Optimal linear granulometric estimation for random sets

摘要

This paper addresses two pattern-recognition problems in the context of random sets. For the first, the random set law is known and the task is to estimate the observed pattern from a feature set calculated from the observation. For the second, the law is unknown and we wish to estimate the parameters of the law. Estimation is accomplished by an optimal linear system whose inputs are features based on morphological granulometries. In the first case these features are granulometric moments; in the second they are moments of the granulometric moments. For the latter, estimation is placed in a Bayesian context by assuming that there exists a prior distribution for the parameters determining the law. A disjoint random grain model is assumed and the optimal linear estimator is determined by using asymptotic expressions for the moments of the granulometric moments. In both cases, the linear approach serves as a practical alternative to previously proposed nonlinear methods. Granulometric pattern estimation has previously been accomplished by a nonlinear method using full distributional knowledge of the random variables determining the pattern and granulometric features. Granulometric estimation of the law of a random grain model has previously been accomplished by solving a system of nonlinear equations resulting from the granulometric asymptotic mixing theorem. Both methods are limited in application owing to the necessity of performing a nonlinear optimization. The new linear method avoids this. It makes estimation possible for more complex models.