Bayesian Hypothesis Testing for Boolean Random Sets with Radial Convex Primary Grains Using Morphological Skeleton Transforms

Abstract

We consider the problem of binary hypothesis testing for planar Boolean random sets with radial convex primary grains. We show that this problem is equivalent to the problem of binary hypothesis testing for Poisson points on a subset of R cube . The log-likelihood ratio for Poisson points can therefore be applied to observation points on this subset of R cube. Several interesting results pertaining to the asymptotic performance of the log-likelihood ratio for Poisson points are known. A major difficulty with this approach is that the test is based on observation points on a subset of R cube, and is not directly given in terms of the observation of a realization of a Boolean random set. An efficient means of mapping realizations of planar Boolean random sets to corresponding realizations of Poisson point processes on this subset of R cube is needed in order to implement the test. We show that this can be achieved via a class of morphological transformations known as morphological skeleton transforms. These transforms are flexible shape-size analysis tools based on elementary morphological and set-theoretic operations. This is the principal contribution of this paper.