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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Image restoration: Total variation, wavelet frames, and beyond
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by Jian-Feng Cai, Bin Dong, Stanley Osher and Zuowei Shen PDF
J. Amer. Math. Soc. 25 (2012), 1033-1089 Request permission

Abstract:

The variational techniques (e.g. the total variation based method) are well established and effective for image restoration, as well as many other applications, while the wavelet frame based approach is relatively new and came from a different school. This paper is designed to establish a connection between these two major approaches for image restoration. The main result of this paper shows that when spline wavelet frames of are used, a special model of a wavelet frame method, called the analysis based approach, can be viewed as a discrete approximation at a given resolution to variational methods. A convergence analysis as image resolution increases is given in terms of objective functionals and their approximate minimizers. This analysis goes beyond the establishment of the connections between these two approaches, since it leads to new understandings for both approaches. First, it provides geometric interpretations to the wavelet frame based approach as well as its solutions. On the other hand, for any given variational model, wavelet frame based approaches provide various and flexible discretizations which immediately lead to fast numerical algorithms for both wavelet frame based approaches and the corresponding variational model. Furthermore, the built-in multiresolution structure of wavelet frames can be utilized to adaptively choose proper differential operators in different regions of a given image according to the order of the singularity of the underlying solutions. This is important when multiple orders of differential operators are used in various models that generalize the total variation based method. These observations will enable us to design new methods according to the problems at hand, hence, lead to wider applications of both the variational and wavelet frame based approaches. Links of wavelet frame based approaches to some more general variational methods developed recently will also be discussed.
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Additional Information
  • Jian-Feng Cai
  • Affiliation: Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242-1419
  • Email: jianfeng-cai@uiowa.edu
  • Bin Dong
  • Affiliation: Department of Mathematics, The University of Arizona, 617 North Santa Rita Avenue, Tucson, Arizona 85721-0089
  • Email: dongbin@math.arizona.edu
  • Stanley Osher
  • Affiliation: Department of Mathematics, University of California, Los Angeles, 405 Hilgard Avenue, Los Angeles, California 90095-1555
  • Email: sjo@math.ucla.edu
  • Zuowei Shen
  • Affiliation: Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, Singapore 119076
  • MR Author ID: 292105
  • Email: matzuows@nus.edu.sg
  • Received by editor(s): April 3, 2011
  • Received by editor(s) in revised form: March 26, 2012
  • Published electronically: May 17, 2012
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 25 (2012), 1033-1089
  • MSC (2010): Primary 35A15, 41A25, 42C40, 45Q05, 65K10, 68U10
  • DOI: https://doi.org/10.1090/S0894-0347-2012-00740-1
  • MathSciNet review: 2947945