  This paper introduces several depths for random sets with possibly non-convex
realisations, proposes ways to estimate the depths based on the samples and
compares them with existing ones. The depths are further applied for the
comparison between two samples of random sets using a visual method of DD-plots
and statistical testing. The advantage of this approach is identifying sets
within the sample that are responsible for rejecting the null hypothesis of
equality in distribution and providing clues on differences between
distributions. The method is justified using a simulation study and applied to
real data consisting of histological images of mastopathy and mammary cancer
tissue.
