Tensor Nuclear Norm-Based Low-Rank Approximation With Total Variation Regularization

Some existing low-rank approximation approaches either need to predefine the rank values (such as the matrix/tensor factorization-based methods) or fail to consider local information of data (e.g., spatial or spectral smooth structure). To overcome these drawbacks, this paper proposes a new model called the tensor nuclear norm-based low-rank approximation with total variation regularization (TLR-TV) for color and multispectral image denoising. TLR-TV uses the tensor nuclear norm to encode the global low-rank prior of tensor data and the total variation regularization to preserve the spatial-spectral continuity in a unified framework. Including the hyper total variation (HTV) and spatial-spectral total variation (SSTV), we propose two TLR-TV-based algorithms, namely TLR-HTV and TLR-SSTV. Using the alternating direction method of multiplier, we further propose two simple algorithms to solve TLR-HTV and TLR-SSTV. Extensive experiments on simulated and real-world noisy images demonstrate that the proposed TLR-HTV and TLR-SSTV outperform the state-of-the-art methods in color and multispectral image denoising in terms of quantitative and qualitative evaluations.

[1]  Yongli Wang,et al.  Augmented Lagrangian alternating direction method for low-rank minimization via non-convex approximation , 2017, Signal Image Video Process..

[2]  Misha Elena Kilmer,et al.  Novel Methods for Multilinear Data Completion and De-noising Based on Tensor-SVD , 2014, 2014 IEEE Conference on Computer Vision and Pattern Recognition.

[3]  John Wright,et al.  Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Matrices via Convex Optimization , 2009, NIPS.

[4]  Liangpei Zhang,et al.  Hyperspectral Image Restoration Using Low-Rank Matrix Recovery , 2014, IEEE Transactions on Geoscience and Remote Sensing.

[5]  Angshul Majumdar,et al.  Hyperspectral Image Denoising Using Spatio-Spectral Total Variation , 2016, IEEE Geoscience and Remote Sensing Letters.

[6]  Dacheng Tao,et al.  GoDec: Randomized Lowrank & Sparse Matrix Decomposition in Noisy Case , 2011, ICML.

[7]  Yongli Wang,et al.  Robust PCA Using Matrix Factorization for Background/Foreground Separation , 2018, IEEE Access.

[8]  M. Kilmer,et al.  Factorization strategies for third-order tensors , 2011 .

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

[10]  Soon Ki Jung,et al.  Decomposition into Low-rank plus Additive Matrices for Background/Foreground Separation: A Review for a Comparative Evaluation with a Large-Scale Dataset , 2015, Comput. Sci. Rev..

[11]  R. Courant.  Variational methods for the solution of problems of equilibrium and vibrations , 1943 .

[12]  Tamara G. Kolda,et al.  Tensor Decompositions and Applications , 2009, SIAM Rev..

[13]  Jiangjun Peng,et al.  Hyperspectral Image Restoration Via Total Variation Regularized Low-Rank Tensor Decomposition , 2017, IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing.

[14]  Jun Huang,et al.  Hyperspectral image denoising using the robust low-rank tensor recovery. , 2015, Journal of the Optical Society of America. A, Optics, image science, and vision.

[15]  D. Shen,et al.  LRTV: MR Image Super-Resolution With Low-Rank and Total Variation Regularizations , 2015, IEEE Transactions on Medical Imaging.

[16]  Donald Goldfarb,et al.  Robust Low-Rank Tensor Recovery: Models and Algorithms , 2013, SIAM J. Matrix Anal. Appl..

[17]  Larry S. Davis,et al.  Learning Structured Low-Rank Representations for Image Classification , 2013, 2013 IEEE Conference on Computer Vision and Pattern Recognition.

[18]  Eric L. Miller,et al.  Tensor-Based Formulation and Nuclear Norm Regularization for Multienergy Computed Tomography , 2013, IEEE Transactions on Image Processing.

[19]  Wensheng Zhang,et al.  The Twist Tensor Nuclear Norm for Video Completion , 2017, IEEE Transactions on Neural Networks and Learning Systems.

[20]  Pan Zhou,et al.  Tensor Factorization for Low-Rank Tensor Completion , 2018, IEEE Transactions on Image Processing.

[21]  Lei Zhang,et al.  Weighted Nuclear Norm Minimization and Its Applications to Low Level Vision , 2016, International Journal of Computer Vision.

[22]  Hongyu Zhao,et al.  Low-Rank Modeling and Its Applications in Image Analysis , 2014, ACM Comput. Surv..

[23]  Lei Zhang,et al.  Multispectral Images Denoising by Intrinsic Tensor Sparsity Regularization , 2016, 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR).

[24]  Wei Liu,et al.  Tensor Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Tensors via Convex Optimization , 2016, 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR).

[25]  Zhao Kang,et al.  Robust PCA Via Nonconvex Rank Approximation , 2015, 2015 IEEE International Conference on Data Mining.

[26]  Yin Zhang,et al.  Solving a low-rank factorization model for matrix completion by a nonlinear successive over-relaxation algorithm , 2012, Mathematical Programming Computation.

[27]  Liangpei Zhang,et al.  Total-Variation-Regularized Low-Rank Matrix Factorization for Hyperspectral Image Restoration , 2016, IEEE Transactions on Geoscience and Remote Sensing.

[28]  Dong Wang,et al.  Denoising of Hyperspectral Images Using Nonconvex Low Rank Matrix Approximation , 2017, IEEE Transactions on Geoscience and Remote Sensing.

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

[30]  Yunming Ye,et al.  Low-Rank Tensor Completion with Total Variation for Visual Data Inpainting , 2017, AAAI.

[31]  Yi Ma,et al.  The Augmented Lagrange Multiplier Method for Exact Recovery of Corrupted Low-Rank Matrices , 2010, Journal of structural biology.

[32]  Hidekata Hontani,et al.  Simultaneous Visual Data Completion and Denoising Based on Tensor Rank and Total Variation Minimization and Its Primal-Dual Splitting Algorithm , 2017, 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR).

[33]  G. Sapiro,et al.  A collaborative framework for 3D alignment and classification of heterogeneous subvolumes in cryo-electron tomography. , 2013, Journal of structural biology.

[34]  Lucien Wald,et al.  Data Fusion. Definitions and Architectures - Fusion of Images of Different Spatial Resolutions , 2002 .

[35]  R. Tibshirani.  Regression Shrinkage and Selection via the Lasso , 1996 .

[36]  Yuan Xie,et al.  Hyperspectral Image Restoration via Iteratively Regularized Weighted Schatten $p$-Norm Minimization , 2016, IEEE Transactions on Geoscience and Remote Sensing.

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

[38]  Liangpei Zhang,et al.  Hyperspectral Image Denoising via Noise-Adjusted Iterative Low-Rank Matrix Approximation , 2015, IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing.

[39]  Jean-Michel Morel,et al.  Multiscale Image Blind Denoising , 2015, IEEE Transactions on Image Processing.

[40]  L. Tucker,et al.  Some mathematical notes on three-mode factor analysis , 1966, Psychometrika.

[41]  Stephen P. Boyd,et al.  Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers , 2011, Found. Trends Mach. Learn..

[42]  Hanghang Tong,et al.  Robust bilinear factorization with missing and grossly corrupted observations , 2015, Inf. Sci..

[43]  Jieping Ye,et al.  Tensor Completion for Estimating Missing Values in Visual Data , 2009, IEEE Transactions on Pattern Analysis and Machine Intelligence.

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

[45]  Wotao Yin,et al.  Parallel matrix factorization for low-rank tensor completion , 2013, ArXiv.

[46]  Marc Teboulle,et al.  Fast Gradient-Based Algorithms for Constrained Total Variation Image Denoising and Deblurring Problems , 2009, IEEE Transactions on Image Processing.

[47]  Ying Wu,et al.  A unified approach to salient object detection via low rank matrix recovery , 2012, 2012 IEEE Conference on Computer Vision and Pattern Recognition.

[48]  John Wright,et al.  Provable Low-Rank Tensor Recovery , 2014 .