Granulometric parametric estimation for the random Boolean model using optimal linear filters and optimal structuring elements

Morphological granulometries have been used successfully to discriminate textures in the context of classical feature-based classification, and more recently they have been used as observation variables for estimators in the context of random sets. This paper considers optimal linear parametric estimation of the law of a random set. It is set in a Bayesian framework in that estimation is of the law of a random Boolean model whose parameters satisfy prior distributions. It is assumed that the distribution of the primary grain has unknown parameters, and the task is to estimate these model parameters, along with the intensity of the Boolean model. The estimator will be the optimal linear filter for an input vector of granulometric moments. Furthermore, it will be assumed that the structuring elements generating the granulometry are themselves parameterized, and that the optimal parameters (those giving rise to the best optimal filter) are to be determined. It is experimentally observed that choosing optimal structuring elements can give much better results than arbitrarily choosing structuring element parameters.