Optimal mean-square N-observation digital morphological filters : II. Optimal gray-scale filters

Abstract

The second part of this two-part study interprets digital gray-scale morphological operations as numerical functionals on integer-valued vectors. The result is a gray-scale morphological statistical estimation theory based on N observation random variables and a consequent theory of mean-square optimization. Whereas in the binary setting the search for optimal structuring elements is clearly restricted by the requirement of binary N-vector structuring elements, gray-scale structuring elements are N-vectors of integers (not confined to the range of the images). Thus, it is necessary to find a set of structuring elements from which both singleand multiple-erosion filters can be optimized, the latter being characterized within the framework of the gray-scale Matheron representation. Consequently, a significant part of the paper is concerned with the derivation of the fundamental set, this set being a minimal family of structuring elements from which it is always possible to select the erosions comprising an optimal filter. As in the case of binary optimization, we treat constrained optimality; however, here we apply it to construction of optimal linear filters. Finally, we demonstrate that erosion optimization is equivalent to dilation optimization.