Nonparametric Statistical Active Contour Based on Inclusion Degree of Fuzzy Sets

In this paper, inclusion degree of fuzzy sets is introduced to image segmentation. The image segmentation problem is novelly modeled as the minimization of the overlapping rates between the inside and outside regions, subject to a constraint on the total length of the region boundaries. Considering the similar properties of fuzzy sets and statistical image domain, we use fuzzy membership functions to represent the inside and outside regions and utilize nonparametric density estimates to estimate them. Then, the inclusion degree of fuzzy sets is adopted to formulate the overlapping rates between the inside and outside regions. We solve the inclusion-degree-based optimization problem by deriving the associated gradient flow and applying curve evolution techniques. Experimental results on both synthetic and real images confirm the effectiveness of the proposed method. Compared with the previous active contour models formulated to solve the same nonparametric statistical segmentation problem, our method performs well in efficiency and evolution time.