Primal dual algorithms for convex models and applications to image restoration, registration and nonlocal inpainting

The main subject of this dissertation is a class of practical algorithms for minimizing convex non-differentiable functionals coming from image processing problems defined as variational models. This work builds largely on the work of Goldstein and Osher [GO09] and Zhu and Chan [ZC08] who proposed respectively the split Bregman and the primal dual hybrid gradient (PDHG) methods for total variation (TV) image restoration. We relate these algorithms to classical methods and generalize their applicability. We also propose new convex variational models for image registration and patch-based nonlocal inpainting and solve them with a variant of the PDHG method. We draw connections between popular methods for convex optimization in image processing by putting them in a general framework of Lagrangian-based alternating direction methods. Furthermore, operator splitting and decomposition techniques are used to generalize their application to a large class of problems, namely minimizing sums of convex functions composed with linear operators and subject to convex constraints. Numerous problems in image and signal processing such as denoising, deblurring, basis pursuit, segmentation, inpainting and many more can be modeled as minimizing exactly such functionals. Numerical examples focus especially on when it is possible to minimize such functionals by solving a sequence of simple convex minimization problems with explicit formulas for their solutions. In the case of the split Bregman method, we point out an equivalence to the classical alternating direction method of multipliers (ADMM) and Douglas Rachford splitting methods. Existing convergence arguments and some minor extensions justify application to common image processing problems. In the case of PDHG, its general convergence is still an open problem, but in joint work with Xiaoqun Zhang and Tony Chan we propose a simple modification that guarantees convergence. We also show convergence of some special cases of the original method. Numerical examples show PDHG and its variants to be especially well suited for large scale problems because their simple, explicit iterations can be constructed to avoid the need to invert large matrices at each iteration. The two proposed convex variational models for image registration and non-local inpainting are novel because most existing variational approaches require minimizing nonconvex functionals.

[1]  H. H. Rachford,et al.  The Numerical Solution of Parabolic and Elliptic Differential Equations , 1955 .

[2]  H. H. Rachford,et al.  On the numerical solution of heat conduction problems in two and three space variables , 1956 .

[3]  G. Minty Monotone (nonlinear) operators in Hilbert space , 1962 .

[4]  J. Moreau Proximité et dualité dans un espace hilbertien , 1965 .

[5]  M. Hestenes Multiplier and gradient methods , 1969 .

[6]  H. Brezis Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert , 1973 .

[7]  J. L. Webb OPERATEURS MAXIMAUX MONOTONES ET SEMI‐GROUPES DE CONTRACTIONS DANS LES ESPACES DE HILBERT , 1974 .

[8]  G. Stephanopoulos,et al.  The use of Hestenes' method of multipliers to resolve dual gaps in engineering system optimization , 1975 .

[9]  B. Mercier,et al.  A dual algorithm for the solution of nonlinear variational problems via finite element approximation , 1976 .

[10]  R. Tyrrell Rockafellar,et al.  Augmented Lagrangians and Applications of the Proximal Point Algorithm in Convex Programming , 1976, Math. Oper. Res..

[11]  I. Ekeland,et al.  Convex analysis and variational problems , 1976 .

[12]  丸山 徹 Convex Analysisの二,三の進展について , 1977 .

[13]  Gregory B. Passty Ergodic convergence to a zero of the sum of monotone operators in Hilbert space , 1979 .

[14]  P. Lions,et al.  Splitting Algorithms for the Sum of Two Nonlinear Operators , 1979 .

[15]  L. Popov A modification of the Arrow-Hurwicz method for search of saddle points , 1980 .

[16]  Berthold K. P. Horn,et al.  Determining Optical Flow , 1981, Other Conferences.

[17]  M. Fortin,et al.  Augmented Lagrangian methods : applications to the numerical solution of boundary-value problems , 1983 .

[18]  D. Gabay Applications of the method of multipliers to variational inequalities , 1983 .

[19]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[20]  Naum Zuselevich Shor,et al.  Minimization Methods for Non-Differentiable Functions , 1985, Springer Series in Computational Mathematics.

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

[22]  Jonathan Eckstein Splitting methods for monotone operators with applications to parallel optimization , 1989 .

[23]  John N. Tsitsiklis,et al.  Parallel and distributed computation , 1989 .

[24]  P. Tseng Applications of splitting algorithm to decomposition in convex programming and variational inequalities , 1991 .

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

[26]  Dimitri P. Bertsekas,et al.  On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators , 1992, Math. Program..

[27]  Jonathan Eckstein,et al.  Nonlinear Proximal Point Algorithms Using Bregman Functions, with Applications to Convex Programming , 1993, Math. Oper. Res..

[28]  Marc Teboulle,et al.  A proximal-based decomposition method for convex minimization problems , 1994, Math. Program..

[29]  Dimitri P. Bertsekas,et al.  Nonlinear Programming , 1997 .

[30]  Michael A. Saunders,et al.  Atomic Decomposition by Basis Pursuit , 1998, SIAM J. Sci. Comput..

[31]  Alexei A. Efros,et al.  Texture synthesis by non-parametric sampling , 1999, Proceedings of the Seventh IEEE International Conference on Computer Vision.

[32]  Gene H. Golub,et al.  A Nonlinear Primal-Dual Method for Total Variation-Based Image Restoration , 1999, SIAM J. Sci. Comput..

[33]  Tony F. Chan,et al.  Active contours without edges , 2001, IEEE Trans. Image Process..

[34]  Michael Patriksson,et al.  On the convergence of conditional epsilon-subgradient methods for convex programs and convex-concave saddle-point problems , 2003, Eur. J. Oper. Res..

[35]  T. Chan,et al.  Image inpainting by correspondence maps: A deterministic approach , 2003 .

[36]  I. Daubechies,et al.  An iterative thresholding algorithm for linear inverse problems with a sparsity constraint , 2003, math/0307152.

[37]  Hiroshi Ishikawa,et al.  Exact Optimization for Markov Random Fields with Convex Priors , 2003, IEEE Trans. Pattern Anal. Mach. Intell..

[38]  Thomas Brox,et al.  High Accuracy Optical Flow Estimation Based on a Theory for Warping , 2004, ECCV.

[39]  Laurent D. Cohen,et al.  Image Registration, Optical Flow and Local Rigidity , 2001, Journal of Mathematical Imaging and Vision.

[40]  M. Nikolova An Algorithm for Total Variation Minimization and Applications , 2004 .

[41]  Wotao Yin,et al.  Second-order Cone Programming Methods for Total Variation-Based Image Restoration , 2005, SIAM J. Sci. Comput..

[42]  Patrick L. Combettes,et al.  Signal Recovery by Proximal Forward-Backward Splitting , 2005, Multiscale Model. Simul..

[43]  E. Candès,et al.  Stable signal recovery from incomplete and inaccurate measurements , 2005, math/0503066.

[44]  Yurii Nesterov,et al.  Smooth minimization of non-smooth functions , 2005, Math. Program..

[45]  Justin Romberg,et al.  Practical Signal Recovery from Random Projections , 2005 .

[46]  T. Chan,et al.  Variational image inpainting , 2005 .

[47]  Tony F. Chan,et al.  Aspects of Total Variation Regularized L[sup 1] Function Approximation , 2005, SIAM J. Appl. Math..

[48]  Mila Nikolova,et al.  Algorithms for Finding Global Minimizers of Image Segmentation and Denoising Models , 2006, SIAM J. Appl. Math..

[49]  Tony F. Chan,et al.  Total Variation Wavelet Inpainting , 2006, Journal of Mathematical Imaging and Vision.

[50]  Hemant D. Tagare,et al.  A Geometric Theory of Symmetric Registration , 2006, 2006 Conference on Computer Vision and Pattern Recognition Workshop (CVPRW'06).

[51]  Antti Oulasvirta,et al.  Computer Vision – ECCV 2006 , 2006, Lecture Notes in Computer Science.

[52]  Horst Bischof,et al.  A Duality Based Approach for Realtime TV-L1 Optical Flow , 2007, DAGM-Symposium.

[53]  Yurii Nesterov,et al.  Dual extrapolation and its applications to solving variational inequalities and related problems , 2003, Math. Program..

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

[55]  Wotao Yin,et al.  Bregman Iterative Algorithms for (cid:2) 1 -Minimization with Applications to Compressed Sensing ∗ , 2008 .

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

[57]  Jian-Feng Cai,et al.  A framelet-based image inpainting algorithm , 2008 .

[58]  Abderrahim Elmoataz,et al.  Nonlocal Discrete Regularization on Weighted Graphs: A Framework for Image and Manifold Processing , 2008, IEEE Transactions on Image Processing.

[59]  Daniel Cremers,et al.  A Convex Formulation of Continuous Multi-label Problems , 2008, ECCV.

[60]  Jan-Michael Frahm,et al.  Fast Global Labeling for Real-Time Stereo Using Multiple Plane Sweeps , 2008, VMV.

[61]  T. Chan,et al.  Variational pde-based image segmentation and inpainting with applications in computer graphics , 2008 .

[62]  T. Chan,et al.  Convex Formulation and Exact Global Solutions for Multi-phase Piecewise Constant Mumford-Shah Image Segmentation , 2009 .

[63]  Junfeng Yang,et al.  An Efficient TVL1 Algorithm for Deblurring Multichannel Images Corrupted by Impulsive Noise , 2009, SIAM J. Sci. Comput..

[64]  A. Chambolle,et al.  An introduction to Total Variation for Image Analysis , 2009 .

[65]  Gilles Aubert,et al.  Efficient Schemes for Total Variation Minimization Under Constraints in Image Processing , 2009, SIAM J. Sci. Comput..

[66]  Xiaoqun Zhang,et al.  A General Framework for a Class of First Order Primal-Dual Algorithms for TV Minimization , 2009 .

[67]  Simon Setzer,et al.  Split Bregman Algorithm, Douglas-Rachford Splitting and Frame Shrinkage , 2009, SSVM.

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

[69]  Daniel Cremers,et al.  An algorithm for minimizing the Mumford-Shah functional , 2009, 2009 IEEE 12th International Conference on Computer Vision.

[70]  Michael K. Ng,et al.  A Fast l1-TV Algorithm for Image Restoration , 2009, SIAM J. Sci. Comput..

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

[72]  Guillermo Sapiro,et al.  A Variational Framework for Non-local Image Inpainting , 2009, EMMCVPR.

[73]  Xue-Cheng Tai,et al.  Global Minimization for Continuous Multiphase Partitioning Problems Using a Dual Approach , 2011, International Journal of Computer Vision.

[74]  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..

[75]  Xavier Bresson,et al.  Bregmanized Nonlocal Regularization for Deconvolution and Sparse Reconstruction , 2010, SIAM J. Imaging Sci..

[76]  Saïd Ladjal,et al.  Exemplar-Based Inpainting from a Variational Point of View , 2010, SIAM J. Math. Anal..

[77]  T. Chan,et al.  WAVELET INPAINTING BY NONLOCAL TOTAL VARIATION , 2010 .

[78]  Ernie Esser,et al.  Applications of Lagrangian-Based Alternating Direction Methods and Connections to Split Bregman , 2009 .

[79]  Stephen J. Wright,et al.  Duality-based algorithms for total-variation-regularized image restoration , 2010, Comput. Optim. Appl..

[80]  Ernie Esser,et al.  A Convex Model for Image Registration , 2010 .

[81]  Stanley Osher,et al.  A Unified Primal-Dual Algorithm Framework Based on Bregman Iteration , 2010, J. Sci. Comput..

[82]  Emmanuel J. Candès,et al.  NESTA: A Fast and Accurate First-Order Method for Sparse Recovery , 2009, SIAM J. Imaging Sci..

[83]  S. Osher,et al.  Global minimization of Markov random fields with applications to optical flow , 2012 .

[84]  K. Schittkowski,et al.  NONLINEAR PROGRAMMING , 2022 .