Image Segmentation with a Sobolev Gradient Method

The most effective methods for finding object boundaries in a digital image involve minimizing a functional over a set of curves or surfaces, where the functional includes internal energy terms for regularization and external energy terms that allign the curves or surfaces with object boundaries. Current practice is to seek critical points of the energy functional by what amounts to a steepest descent iteration with the discretized L2 gradient. Since the functional involves derivatives, a descent method with a discretized Sobolev gradient is likely to be much more efficient. We demonstrate this with test results for an implementation of a variational level set method for edge-based segmentation with active contours in two dimensions.