Image restoration: A general wavelet frame based model and its asymptotic analysis

Bin Dong, Zuowei Shen, Peichu Xie

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


Image restoration is one of the most important areas in imaging science. Mathematical tools have been widely used in image restoration, where the wavelet frame based approach is one of the successful examples. In this paper, we introduce a generic wavelet frame based image restoration model, called the "general model," which includes most of the existing wavelet frame based models as special cases. Moreover, the general model also includes examples that are new to the literature. Motivated by our earlier studies [R. Adams, Sobolev Spaces, Academic Press, New York, 1975; D. Bertsekas and J. Tsitsiklis, Parallel and Distributed Computation: Numerical Methods, Prentice-Hall, New York, 1989; K. Bredies, K. Kunisch, and T. Pock, SIAM J. Imaging Sci., 3 (2010), p. 492], we provide an asymptotic analysis of the general model as image resolution goes to infinity, which establishes a connection between the general model in discrete setting and a new variatonal model in continuum setting. The variational model also includes some of the existing variational models as special cases, such as the total generalized variational model proposed by [J. Cai, R. Chan, L. Shen, and Z. Shen, Advances in Computational Mathematics, 31 (2009), pp. 87-113].

Original languageEnglish (US)
Pages (from-to)421-445
Number of pages25
JournalSIAM Journal on Mathematical Analysis
Issue number1
StatePublished - 2017


  • (tight) wavelet frames
  • Framelets
  • Image restoration
  • Variational method
  • Γ-convergence

ASJC Scopus subject areas

  • Analysis
  • Computational Mathematics
  • Applied Mathematics


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