Dualization of Signal Recovery Problems

Combettes, Patrick L.; Dũng, Đinh; Vũ, Bằng Công

doi:10.1007/s11228-010-0147-7

Cited by 83 publications

(72 citation statements)

References 65 publications

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“…In the following we provide an analysis of the notion of inexactness given in Definition 2.1, which will clarify the nature of these approximations and the scope of applicability. To this purpose, we will make use of the duality technique, an approach that is quite common in signal recovery and image processing applications [18,12,16]. The starting point is the Moreau decomposition formula [41,18] prox λg (y) = y − λprox g * /λ (y/λ), (2.5) where g * : H → R is the conjugate functional of g defined as g * (y) = sup x∈H ( x, y − g(x)).…”

Section: Main Contributionsmentioning

confidence: 99%

“…convex function. The structure (2.7) often arises in regularization for ill-posed inverse problems [13,10,28,53,67,16]. By definition, finding prox λg (y) requires the solution of the minimization problem…”

Section: Main Contributionsmentioning

confidence: 99%

“…A simple choice is the forwardbackward splitting algorithm (called also ISTA [7]). In this case, as done in [16], we get the following algorithm (for an arbitrary initialization v 0 ∈ G)…”

Section: 2mentioning

confidence: 99%

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Accelerated and Inexact Forward-Backward Algorithms

Villa¹,

Salzo²,

Baldassarre³

et al. 2013

SIAM J. Optim.

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Abstract. We propose a convergence analysis of accelerated forward-backward splitting methods for composite function minimization, when the proximity operator is not available in closed form, and can only be computed up to a certain precision. We prove that the 1/k 2 convergence rate for the function values can be achieved if the admissible errors are of a certain type and satisfy a sufficiently fast decay condition. Our analysis is based on the machinery of estimate sequences first introduced by Nesterov for the study of accelerated gradient descent algorithms. Furthermore, we give a global complexity analysis, taking into account the cost of computing admissible approximations of the proximal point. An experimental analysis is also presented.

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Section: Main Contributionsmentioning

confidence: 99%

Section: Main Contributionsmentioning

confidence: 99%

See 1 more Smart Citation

Accelerated and Inexact Forward-Backward Algorithms

Villa¹,

Salzo²,

Baldassarre³

et al. 2013

SIAM J. Optim.

180

224

View full text Add to dashboard Cite

show abstract

“…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%

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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“…Note that the dual forward-backward algorithm proposed in [54] is not directly applicable to Problem (2.8) since the projection onto C is not explicit in general. Dykstra's algorithm [55] to compute the projection onto an intersection of convex sets would not be applicable either, as the matrix DΓD is usually singular.…”

Section: Convex Analysis Toolsmentioning

confidence: 99%

Dual Constrained TV-based Regularization on Graphs

Grady¹,

Najman²,

Pesquet³

et al. 2013

SIAM J. Imaging Sci.

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Abstract. Algorithms based on Total Variation (TV) minimization are prevalent in image processing. They play a key role in a variety of applications such as image denoising, compressive sensing and inverse problems in general. In this work, we extend the TV dual framework that includes Chambolle's and Gilboa-Osher's projection algorithms for TV minimization. We use a flexible graph data representation that allows us to generalize the constraint on the projection variable. We show how this new formulation of the TV problem may be solved by means of fast parallel proximal algorithms. On denoising and deblurring examples, the proposed approach is shown not only to perform better than recent TV-based approaches, but also to perform well on arbitrary graphs instead of regular grids. The proposed method consequently applies to a variety of other inverse problems including image fusion and mesh filtering.

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Dualization of Signal Recovery Problems

Cited by 83 publications

References 65 publications

Accelerated and Inexact Forward-Backward Algorithms

Accelerated and Inexact Forward-Backward Algorithms

An alternating proximal approach for blind video deconvolution

Dual Constrained TV-based Regularization on Graphs

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