Coarse-to-fine segmentation and tracking using Sobolev active contours

Recently proposed Sobolev active contours introduced a new paradigm for minimizing energies defined on curves by changing the traditional cost of perturbing a curve and thereby redefining gradients associated to these energies. Sobolev active contours evolve more globally and are less attracted to certain intermediate local minima than traditional active contours, and it is based on a well-structured Riemannian metric, which is important for shape analysis and shape priors. In this paper, we analyze Sobolev active contours using scale-space analysis in order to understand their evolution across different scales. This analysis shows an extremely important and useful behavior of Sobolev contours, namely, that they move successively from coarse to increasingly finer scale motions in a continuous manner. This property illustrates that one justification for using the Sobolev technique is for applications where coarse-scale deformations are preferred over fine-scale deformations. Along with other properties to be discussed, the coarse-to-fine observation reveals that Sobolev active contours are, in particular, ideally suited for tracking algorithms that use active contours. We will also justify our assertion that the Sobolev metric should be used over the traditional metric for active contours in tracking problems by experimentally showing how a variety of active-contour-based tracking methods can be significantly improved merely by evolving the active contour according to the Sobolev method.