Non-Lipschitz lp-regularization and box constrained model for image restoration

Xiaojun Chen*, Kwok Po Ng, Chao Zhang

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

96 Citations (Scopus)


Nonsmooth nonconvex regularization has remarkable advantages for the restoration of piecewise constant images. Constrained optimization can improve the image restoration using a priori information. In this paper, we study regularized nonsmooth nonconvex minimization with box constraints for image restoration. We present a computable positive constant θ for using nonconvex nonsmooth regularization, and show that the difference between each pixel and its four adjacent neighbors is either 0 or larger than θ in the recovered image. Moreover, we give an explicit form of θ for the box-constrained image restoration model with the non-Lipschitz nonconvex l p-norm (0<p<1) regularization. Our theoretical results show that any local minimizer of this imaging restoration problem is composed of constant regions surrounded by closed contours and edges. Numerical examples are presented to validate the theoretical results, and show that the proposed model can recover image restoration results very well.

Original languageEnglish
Article number6307860
Pages (from-to)4709-4721
Number of pages13
JournalIEEE Transactions on Image Processing
Issue number12
Publication statusPublished - Dec 2012

Scopus Subject Areas

  • Software
  • Computer Graphics and Computer-Aided Design

User-Defined Keywords

  • Box constraints
  • image restoration
  • non-Lipschitz
  • nonsmooth and nonconvex
  • regularization


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