Analysis of the statistical properties of 1-D morphological filters

Nonlinear morphology filters provide a useful tool for image analysis and computer vision, specially for treating noisy images. Serra (1989) has defined morphological filters as idempotent increasing operators. Two most important classes of morphological filters are openings and closings. Usually, erosions and dilations are also considered as morphological filters because they are basic operations of mathematical morphology. The authors concentrate mainly on the statistical properties of multilevel erosions and openings, as dilations and closings are respectively the dual filters of erosions and openings. Very simple output distribution formulae for 1-D erosions and openings are derived in the case of independent non-identically distributed inputs. These distribution formulae are then applied to illustrate the noise suppression and edge preservation performances of morphological filters