Iterative Support Detection-Based Split Bregman Method for Wavelet Frame-Based Image Inpainting

The wavelet frame systems have been extensively studied due to their capability of sparsely approximating piecewise smooth functions, such as images, and the corresponding wavelet frame-based image restoration models are mostly based on the penalization of the ℓ1 norm of wavelet frame coefficients for sparsity enforcement. In this paper, we focus on the image inpainting problem based on the wavelet frame, propose a weighted sparse restoration model, and develop a corresponding efficient algorithm. The new algorithm combines the idea of iterative support detection method, first proposed by Wang and Yin for sparse signal reconstruction, and the split Bregman method for wavelet frame ℓ1 model of image inpainting, and more important, naturally makes use of the specific multilevel structure of the wavelet frame coefficients to enhance the recovery quality. This new algorithm can be considered as the incorporation of prior structural information of the wavelet frame coefficients into the traditional ℓ1 model. Our numerical experiments show that the proposed method is superior to the original split Bregman method for wavelet frame-based ℓ1 norm image inpainting model as well as some typical ℓp(0 ≤ p <; 1) norm-based nonconvex algorithms such as mean doubly augmented Lagrangian method, in terms of better preservation of sharp edges, due to their failing to make use of the structure of the wavelet frame coefficients.

[1]  Wei Lu,et al.  Regularized Modified BPDN for Noisy Sparse Reconstruction With Partial Erroneous Support and Signal Value Knowledge , 2010, IEEE Transactions on Signal Processing.

[2]  Robert D. Nowak,et al.  An EM algorithm for wavelet-based image restoration , 2003, IEEE Trans. Image Process..

[3]  Raymond H. Chan,et al.  Convergence analysis of tight framelet approach for missing data recovery , 2009, Adv. Comput. Math..

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

[5]  G. Sapiro,et al.  Geometric partial differential equations and image analysis [Book Reviews] , 2001, IEEE Transactions on Medical Imaging.

[6]  Wotao Yin,et al.  Sparse Signal Reconstruction via Iterative Support Detection , 2009, SIAM J. Imaging Sci..

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

[8]  Zuowei Shen.  Wavelet Frames and Image Restorations , 2011 .

[9]  YinWotao,et al.  Sparse Signal Reconstruction via Iterative Support Detection , 2010 .

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

[11]  Jong Chul Ye,et al.  k‐t FOCUSS: A general compressed sensing framework for high resolution dynamic MRI , 2009, Magnetic resonance in medicine.

[12]  M. J. D. Powell,et al.  A method for nonlinear constraints in minimization problems , 1969 .

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

[14]  Junzhou Huang,et al.  Compressive Sensing MRI with Wavelet Tree Sparsity , 2012, NIPS.

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

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

[17]  I. Daubechies,et al.  Framelets: MRA-based constructions of wavelet frames☆☆☆ , 2003 .

[18]  Wei Lu,et al.  Modified-CS: Modifying compressive sensing for problems with partially known support , 2009, 2009 IEEE International Symposium on Information Theory.

[19]  P. Lions,et al.  Image recovery via total variation minimization and related problems , 1997 .

[20]  Bhaskar D. Rao,et al.  Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimum norm algorithm , 1997, IEEE Trans. Signal Process..

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

[22]  Andy M. Yip,et al.  Total Variation Image Restoration: Overview and Recent Developments , 2006, Handbook of Mathematical Models in Computer Vision.

[23]  Junzhou Huang,et al.  The benefit of tree sparsity in accelerated MRI , 2014, Medical Image Anal..

[24]  S. Frick,et al.  Compressed Sensing , 2014, Computer Vision, A Reference Guide.

[25]  Robert D. Nowak,et al.  A bound optimization approach to wavelet-based image deconvolution , 2005, IEEE International Conference on Image Processing 2005.

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

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

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

[29]  Jian-Feng Cai,et al.  Linearized Bregman Iterations for Frame-Based Image Deblurring , 2009, SIAM J. Imaging Sci..

[30]  Bin Dong,et al.  MRA-based wavelet frames and applications , 2013 .

[31]  Laurent Jacques,et al.  A short note on compressed sensing with partially known signal support , 2009, Signal Process..

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

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

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

[35]  David H. Wolpert,et al.  No free lunch theorems for optimization , 1997, IEEE Trans. Evol. Comput..

[36]  A. Cohen.  Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 61, I. Daubechies, SIAM, 1992, xix + 357 pp. , 1994 .

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

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

[39]  Bin Dong,et al.  ℓ0 Minimization for wavelet frame based image restoration , 2011, Math. Comput..

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

[41]  David L. Donoho,et al.  De-noising by soft-thresholding , 1995, IEEE Trans. Inf. Theory.

[42]  Jean-Luc Starck,et al.  Sparse representations and bayesian image inpainting , 2005 .

[43]  Kenneth E. Barner,et al.  Iterative hard thresholding for compressed sensing with partially known support , 2011, 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

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

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

[46]  Jian-Feng Cai,et al.  Simultaneous cartoon and texture inpainting , 2010 .

[47]  Bin Dong,et al.  An Efficient Algorithm for ℓ0 Minimization in Wavelet Frame Based Image Restoration , 2013, J. Sci. Comput..

[48]  Pierre Kornprobst,et al.  Mathematical problems in image processing - partial differential equations and the calculus of variations , 2010, Applied mathematical sciences.

[49]  Michael Elad,et al.  Submitted to Ieee Transactions on Image Processing Image Decomposition via the Combination of Sparse Representations and a Variational Approach , 2022 .

[50]  S. Osher,et al.  Image restoration: Total variation, wavelet frames, and beyond , 2012 .

[51]  R. Chan,et al.  A Framelet-Based Approach for Image Inpainting , 2005 .

[52]  I. Daubechies,et al.  Iteratively solving linear inverse problems under general convex constraints , 2007 .

[53]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[54]  Raymond H. Chan,et al.  Wavelet Algorithms for High-Resolution Image Reconstruction , 2002, SIAM J. Sci. Comput..

[55]  Zuowei Shen.  Affine systems in L 2 ( IR d ) : the analysis of the analysis operator , 1995 .

[56]  Joachim Weickert,et al.  Rotationally Invariant Wavelet Shrinkage , 2003, DAGM-Symposium.

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

[58]  Kenneth E. Barner,et al.  Iterative algorithms for compressed sensing with partially known support , 2010, 2010 IEEE International Conference on Acoustics, Speech and Signal Processing.

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

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

[61]  A. Ron,et al.  Affine Systems inL2(Rd): The Analysis of the Analysis Operator , 1997 .

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

[63]  Mohamed-Jalal Fadili,et al.  Inpainting and Zooming Using Sparse Representations , 2009, Comput. J..

[64]  Ronald Fedkiw,et al.  Level set methods and dynamic implicit surfaces , 2002, Applied mathematical sciences.

[65]  Tong Zhang,et al.  Analysis of Multi-stage Convex Relaxation for Sparse Regularization , 2010, J. Mach. Learn. Res..

[66]  D. Donoho,et al.  Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA) , 2005 .

[67]  Hassan Mansour,et al.  Recovering Compressively Sampled Signals Using Partial Support Information , 2010, IEEE Transactions on Information Theory.

[68]  Wotao Yin,et al.  Iteratively reweighted algorithms for compressive sensing , 2008, 2008 IEEE International Conference on Acoustics, Speech and Signal Processing.