International Journal of Computer Vision

, Volume 50, Issue 3, pp 271–293

A Multiphase Level Set Framework for Image Segmentation Using the Mumford and Shah Model

  • Luminita A. Vese
  • Tony F. Chan

DOI: 10.1023/A:1020874308076

Cite this article as:
Vese, L.A. & Chan, T.F. International Journal of Computer Vision (2002) 50: 271. doi:10.1023/A:1020874308076


We propose a new multiphase level set framework for image segmentation using the Mumford and Shah model, for piecewise constant and piecewise smooth optimal approximations. The proposed method is also a generalization of an active contour model without edges based 2-phase segmentation, developed by the authors earlier in T. Chan and L. Vese (1999. In Scale-Space'99, M. Nilsen et al. (Eds.), LNCS, vol. 1682, pp. 141–151) and T. Chan and L. Vese (2001. IEEE-IP, 10(2):266–277). The multiphase level set formulation is new and of interest on its own: by construction, it automatically avoids the problems of vacuum and overlap; it needs only log n level set functions for n phases in the piecewise constant case; it can represent boundaries with complex topologies, including triple junctions; in the piecewise smooth case, only two level set functions formally suffice to represent any partition, based on The Four-Color Theorem. Finally, we validate the proposed models by numerical results for signal and image denoising and segmentation, implemented using the Osher and Sethian level set method.

energy minimization multi-phase motion image segmentation level sets curvature PDE's denoising edge detection active contours 

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Luminita A. Vese
    • 1
  • Tony F. Chan
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaLos AngelesUSA