Minimization of an Edge-Preserving Regularization Functional by Conjugate Gradient Type Methods

Jian-Feng CAI; Raymond CHAN; Benedetta MORINI

doi:10.1007/978-3-540-33267-1_7

Minimization of an Edge-Preserving Regularization Functional by Conjugate Gradient Type Methods

Jian-Feng CAI, Raymond CHAN, Benedetta MORINI

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

Abstract

Recently, a powerful two-phase method for removing impulse noise has been developed. It gives a satisfactory result even for images with 90% pixels corrupted by impulse noise. However, the two-phase method is not computationally efficient, because it requires the minimization of a non-smooth functional in the second phase, which is done by a relaxation-based method. In this paper, we remove the non-smooth term from the functional, and call for the minimization of a smooth one. The minimizer is then found by using a conjugate gradient method proposed by J. Sun and J. Zhang. We prove the global convergence of the conjugate gradient type method applied to our functional. Simulation results show that our method is several times faster than the relaxation-based method when the noise ratio is high.

Original language	English
Title of host publication	Image Processing Based on Partial Differential Equations: Proceedings of the International Conference on PDE-Based Image Processing and Related Inverse Problems, CMA, Oslo, August 8-12, 2005
Editors	Xue-Cheng TAI, Knut-Andreas LIE, Tony F. CHAN, Stanley OSHER
Publisher	Springer Berlin Heidelberg
Pages	109-122
Number of pages	14
ISBN (Electronic)	9783540332671
ISBN (Print)	9783540332664, 9783642069901
DOIs	https://doi.org/10.1007/978-3-540-33267-1_7
Publication status	Published - 2007
Externally published	Yes

Publication series

Name	Mathematics and Visualization
Publisher	Springer Berlin, Heidelberg
ISSN (Print)	1612-3786
ISSN (Electronic)	2197-666X

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10.1007/978-3-540-33267-1_7

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CAI, J.-F., CHAN, R., & MORINI, B. (2007). Minimization of an Edge-Preserving Regularization Functional by Conjugate Gradient Type Methods. In X.-C. TAI, K.-A. LIE, T. F. CHAN, & S. OSHER (Eds.), Image Processing Based on Partial Differential Equations: Proceedings of the International Conference on PDE-Based Image Processing and Related Inverse Problems, CMA, Oslo, August 8-12, 2005 (pp. 109-122). (Mathematics and Visualization). Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-540-33267-1_7

CAI, Jian-Feng ; CHAN, Raymond ; MORINI, Benedetta. / Minimization of an Edge-Preserving Regularization Functional by Conjugate Gradient Type Methods. Image Processing Based on Partial Differential Equations: Proceedings of the International Conference on PDE-Based Image Processing and Related Inverse Problems, CMA, Oslo, August 8-12, 2005. editor / Xue-Cheng TAI ; Knut-Andreas LIE ; Tony F. CHAN ; Stanley OSHER. Springer Berlin Heidelberg, 2007. pp. 109-122 (Mathematics and Visualization).

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title = "Minimization of an Edge-Preserving Regularization Functional by Conjugate Gradient Type Methods",

abstract = "Recently, a powerful two-phase method for removing impulse noise has been developed. It gives a satisfactory result even for images with 90% pixels corrupted by impulse noise. However, the two-phase method is not computationally efficient, because it requires the minimization of a non-smooth functional in the second phase, which is done by a relaxation-based method. In this paper, we remove the non-smooth term from the functional, and call for the minimization of a smooth one. The minimizer is then found by using a conjugate gradient method proposed by J. Sun and J. Zhang. We prove the global convergence of the conjugate gradient type method applied to our functional. Simulation results show that our method is several times faster than the relaxation-based method when the noise ratio is high.",

author = "Jian-Feng CAI and Raymond CHAN and Benedetta MORINI",

year = "2007",

doi = "10.1007/978-3-540-33267-1_7",

language = "English",

isbn = "9783540332664",

series = "Mathematics and Visualization",

publisher = "Springer Berlin Heidelberg",

pages = "109--122",

editor = "Xue-Cheng TAI and Knut-Andreas LIE and CHAN, {Tony F.} and Stanley OSHER",

booktitle = "Image Processing Based on Partial Differential Equations: Proceedings of the International Conference on PDE-Based Image Processing and Related Inverse Problems, CMA, Oslo, August 8-12, 2005",

}

CAI, J-F, CHAN, R & MORINI, B 2007, Minimization of an Edge-Preserving Regularization Functional by Conjugate Gradient Type Methods. in X-C TAI, K-A LIE, TF CHAN & S OSHER (eds), Image Processing Based on Partial Differential Equations: Proceedings of the International Conference on PDE-Based Image Processing and Related Inverse Problems, CMA, Oslo, August 8-12, 2005. Mathematics and Visualization, Springer Berlin Heidelberg, pp. 109-122. https://doi.org/10.1007/978-3-540-33267-1_7

Minimization of an Edge-Preserving Regularization Functional by Conjugate Gradient Type Methods. / CAI, Jian-Feng; CHAN, Raymond; MORINI, Benedetta.
Image Processing Based on Partial Differential Equations: Proceedings of the International Conference on PDE-Based Image Processing and Related Inverse Problems, CMA, Oslo, August 8-12, 2005. ed. / Xue-Cheng TAI; Knut-Andreas LIE; Tony F. CHAN; Stanley OSHER. Springer Berlin Heidelberg, 2007. p. 109-122 (Mathematics and Visualization).

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

TY - GEN

T1 - Minimization of an Edge-Preserving Regularization Functional by Conjugate Gradient Type Methods

AU - CAI, Jian-Feng

AU - CHAN, Raymond

AU - MORINI, Benedetta

PY - 2007

Y1 - 2007

N2 - Recently, a powerful two-phase method for removing impulse noise has been developed. It gives a satisfactory result even for images with 90% pixels corrupted by impulse noise. However, the two-phase method is not computationally efficient, because it requires the minimization of a non-smooth functional in the second phase, which is done by a relaxation-based method. In this paper, we remove the non-smooth term from the functional, and call for the minimization of a smooth one. The minimizer is then found by using a conjugate gradient method proposed by J. Sun and J. Zhang. We prove the global convergence of the conjugate gradient type method applied to our functional. Simulation results show that our method is several times faster than the relaxation-based method when the noise ratio is high.

AB - Recently, a powerful two-phase method for removing impulse noise has been developed. It gives a satisfactory result even for images with 90% pixels corrupted by impulse noise. However, the two-phase method is not computationally efficient, because it requires the minimization of a non-smooth functional in the second phase, which is done by a relaxation-based method. In this paper, we remove the non-smooth term from the functional, and call for the minimization of a smooth one. The minimizer is then found by using a conjugate gradient method proposed by J. Sun and J. Zhang. We prove the global convergence of the conjugate gradient type method applied to our functional. Simulation results show that our method is several times faster than the relaxation-based method when the noise ratio is high.

U2 - 10.1007/978-3-540-33267-1_7

DO - 10.1007/978-3-540-33267-1_7

M3 - Conference paper (refereed)

SN - 9783540332664

SN - 9783642069901

T3 - Mathematics and Visualization

SP - 109

EP - 122

BT - Image Processing Based on Partial Differential Equations: Proceedings of the International Conference on PDE-Based Image Processing and Related Inverse Problems, CMA, Oslo, August 8-12, 2005

A2 - TAI, Xue-Cheng

A2 - LIE, Knut-Andreas

A2 - CHAN, Tony F.

A2 - OSHER, Stanley

PB - Springer Berlin Heidelberg

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CAI JF, CHAN R, MORINI B. Minimization of an Edge-Preserving Regularization Functional by Conjugate Gradient Type Methods. In TAI XC, LIE KA, CHAN TF, OSHER S, editors, Image Processing Based on Partial Differential Equations: Proceedings of the International Conference on PDE-Based Image Processing and Related Inverse Problems, CMA, Oslo, August 8-12, 2005. Springer Berlin Heidelberg. 2007. p. 109-122. (Mathematics and Visualization). doi: 10.1007/978-3-540-33267-1_7

Minimization of an Edge-Preserving Regularization Functional by Conjugate Gradient Type Methods

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