Continuous Choquet integrals with respect to random sets

An interpretation of continuous Choquet integrals with respect to random sets is given in the context of image filtering and shape detection. In this context, random sets represent random shapes defined on the plane. Random sets are characterized by their capacity functionals. Capacity functionals are fuzzy measures. Thus, input images can be integrated with respect to random sets. In this paper, input images are represented as fuzzy sets. The integration is interpreted in the context of mathematical morphology as the average a generalization morphological dilation or erosion. Specifically, the integrals represent the average probability that sets either intersect or are contained in the random sets, the average being over the alpha cuts of the input image. This interpretation has the potential for deriving new learning algorithms for using Choquet integrals in shape detection.