Directional Total Variation

This paper introduces a “directional total variation” (TV) where the gradients are weighted depending on their direction. The introduced directional TV has increased (and tunable) sensitivity to variations at a selected direction. In order to demonstrate the utility of the directional TV, we consider an image denoising formulation. This formulation requires the realization of the “proximal map” of the directional TV. Therefore, it is relevant for more general inverse problem settings as well. We derive an algorithm that solves the problem and use the algorithm to study the effects of the parameters of the directional TV.

[1]  Daniel Cremers,et al.  Introducing total curvature for image processing , 2011, 2011 International Conference on Computer Vision.

[2]  Antonin Chambolle,et al.  A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging , 2011, Journal of Mathematical Imaging and Vision.

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

[4]  Patrick L. Combettes,et al.  Proximal Splitting Methods in Signal Processing , 2009, Fixed-Point Algorithms for Inverse Problems in Science and Engineering.

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

[6]  José M. Bioucas-Dias,et al.  A New TwIST: Two-Step Iterative Shrinkage/Thresholding Algorithms for Image Restoration , 2007, IEEE Transactions on Image Processing.

[7]  Antonin Chambolle,et al.  Total Variation Minimization and a Class of Binary MRF Models , 2005, EMMCVPR.

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

[9]  Anhua Lin,et al.  On the Distance between Two Ellipsoids , 2002, SIAM J. Optim..

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

[11]  Benjamin Berkels,et al.  Cartoon Extraction Based on Anisotropic Image Classification Vision , Modeling , and Visualization Proceedings , 2006 .

[12]  M. Burger,et al.  Joint denoising and anisotropy estimation: original image, anisotropic cartoon and estimated orientation. Cartoon Extraction Based on Anisotropic Image Classification , 2006 .

[13]  Patrick L. Combettes,et al.  Signal Recovery by Proximal Forward-Backward Splitting , 2005, Multiscale Model. Simul..

[14]  A. Chambolle.  An Algorithm for Total Variation Minimization and Applications , 2004, Journal of Mathematical Imaging and Vision.

[15]  N. Sochen.  Affine Invariant Flows in the Beltrami Framework , 2004, Journal of Mathematical Imaging and Vision.