Nested Iterative Algorithms for Convex Constrained Image Recovery Problems

Chaux, Caroline; Pesquet, Jean-Christophe; Pustelnik, Nelly

doi:10.1137/080727749

Cited by 70 publications

(75 citation statements)

References 43 publications

(87 reference statements)

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“…Some authors have studied the use of nested algorithms to solve (1), for practical imaging problems [35][36][37]. This approach consists in embedding an iterative algorithm as an inner loop inside each iteration of another iterative method.…”

Section: Relationship To Existing Optimization Methodsmentioning

confidence: 99%

“…This implies that ST (z) = ST (Sz) = Sz, so that Sz is a fixed point of T ′ and fix(T ′ ) = . Conversely, by (35), z ∈ fix(T ′ ) ⇒ P T (z) = P z ⇒ T (z) ∈ zer(A). All together with assumption (ii), the conditions are met to apply Lemma 4.1 to the iteration (34), after a change of variables like in the proof of Lemma 4.2, so that (z…”

Section: Proof Of Theorem 32 For Algorithm 31mentioning

confidence: 99%

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A Primal–Dual Splitting Method for Convex Optimization Involving Lipschitzian, Proximable and Linear Composite Terms

Condat

2012

J Optim Theory Appl

740

926

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To cite this version: , vol. 158, no. 2, pp. 460-479, 2013. AbstractWe propose a new first-order splitting algorithm for solving jointly the primal and dual formulations of large-scale convex minimization problems involving the sum of a smooth function with Lipschitzian gradient, a nonsmooth proximable function, and linear composite functions. This is a full splitting approach, in the sense that the gradient and the linear operators involved are applied explicitly without any inversion, while the nonsmooth functions are processed individually via their proximity operators. This work brings together and notably extends several classical splitting schemes, like the forward-backward and Douglas-Rachford methods, as well as the recent primal-dual method of Chambolle and Pock designed for problems with linear composite terms.

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Section: Relationship To Existing Optimization Methodsmentioning

confidence: 99%

Section: Proof Of Theorem 32 For Algorithm 31mentioning

confidence: 99%

A Primal–Dual Splitting Method for Convex Optimization Involving Lipschitzian, Proximable and Linear Composite Terms

Condat

2012

J Optim Theory Appl

740

926

View full text Add to dashboard Cite

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“…Seeking an estimate of α as a minimizer of F , it is possible to apply an algorithm as the one proposed in [46], where the proximity operator of the sum of two convex functions is derived from a Douglas Rachford Algorithm. We present such an algorithm with the "ISTA" framework in Algorithm 4, but it can be embedded in FISTA instead.…”

Section: Algorithms For Social Sparsitymentioning

confidence: 99%

Social Sparsity! Neighborhood Systems Enrich Structured Shrinkage Operators

Kowalski

Siedenburg

Dörfler

2013

IEEE Trans. Signal Process.

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Abstract-Sparse and structured signal expansions on dictionaries can be obtained through explicit modeling in the coefficient domain. The originality of the present article lies in the construction and the study of generalized shrinkage operators, whose goal is to identify structured significance maps and give rise to structured thresholding. These generalize Group Lasso and the previously introduced Elitist Lasso by introducing more flexibility in the coefficient domain modeling, and lead to the notion of social sparsity. The proposed operators are studied theoretically and embedded in iterative thresholding algorithms. Moreover, a link between these operators and a convex functional is established. Numerical studies on both simulated and real signals confirm the benefits of such an approach.Index Terms-Structured Sparsity, Iterative Thresholding, Convex Optimization I. INTRODUCTIONA wide range of inverse problems arising in signal processing have benefited from sparsity. Introduced in the mid 90's by Chen, Donoho and Saunders [1], the idea is that a signal can be efficiently represented as a linear combination of elementary atoms chosen from an appropriate dictionary. Here, efficiently may be understood in the sense that only few atoms are needed to reconstruct the signal. The same idea appeared in the machine learning community [2], where often only few variables are relevant in inference tasks based on observations living in very high dimensional spaces.The natural measure of the cardinality of a support set, and hence its sparsity, is the 0 "norm" which counts the number of non-zero coefficients. Minimizing such a penalty leads to a combinatorial problem which is usually relaxed into a 1 norm which is convex.Solving an inverse problem by using the sparse principle can be done by the following steps:• Choose a dictionary where the signal of interest is supposed to be sparse. Such a choice is driven by the nature

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“…Since p ′ is a quadratic polynomial with minimimum at − z 17 , we have only to show that p ′ − z 17 = 2 24 17 z 2 − 8ζ − 2x > 0. Plugging in (8) and (9) The sequence (t k ) k∈N generated by Newton's algorithm is such that there exists k 0 ∈ N \ {0} such that t k0 ≥ t − . Otherwise, (t k ) k∈N would be an increasing sequence which would necessarily converge to t − and there would exist k 1 ∈ N such that p is convex over [t k1 , +∞[.…”

Section: Notationmentioning

confidence: 99%

“…Note that this method appears mainly to be well-founded for denoising problems. When a linear degradation operator H is present, a better approach consists of adopting a variational framework [8,13] where one minimizes a data fidelity term…”

Section: Introductionmentioning

confidence: 99%

Epigraphical Projection for Solving Least Squares Anscombe Transformed Constrained Optimization Problems

Harizanov

Pesquet

Steidl

2013

Lecture Notes in Computer Science

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Abstract. This papers deals with the restoration of images corrupted by a non-invertible or ill-conditioned linear transform and Poisson noise. Poisson data typically occur in imaging processes where the images are obtained by counting particles, e.g., photons, that hit the image support. By using the Anscombe transform, the Poisson noise can be approximated by an additive Gaussian noise with zero mean and unit variance. Then, the least squares difference between the Anscombe transformed corrupted image and the original image can be estimated by the number of observations. We use this information by considering an Anscombe transformed constrained model to restore the image. The advantage with respect to corresponding penalized approaches lies in the existence of a simple model for parameter estimation. We solve the constrained minimization problem by applying a primal-dual algorithm together with a projection onto the epigraph of a convex function related to the Anscombe transform. We show that this epigraphical projection can be efficiently computed by Newton's methods with an appropriate initialization. Numerical examples demonstrate the good performance of our approach, in particular, its close behaviour with respect to the I-divergence constrained model.

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Nested Iterative Algorithms for Convex Constrained Image Recovery Problems

Cited by 70 publications

References 43 publications

A Primal–Dual Splitting Method for Convex Optimization Involving Lipschitzian, Proximable and Linear Composite Terms

A Primal–Dual Splitting Method for Convex Optimization Involving Lipschitzian, Proximable and Linear Composite Terms

Social Sparsity! Neighborhood Systems Enrich Structured Shrinkage Operators

Epigraphical Projection for Solving Least Squares Anscombe Transformed Constrained Optimization Problems

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