Minimal Surfaces: A Three Dimensional Segmentation Approach

A novel geometric approach for three dimensional object segmentation is presented. The scheme is based on geometric deformable surfaces moving towards the objects to be detected. We show that this model is equivalent to the computation of surfaces of minimal area (local minimal surfaces) in a Riemannian space. This space is deened by a metric induced from the three dimensional image in which the objects are to be detected. The model also shows the relation between classical deformable surfaces obtained via energy minimization, and geometric ones derived from curvature ows in the surface evolution framework. The new approach is stable, robust, and automatically handles changes in the surface topology during the deformation. Results related to existence, uniqueness, stability, and correctness of the solution to this geometric deformable model are presented as well. Based on an eecient numerical algorithm for surface evolution, we present a number of examples of object detection in real and synthetic images.