A multi-resolution scheme for distortion-minimizing mapping between human subcortical structures based on geodesic construction on Riemannian manifolds

In this paper, we deal with a subcortical surface registration problem. Subcortical structures including hippocampi and caudates have a small number of salient features such as heads and tails unlike cortical surfaces. Therefore, it is hard, if not impossible, to perform subcortical surface registration with only such features. It is also non-trivial for neuroanatomical experts to select landmarks consistently for subcortical surfaces of different subjects. We therefore present a landmark-free approach for subcortical surface registration by measuring the amount of mesh distortion between subcortical surfaces assuming that the surfaces are represented by meshes. The input meshes can be constructed using any surface modeling tool available in the public domain since our registration method is independent of a surface modeling process. Given the source and target surfaces together with their representing meshes, the vertex positions of the source mesh are iteratively displaced while preserving the underlying surface shape in order to minimize the distortion to the target mesh. By representing each surface mesh as a point on a high-dimensional Riemannian manifold, we define a distance metric on the manifold that measures the amount of distortion from a given source mesh to the target mesh, based on the notion of isometry while penalizing triangle flipping. Under this metric, we reduce the distortion minimization problem to the problem of constructing a geodesic curve from the moving source point to the fixed target point on the manifold while satisfying the shape-preserving constraint. We adopt a multi-resolution framework to solve the problem for distortion-minimizing mapping between the source and target meshes. We validate our registration scheme through several experiments: distance metric comparison, visual validation using real data, robustness test to mesh variations, feature alignment using anatomic landmarks, consistency with previous clinical findings, and comparison with a surface-based registration method, LDDMM-surface.