Convergence analysis of tight framelet approach for missing data recovery

Jian Feng Cai, Raymond H. Chan, Lixin Shen, Zuowei Shen

Research output: Contribution to journalArticle

58 Scopus citations

Abstract

How to recover missing data from an incomplete samples is a fundamental problem in mathematics and it has wide range of applications in image analysis and processing. Although many existing methods, e.g. various data smoothing methods and PDE approaches, are available in the literature, there is always a need to find new methods leading to the best solution according to various cost functionals. In this paper, we propose an iterative algorithm based on tight framelets for image recovery from incomplete observed data. The algorithm is motivated from our framelet algorithm used in high-resolution image reconstruction and it exploits the redundance in tight framelet systems. We prove the convergence of the algorithm and also give its convergence factor. Furthermore, we derive the minimization properties of the algorithm and explore the roles of the redundancy of tight framelet systems. As an illustration of the effectiveness of the algorithm, we give an application of it in impulse noise removal.

Original languageEnglish (US)
Pages (from-to)87-113
Number of pages27
JournalAdvances in Computational Mathematics
Volume31
Issue number1-3
DOIs
StatePublished - Oct 1 2009

Keywords

  • Impulse noise
  • Inpainting
  • Missing data
  • Tight frame

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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