Advances in Computational Mathematics

, Volume 31, Issue 1, pp 87–113

Convergence analysis of tight framelet approach for missing data recovery

  • Jian-Feng Cai
  • Raymond H. Chan
  • Lixin Shen
  • Zuowei Shen
Article

DOI: 10.1007/s10444-008-9084-5

Cite this article as:
Cai, JF., Chan, R.H., Shen, L. et al. Adv Comput Math (2009) 31: 87. doi:10.1007/s10444-008-9084-5

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.

Keywords

Tight frameMissing dataInpaintingImpulse noise

Mathematics Subject Classifications (2000)

42C4065T6068U1094A08

Copyright information

© Springer Science+Business Media, LLC. 2008

Authors and Affiliations

  • Jian-Feng Cai
    • 1
  • Raymond H. Chan
    • 2
  • Lixin Shen
    • 3
  • Zuowei Shen
    • 1
  1. 1.Temasek Laboratories and Department of MathematicsNational University of SingaporeSingaporeSingapore
  2. 2.Department of MathematicsThe Chinese University of Hong KongHong KongPeople’s Republic of China
  3. 3.Department of MathematicsSyracuse UniversitySyracuseUSA