  Convolutional Neural Networks (CNNs) have been providing the state-of-the-art
performance for learning-related problems involving 2D/3D images in Euclidean
space. However, unlike in the Euclidean space, the shapes of many structures in
medical imaging have a spherical topology in a manifold space, e.g., brain
cortical or subcortical surfaces represented by triangular meshes, with large
inter-subject and intrasubject variations in vertex number and local
connectivity. Hence, there is no consistent neighborhood definition and thus no
straightforward convolution/transposed convolution operations for
cortical/subcortical surface data. In this paper, by leveraging the regular and
consistent geometric structure of the resampled cortical surface mapped onto
the spherical space, we propose a novel convolution filter analogous to the
standard convolution on the image grid. Accordingly, we develop corresponding
operations for convolution, pooling, and transposed convolution for spherical
surface data and thus construct spherical CNNs. Specifically, we propose the
Spherical U-Net architecture by replacing all operations in the standard U-Net
with their spherical operation counterparts. We then apply the Spherical U-Net
to two challenging and neuroscientifically important tasks in infant brains:
cortical surface parcellation and cortical attribute map development
prediction. Both applications demonstrate the competitive performance in the
accuracy, computational efficiency, and effectiveness of our proposed Spherical
U-Net, in comparison with the state-of-the-art methods.
