Volume 30, Issue 2
Homotopy Curve Tracking for Total Variation Image Restoration

Fenlin Yang, Ke Chen, & Bo Yu

J. Comp. Math., 30 (2012), pp. 177-196

Published online: 2012-04

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  • Abstract

The total variation (TV) minimization problem is widely studied in image restoration. Although many alternative methods have been proposed for its solution, the Newton method remains not usable for the primal formulation due to no convergence. A previous study by Chan, Zhou and Chan \cite{cc95} considered a regularization parameter continuation idea to increase the domain of convergence of the Newton method with some success but no robust parameter selection schemes. In this paper, we consider a homotopy method for the same primal TV formulation and propose to use curve tracking to select the regularization parameter adaptively. It turns out that this idea helps to improve substantially the previous work in efficiently solving the TV Euler-Lagrange equation. The same idea is also considered for the two other methods as well as the deblurring problem, again with improvements obtained. Numerical experiments show that our new methods are robust and fast for image restoration, even for images with large noisy-to-signal ratio.

  • Keywords

Image restoration Total variation Newton method Homotopy method Correction and curve tracking

  • AMS Subject Headings

65N06 65B99.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-30-177, author = {Fenlin Yang, Ke Chen, and Bo Yu}, title = {Homotopy Curve Tracking for Total Variation Image Restoration}, journal = {Journal of Computational Mathematics}, year = {2012}, volume = {30}, number = {2}, pages = {177--196}, abstract = { The total variation (TV) minimization problem is widely studied in image restoration. Although many alternative methods have been proposed for its solution, the Newton method remains not usable for the primal formulation due to no convergence. A previous study by Chan, Zhou and Chan \cite{cc95} considered a regularization parameter continuation idea to increase the domain of convergence of the Newton method with some success but no robust parameter selection schemes. In this paper, we consider a homotopy method for the same primal TV formulation and propose to use curve tracking to select the regularization parameter adaptively. It turns out that this idea helps to improve substantially the previous work in efficiently solving the TV Euler-Lagrange equation. The same idea is also considered for the two other methods as well as the deblurring problem, again with improvements obtained. Numerical experiments show that our new methods are robust and fast for image restoration, even for images with large noisy-to-signal ratio.}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1107-m3423}, url = {http://global-sci.org/intro/article_detail/jcm/8424.html} }
TY - JOUR T1 - Homotopy Curve Tracking for Total Variation Image Restoration AU - Fenlin Yang, Ke Chen, & Bo Yu JO - Journal of Computational Mathematics VL - 2 SP - 177 EP - 196 PY - 2012 DA - 2012/04 SN - 30 DO - http://doi.org/10.4208/jcm.1107-m3423 UR - https://global-sci.org/intro/article_detail/jcm/8424.html KW - Image restoration KW - Total variation KW - Newton method KW - Homotopy method KW - Correction and curve tracking AB - The total variation (TV) minimization problem is widely studied in image restoration. Although many alternative methods have been proposed for its solution, the Newton method remains not usable for the primal formulation due to no convergence. A previous study by Chan, Zhou and Chan \cite{cc95} considered a regularization parameter continuation idea to increase the domain of convergence of the Newton method with some success but no robust parameter selection schemes. In this paper, we consider a homotopy method for the same primal TV formulation and propose to use curve tracking to select the regularization parameter adaptively. It turns out that this idea helps to improve substantially the previous work in efficiently solving the TV Euler-Lagrange equation. The same idea is also considered for the two other methods as well as the deblurring problem, again with improvements obtained. Numerical experiments show that our new methods are robust and fast for image restoration, even for images with large noisy-to-signal ratio.
Fenlin Yang, Ke Chen, & Bo Yu. (1970). Homotopy Curve Tracking for Total Variation Image Restoration. Journal of Computational Mathematics. 30 (2). 177-196. doi:10.4208/jcm.1107-m3423
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