VARIATIONAL ASYMPTOTIC DERIVATION OF AN ELASTIC MODEL ARISING FROM THE PROBLEM OF 3D AUTOMATIC SEGMENTATION OF CARDIAC IMAGES
Résumé
Segmentation of 3D cardiac images using a deformable elastic model of the heart proved to be significantly improved by applying special boundary conditions on the elastic model [15]. The purpose of this paper is to derive those boundary conditions by means of a rigourous convergence result. We consider a simplified two-layer elastic shell model and show that when the thickness ε of the thin external fibrous layer tends to 0 it can be replaced by the above mentioned boundary conditions on the internal layer. A mixed variational formulation of the problem in curvilinear coordinates is introduced. This formulation is then scaled in order to be defined over an ε-independent domain. Finally, several a priori estimations on the solution are obtained which enable us to pass to the limit and prove our result.