisé des images MR histologiques et microscopiques pour créer l’atlas et les images sont alignées en utilisant les logiciels automatisés produits à LONI. La Figure 1 - 6 illustre un exemple extrait de l’atlas Map.

Figure 1 - 6. Un exemple de section du cerveau de la souris, dans le cadre du projet MAP

(http://www.loni.ucla.edu/MAP)

Le projet « Caltech µMRI Atlas of Mouse Development » supporté par l’institut de Beckman à Caltech ( http://mouseatlas.caltech.edu ) a été réalisé en collaboration avec le projet Emap. Le but de ce projet d'atlas est de produire : (1) un outil pour apprendre l'embryogenèse de souris, (2) un modèle standard pour comparer les sujets d'expérience, ( 3 ) une base de données d’images IRM pour l’embryon de la souris, (4) un cadre pour l’intégration de la connaissance au sujet du développement de la souris, (5) un atlas construit en utilisant des images de résonance magnétique microscopique (µMRI) qui fournit des images tridimensionnelles tout en préservant la géométrie normale de l’embryon ( Figure 1 - 7 ). Les images ont été segmentées manuellement utilisant le logiciel « Amira (http://www.amiravis.com) ».


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Figure 1 - 7. Le modèle de l’embryon de la souris développé dans le cadre du projet « Caltech µMRI Atlas

of Mouse Development (http://mouseatlas.caltech.edu) » ; (a) un modèle 3D de l’embryon (b) différentes

étapes du développement de l’embryon

R. E. Jacobs et ses collègues dans [JAC' 99] ont présenté une méthode pour construire un atlas anatomique pour un seul embryon de souris 6.5-16 d.p.c ( Day Post Coception ) afin d’étudier son développement. Ils ont utilisé une série d'image MR, la segmentation était manuelle pour les petites structures et semi automatique pour déterminer les grosses structures. M. Taguchi et autres présentent leurs travaux dans [TAG' 03] pour construire une structure tridimensionnelle des ventricules cérébrals d'une seule souris âgée de 8 semaines, à partir de 88 images 2D histologiques segmentées manuellement. Dans [BRU' 99], les auteurs présentent leurs travaux pour construire un modèle tridimensionnelle de l'embryon de souris au 9ème jour. A partir de 307 images histologiques pour un seul embryon, avec une segmentation manuelle, le modèle anatomique était capable de présenter la plupart des structures anatomiques de l'embryon en 2D et 3D et les relations entre les structures (Figure 1 - 8). Nous trouvons dans [DHE' 01] une construction d'atlas anatomique pour l'embryon de souris de 6 à 11.5 d.p.c, à partir de séries d'images µMR, la segmentation et le recalage étant manuelles. Pour distinguer les différentes structures, ils ont attribué à chaque organe une couleur artificielle spécifique, que l’on retrouve dans chaque image 2D, et dans la visualisation 3D, générée par le logiciel « Amira » (Figure 1 - 9).


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Figure 1 - 8. Vues 3D de l’embryon de la souris au 9ème jours où l’amnion a été enlevé [BRU' 99]. (a) la

surface de l’ embryon (b) tissu neural (jaune), vésicules optiques (orange), trou optique (orange pâle),

métamère (vert foncé), artères (rouge orange), veines (bleu foncé), coeur (lilas), ventricule primitif

(magenta), cordis de bulbus (rouge), et intestin (bleu clair).

A B C D

Figure 1 - 9. Les modèles 3D de l’embryon de la souris [DHE' 01] (a) toutes les organes, (b) système

nerveux central, (c) squelette, (d) toutes les combinaisons des systèmes (dans ce cas-ci, entérique,

pulmonaire, circulatoire, etc.)


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1.6 Conclusion

Les processus de construction de quelques atlas anatomiques récents présentés précédemment dans ce chapitre, peuvent se résumer en quatre étapes principales : (a) acquisition des images, (b) segmentation, (c) recalage des images et (d) reconstruction du modèle tridimensionnel. Les atlas anatomiques consistent en une représentation étiquetée d'une partie de l'anatomie. Le choix de la représentation est différent selon l'application visée. Les structures anatomiques sont présentées par des dessins de couleur dans quelques atlas, et par des images médicales 2D et 3D issues de différentes modalités d’imagerie médicales pour d’autres atlas. Le maillage surfacique est un autre type de représentation utilisé pour modéliser l’anatomie humaine comme dans les projets Visible human, Voxel Man et CaveMan, et l’anatomie embryonnaire chez les souris comme les projets Emap, MAP, …etc. Les observations précédentes nous ont amenés à conclure que la représentation surfacique (maillage) donne plus de possibilité pour l’utilisation et l’enrichissement de modèle dans des applications de haut niveau. Dans le chapitre suivant nous nous concentrerons sur les méthodes existantes pour la construction de modèle surfacique (modèle maillé) à partir des données volumiques, le concept d’atlas numérique moyen (modèle moyen) et l’enrichissement de modèle.

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Ch 2 : Construction du modèle moyen enrichi

37

Ch.2. Construction du modèle moyen

enrichi


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2.1 Introduction

Compte tenu de la terminologie introduite dans le chapitre précèdent, nous nous concentrons dans ce chapitre sur la construction d’atlas numérique moyen enrichi (modèle moyen enrichi). Tout d’abord, nous résumerons de façon non exhaustive les méthodes existantes d’extraction de représentation surfacique des données volumiques (images 3D). Ensuite, nous analyserons quelques travaux existants concernant la construction d’atlas numérique moyen (modèle moyen) à partir d’ensembles d’apprentissage composés de plusieurs modèles représentant la même structure anatomique. Enfin, nous expliquerons la nécessité d’enrichir le modèle surfacique simple afin d’utiliser ce modèle enrichi dans une application de haut niveau. Une vue générale sur les différents types d’enrichissement ainsi que quelques travaux dans ce domaine sera présenté à la fin de ce chapitre.

2.2 Problème de reconstruction

Une image 3D est une grille tridimensionnelle où une intensité (ou niveau de gris) est associée à chacun des nœuds de la grille, ces nœuds sont appelée voxels. Les images numériques ne sont généralement appropriées ni à l’archivage et au transfert de données, ni à la représentation d’une géométrie complexe en 3D. Il faut donc extraire des représentations surfaciques des constituants de l'image, sous forme de surfaces ou de volumes. Les caractéristiques géométriques, topologiques, physiques ou statistiques des différentes composantes de l'image sont déduites de ces représentations. Cette extraction nécessite deux opérations essentielles, qui peuvent être disjointes ou couplées en un seul processus : la segmentation, qui réalise une partition de l'image en ses composantes, et la reconstruction, qui transforme les composantes détectées en structures surfaciques. L'ensemble de ces transformations sera appelé processus de segmentation/reconstruction. On distingue deux approches différentes au problème de la segmentation : l’approche région et l’approche frontière. Le fait d’opter pour l’une ou l’autre des ces approches détermine souvent si la processus segmentation/reconstruction est réalisée en deux étapes distinctes ou non :

L’approche région se base sur les caractéristiques propres des constituants (i.e. répartition des niveaux de gris, texture, homogénéité) pour construire une zone dont les éléments appartiennent à la composante que l’on recherche. On dit que les voxels ont été classifiés. Cette approche nécessite des informations a priori sur les constituants pour pouvoir les différencier. Ensuite, un ensemble de zones (homogènes par exemple) est déterminée. Ces zones visent à définir les composantes de l'image. En collectant l'ensemble des voxels de chaque composante, une structure géométrique peut être extraite directement à partir de cet ensemble. Les propriétés (géométriques ou physiques) que l'on pourrait attribuer à cette structure n'influent donc pas sur la forme extraite : seule l'étape de segmentation différencie les constituants de l'image. Des algorithmes discrets sont donc employés pour construire


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rapidement une représentation géométrique à partir de la classification des voxels (Figure 2 - 1 (A)).

Données volumétriques

Segmentation / Classification

Reconstruction discrète

Application

Segmentation / Reconstructionpar modèle déformable

A B

Figure 2 - 1. extraction de formes composant une image volumétrique par : (A) classification de

ses voxels puis reconstruction discrète, (B) modèle déformable.

L'approche frontière exploite les différences entre régions de l'image pour détecter les bords de chaque constituant. En conséquence, cette approche est plus indépendante des caractéristiques propres de chaque constituant. Elle est en revanche dépendante de la netteté des bords entre chaque constituant. On constate que cette approche ne construit pas une classification des voxels et la segmentation des données est partielle après cette étape. L'étape de reconstruction intègre donc une importante partie de l'étape de segmentation car elle doit extraire des structures géométriques à partir des données brutes et des données disparates de bords. Pour ce faire, il est indispensable d'introduire un modèle (géométrique, topologique, physique et/ou statistique) pour guider la reconstruction. Les propriétés associées au modèle sont exploitées pour combler l'information manquante. Naturellement, cette approche de la segmentation/reconstruction ne peut en général se faire de manière directe. Elle résulte de la convergence d'un processus qui, en modifiant progressivement les paramètres du modèle, cherche à approcher des formes de l'image selon certains critères associés au modèle. On parle donc de modèles déformables (Figure 2 - 3 (B)).

2.3 Méthode discrète de reconstruction

2.3.1 Segmentation et classification des données

La segmentation d’image est une opération de traitement d’images de bas niveau qui consiste à localiser dans une image les régions (ensembles de pixels) appartenant à une même structure


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(objets ou scène imagées). Cette opération est à la base de nombreuses applications tant en vision industrielle, qu’en imagerie médicale. De nombreuses recherches ont eu lieu dans le passé sur les méthodes de segmentation. Plusieurs algorithmes de segmentation d’image, de filtrage et d’amélioration de l’image peuvent être combinés afin d’extraire la région d’intérêt des données brutes. Cette thématique a été un thème de recherche très actif pendant de nombreuses années et a conduit au développement d’algorithmes performants (c.f. [ZOU' 04] pour une synthèse de techniques de segmentation d’image) et ne sera pas détaillé ici.

2.3.2 Extraction directe de modèle surfacique

Extraire un modèle surfacique à partir d'un calcul direct et local sur une image volumétrique peut servir à deux objectifs : d'une part, à la construction et la visualisation rapide de certaines composantes de l'image, d'autre part à la fabrication d'une représentation surfacique à partir d'images dont les voxels ont été classifiés par un processus de segmentation antérieur. Le premier objectif induit des algorithmes qui rassemblent le processus de segmentation et le processus de reconstruction. Comme ce processus calcule en une seule passe et localement le résultat, la segmentation produite doit être relativement élémentaire, en général, seul un simple seuillage est possible. Le modèle reconstruit peut prendre différentes formes suivant que l'on privilégie la réutilisabilité du modèle ou sa rapidité d'obtention. L'application principale est de permettre au praticien d'explorer l'image de manière quasi-interactive et de lui fournir un premier aperçu des données. La segmentation n'est souvent pas assez précise pour que le modèle reconstruit soit exploité directement dans d'autres applications (planification et simulation chirurgicale par exemple). En revanche, il peut servir d'initialisation à un processus type modèle déformable [MCI' 96]. Le deuxième objectif impose des algorithmes qui traitent en entrée des images dont les voxels ont été pré-segmentés. Ces algorithmes permettent alors d'extraire rapidement un modèle surfacique à partir d'une image binaire. En fait, les mêmes algorithmes sont utilisés indifféremment pour ces deux objectifs : visualisation rapide, ou extraction d'un modèle surfacique exploitable dans d'autres applications. Nous les classons suivant le modèle reconstruit : une surface triangulée, une surface digitale, une surface sous forme de B-rep « boundary representation ». Selon le but recherché par l'utilisateur, l'une ou l'autre est préférée.

Reconstruction sous forme de surface triangulée

Ces algorithmes sont en général utilisés sur des images volumiques de niveaux de gris, même s'ils peuvent directement être appliqués sur des images binaires, la valeur de seuillage étant alors implicite. Avec les images 3D en niveaux de gris (comme celle fournies par les imageurs IRM), la fonction de densité peut être vue comme une fonction implicite discrète dont la valeur est connue


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uniquement aux points ( , , )P x y z d’échantillonnage (les voxels). L'utilisateur fournit un paramètreπ , appelé iso-valeur. Une iso-surface est définie comme l'ensemble des points tels que ( , , )P x y z π= . La surface extraite par ces algorithmes, formée de triangles, doit alors approcher au mieux l'iso-surface définie. L'algorithme le plus répandu est certainement le « Marching-Cubes », développé par Lorensen et Cline [LOR' 87]; une version très proche a été développée indépendamment par Wyvill et al. [WYV' 86]. Son principe est de balayer l'image par cubes de huit voxels adjacents et de déterminer l’intersection de l’iso-surface avec ce cube. Après cette opération locale, il se déplace (marche) vers le bloc de huit voxels suivant. L’intersection de la surface avec un cube est calculée en attribuant le code 1 aux sommets du cube qui ont un niveau de gris supérieur ou égal au seuil prédéfini (iso-valeur) et zéro dans le cas contraire. Les sommets avec le code 1 sont à l’intérieur de la surface, ceux avec le code zéro sont à l’extérieur de la surface. On en déduit que l’iso surface coupe les arêtes qui relient un sommet intérieur à la surface (code 1) à un sommet extérieur (code 0). Nous avons 8 sommets par cube et deux états possibles pour les sommets. Il s’en suit 28 = 256 façons de couper le cube par l’iso surface. Toutefois, en tenant compte de configurations équivalentes qui se déduisent l’une de l’autre par symétrie, il y a seulement 15 configurations canoniques, notées de 0 à 14. A l'origine, cet algorithme pouvait produire des surfaces triangulées non fermées [DUR' 88] ce qui contredisait la définition de l'iso-surface dans un champ potentiel continu. Dans le cadre d'une simple visualisation, ce problème n'a quasiment aucune incidence sur le résultat affiché. En revanche, la correction de ce problème de fermeture devient nécessaire si l'on veut exploiter, au-delà de la simple visualisation, le modèle géométrique construit. De nombreuses méthodes ont été proposées pour modifier l'algorithme des marching-cubes de façon à ce que les surfaces construites soient fermées [GEL' 94, LAC' 96, NAT' 94, NIE' 91, ROL' 95, WYV' 86]. De nombreuses optimisations ont été rajoutées à l'algorithme initial, soit pour accélérer l'extraction de la surface [JUN' 95, SHU' 95, WIL' 92, ZHO' 95], soit pour construire efficacement l'adjacence entre les triangles [MIG' 97, ZHO' 95]. L'algorithme est également facilement parallélisable [MIG' 97]. D'autres auteurs [FRE' 96] ont profité des propriétés des décompositions simples de l'espace pour proposer un algorithme d'extraction d'iso-surfaces basé sur un balayage de tous les tétraèdres de l'image (une sorte de marching tetrahedra où tout cube de 8 voxels de l'image est décomposé en six tétraèdres). La surface générée est effectivement fermée. Dans tous les cas, le but est de produire une surface triangulée, fermée et orientable. Ces algorithmes permettent d'extraire assez rapidement un modèle surfacique d'un volume de données, même si le critère de segmentation est naïf (simple seuillage). Dans certaines modalités d'acquisition (e.g. IRM, échographie), un prétraitement est indispensable, par exemple sous la forme d'une classification. Enfin, si l'utilisateur ne souhaite que visualiser une iso-surface et non obtenir un modèle, certains algorithmes dédiés peuvent être préférés : génération d'un nuage de points [CLI' 88],


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construction d'un treillis approchant la surface[LAS' 92, THI, 96 b 96 b]. Enfin Lee et al. [LEE' 05] ont proposé l’algorithme de « Modified Marching-Cubes MMC » pour avoir des maillages plus réguliers que ceux des Marching-Cubes.

Reconstruction sous forme de surface digitale

Cette approche vise à extraire une surface, appelée surface digitale, formé d’éléments appelés surfels (surface elements). En topologie digitale, un surfel est l’intersection de deux voxels adjacents (face de voxel) [HER' 92]. Les algorithmes extraient les surfels qui sont à la frontière de l’objet. En pratique, l’utilisateur fournit une valeur de seuillage et l’algorithme recherche les éléments de surface sur le volume de données ainsi seuillé. Similairement au marching-cubes, les informations de niveaux de gris permettent de calculer les normales à la surface, utiles par exemple pour la visualisation. Différents algorithmes existent pour calculer ces surfaces. Leur principale spécificité est de ne pas balayer l’image mais plutôt de déterminer les surfels connexes à un surfel donné en initialisation. Ils se basent sur une notion d’adjacence entre les surfels qui ont une arête commune [UDU' 94] ou un sommet commun [LAC' 98]. L’approche étant surfacique, les algorithmes permettent donc d’extraire extrêmement rapidement (e.g. plus rapidement que le marching-cubes) une composante de surface de l’image 3D. Les surfaces extraites sont formées de faces de voxels, topologiquement connectées ou non suivant l’application. Le calcul et le rendu de telles surfaces sont très rapides. En visualisation, on peut même s’affranchir du problème des « marches d’escalier » en exploitant l’information de gradient de l’image pour approcher les normales. En revanche, leur utilisation dans un post-traitement et plus problématique que les surfaces triangulées.

Reconstruction sous forme de B-rep

Une autre approche a été proposée par Kalvin et al.[KAL' 91]. L'algorithme, appelé « alligator », construit une surface en examinant l'image coupe par coupe. A partir d'une image volumétrique binaire, une Boundary representation ou plus communément B-rep est construite par additions successives de morceaux de surfaces. L'algorithme se sert d'opérations classiques sur les B-rep, comme l'addition de nouveaux morceaux de surfaces, pour garantir la consistance topologique du résultat. De même, la décimation des facettes coplanaires est implicite dans cette représentation. L'inconvénient est que la structure manipulée pendant l'extraction est plus onéreuse en mémoire et en temps de calcul que les simples surfaces triangulées. Si elle construit effectivement un modèle géométrique utilisable dans un post-traitement, son intérêt pour la visualisation est moindre.

2.4 Segmentation/reconstruction par modèle déformable

Au contraire des approches précédentes, les modèles décrits dans cette section intègrent des paramètres géométriques, mais aussi des paramètres physiques, dynamiques ou statistiques dans


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leur formulation. Ceux-ci fournissent un ensemble de lois et de contraintes qui permet au processus d’extraction d’utiliser un mieux les données disponibles et de combler l’information manquante. Pour extraire effectivement les composantes d’une image brute ou d’une image de contours, il faut déterminer les paramètres d’un modèle de façon à ce que sa géométrie corresponde au mieux aux données. Les paramètres associés au modèle sont souvent dérivés de modèles physiques. De même, les paramètres « optimaux » du modèle ne seront pas déterminés en une seule passe, mais ils seront plutôt corrigés de manière itérative. C’est pourquoi le terme modèle déformable s’est imposé. Les modèles déformables sont des contours ou des surfaces qui évoluent d’un point de départ vers un état final qui doit correspondre à l’objet que l’on souhaite segmenter [KAS' 87, TER' 88]. Leur évolution est généralement régie par deux types d’information : un terme d’attache aux données qui attire le modèle vers les contours de l’image et un terme de régularisation. Leur évolution correspond à la minimisation de l’énergie suivante:

int( ) ( ) ( )extE X E X E Xα β= + (1)

où X est le contour ou la surface déformable, int ( )E X est l’énergie interne qui contrôle la régularité de la surface et ( )extE X est l’énergie externe qui l’attire vers les contours de l’objet recherché. D'autres forces peuvent enfin être ajoutées comme les forces de ballon [COH' 91] permettant d'éviter que le modèle ne se rétracte en un point. Ces méthodes ont tout d'abord été développées en déformant une forme de base circulaire ou sphérique. Les poids relatifs de ces énergies sont ensuite fixés par les paramètres α et β dans l’équation, permettant de régler l'influence de chaque énergie. Pour la segmentation d'organes dont la forme est globalement connue, de nombreuses méthodes, comme [MON' 99], introduisent un a priori plus fort sur la forme de la structure, en utilisant des modèles dont la forme est proche de celle recherchée dans l'image. [MON' 99] introduit également une force de rappel à une forme prédéfinie, permettant une meilleure segmentation de la forme finale. Toutes ces méthodes ne permettent de segmenter en général qu'un organe précis à la fois. Des méthodes de modèles déformables couplés ont donc été développées dans la littérature, comme par exemple [COO' 94] ou encore [CIO' 06], utilisant des ensembles de niveau en compétition et permettant d'introduire un a priori sur la position relative de structures. Dans l'objectif de la segmentation/reconstruction, les modèles représentant des surfaces sont les plus couramment employés, d'une part parce qu'ils sont moins coûteux en temps de calcul que les modèles purement volumiques, d'autre part parce qu'ils sont mieux adaptées à l'approche frontière de la segmentation. Une surface sans bord pourra représenter efficacement le bord d'une composante de l'image.

2.5 Atlas numérique moyen


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Les atlas numériques moyens (modèles moyens), visent à modéliser une anatomie moyenne. De nombreuses méthodes ont été explorées dans la littérature afin de créer ce type d’atlas. Très souvent, celles-ci se concentrent sur la création d'un atlas du cerveau. Certaines méthodes s'intéressent spécifiquement à la création d'une image moyenne à partir d'une base d'images segmentées. Toutes les approches dans la littérature se basent sur l'utilisation d'un algorithme de recalage non linéaire (élastique) afin d'amener toutes les images dans un référentiel commun. Le principal problème est alors le choix de ce référentiel. Choisir l'une des images de l’ensemble d’apprentissage comme référentiel introduit nécessairement un biais dans la construction de l'atlas. Certains articles [GUI' 00, JOS' 04, LOR' 05] ont développés des méthodes permettant d'éviter cette contrainte. Ces méthodes alternent entre un recalage des images sur une référence et l'application de la moyenne des transformations à l'image moyenne. Guimond et al.[GUI' 00] ont proposés une méthode pour construire un atlas numérique moyen (modèle moyen) du cerveau humain en utilisant un ensemble d’images de résonance magnétique obtenues chez des sujets sains. Ils ont montrés que cette méthode permet d'être indépendante du choix de la référence parmi la base d'images. Le modèle moyen (atlas) construit possède deux importantes propriétés généralement absentes dans les autres atlas numérique : une intensité moyenne et une forme moyenne des tissus du cerveau. Un atlas symétrique peut être généré directement à partir d'une base d'images [GRA' 06], cette méthode est basée également sur la méthode proposé dans [GUI' 00] mais utilise à chaque itération des recalages effectués à partir des images et de leurs symétriques. D'un autre point de vue, Marsland et al. [MAR' 03] sélectionnent l'image de référence comme étant le sujet qui minimise la somme des distances entre elle-même et les autres images. Park et al. [PAR' 05] procèdent d'une manière similaire en effectuant tous les recalages deux à deux entre les images de l’ensemble d’apprentissage. Cette méthode permet alors de calculer les distances entre les images (distance cohérente avec l'algorithme de recalage utilisé) et de choisir l'image de référence permettant d'avoir le moins de biais possible dans la construction de l'atlas. Bondiau [BON' 04] a étudié dans le cadre du cerveau, la création d’un atlas selon différentes méthodes. Ainsi, au cours de ses travaux, Bondiau a défini successivement trois atlas :

Un premier atlas a tout d'abord été défini comme étant l'image d'un sujet sain dont les structures ont été segmentées manuellement. Ceci introduit malheureusement un biais lié à la sélection du sujet utilisé pour l'atlas,

Par la suite, un second atlas avait été utilisé, créé à partir d'une IRM simulée d'une anatomie moyenne issue du BrainWEB (http://www.bic.mni.mcgill.ca/brainweb) [COC' 97, COL' 98, KAW' 96]. Cette anatomie moyenne permet de minimiser la distance entre l'anatomie de l'atlas et celle du patient. Un problème reste cependant présent, lié à l'asymétrie de l'atlas.

Enfin, un atlas moyen symétrisé a été développé afin de résoudre tous ces problèmes.


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Commowick [COM' 07] a proposé récemment une méthode permettant d'estimer conjointement les segmentations moyennes ainsi que l'image moyenne à partir d'images de la sphère ORL segmentées manuellement par un expert. La méthode de construction est composée de trois étapes principales (dans l'ordre où elles sont effectuées dans le processus de construction) :

1. la construction d'une image moyenne à partir des images de l’ensemble d’apprentissage [GUI' 00],

2. le calcul des segmentations moyennes à partir des segmentations manuelles individuelles ramenées dans le référentiel de l'image moyenne selon une méthode similaire à STAPLE [WAR' 04],

3. la symétrisation de l'atlas obtenu [BON' 04, PRI' 02].

Bailleul et al. [BAI' 03] ont proposés une méthode automatique afin de créer un modèle moyen de forme PDM 3D (Point Distribution Model) [COO' 95] de 4 structures anatomiques cérébrales (Thalami, Putamens, Noyaux Caudés, Hippocampes). Un modèle PDM 3D consiste en un modèle moyen complété par des modes et des amplitudes de variation de forme. Le rôle du modèle PDM 3D est de réguler une procédure de segmentation de structures d’intérêt en IRM 3D. Il s’agit de positionner correctement le modèle moyen de la structure sur l’image à segmenter, puis d’exploiter autant que possible les informations de l’image pour déformer le modèle itérativement jusqu’à une certaine convergence. Les modèle moyens sont crées à partir d’un atlas anatomique volumique de structure d’intérêt (Figure 2 - 2) et de série d’image IRMs anatomique de patients (27 volumes). Les principales étapes du processus proposé afin de construire un modèle moyen de forme PDM 3D pour chaque structure d’intérêt sont :

Segmentation du cortex cérébral. Afin que les recalages à venir ne soient pas perturbés par les intensités des formes osseuses et des tissus situés hors du cortex cérébral, ce dernier est segmenté sur chaque volume par la méthode de Ruan et al. [RUA' 02].

Création d’un ensemble volumique d’apprentissage. Un ensemble d’apprentissage pour chaque structure est construit en effectuant un recalage non-rigide de l’atlas anatomique (structures d’intérêts) vers chaque volume de patient (Figure 2 - 2).


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Figure 2 - 2. création d’ensemble d’apprentissage volumique [BAI' 03]

Création d’un ensemble surfacique d’apprentissage. Chaque volume d’un ensemble d’apprentissage est transformé en une surface triangulée, d’une part pour simplifier la recherche automatique de points caractéristiques, et d’autre part pour diminuer le temps de calcul.

Recalage rigide des surfaces. Tous les modèles de l’ensemble d’apprentissage surfacique sont recalés rigidement entre eux en utilisant la méthode ICP « Iterative Closest Point » [BES' 92].

Annotation de modèles. Chaque ensemble d’apprentissage surfacique est soumis au programme d’annotation développée par Kildely [KIL' 02]. Un triangle de référence est désigné manuellement sur chaque surface afin d’expliciter une orientation commune. L’objectif est de détecter des points correspondants, ou caractéristiques « landmark » sur chaque modèle d’un ensemble d’apprentissage. Un point de correspondance est un point labellisé, déposé automatiquement sur le même locus (endroit de forme) pour tout modèle de l'ensemble d'apprentissage.

Détermination du modèle moyen. En utilisant l’analyse de Procruste généralisée, connu sous l’acronyme GPA « Generalized Procrustes Analysis » [GOW' 75] (c.f. § 5.2), un modèle moyen est déterminé via un processus itératif de recalage rigide utilisant les points correspondants.

J. Lötjönen et al. [LOT' 04] ont introduit un modèle statistique de cœur comprenant les deux ventricules, les oreillettes et le péricarde. Ce modèle a été développé à des fins de segmentation automatique d’images cardiaques avec a priori statistique de forme. Les contours des structures ont été extraits manuellement à partir des données issues de 25 cas sains en IRM, et les surfaces maillées correspondantes reconstruites. Après recalage des surfaces sur une surface choisie comme référence, la référence est recalée non rigidement sur chaque segmentation. Un modèle de forme moyen est calculé à partir de la transformation moyenne.


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Dans le travail de Flores et al. [FLO' 05], nous trouvons une méthode pour construire un modèle moyen d’une tête humaine, avec les configurations faciales, à partir d’un ensemble de modèle d’une population normale. La méthode proposée a été appliquée pour construire une forme moyenne de l'oreille externe à partir d’une base de donnée composée de 24 modèles. D’abord, un recalage affine basé sur des lignes caractéristiques (lignes de crêtes) était fait pour amener tous les modèles dans un référentiel commun. Ensuite, les modèles recalés ont été transformés en volumes (voxelization) afin de calculer pour chaque modèle une carte de distance signée. Une nouvelle carte a été construite par l’addition de chaque carte de distance. Des facteurs de pondération ont ajoutés à la création de la dernière carte afin de donner plus d’importance à un modèle par rapport aux autres. L’iso-surface correspondant à l’iso-valeur 0 est considérée comme le modèle moyen.

2.6 Enrichissement du modèle maillé

Un modèle 3D (surface) est traditionnellement représenté par un maillage triangulaire. Il existe de nombreuses notations pour décrire la connectivité et la géométrie du modèle maillé (Maillage) qui présente une structure anatomique. Alors que la connectivité décrit les liaisons entre les sommets, la géométrie donne les coordonnées des sommets dans l’espace 3D. Une présentation de maillage surfacique par la géométrie et la connectivité seulement reste cependant trop simple pour modéliser une structure anatomique. Il faut donc des modèles plus riches si l’on veut utiliser ces modèles dans une application de haut niveau. L’enrichissement du modèle dépendra des applications envisagées par ce modèle. A partir du modèle lui-même nous pouvons localiser des caractéristiques ponctuelles sur la surface comme, par exemple, les points caractéristiques (e.g. les points extrémaux [TIE' 06, YOO' 05]), ou des caractéristiques linéaires comme les lignes de crêtes et de vallées [STY' 03, SUB' 95] (Figure 2 - 3). Aussi, à partir du modèle lui-même nous pouvons calculer d’autres informations globales issues sois d’analyses statistiques sur l’ensemble des points du modèle (Axes principaux, centre de gravité….) soit de mesures sur le modèle (volume, aire de surface …).


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Figure 2 - 3. Un exemple de lignes de crêtes et de vallées sur le cerveau [SUB' 95], lignes en bleu

correspondants aux lignes de crêtes et en rouge aux lignes de vallées.

Pour une autre application, où l’objectif est de présenter les différents composants de structure anatomique présenté par un modèle maillé, l’enrichissement peut être faites par la coloration de chaque composante par une couleur différente (Figure 2 - 4). On peut citer aussi les travaux de [BRU' 99], [DHE' 01] pour présenter les structures anatomiques de l’embryon de la souris (Figure 1 - 8, Figure 1 - 9).

Figure 2 - 4. Un exemple des possibilités de

l’atlas anatomique Voxel-Man commercialisé par

l’éditeur allemande Springer-Verlag :

visualisation tridimensionnelle avec colorisation

artificielle de chaque composant du crâne.

Bailleul et al. [BAI' 03] a utilisé les variations de points caractéristiques calculées à partir de l’ensemble d'apprentissage pour enrichir le modèle moyen. L'ensemble d'apprentissage annoté de n points caractéristiques aligné est considéré comme un nuage de points dans un espace 3D, dont l'origine est le modèle moyen. L'ACP (Analyse en composantes principales) [FUK' 90] est utilisé pour calculer les vecteurs et valeurs propres normalisées de la matrice de covariance de


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chaque point caractéristique détectée sur l'ensemble d'apprentissage aligné. Chaque vecteur propre décrit un mode de variation de modèle moyen, et la valeur propre correspondante indique l'importance que représente ce mode par rapport à la dispersion totale.

2.7 Conclusion

Dans ce chapitre, nous avons d’abord donné un aperçu des méthodes existantes pour créer un modèle surfacique à partir de données volumiques. La création de ce modèle est effectuée différemment suivant la nature de données volumiques :

Si les données sont adaptées à une approche région de classification ou si leur nature autorise un simple seuillage pour approcher leurs composantes, on choisira un algorithme discret d’extraction d’une surface (c.f. § 2.3).

Si les données sont brutes ou que leur nature impose une approche frontière à l’extraction de leurs composantes, une extraction par modèle déformable sera préférée. Le processus de segmentation sera indissociable de la construction de la surface (c.f. § 2.4).

Ensuite, la notion d’atlas numérique moyen (modèle moyen) et quelques méthodes existantes pour construire cet atlas à partir d’un ensemble d’apprentissage (base de données) ont été introduites. Enfin, nous avons brièvement décrit les principes de l’enrichissement du modèle représenté par un maillage surfacique.


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Conclusion de la première partie

A travers cette partie, un état de l’art, non exhaustif, sur l’atlas anatomique numérique est présenté dans le premier chapitre. Ensuite, dans le deuxième chapitre nous avons présenté les méthodes existantes pour créer un modèle surfacique à partir de données volumiques, puis nous nous sommes concentré sur la construction de modèle moyen à partir d’un ensemble d’apprentissage. Enfin, quelques exemples d’enrichissement de modèle surfacique sont présentés à la fin du deuxième chapitre. Nous avons vu que le type du modèle moyen dépend de l’origine des modèles de l’ensemble d’apprentissage. Autrement dit, si la base de données a été construite à partir d’images médicales, le modèle moyen sera une image moyenne (une intensité moyenne pour chaque pixel de l’image) et une forme moyenne [BON' 04, COM' 07, GUI' 00, MAR' 03, PAR' 05]. De la même façon, un ensemble d’apprentissage composé de maillages donne un modèle moyen surfacique [BAI' 03, FLO' 05, LOT' 04]. Nous situons nos travaux dans le cadre de la construction d’un modèle moyen enrichi à partir d’un ensemble d’apprentissage de modèles surfacique. Parmi les méthodes existantes, les méthodes proposées par Bailleul et al. [BAI' 03] et Flores et al. [FLO' 05] sont les plus répandues pour réaliser cet objectif. Nous pouvons diviser le processus de construction en deux phases principales :

phase de préparation : la mise en correspondance de chaque modèle de l’ensemble d’apprentissage avec un modèle de référence (initial) est la première étape de la construction d’un modèle moyen. Bailleul et al. [BAI' 03] ont utilisés un recalage non rigide de l’atlas anatomique (volume) vers chaque volume de patient pour construire un ensemble d’apprentissage volumique. Puis, chaque volume de l’ensemble d’apprentissage est transformé en une surface triangulée pour construire un ensemble d’apprentissage surfacique. Un recalage rigide basé sur la méthode ICP (Iterative Closest Point) est utilisé pour réaligner les modèles afin d’étudier leurs formes. Flores et al. [FLO' 05] ont utilisé une méthode proposée par Subsol [SUB' 95] afin de recaler les modèle d’oreilles externes avec un modèle de référence (un modèle choisi parmi les modèles d’ensemble d’apprentissage). L’idée de base est d’extraire des lignes caractéristiques (lignes de crêtes) sur tous les modèles, puis d’utiliser ces lignes pour effectuer un recalage grossier nécessaire à l’algorithme ICP. Ensuite, un recalage rigide basé sur la méthode ICP est utilisé pour diminuer les erreurs entre les modèles recalés.

phase de construction : Bailleul et al. [BAI' 03] ont utilisés un processus itératif de recalage rigide basé sur GPA (Generalized Procrustes Analysis) pour créer un modèle moyen pour chaque structure étudiée à partir de quelques points de correspondance détectés, de manière semi-automatique, sur les modèles (Annotation de modèles). Une autre méthode a été


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proposée dans [FLO' 05] pour créer un modèle moyen de l’oreille externe. Le point de départ était de transformer chaque modèle surfacique en volume binaire (voxelization) pour créer une carte de distance signée pour chaque modèle de l’ensemble d’apprentissage. Puis, à partir de ces cartes, une iso-surface est extraite correspondant de l’iso-valeur (0) représentant le modèle moyen recherché.

Dans la première phase, un processus de recalage de surface robuste est nécessaire pour amener les modèles dans un référentiel commun. L’algorithme ICP introduit par Besel en 1992 dans [BES' 92] est la plus connu pour le recalage de surfaces. Cet algorithme nécessite une initialisation préalable de la transformation entre les deux surfaces à recaler. Flores et al. [FLO' 05] utilisent les lignes de crêtes pour réaliser cette initialisation. Les lignes de crêts sont utilisables si la morphologie des surfaces permet d’extraire ces lignes (modèle de cerveaux, crâne, oreille externe, …). Nous proposons à travers le quatrième chapitre une solution afin de réaliser cette initialisation de manière entièrement automatique, utilisant des points caractéristiques stables calculable sur toutes les surfaces. La construction de modèle moyen utilisant des cartes de distances [FLO' 05] nécessite, d’une part une intégration de facteurs de pondération pour prendre en compte l’importance de chaque modèle, et d’autre part une iso-valeur pour détecter l’iso surface correspondant au modèle moyen. La méthode proposée dans [BAI' 03] est d’utiliser les points caractéristiques détectés de façon semi-automatique sur chaque modèle de l’ensemble d’apprentissage, afin de créer un modèle moyen par l’analyse généralisée de Procruste (GPA). Dans le cinquième chapitre, nous utilisons la méthode GPA pour créer un modèle moyen affine et élastique (GPA basé sur un recalage itératif affine et élastique). Ensuite, nous proposons une nouvelle méthode pour créer un volume binaire moyen (forme moyenne) et le modèle moyen correspondant. L’idée de base de cette méthode est de transformer chaque modèle d’apprentissage en volume binaire en utilisant une méthode de voxelization solide. Puis, un volume moyen et un modèle surfacique correspondant sont calculés à partir de ces volumes binaires en utilisant des opérations morphologiques. Enfin, une combinaison de la méthode GPA avec la méthode proposée est effectuée afin de prendre en compte les avantages des deux méthodes. Enfin, nous avons vu que l’enrichissement dépend de l’application envisagée. Nous envisagerons d’utiliser le modèle moyen enrichi, d’une part comme un modèle de comparaison dans un processus de phénotypage pour la classification du petit animal (modèles de crânes de souris), et d’autre part comme un modèle de référence dans un processus de segmentation a priori. Dans le sixième chapitre nous présenterons deux types d’enrichissement de modèle moyen. Le premier est l’enrichissement par des lignes caractéristiques (lignes de crêtes) extraites du modèle moyen lui-même. Le deuxième type introduit des informations de variabilités représentées sous forme de cartes de distances calculées sur l’ensemble d’apprentissage. Les cartes de distances sont présentées sur deux formats, volumique pour enrichir le volume moyen et surfacique pour enrichir le modèle (surface) moyen.


53

Dans la partie suivante, nous détaillons chaque étape de la méthodologie proposée.

54

Partie 2 : Anatomical Average model Construction Methodology

55

II. Anatomical Average Model

Construction Methodology

Partie 2 : Anatomical Average model Construction Methodology

56

Introduction to the second part

In the previous part we showed some examples how to construct an average digital atlas (image or meshed surface) using a set of training models. According to our survey, there is no actual general methodology to construct an enriched average model from a set of triangulated surfaces. In this part we shall detailed our proposed methodology to construct such a model using a set of triangulated models (3D triangular meshes). This part of the dissertation is organized as follows. An overview of our proposed methodology will be briefly introduced in chapter 3. Then, the surface registration problem and its basic theoretical concepts, which will be used in all steps of the methodology, will be presented in the first part of chapter 4, whereas our proposed method to register two surfaces using some feature points will be detailed in the second part of this chapter. A detailed description of our proposed methodology to construct an average model will be explained and discussed in chapter 5, followed by a description of the procedures in order to enrich the constructed average model in chapter 6. In the following four chapters, we will explain and detail each step of our proposed methodology.

Ch 3: Overview of the proposed methodology

57

Ch.3. Overview of the proposed

methodology


58

3.1 Introduction

In this chapter, a general description of the proposed methodology to construct an enriched average model is briefly introduced and illustrated via some schematic diagrams. The details of each step of the methodology will be explained in the following chapters of this part.

3.2 General description

The following work assumes that the input of our methodology is a set of models (triangulated meshes) built using the basic image processing algorithms (segmentation, reconstruction …), and the output is an enriched average model. The enriched average model is composed of the average model (average surface + corresponding binary volume) and its enrichments. Our methodology consists of two principal stages: average model construction (c.f. § Ch.5) and model enrichments (c.f. § Ch.6). A schematic diagram of the methodology is shown in Figure 3 - 1.

Average model construction (§ ch. 5)

Input

Model enrichments (§ ch. 6)

Output

Applications (§ ch. 7)

Training Set (TS)(triangulated meshes)

Enriched average model

average surfaceaverage binary volume

registeredtraining set

trainingbinary volumes

Figure 3 - 1. Schematic diagram of the proposed methodology.

In the following sections, we will briefly describe the two stages of the proposed methodology.


59

3.3 Average model construction

The objective of this stage is to build an average model using a set of training models (triangulated meshes). All the models in the training set represent the same anatomical structure obtained from normal subjects.

Affine GPA (§ 5.2)Voxelization (§ 5.3.2)

3D Morphology (§ 5.3.4)

Model Construction (§ 5.3.5)

TS

Average volume

Average surface

Rigid registration (ch. 4)

training binary volumesRM

Registered training set

OU

TPU

T

abc

d

INPU

T

Figure 3 - 2. The pipeline of the proposed method to construct an average model.

Figure 3 - 2 illustrates the proposed method to construct the average model which can be summarized in two steps as follows:

Initialization: the goal of this step is to eliminate the differences between all models of the input training set (TS) due to the translation and rotation. Each model of the training set is rigidly registered to a reference model (RM). The reference model is constructed using a GPA (Generalized Procrustes Analysis) algorithm based on iterative process of affine registration (c.f. § 5.2). Next, all models of the input training set are rigidly registered to RM. A robust feature points-based surface registration, introduced in chapter 4, is used to realize the registration in all steps of the methodology. In the output of this stage, all models of the training set are rigidly aligned and ready to use in the construction step.

Construction: first, training binary volumes are constructed by converting each model of the registered training set into a binary volume (voxelization) (§ 5.3.2). Next, an average volume is extracted from the training binary volumes using a set of basic 3D morphological operation (dilation, erosion……etc.) (§ 5.3.4). Finally, an average surface (triangulated mesh) is constructed from the extracted average volume (§ 5.3.5).

The outputs of this stage are: a registered training set (a), a set of training binary volumes corresponding to the registered models (b) and an average model (average binary volume (c) + average surface (d)). The average model will be used directly in the application and the other outputs (training binary volumes and registered training set) are used in the model enrichments stage to enrich the average model with some information extracted from all models from which it was constructed.


60

3.4 Model enrichments

As introduced in the first part, the choice of the suitable enrichments of the model depends on the envisaged applications using this model. In our work, we plan to utilise the enriched average model in two applications. In the first, this model will be used as a comparing model in a phenotyping process of small animals (mice). The enriched average model will be used also as a reference model in image segmentation process using a priori knowledge. To enrich the average model we can distinguish two types of added information: (a) information extracted directly from the average model itself, (b) information extracted from all models of the training set from which it was constructed. In order to add useful information to the average model for our envisaged applications, we propose to use four principal kinds of enrichments (Figure 3 - 3):

Features Lines (§ 6.2)Distance Maps (volumes) (§ 6.4)

Probability Map (volume) (§ 6.3) Measurements (§ 6.5)

Distance Maps (surface) (§ 6.3)

Average surface enrichments(ch. 6)

Average volume enrichments(ch. 6)

Enriched average volume Enriched average surface

Output

Registeredtraining set

Trainingbinary volumes

Averagebinary volume

Averagesurface

Enrichments

Figure 3 - 3. Average model enrichments

Feature lines-based model enrichments (§ 6.2): in this procedure, we try to extract some local features from the geometrical structure of the average surface itself. The crest and valley lines are some examples of such feature lines which can be extracted directly from the surface. These lines can be used in some surface processing applications such as registration, simplification …. etc.

Distance maps-based model enrichments (§ 6.3): we used two types of distance maps: ♦ Distance maps computed from the average surface and the registered training set

models. These maps are presented as scalar values added to each face (triangle) of the average surface. The gaol of these maps is to associate some statistical information (min, max, mean, standard deviation…) about the variation of measured


61

distances from this face to all other models in the training set. These kinds of enrichments are used in the phenotyping process ( Ch.7).

♦ Signed distance maps computed from the average binary volume itself and all volumes of the training binary volumes. These distance maps can be used in image segmentation using a priori knowledge.

Probability map (§ 6.3.3): This map is presented as a volume where each voxel has an intensity value in range (0, 1). This value is computed by counting the number of times the voxel under consideration was a foreground voxel in the training set volumes divided by the number of volumes in the training set. This map can be used also to improve the image segmentation using a priori knowledge.

Some measurements (§ 6.4): Some global descriptors of the models (volume size, area … etc) were measured from all training models in order to study the variation of these quantities.

3.5 Conclusion

In this chapter, we have briefly presented an overview of the proposed methodology to construct an enriched average model from a training set composed of triangulated meshes. In the following chapters we will detailed each step of this methodology.

62

Ch 4: Surface registration

63

Ch.4. Surface registration


64

4.1 Introduction

Registration process is an integral part of computer and robot vision systems and still a topic of high interest in both fields. The problem of surface registration arises in a number of different contexts, including object modelling (registering multiple surface scans into a common coordinate system to build a complete object model) and 3D recognition (matching a measured surface with a database of shapes). The registration of 3D surfaces has an important role in all the stages of our proposed methodology to build an enriched average model from a set of training models. This chapter is organised as follows. First, in section 4.2, the surface registration problem and its basic theoretical concepts are presented. Focus on the iterative registration methods is done and special attention is paid also on the ICP (iterative Closest Point) that is described on details because it is the principal registration technique used in our work. Then, some solutions of overcoming the main limitations of the ICP algorithm are presented. Second, in section 4.3, our proposed method to register two surfaces using some feature points is introduced and some examples of the registration are shown. The registration quality measuring problem is discussed in the end of this section. Third, the registration based on the elastic transformations which can be used to evaluate the local deformation of the surfaces is presented in section 0. Finally, we summarize the main ideas and the content of this chapter.

4.2 Background

4.2.1 Definition

The objective of the registration task is to find the transformation that best aligns two or more surfaces by minimizing some distance between them. In fact, how to efficiently estimate this transformation is one of the main issues in registration [ROD' 02]. Mathematically, the surface registration is defined as follows: given two 3D point sets

{ }jA a= (source) and { }iB b= (target) with { }1,....,i n= and { }1,....,j m= , the objective is to estimate the 3D motion, presented by the transformation Tr that best aligns the points in A with the points in B. We can define Tr by its transformation matrix M. Generally, the parameters of the transformation matrix M are found by using some minimization techniques over a set of matched data points. It is natural to use a geometric distance as an error measurement to evaluate the alignment quality in the iterations of registration process. The approximation error between two surfaces can be defined as the distance between its corresponding points. The distance ( , )d p S between a point p and a surface S can be defined as:


65

( , ) min ( , )d p S E p pp S

′=∈′

(2)

Where: ( , )E p p ′ is the Euclidean distance between two points ( , )p p ′ in R3. The one-side distance between two surfaces S1, S2 is then defined as:

11 2 2( , ) max ( , )

p SD S S d p S

∈=

(3)

This definition of distance is not symmetric. There exist surfaces such that 1 2 2 1( , ) ( , )D S S D S S≠ . A two-sided distance (Hausdorff distance) may be taking the maximum of 1 2( , )D S S and 2 1( , )D S S (see § 4.3.4):

( ) ( )[ ]1 2 1 2 2 1( , ) max , , ,H S S D S S D S S= (4)

The mean distance Dm between two surfaces is defined as the surface integral of the distance divided by the area of S1:

( )1

1 2 2

1

1( , ) , .m

s

D S S d p S dsS

= ∫ (5)

where: 1S denotes the area of S1. From this, the definition of a root mean square distance (error) Drmse follows naturally:

( )1

2

1 2 21

1( , ) , .rmse

s

D S S d p S dsS

= ∫ (6)

If the surfaces are orientable, we can extend the definition of distance between a point p of S1 and the surface S2 to a positive or negative distance. The distance 2( , )d p S is positive if the nearest point 2p S′∈ is in the outer space with respect to S1, and otherwise negative (Figure 4 -1). In other words, if Np is the vector normal to S1 in p and 2p S′∈ is the nearest point, then the sign of our distance measure is the sign of ( ).pN p p′ − .


66

p1

p2

p'1

p'2

S1

S2 1n

2nd1

d2

Figure 4 -1. Signed distance evaluation; distance is positive in p2 and negative in p1.

The signed distance gives an independent evaluation to the section of the first surface which locates in the interior or in the exterior space with respect to the second surface.

4.2.2 Classification of registration

Transformations-based classification

According to the transformation model from the source surface space to the target surface space, surface registration algorithms can be classified in two categories. The first broad category of transformation models includes the linear transformations which are a combination of translation, rotation, global scaling, shear and perspective components. Linear transformations are global in nature, thus not being able to use for local model deformations. There are two principal types of registration based on linear transformation: rigid registration (translation and rotation) and affine registration (translation, rotation and scaling). The second category includes “elastic” or “Non-rigid” transformations. These transformations allow local warping of surface features and thus provide support to local deformations. Non-rigid transformation approaches include polynomial wrapping, interpolation of smooth basis functions (thin-plate Splines and wavelets), and physical continuum models (viscous fluid models and large deformation diffeomorphisms).

Approaches-based classification

A popular type of approaches to solve the 3D registration problem is the iterative approach [BES' 92, CHE '92 b]. Iterative approaches have the advantages that they are fast and easy-to-implement. However, the drawbacks are that: (a) they require a good initial estimate to prevent the iterative process from being trapped in local minimum; (b) There is no guarantee of getting the correct solution for noiseless case.

Another popular type of approaches is the feature-based approach. Feature-based approaches extract invariant local features first and then find the correct correspondences of features for estimation of the rigid transformation between two partially-overlapping 3D data sets


67

[CHE' 04, CHU' 96, FEL' 96, HIG' 95, HUB' 03, JOH' 99, JOH' 00, SHA' 02, THI' 96 a, WYN' 02, WYN' 99, YAN' 98]. Feature-based approaches have the advantage that they do not require initial estimates of the rigid-motion parameters. Their drawbacks are mainly that: (a) they can not solve the problem in which the 3D data sets contain no local features; (b) a large percentage of the computation time is usually spent on pre-processing, which includes extraction of invariant features and organization of the extracted feature-primitives.

Techniques-based classification

The registration techniques differ as to whether initial information is required, so that a rough registration can only be estimated without initial guess. If an estimated transformation between surfaces is available, a fine registration can be computed. In coarse registration the goal is to compute an initial estimation of the transformation between two 3D point sets (surfaces) using correspondences between them. Most of these methods are based on finding correspondences between distinctive features that may be present in the two data sets. The principal term to classify the coarse registration methods is the kind of corresponding. In general, the most common correspondence method used is point-to-point, such as the point signature [CHU' 97], and the method of spin-image [JOH' 97]. However, there are other methods that align lines, like methods of bitangent curves [WYN' 02, WYN' 99] and other that match the surface directly, like algebraic surface model [TAR' 98]. In general, coarse registration methods are iterative. However, a few coarse registration methods provide linear solutions, like the methods based on principal component analysis (PCA) [KIM' 03] or other algebraic surface model [TAR' 98]. In fine registration, the goal is to obtain the most accurate solution as possible. The fine registration approaches are based on the assumption that a good initial transformation (i.e. close to the solution) was previously obtained. Then, precise alignments may be obtained with reliable criteria to measure the quality of the refined transformations. The best-known methods for fine surface registration are variations of the Iterative Closest Point (ICP) algorithm [BES' 92]. ICP is an iterative descent procedure which seeks to minimize the Mean Squared Error (MSE), computed by summing the squared distances between points in one surface (source) and their closest points, respectively, in another surface (target). Since ICP is an iterative descent algorithm, it requires a good initial estimate in order to converge to the global minimum. Additionally, a number of registration methods were proposed to combine both techniques, a coarse registration and a following fine registration, to achieve automatic and precise registration results [CHE' 99, HUB' 03, SAP' 01].

4.2.3 Iterative closest point ICP

The ICP (originally Iterative Closest Point, though Iterative Corresponding Point is perhaps a better expansion for the abbreviation [RUS' 01] ), which was first introduced by Chen and Medioni [CHE


68

'92 b] and Besl and McKay [BES' 92], has become the dominant method for 3-D surface registration. Since the introduction of ICP, many variants have been introduced into the basic ICP concept.

Given two 3D point sets (surfaces) { }jS s= (source) and { }iT t= (target) with

{ }1,....,i n= and { }1,....,j m= , the goal of the iterative process is to minimize the Mean Squared

Error (MSE), computed by summing the squared distances between points in S and their closest

points, respectively, inT .

21.

1

nt M sj in i

MSE ∑ −=

= (7)

where: jt is the closest point in T to the point is S∈ , M is the transformation matrix. The

complexity of ICP is ( . . )O i m n (where i is the number of iterations).

ICP starts with two surfaces and an initial guess for their relative rigid-body transform, and iteratively refines the transformation by repeatedly generating pairs of corresponding points on the surfaces and minimizing an error metric (MSE). ICP iterative algorithm has three basic steps:

Pairing each point of S to the closest point in T. Computing the transformation that minimizes the mean square error between the paired

points. Applying the transformation to S and updating the mean square error.

The major drawback of an ICP based algorithm is that it needs a good initial guess of the true transformation. Many methods are proposed in the literature in order to solve this problem. Most of these methods are based on finding correspondences between distinctive features that may be present in the point sets. The basic procedure involves the identification of features, assignment of feature correspondences, and alignment based on these correspondences. There are many different features that can be used to initialize the iterative registration process. Chaua and Jarvis [CHU' 96] used principal curvatures to constraint a heuristic search of correspondences. Feldmar and Ayache [FEL' 96] performed affine registration by minimizing the combined distance between positions, surface normals and curvatures. Thirion [THI' 96 a] used crest lines to extract extreme points and their associated Darboux frames, which are matched in an ICP-like fashion. The spin image presented by Johnson and Hebert [JOH' 99], which is a data level shape descriptor, has been used by Hubber et al. [HUB' 03] in an automatic registration process. The RANSAC-based data-aligned rigidity-constrained exhaustive search algorithm (DARCES) [CHE' 99] can solve the rigid registration problem without any initial estimation by using rigidity constraints to find the corresponding points. First, three points (primary, secondary,


69

and auxiliary) in the reference surface are selected. Then, each point on the test surface is assumed to be in correspondence to the primary point and the other two corresponding points are found based on the rigidity constraints. For every corresponding triangle, a rigid transformation is computed and the transformation with the maximum number of overlapping points is chosen as the solution.

4.2.4 Discussion

The goal of surface registration in the average model construction methodology is to eliminate some anatomically non significant differences between the models of the training set. First, a rigid registration will be used to minimize the distances between all surfaces of the training set due to translation and rotation. Next, affine and elastic registrations will be used in the GPA iterative process (c.f. § 5.2) to construct affine and elastic average model. In order to realize this goal we need a fast and robust method which can align two surfaces together without initial guess. Iterative approaches have the advantages that they are fast and easy-to-implement but they require a good initial estimate to prevent the iterative. The feature-based approaches have the advantage that they do not require initial estimates of the rigid-transformation parameters. Their main drawback is that they can not solve the problem in which the 3D data sets contain no local features. The RANSAC-based data-aligned rigidity-constrained exhaustive search algorithm (DARCES) [CHE' 99] can solve this problem well for the rigid registration. Due to the exhaustive nature of the search, the solution it finds is the “true” one. Theoretically it is a good method for the rigid registration. However, the precision depends on the resolution of the surface and the time increases concededly with the number of points, so it can be only used in application where time is not critical and the number of points in each model is relatively small. The combination of the iterative and feature-based approaches can solve our problem to achieve automatic and precise registration results. In the combination, we can overcome the drawbacks of the two approaches. In the following section we propose a feature points-based surface registration method composed of two principal stages. First, a coarse registration based on feature points is performed to bring the source model close to the target. Next, a fine registration based on ICP is used to achieve precise registration results.

4.3 Feature points-based surface registration

Our proposed method to register two surfaces (source and target) based on linear transformation consists of two stages. An initial registration (coarse registration) and a final registration (fine registration). The first one will help the convergence of the registration and will get an initial estimate of the relative pose of the final registration (ICP algorithm). In the following sub sections we will detail all steps of the proposed method.


70

4.3.1 Overview of the proposed method

The inputs of our algorithm are two meshed surfaces S, T (source, target) and the output is a matrix Mfin of the transformation which best registers the source with the target and minimizes an objective function D(S, T). The basic idea is to use a coarse registration (initial registration) to bring the source surface in closing to the target, “if necessary”. Then, we perform a finer registration (final registration) in order to get a precise alignment of the two surfaces. As shown in Figure 4 - 2, we can summarize the initial registration phase as follows:

Surfaces normalization (§ 4.3.2): bring the two surfaces in an isotropic normalized pose using anisotropic scale transformations [KAZ' 04]. The outputs M1, M2 are the transformation matrices for source and target respectively. The two surfaces S, T are transformed using M1, M2 respectively to the two isotropic normalized poses M1S, M2T of source and target, respectively.

Feature points extraction (§ 4.3.3): extract the feature points of both M1S, M2T. The feature points are the vertices where a chosen function f(v) has a local maximum or minimum in the neighborhood of the vertex v (k-ring neighbors of v). The chosen function f(v) is the sum of all pairs shortest path distance over the vertex v [COR' 01]. The feature points consist of two sets, the first is the local minimum points (FmnPS, for M1S, FmnPT for M2T) and the second is the local maximum points (FmxPS for M1S, FmxPT for M2T) (§ 4.3.3).

Feature points matching (§ 4.3.4): the objective of this step is to obtain the best transformation matrix M3 which minimizes some distance function H(M3.M1.S, M2T), where H is the approximated sum of the Hausdorff distance between the registered source and the target. Then the searched initial transformation matrix Mini is:

.1 .2 3 1M M M Mini−= (8)

M1S

FmnPS FmxPS FmnPT FmxPT

M2T

Source (S) Target (T)

Surface Normalization Surface Normalization

Feature Points Extraction Feature Points Extraction

Feature Points Matching

M3.M1SMini=M2

-1.M3.M1

S

T

Mini

Figure 4 - 2. Diagram of the proposed initial registration stage


71

The final registration (Figure 4 - 3) can be summarized as follows: Use Mini as the initial transformation matrix of coarse registration to obtain MiniS , the initial

pose of the source S. Align MiniS with T Using the ICP registration algorithm. The output of this step is M4 , the

matrix of best transformation which modifies the source surface MiniS to be similar to the target surface T with the minimum error.

Finally, output the matrix M4 of the transformation which best register the source surface MiniS with the target surface T. then the final transformation matrix M fin is given by:

4 .M M Minifin = (9)

Surface Transformation

Mfin

SMini

MiniS

T

M4

Initial Registration1

ICP2M4Mini 1 2

Mfin=M4.Mini

S TFinal Transformation

Figure 4 - 3. Diagram of the proposed final registration stage

4.3.2 Surface normalization

The key idea of anisotropic scale transformation is based on the observation that much of the challenge of surface registration is in the establishing of correspondences and if the two models are both isotropic, then, it is easier to establish correspondences between them. In order to separate anisotropy from the surface registration matrix, we propose first to remove the anisotropy from each of the two models. Given two 3D point sets { }21 , ,..., nS s s s= , with

{ }, ,ii i ix y zs s s s= and { }1 2, ,..., nT t t t= , with { }, ,i

i i ix y zt t t t= , the sum of squared differences between

the two corresponding points of the point sets is given by:


72

2 2 2( , ) {( ) ( ) ( ) }

1

n i i i i i iSD S T s t s t s tx x y y z zi

∑= − + − + −=

(10)

Kazhdan [KAZ' 04] in his work found that the squared difference between the two models is minimized if the two point sets are uniformly rescaled (normalized), so that their variance in each direction is equal to 1. In order to transform an arbitrary point set into one that has unit variance in any direction, it suffices to compute its covariance matrix C and then transform the point set using the transformation matrix

12C

−

. The covariance matrix C of a point set { }1 1, ,..., nS s s s= is given by the following formula:

( ) ( )1

1 .n

Ti m i m

iC s c s c

n =

= − −∑ (11)

where 1

1 n

m ii

c sn =

= ∑ is the centre of gravity of the point set. C is a kxk symmetric matrix. Then the

covariance matrix C can be expressed in terms of its k eigenvalue-eigenvector pairs ( iλ , ei) as

follows [KAZ' 04]:

1. .

k TC e ei i iiλ= ∑

= (12)

The inverse square root of C:

1 2 1 . .1

k TC e ei ii iλ− = ∑

= (13)

In our case k=3, then the inverse square root of C is a 3x3 matrix. The transformation matrices M1 and M2 which will transform S, T into their isotropic poses, are the inverse square roots of CS, CT (source covariance matrix, target covariance matrix):

1 2 1 2,1 2M C M CTS− −= = (14)

Then any surface model can be normalized by transforming it using the inverse square root

of its covariance matrix as a transformation matrix.

The difficulty in applying this method directly to triangulated models is that the

transformation 12C

− rescales area patches as a function of their normal direction. Thus, points that

are uniformly distributed along the untransformed model no longer need to be uniformly

distributed on the anisotropically rescaled model. In order to address this issue, Kazhdan [KAZ' 04]

proposed an iterative approach to transform the model. At each step of the iteration, the model is


73

first transformed so that its centre of mass is at the origin, then the covariance matrix is computed,

and finally the model is rescaled by the inverse square root of the covariance matrix.

Figure 4 - 4 shows an ellipsoid model and its poses after several steps of the anisotropic scaling iterative process. Note that, after the first iteration, the transformed model is still not isotropic, though, as the figure indicates, the iterative process converges quickly to an isotropic model after 3 iterations.

1st iteration 2nd iteration 3rd iterationInitial

Figure 4 - 4. A visualization of an ellipsoid model after one, two and three iterations of anisotropic scaling

shown at the right. Note that, the transformed ellipsoid is very isotropic after the third iterations of

anisotropic scaling (converging to sphere).

Figure 4 - 5 demonstrates the anisotropic transformation process for a tooth model after 3 iterations. The second step of the proposed method is to extract some feature points which will be used as landmarks in the coarse registration. The feature points will be extracted in both anisotropic source and target models.

1st iteration 2nd iteration 3rd iterationInitial

Figure 4 - 5. A visualization of a tooth model transformed using an iterative anisotropic scaling

transformation.

4.3.3 Feature Points Extraction


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Since the inputs of the registration algorithm are usually large, a common speedup technique is to pick a set of feature points on the inputs surfaces based on the computed descriptor values. Next, the registration is performed only with respect to the feature points, which results in significant reduction of the size of the search space. Several functions have been proposed by the computer graphics community to extract the feature points from 3D models. In all the proposed functions a metric system is needed. We will distinguish two main systems: (a) Euclidean system; (b) Geodesic system. Geodesic metrics can be viewed as a generalization of Euclidean metrics in curved space theory. Rotation and translation invariance properties can be obtained in both systems, scaling invariance property can also be obtained in both systems using normalizes values. The geodesic metrics are invariant to model pose. The geometrical feature points are mesh vertices located on extremities of prominent components of the surface. These points are generated by defining a function (descriptor) f(v) over each vertex v. Hilaga et al. [HIL' 01] used a height function where any vertex is associated its own z-coordinate in a suitable coordinate system. The height function turns out to be good for terrain data. Mortara and Pantanè [MOR' 02] proposed to use the vertices where Gaussian curvature exceeds a given threshold as feature points. Katz et al [KAT' 05] developed an algorithm for mesh segmentation using feature points: first, the mesh vertices are transformed into a pose-invariant representation, based on the theory of a multi-dimensional scaling (MDS), next, the prominent feature points in the convex hull of the MDS transformed mesh are extracted. In our proposed method, to guarantee the invariance to rotation, translation, scaling and model pose, we use the geodesic metrics. In addition, using a geodesic distance provides robustness against noise. The extraction of feature points is performed over the normalized surfaces (source and target) so it is also invariant to scaling. The chosen function (descriptor) f is the sum of all-pairs shortest geodesic paths at each vertex v [COR' 01]. The function f at a vertex v on a surface S is constructed as follows:

( ) ( , )f v g v p dsp S

= ∫∈

(15)

where the function ( , )g v p returns the geodesic distance between vertices v and p on S . Since ( )f v is defined as a sum of geodesic distance from v to all points in the surface, a small value of ( )f v means that the point v is nearer to the centre of the object. The function ( )f v is discretely

approximated by:

( ) ( , )1

pn

f v g v ii

= ∑=

(16)

where n is the number of vertices on the surface. To formally define the feature points we require that these vertices satisfy the conditions described below.


75

v S∀ ∈ , let ( )1 2, ,....,v nN v v v= be the set of neighbouring vertices of vertex v ( m rings of neighbour’s vertices). The local condition that a feature point should satisfy is:

( ) ( )

( ) ( )

:

:

i v i

i v i

v N f v f vorv N f v f v

∀ ∈

∀ ∈ ≺

in other words, a feature point is a vertex which has a local maximum or local minimum value of the function f . The value of the function f at each vertex is calculated using all-pairs shortest path algorithm introduced in [COR' 01] based on Dijikstra’s algorithm [DIJ' 59, SKI' 90]. Figure 4 - 6 presents some feature points extracted on 3d models using the proposed method.

Figure 4 - 6. Vertebra and foot models with some of its feature points (green balls present some local

maximum feature points, blue ball presents a local minimum feature point)

One of the most important properties of the used function is that for each surface there exist at least two feature points (one local minimum and one local maximum feature points). The computation cost of the integral of geodesic distance calculation is quite high for surfaces with a high number of points. So, in this case, we can simplify the surface to decrease the computation cost, for the feature points extraction process. We used the algorithm introduced by Hoppe [HOP' 99] to simplify the two surfaces. Moreover, in order to select feature points we observe the geodesic gradient behaviours of the used function. Consequently, we can state that the used algorithm is robust against the mesh sampling variations as illustrated in Figure 4 - 7. Note that, the feature points do not appear exactly on the same vertices but in the same geodesic neighborhood. So, these points will be used in the phase of coarse registration.


76

(a) 36616 points (b) 3000 points (c) 1000 points (d) 200 points

Figure 4 - 7. Feature points on a hand model (a) with different resolution (b, c, d), green balls present some

local maximum feature points, blue ball presents a local minimum feature point.

4.3.4 Feature points matching

Distance metrics:

Given a set P composed of n feature points extracted from the source surface S , and a set Q composed of m feature points extracted from the target surfaceT , the goal is to find, for each

ip P∈ , a point iq Q∈ , which is the best match of ip . To evaluate the corresponding points matching results we use the approximated Hausdorff distance function [HU' 06]. The Hausdorff distance is a shape comparison method, which is based on a distance measure between two objects. A major advantage of this distance measure is that it can be calculated without an explicit point correspondence in their respective data sets. Given the surfaces ,S T the Hausdorff distance is defined as follows:

( , ) max( ( , ), ( , ))H S T h S T h T S= (17)

where: max max( , ) i js S t Ti j

h S T s t∈ ∈

=⎛ ⎞

−⎜ ⎟⎜ ⎟⎝ ⎠

, and i js t− is some underlying norm on the surfaces ,S T

(e.g. Euclidean norm). The function ( , )h S T is called the directed Hausdorff distance from S toT .

Correspondence search:

The goal is to find the transformation matrix 3M which best registers the isotropic source surface

1M S with the isotropic target surface 2M T using the feature points extracted from each surface. For each surface, the extracted feature points are divided into two sets. So we have two subsets of feature points for source S and two subsets for targetT . Local minimum feature points mn sF P ,

mn TF P and local maximum feature points mx sF P , mx TF P for source and target, respectively. Next, the feature points in the subsets are sorted according to their functional value ( )f v . The transformation matrix 3M is obtained as follows:


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1. Compute the centre of mass of 1M S using the following formula:

1.

1s

ncC C Aci iiA

= ∑=

(18)

2. where Cci is the centre of mass of the thi source surface face, iA is its area, A is the total area of 1M S , cn is the number of faces in 1M S .

3. Select three points from 1M S : 3mn S 2 mxp , p , p C1 sF P F P∈ ∈ = . There are totally ( mn mx.ks ks ) triplets, where: mn mx,ks ks are the numbers of points of

mn mxP , PS SF F respectively. 4. Go to the next step if the three selected points are not aligned; otherwise, reselect a

new triplet. 5. Compute the normal vector n (Figure 4 - 8) of the plan defined by the three

selected points. 6. Translate the point p1, p2 and p3 with the normal vector n in order to take into

account the normal orientation in the registration process. We obtain three new corresponding points ( )/ / /

1 2 3, ,p p p as illustrated in Figure 4 - 8.

P3 P2

P1

/1p

/3p /

2p

n

Figure 4 - 8. The three selected points (p1, p2, p3) and their corresponding points

7. Also apply the steps 1 to 5 on the transformed target 2M T to get another six points set.

8. Now we have six landmark (feature) points from 1M S and six landmark points from 2M T . Calculate the transformation matrix M which best registers the source landmarks with the target landmarks using quaternion-based closed-form solution algorithm [HOR' 87]. Then compute the approximated Hausdorff distance function ( )1 2,H MM S M T .

The transformation matrix which makes the approximated Hausdorff distance function

( )1 2,H MM S M T has the smallest value, is just what the transformation matrix 3M searched for. The cost of our method to obtain the transformation matrix 3M is ( )mn mx mn mx. . .O ks ks kt kt . Figure 4 - 9 shows two registration results using the same pair of surfaces without initialization. As can be seen, without initialization ICP converges to a completely erroneous result, but the proposed feature point matching converges to a correct result.


78

(a) (b) (c)

Figure 4 - 9. Rigid registration Results for a pair of vertebra models: (a) initial models positions without any

pre-alignment; (b) the ICP registration result starting from the models positioned in (a); (c) the initial

registration result using the proposed feature point matching.

4.3.5 Final Registration

Using an initial registration (coarse registration) will help the registration and also will get an initial estimate of the relative pose of the final registration (fine registration). This phase is based on the ICP algorithm using the Hausdorff distance as quality alignment error measurements. The inputs of this phase are the original source surface S , the original target surface T and the initial transformation matrix iniM (Figure 4 - 3). The registration process is based on the ICP algorithm which will align the transformed source surface using the initial transformation matrix

iniM S with the target surfaceT . The output of ICP registration is the transformation matrix 4M . The matrix of the best transformation which modifies the source surface to be similar to the target surface with the minimum error is given by equation (19).

4 .M M Minifin = (19)

4.3.6 Qualitative and quantitative evaluation of the registration results

The evaluation functions to measure the quality of registration are one of the main differences among the various registration approaches. Accuracy, error, or uncertainty measurements play an important role in the evaluation of the surface registration methods. The local error measurements on 3D surfaces indicate the local differences between the two registered surfaces and determine the visual impression. A number of researchers have reported their work on algorithmic and quantitative approaches to measure the error between two surfaces which can be used to evaluate the results of the registration [ASP' 02, CIG' 98, ZHO' 01]. Several metrics can be used to quantify the error between the two registered surfaces. All metrics are based on the measuring of the geometric distance to evaluate the alignment quality of the registration (see § 4.2.1).


79

Visual comparison can provide qualitative evaluation of registration. Dalley and Flynn [DAL' 01] suggested that a good registration must preset a large “splotchy” surface, which is the visual result of mapping two surfaces to each other, each surface is rendered in a different colour, crossing over each other repeatedly in the overlapping area. This effect can be described as the interpenetration of the two surfaces. However, it is impossible to measure the degree of interpenetration by visual inspection alone because the resulted image depends on a variety of factors, such as rendering resolution, illumination, image sampling, surface representation, etc. At best we gain a qualitative assessment. Silva et al. [SIL' 05] used a novel evaluation metric called the Surface Interpenetration Measure (SIM), to calculate the overlapping area between views and precisely measure the quality of the alignments from range image registration. In order to quantify the differences between the registered and target models , we developed a distance measuring tool between two surfaces based on the work of Nicolas Aspert [ASP' 02]. The developed measuring tool (called MESH Measuring Tool (Figure 4- 10)) gives the surface to surface distances between two triangle meshes using specific uniform sampling. Beside its visualization of the colour-coded distribution of Euclidean distances (Absolute and signed) between the two surfaces (source and target) using VTK library (http://www.vtk.org/), it also provides useful histogram and statistical information such as relative (%) minimum (Min), maximum (Max), mean and root mean square (RMS) Hausdorff distances between the two surfaces (compared to the length of the diagonal oriented bounding box of the target surface). MESH measuring Tool will be used also to compute the distances between the constructed average model and all models of the training set from which it was constructed. The measured values will be used to enrich the average model with some information extracted from all models of the used training set (§ 6.3.1).


80

Figure 4- 10. Screenshot of MESH Measuring Tool.

4.3.7 Results

Figure 4 - 11 presents the initial and final phases results of the proposed registration method for two femur models, where the source model (a) is obtained from the target model (b) after an affine transformation.


81

S

Initial registration

d

g

fi

b

c

f

i

j

e

Final registration

S T

T

S = Source

a

T = Target

h

Figure 4 - 11. Initial and final registration for two femur models. (a) source, (b) target , (c) source and

target in the same rendering windows, (d) initial registered source, (e) source and target after initial

registration, (f) distributed Hausdorff distances on the target after initial registration, (g) final registered

source, (h) source and target after final registration, (i) distributed Hausdorff distances on the target after

final registration, (j) the scalar bars corresponding to the Hausdorff distance distribution on the target

model in (e), (f).

To evaluate the proposed method results we use two types of evaluations:


82

Qualitative evaluation: The visual result of initial and final registration is illustrated in (e, h) of Figure 4 - 11, where the source is rendered in red and the target in green. A good registration must preset a large “splotchy” surface according to the suggestion of [DAL' 01], t (see § 4.3.6). If one compares the visual result of the initial registration (e) with that of the final registration (h), it is clear that the splotchy in (h) is better than that in (e).

Quantitative evaluation: In Table 4 - 1, we present the relative absolute Hausdorff distances (% of the diagonal diameter length of target oriented bounding box) between the registered source and target models in the two phases of the proposed registration method. This values are computed using MESH Measuring Tool (c.f. § 4.3.6). The distribution of the forward Euclidean distances on the faces of the target model is illustrated in (f, e) for the initial and final registration phases respectively.

Relative (%) Distance Initial registration Final registration

Minimum distance 0.000044 0.0000

Maximum distance 1.484831 0.000137

Mean distance 0.196477 0.000009

RMS 0.287089 0.000017

Minimum distance 0.000099 0.00000

Maximum distance 0.1269200 0.000145

Mean distance 0.191594 0.000010

RMS 0.274407 0.000017

Relative distance measuredform registered model to

target

Relative distance measuredform target to registered

model

Table 4 - 1. The relative distances between the registered source and target models in the two phases of the

proposed registration method, illustrated in Figure 4 - 11.

The results, in Table 4 - 1, show that the registration process leads globally to high satisfactory results.

4.4 Elastic Registration

The elastic registration is a non-rigid registration technique based on elastic transformation that allows non-uniform scaling and displacement of each point in the surface. These transformations allow local warping of surface features, thus provide support to local deformations. We will use an elastic registration based on the thin-plate Splines algorithm [BOO' 89, ROH' 01] to evaluate residual variations due to pure morphological differences after an affine registration (if necessary). The inputs of the elastic registration process are the source model S , which will be elastically registered to the target model T using a set of k points as landmarks. The choice of landmarks is very important. We can distinguish two types of landmarks:


83

Global landmarks: a set of k points scattered almost equally on S as source landmarks, and another set of k points on T as target landmarks that are closest to the source landmarks.

Local landmarks: a set of feature points on S , and the corresponding closest points onT . There are many methods to get feature points on the 3D surfaces presented by triangular meshes (§ 4.3.3). Among the existed methods to extract feature points, the points of crest and valley lines extracted on both source and target surfaces (§ 6.2) are one of the best feature points which can be used as landmarks for local elastic registration. Crest (valley) line is a locus of surface points whose maximal (minimal) principal curvature is a local maximum (minimum) in the corresponding principal direction. The Figure 4 - 12 illustrates a registrations procedure for a sphere surface as source (a) and a cube surface as target (b). First, an affine registration between the two models is made. The affine registered sphere surface (d) is rendered in the same windows with the target surface (e). Using the elastic registration (based on local and global landmarks) between the affine-registered sphere and the target model, we see clearly the modification in the sphere’s form (g). The quantitative evaluation using the MESH Measuring Tool presented by the distribution of the absolute Euclidean distances on the registered model after the affine registration (f) and the elastic registration (i) gives us a local view of the modification of the sphere’s form.


84

Affine Registration

Elastic Registration

S T

e

h

S

T

j

f

i

a

c

b

d

g

i f

Figure 4 - 12. Affine and elastic registration for sphere and cube.


85

4.5 Conclusion

In this chapter, first, an overview of the surface registration concepts has been presented. Next, we have classified the surface registration according to the used transformations, the used approaches and the used techniques focusing on the iterative approaches (ICP algorithm). Second, we have proposed a method combining the feature-based and iterative approaches to achieve precise registration results. Then, we have presented the MESH Measuring Tool in order to evaluate the registration quality, which will be used later in the enrichment stage of the enriched average model construction methodology. Finally, the elastic registration based on thin-plate Splines algorithm, to evaluate residual variations due to pure morphological differences between the source and target models, is presented at the end of this chapter. This proposed registration method and MESH Measuring Tool will be used later in the following chapters. In the next chapter, the average model construction methodology will be presented.

86

Ch 5: Anatomical average model construction

87

Ch.5. Anatomical average model

construction


88

5.1 Introduction

In this chapter, two methods to construct an average model from of set training models (triangulated meshes) are presented. First the Generalized Procrustes Analysis GPA [BOO' 97 b, COO' 01, DRY' 98, GOW' 75, HOR' 87] is used to construct an affine and elastic average models. Some results of this method are analysed and discussed. According to the analysis of these results, some drawbacks are detected. Second, in order to overcome these drawbacks, a new method based on morphological operations to construct an average model is proposed. The outputs of the new method are an average binary volume (average form) and a corresponding average model (triangulated mesh). Finally, analysing the advantages and the drawbacks to the two methods, we propose to combine the two methods together in order to take into account the advantages of the two methods.

5.2 Generalized Procrustes Analysis GPA

The Procrustes distance is the square root of the sum of squared differences in the positions of the landmarks in two shapes [DRY' 98, MAR' 96]. Procrustes analysis is a method to fit a set of points to another by minimizing the Procrustes distance between the corresponding landmarks of two data sets. The Generalized Procrustes Analysis (GPA) [BOO' 97 b, COO' 01, DRY' 98, GOW' 75, HOR' 87] is a procedure applying the aforementioned Procrustes analysis method to align a population of shapes instead of only two shape instances. The GPA algorithm can be summarized as follows:

1. Choose a reference shape to be the approximate average shape. 2. Align all other shapes to current reference. 3. Calculate the new approximate average shape from the aligned shapes. 4. If the approximate average shapes steps 2 and 3 are different then change the current

reference model to be the new approximated average model and return to step 2, otherwise you have found the true average shape of the set.

The registration type which will be used in the second step of GPA will determine the type of the constructed average shape. We will apply this algorithm to build the average 3D model from a set of meshed models. In order to take into account the affine and elastic deformation in the constructed average model, we propose using two types of registration in the GPA process (affine and elastic registration). The construction process of the average model using the GPA algorithm is performed in two successive stages:

Build an affine average model (AAM) using the generalized Procruste analysis algorithm (GPA) based on an iterative affine surface registration (§ 5.2.1).

Build an elastic average model (EAM) using the affine average model AAM as the reference model of the generalized Procrustes analysis algorithm (GPA). The iterative registration is performed using an elastic surface registration (§ 4.4).


89

In the following we will detail each step of the average model construction methodology based on GPA algorithm.

5.2.1 Affine average model construction (AAM)

The first stage of the construction method based on GPA method is to construct an average model which takes into account the affine deformations of all the training set models. We create an affine average model (AAM) through an iterative process using all of the training set models. First, we select an arbitrary model from the N training set models to be the initial average model. Second, we start the iterative process. We begin by an affine registration between the initial average model as a source and each model of the training set as a target. The proposed feature point-based registration introduced in chapter 4 is used in this step. N affine transformations ( )1,...., NT T are produced. Using these transformations, we compute an average transformation as follows:

1

1 N

ii

T TN =

= ∑ (20)

Finally, we transform the initial average model using the average transformation to get the average model for this iteration. The process is repeated for a chosen number of iterations, or until no change in the average model is produced. The GPA-based average model construction algorithm is given in pseudo-code presented in Figure 5- 1.

1 Set initial average model 0M = arbitrary training model

2 Rigidly register each model of the training set to 0M

3 Repeat for 0...... ITERSt N=

4 Register (affine) tM to each registered training set model, producing transformations

( )1,...., NT T .

5 Compute average transformation1

1 N

ii

T TN =

= ∑ .

6 Set temporary average model 1t tM T M+ =

Figure 5- 1. Pseudo code of the iterative algorithm to compute the average affine model AAM.

In order to test this algorithm, we generate a training set composed of five models from a base model (tooth model). All models of the training set have been submitted to different types of controlled affine transformations, where the geometrical average of the scale factors in each


90

direction is one, the average translation is zero and the average rotation angle around each axe is zero. Then, using the construction method based on affine GPA algorithm, we construct an affine average model from this training set. The test procedure and its results are presented in Figure 5- 2. To compare and evaluate the resulting affine average model AAM with the real average model of the training set (the base model), we measure the distances between the base and the extracted affine average model AAM using MESH Measuring Tool (§ 4.3.6). In Table 5- 1, we present the measured relative absolute distances between the extracted AAM model and the base model.

Min (%) Max (%) Mean (%) RMS (%)

AAM to Base 0.000000 0.000696 0.000147 0.000194

Base to AAM 0.000000 0.000658 0.000149 0.000197

Table 5- 1. Relative measured distances between AAM and base model surfaces.

The qualitative evaluation of the results is also presented in Figure 5- 2. First, the constructed affine average model AAM and the base model are mapped together. Second, the distributed relative absolute Euclidean distances (Errors) between the AAM model and the base model are illustrated on the base model.


91

Training set generation Affine GPA

1245

Training SetReference

Mapping MESH Measuring Tool

Base AAM

3

0.000 0.001310.000656

Figure 5- 2. The affine average model AAM construction test. The base model composed of 5650 points, the

number of iterations of GPA algorithm is 2.

5.2.2 Elastic average model construction EAM

If necessary, the GPA algorithm based on elastic registration is used after the affine GPA to evaluate residual variations due to pure morphological difference. In this stage, first, an affine average model AAM will be computed to be the approximated average model in the first step of the GPA algorithm. Next, all training set models will be rigidly registered to the affine average model. Then, we start the iterative process, using the same technique presented in the affine average model construction, but instead of the affine registration we used an elastic registration based on the thin-plate spline algorithm presented in chapter 4 (§ 4.4).


92

The affine average model is elastically registered to each model of the registered training set, and then the mean position of each vertex of after the registration is computed. Figure 5- 3 shows the results of the elastic GPA algorithm (using general and local elastic registration (§ 4.4) to construct an average model from a set of 3 models (sphere, ellipsoid and cube).

Elastic GPAReference

TS EAM

Figure 5- 3. The elastic average model of a training set composed of 3 models (sphere, ellipsoid and cube),

where the reference is the sphere model. The number iterations in GPA is 3 iterations, the number of selected

landmarks in each iteration of the general elastic registration is 100 landmarks.

5.2.3 Results and Discussions

To analyse the results of the construction method based on the GPA algorithm, we must take into account two factors: (a) the effect of changing the reference model which will be used as the first approximation of the average model, (b) the number of iterations needed to achieve the convergence towards the average model. First, we will study the effects of the two factors if all models in the training set have the same shape after removing the affine deformations. To realize this study, we use a training set composed of five models (M0, M1, M2, M3, M4) generated from a vertebra model (Figure 5- 4). All models in the training set are submitted to different affine deformations (translation, rotation and anisotropic scaling). The used procedure in this study can be summarized as follows:

Select a model Mi ( 0,.., 4where i = ) of the training set to be the reference model of the GPA algorithm (Figure 5- 4 A).

Compute the affine average model of the training set using the previous reference for three successive iterations AAMik ( 1, 2,3where k = is the number of iterations).

To study the effects of number of iterations, we compute the maximum, mean and RMS relative absolute distances between the actual average and the average model of the previous iteration using MESH Measuring Tool ( )1,ik ikH AAM AAM − ,

0,.., 4, 0,1,2,3where i k = = (AAMi0 is the used source model for the measurements) (Figure 5- 4 B).


93

To study the effects of changing the reference model, we use MESH Measuring Tool to compare the constructed average models in the third iteration AAMi3 with AAM03

( )3 03, , 0,.., 4iH AAM AAM where i = (Figure 5- 4 C).

Iteration 1 Iteration 2 Iteration 3Reference

0.0000

0.0000

0.0000

H(AAMi3,AAM03):i=0..4

0.000811

0.000057

0.000085

0.000811

0.000057

0.000085

0.000529

0.000040

0.000060

0.000757

0.000040

0.000056

6.732 0.00628 0.000453

2.028 0.00035 0.000026

2.487 0.00062 0.000046

AAM41 AAM43AAM42

8.1009 0.00628 0.000453

2.5472 0.00035 0.000026

3.0991 0.00062 0.000046

AAM31 AAM33AAM32

9.600 0.00573 0.000524

3.027 0.00032 0.000029

3.661 0.00058 0.000047

AAM21 AAM23AAM22

7.089 0.0057 0.000488

1.972 0.000316 0.000028

2.544 0.000567 0.000046

AAM11 AAM13AAM12

14.494 0.000533 0.000476

3.609 0.000031 0.000028

4.354 0.000057 0.000047

AAM01 AAM03AAM02

1.72e-7

0.000145

1.72e-7

0.000145

0.0000

9.09e-5

1.23e-7

0.00011

0.0

0.0

A B C

Relativedistance (%)

Max.

Mean

RMS.

M0=AAM00


Max.

Mean

RMS.

M1=AAM10


Max.

Mean

RMS.

M2=AAM20


Max.

Mean

RMS.

M3=AAM30


Max.

Mean

RMS.

M4=AAM40

Figure 5- 4. Effects of changing the reference model and the number of iterations on the average model

construction method based on affine GPA algorithm.


94

M0M1 M2

AAM0AAM1 AAM2

EAM1 EAM0EAM2

Affine GPA

TSM1

Affine GPA

TSM0

Affine GPA

TSM2

Elastic GPA

Affine TSAAM1

Elastic GPA

Affine TSAAM0

Elastic GPA

Affine TSAAM2

ref ref ref

ref ref ref

Mapping Mapping

Mapping Mapping

0.00102 0.0161 0.0312 0.000474 0.00160 0.0315

MESH Measuring Tool MESH Measuring Tool

3.36 e-5 0.00368 0.007129.75 e-5 0.00361 0.00712

MESH Measuring Tool MESH Measuring Tool

Mapping Mapping

Figure 5- 5. The affine average models (AAM0, AAM1 and AAM2) and the elastic average models (EAM0,

EAM1 and EAM2) constructed form the training set models TS (M0, M1 and M2).

Second, we will study the effect of changing the reference model on the results of GPA algorithm if the training models have different geometrical shapes. Two examples will be presented.


95

In the first example (Figure 5- 5), we use a training set TS composed of 3 models (M0, M1 and M2). We compute three affine average models ( : 0,1, 2iAAM i = ). Each affine average model

iAAM is constructed form the training set TS using a selected reference model iM on the GPA process. Next, we use the three constructed affine average models (AAM1, AAM2 and AMM3) as an input training set (affine TS) for another GPA process based on elastic registration (c.f. § 4.4). Similarly, we compute three elastic average models ( : 0,1, 2iEAM i = ) from the input affine training set using a selected affine average model iAAM as a reference model for the GPA process. We used the same number of global landmarks (100 landmarks) for each elastic registration process. To evaluate the differences between the constructed average models, we use the MESH Measuring Tool where the target model is AAM0 for the affine average models and EAM0 for the elastic average models (Figure 5- 5). In the second example (Figure 5- 6), we test the effect of changing the reference model using a training set TS composed of two rigidly registered models (M0 and M1). The elastic average model EAM0 (EAM1) is computed using M0 (M1) as reference model of the GPA process using an elastic registration based on the thin-plate Splines algorithm (c.f. § 4.4).

M0

Elastic GPA

M0

refElastic GPA

TSM1

ref

M1

EAM0 EAM1

Mapping

Mapping

TS

Figure 5- 6. The affine average models (AAM0 and AAM1) and the elastic average models (EAM0 and

EAM1) constructed form the training set models TS (M0 and M1). The average models are constructed using

the same number of iterations (5 iterations) and the same number of landmarks (150 landmarks).

Analysing the results of the first and second analysing procedures, we can summarize our observations as follows:


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The effect of number of iterations: The required number of iterations to construct an average model depends on the number of models in the input training set and the initial pose of the training models (the differences between the models). Regarding the results (differences between the actual average model and the average model of the average model of the previous iteration) presented in Figure 5- 4, we find that one or two iterations should be sufficient in most cases. The results are comparable with the observation of Bookstein [BOO' 97 b].

The effect of changing the reference model: if all models of the training set present the same object submitted to different affine transformations, changing the reference model should produce an identical affine average model (Figure 5- 4). In other words the constructed affine average model is independent to the choice of the reference model in this case. In general case where the models in the training set have a different geometrical shapes, changing the reference model for the GPA algorithm will produce different affine average models (Figure 5- 5). This is a consequence of the fact that the affine registration algorithm uses global transformations (translation, rotation and scaling) to deform the source model. Using an Elastic GPA process (iterative local transformations) after an affine GPA process can reduce the differences between the constructed average models (Figure 5- 5). An example of the dependence of the constructed elastic average models on the choice of the starting reference model using the GAP process is shown in Figure 5- 6. This dependence is due to the connectivity constraints, the number of points of the used meshed models and also the number of used landmarks used to elastically deform the source model to be similar to the target model.

In the following section, we will propose a new method based on some binary image morphological operations to construct an average model (binary volume and its corresponding average surface) form a set of rigidly registered models (triangulated meshes or binary volumes). The new proposed method doesn’t need to choose any reference model form the used training set to be a starting reference model as in GPA process.

5.3 Morphological-based average model construction

5.3.1 Overview of the proposed method

In this section a new method to construct an average model is proposed. The input of our proposed method is a training set TS of N models 1 2, ,........., NM M M representing the group of the subjects under consideration. All models of TS are triangulated meshes. The proposed method can be summarized as follows:

Convert the models into a volumetric representation (binary volumes). Extract the average volume of the set of the volumes using some morphological operations. Convert back the average volume into a triangulated mesh.


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The reason for working in the volume domain is that we have access to a wide array of techniques that have been developed for image processing since volumes have the same regular structures as images but in one higher dimension. Figure 5- 7 shows a schematic diagram of the proposed construction pipeline.

Rigi

d Re

gistr

atio

n

Targ

etSo

urceTS Voxelization

3D Morphology

Model CreationAverage modelAverage volume

Surface Volume

Figure 5- 7. Morphological-based average model construction pipeline.

There are four steps in our proposed method: Align all models of TS into a common reference. The proposed feature point-based

registration introduced in chapter 4 is used to rigidly register each model of TS to one arbitrary selected model of TS (e.g. the first model). The rigid registration is used to eliminate the differences between the models due to translation and rotation.

Convert each model of the registered TS into a volume (3D binary image) (voxelization, § 5.3.2), the same grid is used to voxelize each model of the registered S.

Extract the average volume from the resulting binary volumes set, using a set of basic 3D morphological operation in 3D image processing (dilation, erosion……) (§ 5.3.4).

Convert back the extracted average volume into a triangulated mesh (§ 5.3.5). In the following sections we will detail the basic steps of our proposed.

5.3.2 Background of the voxelization

The volumetric representation plays an important role in computer graphics community as an alternative to traditional geometric representation. Conceptually, a reformulation process is required to generate volumetric representation from geometric object (surface). This stage is typically called voxelization. It accomplishes the conversion from a set of continuous geometric primitives to an array of voxels in the 3D discrete space that approximates the shape of the model as closely as possible. This concept was first introduced by Kaufman [KAU' 87, KAU' 86]. There are many strategies to conduct polygonal voxelization. Basically, they can be classified as surface voxelization and solid voxelization (Figure 5- 8). The surface voxelization of polygonal is one in which only voxels that are near a polygon of the original model have a nonzero


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voxel value (Figure 5- 8 a). Surface voxelization is performed by finding the distance between a given voxel and the nearest polygon of the model [DAC' 00, HUA' 98, LEV' 90, STO' 97a, WAN' 93]. Performing isosurface extraction on such a model would produce a polygonal model that has two surfaces which are near each other. In contrast, all the interior voxels of the object carrying the value of 1 in the solid voxelization (Figure 5- 8 b), so that isosurface extraction yields a single surface [HAU' 02, KOL' 01, NOO' 03, SCH' 94, SRA' 99].

(a) (b)

Figure 5- 8. Slices through a sphere: (a) surface representation, (b) solid representation.

In this stage we aim to create a 3D binary image for each registered model. The used models in our proposed method are manifold and composed of only one part. Therefore, it is necessary to use a solid voxelization method. According to our requirements, we use the extended count parity method [NOO' 03] to voxelize the training models. Nooruddin and Truk [NOO' 03] presented two approaches to classify the voxels for solid voxelization. The first one is known as parity count method of voxel classification which used on manifold polygonal models which have no cracks or holes. For such models, they classify a voxel V by counting the number of times that a ray with its origin at the center of V intersects the model. An odd number of intersections means that V is inside the model, and an even number means that it is outside. This is simply the 3D extension to the parity count method of determining whether a point is interior of a polygon in 2D. The direction of the ray is unimportant for manifold models. This will speed up the voxel classification. In essence, they cast many parallel rays through the polygonal model, and each one of these rays classifies all of the voxels along the ray. The parity count method works well for manifold models. Models which have small cracks or holes in its surfaces are a common problem of the voxelization algorithms. To voxelize such models, they extended the parity count method by using k different directions of orthographic projection and by scan-converting the model once for each direction. Each of the k projections votes on the classification of a voxel (interior or exterior), and the majority vote is the voxel's final classification. The principal step of the voxelization process is to define a 3D uniform grid embedding the polygonal object. For our needs, we must use the same 3D grid to voxelize all models of the training set. Using the same 3D grid, the voxelization process will produce a set of 3D binary images which have the same characteristics (origin, size and spacing).


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The used grid properties are defined in three steps: First, we compute the bounding box of each registered model of the training set. Second, we compute the global bounding box which best encloses all the computed

bounding boxes. Origin point and size of the global bounding box will be used directly for the 3D grid.

Finally, we define the number of voxels along the three axes of the grid. Then the spacing is calculated using the size and the defined dimensions.

Figure 5- 9 shows the results of the voxelization of a mouse brain model using different 3D grid resolution.

213

205

256

106

102

12853

51

64

VoxelizationVoxelizationVoxelization

Figure 5- 9. Volumetric representation of a mouse brain model using different 3D grids resolution.

5.3.3 Background of 3D Morphology

The fundamental properties of the set theory are the intersection ( )∩ and the union ( )∪ of sets. The intersection of two sets is all items that appear in both of the sets. In contrast, the union of sets combines the member of sets together to form a new combined set. If every member of set A is a member of set B , then A is said to be a subset of B , written A B⊆ . On the other hand, the intersection (union) operation in set theory reduces to AND (OR) operation when the variables involved are binary. Terms such as intersection and AND (union and OR) are used interchangeably in the literature to denote general or binary set operations. The principal logic operation used in our method are AND, OR and XOR. Their properties are summarized in Table 5- 2.


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a b a AND b a OR b a XOR b0 0 0 0 00 1 0 1 11 0 0 1 11 1 1 1 0

Table 5- 2. AND, OR and XOR operations table.

These logic operations are performed on pixel by pixel basis between corresponding pixels of two or more images. Because the AND operation of two binary variables is 1 only when both variables are 1, the result at any location in a resulting AND image is 1 only if the corresponding pixels in the two input images are 1. The XOR (exclusive OR) operation yields 1 when one or the other pixel (but not both) is 1, and it yields 0 otherwise. Figure 5- 10 shows various examples of logic operations involving images, where black indicates binary 0 and white indicates binary 1.

AND

OR

XOR

Figure 5- 10. Some logic operations between binary images; black represents a binary 0 and white a binary 1.

Dilation and erosion are the basic operators in the area of mathematical morphology. They are typically applied to binary images. The basic effect of dilation (erosion) operator on binary image is to gradually enlarge (erode away) the boundaries of regions of foreground pixels (i.e. white pixels, typically), thus areas of foreground pixels grow (shrink) in size while holes within those regions become smaller (larger). Dilation of object M by the structural element (also know as a kernel) k is denoted by M k⊕ . Erosion is denoted by M kΘ . As shown in Figure 5- 11, a conditional dilation of a region A called “base” by a structuring element k using a mask M (where A M⊆ ) is a repetition of a dilation of the region A with the structuring element k by an intersection with the mask region M. This is defined as:

( ),i

iCD A M A k M⎛ ⎞= ⊕⎜ ⎟⎝ ⎠

∩ (21)

where i is the smallest integer such that ( ) ( )1, ,i iCD A M CD A M+= .


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On the other hand, the conditional erosion is the operation to erode away a region A called “base” within a boundary M called “mask” where M A⊆ . It is defined as:

( ),i

iCE A M A k M⎛ ⎞= Θ⎜ ⎟⎝ ⎠

∪

(22)

where i is the smallest integer such that ( ) ( )1, ,i iCE A M CE A M+= .

Mapping

(a) (b)

(c) (d)

Mask (M)

Base (A)

Base Mask

Conditional Dilation

Figure 5- 11. The conditional dilation results of a base (A) using a mask (M).(a) after 5 iteration, (b) after 20

iterations, (c) after 30 iterations, (d) after 40 iterations.

In the following section we will present a new method based on morphological operators to construct an average volume from a set of binary volumes.

5.3.4 Average volume extraction using morphological operators

Our proposed morphological-based method aims to construct an average binary volume (average form) from a set of training binary volumes. The constructed average volume presents the average of the morphological variations of all volumes in the training set. The proposed method can be best described by looking at Figure 5- 12 which shows a network of 3 layers to construct an average volume from 8 binary volumes.


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Layer 3

AO

AO

AO

AO

AOAO

AO

v0

v6

v1

v2

v3

v5

v4

v7

Average

Volume

Layer 2 Layer 1 Layer 0

Figure 5- 12. Network for the constructing of an average volume for training set of 8 volumes using an

averaging operator (AO) in each node of the network.

The basic idea of our proposed method is to use a network of 2n log N= layers to

construct an average volume from a set of training volumes; we assume that the number of volumes

in our training set 2nN = for some n∈� . We number the layers from 0 to n where layer 0

contains the output and layer n the input of the network. Layer number i has 2i nodes. Every node

in layer i,0 i n≤ < , consists of an averaging operator (AO) which takes two binary volumes from

the layer i+1 as input and construct an average volume as output. The averaging operator in node

k, 0 2ik≤ < , of layer i receives its input from the nodes k and 2ik + in layer i+1. The reason for

why we choose the number of training modes N a power of 2 is because that this choice will

guarantee that all models of the training set contribute in the construction of the average model in

the same ratio.

The morphological operators constitute the heart of our proposed averaging operator. The

averaging operator is an iterative recursive process. It repeats the same following 6 steps until the

average volume is obtained:

1. Compute the difference volume between the two input volumes ( )1 2,V V . The

difference volume is computed using a logical XOR operator.

1 1 2DIFV V XOR V=

(23)

2. Extract the extreme volumes from the two input volumes ( )1 2,V V (Figure 5- 13). The

extreme volumes are the minimum and maximum volumes ( ),MIN MAXV V . The

minimum (maximum) volume is the result of intersection (union) operation between

the two input volumes (Figure 5- 13).

1 2

1 2

MIN

MAX

V V AND V

V V OR V

=

=

(24)


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MAXV

1V

2V

MINVAND

OR

Figure 5- 13. Extreme volume extraction.

3. Dilate the minimum volume MINV using a conditional dilation for only one iteration.

The base of the conditional dilation is the minimum volume MINV and the mask is the

maximum volume MAXV . The used structure element k in the dilation process is a

square of 2x2 pixels in 2D and a cube of 3x3x3 voxels in 3D. (Figure 5- 14).

( )1

1 ,A MIN MAX MIN MAXV CD V V V k V

= = ⊕

∩ (25)

4. Erode the maximum volume MAXV using conditional erosion by the same structure

element k as in the precedent step also for only one iteration (Figure 5- 14). In contrast

of the previous step, the mask in this step is the minimum volume MINV and the base

is MAXV .

( )1

1 ,B MAX MIN MAX MINV CE V V V k V

= = Θ

∪ (26)

+

-

MINV

MAXV

( )1,

MIN MAXCD V V

( )1 ,MAX MINCE V V

AND

OR

Figure 5- 14. Conditional dilation and erosion diagram.

5. Compute the differences between the dilated volume ( )1 ,A MIN MAXV CD V V= and the

eroded volume ( )1 ,B MAX MINV CE V V= using a logical XOR operator.


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2DIF A BV V XOR V= (27)

7. If the results of steps 1 and 5 are different then replace the input of the averaging operator by the dilated and eroded volumes and return to step 1, otherwise stop the process and the average volume is the dilated volume (or eroded volume).

( )

{

1 2

1

2

DIF DIF

A

B

A B

IF V V

V VV V

ELSEaverage volume V V

≠

=⎧⎨ =⎩

(28)

The averaging operator is symmetric in its tow inputs:

( ) ( )1 2 2 1, ,AO V V AO V V= (29)

Figure 5- 15 shows the extracted average image of two 256 x 256 binary images (circle and rectangle) using the proposed algorithm. Another example of the average image extraction using the proposed method in 2D is also presented in Figure 5- 16.


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I0 AIAO

I1

Edge detectorEdge detector Edge detector

Mapping

Figure 5- 15. The average image AI of two binary images I0, I1 extracted using AO.


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AO AO

AO

Laye

r 2

I02 I13

I0213

C I02 C I13

C I0213

Edge Detector

I0 I2 I02

Edge Detector

I3 I1 I13

Edge DetectorC I13

C I0213

C I02

I2I0 I1 I3

Laye

r 1

Laye

r 0

Figure 5- 16. The two layers average of a set of four 256 x 256 binary images (I0, I1, I2, I3).

In Figure 5- 17 we test the proposed method in 3D case using four centralized meshed models (M0, M1, M2, and M3). Firstly, we voxelize the models to obtain a training binary volume set composed of four 240 256 220x x binary volumes ( )0 1 2 3, , ,v v v v . Then, using the proposed method, we extracted the average volume 0123v which corresponds to 0 1 2 3v v v v sequence of the four voxelized models.


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Voxelization

AO

AO

AO

V0 V1 V2 V3

V0213

M0 M1

Models

Volumes

Averagingnetwork

Averagevolume

M2 M3

Figure 5- 17. Using the morphological-based average model extraction in 3D case.

In section 5.3.6 we will study the effect of changing the order of images (volumes) in the input set on the results of the proposed morphological-based method to extract an average image (volume).


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5.3.5 Model creation

Now, we have seen how to extract an average volume using the volumetric morphology, we turn our attention to convert the average volume back to mesh. There are two steps involved in this: isosurface extraction and mesh post processing. To create a manifold triangulated mesh, we extract an isosurface from our volumetric representation of the average model. We do this using the marching cubes algorithm (§ 2.3). The extracted isosurface has some voxelization artefacts. We use Taubin’s smoothing technique to reduce these artefacts [TAU' 00]. Taubin used a low-pass filter over the position of the vertices to create a new surface that is smoother than the original. One of the design goals of this smoothing method was its ability of reducing the voxelization artefacts that are introduced during the voxelization stage.

Marching Cubes

Post-processing

Binary Volume 240 x 256 x 220

Voxelization

Figure 5- 18. Model creation steps.

The marching cube algorithm generates an excessively large number of triangles to represent an isosurface. This allows us to drastically reduce the triangle count without degrading the model’s quality. To achieve this, we use Garland and Heckbert’s polygon-based simplification method, which is guided by a quadric error measure [GAR' 97].


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Figure 5- 18 shows an example of the model creation steps: voxelization, converting the volume back to meshed model using the isosurface extraction based on marching cube algorithm and the post processing process (smoothing and simplification).

5.3.6 Results and Discussions

In this section, we will focus on the evaluation of the morphological-based average model construction method. Both qualitative and quantitative evaluations are performed to evaluate the effect of changing the order of images (volumes) in the input set on the result of the proposed method. To quantify the differences between two binary images (volumes), we measure the internal and external distortion rates proposed by Goumeidane et al. [GOU' 03]. Let A and B be two binary shapes representing the same average shape placed in common support. The pixels of the region B compared to A can be classified in three types, under detected, detected and over detected pixels (Figure 5- 19).

B A

Under detected pixels

Detected pixels

Over detected pixels

Figure 5- 19. The regions A and B superimposed [GOU' 03].

The internal distortion rate (IDR) of the image B compared to A is given by:

( )

( )

12

1

2

1

100%

K

ui

N

Aj

d iIDR x

d j

=

=

=∑

∑ (30)

where ( )ud i is the Euclidean distance between the under detected pixel i of B and closest pixel of the reference background (external boundary of A), ( )Ad j is the distance between the pixel j of the region A and it’s nearest pixel in the reference background, K1 is the number of the under detected pixels and N is the number of pixels in A. IDR estimates the discrepancy between A and B, induced by the under detected pixels.


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In contrast, the external distortion rate (EDR) of the B compared to A is defined by:

( )

( )

22

1

2

1

100%

K

oi

N

Aj

d iEDR x

d j

=

=

=∑

∑

(31)

In equation (31), ( )od i is the Euclidian distance between the over detected pixel i of B and closest pixel of the reference A, K2 is the number of the over detected. EDR provides a way to evaluate the effects of the over detected pixels on the region B. In order to study the effect of changing the order of images in the training set on the result of our proposed method in 2D case, a set of four 256x256 binary images (I0, I1, I2, I3) is used. We extract an average image for each possible ordering of the four images. The three possible sequences are I0I1I2I3, I0I2I1I3 and I0I2I3I1 (the proposed averaging operator is symmetric in their two inputs). Using the proposed method we obtain three average images ( )0123 0213 0321, ,I I I , each for one sequence set (Figure 5 - 20).


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I3

I1

I2

I0AO

AO

AO

I01

I23

I0123

I3

I2

I1

I0AO

AO

AO

I02

I13

I0213

I0213I02 I13

I0123I01 I23

I0321I03 I21

I2I0 I1 I3

I1

I3

I2

I0AO

AO

AO

I03

I21

I0321

Figure 5 - 20. Average images of the three possible sequences of the four input binary images (I0, I1, I2, I3).

Qualitative evaluation assesses the differences between the average images according to the subjective criteria. The visual comparison between each average image and another selected image is used to perform this type of evaluation. The differences between each pair of the three average images are computed using the logic XOR operation and shown in Figure 5- 21.


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I0213 XOR I0123 I0123 XOR I0321I0213 XOR I0321

Figure 5- 21. Difference between the extracted average images.

On the other hand, the distortion rates (IDR, EDR) [GOU' 03] are used to quantify the difference between the extracted average images. Table 5 - 3 gives some discrepancy measures computed on the average images (I0213, I0123) compared to the third average image I0321.

Image IDR (%)

EDR (%)

Under detected (Pixel)

Over detected(Pixel)

detected(Pixel)

I0123 0.225 0.237 37 38 8983

I0213 0.401 0.135 109 13 8983

Table 5 - 3. Discrepancy measures computed on the average images (I0213, I0123) of Figure 5- 16 compared to

I0321.

In 3D case, we use the same example as in Figure 5- 17, but in this time we extracted three average volumes ( )0213 0123 0321, ,v v v , each average model corresponds to one of the three possible orderings of the four volumes ( )0 1 2 3 0 2 1 3 0 2 3 1, ,v v v v v v v v v v v v . After the isosurface extraction step, in the model creation stage, the same post processing is performed on the three average models. The results of this test are shown in Figure 5 - 22.


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V0321 V0213

V3V2

V0123

V1V0

AO AO

AO

V1V2 V3V0

AO AO

AO

V0321

V3V1 V2V0

AO AO

AO

V0213

V0123

V0V1 V2 V3

Voxelization

Model Creation

M0123

M0M3M2M1

M0321 M0213

Figure 5 - 22. Average models of the three possible ordering of the four input models (M0, M1, M2, M3) using

the proposed average model construction model.

Table 5 - 4 shows the distortion rates (IDR, EDR), under detected, detected and over detected number of voxels of the average volumes (v0213, v0123), compared to the third average volume v0321.


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Volumes IDR (%)

EDR (%)

Under detected (Pixel)

Over detected(Pixel)

detected(Pixel)

V0123 0.571 0.508 10991 7663 721730

V0213 0.661 0.467 14935 6264 721730

Table 5 - 4. Discrepancy measures computed on the average volumes of Figure 5 - 22, using the first average

volume v0321 as a reference volume.

The qualitative and quantitative comparisons of the extracted average models in Figure 5 - 22 are evaluated using MESH Measuring Tool. In Figure 5 - 23 and Table 5- 5 the results of comparison between M0123, M0213 as source M0321 as target are presented.

(a) (b)

(c) (d)0.000119 0.0381

0.000112 0.0328

(c)

(d)

Figure 5 - 23. Visual comparison of the extracted average models in Figure 5 - 22. (a) Mapping of M0123 and

M0321, (b) mapping of M0213 and M0321. Distributed absolute relative Euclidean distances between M0321 and

(c) M0123, (d) M0213.


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Min (%) Max (%) Mean (%) RMS (%)H (M0123, M0321) 0 0.968 0.146 0.183H (M0213, M0321) 0 0.844 0.174 0.212

Table 5- 5. Relative measured distances between results average models and the final average model M0321

(Figure 5 - 22).

Analysing the results of qualitative and quantitative evaluation used to test the effect of changing the order of input subjects on the output of the proposed method, we find that there are very small differences between the extracted average objects (images, volumes and models) that are due to integer rounding in the discrete pixel or voxel space. Consequently, the proposed average model is independent on the ordering of input subjects.

5.4 Combining GPA and morphological-based average model

extraction techniques

Regarding the results of the two used methods to construct an average model from a set of training models, we found that the GPA-based method can construct an average model directly using an iterative registration process in few iterations of registration (two iterations). But the constructed average model in general case depends to the used reference model for the GPA method. This drawback is solved using the morphological-based method. This last one needs an initialization step to bring all input training set models to initial pose using a rigid registration. The initialization step is necessary to eliminate the differences between the two models due to the translation and the rotation. In order to take into account the advantages of the GPA based and morphological-based construction methods, we propose to combine the two methods. The new combined method consists of two stages (Figure 5- 24).


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Rigid Registration

TargetSo

urce Voxelization

3D Morphology

Model Construction

TS (N models) Average model

Initialization Construction

Average volumeAAM

Affine GPA

Figure 5- 24. Combined average method construction pipeline.

In the first stage, all models of the input training set will be rigidly registered to an affine average model constructed using the GPA-based method. Whereas in the second stage, the average model will be constructed using the proposed morphological-based average model extraction method. The affine average model constructed using the GPA algorithm will be used as a target model for the rigid registration process in the first stage. In contrast, the morphological-based average model construction method aims to construct an average model which presents the average of the morphological variations of all models in the input training set. The combination of the two methods will be used in the application part (third part) to create an average model of a set of real anatomical structures representing the normal brain of mouse.

5.5 Conclusion

In this chapter, two methods to construct an average model from a set of training models (triangulated meshes) have been introduced. First, a method based on GPA algorithm with two successive stages has been presented and tested. In the first stage, an affine average model was constructed from the input training set using GPA algorithm based on iterative affine registration process. In the second stage, the built affine average model is used as a reference for another GPA process based on elastic registration to evaluate the variations of the models due to the morphological differences. Second, we proposed a new method to construct an average binary volume (average form) and a corresponding average surface (triangulated mesh) using some morphological image processing techniques. The proposed method consists of three principal steps: voxelization, morphological average binary volume construction and model creation. The proposed method has been tested in 2D and 3D cases and the results are evaluated using quantitative and qualitative evaluations.


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Finally, we proposed to combine the based on GPA algorithm and the proposed morphological-based construction methods. The combined method will take into account the advantages of the two methods to build an average binary volume and a corresponding average surface. We will use this method in the third part of this document for a real application to build an average model of some anatomical structures of small animals. In the following chapter, we will detail the used procedures to enrich the constructed average model.

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Ch 6: Model enrichments

119

Ch.6. Model enrichments


120

6.1 Introduction

Nowadays, high level applications such as pattern recognition, phenotyping…etc require more enriched models. Typically the extracted average model does not contain much information that could support a high level application. Therefore model enrichment is necessary to equip the geometrical data of the average model with additional information. It is important to note that such enrichment information is not typically available in the average model directly. As introduced in the first part, the choice of the suitable enrichments of the model depends on the envisaged applications using this model. Our average model is composed of average surface (triangulated mesh) and average binary volume (average form). In our work, we plan to utilise the enriched average model in two applications. In the first, this enriched average surface will be used as a comparing model in a phenotyping process of small animals (mice). The enriched average binary volume will be used also as a reference model in image segmentation process using a priori knowledge. To enrich the average model we can distinguish two types of added information:

Information extracted directly from each the average model itself (average surface or average volume).

Information extracted from all models of the training set from which it was constructed (from the model (surface) training set or binary volumes training set).

In order to add useful information to the average model for our envisaged applications, we propose to use four principal kinds of enrichments (Figure 6 - 1):

Feature lines-based model enrichments (§ 6.2): in this procedure, some local features lines (crest and valley lines) are extracted directly from the geometrical structure of the average

Features Lines (§ 6.2)Distance Maps (volumes) (§ 6.4)

Probability Map (volume) (§ 6.3) Measurements (§ 6.5)

Distance Maps (surface) (§ 6.3)

Average surface enrichmentsAverage volume enrichments

Enriched average volume Enriched average surface

Output

Registeredtraining set

Trainingbinary volumes

Averagebinary volume

Averagesurface

Enrichments

Figure 6 - 1. Average model enrichments.


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surface itself. These lines can be used in some surface processing applications such as registration, simplification …. etc.

Distance maps-based model enrichments (§ 6.3): we used two types of distance maps: ♦ Distance maps computed from the average surface and the registered training set

models. These maps are presented as scalar values added to each face of the average surface. The goal of these maps is to associate some statistical information (min, max, mean, standard deviation, z-scores …) about the variation of measured distances from this face to all other surfaces in the training set. These kinds of enrichments are used in the phenotyping process ( Ch.7).

♦ Signed distance maps computed from the average binary volume itself and all volumes of the training binary volumes. These distance maps can be used in image segmentation using a priori knowledge ( Ch.7).

Probability map (§ 6.3.3): This map is presented as volume (3D image) where each voxel has an intensity value in range (0, 1). This value is computed by counting the number of times that the voxel under consideration was a foreground voxel in the training set volumes divided by the number of volumes in the training set. This map can be used also to improve the image segmentation using a priori knowledge.

Some measurements (§ 6.4): Some global descriptors of the models (volume size, area … etc) were measured from all training models in order to study the variation of these quantities. We can also integrate some anatomical landmarks added by a medical expert to enrich our average model.

In the following sections, we will detail each average model enrichment procedure.

6.2 Feature lines-based model enrichment

6.2.1 Introduction

Feature Lines are shape descriptors that provide a satisfactory geometrical representation of important physical and geometrical properties. Peaks, pits, valleys, crests, passes and flat regions are some useful topographic features used in surface analysis. A peak is the highest point in its neighborhood. Similarly, in the neighborhood of a pit, there is no lower point. A crest line corresponds to a long, narrow chain of higher points, while valley line corresponds to a chain of points with lower elevations. A pass is a low point on a crest line or between two adjacent peaks (Figure 6 - 2).


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Crest and valley lines are curves on 3D surfaces capturing important shape information (Figure 6 - 3). In this section, we will use the crest and valley lines to enrich the constructed average model. Since the feature lines (crest-valley lines) represent some geometrical properties that are not directly represented in the meshed model, these feature lines can be seen as a form of model enrichment.

Figure 6 - 3. Red crest and blue valley lines detected on Michelangelo’s David head model [OHT' 04].

Crest and valley lines have a nice mathematical background and it is usually easy to identify them automatically. The following subsections explain the computation of the differential properties of the surface represented by a triangulated mesh and focuses on the crest-valley lines extraction algorithm.

6.2.2 Basic background of surface geometry

Surfaces can be described mathematically in 3 types of forms: Explicit form: ( , )z f x y= . Implicit form: ( , , ) 0F x y z = . Parametric form: ( ) ( ) ( ) ( ), , , , , ,r u v x u v y u v z u v= ⎡ ⎤⎣ ⎦ .

Let us consider a surface expressed in the parametric

form ( ) ( ) ( ) ( ), , , , , ,r u v x u v y u v z u v= ⎡ ⎤⎣ ⎦ . The partial derivatives ( 1 ut r dr du= = ) and

Crest Valley FlatPeak Pit

Normal vectors

Figure 6 - 2. Topographic features in triangular meshes.


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( 2 vt r dr dv= = ) are tangent vectors of the curves u const= and v const= respectively. The unit

normal vector of a point P on the surface S is given by 1 2

1 2

( )n Pt tt t

=×

× where 1 2,t t are the two

tangent vectors on the point P (Figure 6 - 4).

n

1ur t=

2vr t=

P

tangent vectors

Normal

( ),r u v

uconst

=

vcon

st=

Figure 6 - 4. Normal and tangents vectors on parametric smooth surface.

Let ( ) ( ),u u s v v s= = be a curve γ on a surface, where s is the curve arc length. According

to the Frenet formulas dt ds km K= = where k is the curvature of the curve, m is the normal of

the curve, and K is the curvature vector. The curvature vector is the sum of its normal and

tangential components n gK K K= + , as seen in Figure 6 - 5 (a). The tangent component is called

the geodesic curvature vector whereas the normal component of the curvature vector called the

normal curvature vector. Let b t n= × , then the geodesic curvature gk is defined by the

equation g gK k b= . Thus singk k θ= , where θ is the angle between the curve normal m and

surface normal n (Figure 6 - 5 (a)).


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n

t

Surfa

ce

Normal

secti

on

( )tγ

n

nK

gK

K

mθt

bγ

(a) (b)

Figure 6 - 5. Definitions of geodesic and normal curvatures, normal section curve.

Consider the intersection curve between the surface and the plane generated by the tangent vector t and the normal vector n . Let us call it the normal section (Figure 6 - 5 (b)). Let the plane passing through a point on the surface be not orthogonal to the tangent plane at the point, that is, the section is not a normal section. Then the curvature of the section at a point is given by cosk θ , where k is the curvature of the normal section and θ is the angle between the normal plane and the oblique plane. If γ is the normal section associated with t we obtain either n m= or n m= − . Let us

define the normal curvature ( )nk t of the normal section γ associated with t :

( )n

k if n mk t

k if n m

⎧ =⎪= ⎨− = −⎪⎩

(32)

When the normal plane rotates around the normal n , the shape of the normal section

changes with its tangent direction t , and so does the normal curvature ( )nk t . The shape returns to

the initial state after a half rotation of the normal plane. The maximum and minimum values

of ( )nk t , maxk and mink , are called the principal curvatures of surface. The associated tangent

directions maxt and mint , along which the external curvatures are attained ( )maxmax nk k t=

and ( )minmin nk k t= , are the principal directions. Finally, max min.G k k= is known as Gaussian

curvature and max min

2k kH +

= is known as mean curvature.

Meusnier’s theorem implies that all curves at a point P with the same tangent vector will have the same normal curvature. Moreover, we have, for any unit vector t in the tangent space,


125

max mincos sint t tθ θ= + (33)

Where max mint and t are the principal directions andθ is the angle between the vectors maxt and t . The normal curvature nk along the unit vector t is given by

2 2max min( ) cos sinnk t k kθ θ= + (34)

This is known as the Euler’s formula. In discrete space, let M= (V, E, F) be a smooth surface in 3 approximated by a dense triangular mesh, where { }| 1V v i nvi= < < denotes the set of vertices, E denotes the set of edges and { }| 1F f i nFk= < < is the list of faces (triangles). We assume M is oriented and consistent. Each vertex iv M∈ is conventionally represented using Cartesian coordinates, denoted by ( ), ,i i i iv x y z= . In the following subsections we present some state of arts and the used methods to estimate the previous different quantities and its derivatives on triangular meshes.

6.2.3 Normal estimation

On a polygonal surface M, consider a vertex iv M∈ . Let { }{ }| 0,1,..., 1MB v jj m∈= ∈ − be the

set of neighbors of iv which are permuted counter clockwise and 0 mv v= . Let jf be the triangle

(face) constructed by the vertices ( )1, ,i j jv v v + (Figure 6 - 6 ). Then for each triangle jf , we can

approximate the normal vector jfn at iv by

( ) ( )( ) ( )

1

1

j

j i j if

j i j i

nv v v v

v v v v

+

+

=− × −

− × − (35)

The w-weight unit normal vector ( )in v at a vertex vi on M is given as follows [CHE' 05]:

1

01

0

( )

.

.

j

j

j f

i

j f

m

jm

j

n v

w n

w n

−

=−

=

=∑

∑ (36)

where jf is the triangle (face) which is constructed by the vertices ( )1, ,i j jv v v + , jfn is the normal

vector of jf and m is the number of the triangles incident to vi and wj is a weight coefficient.


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n

1jv −

jv1jv +

iv

fjn

Figure 6 - 6. One-ring neighborhood of a vertex vi.

The choice of weights is an important task in the weighted normal method. Chen et al. [CHE' 05] discussed the different types of weights and suggested an improved weight based on the theory of gravity.

2f j

g vj iw j

−= (37)

Where f j denotes the area of the triangle jf , jg denotes the centroid point of jf .

6.2.4 Curvature estimation

From the theoretical point of view, triangular meshes don’t have any curvature at all, since all faces are flat and the curvature isn’t properly defined along edges and vertices because the surface is not C2 differentiable. But if we consider the triangular mesh as a piecewise linear approximation of an unknown smooth surface, we can try to estimate the curvatures of the unknown surface using only the information given by the triangular mesh itself. There are three basic approaches to calculate curvature on mesh:

Fitting methods: surface fitting involves finding an analytical function that fits the mesh locally. The curvature of the analytic function is well-defined [ABD' 89, DES' 02, FLY' 89, GAT' 03, GOL' 04, MCL' 97, SHE' 02]. The primary discriminator between fitting methods is the function chosen to model the local surface shape. Function may be parametric, requiring a local parameterization of the surface near each vertex, or implicit. The chosen function is fit separately at each vertex of the mesh. A local 3D coordinate frame, centred at the vertex, is used to simplify the estimation of the chosen function coefficients. The second discriminator is the number of vertices fit by the function. If too few vertices are fit, the problem is under-constrained. Therefore, a minimum number of vertices, based on the number of function coefficients, should be supplied. Fitting the minimum number of vertices defines an interpolating function where the function goes through each vertex. Fitting more than minimum number of vertices leads to approximating function, which minimizes some


127

measure of distance from the function to the vertices, for example, a least-squares minimization.

Discrete methods: one of the main motivations for discrete methods is to avoid the computational cost associated with fitting algorithms. These methods don’t involve solving a least square problem and are very fast. However, many of these methods only provide a subset of Gaussian, mean, and principal curvature directions (unlike a surface fit, from which any of these data can be calculated) [CSA' 00, GAT' 03, HAM' 02, MEE' 00, MEY' 02, TAN' 99].

Estimating the curvature tensor: curvature tensor estimation is similar to the discrete methods, except that instead of estimating the curvature directly; a discrete estimation of the curvature tensor is created; and curvatures and principal directions are calculated from the curvature tensor. These methods tend to have computational complexity lower than the fitting methods, but slightly higher than the discrete methods [RUS' 04, THE' 04].

The different estimation methods are analyzed and compared by Gatzke and Grimm [GAT' 06], the results showed that the fitting methods based on one or two ring neighborhood are recommended. But these methods have a great computational cost and are very dependent on the type of surface being fit. On the other hand, the discrete curvature methods are appealing because of their speed. These methods are recommended to estimate the curvature on regular meshes for which either noise is absent or smoothing has been applied. Chen and Schmitt [CHE' 92 a] estimated normal curvature and principal directions by Euler formula. Their main idea is to use three or more circular fits through a vertex and two of its neighbors to solve the coefficients of the Dupin indicatrix [HOS' 93]. Since there are many triples of points that can be used to create circular fits, the ones forming curves closest to normal section (intersection of the surface with a plane containing the normal vector) are used. Dong and Wang [DON' 05] developed a new method to estimate the normal curvature from which we can calculate the two principal curvatures; they are also simplifying the Chen and Schmitt's method to estimate the principal directions. Their new method is very fast and gets good results both in principal curvature values and directions estimation. In our work, we use the algorithm developed by Dong and Wang [DON' 05] to estimate the principal curvatures values ,max mink k and its corresponding directions max min,t t . Figure 6 - 7 illustrates the maximum and minimum curvature on the brain model using Dong’s curvature estimator.


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Figure 6 - 7. Curvature measures for a brain model using Dong’s curvature estimator [DON' 05]: (a)

minimum curvature values , (b) maximum curvature.

6.2.5 Feature lines on surface

Olivier Monga et al. had shown in [MON' 95] that crests (that are lines where the maximal curvature is locally maximal) can be characterized as the zero crossing of a coefficient maxe , which they called the extremality. maxe is the directional derivative of the maximum principal curvature ( maxk ) in the corresponding principal direction ( maxt ). Corresponding to the two principle directions, there are also two extremality coefficients:

( )( )

m axm ax m ax

m inm in m in

,

,

e k t

e k t

= ∇

= ∇ (38)

Note that extremalities are not well-defined functions on the surface, because the ambiguity of choosing the sign of principal curvature directions ( m ax m in,t t ) induces an ambiguity of the sign of extremalities ( m ax m in,e e ). Still, in a local neighborhood of non-umbilic points (principal curvatures values are different), one can choose the sign of principal direction in such a way that the corresponding extremality becomes locally a smooth function [THI, 96 b 96 b]. The full set of the principal curvatures extrema along their corresponding curvature directions are called the crest-valley lines of the surface M. These feature lines carry significant information about characteristics of the surface. In [HAL' 99, OHT' 04, PRO' 94, STY' 03, THI, 96 b 96 b, YOS' 05] crest and valley lines are salient ridge lines that fulfill two additional requirements, respectively for m axe and m ine :


129

maxmax max min

max0 : 0, | |, ( ),

k

ee k k crests= < >

∂∂

(39)

maxmin maxmin

min0 : 0, | |, ( ).

k

ee k k valley= < > −

∂∂

(40)

If the orientation of the surface gets flipped, equations (39) and (40) exchange their roles. Since the crests and valleys turn into each other as surface orientation is changed, without loss of generality we can consider only the crest lines.

6.2.6 Feature lines extraction

The principal difficulty in detecting the crest lines and similar features on discrete surfaces consists of achieving an accurate estimation of surface curvatures and their derivatives. Different approaches are used in the crest lines estimation to estimate the principal curvatures and their directions. The polynomial fitting was used in the works of [CAZ' 06 b, CAZ' 06 a, KIM' 06], Hildebarndt et al. [HIL' 05] used discrete differential operators to estimate the principal curvature values and its directions, and various combinations of continuous and discrete techniques were used in [OHT' 04, YOS' 05]. While estimating surface curvatures and their derivatives with geometrically inspired discrete differential operators [HIL' 05, RUS' 04] is much more elegant than using fitting methods, the former usually requires noise elimination and the latter seems more robust. On the other hand, as pointed out in [RUS' 04], fitting methods incorporate a certain amount of smoothing in the curvature and curvature derivative estimation processes and that amount is very difficult to control. Another limitation of fitting schemes consists of their relatively low speed to compare with discrete differential operators. Further, since predefined local primitives are used for fitting, one cannot expect a truly faithful estimation of surface differential properties.

Once the weighted normal vector in each vertex of the mesh is estimated using the proposed method in [CHE' 05], the principal curvature values and its directions are estimated using the proposed algorithm in [DON' 05]. Next we calculate the extremality coefficients m axe at each vertex using equation (35), where m axk∇ (the gradient of maxk along its direction maxt ) is calculated using the proposed formula of Guoliang [GUO' 04] for the discretization of gradient via linear approximation. Finally, we inspect each edge of M and check whether it contains a crest point using the proposed technique [OHT' 04], for the edge [ ]1 2,v v which contains a crest point we calculate the coordinates of the extracted point P using the following formula [OHT' 04]:


130

max 2 1 max 1 2

max 2 max 1

( ) ( )( ) ( )

e v v e v vP

e v e v+

=+

(41)

The crest vertices are connected together according to the proposed procedure in [OHT' 04]. In order to reduce the fragmentation of the crest lines we inspect the mesh vertices and their one-ring neighbourhoods to add a new crest point to connect two crest segments if necessary. The used technique in the previous step is inspired by [YOS' 05]. Some examples of crest and valley lines (feature point clouds) extracted on triangular meshes are shown in Figure 6 - 8.

6.2.7 Feature lines thresholding

For real applications dealing with coarse meshes, or meshes featuring degenerate regions or sharp features, or meshes conveying some amount of noise we need to filter the feature lines to eliminate the non significant lines. To classify the feature lines, we first measure the strength of each line, next we threshold the lines using a defined threshold T. The first step is to form one feature line of each set of connected feature point. To do that, we use an algorithm inspired by the region growing segmentation technique to mark the connected feature points by the same label. Each line is composed of a set of connected feature points which have the same label. The strength of each line can be measured by the following formula [OHT' 04]:

Nmax i max i 1

i i 1i 1

k ( p ) k ( p )f p p

2+

+=

+= −∑ (42)

where N is the number of feature points in the line, ip and i 1p + are successive points of the feature line. The maximum curvature at the crest points is estimated using a linear interpolation as follows:

( ) max 2 max 1 max 1 max 2max

max 2 max 1

e ( v ) k ( v ) e ( v ) k ( v )k p

e ( v ) e ( v )+

=+ (43)

where [ ]1 2,v v is the mesh edge containing P. We ignore those crest lines for which the strength f is less than a user-specified value of threshold T. Figure 6 - 9, Figure 6 - 10 show the crest and valley points extracted on Fandisk, horse model respectively and the effect of thresholding the extracted feature lines using a defined threshold.


131

(a) (b)

(a 1) (b 1)

(c)

(c 1)

(d) (e) (f)

Figure 6 - 8. Crest-valley points detected on various models. (a) crest points,(b) valley points, detected on

standard bunny model, (c) crest points detected on pyramid model, (a1, b1, c1) crest, valley points alone are

sufficient for recognizing the models, (d) crest points on cube model, (e) valley points on the Max-Planck

model, (f) valley points detected on Feline model.


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Vally lines force

0

5

10

15

20

25

30

35

40

45

50

1 37 73 109 145 181 217 253 289 325 361 397 433 469 505 541 577 613 649 685 721 757

Valley Line No.

Forc

e

Crest Lines Force

0

20

40

60

80

100

120

140

160

180

200

1 19 37 55 73 91 109 127 145 163 181 199 217 235 253 271 289 307 325 343 361

Crest line No.

Forc

e

(a)

(b)

Figure 6 - 9. Crest and valley points on Fandisk model (a) (left) crest points before the thresholding (5356

points) (top) and after the thresholding (T=1.5) (bottom), (right) force of crest lines, (b) (left) valley points

before the thresholding (5454 points) (top) and after the thresholding (T=2.0) (bottom), (right) force of

valley lines.


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Crest Lines force

0

5

10

15

20

25

30

35

40

1 48 95 142 189 236 283 330 377 424 471 518 565 612 659 706 753 800 847 894 941 988

Crest line No,

Forc

e

Valley Lines Force

0

5

10

15

20

25

1 20 39 58 77 96 115 134 153 172 191 210 229 248 267 286 305 324 343 362 381 400

Valley line No,

Forc

e

(a)

(b)

Figure 6 - 10. Crest and valley points on Horse model (a) (left) crest points before the thresholding (5356

points) (top) and after the thresholding (T=2.0) (bottom), (right) force of crest lines, (b) (left) valley points

before the thresholding (5454 points) (top) and after the thresholding (T=1.0) (bottom), (right) force of

valley lines.


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6.3 Distance map-based model enrichments

Distance maps provide a good method to enrich its associated model by visualizing some additional information non-visible directly on the model. In the following subsections we distinguish two types of distance maps. First, we compute distance maps from all training set models in order to enrich the average surface. Second, we compute some distance maps from the average binary volume itself and from all training binary volumes.

6.3.1 Distance maps-based average surface enrichments

After the construction of the average surface, we rigidly register each surface of the training set to the average surface. Next, Using MESH Measuring Tool (§ 4.3.6), we compute the signed Euclidean distances between the constructed average surface as source and each surface of the registered training set as target. We will obtain a set of signed Euclidean distances distribution associated to the average surface. Each face of the average surface has a set of signed Euclidean distances. Then, we compute the minimum, maximum, mean and standard deviation of these values for each face of the average surface using its set of signed distances. The computed values give us four distributions of scalar values associated with the average surface faces. We call these distributions as distance maps (min, max, mean and standard deviation distance maps, respectively). To quantify the range of distance variation of each face of the average surface, we compute the two extreme z-scores distances for each face using the four computed distance maps. The z-score indicates by how many standard deviations σ a measured Euclidean distance at a face of the average model differs from the mean distance value μ computed at same face.

i ii

i

xz μσ−

= (44)

where xi being the signed minimum (maximum) Euclidean distance measured at the face i (taken from the min and the max distance maps, respectively), i iandμ σ being, respectively, the mean and the standard deviation values at the same face (taken from the mean and the standard deviation distance maps, respectively). Having a min and max z-score values at each face of the average surface provide two additional distributions of measured distances associated to the average surface. We called the two new distributions as Min and Max z-scores distance maps. The enriched average surface is composed of the average surface itself and four additional distance maps (the mean, the standard deviation and the two extreme z-scores distance maps). The training set surfaces and its enriched average surface are represented schematically in Figure 6 - 11, where lines between models represent the forward (form average surface to each


135

surface of the registered training set ) signed Euclidian distance distributions on the average surface labelled EDDi (i=0,.., n) measured using MESH Measuring Tool.

Average ModelConstruction

Trainingsurfaces

RegisteredTraining surfaces

Distance mapscomputation

RigidRegistration

Extremes z-scoresMaps computation

MESH MeasuringTool

EDDN_0

EDDN_1

EDDN_2

EDDN_n

Mean DM

Min z-scores Map

SD DM

Max z-scores Map

Source

Source

Target

Average surface

Enriched Average SurfaceM_0

M_n

M_2

M_1

R M_0

R M_n

R M_2

R M_1

Figure 6 - 11. Schematic diagram indicating training models (M0, M1…, Mn) and the constructed model.

The mean distance map presents the central tendency of the measured distances in each face of the average model. In contrast, the standard deviation distance map provides some information about the statistical dispersion of the measured distances in each face of the average surface. If the measured distances associated to one face of the average surface are close to the mean, then the standard deviation is small; in contrast, if these distances are far from the mean, then the standard deviation is large. The faces with a high standard deviation can be used as landmarks on the average model in order to construct a statistical shape model to describe the shape variations of the studied anatomical structure as in [BAI' 03, COO' 95, COO' 94]. The min and max z-scores distance maps allow to local comparisons across surfaces. Figure 6 - 12 shows an example of the enriched average surface computed from a four training set (M0, M1, M2, and M3).


136

M1 M2 M3AM

MESH Measuring ToolSource

Distance maps-based model enrichmentEuclidean

Distance Distribution

M0

EDD0 EDD1 EDD2 EDD3

Mean. DM SD. DM Min. z-scores Map Max. z-scores Map

-0.303 -0.0828 0.137-0.161 0.0284 0.218-0.233 0.00742 0.218-0.183 0.0692 0.322-0.0291 -0.00283 0.2340.00552 0.125 0.245

-1.50 -0.837 -0.1740.301 0.900 1.5

EDD0

EDD1

EDD2

EDD3Mean. DMSD. DMMin. z-scores MapMax. z-scores Map

Figure 6 - 12. Enriched average surface computed from four training surfaces (M0, M1, M2, M3).

In the following chapter ( Ch.7), we will use the enriched average surface in a phenotyping process application. The basic idea is to classify a new model as normal or abnormal compared to the enriched average normal surface. First, an average surface is constructed from a training set


137

representing the same normal anatomical structure. Four distance maps (mean, standard deviation and the two extreme z-scores distance maps) are computed to enrich the average surface. Second, for a new surface out of the training set, we use MESH Measuring Tool to measure the distance between each face (cell) of the average surface and a the new surface. For each face of the average surface, we compute a new z-score value using the new measured distance, mean and standard deviation values associated with this face (taken from the mean and standard deviation distance maps, respectively). Next, the new z-score value will compared to the two extremes values of z-scores to detect and quantify the differences between the two surfaces.

6.3.2 Distance maps-based average volume enrichments

In order to enrich the average binary volume, some addition Euclidean distance maps are generated from the training binary volumes (voxelized training set). First, we compute one 3D image from all volumes of the voxelized training set. The new image is composed of three regions, namely, internal, external and middle regions (Figure 6 - 13 B). The internal region presents the foreground voxels of the extreme minimum volume (intersection of all voxelized training set volumes), where a (-1) value is assigned to each voxel in this region. In contrast, the external region is the background voxels of the extreme maximum volume (union of all volumes), all voxels of this region have the intensity value (+1). The middle region is composed of all voxels locating between the two regions. These voxels (labelled as middle voxels) have the intensity value (0). Next, from the precedent image, we compute a signed Euclidean distance map [CUI' 99] by assigning to each voxel its Euclidean distance to the nearest voxel of the middle region (Figure 6 - 13 C). The measured distances are negative for the internal voxels and positive for the external voxels. Finally, another signed distance map is computed from the constructed average volume (Figure 6 - 13 A). Figure 6 - 13 A, Figure 6 - 14 illustrate the computed Euclidean distance maps in 2D, 3D respectively. These distance maps can be used beside the average binary volume in some segmentation process using a priori knowledge such as the method proposed by Rose et al. [ROS' 07] to integrate the shape priori in a image segmentation based on region growing method.


138

Mapping

+1 (Out region)

0 (Middle region)

-1 (In region)

AveragingOperator

Average Image (AI)

Average image signed distance map In, out and middle regions In-Out signed distance map

External Region Middle Region Internal Region

I0 I1

In, middle and out regions -1.0 0 1.0

In-Out signed distance map -36.0 47.5 131

Average image signed distance map -43.0 45.0 143.0

0.0

NOR XOR AND

(A) (B) (C)

Figure 6 - 13. Examples of some Euclidean distance maps (2D) computed from two binary images (I0, I1) and

its average image.


139

Slice XY

M0 M1 M2 M3AM

In, middle and out regions 1.0 2.0 3.0

In-Out signed distance map -42.0 46.0 134

Average image signed distance map -48.0 50.5 149

0.0

V0 V1 V2 V3AV

Slice YZSlice XZ

Models

Volumes

In, out and middle regions

In-Out signed distance map

Average image signed distance map

Figure 6 - 14. The computed distance maps from four training models (M0, M1, M2, M3) and its average

model AM.


140

6.3.3 Probability map

This map is presented as volume (3D image) where each voxel has an intensity value in range {0, 1}. The intensity value associated to each voxel of this volume is computed by counting the number of times that the voxel under consideration was a foreground voxel in the training set volumes divided by the number of volumes in the training set. This map can be used also to improve the image segmentation using a priori knowledge. Example of such type of enrichments was presented [LYN' 06]. The authors proposed to couple a density probability function (PDF) in a level-set segmentation process for left-ventricle myocardium segmentation. The PDF is constructed by aligning the binary manually segmented boundary images and summing the boundary elements. Figure 6 - 15 presents the probability map of the training set used in Figure 6 - 14.

Slice XY

0.0 1.00.25 0.5 0.75

Probability map

Figure 6 - 15. Probability map of the training set (M0, M1, M2, M3) used in Figure 6 - 14.


141

6.4 Additional measurements

Volume size, surface area and diagonal diameter length of the oriented boundary box and its variations are some measured quantities which can be used to add another type of enrichment to the average model.

6.5 Conclusion

In this chapter, we proposed to enrich the constructed average model using two principal types of information. Some information extracted directly from the average model itself and other information extracted from all models from which the average model was constructed. The average model is presented in two associated forms: average surface and average binary volume (average form), so we proposed to enrich the two parts of the average model. First, we used a set of existing algorithms to extract some local geometrical characteristics (crest and valley lines) on the average surface itself. The detected feature lines have geometrical characteristics which make of the average surface an enriched reference in such application based on these types of characteristics (a priori segmentation (snake), surface registration …etc). Second, we compute some distance maps to enrich both average surface and average volume. The MESH Measuring Tool (§ 4.3.6) is used to obtain for each average surface four additional distance maps (mean, standard deviation and two extremes z-scores distance maps) in order to enrich the average surface. These distance maps are four distributions of scalar values associated with the faces of the average surface used. Another distance maps are computed from either average binary volume itself or training binary volumes (voxelized training set) to enrich the average volume. In this chapter, the proposed enrichments of the average model are presented directly without any related applications. Next, in the following part, two applications based on the enriched average model (average model + enrichments) will be introduced.

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Conclusion of the second part

In this part, we have presented our proposed methodology to construct an enriched average model from a set of training models (triangulated meshes). This part was composed of four chapters. In chapter 3, we have presented a brief overview of the proposed method. Next, in the first part of the chapter 4, we have introduced the surface registration problem and its basic theoretical concepts. Then, in the next part, we have proposed a feature points-based surface registration to register two surfaces. The proposed surface registration method is used later in all steps of the average model construction methodology. The proposed methodology to construct an average model was presented and detailed in chapter 5. First, we tested a method based on the GPA algorithm (generalized Procrustes Analysis) to construct an affine and elastic average model from a set of triangulated meshes. The results of this method was analysed using two factors: the effect of changing the reference model in the input of the GPA iterative process and the number of necessary iterations to get a stable average model. According to the analysis of the results, we found that this method gives the same results if all models in the training set have the same geometrical shapes. But there are significant differences between the constructed average models if the training set is composed of some models with different geometrical shapes (sphere, cube… etc). Next, in the second part of this chapter, we have proposed a new method based on morphological operations to construct an average model. The constructed average model using the new proposed method is composed of two parts: average surface and its associated binary volume (average binary volume). This method was tested and evaluated using some synthetic models. Finally, we proposed to combine the two methods together in order to take into accounts the advantages of the two methods. In chapter 6, the problem of average model enrichments is investigated and some propositions are detailed. In this chapter, only some examples of the enrichments procedures are presented without any related applications. The following part will present some of these enrichments used in applications and the construction of an enriched average model.

Part 3: Introduction of the third part

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III. Applications

Part 3: Introduction of the third part

144

Introduction of the third part

In this part, we apply our proposed methodology to construct an enriched average model in some real applications using intracranial models of small animals (Mastomys mice). The part is composed of two chapters. In the first chapter, we will use the constructed enriched average model (average surface) of some normal model as a reference model in a phenotyping process. The goal of this application is to study the anatomical differences between the intracranial models of normal and abnormal intracranial Mastomys mice. The abnormal mice are infected by the Lassa virus (Lassa +). First, we will construct an average model, using the proposed methodology, from a set of normal mice models (Lassa-). The average model will be enriched using some distance maps computed using all normal models (as explained in section 6.3.1). Next, we will test the ability of the enriched average model to describe the population from which it was constructed. Similarly, its ability to differentiate a model that belongs to different population will be tested. Finally, we will introduce a phenotyping process based on the normal average model and its enrichments to classify the models as normal or abnormal. In the second chapter, another application of the average model (average binary volume) as a reference model in a segmentation process using a priori knowledge will be presented. First, a general background about the model-based segmentation will be introduced. Next, the RGISP (Region Growing method Integrating Shape Prior) segmentation method [ROS' 07] will be briefly presented. Finally, two examples of how we can use the enriched average model (average volume) as a reference model in RGIPS method integrating shape prior information will be presented.

Ch 7: Average model in mouse phenotyping process

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Ch.7. Average model in mouse

phenotyping process


146

7.1 Introduction

A living organism is defined as the outward physical manifestation of internally coded, inheritable, information. In this definition, there are two parts [BLA' 94]:

The "outward, physical manifestation" of the organism is known as the phenotype. These are the physical parts, the sum of the atoms, molecules, macromolecules, cells, structures, metabolism, energy utilization, tissues, organs, reflexes and behaviours; anything that is part of the observable structure, function or behaviour of a living organism.

The "internally coded, inheritable information" carried by all living an organism is known as the genotype. This stored information is used as a "blueprint" or set of instructions for building and maintaining a living creature. These instructions are found within almost all cells (the "internal" part), they are written in a coded language (the genetic code), they are copied at the time of cell division or reproduction and are passed from one generation to the next ("inheritable"). These instructions are intimately involved with all aspects of the life of a cell or an organism. They control everything from the formation of protein macromolecules, to the regulation of metabolism and synthesis.

Genotyping and phenotyping are two sides of a coin. Genotyping is a series of procedures to help determine the unique set of genes that make up an individual’s DNA (deoxyribonucleic acid) whereas phenotyping is a set of procedures which aim to characterize the differences in biological organism compared to a control one. Genetically modified animals, especially mice, have became key models in biomedical research for understanding the biology of development and disease processes. The study of these animals introduces a need for efficient mouse characterization. Such characterization must be as objective as possible, particularly since individual genetic mutations may produce unknown alterations in several developmental or physiological processes. Mice are anatomically and phylogenetically closest to the human. They are useful tools for understanding human diseases. Transgenic mice (tg) are costly and sometimes difficult to generate; their availability, at least while establishing the initial population, is in most instances limited, since animals are blocked as breeders. Therefore, it would represent a great advantage if the phenotype of such animals at the organ-level could be already established in the initially available animals by non-invasive methods [BEC' 01]. The anatomical phenotyping studies aim to characterize differences in an anatomically normal control group and a mutant (or abnormal) group. These studies require an anatomical comparison of the models in the two groups. It is important that there be several mice in each group since anatomical differences should be assessed against inherent biological variability [NIE' 06 a]. The imaging techniques such as computed tomography (CT) and magnetic resonance imaging (MRI) have been applied to the anatomical phenotyping of transgenic mouse embryos [DUL' 07, JOH' 02, SCH' 04] as well as in the brain and skulls of mouse models [NIE' 06 a, SRA' 06]. The measurement of 3-D coordinates as biological landmarks on the skull was used to analyze


147

craniofacial phenotypes in mouse models for Down syndrome [RIC' 02]. Similarly, metabolic profiling of cardiac tissue through high-resolution nuclear magnetic resonance spectroscopy in conjunction with multivariate statistics was used to classify mouse models of cardiac disease [JON' 05]. In this application, we will see how to use the enriched average model as a reference model in an anatomical-based phenotyping process. The anatomical phenotyping is performed using 72 intracranial models of Mastomys natalensis mice divided into two groups. The first group included 28 models obtained from mice infected by Lassa virus (Lassa+), whereas the 44 models of the second group were obtained from not-infected mice (Lassa-). Each group is also divided into subgroups according to age and sex. Our proposed phenotyping approach consists of an initial comprehensive evaluation, followed by more focused or targeted evaluation based on information derived from initial evaluation. In the initial evaluation, we will compare some global measured features (volume, area….etc) between the intracranial models of normal (Lassa -) and abnormal (Lassa +) mice. The goal of this comparative study is to see if there are significant differences between the measured values of the two groups. According to the results of this study, we will choose some subgroups which have significant differences between the normal and abnormal models to be used later in our application. In the second evaluation, we will construct an enriched average model form the selected normal subgroup models. Next, we will test the ability of the constructed enriched average model to describe the population from which it was constructed (normal models) and to differentiate a model that belongs to different population (abnormal models) using a proposed anatomical-based phenotyping process. Finally, we will classify 24 models (16 normal and 8 abnormal) using the proposed phenotyping process. Some results of this application will be presented and discussed in the end of this chapter.

7.2 Material and data acquisition methods

7.2.1 Animals

For this study, we use 72 intracranial models of Mastomys natalensis mice in total (36 models from male and 36 models from female mice): 44 of these models were obtained from normal mice (Lassa -), and the other 28 models were obtained from mice infected by Lassa virus (Lassa +). Depending on the age of the mice, the models were classified into three age classes (from younger to older). For descriptive statistics of the used models, see Table 7 - 1:


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Nromal (Lassa -) Abnromal (Lassa+) Nromal (Lassa -) Abnromal (Lassa+)1 2 2 2 22 4 4 4 43 16 8 16 8

Age classMale Female

Table 7 - 1. Collection of Mastomys mice intracranial models.

7.2.2 Imaging and segmentation

Before the imaging stage all skulls of used mice were isolated and all intracranial cavity holes were filled using a pate to enhance the results of the used segmentation algorithm. In this study the in-vivo SkyScan 1076 micro CT-scanner (ANIMAGE, CERMEP, Hôpital Neuro-Cardiologique, Lyon, France) has been used to scan the skulls of used mice. All datasets were acquired with the same protocol (100 kv, 100 μA, rotation 360°, rotation setp 0.5°, camera pixel size 11.7 μm). The datasets are reconstructed using NRecon volumetric reconstruction software (SkyScan) (Image pixel size 36 μm, depth 16 bit, reconstruction duration per slice 1.198853 second). The segmentation of each mouse intracranial cavity is done using a multi-resolution active contours (snake) algorithm [GOUA' 03]. The following figure (Figure 7- 1) shows a schematic diagram of the imaging and segmentation stage.

Holes fillingSegmentation

Figure 7- 1. Imaging / segmentation stage

Representative examples of the intracranial models from normal mice in comparison to abnormal mice for different sexes and ages are shown in Figure 7- 2.


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7.3 Initial comprehensive analysis of data

7.3.1 Data analysis

Overall, the skull models of abnormal mice were observed to be smaller than their normal counterparts. To evaluate the differences between the constructed intracranial models, first, we compare some global quantitative features (properties) between the normal and abnormal models. Some examples of used measurable features in our application are volume (V), surface area (A), and oriented bounding box diagonal diameter length (OBBDD) [GOT' 96]. Since these features can only describe a model very roughly they are not very discriminative but they can be used as an initial filtering stage in a phenotyping application. We used the method proposed in [ZHA' 01] to compute the volume and surface area of 3D models. The measurements results of the previous features performed on the intracranial models are

given in Figure 7- 3, Figure 7- 4 for male and female mice, respectively. Some statistical

parameters (Mean, Standard deviation (SD), Abnormal

Normal

Meanvariation (%var=100 100)Mean

− ) are

computed using the measured values also presented in the following figures.

The statistical significance of the differences between two lists (normal and abnormal models) of measured quantities is assessed using the p-value [FEI' 02]. The alternative hypothesis is that the two compared lists of values are not different, so a two sided p-value (two tail p-value) is

Age 2

Age 1

Age 3

Male Female

Nromal Abnormal Nromal Abnormal

Figure 7- 2. Examples of intracranial model of Mastomys natalensis mice


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computed using the non-parametric Mann-Whitney U-test [FEI' 02]. Table 7 - 2 summarizes the set of p-value for statistical significance used in our analysis:

p-value Wording Summary> 0.05 Not significant NS

0.01 to 0.05 Significant *0.001 to 0.01 very significant **

< 0.001 Extremely significant ***

Table 7 - 2. Table summarizes the p-value using the words in the middle column

Abnormal groups (Lassa +) of male Mastomys mice display a reduced mean values of the measured features (volume, area and OBBDD length) in comparison to the normal groups (Lassa -) for the three age classes (Figure 7- 3 A). The comparative analysis of younger mice (Age classes 1, 2) determined no significant differences (p-value > 0.05) between the intracranial models of normal and abnormal male mice (Figure 7- 3 B, C, D). Note that, the numbers of the used younger mice (1, 2 age classes) are very small to validate the statistical comparison (2, 4 for age classes 1, 2,

Normal Abnormal Normal Abnormal Normal Abnormal

557,36 545,48 374,16 367,68 18,49 18,49541,44 481,16 372,34 336,02 18,12 17,38

Mean 549,40 513,32 373,25 351,85 18,30 17,93SD 11,26 45,49 1,29 22,39 0,26 0,79

%Var652,84 680,84 417,55 434,20 19,31 19,77687,15 617,07 433,44 399,40 19,82 19,18672,10 608,66 427,88 399,27 19,50 19,22637,34 609,96 408,42 399,59 19,54 18,97

Mean 662,36 629,13 421,82 408,11 19,54 19,29SD 21,80 34,67 11,10 17,39 0,21 0,34

%Var773,25 682,76 475,62 447,15 20,38 19,90791,05 669,13 486,29 423,94 20,71 19,86820,50 680,15 487,59 431,10 20,96 19,63742,37 659,16 466,48 417,95 20,73 19,72742,71 664,19 455,75 417,22 20,29 19,95693,31 665,20 446,68 435,43 20,01 19,83743,24 589,95 458,29 390,18 20,29 18,80757,46 617,06 467,27 406,60 20,58 19,07681,09 438,92 19,83644,13 410,96 19,38791,93 488,22 20,85750,10 464,35 20,33722,29 456,14 20,19683,08 428,70 19,83640,37 412,92 19,18682,21 436,58 19,90

Mean 728,69 653,45 455,05 421,20 20,21 19,60SD 53,50 32,64 24,52 17,66 0,51 0,43

%Var

Male

Age Class V (mm3) A (mm2) OBBDD (mm)

1

-6,57 -5,74 -2,02

2

-5,02 -3,25 -1,30

-10,33 -7,44 -3,06

3

Mean Intracranial Volume (Male)

450

500

550

600

650

700

750

800

Mean Age 1 Mean Age 2 Mean Age 3

Volu

me

(mm

3 )

Normal Abnormal

Age 1 : p=0,666 ( NS )Age 2 : p=0,200 ( NS )

Age 3 : p=0,001 (** )NS

NS **

B

Mean Intracranial Area (Male)

300

340

380

420

460

500


Are

a (m

m2 )

Normal Abnormal

Age 1 : p=0,33 3 ( NS )Age 2 : p=0,324 ( NS )

Age 3 : p=0,003 ( **)

NS

NS**

C

D

A

Mean Intracranial OBBDD length (Male)

17

18

19

20

21


OB

B_D

D (m

m)

Normal Abnormal

Age 1 : p=0,999 ( NS )Age 2 : p=0,200 ( NS )

Age 3 : p=0,005 ( ** )NS

NS **

D

Figure 7- 3. Male mice intracranial models characteristics


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respectively). In contrast, for the older mice (Age class 3), the abnormal male mice demonstrate a very statistically significant reduced mean values of the measured features (Table 7 - 3).

Normal Abnormal p-value

728,69 ± 13,7 653,45 ± 11,54 0,001 ( ** )455,05 ± 6,13 421,20 ± 6,24 0,003 ( ** )

20,21 ± 0,13 19,6 ± 0,15 0,005 ( ** )OBBDD (mm)

Male ( Age 3)FeatureV (mm3) A (mm2)

Table 7 - 3. Statistical comparison between the measured features of the intracranial models of male mice

(3rd age class) (Mean ± SEM (standard error of mean)), the corresponding p-values.

The comparative analysis of the measured features values of female mice models demonstrates a reduced mean values for the younger abnormal mice models (Age classes 1, 2) in comparison with the normal mice of the same ages (Figure 7- 4). In contrast, the differences between these values are very small in the third age class. The differences between the measured features values for female mice intracranial models are not statistically significant for all age classes (p-value > 0.05) (Figure 7- 4 B, C, D).

In Figure 7- 5, we compare the mean values of the measured features (volume, area, OBBDD length) of both male and female normal intracranial models.

Normal Abnormal Normal Abnormal Normal Abnormal

614,11 572,65 399,13 381,43 19,04 18,61635,43 548,89 406,71 376,12 19,09 18,30

Mean 624,77 560,77 402,92 378,78 19,07 18,45SD 15,07 16,81 5,36 3,75 0,03 0,22

%Var642,30 631,31 417,87 409,42 19,30 19,19684,80 624,70 437,73 408,40 19,81 19,43656,49 594,13 410,35 396,80 19,47 19,00646,30 642,43 410,80 419,34 19,40 19,33

Mean 657,47 623,14 419,19 408,49 19,50 19,24SD 19,17 20,68 12,83 9,23 0,22 0,18

%Var700,63 735,70 435,94 461,06 20,07 20,33764,19 680,04 471,80 430,38 20,70 19,58809,41 757,81 481,76 459,27 20,78 20,21788,64 747,46 475,54 463,03 20,99 20,54653,11 648,55 414,13 420,86 19,31 19,65762,68 825,91 464,45 485,30 20,38 21,25726,47 638,33 457,54 421,99 20,19 19,59607,56 691,03 399,53 441,28 19,22 19,86750,07 464,05 20,58763,40 472,61 20,48687,49 434,80 19,73723,95 456,84 20,32653,70 424,32 19,51752,85 464,63 20,66699,00 438,02 19,88711,55 447,37 19,91

Mean 722,17 715,60 450,21 447,90 20,17 20,13SD 54,03 62,91 23,73 22,94 0,54 0,58

%Var

Female

Age Class V (mm3) A (mm2) OBBDD (mm)

1

-10,24 -5,99 -3,22

2

-5,22 -2,55 -1,31

3

-0,91 -0,51 -0,21

Mean Intracranial Volume (Female)

500550600650700750800


Volu

me

(mm

3 )

Normal Abnormal

Age 1 : p=0,333 ( NS )Age 2 : p=0,057 ( NS )Age 3 : p=0,568 ( NS )

NS

NS

NS

B

Mean Intracranial Area (Female)

350

375

400

425

450

475

500


Are

a (m

m2 )

Normal Abnormal

Age 1 : p=0,333 ( NS )Age 2 : p=0,200 ( NS )Age 3 : p=0,652 ( NS )

NS

NSNS

C

Mean Intracranial OBBDD (Female)

18

19

20

21


OB

B_D

D (m

m)

Normal Abnormal

NS

NS

NSAge 1 : p=0,333 ( NS )Age 2 : p=0,200 ( NS )Age 3 : p=0,696 ( NS )

D

A

Figure 7- 4. female mice intracranial models characteristics


152

Regarding the results presented in Figure 7- 5, it can be seen that the differences between the measured features, for all age classes, between normal male and female Mastomys mice intracranial models are not significant (p-value > 0.05).

7.3.2 Results

Analysing the results presented in this section, we found that the measured features of the abnormal intracranial models are generally smaller than those of the normal models. These differences are significant (p-value < 0.05) only for the intracranial models of the 3rd age male models. In the next section, we will focus our study on the 3rd age class male mice. First, an enriched average model will be constructed from the normal models. Next, the constructed average model will be used as a reference model in a proposed anatomical phenotyping process to identify all the regions of differences between the normal and abnormal models. Finally, the ability of the enriched average model to describe the models from which it was constructed (normal models) and

Mean Intracranial Volume (Lassa -)

500

550

600

650

700

750

800

Mean Age 1 Mean Age 2 Mean Age 3Vo

lum

e (m

m3 )

M alel Femalel

Age 1 : p=0,333 (NS)Age 2 : p=0,885 (NS)Age 3 : p=0,955 (NS)

NS

NS

NS

Mean Intracranial Area (Lassa -)

300

340

380

420

460

500


Are

a (m

m2 )

M ale Femalel


NS NS

NS

Mean Intracranial OBBDD length (Lassa -)

17

18

19

20

21


OB

B DD (m

m)

M alel Femalel


NSNS

NS

Figure 7- 5. Male-female normal mice characteristics


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to differentiate models that belong to different group (abnormal models) will be tested. Some results will be presented in the end of the next section.

7.4 Average model-based phenotyping analysis

In order to study the difference between the normal and abnormal intracranial models of the 3rd age class male mice, we propose to use a phenotyping process consists of the two following steps (Figure 7- 6):

Enriched normal average model construction from the normal training models (Lassa-) (§ 7.4.1).

Phenotyping process to test the ability of the enriched average model to classify unknown models as normal or abnormal model (§ 7.4.2).

Enriched Average model construction

Phenotyping process

Normal ModelsUnknown model

AMMean DM

SDDM

Minz-scores map

Maxz-scores map

Decision (Normal / Abnormal)

Figure 7- 6. Average model-based phenotyping analysis pipeline

In the following subsections we will detail each step of the phenotyping process.

7.4.1. Enriched normal average model construction

First, an average model (average surface) is constructed by the method described previously ( Ch.5). The used training set is composed of 16 models representing the normal intracranial models of the 3rd age class of male mice. Next, the normal average surface is enriched using distance maps computed from all models of the input training set (§ 6.3.1). Figure 7- 7 shows different views of the constructed average model (average surface) and its enrichments.


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Average surface Mean DM SD DM Min z-scores DM Max z-scores DM

Max z-scores DM 2.882.341.81.260.723-0.784-1.35-1.91-2.47-3.03Min z-scores DM

0.0306-0.0231-0.0768-0.130-0.184 (mm)Mean DM0.0769 0.186 0.295 0.404 0.513 (mm)SD DM

Figure 7- 7. Enriched average model of the normal 3rd age class (male mice).

One of the main difficulties in the construction of an anatomical average model is its assessment, i.e. how well does the average model describe normality and how sensitive is it at detecting abnormalities? To test the ability of the average model to describe the population from which it was constructed, we computed the face-by-face z-scores distance (§ 6.3.1) from the average surface to


155

each surface of the normal training models ( _ : 0,1,2,...,15iN M i = ) (Figure 7- 8 (A)). As further test of how well the average model represents the population, two additional 3rd age class intracranial male models (new normal models) were rigidly registered to the average model. These models were also normal models but were not used to construct the average model. The z-scores distributions for these models (Figure 7- 8 B) are also correlated with the distributions of the normal models used to construct the average model.

Percentage of average model cells based on their absolute z-score value (Normal models)

Model

N_M_00 77.2 % 98.7 % 100 %N_M_01 82.9 % 99.4 % 100 %N_M_02 82.0 % 99.0 % 100 %N_M_03 79.6 % 99.2 % 100 %N_M_04 81.4 % 98.8 % 99.9 %N_M_05 72.3 % 96.1 % 100 %N_M_06 77.7 % 99.3 % 100 %N_M_07 80.6 % 99.9 % 100 %N_M_08 65.2 % 93.5 % 99.9 %N_M_09 44.07 % 86.5 % 99.8 %N_M_10 78.5 % 99.9 % 100 %N_M_11 78.8 % 96.0 % 99.9 %N_M_12 86.4 % 100 % 100 %N_M_13 69.4 % 92.7 % 98.6 %N_M_14 53.5 % 75.5 % 92.7 %N_M_15 60.4 % 90.1 % 99.7%

1z σ≤ 2z σ≤ 3z σ≤

A

B Percentage of average model cells based on their absolute z-score value (New normal models)

Model

New N_M_00 77.7 % 97.1 % 99.2 %

New N_M_01 80.3 % 98.3 % 99.9 %

1z σ≤ 2z σ≤ 3z σ≤

Figure 7- 8. The percentage of average model faces, for different normal models, (A) normal models used to

construct the average model, (B) new normal models not used in the average model construction stage

These z-scores histograms (Figure 7- 8) indicate that the measured z-scores values have a normal distribution (About 68% of values are within one standard deviation and about 95% of the values are within two standard deviations (68-95-99.7 rule), which assess that the average model represents the normal models well. The second validation of the average model focused on its ability to differentiate a model that belongs to different population. This was tested using the abnormal 8 models (Lass+)


156

( _ : 1,2,...,8iAb M i = ). All abnormal models were rigidly registered to the average model, and its corresponding z-score values were calculated. Those results are reported in Figure 7- 9 (A).

Percentage of average model cells based on their z-score distance (Abnormal models)

Model

Ab_M1 62 % 89 % 96 %Ab_M2 68 % 94 % 99 %Ab_M3 64 % 93 % 98 %Ab_M4 55 % 81 % 92 %Ab_M5 60 % 90 % 96 %Ab_M6 56 % 84 % 96 %Ab_M7 27 % 53 % 73 %Ab_M8 39 % 71 % 88 %

A

1z σ≤ 2z σ≤ 3z σ≤

statistical significance (p value)

statistical significance

p-value 0.0022 ( * * ) 0.0006 ( * * * ) < 0.0001 ( * * * )

B 1z σ≤ 2z σ≤ 3z σ≤

Figure 7- 9. (A) The percentage of average model faces, for different abnormal normal models, (B) Table of

the computed p-values to evaluate the stochastic significance between normal and abnormal models.

In order to assess the global statistical significance of the difference between the normal and abnormal models, we compared its z-scores histograms (Figure 7- 8 (A) and Figure 7- 9 (A)). The alternative hypothesis is that the two z-scores histograms are different, so a two sided p-value (two tail p-value ) is computed using the non-parametric Mann-Whitney U-test [FEI' 02]. The results reported in Figure 7- 9 (B) demonstrate very significant difference in the computed z-score histograms between normal and abnormal models. These results are correlated with the data analysis results of the global descriptors (features) (volume, area and OBBDD length) introduced in section 7.3.

7.4.2. Phenotyping process

The objective of the phenotyping process is to classify an unknown model (triangulated mesh) as normal or abnormal using an enriched average model (average surface) as a reference in a comparing process. We assume that all models used to construct the enriched average model are normal models, but the normality and abnormality are relative. Such statements can only be true to certain degree.


157

MESH Measuring Tool

z-scores Map computation

EDD

Mean DMSD DM

Min Z Scores Map

Max Z Scores Map

Rigid registrationNormalaverage model

Unknown model

Faces classification

z-scores map

Decision(Normal / Abnormal)

Nor

mal

Enr

iche

d A

vera

geM

odel

Targ

et Source

Model Classification

thr1

thr2

Visualization

Target

Sour

ce

Figure 7- 10. Phenotyping process

The proposed phenotyping process, illustrated in Figure 7- 10, consists of five principal steps:

1. Rigid registration: the unknown model (source) will be rigidly registered to the normal average model (target). This step is necessary to eliminate the differences between the two models due to the translation and the rotation.

2. Signed Euclidean distance distribution (EDD) computation: using MESH Measuring Tool (§ 4.3.6) a signed Euclidean distance between each face of the normal average model (source) and the registered unknown model (target) will be computed.

3. Z-score distance map computation: a z-score distance map will be computed using EDD, the mean and standard deviation distance maps provided as enrichment of the normal average model.

i ii

i

xz μσ−

= (45)

where iz is the measured z-score value at the ith face of the average model, i iandμ σ being, respectively, the mean and the standard deviation values at the same face (taken from the mean and the standard deviation distance maps, respectively).

4. Classification of the normal average model faces: each face of the average model will be classified into normal or abnormal face using its z-score value computed from this face to


158

the registered unknown model. First, we measure the differences between the z-score value of the face and its two corresponding extreme values:

min min

max max

i i i

i i i

z z z

z z z

Δ = −

Δ = − (46)

where iz is the measured z-score value at the ith face of the average model , min maxi iz and z are its two

corresponding extreme z-score values taken directly from the minimum and maximum z-score

distance maps, respectively. Next, we can classify the faces in three categories depending on

min maxi iz and zΔ Δ . Each face hasmaxizΔ larger than a given threshold ( 1thr ) is located outside the

maximum outlier of the normal models (abnormal face), similarly, each face has minizΔ smaller than

negative threshold ( 1thr− ) is located inside the minimum outlier of the normal models (abnormal

face), otherwise, the face is located in the normal range of the average models (normal face).

( )min max1 1

th

th

( )

(the i face is normal)

(the i face is abnormal)

i iif z thr and z thr

else

Δ ≥ − Δ ≤

5. Model classification: One example of the used classification creterion is the percentage of abnormal classified faces of the total average model faces number, when this percentage is more than a given threshold ( 2thr ), we consider the unknown model as abnormal and normal otherwise.

In the following subsection, we will test our phenotyping process to classify the intracranial of 3rd age class male mice.

7.4.3. Results

We test our proposed phenotyping process using 24 intracranial models of 3rd age class male mice used previously (16 normal and 8 abnormal models). The enriched average model is constructed from the normal models (16 models) to be the reference model of the phenotyping process. Each model must be examined separately using the five phenotyping process steps (§ 7.4.2) for the same two thresholds 1thr and 2thr . Figure 7- 10 shows the percentage of average model faces classified as normal (4th step results of the phenotyping process) for the 8 abnormal models ( _ : 1,2,...,8iAb M i = ) using different values of ( )( )1 0.0, 0.1, .......,1.0thr σ . Knowing that for the 16 normal models, all faces of the average model are classified as normal faces (100%) for all used values of 1thr .


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Using the proposed phenotyping process, we classified the used 24 intracranial models into normal or abnormal using different values of the first and second thresholds ( ( )1 2:0.0, 0.1, .......,1.0 , :10%, 9%, .......,1%thr thrσ ). The results of our experiments reported in Table 7 - 4 are computed in following manner: If the model is normal (obtained from normal mouse) and it is classified as normal model by the phenotyping process, we count one true negative (TN). Similarly, if the model is abnormal and the phenotyping process classified it as abnormal, we count one true positive (TP). If the phenotyping process classified a normal model as abnormal, we count a false negative (FN). Conversely, if an abnormal model is classified as normal model, we count one false positive (FP). We summed the number of TP occurrences together. Similarly, we summed the TN occurrences, the FP occurrences, and the FN occurrences.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Ab_M1 88,1 % 89,9 % 91,6 % 92,9 % 94 % 94,9 % 95,9 % 96,7 % 97,6 % 98,1 % 98,5 %Ab_M2 93,8 % 95,1 % 96,6 % 97,6 % 98,3 % 98,8 % 99,2 % 99,4 % 99,6 % 99,8 % 99,8 %Ab_M3 91,2 % 92,8 % 94,2 % 95,6 % 96,3 % 96,9 % 97,3 % 97,6 % 97,8 % 98 % 98,2 %Ab_M4 81 % 83 % 85 % 87,1 % 88,9 % 90,4 % 91,6 % 92,9 % 94 % 94,8 % 95,7 %Ab_M5 90 % 92,1 % 93,6 % 94,9 % 96,3 % 97,1 % 97,7 % 98,1 % 98,5 % 98,8 % 98,9 %Ab_M6 85,1 % 87,6 % 89,6 % 91 % 92,4 % 93,8 % 95,2 % 96,4 % 97,1 % 97,6 % 98 %Ab_M7 57,5 % 60,6 % 63,6 % 66,3 % 69 % 71,7 % 74,3 % 76,7 % 79,1 % 81,1 % 83,2 %Ab_M8 73,7 % 76,6 % 79,3 % 82,1 % 84,8 % 87,1 % 89,1 % 90,7 % 92 % 92,9 % 93,9 %

1( )thr σ

Percentage of the average model faces number classified as normal faces ( )min max1 1( )i iz thr and z thrΔ ≥ − Δ ≤

50556065707580859095

100

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0

t

% n

umbe

r of a

vera

ge m

odel

face

s cl

assi

fied

as n

orm

al c

ells Ab_M1

Ab_M2

Ab_M3

Ab_M3

Ab_M4

Ab_M5

Ab_M6

Ab_M7

1thr ( σ)

Figure 7- 11. The percentage of the average model faces classified as normal faces for the abnormal models.


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0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

10%

TN 16 16 16 16 16 16 16 16 16 16 16FN 2 3 4 5 5 6 6 7 7 7 7

TP 6 5 4 3 3 2 2 1 1 1 1FP 0 0 0 0 0 0 0 0 0 0 0

9%

TN 16 16 16 16 16 16 16 16 16 16 16

FN 2 3 4 4 5 5 6 6 7 7 7TP 6 5 4 4 3 3 2 2 1 1 1

FP 0 0 0 0 0 0 0 0 0 0 0

8%

TN 16 16 16 16 16 16 16 16 16 16 16

FN 1 3 3 4 5 5 5 6 6 7 7TP 7 5 5 4 3 3 3 2 2 1 1FP 0 0 0 0 0 0 0 0 0 0 0

7%

TN 16 16 16 16 16 16 16 16 16 16 16FN 1 1 3 3 4 5 5 5 6 6 7

TP 7 7 5 5 4 3 3 3 2 2 1FP 0 0 0 0 0 0 0 0 0 0 0

6%

TN 16 16 16 16 16 16 16 16 16 16 16FN 0 1 2 3 3 4 5 5 5 6 6TP 8 7 6 5 5 4 3 3 3 2 2

FP 0 0 0 0 0 0 0 0 0 0 0

5%

TN 16 16 16 16 16 16 16 16 16 16 16

FN 0 1 1 2 3 3 5 5 5 5 6TP 8 7 7 6 5 5 3 3 3 3 2

FP 0 0 0 0 0 0 0 0 0 0 0

4%

TN 16 16 16 16 16 16 16 16 16 16 16FN 0 0 1 1 3 3 3 5 5 5 5

TP 8 8 7 7 5 5 5 3 3 3 3FP 0 0 0 0 0 0 0 0 0 0 0

3%

TN 16 16 16 16 16 16 16 16 16 16 16FN 0 0 0 1 1 2 2 3 5 5 5

TP 8 8 8 7 7 6 5 5 3 3 3FP 0 0 0 0 0 0 0 0 0 0 0

2%

TN 16 16 16 16 16 16 16 16 16 16 16

FN 0 0 0 0 1 1 1 2 2 3 4TP 8 8 8 8 7 7 7 6 6 5 4

FP 0 0 0 0 0 0 0 0 0 0 0

1%

TN 16 16 16 16 16 16 16 16 16 16 16

FN 0 0 0 0 0 0 1 1 1 1 1TP 8 8 8 8 8 8 7 7 7 7 7FP 0 0 0 0 0 0 0 0 0 0 0

1thr2

thr

Table 7 - 4. The results of the proposed phenotyping process classification of 24 models (16 normal (Lassa -)

and 8 abnormal (Lassa +)) using different values of thresholds.

The optimal thresholds ( )1 2,thr thr for discriminating abnormal from normal models were determined using the medical statistics formulated in equations (47-50) [PET' 05]:


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TPSensitivityTP FN

=+

(47)

TNSpecificityTN FP

=+

(48)

( ) TPPosisitve PredectiveValue PPVTP FP

=+

(49)

( ) TNNegative PredectiveValue NPVTN FN

=+

(50)

The computed medical statistics (Sensitivity, Specificity, PPV, and NPV), based on the classification results presented in Table 7 - 4, are reported in Table 7 - 5. Ideally, a phenotyping process should have a 100% score for all these statistics. A100% sensitivity would indicate that it detects all abnormal models. A 100% specificity would indicate that it never classifies abnormal model as normal one. A 100% positive predictive value (PPV) would indicate that, when the phenotyping process detects an abnormal model, the model is actually abnormal. Similarly, a 100% negative predictive value (NPV) would indicate that, when the phenotyping process detects a normal model, the model is actually normal. In Table 7 - 4 and Table 7 - 5, we can determine the ideal thresholds for this application using the red framed area.


162

Finally, in order to see if the detcted abnormal regions are located in a specific area of the

abnormal intracranial models, the classified faces of the average model (4th step results of the

phenotyping process) are color-coded according to its corresponding computed z-score value

(Figure 7- 12). Figure 7- 12 shows the classification results of the 8 abnormal models

( _ : 1,2,...,8iAb M i = ) using the proposed phenotyping process for two chosen

thresholds ( )1 20.3, 3%thr thr= = . For each abnormal model, four images framed together present:

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

10%

Sensitivity 75,0% 62,5% 50,0% 37,5% 37,5% 25,0% 25,0% 12,5% 12,5% 12,5% 12,5%Specificity 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0%

PPV 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0%NPV 88,9% 84,2% 80,0% 76,2% 76,2% 72,7% 72,7% 69,6% 69,6% 69,6% 69,6%

9%


PPV 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0%NPV 88,9% 84,2% 80,0% 80,0% 76,2% 76,2% 72,7% 72,7% 69,6% 69,6% 69,6%

8%


PPV 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0%NPV 94,1% 84,2% 84,2% 80,0% 76,2% 76,2% 76,2% 72,7% 72,7% 69,6% 69,6%

7%


PPV 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0%NPV 94,1% 94,1% 84,2% 84,2% 80,0% 76,2% 76,2% 76,2% 72,7% 72,7% 69,6%

6%


PPV 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0%NPV 100,0% 94,1% 88,9% 84,2% 84,2% 80,0% 76,2% 76,2% 76,2% 72,7% 72,7%

5%


PPV 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0%NPV 100,0% 94,1% 94,1% 88,9% 84,2% 84,2% 76,2% 76,2% 76,2% 76,2% 72,7%

4%


PPV 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0%NPV 100,0% 100,0% 94,1% 94,1% 84,2% 84,2% 84,2% 76,2% 76,2% 76,2% 76,2%

3%


PPV 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0%NPV 100,0% 100,0% 100,0% 94,1% 94,1% 88,9% 88,9% 84,2% 76,2% 76,2% 76,2%

2%


PPV 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0%NPV 100,0% 100,0% 100,0% 100,0% 94,1% 94,1% 94,1% 88,9% 88,9% 84,2% 80,0%

1%


PPV 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 100,0%NPV 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 94,1% 94,1% 94,1% 94,1% 94,1%

1thr2

thr

Table 7 - 5. The medical statistics of the classification results in Table 7 - 4


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(1) the z-scores distance maps on tow views of the average model (the two left images), (2) its

corresponding classified faces (the two right images), the normal faces are shown in green colour

whereas the abnormal faces are shown in two colours: red for min

0.3izΔ < − , and blue for

max0.3izΔ > .

Ab_M3

Ab_M6

Ab_M7

Ab_M2

Ab_M4 Ab_M5

Ab_M8

TP

TPFNTP

TPTP

TP TP

Ab_M1

-4.11 -0.197 3.71

-4.26 +0.115 6.56

-5.42 -1.33 2.76

-4.34 -1.44 1.46

-6.57 -2.07 2.43

-4.48 -1.05 2.38

Ab_M1

Ab_M2

Ab_M3

Ab_M4

Ab_M5

Ab_M6

Ab_M7

Ab_M8

-3.06 -0.371 2.86

-3.62 -0.399 2.83

Figure 7- 12. Phenotyping process classification results for the abnormal models. For each model: (left) the

z-scores distance maps, (right) the detected abnormal regions (right). Red, blue regions indicate the regions

where ( )1 20.3, 3%thr thr= =


164

Color-coded phenotyping process results ( )1 20.3, 3%thr thr= = presnted in Figure 7- 12 shows that, the detcted abnormal regions are not located in a specific region of the intracranial models constructed from mice infected by the Lassa virus (Lassa+).

7.5 Conclusion and discussion

In this chapter, we have presented an application of the enriched average model (average surface) to study the differences between some normal and abnormal intracranial models of Mastomys natalensis mice. The principal goal of this application was to study the ability of the constructed enriched average model to describe the population from which it was constructed (normal models) and to differentiate a model that belongs to different population (abnormal models) using a real data. First, we introduced the phenotyping problem and the goal of this application. Next, we compared some global measured quantitative features between the constructed intracranial models from normal and abnormal mice. According to the results of this initial comprehensive analysis of data, we found that abnormal intracranial models are smaller than the normal models. The differences are very significant (p-value < 0.05) only for the intracranial models of the 3rd age male models, so we focused our study on this age class models. Second, we proposed an anatomical-based phenotyping process in order de detect and localize the differences between the normal and abnormal models of the 3rd age class male mice. We used the proposed phenotyping process to classify 24 intracranial models (16 normal, 8 abnormal). An enriched average model was constructed from the normal models. The enriched average model greatly reduced the complications (local anatomical comparison intra-models) of analyzing multiple anatomical models in representing the normal populations by one enriched model. Finally, some results of this phenotyping process are presented in the end of this chapter. Also, the optimal values of some used parameters (thresholds) are determined using usual medical statistical methods. In high level application of the mice phenotyping, the phenotyping process will be more complex than what we have proposed here. But this is an example of an anatomical phenotyping process using an enriched average model which will be enhanced in the future works by integrating additional normality criterions.

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Ch.8. Enriched average model-based

segmentation

166

8.1 Introduction

In this chapter, we present two examples of how we can use the enriched average model (average volume) as a reference model in image segmentation process integrating shape prior information. First, a general background about the model-based segmentation will be introduced. Next, the RGISP (Region Growing method Integrating Shape Prior) segmentation method developed by Rose et al. [ROS' 07] will be briefly presented. Finally, two examples of the segmentation using the RGISP process integrating shape prior information via the enriched average model will be illustrated and discussed in the end of this chapter. First example aims to present the advantage of using a reference model in a segmentation process using a synthesized 2D images whereas the second is an example of RGISP method performed on micro-CT images of mouse skull.

8.2 Model-based segmentation

Segmentation of anatomic structures in medical images and reconstructing their shape represents a particularly challenging problem due to the complexity and variability of human anatomy. Several classical image processing techniques (e.g. multiple thresholding, region growing, morphologic filtering, active contours) have been applied to try to solve this problem, with variable outcomes [SUR' 01]. However, the above techniques are all grey intensity based. They tend to be unreliable when segmenting a structure that is surrounded or occluded by other structures with similar image intensity (e.g. low-contrast structures). Therefore, in these cases, prior shape information is needed to successfully segment the structures. The process of segmentation with the incorporation of shape information is called shape prior segmentation [CHA' 05]. Moreover, the classical image segmentation techniques typically make very limited use of the available knowledge of the underlying anatomy. User intervention is usually required to separate adjacent structures of similar image intensity. Recently, several model-based image segmentation methods have been investigated and successfully implemented. The common goal of all these methods is to integrate available a priori knowledge with sophisticated image processing techniques. In general, these types of frameworks are somewhat successful as automatic or semiautomatic segmentation tools and have countless applications in medical image analysis.

There are many works on shape prior segmentation in the literature [CAS' 97, CHA' 05, CHE' 02, CRE' 02, LEV' 00, LYN' 06, NG' 06, ROS' 07, TSA' 01]. The common point between all the proposed methods is to use a reference model as some types of priori knowledge. In medical imaging the reference model can be initialized from a numerical atlas [LEV' 00], can be defined interactively by an operator, or deduced from previous segmentation of a reference image. Recently, Rose et al. [ROS' 07] proposed a new automated region growing method integrating shape prior (RGISP) in order to include global shape information in the process of region growing. The prior knowledge was given by a reference model (binary image).

167

In the following work, we will use our proposed methodology to construct an enriched

average model (enriched average binary volume) from a set of training models to be the reference model of the RGISP segmentation method. First, we will briefly introduce RGISP method in the following section followed by the segmentation process framework used to segment the images in this chapter.

8.3 Region growing method integrating shape prior (RGISP) [ROS'

07]

Let us consider an image I of spatial domainΩ and ( )xI the grey intensity value of the voxel x ( Ω∈x a voxel of the image). Using a reference model (binary image), the reference region (white region of the reference model) is noted refR and the evolving region inR . The boundaries of inR and

refR are denoted by Γ , refΓ , respectively. The signed Euclidean distance between a voxel x of the image and the reference contour refΓ is noted as ( )refxd Γ, and defined as follows (equation (51)):

( )( )( )( )

0 ,

( , ) , 0 ,

, 0 ,

ref

ref

ref

ref ref ref

y

ref

y

x

d x Min d x y x R

Min d x y x R∈Γ

∈Γ

⎧⎪ ∈Γ⎪

⎡ ⎤Γ = − < ∈ −Γ⎨ ⎣ ⎦⎪⎪ ⎡ ⎤+ > ∈ Ω −⎣ ⎦⎩

(51)

where ( )yxd , is the Euclidean distance between two voxels x and y.

In region growing approaches [ZUC' 76], the merging of a pixel to the evolving region is governed by an aggregation criterion which must be satisfied. At each step, a set of candidate pixels not belonging but neighbouring to the evolving region are tested. Candidate pixels which satisfy the aggregation criterion are added to the evolving region, which results in a new region. A studied pixel is defined as a pixel belonging to the evolving region and located on its contour. Let

( ) [ ]0,1xϕ ∈ be the function used for assessing the aggregation criterion for a pixel x. The aggregation criterion is true when:

( ) δϕ ≥x (52)

where [ ]1,0∈δ is a given threshold. The aggregation criterion ( )xϕ computed from two terms

( )xregionϕ and ( )( )refshape xd Γ,ϕ . ( )xϕ is expressed in the following:

( ) ( ) ( ).region shapex x xϕ ϕ ϕ= (53)

where ( )region xϕ corresponds to the force of image data and ( )shape xϕ is driven by the average model. The first term ( ) [ ]0,1region xϕ ∈ defined in (54) corresponds to a measure of similarity (or

168

homogeneity) between ( )xI and inR . We assume that the underlying distribution of grey levels in inR is gaussian with mean μ and standard deviationσ .

( )2

2( ( ) )

2I x

region x eμ

σϕ− −

= − (54)

The second term ( )shape xϕ takes into account geometrical features and the proximity of x to

the reference model as shown in equation (55):

( )( )( )1tan ,

2ref

x

shape

d xx

π λϕ

π

−⎛ ⎞ − × Γ⎜ ⎟⎝ ⎠= (55)

where λ is a tuning parameter and ( )refxd Γ, is the signed Euclidean distance between a voxel x of the image and the reference contour refΓ . Parameter xλ is related to the magnitude of integrating shape prior in the segmentation as shown in equation (56).

min max min max. ( ).2 2x GVFP xλ λ λ λ

λ− +⎛ ⎞ ⎛ ⎞= ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ (56)

where minλ and maxλ are the bounds of value xλ . [ ]( ) 1, 1GVFP x ∈ − + is the scalar product between the Gradient Vector Flow (GVF) direction v and the growing direction D . If a high GVF vector v is defined at one pixel and the growing direction D is similar, ( )GVFP x is close to 1 so xλ value can be relaxed to limit the shape prior.

8.4 Segmentation Process framework

In this application, our proposed method will be used to construct an average enriched binary volume (average volume + signed Euclidean distance map) to be reference model for a segmentation process based on RGISP algorithm (Figure 8- 1).

169

RGISP

Training Set

Morphological-based averagemodel construction § 5.3

Model Enrichments § 6.3.2

New image

Enriched average volume Segmented Image

average binary volume

Average volume

Distance map

Figure 8- 1. Flow diagram of the segmentation process using RGISP

The prior knowledge will be presented using a signed Euclidean distance map computed from the average binary volume as presented in chapter 6 (c.f. § 6.3.2). This distance map will be used directly to calculate ( )shape xϕ (c.f. equation (55)). This framework is applied in the two following examples.

8.5 Examples

The first example, illustrated in Figure 8- 2, aims to present the advantage of using an average enriched model as a reference in a shape prior-based segmentation process. First, an average model (average binary image) I1324 is constructed from a training set composed of 4 synthetic binary images (I1, I2, I3, I4) using the morphological based average model construction method (c.f. § 5.3). Next, the average image is enriched by a signed Euclidean distance map computed from the average image (c.f. § 6.3.2).

In this example, the objective is to detect a target object TO in a 2D noisy synthesized

image I (Figure 8- 2). A classical region growing segmentation process (without shape prior) is used to segment the input image I, the result of the segmentation is illustrated by a red contour that delineates the segmented region (R1). Next, the RGISP algorithm integrating a shape prior ( 20, 0.5xλ = ∂ = ) based on the enriched average model (average image + signed distance map) is performed on the same image; the red contour in the image R2 (Figure 8- 2) detect the segmented region.

The visual comparison between the two detected regions in R1, R2, on one side and the TO on the other side, shows that the RGISP method using the enriched average model gives a very similar result than the initial target object.

170

Averaging Operator§ 5.3

RGISP [ROS' 07]

ClassicalRegion Growing

I1 I2 I3 I4

Signed distance mapAverage volume

I13 I24



I1324

Seed

-59

-25

8.5

42

76

Signed EuclideanDistance Map

Computation § 6.3..2

Edge Detector

I2 I4I24

Edge Detector

I13 I24I1324

Edge Detector

I1 I3I13

Enriched average model

TO I R 1 R 2

Figure 8- 2. Example of suing enriched average model as a reference model in a segmentation process based

on RGIPS method [ROS' 07]. (I1, I2, I3, I4) are the training set from which the average image I1324 will be

constructed, TO is the target object, I is the noised input image (Gaussian noise 20Nσ = ). R1, R2 are the

detected regions using the classical region growing method (without shape prior), RGISP method

( 20, 0.5λ = ∂ = ), respectively.

Figure 8- 3 illustrates the results using RGISP algorithm to segment one volume of intracranial Mastomys mouse cavity using the enriched average model presented in chapter 7 as a

171

reference model. In this example, it appears that the RGISP segmentation method using the enriched average model as a prior shape successfully extracts the region of interest from the image (red contour delineates the segmented region). Without shape information, the region growing will spread through the leak points (indicated by green flashes in Figure 8- 3) in the extra-cranial region.

(a) (b)

(c) (d)

Figure 8- 3. Segmentation results on micro-CT images of mouse skull.

8.6 Conclusion

This chapter show how the enriched average model constructed by the proposed methodology can be used as a reference model in a segmentation process integrating a priori knowledge. First, a general introduction about the model-based segmentation is introduced. Next, we briefly presented the RGISP method developed in our laboratory to integrate a priori knowledge in segmentation process followed. Finally, two examples are illustrates in the end of this chapter to show the ability of using our proposed methodology in this kind of applications. The results demonstrate the advantages of the enriched average model (average volume + distance map) used as reference model in a segmentation process where the classical region growing segmentation process (without shape prior) methods will fail to get the region of interest.

172

Conclusion of the third part

Two applications of the enriched average model were presented in this part of dissertation. In the first application (chapter 7), the goal was to study the ability of the constructed enriched average model to describe the normal models from which it was constructed and to differentiate a model that belongs to abnormal models. We used 72 intracranial models of Mastomys natalensis mice in total (36 models from male and 36 models from female mice): 44 of these models were obtained from normal mice (Lassa -), and the other 28 models were obtained from mice infected by Lassa virus (Lassa +). Depending on the age of the mice, the models were classified into three age classes (from younger to older). First, we compared some measured global quantitative features (volume, area… etc) between normal and abnormal models. According to the results of this initial comprehensive analysis of data, we found that the measured features of the abnormal intracranial models are smaller than those of the normal models. These differences are significant (p-value < 0.05) only for the intracranial models of the 3rd age male models, so we focused our study on the 3rd age class male mice. Next, an enriched average model was constructed from the 3rd age intracranial male mice models to be a reference model in a proposed phenotyping process. We used the proposed phenotyping process to classify 24 intracranial models (16 normal, 8 abnormal). Finally, some results of this phenotyping process are presented in the end of this chapter. Also, the optimal values of some used parameters (thresholds) are determined using usual medical statistical methods. In the second application (chapter 8), an average model (average binary image) constructed using our proposed methodology was used as a reference model in a model-based segmentation. Two examples were realised using RGIPS [ROS' 07] method. The results of this application show the advantage of using the average model as a reference in a model-based segmentation process.

Conclusion and future works

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One of the important features of a good anatomical atlas is that it includes information from a set of models rather than being based on a single chosen model. Average 3D digital atlases (average models) are important in various aspects of computational anatomy. Typical applications of these atlases are the providing of a template anatomy for segmentation and registration, and the establishment of a standard coordinate system for statistical analysis of structural and functional variability observed in the population. The aim of this thesis was to propose a general methodology for the creation of an enriched average model which represents a set of models, as well as its variability. In the first part of this dissertation, we studied some existing approaches to construct such type of models (average image or meshed surface). In the second part, we presented a new method to construct an enriched average model using a set of models (3D triangular meshes) based on morphological operations. In the third part, the proposed methodology to construct an average model was applied in two applications. Thanks to the first application, we demonstrated that the constructed enriched average model is able to describe the training models from which it was constructed and to differentiate a model that belongs to a different population. In the second application, we illustrated how we can use the enriched average model as a reference in a model-based segmentation. The main contribution of this thesis is the proposition of a general methodology to construct an enriched average model using image and surface processing operations. We can summarize the principal contributions as follows:

First, we have proposed a new method for coarse registration based on feature points detected on both source and target models. This method can be used for obtaining an initial pose in a registration process based on the ICP algorithm.

Second, we have proposed a new method to construct an average model (average surface + average binary volume) from a set of triangulated meshes. The average model is constructed through three main steps: (a) conversion of the input training set models (triangulated meshes) into training binary volumes (voxelization), (b) extraction of the average volume (average form) from the voxelized training set using morphological operations, (c), then conversion of the average volume into a triangulated mesh.

Third, we have proposed a phenotyping process based on the enriched average normal model to classify the models into normal and abnormal models.

There are also some points that could be enhanced in the average model construction methodology. The first point that can be enhanced is the construction of an average grey-scale volume corresponding to the constructed average model. An idea to enhance this weak point is to register a pre-segmented volume to the constructed average model (average binary volume or average surface). Next, the average grey-scale volume can be obtained by deforming the original


174

grey-scale volume using the computed registration parameters (transformation matrix). Another idea is to simulate a grey-scale volume corresponding to the average model using a medical image simulator such as the simulator of magnetic resonance images SIMRI [BEN' 05]. Another weak point of the proposed methodology is the assumption that the number of models in the input training set is 2n , which is not always possible. One idea to overcome this weak point is to repeat (eliminate) some models of the training set. The repeated (eliminated) models will be chosen randomly or using some importance criterions in accordance with the resemblance between the models of the training set. In future works, we will try to overcome these weak points in order to enhance our proposed methodology to construct an enriched average anatomical model. The work developed in this thesis can be used as a starting point for further researches related to the anatomical average model. Hereafter, we list two possible future applications using the anatomical average model.

The use of the average model in a registration-based segmentation process: we can create a grey-scale volume (simulated or extracted from the original training set volumes) corresponding to the average model (average binary volume + average surface). The average grey-scale volume can be registered to the new anatomical 3D image to be segmented. When the registration is correctly performed, segmentation of the corresponding anatomical structure will be done by deforming the average binary volume using the same registration transformation.

The construction of an anatomical pathology model: using the proposed methodology we can construct an average normal model for a specific anatomical structure. Then, the constructed model can be deformed using some anatomical landmarks to modulate a pathological case of the studied organ. This kind of applications is useful in surgery teaching (using virtual patient) or other related medical application.

In conclusion, always the eternal “why” and “how” questions will be asked again, whether in image processing, computer graphics, medical applications or any other scientific discipline. We hope that the presented work in this thesis will motivate more research on methods for constructing an enriched average anatomical model. The constructed average models can be used as reference models in high level medical applications as phenotyping, virtual surgery, modelling, medical image segmentation…..

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FOLIO ADMINISTRATIF

THESE SOUTENUE DEVANT L'INSTITUT NATIONAL DES SCIENCES APPLIQUEES DE LYON

NOM : KHADRA DATE de SOUTENANCE : 26 juin 2008 Prénoms : Yasser TITRE : Méthodologie de Réalisation de Modèles Anatomique Maillés : Application à l’Imagerie du Petit Animal NATURE : Doctorat Numéro d'ordre : 2008 ISAL 0034 Ecole doctorale : Electronique, Electrotechnique, Automatique

Spécialité : Images et Systèmes Cote B.I.U. - Lyon : T 50/210/19 / et bis CLASSE :

RESUME : En imagerie médicale, la robustesse de l’analyse et de la segmentation d’image est améliorée grâce à l’utilisation de connaissances à priori. Les atlas anatomiques constituent une base de connaissances à priori utiles pour localiser certains organes, ainsi que certaines structures très difficiles à distinguer sur des images complexes. Les atlas construits à partir d’un seul jeu de données ne permettent pas de prendre en compte les variations morphologiques et pathologiques inter-individus contrairement aux atlas moyens (construits à partir de plusieurs jeux de données). Dans cette thèse, nous nous intéressons à la construction d’un modèle moyen enrichi à partir des jeux de données surfaciques (modèles maillés). Ce modèle enrichi consiste en un modèle moyen (élément de base pour un atlas moyen) complété par des informations de variations géométriques de la structure anatomique étudiée (enrichissement). Pour atteindre cet objectif, nous construisons d’abord un modèle moyen déduit d’un ensemble d’apprentissage. Ensuite, nous procédons à l’enrichissement du modèle par des informations quantitatives, statistiques et géométriques extraites à partir de modèle moyen lui-même et de tous les modèles utilisés pour le construire. Les informations d’enrichissement du modèle moyen permettent ainsi la caractérisation de la variabilité d’une structure anatomique pathologique. La simplicité ou la complexité de l’enrichissement du modèle dépendront de l’application envisagée. Dans le cadre de cette thèse, nous proposons deux applications basées sur l’utilisation de ce modèle moyen enrichi : - Modèle de comparaison pour un processus de phénotypage anatomique du petit animal. - Modèle de référence pour la segmentation des images médicales in vivo intégrant des a priori sur la forme de la structure anatomique à segmenter. MOTS-CLES : atlas moyen, modèle maillé, recalage de surface, modèle moyen, modèle de référence,enrichissement de modèle, géométrie différentielle, lignes de crête, carte de distance, analyse statistique, phénotypage. Laboratoire (s) de recherche : CREATIS-LRMN, UMR CNRS 5520, U630 Inserm

Directeur de thèse: Christophe ODET Président de jury : Hugues-Benoit CATTIN Composition du jury : Fabrice MÉRIAUDEAU Jean-Charles PINOLI Marc JANIER Hugues Benoit-CATTIN Christophe ODET

Documents

Méthodologie de réalisation de modèles anatomiques maillés ...theses.insa-lyon.fr/publication/2008ISAL0034/these.pdfCNRS UMR 5558 Université Claude Bernard Lyon 1 Bât G. Mendel