Morphological granulometric estimation of random patterns in the context of parameterized random sets

Morphological features are used to estimate the state of a random pattern (set) governed by a multivariate probability distribution. The feature vector is composed of granulometric moments and pattern estimation involves feature-based estimation of the parameter vector governing the random set. Under such circumstances, the joint density of the features and parameters is a generalized function concentrated on a solution manifold and estimation is determined by the conditional density of the parameters given an observed feature vector. The paper explains the manner in which the joint probability mass of the parameters and features is distributed and the way the conditional densities give rise to optimal estimators according to the distribution of probability mass, whether constrained or not to the solution manifold. The estimation theory is applied using analytic representation of linear granulometric moments. The e!ects of random perturbations in the shape-parameter vector is discussed, and the theory is applied to random sets composed of disjoint random shapes. The generalized density framework provides a proper mathematical context for pattern estimation and gives insight, via the distribution of mass on solution manifolds, to the manner in which morphological probes discriminate random sets relative to their distributions, and the manner in which the use of additional probes can be bene"cial for better estimation. 2001 Pattern Recognition Society. Published by Elsevier Science Ltd. All rights reserved.