Continuation method for total variation denoising problems

Tony F. CHAN; H. M. ZHOU; Raymond H. CHAN

doi:10.1117/12.211408

Continuation method for total variation denoising problems

Tony F. CHAN, H. M. ZHOU, Raymond H. CHAN

Research output: Book Chapters | Papers in Conference Proceedings › Conference paper (refereed) › Research › peer-review

42 Citations (Scopus)

Abstract

The denoising problem can be solved by posing it as a constrained minimization problem. The objective function is the TV norm of the denoised image whereas the constraint is the requirement that the denoised image does not deviate too much from the observed image. The Euler-Lagrangian equation corresponding to the minimization problem is a nonlinear equation. The Newton method for such equation is known to have a very small domain of convergence. In this paper, we propose to couple the Newton method with the continuation method. Using the Newton-Kantorovich theorem, we give a bound on the domain of convergence. Numerical results are given to illustrate the convergence.

Original language	English
Title of host publication	Proceedings of SPIE : Advanced Signal Processing Algorithms
Editors	Franklin T. LUK
Publisher	SPIE
Pages	314-325
Number of pages	12
ISBN (Print)	9780819419224
DOIs	https://doi.org/10.1117/12.211408
Publication status	Published - 7 Jun 1995
Externally published	Yes
Event	SPIE 1995 International Symposium on Optical Science, Engineering, and Instrumentation - San Diego, United States Duration: 9 Jul 1995 → 14 Jul 1995

Publication series

Name	Proceedings of SPIE : The International Society for Optical Engineering
Publisher	SPIE
Volume	2563
ISSN (Print)	0277-786X

Conference

Conference	SPIE 1995 International Symposium on Optical Science, Engineering, and Instrumentation
Country/Territory	United States
City	San Diego
Period	9/07/95 → 14/07/95

Bibliographical note

Acknowledgment: The first author would like to acknowledge the hospitality of the Department of Mathematics at the Chinese University of Hong Kong where this work was initiated during a visit.
Publisher Copyright: © 2015 SPIE. All Rights Reserved.

Keywords

Denoising
Fixed-point method
Newton method
Total-variation

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10.1117/12.211408

Cite this

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title = "Continuation method for total variation denoising problems",

abstract = "The denoising problem can be solved by posing it as a constrained minimization problem. The objective function is the TV norm of the denoised image whereas the constraint is the requirement that the denoised image does not deviate too much from the observed image. The Euler-Lagrangian equation corresponding to the minimization problem is a nonlinear equation. The Newton method for such equation is known to have a very small domain of convergence. In this paper, we propose to couple the Newton method with the continuation method. Using the Newton-Kantorovich theorem, we give a bound on the domain of convergence. Numerical results are given to illustrate the convergence.",

keywords = "Denoising, Fixed-point method, Newton method, Total-variation",

author = "CHAN, {Tony F.} and ZHOU, {H. M.} and CHAN, {Raymond H.}",

note = "Acknowledgment: The first author would like to acknowledge the hospitality of the Department of Mathematics at the Chinese University of Hong Kong where this work was initiated during a visit. Publisher Copyright: {\textcopyright} 2015 SPIE. All Rights Reserved.; SPIE 1995 International Symposium on Optical Science, Engineering, and Instrumentation ; Conference date: 09-07-1995 Through 14-07-1995",

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doi = "10.1117/12.211408",

language = "English",

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series = "Proceedings of SPIE : The International Society for Optical Engineering",

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editor = "LUK, {Franklin T.}",

booktitle = "Proceedings of SPIE : Advanced Signal Processing Algorithms",

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CHAN, TF, ZHOU, HM & CHAN, RH 1995, Continuation method for total variation denoising problems. in FT LUK (ed.), Proceedings of SPIE : Advanced Signal Processing Algorithms. Proceedings of SPIE : The International Society for Optical Engineering, vol. 2563, SPIE, pp. 314-325, SPIE 1995 International Symposium on Optical Science, Engineering, and Instrumentation, San Diego, California, United States, 9/07/95. https://doi.org/10.1117/12.211408

Continuation method for total variation denoising problems. / CHAN, Tony F.; ZHOU, H. M.; CHAN, Raymond H.
Proceedings of SPIE : Advanced Signal Processing Algorithms. ed. / Franklin T. LUK. SPIE, 1995. p. 314-325 (Proceedings of SPIE : The International Society for Optical Engineering; Vol. 2563).

Research output: Book Chapters | Papers in Conference Proceedings › Conference paper (refereed) › Research › peer-review

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N2 - The denoising problem can be solved by posing it as a constrained minimization problem. The objective function is the TV norm of the denoised image whereas the constraint is the requirement that the denoised image does not deviate too much from the observed image. The Euler-Lagrangian equation corresponding to the minimization problem is a nonlinear equation. The Newton method for such equation is known to have a very small domain of convergence. In this paper, we propose to couple the Newton method with the continuation method. Using the Newton-Kantorovich theorem, we give a bound on the domain of convergence. Numerical results are given to illustrate the convergence.

AB - The denoising problem can be solved by posing it as a constrained minimization problem. The objective function is the TV norm of the denoised image whereas the constraint is the requirement that the denoised image does not deviate too much from the observed image. The Euler-Lagrangian equation corresponding to the minimization problem is a nonlinear equation. The Newton method for such equation is known to have a very small domain of convergence. In this paper, we propose to couple the Newton method with the continuation method. Using the Newton-Kantorovich theorem, we give a bound on the domain of convergence. Numerical results are given to illustrate the convergence.

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Continuation method for total variation denoising problems

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