  The problem of finding a time-dependent vector field which warps an initial
set of points to a target set is common in shape analysis. It is an example of
a problem in the diffeomorphic shape matching regime, and can be thought of as
a spatial discretization of diffeomorphic image matching. In this paper, we
consider landmark matching modified by restricting the set of available vector
fields in the sense that vector fields are parametrized by a set of controls.
We determine the geometric setting of the problem, referred to as
sub-Riemannian landmark matching, and derive the equations of motion for the
controls. We provide two computational algorithms and demonstrate them in
numerical examples. In particular, the experiments highlight the importance of
the regularization term. A strong motivation is that sub-Riemannian landmark
matching have connections with neural networks, in particular the
interpretation of residual neural networks as time discretizations of
continuous control problems. It allows shape analysis practitioners to think
about neural networks in terms of diffeomorphic landmark matching, thereby
providing a bridge between the two fields.
