Algorithms for computing the group exponential of diffeomorphisms: Performance evaluation

In computational anatomy variability among medical images is encoded by a large deformation diffeomorphic mapping matching each instance with a template. The set of diffeomorphisms is usually endowed with a Riemannian manifold structure and parameterized by non-stationary velocity vector fields. An alternative parameterization based on stationary vector fields has been proposed, where paths of diffeomorphisms are the one-parameter subgroups, identified with the group exponential map. A log-Euclidean framework was proposed to compute statistics on finite dimensional Lie groups and later extended to diffeomorphisms. A fast algorithm based on the scaling and squaring (SS) method for the matrix exponential was applied to compute the exponential of diffeomorphisms. In this work we evaluate the performance of different approaches to compute the exponential in terms of accuracy and computational time. These approaches include forward Euler method, Taylor expansion, iterative composition, SS method, and a combination of interpolation and SS. In our results the SS method obtained the best performance trade-off, as it is accurate, fast and robust, but it has an intrinsic lower bound in accuracy. This lower bound can be partially overcome by oversampling the grid, at the expense of increased memory and time requirements. The Taylor expansion provided a fast alternative when spatial frequencies are small, and particularly for low ambient dimensions, but its convergence is not guaranteed in general.