OPTIMAL MORPHOLOGICAL PATTERN RESTORATION FROM NOISY BINARY IMAGES

A theoretical analysis of morphological filters for the "optimal" restoration of noisy binary images is presented. The problem is formulated in a general form and an "optimal" solution is obtained by using fundamental tools from mathematical morphology and decision theory. We consider the set-difference distance function as a measure of comparison between images, and, by using this function, we introduce the mean difference function as a quantitative measure of the degree of geometrical and topological distortion introduced by morphological filtering. We prove that the class of alternating sequential filters is a set of parametric, smoothing morphological filters that "best" preserve the crucial structure of input images in the least mean difference sense. A theory is also presented that demonstrates some important properties of the class of alternating filters and the class of alternating sequential filters and provides a theoretical justification for their use in morphological image analysis applications. A minimax estimation procedure is also proposed that allows us to obtain the "optimal" alternating sequential filter. This filter "optimally" eliminates the rough characteristics of the degradation noise while it "optimally" preserves the crucial geometrical and topological features of the noise-free pattern under consideration.