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

Xavier Bresson, Tony F. Chan

Research output: Contribution to journalArticlepeer-review

302 Scopus citations


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 regular- ization algorithm. Finally, and maybe more importantly, the proposed VTV minimization scheme can be easily extended to many standard applications. We apply this L1 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 de- composition, color image deblurring, and color denoising on manifolds. Gen- erally speaking, this VTV minimization scheme can be used in problems that required vector field (color, other feature vector) regularization while preserv- ing discontinuities.

Original languageEnglish (US)
Pages (from-to)455-484
Number of pages30
JournalInverse Problems and Imaging
Issue number4
StatePublished - 2008
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2008 American Institute of Mathematical Sciences.


  • BV space
  • Chromaticity-brightness color representation
  • Denoising on manifold
  • Dual formulation
  • Image deblurring
  • Image decomposition
  • Image denoising
  • Image inpainting
  • Inverse scale space
  • ROF model
  • Vector-valued TV norm
  • Wavelet shrinkage

ASJC Scopus subject areas

  • Analysis
  • Modeling and Simulation
  • Discrete Mathematics and Combinatorics
  • Control and Optimization


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