Applications of random sets in image analysis. How to average a cat and a dog?

The expectation (or average) is one of the basic concepts in probability theory and statistical inference. However, this rather basic concept becomes fairly complicated for random elements in non-linear spaces. The family of compact sets is a typical example of such non-linear space, alongside with numerous possible examples that involve curved spaces. To highlight various complications, it is possible to consider a random compact set that with equal probabilities is the unit interval or its end-points, or a set that is either the circle or the disk bounded by it, or a set that takes values {0} or {0, 1}. Various definitions of expectations respect the shape of the set, its cardinality or measure, or its connectivity properties

On the integration of fuzzy level sets

International audienceWe study the integration of fuzzy level sets associated with a fuzzy random variable when the underlying space is a separable Banach space or a weak star dual of a separable Banach space. In particular, the expectation and the conditional expectation of fuzzy level sets in this setting are presented. We prove the SLLN for pairwise independent identically distributed fuzzy convex compact valued level sets through the SLLN for pairwise independent identically distributed convex compact valued random set in separable Banach space. Some convergence results for a class of integrand martingale are also presented