Two-Phase Image Segmentation by the Allen-Cahn Equation and a Nonlocal Edge Detection Operator

Zhonghua Qiao, Qian Zhang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

8 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)1147-1172
Number of pages26
JournalNumerical Mathematics-Theory Methods and Applications
Issue number4
Publication statusPublished - 1 Nov 2022


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

ASJC Scopus subject areas

  • Modelling and Simulation
  • Control and Optimization
  • Computational Mathematics
  • Applied Mathematics


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