Fast dual minimization of the vectorial total variation norm and applications to color image processing

Xavier Bresson; Tony F. Chan

doi:10.3934/ipi.2008.2.455

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2008, Volume 2, Issue 4: 455-484. Doi: 10.3934/ipi.2008.2.455

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Fast dual minimization of the vectorial total variation norm and applications to color image processing

Xavier Bresson^1, and
Tony F. Chan^1,

1.
Department of Mathematics, University of California, Los Angeles, CA 90095-1555

Received: June 30, 2008

Revised: September 30, 2008

Published: October 2008

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Abstract

We propose a regularization algorithm for color/vectorial images which is fast, easy to code and mathematically well-posed. More precisely, the regularization model is based on the dual formulation of the vectorial Total Variation (VTV) norm and it may be regarded as the vectorial extension of the dual approach defined by Chambolle in [13] for gray-scale/scalar images. The proposed model offers several advantages. First, it minimizes the exact VTV norm whereas standard approaches use a regularized norm. Then, the numerical scheme of minimization is straightforward to implement and finally, the number of iterations to reach the solution is low, which gives a fast regularization algorithm. Finally, and maybe more importantly, the proposed VTV minimization scheme can be easily extended to many standard applications. We apply this $L^1$ vectorial regularization algorithm to the following problems: color inverse scale space, color denoising with the chromaticity-brightness color representation, color image inpainting, color wavelet shrinkage, color image decomposition, color image deblurring, and color denoising on manifolds. Generally speaking, this VTV minimization scheme can be used in problems that required vector field (color, other feature vector) regularization while preserving discontinuities.

Keywords:

Vector-valued TV norm,
dual formulation,
BV space,
image denoising,
ROF model,
inverse scale space,
chromaticity-brightness color representation,
image decomposition,
image inpainting,
image deblurring,
wavelet shrinkage,
denoising on manifold.

Mathematics Subject Classification: 68U10, 65K10.

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