Region tracking on surfaces deforming via level-sets methods

Since the work by Osher and Sethian on level-sets algorithms for numerical shape evolutions, this technique has been used for a large number of applications in numerous fields. In medical imaging, this numerical technique has been successfully used for example in segmentation and cortex unfolding algorithms. The migration from a Lagrangian im- plementation to an Eulerian one via implicit representations or level-sets brought some of the main advantages of the technique, mainly, topology independence and stability. This migration means also that the evolution is parametrization free, and therefore we do not know exactly how each part of the shape is deforming, and the point-wise correspondence is lost. In this note we present a technique to numerically track regions on sur- faces that are being deformed using the level-sets method. The basic idea is to represent the region of interest as the intersection of two implicit surfaces, and then track its deformation from the deformation of these surfaces. This technique then solves one of the main shortcomings of the very useful level-sets approach. Applications include lesion localization in medical images, region tracking in functional MRI visualization, and geometric surface mapping.