, Volume 31, Issue 1-3, pp 87-113

Convergence analysis of tight framelet approach for missing data recovery

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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.

Communicated by Yuesheng Xu.
The research was supported by US National Science Foundation under grant DMS-0712827, and in part by HKRGC Grant 400505, CUHK DAG 2060257, and Grant R-146-000-060-112 at the National University of Singapore.