A Globally Convergent Algorithm for a Constrained Non-Lipschitz Image Restoration Model

Weina Wang, Chunlin Wu*, Xue Cheng Tai

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

12 Citations (Scopus)


In this paper, we study a non-Lipschitz and box-constrained model for both piecewise constant and natural image restoration with Gaussian noise removal. It consists of non-Lipschitz isotropic first-order ℓp (0 < p< 1) and second-order ℓ1 as regularization terms to keep edges and overcome staircase effects in smooth regions simultaneously. The model thus combines the advantages of non-Lipschitz and high order regularization, as well as box constraints. Although this model is quite complicated to analyze, we establish a motivating theorem, which induces an iterative support shrinking algorithm with proximal linearization. This algorithm can be easily implemented and is globally convergent. In the convergence analysis, a key step is to prove a lower bound theory for the nonzero entries of the gradient of the iterative sequence. This theory also provides a theoretical guarantee for the edge preserving property of the algorithm. Extensive numerical experiments and comparisons indicate that our method is very effective for both piecewise constant and natural image restoration.

Original languageEnglish
Article number14
Number of pages29
JournalJournal of Scientific Computing
Issue number1
Publication statusPublished - 1 Apr 2020

Scopus Subject Areas

  • Software
  • Theoretical Computer Science
  • Numerical Analysis
  • Engineering(all)
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics

User-Defined Keywords

  • Box-constrained
  • High order regularization
  • Image restoration
  • Kurdyka–Łojasiewicz property
  • Lower bound theory
  • Non-Lipschitz optimization
  • Support shrinking


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