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

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. The log-likelihood ratio for Poisson points can therefore be applied to observation points on this subset of R. Several interesting results pertaining to the asymptotic performance of the log-likelihood ratio for Poisson points are known. A major di culty with this approach is that the test is based on observation points on a subset of R, and is not directly given in terms of the observation of a realization of a Boolean random set. An efcient means of mapping realizations of planar Boolean random sets to corresponding realizations of Poisson point processes on this subset of R 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 exible shape-size analysis tools based on elementary morphological and settheoretic operations. This is the principal contribution of this paper. Research partially supported by NSF grant NSFD CDR 8803012, through the Engineering Research Centers Program Also with the Department of Electrical Engineering Also with the Department of Electrical Engineering Also with the Department of Mathematics