Single molecule localization by ℓ2−ℓ0\ell_2-\ell_0 constrained optimization

Single Molecule Localization Microscopy (SMLM) enables the acquisition of high-resolution images by alternating between activation of a sparse subset of fluorescent molecules present in a sample and localization. In this work, the localization problem is formulated as a constrained sparse approximation problem which is resolved by rewriting the $\ell_0$ pseudo-norm using an auxiliary term. In the preliminary experiments with the simulated ISBI datasets the algorithm yields as good results as the state-of-the-art in high-density molecule localization algorithms.Comment: In Proceedings of iTWIST'18, Paper-ID: 13, Marseille, France, November, 21-23, 201

Two constrained formulations for deblurring poisson noisy images

International audienceDeblurring noisy Poisson images has recently been subject of an increasingly amount of works in many areas such as astronomy or biological imaging. Several methods have promoted explicit prior on the solution to regularize the ill-posed inverse problem and to improve the quality of the image. In each of these methods, a regularizing parameter is introduced to control the weight of the prior. Unfortunately, this regularizing parameter has to be manually set such that it gives the best qualitative results. To tackle this issue, we present in this paper two constrained formulations for the Poisson deconvolution problem, derived from recent advances in regularizing parameter estimation for Poisson noise. We first show how to improve the accuracy of these estimators and how to link these estimators to constrained formulations. We then propose an algorithm to solve the resulting optimization problems and detail how to perform the projections on the constraints. Results on real and synthetic data are presented

Regularizing parameter estimation for Poisson noisy image restoration

International audienceDeblurring images corrupted by Poisson noise is a challenging process which has devoted much research in many applications such as astronomical or biological imaging. This problem, among others, is an ill-posed problem which can be regularized by adding knowledge on the solution. Several methods have therefore promoted explicit prior on the image, coming along with a regularizing parameter to moderate the weight of this prior. Unfortunately, in the domain of Poisson deconvolution, only a few number of methods have been proposed to select this regularizing parameter which is most of the time set manually such that it gives the best visual results. In this paper, we focus on the use of l1-norm prior and present two methods to select the regularizing pa- rameter. We show some comparisons on synthetic data using classical image fidelity measures

Régularité et parcimonie pour les problèmes inverses en imagerie : algorithmes et comparaisons

National audienceThis article is a survey on regularization techniques for inverse problems based on l1 criteria. We split these criteria in two categories : those which promote regularity of the signal (e.g. total variation) and those which express the fact that a signal is sparse in some dictionnary. In the first part of the paper, we give guidelines to choose a prior and propose a comparative study of these two priors on standard transforms such as total variation, redundant wavelets, and curvelets. In the second part of the paper, we give a sketch of different first order algorithms adpated to the minimization of these l1-terms.Dans cet article, nous nous intéressons à la régularisation de problèmes inverses reposant sur des critères l1 . Nous séparons ces critères en deux catégories : ceux qui favorisent la régularisation des signaux (à variation totale bornée par exemple) et ceux qui expriment le fait qu'un signal admet une représentation parcimonieuse dans un dictionnaire. Dans une première partie, nous donnons quelques éléments de comparaisons théoriques et pratiques sur les deux a priori, pour aider le lecteur à choisir l'un ou l'autre en fonction de son problème. Pour cette étude, nous utilisons les transformées communément utilisées telles que la variation totale, les ondelettes redondantes ou les curvelets. Dans une deuxième partie, nous proposons un état des lieux des algorithmes de premier ordre adaptés à la minimisation de ces critères

ERRATUM: A Continuous Exact l0 penalty (CEL0) for least squares regularized problem

International audienceLemma 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 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 supplya 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

Formulation contrainte pour la déconvolution de bruit de Poisson

National audienceWe focus here on the restoration of blurred and Poisson noisy images. Several methods solve this problem by minimizing a convex cost function composed of a data term and a regularizing term chosen from the prior that one have on the image. One of the recurrent problems of this approach is how to choose the regularizing paramater which controls the weight of the regularization term in front of the data term. One method consists in solving the minimization problem for several values of this parameter and by keeping the value which gives an image verifying a quality criterion (either qualitative or quantitative). This technique is obviously time consuming when one deal with high dimensional data such as in 3D microscopy imaging. We propose to formulate the blurred and Poisson noisy images restoration problem as a constrained problem on the antilog of the Poisson likelihood and propose an estimation of the bound from the works of Bertero et al. on the discrepancy principle for the estimation of the regularizing parameter for Poisson noise. We show results on synthetic and real data and we compare these results to the one obtained with the unconstrained formulation using the Gaussian approximation of the Poisson noise for the estimation of the regularizing parameter.Nous considérons le problème de la restauration d'image floue et bruitée par du bruit de Poisson. De nombreux travaux ont proposé de traiter ce problème comme la minimisation d'une énergie convexe composée d'un terme d'attache aux données et d'un terme de régularisation choisi selon l'a priori dont on dispose sur l'image à restaurer. Un des problèmes récurrents dans ce type d'approche est le choix du paramètre de régularisation qui contrôle le compromis entre l'attache aux données et la régularisation. Une approche est de choisir ce paramètre de régularisation en procédant à plusieurs minimisations pour plusieurs valeurs du paramètre et en ne gardant que celle qui donne une image restaurée vérifiant un certain critère (qu'il soit qualitatif ou quantitatif). Cette technique est évidemment très couteuse lorsque les données traitées sont de grande dimension, comme c'est le cas en microscopie 3D par exemple. Nous proposons ici de formuler le problème de restauration d'image floue et bruitée par du bruit de Poisson comme un problème contraint sur l'antilog de la vraisemblance poissonienne et proposons une estimation de la borne à partir des travaux de Bertero et al. sur le principe de discrepancy pour l'estimation du paramètre de régularisation en présence de bruit de Poisson. Nous montrons des résultats sur des images synthétiques et réelles et comparons avec l'écriture non-contrainte utilisant une approximation gaussienne du bruit de Poisson pour l'estimation du paramètre de régularisation

Complex wavelet regularization for 3D confocal microscopy deconvolution

Confocal microscopy is an increasingly popular technique for 3D imaging of biological specimens which gives images with a very good resolution (several tenths of micrometers), even though degraded by both blur and Poisson noise. Deconvolution methods have been proposed to reduce these degradations, some of them being regularized on a Total Variation prior, which gives good results in image restoration but does not allow to retrieve the thin details (including the textures) of the specimens. We first propose here to use instead a wavelet prior based on the Dual-Tree Complex Wavelet transform to retrieve the thin details of the object. As the regularizing prior efficiency also depends on the choice of its regularizing parameter, we secondly propose a method to select the regularizing parameter following a discrepancy principle for Poisson noise. Finally, in order to implement the proposed deconvolution method, we introduce an algorithm based on the Alternating Direction technique which allows to avoid inherent stability problems of the Richardson-Lucy multiplicative algorithm which is widely used in 3D image restoration. We show some results on real and synthetic data, and compare these results to the ones obtained with the Total Variation and the Curvelets priors. We also give preliminary results on a modification of the wavelet transform allowing to deal with the anisotropic sampling of 3D confocal images

An Elementary Proof of the Equivalence between 2D and 3D Classical Snakes and Geodesic Active Contours

Recently, Caselles et al. have shown in the equivalence between a classical snake problem of Kass et al. and a geodesic active contour model. The PDE derived from the geodesic problem gives an evolution equation for active contours which is very powerfull for image segmentation since changes of topology are allowed using the level set implementation. However in Caselles' paper the equivalence with classical snake is only shown for 2D images with 1D curves, by using concepts of Hamiltonian theory which have no meanings for active contours. This paper propose a proof using only elementary calculus of mathematical analysis. This proof is also valid in the 3D case for active surfaces

Sparse Poisson Noisy Image Deblurring

International audienceDeblurring noisy Poisson images has recently been subject of an increasingly amount of works in many areas such as astronomy or biological imaging. In this paper, we focus on confocal microscopy which is a very popular technique for 3D imaging of biological living specimens which gives images with a very good resolution (several hundreds of nanometers), even though degraded by both blur and Poisson noise. Deconvolution methods have been proposed to reduce these degradations and we focus in this paper on techniques which promote the introduction of explicit prior on the solution. One difficulty of these techniques is to set the value of the parameter which weights the trade-off between the data term and the regularizing term. Actually, only few works have been devoted to the research of an automatic selection of this regularizing parameter when considering Poisson noise so it is often set manually such that it gives the best visual results. We present here two recent methods to estimate this regularizing parameter and we first propose an improvement of these estimators which takes advantage of confocal images. Following these estimators, we secondly propose to express the problem of Poisson noisy images deconvolution as the minimization of a new constrained problem. The proposed constrained formulation is well suited to this application domain since it is directly expressed using the anti log-likelihood of the Poisson distribution and therefore does not require any approximation. We show how to solve the unconstrained and constrained problem using the recent Alternating Direction technique and we present results on synthetic and real data using well-known priors such as Total Variation and wavelet transforms. Among these wavelet transforms, we specially focus on the Dual-Tree Complex Wavelet transform and on the dictionary composed of Curvelets and undecimated wavelet transform