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 vecto r is composed of granulometric moments and pattern estimation involves feat ure-based estimation of the parameter vector governing the random set. Unde r 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 probabilit y mass of the parameters and features is distributed and the way the condit ional densities give rise to optimal estimators according to the distributi on of probability mass, whether constrained or not to the solution manifold . The estimation theory is applied using analytic representation of linear granulometric moments. The effects of random perturbations in the shape-par ameter vector is discussed, and the theory is applied to random sets compos ed of disjoint random shapes. The generalized density framework provides a proper mathematical context for pattern estimation and gives insight via th e distribution of mass on solution manifolds, to the manner in which morpho logical probes discriminate random sets relative to their distributions, an d the manner in which the use of additional probes can be beneficial for be tter estimation (C) 2001 Pattern Recognition Society. Published by Elsevier Science Ltd. All rights reserved.