A kernel class allowing for fast computations in shape spaces induced by diffeomorphisms

Reproducing kernel Hilbert spaces play an important role in diffeomorphic matching of shapes and in which they intervene in the construction of Riemannian metrics on diffeomorphisms and shape spaces. In such contexts, they are directly involved in the expressions of geodesic equations, and in their numerical solutions via particle evolutions. Solving such equations, however, involves computing kernel sums over irregular grids which can be a big computational overhead if the number of particles is large. In this paper we introduce and establish properties of a finitely generated kernel class in which the kernel is defined using a double interpolation from a discrete kernel supported by a regular grid covering the domain of the system of particles under consideration. It not only speeds up the calculations by utilizing standard algorithms for faster computations over regular grids, but also maintains the exactness and consistency of the system. We provide experimental results in support of this, comparing in particular the computation time and accuracy to similar competing methods.