Journal of Mathematical Imaging and Vision

, Volume 37, Issue 2, pp 98–111 | Cite as

Multiphase Soft Segmentation with Total Variation and H 1 Regularization

  • Fang LiEmail author
  • Chaomin Shen
  • Chunming Li
Open Access


In this paper, we propose a variational soft segmentation framework inspired by the level set formulation of multiphase Chan-Vese model. We use soft membership functions valued in [0,1] to replace the Heaviside functions of level sets (or characteristic functions) such that we get a representation of regions by soft membership functions which automatically satisfies the sum to one constraint. We give general formulas for arbitrary N-phase segmentation, in contrast to Chan-Vese’s level set method only 2 m -phase are studied. To ensure smoothness on membership functions, both total variation (TV) regularization and H 1 regularization used as two choices for the definition of regularization term. TV regularization has geometric meaning which requires that the segmentation curve length as short as possible, while H 1 regularization has no explicit geometric meaning but is easier to implement with less parameters and has higher tolerance to noise. Fast numerical schemes are designed for both of the regularization methods. By changing the distance function, the proposed segmentation framework can be easily extended to the segmentation of other types of images. Numerical results on cartoon images, piecewise smooth images and texture images demonstrate that our methods are effective in multiphase image segmentation.


Chan-Vese model Level set Total variation regularization H1 regularization Soft membership function 


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© The Author(s) 2010

Open AccessThis is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (, which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Authors and Affiliations

  1. 1.Department of MathematicsEast China Normal UniversityShanghaiChina
  2. 2.Department of Computer ScienceEast China Normal UniversityShanghaiChina
  3. 3.Institute of Imaging ScienceVanderbilt UniversityNashvilleUSA

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