  In this paper we present the methodology for detecting outliers and testing
the goodness-of-fit of random sets using topological data analysis. We
construct the filtration from level sets of the signed distance function and
consider various summary functions of the persistence diagram derived from the
obtained persistence homology. The outliers are detected using functional
depths for the summary functions. Global envelope tests using the summary
statistics as test statistics were used to construct the goodness-of-fit test.
The procedures were justified by a simulation study using germ-grain random set
models.
