Using Deformable Surfaces to Segment 3-D Images and Infer Differential Structures

Abstract

In this paper, we generalize the deformable model [4, 7] to a 3-D model, which evolves in 3-D images, under the action of internal forces (describing some elasticity properties of the surface), and external forces attracting the surface toward some detected edgels. Our formalism leads to the minimization of an energy which is expressed as a functional. We use a variational approach and a finite element method to actually express the surface in a discrete basis of continuous functions. This leads to a reduced computational complexity and a better numerical stability. The power of the present approach to segment 3-D images is demonstrated by a set of experimental results on various complex medical 3-D images. Another contribution of this approach is the possibility to infer easily the differential structure of the segmented surface. As we end-up with an analytical description of the surface, this allows to compute for instance its first and second fundamental forms. From this, one can extract a curvature primal sketch of the surface, including some intrinsic features which can be used as landmarks for 3-D image interpretation.