A Low Patch-Rank Interpretation of Texture

We propose a novel cartoon-texture separation model using a sparse low-rank decomposition. Our texture model connects the separate ideas of robust principal component analysis (PCA) [E. J. Candes, X. Li, Y. Ma, and J. Wright, J. ACM, 58 (2011), 11], nonlocal methods [A. Buades, B. Coll, and J.-M. Morel, Multiscale Model. Simul., 4 (2005), pp. 490--530], [A. Buades, B. Coll, and J.-M. Morel, Numer. Math., 105 (2006), pp. 1--34], [G. Gilboa and S. Osher, Multiscale Model. Simul., 6 (2007), pp. 595--630], [G. Gilboa and S. Osher, Multiscale Model. Simul., 7 (2008), pp. 1005--1028], and cartoon-texture decompositions in an interesting way, taking advantage of each of these methodologies. We define our texture norm using the nuclear norm applied to patches in the image, interpreting the texture patches to be low-rank. In particular, this norm is easier to implement than many of the weak function space norms in the literature and is computationally faster than nonlocal methods since there is no explicit weight ...

[1]  Wotao Yin,et al.  An Iterative Regularization Method for Total Variation-Based Image Restoration , 2005, Multiscale Model. Simul..

[2]  Guy Gilboa,et al.  Nonlocal Operators with Applications to Image Processing , 2008, Multiscale Model. Simul..

[3]  Guillermo Sapiro,et al.  Simultaneous structure and texture image inpainting , 2003, 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2003. Proceedings..

[4]  L. Lieu,et al.  Image Restoration and Decomposition via Bounded Total Variation and Negative Hilbert-Sobolev Spaces , 2008 .

[5]  Guillermo Sapiro,et al.  Simultaneous structure and texture image inpainting , 2003, IEEE Trans. Image Process..

[6]  Guillermo Sapiro,et al.  Non-local sparse models for image restoration , 2009, 2009 IEEE 12th International Conference on Computer Vision.

[7]  S. Osher,et al.  Multi-energy CT based on a prior rank, intensity and sparsity model (PRISM) , 2011, Inverse problems.

[8]  Guillermo Sapiro,et al.  Non-Parametric Bayesian Dictionary Learning for Sparse Image Representations , 2009, NIPS.

[9]  Guillermo Sapiro,et al.  Online Learning for Matrix Factorization and Sparse Coding , 2009, J. Mach. Learn. Res..

[10]  Stanley Osher,et al.  Image Decomposition and Restoration Using Total Variation Minimization and the H1 , 2003, Multiscale Model. Simul..

[11]  Hongkai Zhao,et al.  Robust principal component analysis-based four-dimensional computed tomography , 2011, Physics in medicine and biology.

[12]  Jean-François Aujol,et al.  Color image decomposition and restoration , 2006, J. Vis. Commun. Image Represent..

[13]  S. Osher,et al.  Fast Singular Value Thresholding without Singular Value Decomposition , 2013 .

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

[15]  Lei Zhang,et al.  Sparsity-based image denoising via dictionary learning and structural clustering , 2011, CVPR 2011.

[16]  Michael Elad,et al.  Sparse Representation for Color Image Restoration , 2008, IEEE Transactions on Image Processing.

[17]  Antonin Chambolle,et al.  Image Decomposition into a Bounded Variation Component and an Oscillating Component , 2005, Journal of Mathematical Imaging and Vision.

[18]  Jean-François Aujol.  Contribution à l'analyse de textures en traitement d'images par méthodes variationnelles et équations aux dérivées partielles , 2004 .

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

[20]  S. Osher,et al.  IMAGE DECOMPOSITION AND RESTORATION USING TOTAL VARIATION MINIMIZATION AND THE H−1 NORM∗ , 2002 .

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

[22]  Jean-Michel Morel,et al.  Neighborhood filters and PDE’s , 2006, Numerische Mathematik.

[23]  Cordelia Schmid,et al.  A sparse texture representation using local affine regions , 2005, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[24]  Tony F. Chan,et al.  Combining geometrical and textured information to perform image classification , 2006, J. Vis. Commun. Image Represent..

[25]  Zuowei Shen,et al.  Robust video denoising using low rank matrix completion , 2010, 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[26]  Guy Gilboa,et al.  Nonlocal Linear Image Regularization and Supervised Segmentation , 2007, Multiscale Model. Simul..

[27]  John Wright,et al.  RASL: Robust alignment by sparse and low-rank decomposition for linearly correlated images , 2010, 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[28]  Stanley Osher,et al.  Modeling Textures with Total Variation Minimization and Oscillating Patterns in Image Processing , 2003, J. Sci. Comput..

[29]  Jean-Michel Morel,et al.  A Review of Image Denoising Algorithms, with a New One , 2005, Multiscale Model. Simul..

[30]  Jérôme Gilles,et al.  Noisy Image Decomposition: A New Structure, Texture and Noise Model Based on Local Adaptivity , 2007, Journal of Mathematical Imaging and Vision.

[31]  Michael Elad,et al.  Image Denoising Via Sparse and Redundant Representations Over Learned Dictionaries , 2006, IEEE Transactions on Image Processing.

[32]  L. Vese,et al.  IMAGE RECOVERY USING FUNCTIONS OF BOUNDED VARIATION AND SOBOLEV SPACES OF NEGATIVE DIFFERENTIABILITY , 2009 .

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

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

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

[36]  Yi Ma,et al.  TILT: Transform Invariant Low-Rank Textures , 2010, ACCV.

[37]  Yves Meyer,et al.  Properties of $BV-G$ Structures $+$ Textures Decomposition Models. Application to Road Detection in Satellite Images , 2010, IEEE Transactions on Image Processing.