Maintaining the point correspondence in the level set framework

In this paper, we propose a completely Eulerian approach to maintain a point correspondence during a level set evolution. Our work is in the spirit of some recent methods (D. Adalsteinsson, J. Sethian, Transport and diffusion of material quantities on propagating interfaces via level set methods, Journal of Computational Physics 185(1) (2003) 271–288; J.-J. Xu, H.-K. Zhao, An Eulerian formulation for solving partial differential equations along a moving interface, Journal of Scientific Computing 19 (2003) 573–594) for handling interfacial data on moving level set interfaces. Our approach maintains an explicit backward correspondence from the evolving interface to the initial one, by advecting the initial point coordinates with the same velocity as the level set function. It leads to a system of coupled Eulerian partial differential equations. We describe in detail a robust numerical implementation of our approach, in accordance with the narrow band methodology. We show in a variety of numerical experiments that it can handle both normal and tangential velocities, large deformations, shocks, rarefactions and topological changes. The possible applications of our approach include scientific visualization, computer graphics and image processing.