We consider linear inverse problems where the solution is assumed to have a sparse expansion on an arbitrary preassigned orthonormal basis. We prove that replacing the usual quadratic regularizing penalties by weighted p-penalties on the coefficients of such expansions, with 1 ≤ p ≤ 2, still regularizes the problem. Use of such p-penalized problems with p < 2 is often advocated when one expects the underlying ideal noiseless solution to have a sparse expansion with respect to the basis under consideration. To compute the corresponding regularized solutions, we analyze an iterative algorithm that amounts to a Landweber iteration with thresholding (or nonlinear shrinkage) applied at each iteration step. We prove that this algorithm converges in norm. © 2004 Wiley Periodicals, Inc.

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An Iterative Thresholding Algorithm for Linear Inverse Problems with a Sparsity Constraint

We consider linear inverse problems where the solution is assumed to have a sparse expansion on an arbitrary pre-assigned orthonormal basis. We prove that replacing the usual quadratic regularizing penalties by weighted l^p-penalties on the coefficients of such expansions, with 1 < or = p < or =2, still regularizes the problem. If p < 2, regularized solutions of such l^p-penalized problems will have sparser expansions, with respect to the basis under consideration. To compute the corresponding regularized solutions we propose an iterative algorithm that amounts to a Landweber iteration with thresholding (or nonlinear shrinkage) applied at each iteration step. We prove that this algorithm converges in norm. We also review some potential applications of this method.

An iterative thresholding algorithm for linear inverse problems with a sparsity constraint

We show that various inverse problems in signal recovery can be formulated as the generic problem of minimizing the sum of two convex functions with certain regularity properties. This formulation makes it possible to derive existence, uniqueness, characterization, and stability results in a unified and standardized fashion for a large class of apparently disparate problems. Recent results on monotone operator splitting methods are applied to establish the convergence of a forward-backward algorithm to solve the generic problem. In turn, we recover, extend, and provide a simplified analysis for a variety of existing iterative methods. Applications to geometry/texture image decomposition schemes are also discussed. A novelty of our framework is to use extensively the notion of a proximity operator, which was introduced by Moreau in the 1960s.

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Signal recovery by proximal forward-backward splitting ∗

Monthly Notices is one of the three largest general primary astronomical research publications. It is an international journal, published by the Royal Astronomical Society. This article 1 describes its publication policy and practice.

Monthly Notices of the Royal Astronomical Society

Iterative shrinkage/thresholding (1ST) algorithms have been recently proposed to handle a class of convex unconstrained optimization problems arising in image restoration and other linear inverse problems. This class of problems results from combining a linear observation model with a nonquadratic regularizer (e.g., total variation or wavelet-based regularization). It happens that the convergence rate of these 1ST algorithms depends heavily on the linear observation operator, becoming very slow when this operator is ill-conditioned or ill-posed. In this paper, we introduce two-step 1ST (TwIST) algorithms, exhibiting much faster convergence rate than 1ST for ill-conditioned problems. For a vast class of nonquadratic convex regularizers (lscrP norms, some Besov norms, and total variation), we show that TwIST converges to a minimizer of the objective function, for a given range of values of its parameters. For noninvertible observation operators, we introduce a monotonic version of TwIST (MTwIST); although the convergence proof does not apply to this scenario, we give experimental evidence that MTwIST exhibits similar speed gains over IST. The effectiveness of the new methods are experimentally confirmed on problems of image deconvolution and of restoration with missing samples.

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A New TwIST: Two-Step Iterative Shrinkage/Thresholding Algorithms for Image Restoration

https://perso-etis.ensea.fr/nguyen-verger/publi_web/nguyenSigPro03.pdf

Wavelets and curvelets for image deconvolution: a combined approach

Radon transforms defined on smooth curves are well known and extensively studied in the literature. In this paper, we consider a Radon transform defined on a discontinuous curve formed by a pair of half-lines forming the vertical letter V. If the classical two-dimensional Radon transform has served as a work horse for tomographic transmission and/or emission imaging, we show that this V-line Radon transform is the backbone of scattered radiation imaging in two dimensions. We establish its analytic inverse formula as well as a corresponding filtered back projection reconstruction procedure. These theoretical results allow the reconstruction of two-dimensional images from Compton scattered radiation collected on a one-dimensional collimated camera. We illustrate the working principles of this imaging modality by presenting numerical simulation results.

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On the V-line radon transform and its imaging applications

Morphological Component Analysis (MCA) is a new method which takes advantage of the sparse representation of structured data in large overcomplete dictionaries to separate features in the data based on the diversity of their morphology. It is an efficient technique in such problems as separating an image into texture and piecewise smooth parts or for inpainting applications. The MCA algorithm consists of an iterative alternating projection and thresholding scheme, using a successively decreasing threshold towards zero with each iteration. In this article, the MCA algorithm is extended to the analysis of spherical data maps as may occur in a number of areas such as geophysics, astrophysics or medical imaging. Practically, this extension is made possible thanks to the variety of recently developed transforms on the sphere including several multiscale transforms such as the undecimated isotropic wavelet transform on the sphere, the ridgelet and curvelet transforms on the sphere. An MCA-inpainting method is then directly extended to the case of spherical maps allowing us to treat problems where parts of the data are missing or corrupted. We demonstrate the usefulness of these new tools of spherical data analysis by focusing on a selection of challenging applications in physics and astrophysics.

https://hal.archives-ouvertes.fr/hal-00196082/document

Morphological Component Analysis and Inpainting on the Sphere: Application in Physics and Astrophysics

CMB data analysis and sparsity

A new circular-arc Radon transform arising from the mathematical modeling of image formation in a new modality of Compton scattering tomography is introduced. We describe some of its properties and establish its analytic inverse formula. This result demonstrates the feasibility of image reconstruction from Compton scattered radiation in Compton scattering tomography. We also show that it belongs to a larger class of Radon transforms on algebraic curves, which remain invariant under a specific geometric inversion.

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Mai K. Nguyen

Papers

Wavelets and curvelets for image deconvolution: a combined approach

On the V-line radon transform and its imaging applications

Morphological Component Analysis and Inpainting on the Sphere: Application in Physics and Astrophysics

CMB data analysis and sparsity

Inversion of a new circular-arc Radon transform for Compton scattering tomography