A Model for Elastic Evolution on Foliated Shapes

We study a shape evolution framework in which the deformation of shapes from time t to t+ dt is governed by a regularized anisotropic elasticity model. More precisely, we assume that at each time shapes are infinitesimally deformed from a stress-free state to an elastic equilibrium as a result of the application of a small force. The configuration of equilibrium then becomes the new resting state for subsequent evolution. The primary motivation of this work is the modeling of slow changes in biological shapes like atrophy, where a body force applied to the volume represents the location and impact of the disease. Our model uses an optimal control viewpoint with the time derivative of force interpreted as a control, deforming a shape gradually from its observed initial state to an observed final state. Furthermore, inspired by the layered organization of cortical volumes, we consider a special case of our model in which shapes can be decomposed into a family of layers (forming a foliation). Preliminary experiments on synthetic layered shapes in two and three dimensions are presented to demonstrate the effect of elasticity.