3d morphological modelling of a random fibrous network

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  • Image Anal Stereol 2009;28:129-141Original Research Paper

    3D MORPHOLOGICAL MODELLING OF A RANDOM FIBROUSNETWORK

    CHARLES PEYREGA1, DOMINIQUE JEULIN1, CHRISTINE DELISEE2 AND JEROMEMALVESTIO21Centre de Morphologie Mathematique, Mathematiques et Systemes, Mines ParisTech, 35 rue Saint Honore,77300 Fontainebleau, France, 2Unite des Sciences du Bois et des Biopolymeres, 351, cours de la Liberation -33405 Talence CEDEX, Francee-mail: charles.peyrega@mines-paristech.fr, dominique.jeulin@ensmp.fr, delisee@us2b.pierroton.inra.fr,malvestio@us2b.pierroton.inra.fr(Accepted September 14, 2009)

    ABSTRACT

    In the framework of the Silent Wall ANR project, the CMM and the US2B are associated in order tocharacterize and to model fibrous media studying 3D images acquired with an X-Ray tomograph used bythe US2B. The device can make 3D images of maximal 23043 voxels with resolutions in the range of2 m to 15 m. Using mathematical morphology, measurements on the 3D X-Ray CT images are usedto characterize materials. For example measuring the covariance on these images of an acoustic insulatingmaterial made of wooden fibres highlights the isotropy of the fibres orientations in the longitudinal planeswhich are perpendicular to the compression Oz axis. Moreover, it is possible to extract other morphologicalproperties from these image processing methods such as the size distribution either of the fibres or of thepores by estimating the morphological opening granulometry of the considered medium. Using the theoryof random sets introduced by Georges Matheron in the early 1970s, the aim of this work is to model sucha fibrous material by parametric random media in 3D according to the prior knowledge of its morphologicalproperties (covariance, porosity, size distributions, etc.). A Boolean model of random cylinders in 3D stackedin planes parallel to each other and perpendicular to the Oz compression axis is first considered. Thegranulometry results provide gamma distributions for the radii of the fibres. In addition, a uniform distributionof the orientations is chosen, according to the experimental isotropy measurements in the longitudinal planes.Finally the third statistical factor is the length distribution of the fibres which can be fitted by an exponentialdistribution. Thus it is possible to estimate the validity of this model first by trying to fit the experimentaltransverse and longitudinal covariances of the pores with the theoretical ones taking into account the statisticaldistributions of the dimensions of the random cylinders. The second method to validate the model consistsin comparing morphological measurements (density profiles, covariance, opening granulometry, tortuosity,specific surface area) processed on real and on simulated media.

    Keywords: 3D images, Boolean model, fibrous media, mathematical morphology, random media.

    INTRODUCTION

    Fibrous materials are commonly used for thermaland acoustical insulating in buildings. The Silent WallANR1 projects objective is to build an acousticalinsulating wall made of fibrous media, with innovatingacoustical properties, developed in a context ofenvironmental efficiency and competitiveness by usingnatural raw materials such as wood and other cellulosefibres. In this purpose, morphological measurementswere performed (Peyrega et al., 2009a) to characterizethe Thermisorel, which is a wooden fibre board100% naturally papermaking processed from recycledwood, used as a reference material in the SilentWall project. This paper focuses on modelling themicrostructure of such a fibrous material.

    When 3D images were not available yet, 2D

    images from confocal microscope acquisitions wereused to fit models of random fibrous media. Themethod proposed by Jeulin (2000), Castera et al.(2000), Michaud et al. (2000) and Delisee et al.(2001) consists in generating a Boolean model ofPoisson lines, dilated by spheres with a randomradius (Matheron, 1967; Serra, 1982). This method iseffective to simulate 3D stacks of very long fibres afteridentification from 2D projected images, when theirlength is large compared to the images. Modelling afibrous medium with 3D Poisson lines was also madelater by Schladitz et al. (2006) to study its acousticalproperties. However, the dimensions of the fibres ofThermisorel in the present paper are finite comparedto those of the sample (about 600 600 360 voxelswith resolutions ranging from 2 m to 15 m). That iswhy a different model is proposed here.

    1http://us2b.pierroton.inra.fr/Projets/Silent Wall/description.htm

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  • PEYREGA C ET AL: 3D morphological modelling of a random fibrous network

    As the fibres of Thermisorel have finitedimensions, they could be modelled by a Booleanmodel of random cylinders whose radii, lengthsand orientations are estimated by the morphologicalmeasurements performed by Peyrega et al. (2009a).Moreover, these cylinders are stacked in the xOyplanes perpendicular to the axis of compression Oz,according to the industrial papermaking process ofthis material. The covariance measurements of thefibres (Matheron, 1967; Serra, 1982) highlight theglobal anisotropy of the material, and the isotropyof their orientations in the xOy planes. Thus, itseems to be realistic to simulate uniformly distributedorientations between 0 and in these planes. Theopening granulometry of the fibres (Matheron, 1967;Serra, 1982) allows us to extract the distribution oftheir radii. Automatic morphological fibre analysisusing 2D images with the MorFi system2 brings thedistribution of their lengths out.

    After having presented the 3D X-Ray CTimages of Thermisorel used for the morphologicalcharacterization, the Boolean model of randomcylinders will be introduced. Afterwards two methodsto extract the distribution of the radii will be presented(the first one from the granulometry, and the secondone from a minimization of the mean square errorusing the transverse covariance of the external pores).At last an estimation of the distribution of the lengthswill be introduced.

    3D IMAGES OF THERMISOREL

    The 3D images are obtained with the US2B X-Ray tomograph, with resolutions equal to 2 m, 5 m,9.36 m and 15 m per voxel, depending on thewanted observation scale. The grey level images aresegmented by manual thresholding, after smoothingthe noise by a low-pass filter (Fig. 1). Severalstructures are observable in the Thermisorel (Fig.2a). The fibrous phase is composed of isolated fibresand of clusters, the sticks. The external porosity andthe internal one, i.e., the lumens into the fibres wherethe sap flows in the tree, compose the porous phase.

    With a 9.36 m resolution, the lumens are notobservable, but for 5 m they should be filled in orderto isolate the external porosity. An algorithm to fillthe lumens is proposed by Lux (2005). It consistsin the succession of morphological operations onthe segmented image. The final result is representedin Fig. 2b on an image acquired in Grenoble onthe beamline ID 19 of the European Synchrotron

    Radiation Facility (ESRF) with a resolution of5 m/voxel.

    (a) (b)

    Fig. 1. Binarized X-Ray CT images of Thermisorel;(a) Source: ESRF; resolution: 5 m/voxel;dimensions: 5.9 mm 5.9 mm 1.68 mm, (b) Source:US2B; resolution: 9.36 m/voxel; dimensions:5.6 mm 5.6 mm 3.4 mm.

    (a) (b)

    Fig. 2. Lumen filling of Thermisorel (resolution:5 m/voxel; source: ESRF ID19). (a) Segmented 2Dslice of Thermisorel in the xOy plane, (b) Filledlumens.

    As the production process of Thermisorel isa papermaking one, the fibrous mat is compressedalong the Oz axis. The fibres are thus isotropicallyoriented in the xOy planes perpendicular to Oz. Thisis observable on the measurements of the covarianceC(h) of the fibres (Peyrega et al., 2009a), showingidentical covariances in the xOy planes, whatever theorientation of the vector~h, which is a consequence ofthe transverse isotropy. However, the fibrous mediumis anisotropic in the other directions of 3D space, sincethe correlation length in the Oz direction is shorter thanin the xOy plane.

    2http://cerig.efpg.inpg.fr/dossier/EFPG-innovations/page10.htm

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  • Image Anal Stereol 2009;28:129-141

    BOOLEAN MODEL OF RANDOMCYLINDERS

    THE BOOLEAN MODEL

    In order to model a Thermisorel-like fibrousmedium, with fibres having finite lengths, the Booleanmodel with random cylinders as primary grains seemsto be suitable. The Boolean model was introducedby Matheron (1967) and used by Jeulin (2000) in amodelling of random textures for materials. The firststep consists in generating a Poisson point process.

    Random primary grains A are implanted onPoisson points xk with the intensity i.e., the averagenumber of generated points per unit volume. Theoverlap of these grains is possible. Let A be the randomset generated by the grains (Eq. 1),

    A =xk

    Axk . (1)

    REMINDER ON THEORETICALPROPERTIES OF THE BOOLEANMODEL

    A is the set of fibres with p = P(x A). Theprobability Q(K) for a compact set K to be includedin the set Ac (with q = P(x Ac) = 1 p) is given byMatheron (1967) and Serra (1982) as a function of theerosion and dilation operations in Eq. 2 and Eq.3. Let n be the average Lebesgue measure (averagevolume in 3D), and K = {x,x K} be the transposedset of K.

    Q(K) = P(K Ac) = P(x Ac K) , (2)

    Q(K) = e n(AK) = q

    n(AK)n(A) . (3)

    The covariance (Q(h)), linear erosions (Q(l)), anderosions by balls (Q(B(r))) are particular cases for Kof the expression given in Eq. 2 and Eq. 3.

    Covariance Q(h)

    A is the set of fibres. Let Q(h) (Eq. 4) be thecovariance of the porous medium, i.e., of the set Ac.The Eq. 5 gives the theoretical expressions of Q(h)for the Boolean model. Let K(h) be the geometricalcovariogram of the grain A, K(h) = n(A Ah).The normalized covariogram r(h) is defined by r(h) =K(h)/K(0), with K(0) = n(A),

    Q(h) = P(x Ac,x+h Ac) , (4)QBooleanModel(h) = q2 e(K(h)) = q2r(h) . (5)

    From Eq. 5 it is then possible to deduce theporosity q (Eq. 6), which leads to the intensity of thePoisson point process from which the Boolean modelhas been generated (Eq. 7).

    q = en(A) , (6)

    = ln(q)n(A)

    . (7)

    Linear erosions Q(l)Considering the porous phase, the theoretical

    expression Q(l) is given by Eq. 8 for a random convexgrain A in which we consider r(0) = (dr/dh)h=0.

    QBooleanModel(l) = e n(Al) = q1l r

    (0) . (8)

    Erosions by balls Q(B(r))The probability for a ball with radius r, B(r) to be

    included in the pores is given by Eq. 9:

    QBooleanModel(r) = e n(AB(r)) . (9)

    In the case of convex grains, like random cylinders,the n(A B(r)) can be expanded according tothe Steiners formula involving the Minkowskisfunctionals of A and of B(r). It is therefore apolynomial of degree 3 in r. According to the Steinersformula, it is possible to write Eq. 10 which impliesEq. 11. Let us consider V (K) the volume of a compactset K, A(K) its surface area, and M(K) its integralmean curvature, with R = E[R] and L = E[L].

    n(AB(r)) = V (A)+M(B(r))S(A)

    4

    +M(A)S(B(r))

    4+V (B(r)) ,

    (10)

    n(AB(r)) =43

    r3 +(L+R

    )r2

    +[2R(R+L)

    ]r +E

    [R2]L .

    (11)

    These expressions, as well as the exponentialbehavior of QBooleanModel(l) are used as to test theassumption of a Boolean model with convex grains.

    TEST OF THE VALIDITY OF A BOOLEANMODEL FOR THE FIBROUS NETWORKIn order to validate the assumption of a

    Boolean model of random cylinders to simulate aThermisorel-like material, several morphologicalcharacteristics of the fibrous network have beencompared to the theoretical results. As a first step,these measurements are the linear erosions Q(l)

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  • PEYREGA C ET AL: 3D morphological modelling of a random fibrous network

    (Fig. 3 with a logarithmic scale) and the erosions byrhombicuboctaedra Q(B(r)) (Fig. 4 with a logarithmicscale, and Fig. 7).

    1e-006

    1e-005

    0.0001

    0.001

    0.01

    0.1

    1

    0 100 200 300 400 500 600 700 800 900

    Fra

    cti

    on

    of

    the

    vo

    lum

    e o

    f P

    OR

    ES

    (L

    OG

    Sc

    ale

    )

    Sizes l of LINEAR EROSIONS (MICRONS)

    Q(l) of REAL Thermisorel after LINEAR EROSIONS

    Q(l)_XQ(l)_YQ(l)_Z

    Fig. 3. Linear erosions of Thermisorel in the Ox, Oyand Oz directions, shown in logarithmic scale (source:sample Fig. 1b).

    In the Ox and Oy longitudinal directions log(Q(l))cannot be fitted exactly by a straight line (Fig. 3).This result is due to the fact that fibres are not strictlyconvex. They should be cylinders with some slightbending, which modifies the behavior of log(Q(l)) inthe directions of the xOy planes. In the Oz direction,log(Q(l)) can be fitted by a straight line. However, wewill neglect these points for those fibres, and use theperfect cylinders assumption.

    1e-007

    1e-006

    1e-005

    0.0001

    0.001

    0.01

    0.1

    1

    0 50 100 150 200 250

    Fra

    cti

    on

    of

    the

    vo

    lum

    e o

    f P

    OR

    ES

    (L

    OG

    Sc

    ale

    )

    Sizes r of EROSIONS by RhombiC 24-C (MICRONS)

    Q(r) of REAL Thermisorel after EROSIONS by RC

    Q(r)_Thermisorel

    Fig. 4. Erosions by rhombicuboctaedra B(r) ofThermisorel, shown in logarithmic scale (source:sample Fig. 1b).

    In Fig. 4, the experimental curve log(Q(B(r)))(B(r) being a ball of radius r) should be fitted by atheoretical polynomial of degree 3 in r (Eq. 11).

    BOOLEAN MODEL OF RANDOMCYLINDERS WITH A TRANSVERSEISOTROPY

    We consider now as primary grains a population ofcylinders with a random radius R and a random lengthL. A given grain is completely known from these twocharacteristics and from its orientation. With respectto the already mentioned transverse isotropy of thenetwork, we will consider fibres orthogonal to the Ozaxis, and with a uniform distribution of orientationsin the xOy planes. In this context, the most generalrandom fibre model would require the knowledgeof the bivariate distribution of R and L, f (r, l). Forsimplification, and in absence of simultaneous data onthe same fibres, we will assume here that these tworandom variables are independent.

    Knowing the distributions for R and for L, it ispossible to compute the theoretical covariance in thetransverse direction (along Oz), or in any horizontaldirection in the xOy planes (longitudinal covariance).The input of the observed size distributions enables usto check the validity of the model, by comparisonof the measured and calculated covariances.Alternatively, we can also fit the parameters of thedistribution from the experimental covariances, by aleast squares minimization.

    Consider first the transverse covariance of thepores, QZ(h). From Eq. 5, it is expressed as afunction of the transverse reduced covariogram of thecylindrical fibre, rZCylinder(h).

    (a) (b)

    Fig. 5. Transverse geometrical covariogram of thecylinder. (a) Cylindrical primary grain A, (b)Geometrical covariogram of the disc (red area).

    Let f1(r) be the distribution of the random radiiof the fibres. The transverse geometrical covariogramof a fibre is given as a function of the average lengthLfibres and of the average geometrical covariogram of apopulation of discs with a random radius R followingthe distribution f1(r) by KZCylinder(h) = Lfibres KZDisc(h).Therefore we have:

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  • Image Anal Stereol 2009;28:129-141

    rZCylinder(h) =KZDisc(h)KZDisc(0)

    . (12)

    The normalized transverse geometricalcovariogram of the fibres does not depend on the...

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