Color Bregman TV

In this paper we present a novel iterative procedure for multichannel image and data reconstruction using Bregman distances. The motivation for our approach is that in many applications multiple channels share a common subgradient with respect to a suitable regularization. This implies desirable properties such as a common edge set (and a common direction of the normals to the level lines) in the case of the total variation (TV). Therefore, we propose to determine each iterate by regularizing each channel with a weighted linear combination of Bregman distances to all other image channels from the previous iteration. In this sense we generalize the Bregman iteration proposed by Osher et al. in [Multiscale Model. Simul., 4 (2005), pp. 460--489] to multichannel images. We prove the convergence of the proposed scheme, analyze stationary points, and present numerical experiments on color image denoising, which show the superior behavior of our approach in comparison to TV, TV with Bregman iterations on each ch...

[1]  Jahn Müller,et al.  Total Variation Processing of Images with Poisson Statistics , 2009, CAIP.

[2]  Laura Igual,et al.  A Variational Model for P+XS Image Fusion , 2006, International Journal of Computer Vision.

[3]  Jahn Müller,et al.  Higher-Order TV Methods—Enhancement via Bregman Iteration , 2012, Journal of Scientific Computing.

[4]  Stanley Osher,et al.  Iterative Regularization and Nonlinear Inverse Scale Space Applied to Wavelet-Based Denoising , 2007, IEEE Transactions on Image Processing.

[5]  Daniel Cremers,et al.  An approach to vectorial total variation based on geometric measure theory , 2010, 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

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

[7]  Michael Möller,et al.  A Variational Approach for Sharpening High Dimensional Images , 2012, SIAM J. Imaging Sci..

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

[9]  Kannappan Palaniappan,et al.  Color image denoising by chromatic edges based vector valued diffusion , 2013, ArXiv.

[10]  Xiaoyi Jiang,et al.  A Variational Framework for Region-Based Segmentation Incorporating Physical Noise Models , 2013, Journal of Mathematical Imaging and Vision.

[11]  Stanley Osher,et al.  Image Super-Resolution by TV-Regularization and Bregman Iteration , 2008, J. Sci. Comput..

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

[13]  Martin Burger,et al.  Ground States and Singular Vectors of Convex Variational Regularization Methods , 2012, 1211.2057.

[14]  Tony F. Chan,et al.  Image processing and analysis , 2005 .

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

[16]  O. Scherzer,et al.  Error estimates for non-quadratic regularization and the relation to enhancement , 2006 .

[17]  Lin He,et al.  Error estimation for Bregman iterations and inverse scale space methods in image restoration , 2007, Computing.

[18]  Simon R. Arridge,et al.  Vector-Valued Image Processing by Parallel Level Sets , 2014, IEEE Transactions on Image Processing.

[19]  Laurent Condat,et al.  Joint demosaicking and denoising by total variation minimization , 2012, 2012 19th IEEE International Conference on Image Processing.

[20]  Gilles Aubert,et al.  A Variational Approach to Removing Multiplicative Noise , 2008, SIAM J. Appl. Math..

[21]  Yonina C. Eldar,et al.  C-HiLasso: A Collaborative Hierarchical Sparse Modeling Framework , 2010, IEEE Transactions on Signal Processing.

[22]  T. Chan,et al.  Fast dual minimization of the vectorial total variation norm and applications to color image processing , 2008 .

[23]  L. Bregman The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming , 1967 .

[24]  S. Bianchini,et al.  A decomposition theorem for $BV$ functions , 2011 .

[25]  L. Ambrosio,et al.  Functions of Bounded Variation and Free Discontinuity Problems , 2000 .

[26]  Jian-Feng Cai,et al.  Linearized Bregman iterations for compressed sensing , 2009, Math. Comput..

[27]  Thomas J. Asaki,et al.  A Variational Approach to Reconstructing Images Corrupted by Poisson Noise , 2007, Journal of Mathematical Imaging and Vision.

[28]  S. Osher,et al.  Nonlinear inverse scale space methods , 2006 .

[29]  Martin Burger,et al.  Primal and Dual Bregman Methods with Application to Optical Nanoscopy , 2011, International Journal of Computer Vision.

[30]  I. Daubechies,et al.  Framelets: MRA-based constructions of wavelet frames☆☆☆ , 2003 .

[31]  Wotao Yin,et al.  Analysis and Generalizations of the Linearized Bregman Method , 2010, SIAM J. Imaging Sci..

[32]  Tony F. Chan,et al.  Color TV: total variation methods for restoration of vector-valued images , 1998, IEEE Trans. Image Process..

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

[34]  K. Deimling Nonlinear functional analysis , 1985 .

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