Averaging of random sets based on their distance functions

A new notion of expectation for random sets (or average of binary images) is introduced using the representation of sets by distance functions. The distance function may be the familiar Euclidean distance transform, or some generalisation. The expectation of a random set X is defined as the set whose distance function is closest to the expected distance function of X. This distance average can be applied to obtain the average of non-convex and non-connected random sets. We establish some basic properties, compute examples, and prove limit theorems for the empirical distance average of independent identically distributed random sets.