Proximal Algorithms for Multicomponent Image Recovery Problems

Briceño-Arias, Luis M.; Combettes, Patrick L.; Pesquet, Jean-Christophe; Pustelnik, Nelly

doi:10.1007/s10851-010-0243-1

Cited by 55 publications

(61 citation statements)

References 44 publications

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“…Examples of applications of these methods can be found in f-MRI reconstruction [1,2], satellite image restoration [3,4], microscopy image deconvolution [5,6], computed tomography [7], Positron Emission Tomography [8,9], texture-geometry decomposition [10,11,12], machine learning [13,14], stereo vision [15], and audio processing [16,17]. Proximal algorithms have gained much popularity in solving large-size optimization problems involving non-differentiable (or even non finite) functions.…”

Section: Introductionmentioning

confidence: 99%

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Epigraphical projection and proximal tools for solving constrained convex optimization problems

Chierchia

Pustelnik

Pesquet

et al. 2014

SIViP

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We propose a proximal approach to deal with convex optimization problems involving nonlinear constraints. A large family of such constraints, proven to be effective in the solution of inverse problems, can be expressed as the lower level set of a sum of convex functions evaluated over different, but possibly overlapping, blocks of the signal. For this class of constraints, the associated projection operator generally does not have a closed form. We circumvent this difficulty by splitting the lower level set into as many epigraphs as functions involved in the sum. A closed half-space constraint is also enforced, in order to limit the sum of the introduced epigraphical variables to the upper bound of the original lower level set.In this paper, we focus on a family of constraints involving linear transforms of ℓ1,p balls. Our main theoretical contribution is to provide closed form expressions of the epigraphical projections associated with the Euclidean norm (p = 2) and the sup norm (p = +∞). The proposed approach is validated in the context of image restoration with missing samples, by making use of TV-like constraints. Experiments show that our method leads to significant improvements in term of convergence speed over existing algorithms for solving similar constrained problems.

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Section: Introductionmentioning

confidence: 99%

“…By using now the fact that prox 12]) and by invoking Proposition 2.8, the expression of the optimal solution in (35) follows.…”

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confidence: 99%

Epigraphical projection and proximal tools for solving constrained convex optimization problems

Chierchia

Pustelnik

Pesquet

et al. 2014

SIViP

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“…Note that, if A is the identity matrix, one recovers the usual proximity operator prox f : R N → R N , which is at the core of numerous convex optimization algorithms (see [33,34,35] for tutorials and use for multicomponent image processing). 1 We are now ready to provide Algorithm 1 for the minimization of function F :…”

Section: Minimization Strategymentioning

confidence: 99%

“…16) where the components of s(z) ∈ R N are given by (35). Therefore, for every 17) where, for every z ∈ R N , matrix A log (z) is expressed by (34) where Ω = Ω H +Ω V and the elements of Ω H and Ω V are given by (36).…”

Section: Appendix A2 Log-tvmentioning

confidence: 99%

An alternating proximal approach for blind video deconvolution

Abboud¹,

Chouzenoux

Pesquet

et al. 2019

Signal Processing: Image Communication

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Blurring occurs frequently in video sequences captured by consumer devices, as a result of various factors such as lens aberrations, defocus, relative camerascene motion, and camera shake. When it comes to the contents of archive documents such as old films and television shows, the degradations are even more serious due to several physical phenomena happening during the sensing, transmission, recording, and storing processes. We propose in this paper a versatile formulation of blind video deconvolution problems that seeks to estimate both the sharp unknown video sequence and the underlying blur kernel from an observed video. This inverse problem is ill-posed, and an appropriate solution can be obtained by modeling it as a nonconvex minimization problem. We provide a novel iterative algorithm to solve it, grounded on the use of recent advances in convex and nonconvex optimization techniques, and having the ability of including numerous well-known regularization strategies.

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“…Several methods [19][20][21][22] have been introduced for decomposing a given image into a component with bounded variation, which holds the geometrical information, and an oscillating component, which corresponds to the textural information. A recent work [23] generalized the previous approaches providing a method for decomposing an image in more than two components, while being also able to handle data corrupted with a linear operator and a non-necessarily Gaussian noise.…”

Section: Introductionmentioning

confidence: 99%

Morphological Component Analysis for the Inpainting of Grazing Incidence X-Ray Diffraction Images Used for the Structural Characterization of Thin Films

Tzagkarakis

Pavlopoulou

Fadili

et al. 2013

Oil Gas Sci. Technol. – Rev. IFP Energies nouvelles

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and available online here Cet article fait partie du dossier thématique ci-dessous publié dans la revue OGST, Vol. 69, n°2, et téléchargeable ici D o s s i e rDOSSIER Edited by/Sous la direction de : L. Duval Advances in Signal Processing and Image Analysis for Physico-Chemical, Analytical Chemistry and Chemical SensingProgrès en traitement des signaux et analyse des images pour les analyses physico-chimiques et la détection chimique Résumé -Analyse en composantes morphologiques pour les retouches d'images de diffraction des rayons X en incidence rasante utilisés pour la caractérisation structurale des couches minces -La diffraction des rayons X en incidence rasante (GIXD) est une technique de caractéri-sation souvent utilisée dans l'étude de la structure des couches minces. En ce qui concerne les films organiques, le confinement du film sur le substrat conduit à des structures GIXD anisotropes à deux dimensions, telles celles observées pour les films à base de polythiophène utilisés comme couches actives dans les applications photovoltaiques. D'éventuels dysfonctionnements des détecteurs utilisés peuvent altérer la qualité des images acquises, affectant ainsi le processus d'analyse et l'information structurelle qui en est dérivée. Motivés par le succès des analyses en composantes morphologiques (MCA) en traitement d'images, nous nous attaquons dans cette étude au problème de la récupération de l'information manquante dans les images GIXD due à un dysfonctionnement potentiel des détecteurs. Nous montrons d'abord que les structures géométriques présentes dans les images GIXD peuvent être représentées de façon parcimonieuse en utilisant une combinaison de transformées redondantes, à savoir la transformée en curvelets et en ondelettes non-décimée. Ceci permet une description simple et compacte de l'information contenue dans ces images. Ensuite, l'information manquante est récupérée en appliquant la MCA dans un cadre d'« inpainting », en exploitant la représentation parcimonieuse des données GIXD dans ces deux domaines transformés. L'évaluation expérimentale montre que l'approche proposée est très efficace pour récupérer les informations manquantes lorsqu'elles sont aléatoirement distribuées sur les pixels de l'image ou lorsque des rangées entières sont manquantes, même lorsque la moitié du nombre total de pixels est affectée. Ce résultat indique que la MCA peut être appliquée pour remédier aux potentiels problèmes liés à la performance des détecteurs lors de l'acquisition, ce qui est d'une grande importance dans les expériences synchrotron, puisque le temps de faisceau alloué aux utilisateurs est extrêmement limité et que toute défaillance technique peut être préjudiciable pour le cours du projet expérimental. En outre, nos résultats permettent de réduire le temps d'acquisition ou d'éviter la répétition des mesures, qui donne plus de valeur à l'approche proposée. Abstract -Morphological Component Analysis for the Inpainting of Grazing Incidence X-Ray Diffraction Images Used for the Structural Characterization of Th...

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Proximal Algorithms for Multicomponent Image Recovery Problems

Cited by 55 publications

References 44 publications

Epigraphical projection and proximal tools for solving constrained convex optimization problems

Epigraphical projection and proximal tools for solving constrained convex optimization problems

An alternating proximal approach for blind video deconvolution

Morphological Component Analysis for the Inpainting of Grazing Incidence X-Ray Diffraction Images Used for the Structural Characterization of Thin Films

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