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

Bin Dong, Zuowei Shen, Peichu Xie

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

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
Volume49
Issue number1
DOIs
StatePublished - 2017
Externally publishedYes

Fingerprint

Wavelet Frames
Asymptotic analysis
Image Restoration
Image reconstruction
Asymptotic Analysis
Variational Model
Model
Imaging
Distributed Computation
Parallel Computation
Sobolev Spaces
Imaging techniques
Sobolev spaces
Continuum
Numerical Methods
Infinity
Image resolution
Numerical methods

Keywords

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

ASJC Scopus subject areas

  • Analysis
  • Computational Mathematics
  • Applied Mathematics

Cite this

Image restoration : A general wavelet frame based model and its asymptotic analysis. / Dong, Bin; Shen, Zuowei; Xie, Peichu.

In: SIAM Journal on Mathematical Analysis, Vol. 49, No. 1, 2017, p. 421-445.

Research output: Contribution to journalArticle

@article{479b02d9c8844afcb0d9d8e3f4eff287,
title = "Image restoration: A general wavelet frame based model and its asymptotic analysis",
abstract = "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].",
keywords = "(tight) wavelet frames, Framelets, Image restoration, Variational method, Γ-convergence",
author = "Bin Dong and Zuowei Shen and Peichu Xie",
year = "2017",
doi = "10.1137/16M1064969",
language = "English (US)",
volume = "49",
pages = "421--445",
journal = "SIAM Journal on Mathematical Analysis",
issn = "0036-1410",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "1",

}

TY - JOUR

T1 - Image restoration

T2 - A general wavelet frame based model and its asymptotic analysis

AU - Dong, Bin

AU - Shen, Zuowei

AU - Xie, Peichu

PY - 2017

Y1 - 2017

N2 - 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].

AB - 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].

KW - (tight) wavelet frames

KW - Framelets

KW - Image restoration

KW - Variational method

KW - Γ-convergence

UR - http://www.scopus.com/inward/record.url?scp=85014483335&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85014483335&partnerID=8YFLogxK

U2 - 10.1137/16M1064969

DO - 10.1137/16M1064969

M3 - Article

AN - SCOPUS:85014483335

VL - 49

SP - 421

EP - 445

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 1

ER -