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An augmented Lagrangian based parallel splitting method for separable convex minimization with applications to image processing

Deren Han, Xiaoming Yuan, Wenxing Zhang
2014 Mathematics of Computation  
An algorithm is developed by splitting the augmented Lagrangian function in a parallel way.  ...  splitting methods in that the resulting subproblems could be simple enough to have closed-form solutions for such an application whose functions in the objective are simple.  ...  Acknowledgments We are grateful to the anonymous referees for their valuable suggestions which have helped to improve the presentation of this paper substantially.  ... 
doi:10.1090/s0025-5718-2014-02829-9 fatcat:mqy4nkl3zrcqjkq6askfxq56iq

Augmented-Lagrangian regularization of matrix-valued maps

Guy Rosman, Xue-Cheng Tai, Ron Kimmel, Alfred M. Bruckstein
2014 Methods and Applications of Analysis  
Using an augmented-Lagrangian technique, we formulate a fast and highly parallel algorithm for matrixvalued image regularization.  ...  We demonstrate the applicability of the framework for various problems, such as motion analysis and diffusion tensor image reconstruction, show the formulation of the algorithm in terms of split-Bregman  ...  We present an augmented Lagrangian method for efficient regularization of matrix-valued images, or maps.  ... 
doi:10.4310/maa.2014.v21.n1.a5 fatcat:3syht2y2yjgd7nsf6t5ynue7lq

Fast Regularization of Matrix-Valued Images [chapter]

Guy Rosman, Yu Wang, Xue-Cheng Tai, Ron Kimmel, Alfred M. Bruckstein
2014 Lecture Notes in Computer Science  
Using the augmented Lagrangian framework we separate total-variation regularization of matrix-valued images into a regularization and a projection steps.  ...  We demonstrate the effectiveness of our method for smoothing several groupvalued image types, with applications in directions diffusion, motion analysis from depth sensors, and DT-MRI denoising.  ...  Based on the augmented Lagrangian techniques, we separate the optimization problem into a TVregularization step and a projection step, both of which can be solved in an easy-to-implement and parallel way  ... 
doi:10.1007/978-3-642-54774-4_2 fatcat:yh42b2zbrrdetdmdm2seb7ftty

A Unified Primal-Dual Algorithm Framework Based on Bregman Iteration

Xiaoqun Zhang, Martin Burger, Stanley Osher
2010 Journal of Scientific Computing  
This work was originated from Bregman iteration [41], but we can find the connections to other classical optimization concepts, such as augmented Lagrangian method [44] and proximal point minimization.  ...  In this paper, we propose a unified primal-dual algorithm framework for two classes of problems that arise from various signal and image processing applications.  ...  Zhang would like to thank Ernie Esser for fruitful discussions and valuable suggestions to improve the quality of the paper. M. Burger and S.  ... 
doi:10.1007/s10915-010-9408-8 fatcat:db22izkxrvht5au3hpnzn543ki

DeepSplit: Scalable Verification of Deep Neural Networks via Operator Splitting [article]

Shaoru Chen, Eric Wong, J. Zico Kolter, Mahyar Fazlyab
2022 arXiv   pre-print
The method is modular, scales to very large problem instances, and compromises operations that are amenable to fast parallelization with GPU acceleration.  ...  In this work, we propose a novel operator splitting method that can directly solve a convex relaxation of the problem to high accuracy, by splitting it into smaller sub-problems that often have analytical  ...  Lagrangian Relaxation and Dual Subgradient Method Instead of splitting the neural network equations in (3) with auxiliary variables, an alternative strategy is to directly relax the equations with Lagrangian  ... 
arXiv:2106.09117v2 fatcat:wmuclkb6yzdkzf24pz7s7wbrpe

Optimization in learning and data analysis

Stephen J. Wright
2013 Proceedings of the 19th ACM SIGKDD international conference on Knowledge discovery and data mining - KDD '13  
Accelerated Gradient Methods Accelerate the rate to 1/k 2 for weakly convex, while retaining the linear rate (based on √ κ) for strongly convex case.  ...  Image Processing Natural images are not random! They tend to have large areas of near-constant intensity or color, separated by sharp edges.  ...  (Augmented Lagrangian.) (Sridhar et al., 2013)  ... 
doi:10.1145/2487575.2492149 dblp:conf/kdd/Wright13 fatcat:j6edo6tj65fhjil5sg2hqkpvly

Improved Susceptibility Artifact Correction of Echo-Planar MRI using the Alternating Direction Method of Multipliers

Jan Macdonald, Lars Ruthotto
2017 Journal of Mathematical Imaging and Vision  
We show the superiority of our scheme compared to two state-of-the-art methods both in terms of correction quality and time-to-solution for 13 high-resolution 3D imaging datasets.  ...  We prove the convergence of ADMM for this non-convex optimization problem.  ...  We split the separable and non-separable parts of the objective function by adding an artificial variable and apply ADMM to compute a saddle-point of the associated augmented Lagrangian.  ... 
doi:10.1007/s10851-017-0757-x fatcat:3tdyftcjwvdd7lemrtse3xqxqu

Low patch-rank image decomposition using alternating minimization algorithms

2021 Journal of Nonlinear and Variational Analysis  
Cartoon-texture image decomposition, which refers to the problem of decomposing an image into a cartoon part and a texture component, is one of the most fundamental problems in image processing.  ...  In this paper, we are concerned with the low patch-rank enhanced image decomposition model, which is a convex but nonsmooth optimization problem that could not be solved directly by traditional gradient-based  ...  However, we observe that moderate accuracy solutions are sufficient for image processing in many cases, which motivates us to consider whether the augmented Lagrangian-based splitting methods are the best  ... 
doi:10.23952/jnva.5.2021.3.05 fatcat:gxt6bpsoyfbjfn6uk5mqnwfmyi

High-Speed Compressed Sensing Reconstruction in Dynamic Parallel MRI Using Augmented Lagrangian and Parallel Processing

Çağdaş Bilen, Yao Wang, Ivan W. Selesnick
2012 IEEE Journal on Emerging and Selected Topics in Circuits and Systems  
Various reconstruction algorithms has been proposed for compressed sensing, among which Augmented Lagrangian based methods have been shown to often perform better than others for many different applications  ...  In this paper, we propose new Augmented Lagrangian based solutions to the compressed sensing reconstruction problem with analysis and synthesis prior formulations.  ...  The authors would like to thank Li Feng, Daniel Kim, Ricardo Otazo and Daniel K. Sodickson from NYU Medical Center for their help and support as well as for providing the cardiac MRI dataset.  ... 
doi:10.1109/jetcas.2012.2217032 fatcat:33khmke37jbpvozkis2lyagmvy

A survey on operator splitting and decomposition of convex programs

Arnaud Lenoir, Philippe Mahey
2016 Reserche operationelle  
Operator splitting methods have been designed to decompose and regularize at the same time these kind of models. We review here these models and the classical splitting methods.  ...  We focus on the numerical sensitivity of these algorithms with respect to the scaling parameters that drive the regularizing terms, in order to accelerate convergence rates for different classes of models  ...  Separable Augmented Lagrangian We consider first a general convex minimization problem in IR n :      Minimize p i=1 f i (x) x ∈ S (P 1) where the f i are extended real valued convex functions supposed  ... 
doi:10.1051/ro/2015065 fatcat:aw53v5daljesrdoyai53q7t5te

A Splitting Scheme for Flip-Free Distortion Energies [article]

Oded Stein, Jiajin Li, Justin Solomon
2021 arXiv   pre-print
to deal with the non-convex, non-smooth nature of distortion energies.  ...  We identify and exploit the special structure of distortion energies to employ an operator splitting technique, leading us to propose a novel Alternating Direction Method of Multipliers (ADMM) algorithm  ...  ADMM has been employed in many computer graphics and image processing applications.  ... 
arXiv:2107.05200v1 fatcat:epkagtmjhzhzbomeybuu4fm4fa

Image reconstruction under multiplicative speckle noise using total variation

M. Afonso, J. Miguel Sanches
2015 Neurocomputing  
Applying the Augmented Lagrangian framework and using an iterative alternating minimization method leads to simpler problems involving TV minimization with a least squares term.  ...  The proposed method performs a variable splitting to introduce an auxiliary variable to serve as the argument of the total variation (TV) regularizer term.  ...  , (Constrained) Split Augmented Lagrangian Shrinkage Algorithm [11, 19] , split Bregman method [10] , Fast TV deconvolution (FTVd) [9] , and the Nesterov method based solvers, mxTV [20] , and NESTA  ... 
doi:10.1016/j.neucom.2014.08.073 fatcat:3fq6im2jpnfxzntrdr7toe54x4

Modern Convex Optimization to Medical Image Analysis [article]

Jing Yuan, Aaron Fenster
2018 arXiv   pre-print
These challenging clinical-motivated applications introduce novel and sophisticated mathematical problems which stimulate developments of advanced optimization and computing methods, especially convex  ...  Over the past two decade, it has been recognized that advanced image processing techniques provide valuable information to physicians for diagnosis, image guided therapy and surgery, and monitoring of  ...  Some Applications to Image Processing Total-Variation-Based Image Denoising: For the total-variation-based image denoising, it can be formulated as the following convex optimization problem min u D(u −  ... 
arXiv:1809.08734v1 fatcat:hbretnrgmbgzrb6bnvd25qyava

Algorithms and software for projections onto intersections of convex and non-convex sets with applications to inverse problems [article]

Bas Peters, Felix J. Herrmann
2019 arXiv   pre-print
Other design choices that make the software fast and practical to use, include recently developed automatic selection methods for auxiliary algorithm parameters, fine and coarse grained parallelism, and  ...  The software package, called SetIntersectionProjection, is intended for the regularization of inverse problems in physical parameter estimation and image processing.  ...  Projections onto an intersection also solve set-based formulations for linear image processing problems, possibly combined with simple learning techniques to extract set definitions from example images  ... 
arXiv:1902.09699v2 fatcat:yp3t7tfj6jaulcgtvxe5ejqlkm

Modified hybrid decomposition of the augmented Lagrangian method with larger step size for three-block separable convex programming

Min Sun, Yiju Wang
2018 Journal of Inequalities and Applications  
The Jacobian decomposition and the Gauss-Seidel decomposition of augmented Lagrangian method (ALM) are two popular methods for separable convex programming.  ...  However, their convergence is not guaranteed for three-block separable convex programming.  ...  [13] proposed a partial splitting augmented Lagrangian method for solving three-block separable convex programming, which first updates the primal variables x 1 , x 2 , x 3 in a partially-parallel manner  ... 
doi:10.1186/s13660-018-1863-z pmid:30363783 pmcid:PMC6182414 fatcat:qatkh2yw75hovkssw6ilwggnfy
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