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Volume 15, Issue 4
Two-Phase Image Segmentation by the Allen-Cahn Equation and a Nonlocal Edge Detection Operator

Zhonghua Qiao & Qian Zhang

DOI: 10.4208/nmtma.OA-2022-0008s

Numer. Math. Theor. Meth. Appl., 15 (2022), pp. 1147-1172.

Published online: 2022-10

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  • Abstract

Based on a nonlocal Laplacian operator, a novel edge detection method of the grayscale image is proposed in this paper. This operator utilizes the information of neighbor pixels for a given pixel to obtain effective and delicate edge detection. The nonlocal edge detection method is used as an initialization for solving the Allen-Cahn equation to achieve two-phase segmentation of the grayscale image. Efficient exponential time differencing (ETD) solvers are employed in the time integration, and finite difference method is adopted in space discretization. The maximum bound principle and energy stability of the proposed numerical schemes are proved. The capability of our segmentation method has been verified in numerical experiments for different types of grayscale images.

  • Keywords

Image segmentation, Allen-Cahn equation, nonlocal edge detection operator, maximum principle, energy stability

  • AMS Subject Headings

68U10, 65K10, 65M12, 62H35

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-15-1147, author = {Qiao , Zhonghua and Zhang , Qian}, title = {Two-Phase Image Segmentation by the Allen-Cahn Equation and a Nonlocal Edge Detection Operator}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2022}, volume = {15}, number = {4}, pages = {1147--1172}, abstract = {

Based on a nonlocal Laplacian operator, a novel edge detection method of the grayscale image is proposed in this paper. This operator utilizes the information of neighbor pixels for a given pixel to obtain effective and delicate edge detection. The nonlocal edge detection method is used as an initialization for solving the Allen-Cahn equation to achieve two-phase segmentation of the grayscale image. Efficient exponential time differencing (ETD) solvers are employed in the time integration, and finite difference method is adopted in space discretization. The maximum bound principle and energy stability of the proposed numerical schemes are proved. The capability of our segmentation method has been verified in numerical experiments for different types of grayscale images.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2022-0008s}, url = {http://global-sci.org/intro/article_detail/nmtma/21097.html} }
TY - JOUR T1 - Two-Phase Image Segmentation by the Allen-Cahn Equation and a Nonlocal Edge Detection Operator AU - Qiao , Zhonghua AU - Zhang , Qian JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 1147 EP - 1172 PY - 2022 DA - 2022/10 SN - 15 DO - http://doi.org/10.4208/nmtma.OA-2022-0008s UR - https://global-sci.org/intro/article_detail/nmtma/21097.html KW - Image segmentation, Allen-Cahn equation, nonlocal edge detection operator, maximum principle, energy stability AB -

Based on a nonlocal Laplacian operator, a novel edge detection method of the grayscale image is proposed in this paper. This operator utilizes the information of neighbor pixels for a given pixel to obtain effective and delicate edge detection. The nonlocal edge detection method is used as an initialization for solving the Allen-Cahn equation to achieve two-phase segmentation of the grayscale image. Efficient exponential time differencing (ETD) solvers are employed in the time integration, and finite difference method is adopted in space discretization. The maximum bound principle and energy stability of the proposed numerical schemes are proved. The capability of our segmentation method has been verified in numerical experiments for different types of grayscale images.

Qiao , Zhonghua and Zhang , Qian. (2022). Two-Phase Image Segmentation by the Allen-Cahn Equation and a Nonlocal Edge Detection Operator. Numerical Mathematics: Theory, Methods and Applications. 15 (4). 1147-1172. doi:10.4208/nmtma.OA-2022-0008s
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