In this work, we consider a class of differentiable criteria for sparse image computing problems, where a nonconvex regularization is applied to an arbitrary linear transform of the target image. As special cases, it ...
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In this work, we consider a class of differentiable criteria for sparse image computing problems, where a nonconvex regularization is applied to an arbitrary linear transform of the target image. As special cases, it includes edge-preserving measures or frame-analysis potentials commonly used in image processing. As shown by our asymptotic results, the l(2) - l(0) penalties we consider may be employed to provide approximate solutions to l(0) penalized optimization problems. One of the advantages of the proposed approach is that it allows us to derive an efficient majorize-minimize subspace algorithm. The convergence of the algorithm is investigated by using recent results in nonconvex optimization. The fast convergence properties of the proposed optimization method are illustrated through image processing examples. In particular, its effectiveness is demonstrated on several data recovery problems.
We consider the minimization of a function G defined on , which is the sum of a (not necessarily convex) differentiable function and a (not necessarily differentiable) convex function. Moreover, we assume that G satis...
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We consider the minimization of a function G defined on , which is the sum of a (not necessarily convex) differentiable function and a (not necessarily differentiable) convex function. Moreover, we assume that G satisfies the Kurdyka-Aojasiewicz property. Such a problem can be solved with the Forward-Backward algorithm. However, the latter algorithm may suffer from slow convergence. We propose an acceleration strategy based on the use of variable metrics and of the majorize-minimize principle. We give conditions under which the sequence generated by the resulting Variable Metric Forward-Backward algorithm converges to a critical point of G. Numerical results illustrate the performance of the proposed algorithm in an image reconstruction application.
Nonnegative matrix factorization (NMF) is a powerful blind source separation method that can be used for nonparametric partial volume mixture modeling in a variety of high-dimensional medical imaging experiments. Howe...
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ISBN:
(纸本)9781479923748
Nonnegative matrix factorization (NMF) is a powerful blind source separation method that can be used for nonparametric partial volume mixture modeling in a variety of high-dimensional medical imaging experiments. However, conventional NMF methods can fail to produce meaningful results when the measurements contain substantial non-Gaussian noise. This paper proposes a new NMF modeling approach that is appropriate for noisy MRI magnitude images that follow the noncentral chi (NCC) statistical distribution. We formulate a maximum likelihood optimization problem, which we solve by combining conventional least-squares NMF algorithms with a recent majorize-minimize framework for the NCC distribution. This new approach is applied to real diffusion MRI data, and is demonstrated to yield improved results relative to conventional NMF.
The statistics of noisy MR magnitude and square-root sum-of-squares MR images are well-described by the Rice and non-central chi distributions, respectively. Statistical estimation involving these distributions is com...
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ISBN:
(纸本)9781467364553
The statistics of noisy MR magnitude and square-root sum-of-squares MR images are well-described by the Rice and non-central chi distributions, respectively. Statistical estimation involving these distributions is complicated by the facts that they have first-and second-order moments that depend nonlinearly on the noiseless image, and can have nonconvex negative log-likelihoods. This paper proposes a new majorize-minimize framework to ease the computational burden associated with statistical estimation involving these distributions. We derive quadratic tangent majorants for the negative log-likelihoods, which enables statistical cost functions to be optimized using a sequence of much simpler least-squares or regularized least-squares surrogate problems. We demonstrate the use of this framework in the context of regularized MR image denoising, with both simulated and experimental data.
The statistics of many MR magnitude images are described by the non-central chi (NCC) family of probability distributions, which includes the Rician distribution as a special case. These distributions have complicated...
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The statistics of many MR magnitude images are described by the non-central chi (NCC) family of probability distributions, which includes the Rician distribution as a special case. These distributions have complicated negative log-likelihoods that are nontrivial to optimize. This paper describes a novel majorize-minimize framework for NCC data that allows penalized maximum likelihood estimates to be obtained by solving a series of much simpler regularized least-squares surrogate problems. The proposed framework is general and can be useful in a range of applications. We illustrate the potential advantages of the framework with real and simulated data in two contexts: 1) MR image denoising and 2) diffusion profile estimation in high angular resolution diffusion MRI. The proposed approach is shown to yield improved results compared to methods that model the noise statistics inaccurately and faster computation relative to commonly-used nonlinear optimization techniques.
The Adaptive Ridge Algorithm is an iterative algorithm designed for variable selection. It is also known under the denomination of Iteratively Reweighted Least-Squares Algorithm in the communities of Compressed Sensin...
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The Adaptive Ridge Algorithm is an iterative algorithm designed for variable selection. It is also known under the denomination of Iteratively Reweighted Least-Squares Algorithm in the communities of Compressed Sensing and Sparse Signals Recovery. Besides, it can also be interpreted as an optimization algorithm dedicated to the minimization of possibly nonconvex & ell;q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell <^>q$$\end{document} penalized energies (with 0q$$\end{document} penalty. We will describe in detail how the Adaptive Ridge Algorithm can be numerically implemented and we will perform a thorough experimental study of its parameters. We will also show how the variational formulation of the & ell;q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{d
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