December 15, 1991 / Vol. 16, No. 24 / OPTICS LETTERS 1979

Twin-image elimination in in-line holography of finite-supportcomplex objects

G. Koren and D. Joyeux

Institut d'Optique Th6orique et Appliqu6e, Universit6 de Paris XI, Bat. 503, B.P 147, 91403 Orsay Cedex, France

F. Polack

Laboratoire pour l'Utilisation du Rayonnement Electromagnetique, Centre National de la Recherche Scientifique,Universit6 de Paris XI, Bat. 209 d, 91405 Orsay Cedex, France

Received July 31, 1991

Reconstruction of in-line holograms suffers from the superposition of two twin images having different foci butidentical information content. A simple iterative method of twin-image elimination is presented here. It isbased on the fact that, if the object has a finite support and the recording distance is not too small, the out-of-focus field is known on a large part of the reconstruction plane and is only superposed by the in-focus one insidea restricted support. An iterative procedure is developed to recover the out-of-focus "wave inside the in-focusimage support. Inverse diffraction then gives the reconstructed image. This procedure can be easily auto-mated and works for finite-support real or complexof -1.

In-line holography provides a simple means of high-resolution imaging. The main advantage of themethod is its simplicity: the same beam illumi-nates the object and serves as a reference beam, andno optics is required, which is especially useful withshort wavelengths such as soft x rays.` 3 A draw-back is the superposition of the virtual and realimages in the reconstructed field. This superposi-tion gives a complex interference pattern in whichdetails and structures may be blurred or lost, espe-cially for objects recorded in the near field.

To simplify the analysis, and without loss of gen-erality, it is assumed here that the object is locatedat z = 0 and is characterized by its absorption a(X),which may be complex. It is illuminated by a uni-form plane wave of unit amplitude, and the holo-gram is recorded at a distance z, where, within thescalar approximation, the diffracted field Uh isgiven by the convolution with the propagation func-tion hz(X),

Uh(X) = [1 - a(X)] * * hz(X). (1)

The hologram records the intensity Iz = Uhl12,and, if we assume a linear recording, the transmit-tance of the hologram will be proportional to therecorded intensity,

th(X) = a + MII(X). (2)

To reconstruct the hologram, the record is illumi-nated with a plane wave of the same wavelength,and the reconstructed field at a distance z behindthe hologram is given by

Ur(X) = th(X) * * h,(X). (3)Neglecting insignificant constant factors, we canwrite

Ur(X) = D - a*(X) - a(X) * * h2A(X)

+ la(X) * * hz(X)I2 * * hz(X). (4)

objects, recorded in geometries with a Fresnel number

The different terms in Eq. (4) are the dc component,the in-focus image, the out-of-focus image, and theintermodulation term, respectively. Let us notethat the linear development of I.(X) is valid providedthat la(X)l << 1 (or if the hologram is recorded inthe far field). This means that the objects to berecorded by in-line holography must interact weaklywith the illuminating beam and also that the inter-modulation term is generally negligible.

From Eq. (4) it is clear that the complete informa-tion recorded by the hologram is present both in thein-focus and the out-of-focus images; rejecting oneof them will permit us to retrieve the object. Mostof the work done in digital twin-image eliminationis based on filtering out the out-of-focus imageby using phase retrieval 4 or linear filtering algo-rithms.' 6 These algorithms have been imple-mented only for pure real or pure imaginary objects.For finite support objects, nonlinear processingmethods have been also described. One is a partialsuppression of the in-focus image followed by a simu-lated propagation, as in an optical twin-imagesuppression setup; it can give reasonable results infar-field conditions.7 The second consists of an it-erative reconstruction with clipping of all signalsoutside the object support; the convergence has beenproved theoretically.8

The method presented here can be considered as anumerical and iterative implementation of an opticaltwin-image suppression setup. The in-focus imageis suppressed where it is localized and replaced byan iteratively upgraded estimate of the out-of-focusfield. Propagation of this restored field gives thereconstructed image.

Let us assume that the object has a limited sup-port X, and the hologram is recorded at such a dis-tance that the out-of-focus image size is a few timeslarger than X (which is also the in-focus image size).

0146-9592/91/241979-03$5.00/0 C 1991 Optical Society of America

1980 OPTICS LETTERS / Vol. 16, No. 24 / December 15, 1991

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(3) A first estimate of the out-of-focus wave is nowintroduced,U. ' ( X) - U rXe A'where Ur is an average value of the reconstructedfield over E.

(4) Then this estimate Uo') is diffracted backwardto get a first estimate of the in-focus image in whichthe dc component is included.

U(1)(X) = U"Y)(X) * * h * 2AX)

a)

la

0-

-15.0 -10.0 -5.0 0.0

X [Pm]5.0 10.0 15.0 20.0

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Fig. 1. Original one-dimensional complex object. (a) Ob-ject amplitude, (b) object phase.

Outside I the out-of-focus field D - a * * h2 , is notdifferent from the naturally reconstructed field Ur,whereas inside X it is contaminated by the in-focusimage and remains unknown. However, the knowl-edge of Ur is generally sufficient to evaluate I and toget a first estimate of the out-of-focus wave inside I.An approximation of the in-focus image can then becomputed by backpropagation, and constraints inthis image are applied; the result obtained is for-ward propagated and used to refine the out-of-focuswave estimation in an iterative procedure.

The reconstruction algorithm is therefore imple-mented as follows:

(1) The starting point is the complex amplitude ofthe reconstructed field Ur(X). This field can be ob-tained directly by using phase contrast or equivalenttechniques or from phase-retrieval computation byusing the measured intensity in the real image planeand adding a constant phase condition in the holo-gram plane (thin amplitude hologram). In somecases, it can also be computed from the digitizationof the hologram itself.4' 5

(2) We then need to estimate the support E ofthe in-focus image. The contour line of the objectcan be revealed by its high-frequency content, forinstance, from intensity slopes in the reconstruc-tion plane,

G(X) = 1gradlUr(X)121. (5)Thresholding this function gives the contour line ofE. High values of G inside this support will not betaken into account, as they can reflect high diffrac-tive details in the object. In the presence of lownoise levels this method performs correctly, pro-vided that the signal is smoothed by an averagingwindow. For high noise levels or low-resolutionholograms more refined methods may be required.

(7)

(5) From our assumptions there should not existany signal outside S. However, this is too strong aconstraint, especially in the presence of noise, andwe must allow some fluctuations, but they shouldnot be strong enough to produce contrast reversal.Therefore, outside X, the in-focus amplitude mod-ulus must remain under the dc level D, so the follow-ing constraint is applied:

U2)(X) = {O()(X)X 0 I and !U'()(X)I > Delsewhere

(8)

(6) The resulting field distribution is diffractedforward to form a new estimate of the out-of-focusfield, according to the relation

Ul2)(X) = OR2)(x) * * h2z(X).U(I~~~(9(7) This estimate is better than Uo') inside X only,

so we update the estimate of the out-of-focus field asfollows:

Uo)(X) -= {Ur(X)

ID

E

a)

Cs

(10)X e XX E E,

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10.0 15.0 20.

from a hologram.(b) reconstructed

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December 15, 1991 / Vol. 16, No. 24 / OPTICS LETTERS 1981

S'0

C)

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1 CC

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Fig. 3. Results of twin-image elimination algorithmafter 40 iterations. (a) Reconstructed object amplitude,(b) reconstructed object phase.

With this last estimate the above operations can berepeated from step (4) to create an iterative closedprocedure.

The convergence of this algorithm has not beenproved in general. But, because it is similar to theerror reduction algorithms,9 it is expected that therms error between the object and its reconstructiondecreases or remains constant.

As an example, the reconstruction procedure hasbeen applied to the computer-simulated ideal in-linehologram of the one-dimensional complex objectshown in Fig. 1 (ideal means that film noise, filmresolution, and system aperture have not been takeninto account). This model object has features simi-lar to those of biological objects in soft x rays: alow modulation, soft edges, and correlated amplitudeand phase. The recording (and reconstruction) ge-ometry was also fixed in reference to soft-x-rayholography, i.e., A = 2.5 nm and z = 5 mm. Thesevalues give a Fresnel number Nf = V2/Az = 3.4 foran object support of 6.5 Atm.

The diffracted fields have been computed withthe angular spectrum representation.s The recon-structed hologram (Fig. 2) shows the image recon-structed in the real image plane and the degradationas a result of the in-focus and out-of-focus imagesuperposition. The reconstruction procedure de-scribed above was then applied, with automaticcalculation of the in-focus image boundaries as indi-cated in step (2), Eq. (5). Figure 3 shows the re-constructed object obtained after 40 iterations;the similarity with the original object is highlysatisfactory.

Application of the algorithm has shown that theconvergence speed depends markedly on the Fresnelnumber (i.e., the ratio of in-focus to out-of-focusimage widths), and, for instance, the same rms erroris obtained after three iterations only when Nf = 1.1.The convergence speed is expected to improve fortwo-dimensional objects of equal Nf. The method isalso sensitive to errors in estimating the supportsize, the dc level, and the recording distance. Thenecessary investigations of these have not yet beendone. Finally, it should be noted that the part ofthe information that is located in the in-focus imagesupport is not used. Modification of the algorithmto use it might improve the accuracy. The method,as it is, however, presents some original features:it operates on the field in the real image planeand can reconstruct complex objects recorded in theFresnel region.

References

1. M. R. Howells, M. A. Iarocci, and J. Kirz, J. Opt. Soc.Am. A 3, 2171 (1986).

2. C. Jacobsen, M. R. Howells, J. Kirz, and S. Rothman,J. Opt. Soc. Am. A 7, 1847 (1990).

3. D. Joyeux, S. Lowenthal, R Polack, and A. Bernstein,X-Ray Microscopy II (Springer-Verlag, Berlin, 1988),pp. 246-252.

4. G. Liu and P. D. Scott, J. Opt. Soc. Am. A 4, 159 (1987).5. L. Onural and P. D. Scott, Opt. Eng. 26, 1124 (1987).6. K. A. Nugent, Opt. Commun. 78, 293 (1990).7. A. Lannes, Opt. Commun. 20, 356 (1977).8. K. H. S. Marie, J. C. Bennett, and A. P. Anderson,

Electron. Lett. 15, 241 (1979).9. J. R. Fienup, Appl. Opt. 21, 2758 (1982).

10. J. W Goodman, Introduction to Fourier Optics(McGraw-Hill, New York, 1968), pp. 48-55.

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