Journal of Scientific Computing

, Volume 69, Issue 1, pp 82–106 | Cite as

A Multiphase Image Segmentation Based on Fuzzy Membership Functions and L1-Norm Fidelity

  • Fang LiEmail author
  • Stanley Osher
  • Jing Qin
  • Ming Yan


In this paper, we propose a variational multiphase image segmentation model based on fuzzy membership functions and L1-norm fidelity. Then we apply the alternating direction method of multipliers to solve an equivalent problem. All the subproblems can be solved efficiently. Specifically, we propose a fast method to calculate the fuzzy median. Experimental results and comparisons show that the L1-norm based method is more robust to outliers such as impulse noise and keeps better contrast than its L2-norm counterpart. Theoretically, we prove the existence of the minimizer and analyze the convergence of the algorithm.


Image segmentation Fuzzy membership function L1-norm ADMM Segmentation accuracy 



The research of F. Li was supported by the 973 Program 2011CB707104 and the Science and Technology Commission of Shanghai Municipality (STCSM) 13dz2260400, the research of S. Osher and J. Qin was supported by ONR Grants N00014120838 and N00014140444, NSF Grants DMS-1118971 and CCF-0926127, and the Keck Foundation, the research of M. Yan was supported by NSF Grants DMS-1317602. The work was done when the first author was visiting UCLA Department of Mathematics.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsEast China Normal University, and Shanghai Key Laboratory of Pure Mathematics and Mathematical PracticeShanghaiChina
  2. 2.Department of MathematicsUniversity of CaliforniaLos AngelesUSA
  3. 3.Department of Computational Mathematics, Science and EngineeringMichigan State UniversityEast LansingUSA
  4. 4.Department of MathematicsMichigan State UniversityEast LansingUSA

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