Landmark matching on the sphere using distance functions

Nonlinear registration of 3D surfaces is important in many medical imaging applications, including the mapping of longitudinal changes in anatomy, or of multi-subject functional MRI data to a canonical surface for comparison and integration. To register 3D surfaces, such as the cortical surface of the brain, one approach is to transform them first to planar or spherical objects. Internal landmarks can then be matched on these simpler parameter domains. Here we study the diffeomorphic matching of landmarks on the sphere. Our method builds on the level set technique of Leow et al. for the plane. Both forward and backward matching terms are included, thus ensuring the invertibility of the representation. We demonstrate our technique on a pair of lines on the sphere. The overall approach improves on earlier work in cortical matching by allowing the matching energy to be relaxed along sulcal landmarks, minimizing distortion, and also enables point and curve landmarks to be aligned in the same general framework as densely-defined scalar fields, such as curvature or cortical thickness maps