SHAPE-DISCRIMINATION IN MATHEMATICAL MORP HOLOGY

To any compact planar simply connected shape X, is associated its eros ion curve defined by PSI(X): r is-an-element-of R+ --> A (X - r B) whe re A (X) represents the surface area of X and X - r B the erosion of X by the hall of radius r. This Note presents some answers to the quest ion: characterize all the Y verifying PSI(X) = PSI(Y) for a given X. U nder some regularity conditions, PSI(X) is expressed as an integral of the border distance function q(X) along the skeleton of this shape. I t is shown that the erosion curve is not affected if one bends the arc s of the skeleton : PSI(X) quantifies ''soft'' shapes. Moreover, in th e generic case, PSI(X)'' characterises five possible cases of behaviou r of the skeleton: simple point (local maximum, local minimum or neith er of q(X)), multiple point (local maximum or not of q(X)).