International audience

Super-resolution fluorescence microscopy overcomes blurring arising from light diffraction, allowing the reconstruction of fine scale details in biological structures. Standard methods come at the expense of long acquisition time and/or harmful effects on the biological sample, which makes the problem quite challenging for the imaging of body cells. A promising new avenue is the exploitation of molecules fluctuations, allowing live-cell imaging with good spatio-temporal resolution through common microscopes and conventional fluorescent dyes. Several numerical algorithms have been developed in the literature and used for fluctuant time series. These techniques are developed within the discrete setting, namely the super-resolved image is defined on a finer grid than the observed images. On the contrary, gridless optimisation does not rely on a fine grid and is rather an optimisation of Dirac measures in number, amplitudes and positions. In this work, we present a gridless problem accounting for the independence of fluctuations.

Lemma 4.4 in [E. Soubies, L. Blanc-Féraud and G. Aubert, SIAM J. Imaging Sci., 8 (2015), pp. 1607--1639] is wrong for local minimizers of the continuous exact $\ell_0$ (CEL0) functional. The argument used to conclude the proof of this lemma is not sufficient in the case of local minimizers. In this note, we supply a revision of this lemma where new results are established for local minimizers. Theorem 4.8 in that paper remains unchanged but the proof has to be rewritten according to the new version of the lemma. Finally, some remarks of this paper are also rewritten using the corrected lemma.

Within the framework of the $\ell_0$ regularized least squares problem, we focus, in this paper, on nonconvex continuous penalties approximating the $\ell_0$-norm. Such penalties are known to better promote sparsity than the $\ell_1$ convex relaxation. Based on some results in one dimension and in the case of orthogonal matrices, we propose the continuous exact $\ell_0$ penalty (CEL0) leading to a tight continuous relaxation of the $\ell_2-\ell_0$ problem. The global minimizers of the CEL0 functional contain the global minimizers of $\ell_2 - \ell_0$, and from each global minimizer of CEL0 one can easily identify a global minimizer of $\ell_2 - \ell_0$. We also demonstrate that from each local minimizer of the CEL0 functional, a local minimizer of $\ell_2 - \ell_0$ is easy to obtain. Moreover, some strict local minimizers of the initial functional are eliminated with the proposed tight relaxation. Then solving the initial $\ell_2 - \ell_0$ problem is equivalent, in a sense, to solving it by replacing the $\ell_0$-norm with the CEL0 which provides better properties for the objective function in terms of minimization, such as the continuity and the convexity with respect to each direction of the standard $\mathbb{R}^N$ basis, although the problem remains nonconvex. Finally, recent nonsmooth nonconvex algorithms are used to address this relaxed problem within a macro algorithm ensuring the convergence to a critical point of the relaxed functional which is also a (local) optimum of the initial problem.

In this paper we describe a new variational approach for the detection of fine structures in an image (like filaments in 2D). This approach is based on the computation of the topological gradient associated to a cost function defined from a regularized version of the data (possibly noisy and / or blurred). We get this approximation by solving a fourth order PDE. The study of the topological sensitivity is made in the case of a crack. We give the numerical algorithm to compute this topological gradient and we illustrate our approach by giving several experimental results in 2D and 3D images.

Super-resolution microscopy techniques allow to overstep the diffraction limit of conventional optics. Theses techniques are very promising since they give access to the visualisation of finer structures which is of fundamental importance in biology. In this paper we deal with Multiple-Angle Total Internal Reflection Microscopy (MA-TIRFM) which allows to reconstruct 3D sub-cellular structures of a single layer of ~ 300 nm behind the glass coverslip with a high axial resolution. The 3D volume reconstruction from a set of 2D measurements is an ill-posed inverse problem and a regularization is essential. Our aim in this work is to propose a new reconstruction method for sparse structures robust to Poisson noise and background fluorescence. The sparse property of the solution can be seen as a regularization using the `£° norm'. In order to solve this combinatorial problem, we propose a new algorithm based on smoothed `£° norm' allowing minimizing a non convex energy, composed of the Kullback-Leibler divergence data term and the £° regularization term, in a Graduated Non Convexity framework.

We are interested in blind restoration of optical space variant blurred Poissonian images. For example, blur variation is due to refractive index mismatch in three-dimensional fluorescence microscopy or due to atmospheric turbulence in astrophysical images. In this work, the space variant point spread function (PSF) is approximated by a convex combination of a set of space invariant blurring functions. The latter is jointly estimated with the image by optimizing a given criterion including $l_1$ and $l_2$ norms for regularizing the image and the PSFs. We prove, in the continuous setting, the existence of a solution to this optimization problem. We then propose an alternating optimization algorithm based on a scaled gradient projection method. We show the efficiency of the proposed method on simulated and real optical images.

We are interested in blind image restoration in confocal laser scanning microscopy (CLSM). Two challenging problems in this imaging system are considered: First, spherical aberrations due to refractive index mismatch leads to a depth variant (DV) blur. Second, low illumination leads to a signal dependent Poisson noise. In addition, the DV point spread function (PSF) is unknown, which increases the complexity of the problem considered. Our goal is to remove in a blind framework both the DV blur and the Poisson noise from CLSM images. Using an approximation of the DV PSF, we define in a Bayesian framework a criterion to be jointly minimized w.r.t. the specimen function and the PSF. We then adopt an alternate minimization scheme for the optimization problem. For each elementary minimization, we use the recently proposed scaled gradient projection (SGP) algorithm that has shown a fast convergence rate. Results are shown on simulated and real CLSM images.

This paper contains errata corrige on the paper A formal $\Gamma$-convergence approach for the detection of points in $2$-D images [D. Graziani, L. Blanc-Féraud, and G. Aubert, SIAM J. Imaging Sci., 3 (2010), pp. 578--594].

We propose a new variational method to restore point-like and curve-like singularities in 2-D images. As points and open curves are fine structures, they are difficult to restore by means of first order derivative operators computed in the noisy image. In this paper we propose to use the Laplacian operator of the observed intensity, since it be comes singular at points and curves. Then we propose to restore these singularities by introducing suitable regularization involving the M-norm of the Laplacian operator. Results are shown on synthetic an real data.

We analyze the illumination invariance of the level lines of an image. We show that if the scene surface has Lambertian reflectance and the light is directed, then a necessary and sufficient condition for the level lines to be illumination invariant is that the three-dimensional scene be developable and that its albedo satisfy some geometrical constraints. We then show that the level lines are “almost” invariant for piecewise developable surfaces. Such surfaces fit most of the urban structures. This allows us to devise a fast and simple algorithm that detects changes between pairs of remotely sensed images of urban areas, independently of the lighting conditions. We show the effectiveness of the algorithm both on synthetic OpenGL scenes and real QuickBird images. The synthetic results illustrate the theory developed in this paper. The two real QuickBird images show that the proposed change detection algorithm is discriminant. For easy scenes it achieves a rate of 85% detected changes for 10% false positives, while it reaches a rate of 75% detected changes for 25% false positives on demanding scenes.

The goal of this paper is to develop an algorithm for extracting point features from sequences of aerial infrared images. We propose an efficient method for the detection of threats in a sequence of infrared images by looking for a trajectory which optimizes a regularized criterion. The regularity is introduced by a new concept of total curvature which eliminates too oscillating trajectories while allowing those with ponctual changes of direction. In practice the research of an optimal trajectory is performed with an algorithm of dynamical programming type. Experimental results are also presented.

We propose a new variational model to locate points in 2-dimensional biological images. To this purpose we introduce a suitable functional whose minimizers are given by the points we want to detect. In order to provide numerical experiments we replace this energy with a sequence of more treatable functionals by means of the notion of $\Gamma$-convergence.

We propose an algorithm that equalizes the contrast of grayscale image pairs to simplify the task of change detection. To ensure robustness of the detection under different illumination conditions, some authors recently proposed algorithms that compare the level lines of the images. We show - using ideas from the ldquoshape from shadingrdquo community - that under directed light, a necessary condition for the level lines to be illumination invariant is that the underlying surfaces be developable. The surfaces of cities can be modeled as piece-wise smooth developable surfaces, and it is therefore sensible to make use of the level lines for change detection. Our algorithm is robust and efficient both on synthetic OpenGL scenes and natural Quickbird images.

In this paper we propose a rigorous framework for texture image segmentation relying on region-based active contours (RBAC) and sparse texture representation. Such representations allow to efficiently describe a texture by transforming it in a dictionary of appropriate waveforms (atoms) where the texture representation coefficients are concentrated on a small set. For segmentation purposes. these atoms have to be multiscale and localized both in space and frequency, e.g. the wavelet transform. To discriminate different textures, we measure a ldquodistancerdquo between the non-parametric Parzen estimates of their respective sparse-representation coefficients probability density functions (pdfs). These distance measures are then used within RBAC, and we take benefit from shape derivative tools to derive the evolution speed expression of the RBAC. Our framework is applied to both supervised (with reference textures), and unsupervised texture segmentation. A series of experiments on synthetic textures illustrate the potential applicability of our method.

In this paper, we focus on statistical region-based active contour models where image features (e.g. intensity) are random variables whose distribution belongs to some parametric family (e.g. exponential) rather than confining ourselves to the special Gaussian case. Using shape derivation tools, our effort focuses on constructing a general expression for the derivative of the energy (with respect to a domain) and derive the corresponding evolution speed. A general result is stated within the framework of multi-parameter exponential family. More particularly, when using maximum likelihood estimators, the evolution speed has a closed-form expression that depends simply on the probability density function, while complicating additive terms appear when using other estimators, e.g. moments method. Experimental results on both synthesized and real images demonstrate the applicability of our approach

Active contours are adapted to image segmentation by energy minimization. The energies often exhibit local minima, requiring regularization. Such an a priori can be expressed as a shape prior and used in two main ways: (1) a shape prior energy is combined with the segmentation energy into a trade-off between prior compliance and accuracy or (2) the segmentation energy is minimized in the space defined by a parametric shape prior. Methods (1) require the tuning of a data-dependent balance parameter and methods (1) and (2) are often dedicated to a specific prior or contour representation, with the prior and segmentation aspects often meshed together, increasing complexity. A general framework for category (2) is proposed: it is independent of the prior and contour representations and it separates the prior and segmentation aspects. It relies on the relationship shown here between the shape gradient, the prior-induced admissible contour transformations, and the segmentation energy minimization.

In this paper, we propose to combine formally noise and shape priors in region-based active contours. On the one hand, we use the general framework of exponential family as a prior model for noise. On the other hand, translation and scale invariant Legendre moments are considered to incorporate the shape prior (e.g. fidelity to a reference shape). The combination of the two prior terms in the active contour functional yields the final evolution equation whose evolution speed is rigorously derived using shape derivative tools. Experimental results on both synthetic images and real life cardiac echography data clearly demonstrate the robustness to initialization and noise, flexibility and large potential applicability of our segmentation algorithm.

The paper deals with video and image segmentation using region based active contours. We consider the problem of segmentation through the minimization of a new criterion based on information theory. We first propose to derive a general criterion based on the probability density function using the notion of shape gradient. This general derivation is then applied to criteria based on information theory, such as the entropy or the conditional entropy for the segmentation of sequences of images. We present experimental results on grayscale images and color videos showing the accuracy of the proposed method.

This paper deals with image and video segmentation using active contours. The proposed variational approach is based on a criterion featuring a shape prior allowing free-form deformation. The shape prior is defined as a functional of the distance between the active contour and a contour of reference. We develop the complete differentiation of this criterion. First we propose two applications using only the shape prior term: the first application concerns shape warping and the second concerns video interpolation. Then the shape prior is combined with region-based features. This general framework is applied to interactive segmentation and face tracking on a real sequence.

We consider the problem of image segmentation using active contours through the minimization of an energy criterion involving both region and boundary functionals. These functionals are derived through a shape derivative approach instead of classical calculus of variation. The equations can be elegantly derived without converting the region integrals into boundary integrals. From the derivative, we deduce the evolution equation of an active contour that makes it evolve towards a minimum of the criterion. We focus more particularly on statistical features globally attached to the region and especially to the probability density functions of image features such as the color histogram of a region. A theoretical framework is set for the minimization of the distance between two histograms for matching or tracking purposes. An application of this framework to the segmentation of color histograms in video sequences is then proposed. We briefly describe our numerical scheme and show some experimental results.

A supervised classification model based on a variational approach is presented. This model is specifically devoted to textured images. We want to get a partition of an image, composed of texture regions separated by regular interfaces. Each kind of texture defines a class. We use a wavelet packet transform to analyze the textures, characterized by their energy distribution in each sub-band. In order to have an image segmentation according to the classes, we model the regions and their interfaces by level set functions. We define a functional on these level sets whose minimizers define the optimal classification according to textures. A system of coupled PDEs is deduced from the functional. By solving this system, each region evolves according to its wavelet coefficients and interacts with the neighbour region in order to obtain a partition with regular contours. Experiments are shown on synthetic and real images.

The problem of image segmentation through the minimization of an energy criterion involving both region and boundary functionals is considered. We study the derivation of these functionals using the notion of shape derivative. From the derivative, we deduce the evolution equation of an active contour that will make it evolve towards a minimum of the criterion introduced. We focus on geometric and statistical features globally attached to the boundary or to the region, and we take explicitly into account their evolution in the derivation. First, statistical region-based descriptors using the variance of a region or the distance to a reference region histogram are introduced. Then a geometric prior term is combined with statistical features for homogeneous region segmentation. This geometric prior is introduced to provide a free form deformation from a reference shape. Some experimental results on real images and video sequences show the benefit of combining geometrical and statistical features for segmentation.

We present an anisotropic diffusion equation designed to restore interferometric images. It has two main purposes, the first is to preserve the structures and discontinuities formed by the fringes. The second is to incorporate the noise modeling which is specific to this kind of images. Besides we show that our model formalizes previous related work in interferometry filtering.

We present a supervised classification model based on a variational approach. This model is specifically devoted to textured images. We want to get a partition of an image, composed of texture regions separated by regular interfaces. Each kind of texture defines a class. We use a wavelet packet transform to analyze the textures, characterized by their energy distribution in each sub-band. In order to have an image segmentation according to the classes, we model the regions and their interfaces by level set functions. We define a functional on these level sets whose minimizers define the optimal classification according to texture. A system of coupled PDEs is deduced from the functional. By solving this system, each region evolves according to its wavelet coefficients and interacts with the neighbor regions in order to obtain a partition with regular contours. Experiments are shown on synthetic and real images.

We present a new algorithm for the segmentation of moving objects in a video sequence acquired by a mobile camera using region-based active contours. Indeed, active contours are powerful tools for segmentation. However, as far as segmentation of moving objects is concerned, region-based terms must be incorporated in the evolution equation, in addition to classical boundary-based terms. We consequently propose a general framework for region-based active contours with a new Eulerian method to compute the evolution equation of the active contour. This general framework is then applied to the segmentation of moving objects from a sequence with a moving camera. We propose to jointly perform camera compensation and segmentation by taking directly into account the camera model in the criterion to minimize.

In order to improve efficiency of video coding, temporal redundancy between neighboring frames can be reduced. In MPEG-2, some frames, named interframes, are predicted using a motion estimation based on the conservation of intensity over time. In the new standard MPEG-4, frames are separated into several objects that are transmitted separately. Therefore, the prediction has to be performed on objects instead of frames. So, interframe-objects have to be predicted in order to improve video coding efficiency. The goal of this paper is to propose a new method for object-based prediction using the level sets. First, we propose an efficient object-based motion estimation. The motion of the objects is estimated by assuming the conservation of the level sets function over time, instead of the conservation of the intensity. Secondly, the estimation of the object motion between two frames is used to predict interframe-objects using both forward and backward motion estimations. The method is evaluated on real sequences, illustrating the potential of our approach for an object-based prediction of interframes.

This paper presents a new variational method for the segmentation of a moving object against a still background, over a sequence of [two-dimensional or three-dimensional (3-D)] image frames. The method is illustrated in application to myocardial gated single photon emission computed tomography (SPECT) data, and incorporates a level set framework to handle topological changes while providing closed boundaries. The key innovation is the introduction of a geometrical constraint into the derivation of the Euler-Lagrange equations, such that the segmentation of each individual frame can be interpreted as a closed boundary of an object (an isolevel of a set of hyper-surfaces) while integrating information over the entire sequence. This results in the definition of an evolution velocity normal to the object boundary. Applying this method to 3-D myocardial gated SPECT sequences, the left ventricle endocardial and epicardial limits can be computed in each frame. This space-time segmentation method was tested on simulated and clinical 3-D myocardial gated SPECT sequences and the corresponding ejection fractions were computed.

We propose a new multiresolution variational joint denoising/deblurring approach, involving a priori assumptions on the solution and knowledge of the imaging systems to account for effects due to acquisition noise (edge preservation, degradation noise modeling, bounded noise assumption and spectral control of noise level-whiteness and stationarity). The techniques used are drawn from a variety of areas of modern signal processing, including optimization theory, inverse problem, wavelet packets decomposition and bounded noise assumption.

We present a variational model devoted to image classification coupled with an edge-preserving regularization process. The discrete nature of classification (i.e., to attribute a label to each pixel) has led to the development of many probabilistic image classification models, but rarely to variational ones. In the last decade, the variational approach has proven its efficiency in the field of edge-preserving restoration. We add a classification capability which contributes to provide images composed of homogeneous regions with regularized boundaries, a region being defined as a set of pixels belonging to the same class. The soundness of our model is based on the works developed on the phase transition theory in mechanics. The proposed algorithm is fast, easy to implement, and efficient. We compare our results on both synthetic and satellite images with the ones obtained by a stochastic model using a Potts regularization.

Herein, we present a variational model devoted to image classification coupled with an edge-preserving regularization process. In the last decade, the variational approach has proven its efficiency in the field of edge-preserving restoration. In this paper, we add a classification capability which contributes to provide images compound of homogeneous regions with regularized boundaries. The soundness of this model is based on the works developed on the phase transition theory in mechanics. The proposed algorithm is fast, easy to implement and efficient. We compare our results on both synthetic and satellite images with the ones obtained by a stochastic model using a Potts regularization.

Iterative curve evolution techniques are powerful methods for image segmentation. Classical methods proposed curve evolutions which guarantee close contours at convergence and, combined with the level set method, they easily handled curve topology changes. However, these methods allow only one-way curve evolutions: shrinking or growing of the curve. Thus, the initial curve must encircle all the objects to be segmented or several curves must be used, each one totally inside one object. In this paper, we present a new approach of iterative curve evolution using the level set method based on the variational criterion of an inverse problem. Besides the closing of the final contours and the curve topology change management, our method allows a two-way curve evolution: parts of the curve evolve in the outward direction while others evolve in the inward direction. It offers much more freedom in the initial curve position than with a classical geodesic search method. Our algorithm performs accurate and precise segmentations, with length penalty. Results are shown on damaged images with complex objects (including sharp angles, deep concavities or holes).

Imaging systems involves blur and the coder reduces the binary rate for transmission or storage. These operations remove pertinent information contained by the image, and introduce annoying artifacts. Removing these artifacts allows higher visual quality for the reconstructed data. Unlike usual techniques, which made separately decoding and post-processing, we propose an optimal joint decoding/deblurring method for image reconstruction The goal of this work is to overcome the introduction of these negative effects by taking into account all the acquisition chain model.

In nuclear medical imaging, attenuation maps are images of the set of the linear attenuation coefficients of the observed body region. They are reconstructed from transmission SPECT (single photon emission computed tomography) acquisitions (the projections). The geometrical information of attenuation maps is crucial. We make the reasonable hypothesis that they are composed of homogeneous regions limited by straight edges. Thus, an accurate segmentation of attenuation maps is an important challenge. Instead of reconstructing an attenuation map and then segmenting it, we propose two methods to achieve its segmentation directly from the projections, without reconstructing it. Both are curve evolution algorithms using a level set technique. However, the first one is based on a geodesic search whereas the second one involves a more heuristic evolution speed definition. The methods are applied to 2D attenuation map segmentation from 1D transmission acquisitions in a fan beam geometry.

This paper examines the relationship between quantization noise removal and the variational problem. Traditional transformed and quantized image restoration techniques cannot prevent parasitic effects due to quantization noise. We propose a new method, involving a priori assumptions on the solution and knowledge of the coder (transformation and quantization) to account for effects due to quantization noise. This technique, called MORPHE, can be viewed as an inverse problem with optimization of the transform/quantization/decoding structure. This leads to the study of different ways to solve the constrained optimization problem. Experiments using this nonlinear inverse dynamic filtering demonstrate PSNR gains over standard linear inverse filtering as well as appreciable visual improvements.

This paper deals with edge-preserving regularization for inverse problems in image processing. We first present a synthesis of the main results we have obtained in edge-preserving regularization by using a variational approach. We recall the model involving regularizing functions phi and we analyze the geometry-driven diffusion process of this model in the three-dimensional (3-D) case. Then a half-quadratic theorem is used to give a very simple reconstruction algorithm. After a critical analysis of this model, we propose another functional to minimize for edge-preserving reconstruction purposes. It results in solving two coupled partial differential equations (PDEs): one processes the intensity, the other the edges. We study the relationship with similar PDE systems in particular with the functional proposed by Ambrosio-Tortorelli in order to approach the Mumford-Shah functional developed in the segmentation application. Experimental results on synthetic and real images are presented.

We present a variational approach such that during image restoration, edges detected in the original image are being preserved. We compare the mathematical foundation of this method with respect to some of the well known methods recently proposed in the literature within the class of PDE based algorithms (anisotropic diffusion, mean curvature motion, min/max flow technique). The performance of our approach is carefully examined and compared to the classical methods. Experimental results on synthetic and real images illustrate the capabilities of all the studied approaches.

We present a new approach based on partial differential equations (PDE) to restore noisy blurred images. After studying the methods to denoise images, staying as close as possible to the input image and methods to restore discontinuities, we propose a new scheme which combines all this schemes. A quantified numerical test on a synthetic image demonstrates the efficiency of our scheme and the role of varying the parameters for denoising, enhancement and coupling. A result on a real image is also presented.

This paper deals with segmentation and edge-preserving restoration. We use an active contour method to segment an image which may not be very noisy with regard to its edges. In order to both restore and segment a noisy image, we propose in this paper to derive a system of two partial differential equations, one for edge-preserving restoration (using edges given by segmentation), and the other for segmentation (using the restored image).

Many image processing problems are ill-posed and must be regularized. Usually, a roughness penalty is imposed on the solution. The difficulty is to avoid the smoothing of edges, which are very important attributes of the image. In this paper, we first give conditions for the design of such an edge-preserving regularization. Under these conditions, we show that it is possible to introduce an auxiliary variable whose role is twofold. First, it marks the discontinuities and ensures their preservation from smoothing. Second, it makes the criterion half-quadratic. The optimization is then easier. We propose a deterministic strategy, based on alternate minimizations on the image and the auxiliary variable. This leads to the definition of an original reconstruction algorithm, called ARTUR. Some theoretical properties of ARTUR are discussed. Experimental results illustrate the behavior of the algorithm. These results are shown in the field of 2D single photon emission tomography, but this method can be applied in a large number of applications in image processing.

This paper deals with edge-preserving regularization for image reconstruction. We use a non-quadratic regularization term involving a /spl phi/-function applied on the intensity gradient modulus. During the process, small gradients are smoothed while high gradients are preserved. In order to take into account the noise more specifically, we propose to use the explicit version of the regularization term involving the edge variable. We add a nonlinear constraint on this edge variable, in order to remove the noise. It allows edge enhancement while smoothing the noise, even in the case where the edge and noise generate the same high gradient modulus. We have previously proposed a model composed of two coupled partial differential equations (PDE) on the image intensity and image edges. In this paper, we show that, for a particular regularization intensity function, the two coupled PDE can be slightly modified in order to correspond to the Euler equations associated with the minimization of a global criterion. This new criterion contains a nonlinear regularization term on both the intensity and the edges. We use convergence towards Mumford and Shah (1989) functional to improve our results.

This paper deals with edge-preserving regularization. The definition of the regularizing functions and properties such as edge modeling or stability are studied in the variational approach (by minimizing of a criterion) and in the anisotropic diffusion approach (by solving a PDE). We propose sufficient conditions to define an edge-preserving regularizing function, and analyze comparatively several usual functions. We use the algorithm ARTUR based on the half-quadratic transform to solve the nonlinear equation. The image and the edge map are simultaneously estimated.

Many image processing problems are ill-posed and must be regularized. Usually, a roughness penalty is imposed on the solution. The difficulty is to avoid the smoothing of edges, which are very important attributes of the image. The authors first give sufficient conditions for the design of such an edge-preserving regularization. Under these conditions, it is possible to introduce an auxiliary variable whose role is twofold. Firstly, it marks the discontinuities and ensures their preservation from smoothing. Secondly, it makes the criterion half-quadratic. The optimization is then easier. The authors propose a deterministic strategy, based on alternate minimizations on the image and the auxiliary variable. This yields two algorithms, ARTUR and LEGEND. The authors apply these algorithms to the problem of SPECT reconstruction.>

Many image processing problems are ill-posed and must be regularized. Usually, a roughness penalty is imposed on the solution. The difficulty is to avoid the smoothing of edges, which are very important attributes of the image. We first propose sufficient conditions for the design of such an edge-preserving regularization. Using the Legendre transform, it is then possible to introduce an auxiliary variable which role is twofold. Firstly, it marks the discontinuities and ensures their preservation from smoothing. Secondly, it makes the criterion half-quadratic. The optimization is then easier. We propose a deterministic algorithm, based on alternate minimizations over the image and the auxiliary variable. We apply this algorithm to the problem of SPECT reconstruction.