Unsupervised Learning of Cortical Surface Registration Using Spherical Harmonics

We present novel learning-based spherical registration using the spherical harmonics. Our goal is to achieve a continuous and smooth warp field that can effectively facilitate precise cortical surface registration. Conventional spherical registration typically involve sequential procedures for rigid and non-rigid alignments, which can potentially introduce substantial warp distortion. By contrast, the proposed method aims at joint optimization of both types of alignments. Inspired by a recent study that represents a rotation by 6D parameters as a continuous form in the Euclidean domain, we extend the idea to encode and regularize a velocity field. Specifically, a local velocity is represented by a single rotation with 6D parameters that can vary smoothly over the unit sphere via spherical harmonic decomposition, yielding smooth, spatially varying rotations. To this end, our method can lead to a significant reduction in warp distortion. We also incorporate a spherical convolutional neural network to achieve fast registration in an unsupervised manner. In the experiments, we compare our method with popular spherical registration methods on a publicly available human brain dataset. We show that the proposed method can significantly reduce warp distortion without sacrificing registration accuracy.