The Hamilton-Jacobi Skeleton

The eikonal equation and variants of it are of signii-cant 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 rst introduce a new algorithm for simulating the eikonal equation, which ooers a number of computational and conceptual advantages over the earlier methods when it comes to shock tracking. Next, we introduce a very eecient algorithm for shock detection, where the key idea is to measure the net outward ux of a vector eld 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.