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Volume 14, Issue 4
Accelerated Non-Overlapping Domain Decomposition Method for Total Variation Minimization

Xue Li, Zhenwei Zhang, Huibin Chang & Yuping Duan

DOI: 10.4208/nmtma.OA-2020-0146

Numer. Math. Theor. Meth. Appl., 14 (2021), pp. 1017-1041.

Published online: 2021-09

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  • Abstract

We concern with fast domain decomposition methods for solving the total variation minimization problems in image processing. By decomposing the image domain into non-overlapping subdomains and interfaces, we consider the primal-dual problem on the interfaces such that the subdomain problems become independent problems and can be solved in parallel. Suppose both the interfaces and subdomain problems are uniformly convex, we can apply the acceleration method to achieve an $\mathcal{O}(1 / n^2)$ convergent domain decomposition algorithm. The convergence analysis is provided as well. Numerical results on image denoising, inpainting, deblurring, and segmentation are provided and comparison results with existing methods are discussed, which not only demonstrate the advantages of our method but also support the theoretical convergence rate.

  • Keywords

Non-overlapping domain decomposition method, primal-dual algorithm, total variation, Rudin-Osher-Fatemi model, Chan-Vese model.

  • AMS Subject Headings

68U10, 65M55, 65N55

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-14-1017, author = {Li , XueZhang , ZhenweiChang , Huibin and Duan , Yuping}, title = {Accelerated Non-Overlapping Domain Decomposition Method for Total Variation Minimization}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2021}, volume = {14}, number = {4}, pages = {1017--1041}, abstract = {

We concern with fast domain decomposition methods for solving the total variation minimization problems in image processing. By decomposing the image domain into non-overlapping subdomains and interfaces, we consider the primal-dual problem on the interfaces such that the subdomain problems become independent problems and can be solved in parallel. Suppose both the interfaces and subdomain problems are uniformly convex, we can apply the acceleration method to achieve an $\mathcal{O}(1 / n^2)$ convergent domain decomposition algorithm. The convergence analysis is provided as well. Numerical results on image denoising, inpainting, deblurring, and segmentation are provided and comparison results with existing methods are discussed, which not only demonstrate the advantages of our method but also support the theoretical convergence rate.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2020-0146 }, url = {http://global-sci.org/intro/article_detail/nmtma/19528.html} }
TY - JOUR T1 - Accelerated Non-Overlapping Domain Decomposition Method for Total Variation Minimization AU - Li , Xue AU - Zhang , Zhenwei AU - Chang , Huibin AU - Duan , Yuping JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 1017 EP - 1041 PY - 2021 DA - 2021/09 SN - 14 DO - http://doi.org/10.4208/nmtma.OA-2020-0146 UR - https://global-sci.org/intro/article_detail/nmtma/19528.html KW - Non-overlapping domain decomposition method, primal-dual algorithm, total variation, Rudin-Osher-Fatemi model, Chan-Vese model. AB -

We concern with fast domain decomposition methods for solving the total variation minimization problems in image processing. By decomposing the image domain into non-overlapping subdomains and interfaces, we consider the primal-dual problem on the interfaces such that the subdomain problems become independent problems and can be solved in parallel. Suppose both the interfaces and subdomain problems are uniformly convex, we can apply the acceleration method to achieve an $\mathcal{O}(1 / n^2)$ convergent domain decomposition algorithm. The convergence analysis is provided as well. Numerical results on image denoising, inpainting, deblurring, and segmentation are provided and comparison results with existing methods are discussed, which not only demonstrate the advantages of our method but also support the theoretical convergence rate.

Li , XueZhang , ZhenweiChang , Huibin and Duan , Yuping. (2021). Accelerated Non-Overlapping Domain Decomposition Method for Total Variation Minimization. Numerical Mathematics: Theory, Methods and Applications. 14 (4). 1017-1041. doi:10.4208/nmtma.OA-2020-0146
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