  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.
