A parallel alternating direction method with application to compound l1-regularized imaging inverse problems

We derive a parallel alternating direction method of multipliers (PADMM) and apply it to compound l1-regularized imaging inverse problems. The proposed method is capable of locating the saddle point of large-scale convex minimization problems with the sum of several nonsmooth but proximable terms. Using an operator splitting strategy, the objective is decomposed into subproblems that are conveniently, individually and simultaneously solved. With the assistance of the Moreau decomposition, our method excludes auxiliary variables that exist in the ADMM and possesses a compacter structure. Thus, the proposed method is preferable in distributed computation. The convergence proof and convergence rate analysis are presented. Application to both image restoration and image compressed sensing demonstrates the effectiveness and efficiency of the proposed method.

[1]  David Zhang,et al.  A Survey of Sparse Representation: Algorithms and Applications , 2015, IEEE Access.

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

[3]  Raymond H. Chan,et al.  Parameter selection for total-variation-based image restoration using discrepancy principle , 2012, IEEE Transactions on Image Processing.

[4]  Wotao Yin,et al.  Edge Guided Reconstruction for Compressive Imaging , 2012, SIAM J. Imaging Sci..

[5]  Wei Zhang,et al.  Adaptive shearlet-regularized image deblurring via alternating direction method , 2014, 2014 IEEE International Conference on Multimedia and Expo (ICME).

[6]  Christine Guillemot,et al.  Image Inpainting : Overview and Recent Advances , 2014, IEEE Signal Processing Magazine.

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

[8]  Wei Zhang,et al.  A Fast Adaptive Parameter Estimation for Total Variation Image Restoration , 2014, IEEE Transactions on Image Processing.

[9]  Jie Huang,et al.  Two soft-thresholding based iterative algorithms for image deblurring , 2014, Inf. Sci..

[10]  Boying Wu,et al.  A chaotic iterative algorithm based on linearized Bregman iteration for image deblurring , 2014, Inf. Sci..

[11]  Xuelong Li,et al.  A Variational Approach to Simultaneous Image Segmentation and Bias Correction , 2015, IEEE Transactions on Cybernetics.

[12]  Dianhui Wang,et al.  A fractional-order adaptive regularization primal-dual algorithm for image denoising , 2015, Inf. Sci..

[13]  Heinz H. Bauschke,et al.  Convex Analysis and Monotone Operator Theory in Hilbert Spaces , 2011, CMS Books in Mathematics.

[14]  Raymond H. Chan,et al.  Constrained Total Variation Deblurring Models and Fast Algorithms Based on Alternating Direction Method of Multipliers , 2013, SIAM J. Imaging Sci..

[15]  Jian-Feng Cai,et al.  Split Bregman Methods and Frame Based Image Restoration , 2009, Multiscale Model. Simul..

[16]  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.

[17]  Karl Kunisch,et al.  Total Generalized Variation , 2010, SIAM J. Imaging Sci..

[18]  Weiguo Gong,et al.  Dual-sparsity regularized sparse representation for single image super-resolution , 2015, Inf. Sci..

[19]  José M. Bioucas-Dias,et al.  An Augmented Lagrangian Approach to the Constrained Optimization Formulation of Imaging Inverse Problems , 2009, IEEE Transactions on Image Processing.

[20]  David Cooper,et al.  Multiple regularization based MRI reconstruction , 2014, Signal Process..

[21]  Michael K. Ng,et al.  Solving Constrained Total-variation Image Restoration and Reconstruction Problems via Alternating Direction Methods , 2010, SIAM J. Sci. Comput..

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

[23]  Eero P. Simoncelli,et al.  Image quality assessment: from error visibility to structural similarity , 2004, IEEE Transactions on Image Processing.

[24]  Xuelong Li,et al.  Automatic segmentation of breast lesions for interaction in ultrasonic computer-aided diagnosis , 2015, Inf. Sci..

[25]  Wotao Yin,et al.  A New Detail-Preserving Regularization Scheme , 2014, SIAM J. Imaging Sci..

[26]  Bingsheng He,et al.  On non-ergodic convergence rate of Douglas–Rachford alternating direction method of multipliers , 2014, Numerische Mathematik.

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

[28]  Junfeng Yang,et al.  A Fast Algorithm for Edge-Preserving Variational Multichannel Image Restoration , 2009, SIAM J. Imaging Sci..

[29]  Ke Lu,et al.  Single-image motion deblurring using an adaptive image prior , 2014, Inf. Sci..

[30]  Ting-Zhu Huang,et al.  Image restoration using total variation with overlapping group sparsity , 2013, Inf. Sci..