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Image Segmentation Based on Modified Fractional Allen–Cahn Equation

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  • Dongsun Lee
  • Seunggyu Lee

Abstract

We present the image segmentation model using the modified Allen–Cahn equation with a fractional Laplacian. The motion of the interface for the classical Allen–Cahn equation is known as the mean curvature flows, whereas its dynamics is changed to the macroscopic limit of Lévy process by replacing the Laplacian operator with the fractional one. To numerical implementation, we prove the unconditionally unique solvability and energy stability of the numerical scheme for the proposed model. The effect of a fractional Laplacian operator in our own and in the Allen–Cahn equation is checked by numerical simulations. Finally, we give some image segmentation results with different fractional order, including the standard Laplacian operator.

Suggested Citation

  • Dongsun Lee & Seunggyu Lee, 2019. "Image Segmentation Based on Modified Fractional Allen–Cahn Equation," Mathematical Problems in Engineering, Hindawi, vol. 2019, pages 1-6, January.
    • Handle: RePEc:hin:jnlmpe:3980181
    DOI: 10.1155/2019/3980181
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    Cited by:

    1. Tudor Barbu & Alain Miranville & Costică Moroşanu, 2024. "On a Local and Nonlocal Second-Order Boundary Value Problem with In-Homogeneous Cauchy–Neumann Boundary Conditions—Applications in Engineering and Industry," Mathematics, MDPI, vol. 12(13), pages 1-19, June.
    2. Ham, Seokjun & Kim, Junseok, 2023. "Stability analysis for a maximum principle preserving explicit scheme of the Allen–Cahn equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 207(C), pages 453-465.
    3. Costică Moroşanu & Silviu Pavăl, 2021. "Rigorous Mathematical Investigation of a Nonlocal and Nonlinear Second-Order Anisotropic Reaction-Diffusion Model: Applications on Image Segmentation," Mathematics, MDPI, vol. 9(1), pages 1-23, January.
    4. Sungha Yoon & Darae Jeong & Chaeyoung Lee & Hyundong Kim & Sangkwon Kim & Hyun Geun Lee & Junseok Kim, 2020. "Fourier-Spectral Method for the Phase-Field Equations," Mathematics, MDPI, vol. 8(8), pages 1-36, August.

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