Regularized Non-local Total Variation and Application in Image Restoration

In the usual non-local variational models, such as the non-local total variations, the image is regularized by minimizing an energy term that penalizes gray-levels discrepancy between some specified pairs of pixels; a weight value is computed between these two pixels to penalize their dissimilarity. In this paper, we impose some regularity to those weight values. More precisely, we minimize a function involving a regularization term, analogous to an $$H^1$$H1 term, on weights. Doing so, the finite differences defining the image regularity depend on their environment. When the weights are difficult to define, they can be restored by the proposed stable regularization scheme. We provide all the details necessary for the implementation of a PALM algorithm with proved convergence. We illustrate the ability of the model to restore relevant unknown edges from the neighboring edges on an image inpainting problem. We also argue on inpainting, zooming and denoising problems that the model better recovers thin structures.

[1]  Guillermo Sapiro,et al.  Exemplar-Based Interpolation of Sparsely Sampled Images , 2009, EMMCVPR.

[2]  Jean-François Aujol,et al.  Stability of Over-Relaxations for the Forward-Backward Algorithm, Application to FISTA , 2015, SIAM J. Optim..

[3]  Abderrahim Elmoataz,et al.  Nonlocal Discrete Regularization on Weighted Graphs: A Framework for Image and Manifold Processing , 2008, IEEE Transactions on Image Processing.

[4]  Peyman Milanfar,et al.  Global Image Denoising , 2014, IEEE Transactions on Image Processing.

[5]  Émilie Chouzenoux,et al.  A block coordinate variable metric forward–backward algorithm , 2016, Journal of Global Optimization.

[6]  G. Kanizsa,et al.  Organization in Vision: Essays on Gestalt Perception , 1979 .

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

[8]  Guillermo Sapiro,et al.  A Variational Framework for Exemplar-Based Image Inpainting , 2011, International Journal of Computer Vision.

[9]  A. Chambolle.  Practical, Unified, Motion and Missing Data Treatment in Degraded Video , 2004, Journal of Mathematical Imaging and Vision.

[10]  Leonid P. Yaroslavsky,et al.  Digital Picture Processing , 1985 .

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

[12]  Joachim Weickert,et al.  Universität Des Saarlandes Fachrichtung 6.1 – Mathematik Generalised Nonlocal Image Smoothing Generalised Nonlocal Image Smoothing , 2022 .

[13]  Marc Levoy,et al.  Fast texture synthesis using tree-structured vector quantization , 2000, SIGGRAPH.

[14]  Jean-Michel Morel,et al.  Implementation of the "Non-Local Bayes" (NL-Bayes) Image Denoising Algorithm , 2013, Image Process. Line.

[15]  Jean-Michel Morel,et al.  A Nonlocal Bayesian Image Denoising Algorithm , 2013, SIAM J. Imaging Sci..

[16]  Yunmei Chen,et al.  An Accelerated Linearized Alternating Direction Method of Multipliers , 2014, SIAM J. Imaging Sci..

[17]  Lin He,et al.  Cahn--Hilliard Inpainting and a Generalization for Grayvalue Images , 2009, SIAM J. Imaging Sci..

[18]  Heinz H. Bauschke,et al.  Fixed-Point Algorithms for Inverse Problems in Science and Engineering , 2011, Springer Optimization and Its Applications.

[19]  Damien Garcia,et al.  Robust smoothing of gridded data in one and higher dimensions with missing values , 2010, Comput. Stat. Data Anal..

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

[21]  P. Bickel,et al.  Texture synthesis and nonparametric resampling of random fields , 2006, math/0611258.

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

[23]  Tony F. Chan,et al.  Nontexture Inpainting by Curvature-Driven Diffusions , 2001, J. Vis. Commun. Image Represent..

[24]  Gabriel Peyré,et al.  Sparse Modeling of Textures , 2009, Journal of Mathematical Imaging and Vision.

[25]  David Mumford,et al.  Filtering, Segmentation and Depth , 1993, Lecture Notes in Computer Science.

[26]  Pascal Getreuer,et al.  Enhancement and Recovery in Atomic Force Microscopy Images , 2012 .

[27]  Laurent Condat.  Fast projection onto the simplex and the l1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pmb {l}_\mathbf {1}$$\end{ , 2015, Mathematical Programming.

[28]  A. Bertozzi,et al.  Unconditionally stable schemes for higher order inpainting , 2011 .

[29]  Guillermo Sapiro,et al.  Filling-in by joint interpolation of vector fields and gray levels , 2001, IEEE Trans. Image Process..

[30]  Gabriele Facciolo,et al.  Variational Framework for Non-Local Inpainting , 2015, Image Process. Line.

[31]  L. P. I︠A︡roslavskiĭ.  Digital picture processing : an introduction , 1985 .

[32]  ANTONIN CHAMBOLLE,et al.  An Algorithm for Total Variation Minimization and Applications , 2004, Journal of Mathematical Imaging and Vision.

[33]  Marc Teboulle,et al.  Proximal alternating linearized minimization for nonconvex and nonsmooth problems , 2013, Mathematical Programming.

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

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

[36]  Guillermo Sapiro,et al.  A Variational Framework for Non-local Image Inpainting , 2009, EMMCVPR.

[37]  Yi Liu,et al.  A three-dimensional gap filling method for large geophysical datasets: Application to global satellite soil moisture observations , 2012, Environ. Model. Softw..

[38]  Saïd Ladjal,et al.  Exemplar-Based Inpainting from a Variational Point of View , 2010, SIAM J. Math. Anal..

[39]  Weiyu Xu,et al.  Block Iterative Reweighted Algorithms for Super-Resolution of Spectrally Sparse Signals , 2015, IEEE Signal Processing Letters.

[40]  Yurii Nesterov,et al.  Smooth minimization of non-smooth functions , 2005, Math. Program..

[41]  Laurent Condat,et al.  A Fast Projection onto the Simplex and the l 1 Ball , 2015 .

[42]  Peyman Milanfar,et al.  A General Framework for Regularized, Similarity-Based Image Restoration , 2014, IEEE Transactions on Image Processing.

[43]  J. Aujol,et al.  On the parameter choice for the Non-Local Means , 2010 .

[44]  Luminita A. Vese,et al.  Nonlocal Variational Image Deblurring Models in the Presence of Gaussian or Impulse Noise , 2009, SSVM.

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

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

[47]  Laurent D. Cohen,et al.  Non-local Regularization of Inverse Problems , 2008, ECCV.

[48]  Charles-Alban Deledalle,et al.  Non-local Methods with Shape-Adaptive Patches (NLM-SAP) , 2012, Journal of Mathematical Imaging and Vision.

[49]  Stanley Osher,et al.  Image Recovery via Nonlocal Operators , 2010, J. Sci. Comput..