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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
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