  We present formulations and numerical algorithms for solving diffeomorphic
shape matching problems. We formulate shape matching as a variational problem
governed by a dynamical system that models the flow of diffeomorphism $f_t \in
\operatorname{diff}(\mathbb{R}^3)$. We overview our contributions in this area,
and present an improved, matrix-free implementation of an operator splitting
strategy for diffeomorphic shape matching. We showcase results for
diffeomorphic shape matching of real clinical cardiac data in $\mathbb{R}^3$ to
assess the performance of our methodology.
