  The analysis of the brain surface modeled as a graph mesh is a challenging
task. Conventional deep learning approaches often rely on data lying in the
Euclidean space. As an extension to irregular graphs, convolution operations
are defined in the Fourier or spectral domain. This spectral domain is obtained
by decomposing the graph Laplacian, which captures relevant shape information.
However, the spectral decomposition across different brain graphs causes
inconsistencies between the eigenvectors of individual spectral domains,
causing the graph learning algorithm to fail. Current spectral graph
convolution methods handle this variance by separately aligning the
eigenvectors to a reference brain in a slow iterative step. This paper presents
a novel approach for learning the transformation matrix required for aligning
brain meshes using a direct data-driven approach. Our alignment and graph
processing method provides a fast analysis of brain surfaces. The novel
Spectral Graph Transformer (SGT) network proposed in this paper uses very few
randomly sub-sampled nodes in the spectral domain to learn the alignment matrix
for multiple brain surfaces. We validate the use of this SGT network along with
a graph convolution network to perform cortical parcellation. Our method on 101
manually-labeled brain surfaces shows improved parcellation performance over a
no-alignment strategy, gaining a significant speed (1400 fold) over traditional
iterative alignment approaches.
