The granulometric size density in filter optimization

Binary granulometric filters are constructed from parameterized unions and intersections of morphological openings. They are especially useful in filtering out clutter when an observed image is composed of a signal unioned with the clutter; however, their general theory does not depend on this model. More generally, the granulometric spectral theory provides a set decomposition in which bands within that decomposition are passed or not passed according to the action of a granulometric bandpass filter. A granulometry induces a granulometric size density on a random set and optimal granulometric filtering can be characterized in terms of this size density, as can the robustness of the optimal filter relative to changing states of nature under which the optimal filter has been designed. In this sense, the granulometric size density plays a role analogous to the power spectral density in linear filtering. The paper describes the theory of optimal granulometric filtering as determined by the granulometric size density and it relates this nonlinear granulometric theory to the classical linear theory as applied via the power spectral density