Algorithms for Implicit Deformable Models

This paper presents a framework for implicit deformable models and a pair of new algorithms for solving the nonlinear partial di erential equations that result from this framework. Implicit models o er a useful alternative to parametric models, particularly when dealing with the deformation of higher-dimensional objects. The basic expressions for the evolution of implicit models are relatively straightforward; they follow as a direct consequence of the chain rule for di erentiation. More challenging, however, is the development of algorithms that are stable and e cient. The rst algorithm is a viscosity approximation which gives solutions over a dense set in the range, providing a means of calculating the solutions of embedded families of contours simultaneously. The second algorithm incorporates sparse solutions for a discrete set of contours. This sparseeld method requires a fraction of the computation compared to the rst but o ers solutions only for a nite number of contours. Results from 3d medical data as well as video images are shown.