Upper and Lower Grey-Level Adaptive Morphological Operators

Abstract

Morphological operators designed for grey-scale functions process every points of the space identically whatever their luminance. In many situations however, it is interesting to modulate the amount of processing according to the local grey-level. This leads to the idea of intensity-adaptive morphological operators. A simple way to construct such operators is to threshold the function at every grey-value, then to apply set operators to the level sets obtained in this way, and finally to reconstruct a new transformed function from the transformed level sets. The reconstruction's step is not straightforward since the transformed level sets are not obligatorily nested. Two schemes of stacking reinvestigated in the present paper that lead to two kinds of intensity-adaptive operators: the upper and  lower adaptive operators. Those operators are complementary in the sense that, by coupling, one defines adjunctions and consequently, by composition, one defines intensity-adaptive morphological openings and closings. The theoretical study of grey-level adaptive morphological operators is supplemented of some examples that illustrate the potential of the investigated operators in image filtering applications.