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

Xiaojun Chen, Michael K. Ng, Chao Zhang

Research output: Journal article publicationJournal articleAcademic researchpeer-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 lp-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 - 26 Nov 2012


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

ASJC Scopus subject areas

  • Software
  • General Medicine
  • Computer Graphics and Computer-Aided Design


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