Abstract
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.
Highlights
SOFT VERSUS HARD SEGMENTATIONSegmentation is the key step towards high-level vision modeling and analysis, including object characterization, detection, and classification
We propose a new stochastic-variational soft segmentation model for the following celebrated min E u, Γ | I = min H 1(Γ)+α |∇u|2 +λ (u − I)2, Γ,u where H 1 stands for the 1D Hausdorff measure [7], which is the length when Γ is regular enough
Combining the preceding two sections, we have developed the complete formula for soft Mumford-Shah segmentation with K patterns, that is, to minimize ui − I 2 pi + α
Summary
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. The two can lead to much more powerful novel models and efficient algorithms. 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 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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