A stochastic-variational model for soft Mumford-Shah segmentation
  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. We show that soft segmentation leads to hard segmentation, and hence is more general. The modeling procedure, mathematical analysis, and computational implementation of the new model are explored in detail, and numerical examples of synthetic and natural images are presented. 
Jianhong Shen