Compressive total variation for image reconstruction and restoration

Abstract In this paper, we make use of the fact that the matrix u is (approximately) low-rank in image inpainting, and the corresponding gradient transform matrices D x u , D y u are sparse in image reconstruction and restoration. Therefore we consider that these gradient matrices D x u , D y u also are (approximately) low-rank, and also verify it by numerical test and theoretical analysis. We propose a model called compressive total variation (CTV) to characterize the sparsity and low-rank prior knowledge of an image. In order to solve the proposed model, we design a concrete algorithm with provably convergence, which is based on inertial proximal ADMM. The performance of the proposed model is tested for magnetic resonance imaging (MRI) reconstruction, image denoising and image deblurring. The proposed method not only recovers edges of the image but also preserves fine details of the image. And our model is much better than the existing regularization models based on the TGV, Shearlet-TGV, l 1 − α l 2 TV and BM3D in test for images with piecewise constant regions. And it visibly improves the performances of TV, l 1 − α l 2 TV and TGV, and is comparable to Shearlet-TGV in test for natural images.

[1]  Andrea J. Goldsmith,et al.  Exact and Stable Covariance Estimation From Quadratic Sampling via Convex Programming , 2013, IEEE Transactions on Information Theory.

[2]  Shiqian Ma,et al.  Inertial Proximal ADMM for Linearly Constrained Separable Convex Optimization , 2015, SIAM J. Imaging Sci..

[3]  Ting-Zhu Huang,et al.  Truncated l1-2 Models for Sparse Recovery and Rank Minimization , 2017, SIAM J. Imaging Sci..

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

[5]  Michael K. Ng,et al.  Nonconvex-TV Based Image Restoration with Impulse Noise Removal , 2017, SIAM J. Imaging Sci..

[6]  M. Ng,et al.  ℓ1 − αℓ2 minimization methods for signal and image reconstruction with impulsive noise removal , 2020, Inverse Problems.

[7]  Kevin M. Holt,et al.  Total Nuclear Variation and Jacobian Extensions of Total Variation for Vector Fields , 2014, IEEE Transactions on Image Processing.

[8]  Junfeng Yang,et al.  A Fast Alternating Direction Method for TVL1-L2 Signal Reconstruction From Partial Fourier Data , 2010, IEEE Journal of Selected Topics in Signal Processing.

[9]  Kristian Bredies,et al.  Preconditioned Douglas–Rachford Algorithms for TV- and TGV-Regularized Variational Imaging Problems , 2015, Journal of Mathematical Imaging and Vision.

[10]  Kristian Bredies,et al.  Total Generalized Variation for Manifold-Valued Data , 2017, SIAM J. Imaging Sci..

[11]  T. Pock,et al.  Second order total generalized variation (TGV) for MRI , 2011, Magnetic resonance in medicine.

[12]  D. L. Donoho,et al.  Compressed sensing , 2006, IEEE Trans. Inf. Theory.

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

[14]  Kristian Bredies,et al.  A TGV-Based Framework for Variational Image Decompression, Zooming, and Reconstruction. Part I: Analytics , 2015, SIAM J. Imaging Sci..

[15]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[16]  Tieyong Zeng,et al.  Low Rank Prior and Total Variation Regularization for Image Deblurring , 2016, Journal of Scientific Computing.

[17]  Emmanuel J. Candès,et al.  Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information , 2004, IEEE Transactions on Information Theory.

[18]  S. Osher,et al.  Decomposition of images by the anisotropic Rudin‐Osher‐Fatemi model , 2004 .

[19]  Justin K. Romberg,et al.  An Overview of Low-Rank Matrix Recovery From Incomplete Observations , 2016, IEEE Journal of Selected Topics in Signal Processing.

[20]  Pablo A. Parrilo,et al.  Rank-Sparsity Incoherence for Matrix Decomposition , 2009, SIAM J. Optim..

[21]  Yi Ma,et al.  Compressive principal component pursuit , 2012, 2012 IEEE International Symposium on Information Theory Proceedings.

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

[23]  El-hadi Zahzah,et al.  Robust Matrix Completion through Nonconvex Approaches and Efficient Algorithms , 2016 .

[24]  Dinggang Shen,et al.  Low-Rank Total Variation for Image Super-Resolution , 2013, MICCAI.

[25]  Tieyong Zeng,et al.  Image Deblurring Via Total Variation Based Structured Sparse Model Selection , 2016, J. Sci. Comput..

[26]  Thomas J. Asaki,et al.  A Variational Approach to Reconstructing Images Corrupted by Poisson Noise , 2007, Journal of Mathematical Imaging and Vision.

[27]  Tony F. Chan,et al.  Total variation blind deconvolution , 1998, IEEE Trans. Image Process..

[28]  Yuping Duan,et al.  Total Variation-Based Phase Retrieval for Poisson Noise Removal , 2018, SIAM J. Imaging Sci..

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

[30]  Ivan W. Selesnick,et al.  Improved sparse low-rank matrix estimation , 2016, Signal Process..

[31]  Alessandro Foi,et al.  Image Denoising by Sparse 3-D Transform-Domain Collaborative Filtering , 2007, IEEE Transactions on Image Processing.

[32]  Yonina C. Eldar,et al.  Simultaneously Structured Models With Application to Sparse and Low-Rank Matrices , 2012, IEEE Transactions on Information Theory.

[33]  Kristian Bredies,et al.  Total Generalized Variation in Diffusion Tensor Imaging , 2013, SIAM J. Imaging Sci..

[34]  Jack Xin,et al.  A Weighted Difference of Anisotropic and Isotropic Total Variation Model for Image Processing , 2015, SIAM J. Imaging Sci..

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

[36]  Emmanuel J. Candès,et al.  A Singular Value Thresholding Algorithm for Matrix Completion , 2008, SIAM J. Optim..

[37]  Rick Chartrand,et al.  Exact Reconstruction of Sparse Signals via Nonconvex Minimization , 2007, IEEE Signal Processing Letters.

[38]  Mila Nikolova,et al.  Minimizers of Cost-Functions Involving Nonsmooth Data-Fidelity Terms. Application to the Processing of Outliers , 2002, SIAM J. Numer. Anal..

[39]  Adam M. Oberman,et al.  Anisotropic Total Variation Regularized L^1-Approximation and Denoising/Deblurring of 2D Bar Codes , 2010, 1007.1035.

[40]  Xiaodong Li,et al.  Sparse Signal Recovery from Quadratic Measurements via Convex Programming , 2012, SIAM J. Math. Anal..

[41]  Chao Zeng,et al.  Non-Lipschitz Models for Image Restoration with Impulse Noise Removal , 2019, SIAM J. Imaging Sci..

[42]  Yi Ma,et al.  Robust principal component analysis? , 2009, JACM.

[43]  Emmanuel J. Candès,et al.  Exact Matrix Completion via Convex Optimization , 2009, Found. Comput. Math..

[44]  Stanley Osher,et al.  Image Super-Resolution by TV-Regularization and Bregman Iteration , 2008, J. Sci. Comput..

[45]  David J. Field,et al.  Emergence of simple-cell receptive field properties by learning a sparse code for natural images , 1996, Nature.

[46]  Xuelong Li,et al.  Fast and Accurate Matrix Completion via Truncated Nuclear Norm Regularization , 2013, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[47]  Tony F. Chan,et al.  Mathematical Models for Local Nontexture Inpaintings , 2002, SIAM J. Appl. Math..

[48]  Dorin Comaniciu,et al.  Total variation models for variable lighting face recognition , 2006, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[49]  Yiming Gao,et al.  Infimal Convolution of Oscillation Total Generalized Variation for the Recovery of Images with Structured Texture , 2017, SIAM J. Imaging Sci..