The notions of skeleton and crack, and singularities of the oriented distance function

In problems where a geometric object is the variable, the object can be identified with the oriented distance function which can simultaneously deal with the smooth sets of classical Differential Geometry and sets with a lousy boundary.This paper reviews some properties of the distance function dA to a set A , the oriented distance function bA=dA-dA ( A , the complement of A ), and the associated notions of skeletons , b -crack , and crack . It gives the respective partitions of the boundaries ∂ A and ∂ (∂ A) and the partition of the singularities of ∇ bA . It turns out that the notion of b -crack is possibly too broad since it also includes corners in the core boundary ̄∩∂ Ā of the set A . On one hand, this analysis leads to the notions of crack-free sets and strongly crack-free sets and, on the other hand, to the notion of cracked sets in Image Segmentation and Mathematical Morphology.