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

Abstract

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 $d_A$ to a set $A$, the oriented distance function $b_A=d_A-d_{\complement A}$ ($\complement A$, the complement of $A$), and the associated notions of skeletons, $b$-crack, and crack. It gives the respective partitions of the boundaries $\partial A$ and $\partial (\partial A)$ and the partition of the singularities of $\nabla b_A$. It turns out that the notion of $b$-crack is possibly too broad since it also includes corners in the core boundary $\partial \overline{A}\cap \partial \overline{\complement A}$ 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.