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Volume 15, Issue 2
Multi-Phase Segmentation Using Modified Complex Cahn-Hilliard Equations

Xiakai Wang, Zhongyi Huang & Wei Zhu

DOI: 10.4208/nmtma.OA-2021-0099

Numer. Math. Theor. Meth. Appl., 15 (2022), pp. 442-463.

Published online: 2022-03

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  • Abstract

In this paper, we propose a novel PDE-based model for the multi-phase segmentation problem by using a complex version of Cahn-Hilliard equations. Specifically, we modify the original complex system of Cahn-Hilliard equations by adding the mean curvature term and the fitting term to the evolution of its real part, which helps to render a piecewisely constant function at the steady state. By applying the K-means method to this function, one could achieve the desired multiphase segmentation. To solve the proposed system of equations, a semi-implicit finite difference scheme is employed. Numerical experiments are presented to demonstrate the feasibility of the proposed model and compare our model with other related ones.

  • Keywords

Image segmentation, Cahn–Hilliard equation, semi-implicit finite difference scheme.

  • AMS Subject Headings

68U10, 65K10, 65N06

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-15-442, author = {Wang , XiakaiHuang , Zhongyi and Zhu , Wei}, title = {Multi-Phase Segmentation Using Modified Complex Cahn-Hilliard Equations}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2022}, volume = {15}, number = {2}, pages = {442--463}, abstract = {

In this paper, we propose a novel PDE-based model for the multi-phase segmentation problem by using a complex version of Cahn-Hilliard equations. Specifically, we modify the original complex system of Cahn-Hilliard equations by adding the mean curvature term and the fitting term to the evolution of its real part, which helps to render a piecewisely constant function at the steady state. By applying the K-means method to this function, one could achieve the desired multiphase segmentation. To solve the proposed system of equations, a semi-implicit finite difference scheme is employed. Numerical experiments are presented to demonstrate the feasibility of the proposed model and compare our model with other related ones.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2021-0099}, url = {http://global-sci.org/intro/article_detail/nmtma/20359.html} }
TY - JOUR T1 - Multi-Phase Segmentation Using Modified Complex Cahn-Hilliard Equations AU - Wang , Xiakai AU - Huang , Zhongyi AU - Zhu , Wei JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 442 EP - 463 PY - 2022 DA - 2022/03 SN - 15 DO - http://doi.org/10.4208/nmtma.OA-2021-0099 UR - https://global-sci.org/intro/article_detail/nmtma/20359.html KW - Image segmentation, Cahn–Hilliard equation, semi-implicit finite difference scheme. AB -

In this paper, we propose a novel PDE-based model for the multi-phase segmentation problem by using a complex version of Cahn-Hilliard equations. Specifically, we modify the original complex system of Cahn-Hilliard equations by adding the mean curvature term and the fitting term to the evolution of its real part, which helps to render a piecewisely constant function at the steady state. By applying the K-means method to this function, one could achieve the desired multiphase segmentation. To solve the proposed system of equations, a semi-implicit finite difference scheme is employed. Numerical experiments are presented to demonstrate the feasibility of the proposed model and compare our model with other related ones.

Wang , XiakaiHuang , Zhongyi and Zhu , Wei. (2022). Multi-Phase Segmentation Using Modified Complex Cahn-Hilliard Equations. Numerical Mathematics: Theory, Methods and Applications. 15 (2). 442-463. doi:10.4208/nmtma.OA-2021-0099
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