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International Journal of Biomedical Imaging
Volume 2006 (2006), Article ID 92329, 14 pages

A Stochastic-Variational Model for Soft Mumford-Shah Segmentation

1School of Mathematics, Institute of Technology, University of Minnesota, Minneapolis, MN 55455, USA
2Lotus Hill Institute for Computer Vision and Information Science, E'Zhou, Wuhan 436000, China

Received 20 September 2005; Revised 13 February 2006; Accepted 17 February 2006

Copyright © 2006 Jianhong (Jackie) Shen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


In contemporary image and vision analysis, stochastic approaches demonstrate great flexibility in representing and modeling complex phenomena, while variational-PDE methods gain enormous computational advantages over Monte Carlo or other stochastic algorithms. In combination, the two can lead to much more powerful novel models and efficient algorithms. In the current work, we propose a stochastic-variational model for soft (or fuzzy) Mumford-Shah segmentation of mixture image patterns. Unlike the classical hard Mumford-Shah segmentation, the new model allows each pixel to belong to each image pattern with some probability. Soft segmentation could lead to hard segmentation, and hence is more general. The modeling procedure, mathematical analysis on the existence of optimal solutions, and computational implementation of the new model are explored in detail, and numerical examples of both synthetic and natural images are presented.