Random set estimation using granulometries and application to surface measurement

Shape characterization of random shapes and texture has been a fundamental issue of random sets and pattern recognition. This dissertation addresses basic pattern recognition problems in the context of random sets. In the first case, 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 case, 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 fixing prior distribution for the parameters determining the law. 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.; The Boolean model is the simplest and general random set in which the Poisson point process is coupled with an independent shape or grain process. A typical realization will be a random model with random texture. The model is parameterized and posed in a Bayesian setting; the task is to estimate these model parameters, along with the intensity of the Boolean model. Furthermore, it will be assumed that the structuring elements generating the granulometry are themselves parameterized, and the optimal parameters that give rise to the best optimal filters are then determined. The developed theory is then applied to estimate surface roughness measures, extending the procedure to gray scale. Boolean models with cone primary grains are used to generate different surface roughness textures. Conventional surface roughness measures related to peaks and valleys are later estimated by forming an optimal linear filter with different types of probes and granulometries.