On ℓ1 Data Fitting and Concave Regularization for Image Recovery

Mila Nikolova, Michael K. Ng*, Chi Pan Tam

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

41 Citations (Scopus)
5 Downloads (Pure)


We propose a new family of cost functions for signal and image recovery: they are composed of ℓ1 data fitting terms combined with concave regularization. We exhibit when and how to employ such cost functions. Our theoretical results show that the minimizers of these cost functions are such that each one of their entries is involved either in an exact data fitting component or in a null component of the regularization part. This is a strong and particular property that can be useful for various image recovery problems. The minimization of such cost functions presents a computational challenge. We propose a fast minimization algorithm to solve this numerical problem. The experimental results show the effectiveness of the proposed algorithm. All illustrations and numerical experiments give a flavor of the possibilities offered by the minimizers of this new family of cost functions in solving specialized image processing tasks.

Original languageEnglish
Pages (from-to)A397-A430
Number of pages34
JournalSIAM Journal on Scientific Computing
Issue number1
Publication statusPublished - 24 Jan 2013

Scopus Subject Areas

  • Computational Mathematics
  • Applied Mathematics

User-Defined Keywords

  • ℓ1 data fitting
  • Continuation methods
  • Image recovery
  • Inverse problems
  • MRI
  • Multidimensional shrinkage
  • Nonsmooth and nonconvex analysis
  • Nonsmooth and nonconvex minimization
  • Penalty methods
  • Properties of minimizers
  • Regularization
  • Total variation
  • Variable-splitting
  • Variational methods


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