Skeletons via Shocks of Boundary Evolution

Abstract

In this chapter we develop the Hamiltonian formulation of the eikonal equation and then relate it to the specific case of Blum’s grassfire flow, which gives the level sets of the Euclidean distance function to the boundary. This view provides an explicit association between medial loci and the singularities of this flow. In order to detect these singularities we consider the average outward flux of the gradient of the Euclidean distance function. This measure has very different limiting behaviors depending upon whether the region over which it is computed shrinks to a singular point or a non-singular one. At medial loci the limiting values are related to the object angle. We combine the flux measurement with a homotopy preserving thinning process applied in a discrete lattice. This leads to a robust algorithm for computing skeletons in 2D as well as 3D, which has low computational complexity. We also discuss a related approach to associate medial loci with locations where the gradient of the Euclidean distance function is multi-valued, which uses the object angle as a measure of salience. Both approaches are illustrated with computational examples.