Convergence Rate of Overlapping Domain Decomposition Methods for the Rudin-Osher-Fatemi Model Based on a Dual Formulation

This paper is concerned with overlapping domain decomposition methods (DDMs), based on succes- sive subspace correction (SSC) and parallel subspace correction (PSC), for the Rudin-Osher-Fatemi (ROF) model in image restoration. In contrast to recent attempts, we work with a dual formulation of the ROF model, where one significant difficulty resides in the decomposition of the global con- straint of the dual variable. We introduce a stable "unity decomposition" using a set of "partition of unity functions," which naturally leads to overlapping DDMs based on the dual formulation. The main objective of this paper is to rigorously analyze the convergence of the SSC and PSC algorithms and derive the rate of convergence O(n −1/2 ), where n is the number of iterations. Moreover, we characterize the explicit dependence of the convergence rate on the subdomain overlapping size and other important parameters. To the best of our knowledge, such a convergence rate has not yet been claimed for domain decomposition related algorithms for the ROF model.

[1]  C. Vogel.  A Multigrid Method for Total Variation-Based Image Denoising , 1995 .

[2]  Curtis R. Vogel,et al.  Ieee Transactions on Image Processing Fast, Robust Total Variation{based Reconstruction of Noisy, Blurred Images , 2022 .

[3]  Stanley Osher,et al.  Explicit Algorithms for a New Time Dependent Model Based on Level Set Motion for Nonlinear Deblurring and Noise Removal , 2000, SIAM J. Sci. Comput..

[4]  Zi-Cai Li,et al.  Schwarz Alternating Method , 1998 .

[5]  Carola-Bibiane Schönlieb,et al.  Subspace Correction Methods for Total Variation and 1-Minimization , 2007, SIAM J. Numer. Anal..

[6]  Junfeng Yang,et al.  A New Alternating Minimization Algorithm for Total Variation Image Reconstruction , 2008, SIAM J. Imaging Sci..

[7]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[8]  Yiqiu Dong,et al.  An Efficient Primal-Dual Method for L1TV Image Restoration , 2009, SIAM J. Imaging Sci..

[9]  Tony F. Chan,et al.  A General Framework for a Class of First Order Primal-Dual Algorithms for Convex Optimization in Imaging Science , 2010, SIAM J. Imaging Sci..

[10]  Xuecheng Tai,et al.  A two-level domain decomposition method for image restoration , 2010 .

[11]  Paul Tseng,et al.  Convergence rate analysis of an asynchronous space decomposition method for convex Minimization , 2001, Math. Comput..

[12]  C. Vogel.  Computational Methods for Inverse Problems , 1987 .

[13]  Xuecheng Tai,et al.  Applications of a space decomposition method to linear and nonlinear elliptic problems , 1998 .

[14]  Karl Kunisch,et al.  Total Bounded Variation Regularization as a Bilaterally Constrained Optimization Problem , 2004, SIAM J. Appl. Math..

[15]  J. Mixter.  Fast , 2012 .

[16]  Xue-Cheng Tai,et al.  Rate of Convergence for some constraint decomposition methods for nonlinear variational inequalities , 2003, Numerische Mathematik.

[17]  Xue-Cheng Tai,et al.  Augmented Lagrangian Method, Dual Methods, and Split Bregman Iteration for ROF, Vectorial TV, and High Order Models , 2010, SIAM J. Imaging Sci..

[18]  DongYiqiu,et al.  An Efficient Primal-Dual Method for $L^1$TV Image Restoration , 2009 .

[19]  R. Glowinski,et al.  Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics , 1987 .

[20]  Carola-Bibiane Schönlieb,et al.  Bregmanized Domain Decomposition for Image Restoration , 2013, J. Sci. Comput..

[21]  Curtis R. Vogel,et al.  Iterative Methods for Total Variation Denoising , 1996, SIAM J. Sci. Comput..

[22]  P. Oswald,et al.  Remarks on the Abstract Theory of Additive and Multiplicative Schwarz Algorithms , 1995 .

[23]  Massimo Fornasier,et al.  Domain decomposition methods for compressed sensing , 2009, 0902.0124.

[24]  Xue-Cheng Tai,et al.  Augmented Lagrangian Method, Dual Methods and Split Bregman Iteration for ROF Model , 2009, SSVM.

[25]  Xuecheng Tai,et al.  Rate of Convergence of Some Space Decomposition Methods for Linear and Nonlinear Problems , 1998 .

[26]  Yin Zhang,et al.  A Fast Algorithm for Image Deblurring with Total Variation Regularization , 2007 .

[27]  Barry F. Smith,et al.  Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations , 1996 .

[28]  Mingqiang Zhu,et al.  An Efficient Primal-Dual Hybrid Gradient Algorithm For Total Variation Image Restoration , 2008 .

[29]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[30]  Arjan Kuijper,et al.  Scale Space and Variational Methods in Computer Vision , 2013, Lecture Notes in Computer Science.

[31]  Hugues Talbot,et al.  Globally Optimal Geodesic Active Contours , 2005, Journal of Mathematical Imaging and Vision.

[32]  Tom Goldstein,et al.  The Split Bregman Method for L1-Regularized Problems , 2009, SIAM J. Imaging Sci..

[33]  Olof B. Widlund,et al.  Towards a Unified Theory of Domain Decomposition Algorithms for Elliptic Problems , 2015 .

[34]  Danping Yang,et al.  Domain Decomposition Methods for Nonlocal Total Variation Image Restoration , 2014, J. Sci. Comput..

[35]  ANTONIN CHAMBOLLE,et al.  An Algorithm for Total Variation Minimization and Applications , 2004, Journal of Mathematical Imaging and Vision.

[36]  B. V. Dean,et al.  Studies in Linear and Non-Linear Programming. , 1959 .

[37]  Wotao Yin,et al.  An Iterative Regularization Method for Total Variation-Based Image Restoration , 2005, Multiscale Model. Simul..

[38]  T. Chan,et al.  Domain decomposition algorithms , 1994, Acta Numerica.

[39]  Xuecheng Tai,et al.  DOMAIN DECOMPOSITION METHODS WITH GRAPH CUTS ALGORITHMS FOR IMAGE SEGMENTATION , 2009 .

[40]  Michael Hintermüller,et al.  An Infeasible Primal-Dual Algorithm for Total Bounded Variation-Based Inf-Convolution-Type Image Restoration , 2006, SIAM J. Sci. Comput..

[41]  Michael Hintermüller,et al.  Subspace Correction Methods for a Class of Nonsmooth and Nonadditive Convex Variational Problems with Mixed L1/L2 Data-Fidelity in Image Processing , 2013, SIAM J. Imaging Sci..

[42]  Carola-Bibiane Schönlieb,et al.  A convergent overlapping domain decomposition method for total variation minimization , 2009, Numerische Mathematik.

[43]  Marc Teboulle,et al.  A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems , 2009, SIAM J. Imaging Sci..

[44]  S. Lui,et al.  Domain decomposition methods in image denoising using Gaussian curvature , 2006 .

[45]  J. Müller,et al.  Parallel Total Variation Minimization Diplomarbeit Eingereicht Von , 2008 .

[46]  Ronald H. W. Hoppe,et al.  Finite element methods for Maxwell's equations , 2005, Math. Comput..

[47]  O. Scherzer.  Handbook of mathematical methods in imaging , 2011 .

[48]  Jinchao Xu,et al.  Global and uniform convergence of subspace correction methods for some convex optimization problems , 2002, Math. Comput..

[49]  Jinchao Xu,et al.  Iterative Methods by Space Decomposition and Subspace Correction , 1992, SIAM Rev..

[50]  C. Vogel,et al.  Analysis of bounded variation penalty methods for ill-posed problems , 1994 .

[51]  Antonin Chambolle,et al.  A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging , 2011, Journal of Mathematical Imaging and Vision.

[52]  Xue-Cheng Tai,et al.  Domain decomposition methods with graph cuts algorithms for total variation minimization , 2012, Adv. Comput. Math..

[53]  Jing Qin,et al.  Domain decomposition method for image deblurring , 2014, J. Comput. Appl. Math..

[54]  Michael Hintermüller,et al.  Non-Overlapping Domain Decomposition Methods For Dual Total Variation Based Image Denoising , 2014, Journal of Scientific Computing.