Restoration of images based on subspace optimization accelerating augmented Lagrangian approach

We propose a new fast algorithm for solving a TV-based image restoration problem. Our approach is based on merging subspace optimization methods into an augmented Lagrangian method. The proposed algorithm can be seen as a variant of the ALM (Augmented Lagrangian Method), and the convergence properties are analyzed from a DRS (Douglas-Rachford splitting) viewpoint. Experiments on a set of image restoration benchmark problems show that the proposed algorithm is a strong contender for the current state of the art methods.

[1]  A. Iusem.  Augmented Lagrangian Methods and Proximal Point Methods for Convex Optimization , 1999 .

[2]  E.J. Candes,et al.  An Introduction To Compressive Sampling , 2008, IEEE Signal Processing Magazine.

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

[4]  Shiqian Ma,et al.  An efficient algorithm for compressed MR imaging using total variation and wavelets , 2008, 2008 IEEE Conference on Computer Vision and Pattern Recognition.

[5]  Raymond H. Chan,et al.  Restoration of Chopped and Nodded Images by Framelets , 2008, SIAM J. Sci. Comput..

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

[7]  T. Chan,et al.  Fast numerical algorithms for total variation based image restoration , 2008 .

[8]  Stephen P. Boyd,et al.  Enhancing Sparsity by Reweighted ℓ1 Minimization , 2007, 0711.1612.

[9]  Michael Elad,et al.  Coordinate and subspace optimization methods for linear least squares with non-quadratic regularization , 2007 .

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

[11]  J.-C. Pesquet,et al.  A Douglas–Rachford Splitting Approach to Nonsmooth Convex Variational Signal Recovery , 2007, IEEE Journal of Selected Topics in Signal Processing.

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

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

[14]  José M. Bioucas-Dias,et al.  Fast Image Recovery Using Variable Splitting and Constrained Optimization , 2009, IEEE Transactions on Image Processing.

[15]  José M. Bioucas-Dias,et al.  A New TwIST: Two-Step Iterative Shrinkage/Thresholding Algorithms for Image Restoration , 2007, IEEE Transactions on Image Processing.

[16]  Simon Setzer,et al.  Operator Splittings, Bregman Methods and Frame Shrinkage in Image Processing , 2011, International Journal of Computer Vision.

[17]  Stephen J. Wright,et al.  Sparse Reconstruction by Separable Approximation , 2008, IEEE Transactions on Signal Processing.

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

[19]  D. Donoho,et al.  Sparse MRI: The application of compressed sensing for rapid MR imaging , 2007, Magnetic resonance in medicine.

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

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

[22]  Thomas Serafini,et al.  Non-negatively constrained image deblurring with an inexact interior point method , 2009, J. Comput. Appl. Math..

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

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

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

[26]  Marcel Carbillet,et al.  Iterative methods for the reconstruction of astronomical images with high dynamic range , 2007 .

[27]  A. Chambolle.  Practical, Unified, Motion and Missing Data Treatment in Degraded Video , 2004, Journal of Mathematical Imaging and Vision.

[28]  Michael Elad.  Iterative-Shrinkage Algorithms , 2010 .