Diffeomorphic Matching and Dynamic Deformable Surfaces in 3d Medical Imaging

We consider optimal matching of submanifolds such as curves and surfaces by a variational approach based on Hilbert spaces of diffeomorphic transfor- mations. 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 func- tional consists of two parts measuring the elastic energy of the dynamically deformed surfaces and the quality of the matching. To make the problem computationally ac- cessible, 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 defor- mation 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.