Structuring element adaptation for morphological filters

This paper is devoted to the adaptive optimization of structuring elements of morphological filters in discrete spaces. The adaptation process is similar to the classical least mean square algorithm used for linear filters and its goal is to minimize a statistical criterion such as the mean square (or mean absolute) error between the filter output and a desired signal. The adaptive algorithms rely on an implicit formulation of the basic erosion and dilation. In a first step, it is shown how the structuring element can be optimized in the case of erosion and dilation. Both 3D and flat structuring elements are considered. Then, adaptation formulas for an arbitrary composition of erosions and dilations are derived. This allows optimization in the case of opening, closing, open-close, or alternating sequential filters. Finally, the results are further generalized to filters obtained by a combination of various filter outputs using minimum and maximum operations. This allows adaptation of filters such as maximum of openings or morphological centers. One of the major advantages of the approach is that it leads to a simple adaptation formula. The convergence behaviors of the various algorithms are illustrated using a 1D signal. Then, several practical problems of image processing are addressed. In particular examples of noise cancellation, image restoration, texture defect detection, and shape detection are described.