  Cortical surface registration plays a crucial role in aligning cortical
functional and anatomical features across individuals. However, conventional
registration algorithms are computationally inefficient. Recently,
learning-based registration algorithms have emerged as a promising solution,
significantly improving processing efficiency. Nonetheless, there remains a gap
in the development of a learning-based method that exceeds the state-of-the-art
conventional methods simultaneously in computational efficiency, registration
accuracy, and distortion control, despite the theoretically greater
representational capabilities of deep learning approaches. To address the
challenge, we present SUGAR, a unified unsupervised deep-learning framework for
both rigid and non-rigid registration. SUGAR incorporates a U-Net-based
spherical graph attention network and leverages the Euler angle representation
for deformation. In addition to the similarity loss, we introduce fold and
multiple distortion losses, to preserve topology and minimize various types of
distortions. Furthermore, we propose a data augmentation strategy specifically
tailored for spherical surface registration, enhancing the registration
performance. Through extensive evaluation involving over 10,000 scans from 7
diverse datasets, we showed that our framework exhibits comparable or superior
registration performance in accuracy, distortion, and test-retest reliability
compared to conventional and learning-based methods. Additionally, SUGAR
achieves remarkable sub-second processing times, offering a notable speed-up of
approximately 12,000 times in registering 9,000 subjects from the UK Biobank
dataset in just 32 minutes. This combination of high registration performance
and accelerated processing time may greatly benefit large-scale neuroimaging
studies.
