  Diffeomorphic registration using optimal control on the diffeomorphism group
and on shape spaces has become widely used since the development of the Large
Deformation Diffeomorphic Metric Mapping (LDDMM) algorithm. More recently, a
series of algorithms involving sub-riemannian constraints have been introduced,
in which the velocity fields that control the shapes in the LDDMM framework are
constrained in accordance with a specific deformation model. Here, we extend
this setting by considering, for the first time, inequality constraints, in
order to estimate surface deformations that only allow for atrophy, introducing
for this purpose an algorithm that uses the augmented lagrangian method. We
also provide a version of our approach that uses a weaker constraint in which
only the total volume is forced to decrease. These developments are illustrated
by numerical experiments on brain data.
