The Hamilton-Jacobi Skeleton

The eikonal equation and variants of it are of significant interest for problems in computer vision and image processing. It is the basis for continuous versions of mathematical morphology, stereo, shape-from-shading and for recent dynamic theories of shape. Its numerical simulation can be delicate, owing to the formation of singularities in the evolving front, and is typically based on level set methods. However, there are more classical approaches rooted in Hamiltonian physics, which have received little consideration in computer vision. In this paper we first introduce a new algorithm for simulating the eikonal equation, which offers a number of computational and conceptual advantages over the earlier methods when it comes to shock tracking. Next, we introduce a very efficient algorithm for shock detection, where the key idea is to measure the net outward flux of a vector field per unit volume, and to detect locations where a conservation of energy principle is violated. We illustrate the approach with several numerical examples including skeletons of complex 2D and 3D shapes.