An unconditionally stable hybrid method for image segmentation

Yibao Li, Junseok Kim

Research output: Contribution to journalArticlepeer-review

29 Citations (Scopus)


In this paper, we propose a new unconditionally stable hybrid numerical method for minimizing the piecewise constant Mumford-Shah functional of image segmentation. The model is based on the Allen-Cahn equation and an operator splitting technique is used to solve the model numerically. We split the governing equation into two linear equations and one nonlinear equation. One of the linear equations and the nonlinear equation are solved analytically due to the availability of closed-form solutions. The other linear equation is discretized using an implicit scheme and the resulting discrete system of equations is solved by a fast numerical algorithm such as a multigrid method. We prove the unconditional stability of the proposed scheme. Since we incorporate closed-form solutions and an unconditionally stable scheme in the solution algorithm, our proposed scheme is accurate and robust. Various numerical results on real and synthetic images with noises are presented to demonstrate the efficiency, robustness, and accuracy of the proposed method.

Original languageEnglish
Pages (from-to)32-43
Number of pages12
JournalApplied Numerical Mathematics
Publication statusPublished - 2014 Aug

Bibliographical note

Funding Information:
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2009-0074248 ). The authors also wish to thank the reviewers for the constructive and helpful comments on the revision of this article.


  • Allen-Cahn equation
  • Chan-Vese model
  • Image segmentation
  • Mumford-Shah functional
  • Phase-field method

ASJC Scopus subject areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics


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