A Convex and Exact Approach to Discrete Constrained TV-L1 Image Approximation

We study the TV-L1 image approximation model from primal and dual perspective, based on a proposed equivalent convex formulations. More specifically, we apply a convex TV-L1 based approach to globally solve the discrete constrained optimization problem of image approximation, where the unknown image function u( x ) ∈ { f 1 ,…, f n }, ∀ x ∈ Ω. We show that the TV-L1 formulation does provide an exact convex relaxation model to the non-convex optimization problem considered. This result greatly extends recent studies of Chan et al., from the simplest binary constrained case to the general gray-value constrained case, through the proposed rounding scheme. In addition, we construct a fast multiplier-based algorithm based on the proposed primal-dual model, which properly avoids variability of the concerning TV-L1 energy function. Numerical experiments validate the theoretical results and show that the proposed algorithm is reliable and effective.

[1]  E. Giusti.  Minimal surfaces and functions of bounded variation , 1977 .

[2]  Xavier Bresson,et al.  Fast Global Minimization of the Active Contour/Snake Model , 2007, Journal of Mathematical Imaging and Vision.

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

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

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

[6]  Jérôme Darbon,et al.  Image Restoration with Discrete Constrained Total Variation Part I: Fast and Exact Optimization , 2006, Journal of Mathematical Imaging and Vision.

[7]  Stefano Alliney,et al.  Digital filters as absolute norm regularizers , 1992, IEEE Trans. Signal Process..

[8]  Guy Gilboa,et al.  Nonlinear Inverse Scale Space Methods for Image Restoration , 2005, VLSM.

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

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

[11]  Jérôme Darbon,et al.  Image Restoration with Discrete Constrained Total Variation Part II: Levelable Functions, Convex Priors and Non-Convex Cases , 2006, Journal of Mathematical Imaging and Vision.

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

[13]  Xuecheng Tai,et al.  AUGMENTED LAGRANGIAN METHOD FOR TOTAL VARIATION RESTORATION WITH NON-QUADRATIC FIDELITY , 2011 .

[14]  Wotao Yin,et al.  The Total Variation Regularized L1 Model for Multiscale Decomposition , 2007, Multiscale Model. Simul..

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

[16]  Yves Meyer,et al.  Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures , 2001 .

[17]  Michael K. Ng,et al.  A Fast Total Variation Minimization Method for Image Restoration , 2008, Multiscale Model. Simul..

[18]  Thomas Brox,et al.  On the Equivalence of Soft Wavelet Shrinkage, Total Variation Diffusion, Total Variation Regularization, and SIDEs , 2004, SIAM J. Numer. Anal..

[19]  Kazufumi Ito,et al.  The Primal-Dual Active Set Strategy as a Semismooth Newton Method , 2002, SIAM J. Optim..

[20]  Nikos Paragios,et al.  Handbook of Mathematical Models in Computer Vision , 2005 .

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

[22]  K Fan,et al.  Minimax Theorems. , 1953, Proceedings of the National Academy of Sciences of the United States of America.

[23]  Stefano Alliney,et al.  Recursive median filters of increasing order: a variational approach , 1996, IEEE Trans. Signal Process..

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

[25]  Gabriele Steidl,et al.  Deblurring Poissonian images by split Bregman techniques , 2010, J. Vis. Commun. Image Represent..

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

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

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