AbstractWe consider optimal matching of submanifolds such as curves and
surfaces by a variational approach based on Hilbert spaces of diffeomorphic transformations.
In an abstract setting, the optimal matching is formulated as a minimization
problem involving actions of diffeomorphisms on regular Borel measures considered as
supporting measures of the reference and the target submanifolds. The objective functional
consists of two parts measuring the elastic energy of the dynamically deformed
surfaces and the quality of the matching. To make the problem computationally accessible,
we use reproducing kernel Hilbert spaces with radial kernels and weighted
sums of Dirac measures which gives rise to diffeomorphic point matching and amounts
to the solution of a finite dimensional minimization problem. We present a matching
algorithm based on the first order necessary optimality conditions which include an
initial-value problem for a dynamical system in the trajectories describing the deformation
of the surfaces and a final-time problem associated with the adjoint equations.
The performance of the algorithm is illustrated by numerical results for examples from
medical image analysis.