Automated Regularization Parameter Selection in Multi-Scale Total Variation Models for Image Restoration

Multi-scale total variation models for image restoration are introduced. The models utilize a spatially dependent regularization parameter in order to enhance image regions containing details while still sufficiently smoothing homogeneous features. The fully automated adjustment strategy of the regularization parameter is based on local variance estimators. For robustness reasons, the decision on the acceptance or rejection of a local parameter value relies on a confidence interval technique based on the expected maximal local variance estimate. In order to improve the performance of the initial algorithm a generalized hierarchical decomposition of the restored image is used. The corresponding subproblems are solved by a superlinearly convergent algorithm based on Fenchel-duality and inexact semismooth Newton techniques. The paper ends by a report on numerical tests, a qualitative study of the proposed adjustment scheme and a comparison with popular total variation based restoration methods.

[1]  J. Wolfowitz,et al.  An Introduction to the Theory of Statistics , 1951, Nature.

[2]  E. Gumbel,et al.  Les valeurs extrêmes des distributions statistiques , 1935 .

[3]  I. Ekeland,et al.  Convex analysis and variational problems , 1976 .

[4]  E. Giusti.  Minimal surfaces and functions of bounded variation , 1977 .

[5]  B. R. Hunt,et al.  Digital Image Restoration , 1977 .

[6]  Sidney C. Port,et al.  Probability, Random Variables, and Stochastic Processes—Second Edition (Athanasios Papoulis) , 1986 .

[7]  C. Vogel.  Computational Methods for Inverse Problems , 1987 .

[8]  F. Almgren.  Review: Enrico Giusti, Minimal surfaces and functions of bounded variation , 1987 .

[9]  L. Modica,et al.  Partial Differential Equations and the Calculus of Variations , 1989 .

[10]  Nikolas P. Galatsanos,et al.  Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation , 1992, IEEE Trans. Image Process..

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

[12]  C. Vogel,et al.  Analysis of bounded variation penalty methods for ill-posed problems , 1994 .

[13]  Tony F. Chan,et al.  Spatially and Scale Adaptive Total Variation Based Regularization and Anisotropic Diiusion in Image Processing , 1996 .

[14]  D. Dobson,et al.  Convergence of an Iterative Method for Total Variation Denoising , 1997 .

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

[16]  Aggelos K. Katsaggelos,et al.  Digital image restoration , 2012, IEEE Signal Process. Mag..

[17]  Jerry D. Gibson,et al.  Handbook of Image and Video Processing , 2000 .

[18]  Gjlles Aubert,et al.  Mathematical problems in image processing , 2001 .

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

[20]  I-Liang Chern,et al.  Acceleration Methods for Total Variation-Based Image Denoising , 2003, SIAM J. Sci. Comput..

[21]  T. Chan,et al.  Edge-preserving and scale-dependent properties of total variation regularization , 2003 .

[22]  N. Sochen,et al.  Texture Preserving Variational Denoising Using an Adaptive Fidelity Term , 2003 .

[23]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[24]  Bernard Rougé,et al.  TV Based Image Restoration with Local Constraints , 2003, J. Sci. Comput..

[25]  Rachid Deriche,et al.  Regularizing Flows for Constrained Matrix-Valued Images , 2004 .

[26]  Karl Kunisch,et al.  Total Bounded Variation Regularization as a Bilaterally Constrained Optimization Problem , 2004, SIAM J. Appl. Math..

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

[28]  Eitan Tadmor,et al.  A Multiscale Image Representation Using Hierarchical (BV, L2 ) Decompositions , 2004, Multiscale Model. Simul..

[29]  Michael Hintermüller,et al.  A Second Order Shape Optimization Approach for Image Segmentation , 2004, SIAM J. Appl. Math..

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

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

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

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

[34]  Tony F. Chan,et al.  Scale Recognition, Regularization Parameter Selection, and Meyer's G Norm in Total Variation Regularization , 2006, Multiscale Model. Simul..

[35]  Michael Hintermüller,et al.  An Infeasible Primal-Dual Algorithm for Total Bounded Variation-Based Inf-Convolution-Type Image Restoration , 2006, SIAM J. Sci. Comput..

[36]  Ivan P. Gavrilyuk,et al.  Variational analysis in Sobolev and BV spaces , 2007, Math. Comput..

[37]  I. Gavrilyuk.  Book Review: Variational analysis in Sobolev and BV spaces , 2007 .

[38]  M. Burger,et al.  Asymptotic analysis of an advection-dominated chemotaxis model in multiple spatial dimensions , 2008 .

[39]  L. Vese,et al.  MULTISCALE HIERARCHICAL DECOMPOSITION OF IMAGES WITH APPLICATIONS TO DEBLURRING, DENOISING AND SEGMENTATION ∗ , 2008 .

[40]  Andrés Almansa,et al.  A TV Based Restoration Model with Local Constraints , 2008, J. Sci. Comput..

[41]  Jean-François Aujol,et al.  Irregular to Regular Sampling, Denoising, and Deconvolution , 2009, Multiscale Model. Simul..