  Non-rigid cortical registration is an important and challenging task due to
the geometric complexity of the human cortex and the high degree of
inter-subject variability. A conventional solution is to use a spherical
representation of surface properties and perform registration by aligning
cortical folding patterns in that space. This strategy produces accurate
spatial alignment but often requires a high computational cost. Recently,
convolutional neural networks (CNNs) have demonstrated the potential to
dramatically speed up volumetric registration. However, due to distortions
introduced by projecting a sphere to a 2D plane, a direct application of recent
learning-based methods to surfaces yields poor results. In this study, we
present SphereMorph, a diffeomorphic registration framework for cortical
surfaces using deep networks that addresses these issues. SphereMorph uses a
UNet-style network associated with a spherical kernel to learn the displacement
field and warps the sphere using a modified spatial transformer layer. We
propose a resampling weight in computing the data fitting loss to account for
distortions introduced by polar projection, and demonstrate the performance of
our proposed method on two tasks, including cortical parcellation and
group-wise functional area alignment. The experiments show that the proposed
SphereMorph is capable of modeling the geometric registration problem in a CNN
framework and demonstrate superior registration accuracy and computational
efficiency. The source code of SphereMorph will be released to the public upon
acceptance of this manuscript at https://github.com/voxelmorph/spheremorph.
