Splines for diffeomorphisms.

•This article proposes methods of high order parametric regression for modeling smooth diffeomorphic changes in 4D imaging data.•The acceleration controlled model generalizes the idea of cubic curves to manifold of diffeomorphisms and is capable of modeling nonmonotonic shape changes under the LDDMM setting.•Our shooting based solution to cubic curves enables parametrization of the full regression path using only initial conditions.•The shooting cubic splines smoothly fit complex shape trends while keeping data-independent (finite and few) parameters.•We present a numerically practical algorithm for regression of “non-geodesic” medical imaging data.