Upper and Lower Grey-Level Adaptive Morphological Operators

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.