Proximity algorithms for the L1/TV image denoising model

Charles A. Micchelli, Lixin Shen, Yuesheng Xu, Xueying Zeng

Research output: Contribution to journalArticlepeer-review

78 Scopus citations


This paper introduces a proximity operator framework for studying the L1/TV image denoising model which minimizes the sum of a data fidelity term measured in the ℓ1-norm and the total-variation regularization term. Both terms in the model are non-differentiable. This causes algorithmic difficulties for its numerical treatment. To overcome the difficulties, we formulate the total-variation as a composition of a convex function (the ℓ1-norm or the ℓ2-norm) and the first order difference operator, and then express the solution of the model in terms of the proximity operator of the composition. By developing a "chain rule" for the proximity operator of the composition, we identify the solution as fixed point of a nonlinear mapping expressed in terms of the proximity operator of the ℓ1-norm or the ℓ2-norm, each of which is explicitly given. This formulation naturally leads to fixed-point algorithms for the numerical treatment of the model. We propose an alternative model by replacing the non-differentiable convex function in the formulation of the total variation with its differentiable Moreau envelope and develop corresponding fixed-point algorithms for solving the new model. When partial information of the underlying image is available, we modify the model by adding an indicator function to the minimization functional and derive its corresponding fixed-point algorithms. Numerical experiments are conducted to test the approximation accuracy and computational efficiency of the proposed algorithms. Also, we provide a comparison of our approach to two state-of-the-art algorithms available in the literature. Numerical results confirm that our algorithms perform favorably, in terms of PSNR-values and CPU-time, in comparison to the two algorithms.

Original languageEnglish (US)
Pages (from-to)401-426
Number of pages26
JournalAdvances in Computational Mathematics
Issue number2
StatePublished - Feb 2013


  • Inpainting
  • L1/TV model
  • Moreau envelope
  • Proximity operator

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics


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