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Maximum Likelihood Estimation of Regularization Parameters in High-Dimensional Inverse Problems: An Empirical Bayesian Approach Part I: Methodology and Experiments

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SIAM JOURNAL ON IMAGING SCIENCES 2020年第4期13卷 1945-1989页

作者： Vidal, Ana Fernandez De Bortoli, Valentin Pereyra, Marcelo Durmus, Alain Heriot Watt Univ Maxwell Inst Math Sci Edinburgh EH14 4AS Midlothian Scotland Heriot Watt Univ Sch Math & Comp Sci Edinburgh EH14 4AS Midlothian Scotland Univ Paris Saclay CNRS CMLA Ecole Normale Super Paris Saclay F-94235 Cachan France

Many imaging problems require solving an inverse problem that is ill-conditioned or ill-posed. Imaging methods typically address this difficulty by regularizing the estimation problem to make it well-posed. This often requires setting the value of the so-called regularization parameters that control the amount of regularization enforced. These parameters are notoriously difficult to set a priori and can have a dramatic impact on the recovered estimates. In this work, we propose a general empirical Bayesian method for setting regularization parameters in imaging problems that are convex w.r.t. the unknown image. Our method calibrates regularization parameters directly from the observed data by maximum marginal likelihood estimation and can simultaneously estimate multiple regularization parameters. Furthermore, the proposed algorithm uses the same basic operators as proximal optimization algorithms, namely gradient and proximal operators, and it is therefore straightforward to apply to problems that are currently solved by using proximal optimization techniques. Our methodology is demonstrated with a range of experiments and comparisons with alternative approaches from the literature. The considered experiments include image denoising, nonblind image deconvolution, and hyperspectral unmixing, using synthesis and analysis priors involving the Li, total-variation, total-variation and 4, and total-generalized-variation pseudonorms. A detailed theoretical analysis of the proposed method is presented in our companion paper [V. De Bortoli et al., SIAM J. Imaging Sci., 13 (2020), pp. 1990-2028].

关键词： image processing inverse problems statistical inference empirical Bayes stochastic optimization Markov chain Monte Carlo methods proximal algorithms

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The proximal Augmented Lagrangian Method for Nonsmooth Composite Optimization

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2019年第7期64卷 2861-2868页

作者： Dhingra, Neil K. Khong, Sei Zhen Jovanovic, Mihailo R. Numerica Corp Ft Collins CO 80528 USA Univ Hong Kong Dept Elect & Elect Engn Pokfulam Hong Kong Peoples R China Univ Southern Calif Dept Elect Engn Los Angeles CA 90089 USA

We study a class of optimization problems in which the objective function is given by the sum of a differentiable but possibly nonconvex component and a nondifferentiable convex regularization term. We introduce an auxiliary variable to separate the objective function components and utilize the Moreau envelope of the regularization term to derive the proximal augmented Lagrangian- a continuously differentiable function obtained by constraining the augmented Lagrangian to the manifold that corresponds to the explicit minimization over the variable in the nonsmooth term. The continuous differentiability of this function with respect to both primal and dual variables allows us to leverage the method of multipliers (MM) to compute optimal primal-dual pairs by solving a sequence of differentiable problems. The MM algorithm is applicable to a broader class of problems than proximal gradient methods and it has stronger convergence guarantees and a more refined step-size update rules than the alternating direction method of multipliers (ADMM). These features make it an attractive option for solving structured optimal control problems. We also develop an algorithm based on the primal-descent dual-ascent gradient method and prove global (exponential) asymptotic stability when the differentiable component of the objective function is (strongly) convex and the regularization term is convex. Finally, we identify classes of problems for which the primal-dual gradient flow dynamics are convenient for distributed implementation and compare/contrast our framework to the existing approaches.

关键词： Augmented Lagrangian control for optimization global exponential stability method of multipliers non-smooth optimization primal-dual dynamics proximal algorithms proximal augmented Lagrangian regularization for design structured optimal control

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A simple convergence analysis of Bregman proximal gradient algorithm

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2019年第3期73卷 903-912页

作者： Zhou, Yi Liang, Yingbin Shen, Lixin Duke Univ Dept Elect & Comp Engn Durham NC 27708 USA Ohio State Univ Dept Elect & Comp Engn Columbus OH 43210 USA Syracuse Univ Dept Math Syracuse NY 13244 USA

In this paper, we provide a simple convergence analysis of proximal gradient algorithm with Bregman distance, which provides a tighter bound than existing result. In particular, for the problem of minimizing a class of convex objective functions, we show that proximal gradient algorithm with Bregman distance can be viewed as proximal point algorithm that incorporates another Bregman distance. Consequently, the convergence result of the proximal gradient algorithm with Bregman distance follows directly from that of the proximal point algorithm with Bregman distance, and this leads to a simpler convergence analysis with a tighter convergence bound than existing ones. We further propose and analyze the backtracking line-search variant of the proximal gradient algorithm with Bregman distance.

关键词： proximal algorithms Bregman distance Convergence analysis Line-search

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A Probabilistic Incremental proximal Gradient Method

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IEEE SIGNAL PROCESSING LETTERS 2019年第8期26卷 1257-1261页

作者： Akyildiz, Omer Deniz Chouzenoux, Emilie Elvira, Victor Miguez, Joaquin Univ Warwick Dept Comp Sci London NW1 2DB England Univ Warwick Dept Stat London NW1 2DB England Alan Turing Inst London NW1 2DB England INRIA Saclay CentraleSupelec Ctr Visual Comp F-91190 Gif Sur Yvette France IMT Lille Douai F-59650 Villeneuve Dascq France UMR CNRS 9189 CRIStAL Lab F-59650 Villeneuve Dascq France Univ Carlos III Madrid Dept Signal Theory & Commun Leganes 28912 Spain

In this letter, we propose a probabilistic optimization method, named probabilistic incremental proximal gradient (PIPG) method, by developing a probabilistic interpretation of the incremental proximal gradient algorithm. We explicitly model the update rules of the incremental proximal gradient method and develop a systematic approach to propagate the uncertainty of the solution estimate over iterations. The PIPG algorithm takes the form of Bayesian filtering updates for a state-space model constructed by using the cost function. Our framework makes it possible to utilize well-known exact or approximate Bayesian filters, such as Kalman or extended Kalman filters, to solve large-scale regularized optimization problems.

关键词： Probabilistic optimization stochastic gradient proximal algorithms extended Kalman filtering

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An alternating proximal approach for blind video deconvolution

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SIGNAL PROCESSING-IMAGE COMMUNICATION 2019年 70卷 21-36页

作者： Abboud, Feriel Chouzenoux, Emilie Pesquet, Jean-Christophe Chenot, Jean-Hugues Laborelli, Louis WITBE Le Collines Arche F-92800 Immeuble Opera Puteaux France Univ Paris Est UMR CNRS 8049 LIGM F-77454 Champs Sur Marne France INRIA Saclay CentraleSupelec Ctr Visual Comp F-91190 Gif Sur Yvette France INA F-94366 Bry Sur Marne France

Blurring occurs frequently in video sequences captured by consumer devices, as a result of various factors such as lens aberrations, defocus, relative camera-scene 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.

关键词： Blind deconvolution Video processing Regularization Nonconvex optimization proximal algorithms

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Approximal operator with application to audio inpainting

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SIGNAL PROCESSING 2021年 179卷 107807-107807页

作者： Mokry, Ondrej Rajmic, Pavel Brno Univ Technol Signal Proc Lab Dept Telecommun Fac Elect Engn & Commun Tech 13 Brno 61200 Czech Republic

In their recent evaluation of time-frequency representations and structured sparsity approaches to audio inpainting, Lieb and Stark (2018) have used a particular mapping as a proximal operator. This operator serves as the fundamental part of an iterative numerical solver. However, their mapping is improperly justified. The present article proves that their mapping is indeed a proximal operator, and also derives its proper counterpart. Furthermore, it is rationalized that Lieb and Stark's operator can be understood as an approximation of the proper mapping. Surprisingly, in most cases, such an approximation (referred to as the approximal operator) is shown to provide even better numerical results in audio inpainting compared to its proper counterpart, while being computationally much more effective. (C) 2020 The Authors. Published by Elsevier B.V.

关键词： proximal operator proximal algorithms Approximation Sparsity Audio inpainting

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On the exponential convergence rate of proximal gradient flow algorithms 57

On the exponential convergence rate of proximal gradient flo...

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57th IEEE Conference on Decision and Control (CDC)

作者： Hassan-Moghaddam, Sepideh Jovanovic, Mihailo R. Univ Southern Calif Ming Hsieh Dept Elect Engn Los Angeles CA 90089 USA

ISBN: (纸本)9781538613955

Many modern large-scale and distributed optimization problems can be cast into a form in which the objective function is a sum of a smooth term and a nonsmooth regularizer. Such problems can be solved via a proximal gradient method which generalizes standard gradient descent to a nonsmooth setup. In this paper, we leverage the tools from control theory to study global convergence of proximal gradient flow algorithms. We utilize the fact that the proximal gradient algorithm can be interpreted as a variable-metric gradient method on the forward-backward envelope. This continuously differentiable function can be obtained from the augmented Lagrangian associated with the original nonsmooth problem and it enjoys a number of favorable properties. We prove that global exponential convergence can be achieved even in the absence of strong convexity. Moreover, for in-network optimization problems, we provide a distributed implementation of the gradient flow dynamics based on the proximal augmented Lagrangian and prove global exponential stability for strongly convex problems.

关键词： Distributed optimization forward-backward envelope global exponential stability gradient flow large-scale systems proximal algorithms primal-dual method proximal augmented Lagrangian

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A HYBRID INTERIOR POINT - DEEP LEARNING APPROACH FOR POISSON IMAGE DEBLURRING 30

A HYBRID INTERIOR POINT - DEEP LEARNING APPROACH FOR POISSON...

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30th IEEE International Workshop on Machine Learning for Signal Processing (MLSP)

作者： Galinier, M. Prato, M. Chouzenoux, E. Pesquet, J-C Univ Modena & Reggio Emilia Dipartimento Sci Fis Informat & Matemat Via Campi 213-B I-41125 Modena Italy Univ Paris Saclay Cent Supelec Inria Ctr Vis Numer F-91190 Gif Sur Yvettes France Inst Univ France Paris France

ISBN: (数字)9781728166629

ISBN: (纸本)9781728166629

In this paper we address the problem of deconvolution of an image corrupted with Poisson noise by reformulating the restoration process as a constrained minimization of a suitable regularized data fidelity function. The minimization step is performed by means of an interior-point approach, in which the constraints are incorporated within the objective function through a barrier penalty and a forward-backward algorithm is exploited to build a minimizing sequence. The key point of our proposed scheme is that the choice of the regularization, barrier and step-size parameters defining the interior point approach is automatically performed by a deep learning strategy. Numerical tests on Poisson corrupted benchmark datasets show that our method can obtain very good performance when compared to a state-of-the-art variational deblurring strategy.

关键词： Interior point method proximal algorithms deep unfolding neural network Poisson image restoration

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Space-Time Tomographic Reconstruction of Deforming Objects

Space-Time Tomographic Reconstruction of Deforming Objects

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作者： Zang, Guangming King Abdullah University of Science and Technology

学位级别：博士

X-ray computed tomography (CT) is a popular imaging technique used for reconstructing volumetric properties for a large range of objects. Compared to traditional optical means, CT is a valuable tool for analyzing objects with interesting internal structure or complex geometries that are not accessible with. In this thesis, a variety of applications in computer vision and graphics of inverse problems using tomographic imaging modalities will be presented: The first application focuses on the CT reconstruction with a specific emphasis on recovering thin 1D and 2D manifolds embedded in 3D volumes. To reconstruct such structures at resolutions below the Nyquist limit of the CT image sensor, we devise a new 3D structure tensor prior, which can be incorporated as a regularizer into more traditional proximal optimization methods for CT reconstruction. The second application is about space-time tomography: Through a combination of a new CT image acquisition strategy, a space-time tomographic image formation model, and an alternating, multi-scale solver, we achieve a general approach that can be used to analyze a wide range of dynamic phenomena. Base on the second application, the third one is aiming to improve the tomographic reconstruction of time-varying geometries undergoing faster, non-periodic deformations, by a warp-and-project strategy. Finally, with a physically plausible divergence-free prior for motion estimation, as well as a novel view synthesis technique, we present applications to dynamic fluid imaging (e. g., 4D soot imaging of a combustion process, a mixing fluid process, a fuel injection process, and view synthesis for visible light tomography), which further demonstrates the flexibility of our optimization frameworks.

关键词： tomography X-ray reconstruction computational imaging proximal algorithms motion capture

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Calibration-Less Multi-Coil Compressed Sensing Magnetic Resonance Image Reconstruction Based on OSCAR Regularization

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JOURNAL OF IMAGING 2021年第3期7卷 58-58页

作者： El Gueddari, Loubna Giliyar Radhakrishna, Chaithya Chouzenoux, Emilie Ciuciu, Philippe Univ Paris Saclay CEA NeuroSpin F-91191 Gif Sur Yvette France INRIA Parietal F-91120 Palaiseau France Univ Paris Saclay INRIA OPIS F-91190 Gif Sur Yvette France

Over the last decade, the combination of compressed sensing (CS) with acquisition over multiple receiver coils in magnetic resonance imaging (MRI) has allowed the emergence of faster scans while maintaining a good signal-to-noise ratio (SNR). Self-calibrating techniques, such as ESPiRIT, have become the standard approach to estimating the coil sensitivity maps prior to the reconstruction stage. In this work, we proceed differently and introduce a new calibration-less multi-coil CS reconstruction method. Calibration-less techniques no longer require the prior extraction of sensitivity maps to perform multi-coil image reconstruction but usually alternate estimation sensitivity map estimation and image reconstruction. Here, to get rid of the nonconvexity of the latter approach we reconstruct as many MR images as the number of coils. To compensate for the ill-posedness of this inverse problem, we leverage structured sparsity of the multi-coil images in a wavelet transform domain while adapting to variations in SNR across coils owing to the OSCAR (octagonal shrinkage and clustering algorithm for regression) regularization. Coil-specific complex-valued MR images are thus obtained by minimizing a convex but nonsmooth objective function using the proximal primal-dual Condat-Vu algorithm. Comparison and validation on retrospective Cartesian and non-Cartesian studies based on the Brain fastMRI data set demonstrate that the proposed reconstruction method outperforms the state-of-the-art (l(1)-ESPIRiT, calibration-less AC-LORAKS and CaLM methods) significantly on magnitude images for the T1 and FLAIR contrasts. Additionally, further validation operated on 8 to 20-fold prospectively accelerated high-resolution ex vivo human brain MRI data collected at 7 Tesla confirms the retrospective results. Overall, OSCAR-based regularization preserves phase information more accurately (both visually and quantitatively) compared to other approaches, an asset that can only be assessed on real

关键词： compressed sensing parallel MRI non-Cartesian acquisition cluster norm proximal algorithms

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