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Inline hologram reconstruction with sparsity constraintsLoïc Denis, Dirk Lorenz, Éric Thiébaut, Corinne Fournier, Dennis Trede

To cite this version:Loïc Denis, Dirk Lorenz, Éric Thiébaut, Corinne Fournier, Dennis Trede. Inline hologram reconstruc-tion with sparsity constraints. Optics Letters, Optical Society of America, 2009, 34 (22), pp.3475.�10.1364/OL.34.003475�. �ujm-00397994v2�

Inline hologram reconstruction with sparsity constraints

Loıc Denis,1,2,∗ Dirk Lorenz,3 Eric Thiebaut,4 Corinne Fournier,2 and Dennis Trede5

1Ecole Superieure de Chimie Physique Electronique de Lyon, F–69616 Lyon, France2Laboratoire Hubert Curien, UMR CNRS 5516, Universite de Lyon, F–42000 St-Etienne, France

3TU Braunschweig, D–38092 Braunschweig, Germany4Observatoire de Lyon, Ecole Normale Superieure de Lyon, Universite de Lyon, F–69000 Lyon, France

5Zentrum fur Technomathematik, University of Bremen, D–28334 Bremen, Germany∗Corresponding author: loic.denis@cpe.fr

Compiled October 6, 2009

Inline digital holograms are classically reconstructed using linear operators to model diffraction. It has long beenrecognized that such reconstruction operators do not invert the hologram formation operator. Classical linearreconstructions yield images with artifacts such as distortions near the field-of-view boundaries or twin-images.When objects located at different depths are reconstructed from a hologram, in-focus and out-of-focus imagesof all objects superimpose upon each other. Additional processing, such as maximum-of-focus detection, isthus unavoidable for any successful use of the reconstructed volume. In this letter, we consider inverting thehologram formation model in Bayesian framework. We suggest the use of a sparsity-promoting prior, verifiedin many inline holography applications, and present a simple iterative algorithm for 3D object reconstructionunder sparsity and positivity constraints. Preliminary results with both simulated and experimental hologramsare highly promising. © 2009 Optical Society of America

OCIS codes: 000.0000, 999.9999.

Holography is a two-steps, three dimensional (3D) imag-ing technique. In the first recording step, the wavediffracted by the object(s) is stored as a diffraction orinterference pattern on a planar recording medium (holo-graphic plate or digital camera). The second, reconstruc-tion step, reveals the 3D nature of the holographic image.

Numerical reconstruction techniques almost alwaysaim at simulating optical (diffraction-based) reconstruc-tions of holograms. Several methods have been pro-posed, corresponding to different diffraction models orapproximations (e.g., convolution [1], Fourier [1], frac-tional Fourier [2], continuous [3] or discrete [4] wavelettransforms). These approaches suffer from limited sizesand definitions of digital camera sensors, leading to im-ages with artifacts on the field-of-view boundaries, andfrom the twin-image (i.e., virtual image) problem of in-line holography.

When several objects located at different distancesare considered, the 3D reconstructed volume presentsdiffraction-defocused images of the objects in all but thein-focus plane of each object (Fig. 1a-b). This is dueto the fact that hologram reconstruction is consideredas wave rather than as 3D transmittance reconstructionproblem. We believe that better reconstructions of ob-jects can be achieved by considering the hologram re-construction as an inverse problem, as in [5]. In this let-ter, we use Bayesian framework with a sparsity-enforcingprior for hologram reconstructions.

Inline holography is known to be restricted to dilutemedia, i.e., transmission holograms are usable only if theimaged volume is almost empty (or homogeneous). Thesparsity of transmittance in the object domain is a nat-ural hypothesis in many inline holography applications.Note that sparsity in the transformed domain such as in

the wavelet domain has been found to be relevant in dig-ital holography also, which led to the original auto-focusmethod [6]. Under sparsity condition in the spatial do-main, the hologram formation can be considered as lin-ear [7] (i.e., inter-object interferences can be neglected).The hologram intensity d is then approximated by the(incoherent) summation of diffraction patterns createdby each object. By denoting with H the matrix thatmodels diffraction, with ϑ the unknown transmittancedistribution in the object space, with ε the error account-ing for both the physical and modelling noise and withc a constant, the finite dimensional model is given by:

d = c1−H · ϑ+ ε. (1)

If n is the number of pixels of the hologram, and p thetotal number of pixels of a given sampling of the ob-ject’s volume, then d, 1 and ε are n × 1 matrices (1 isa vector with n “ones”), ϑ is a p × 1 matrix and H an× p matrix. We will consider in the following differentsamplings of the object space, and the corresponding Hmatrices. When sampling in the object space is identi-cal to that of the hologram (i.e., a single plane with thesame pixel grid as the hologram), then H is the linearoperator modeling diffraction at finite distance z, e.g.,the 2D convolution along coordinates (x, y) with kernelhz(x, y) = sin[π(x2 + y2)/(λz)]/(λz) (λ being the wave-length).

The hologram reconstruction problem amounts to de-termining ϑ from data d (1). Due to the missing phase in-formation in the hologram intensity measurements, thisproblem is ill-posed. Digital holograms are classically re-constructed by simulating optical diffraction, which cor-responds to applying the adjoint operator H? to thedata: ϑadj = H?d. Note that the adjoint H? is not

1

the inverse of H (which is not invertible due to phaseloss), and that the reconstruction ϑadj suffers from thelimitations that were pointed out previously (boundaryartifacts, out-of-focus objects, twin-images). These canbe suppressed by considering a regularized solution toequation (1). The above-mentioned sparsity condition ofthe object volume, required for equation (1) to hold,naturally translates into a sparsity-enforcing prior inBayesian framework.

Recently, a considerable attention has been focusedon the theoretical and practical aspects of finding asparse solution to an inverse problem. Basically, thereare two numerical approaches for obtaining a sparse so-lution (i.e., a solution which is mostly zero except insome places): greedy techniques that build up the so-lution incrementally [8], and relaxation techniques thatsearch for the minimum of a `1 norm penalized maximumlikelihood problem [9]. A greedy algorithm has been re-cently proposed for particle holograms [10, 11]. We con-sider here least `1 norm for reconstructing holograms ofmore general objects. The maximum a posteriori esti-mate with a Laplacian sparsity-promoting prior and aGaussian likelihood model leads to the following mini-mization problem:

ϑsparse = arg minϑ,c

½‖c1−Hϑ− d‖22 + τ‖ϑ‖1, (2)

with τ a regularizing parameter, ‖·‖2 and ‖·‖1 the `2 and`1 norms (i.e., ∀v, ‖v‖22 =

∑i v

2i and ‖v‖1 =

∑i |vi|).

Equation (2) is a non-smooth, but convex minimizationproblem. Replacing c by its optimal value yields:

ϑsparse = arg minϑ

½‖Hϑ− d‖22 + τ‖ϑ‖1, (3)

where d = d− 1n11td and H = −H + 1

n11tH.Soft-thresholding iterative algorithms [12] have been

developed in recent years to solve such minimizationproblems. As transmittance is always a positive quan-tity, we consider solving minimization problem (3) un-der a positivity constraint, i.e., ∀i, ϑi ≥ 0. This leadsto the modification of the soft-thresholding iterations asfollows:

ϑ(k+1) = S+τ [ϑ(k) + H?(d− Hϑ(k))], (4)

with S+τ the positive soft-thresholding operator, applied

coordinate-wise:

S+τ (ϑ)i =

{ϑi − τ

2 if ϑi ≥ τ2 ,

0 if ϑi ≤ τ2 .

H and its adjoint H? are not stored in practice sinceonly the computation of the products Hv and H?

w forany p-dimensional vector v and n-dimensional vector wis necessary. These products correspond to 2D convolu-tions which can be efficiently performed by FFTs. Thesoft-thresholding operator can be straightforwardly gen-eralized by considering a spatially varying threshold τ ′.This can be useful to account for the difference between

Fig. 1. Reconstruction of a simulated hologram (b) of twoopaque disks located respectively at 100mm and 110mmfrom the hologram plane (a): (c-d) conventional recon-struction ϑadj; (e-f) sparse reconstruction ϑsparse (200 it-erations). Superimposed is the transmittance profile overthe line between I J signs.

`2 norms of each column of H(dictionaries H whose el-ements are unnormalized). To avoid over-penalizing theelements with small norm, we define τ ′ as τ ′i = τ

∑j H

2ji.

We illustrate the application of our method using bothnumerically simulated (Fig. 1) and experimental holo-grams (Fig. 2). In both cases, τ has been tuned by visualinspection of the results. Following a Bayesian reasoning,we expect τ to scale as the noise variance.

We first simulate a hologram (Fig. 1(b)) of two opaquespherical particles (Fig. 1(a)), which can be consid-ered to be two opaque disks under Fresnel approxima-tion. Conventional reconstructions ϑadj are shown onFig. 1(c-d). Out-of-focus and twin-images are noticeable.Sparse reconstruction gives almost perfect reconstruc-tion (Fig. 1(e-f)): each object appears in a unique planeand the reconstructed value outside objects is exactlyzero. The axial resolution achievable should be compara-ble to the 60µm resolution obtained experimentally when

2

Fig. 2. Reconstruction of an experimental hologram of aplane object: (a-b) the hologram and a truncation with3/4 of the pixels missing; (c-d) conventional reconstruc-tion ϑadj from the full (c) or truncated (d) hologram;(e-f) sparse reconstruction (50 iterations) ϑsparse fromthe full (e) or truncated (f) hologram.

solving problem (3) using a greedy technique [10].An experimental hologram of a planar target is dis-

played on Fig. 2(a), and in Fig. 2(b) after truncation of3/4 of the pixels (PCO Sensicam camera with 1280×1024pixels, pixel size: 6.7µm, double cavity YAG laser withλ = 532 nm, recording distance: 265mm). Figures 2(c-d) show the conventional reconstruction ϑadj from thefull and truncated hologram. The twin-images generatebackground noise that is clearly noticeable in the drawnline profiles. Reconstruction outside the field-of-view ishardly possible with conventional techniques due to avery low signal-to-noise ratio, as illustrated on Fig. 2(d).Sparse reconstruction gives the images Fig. 2(e-f). Twin-image noise is almost completely cleared and the target iscorrectly reconstructed with the marks and dusts of thetarget plate. The target is even partially reconstructedfrom the severely truncated data on Fig. 2(f). Some ele-ments such as the horizontal line of the graduations are

lost since the corresponding horizontal fringes fall out-side of the available data in Fig. 2(b). Vertical lines anddots are restored, even far from the field-of-view bound-aries. This is in agreement with the recent out-of-fielddetection results reported in [11], using a much strongerprior (in fact, parametric).

This letter is the first demonstration of the effec-tiveness of sparsity constraints in reconstructing objecttransmittance from holograms, using both simulated andexperimental data. The images obtained with our itera-tive algorithm are free from the artifacts of the conven-tional methods such as out-of-focus objects and twin-images. Moreover, thanks to the inverse problem ap-proach, field-of-view extrapolation is achieved.

Other sparse representations can be easily imple-mented by changing either the prior or the operatorH. For instance, minimizing the number of significantwavelet coefficients or the total variation is suitable forextended objects. Another possibility is to define H as adictionary of diffraction patterns for a given set of objectshapes. It would be worth comparing this latter approachwith the greedy detection scheme [10,11].

For applications with dense volumes of objects, a non-linear operator H(ϑ) could be considered to account forinterferences more accurately, as was done in [5]. Ouralgorithm can then be adapted, mostly by replacing theadjoint operatorH? by the Jacobian of non-linear modelH(ϑ) in equation (4).

The authors are grateful to P. Refregier for fruitfuldiscussions, and to C. Mennessier and D. Ghosh Roy fortheir careful reading of the manuscript.

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