[Paper Review] Graph Convolutions on Spectral Embeddings: Learning of Cortical Surface Data

Karthik Gopinath, Christian Desrosiers+1 more|arXiv (Cornell University)|Mar 27, 2018

Functional Brain Connectivity StudiesNeuroscience18 references4 citations

TL;DR

This paper introduces a novel graph convolutional framework that leverages spectral graph matching to transfer spectral embeddings across different cortical surface geometries, enabling direct learning of surface data via spectral filters. It achieves 82.2% Dice score in cortical parcellation—comparable to FreeSurfer’s 83%—while reducing computation time from hours to 18 seconds per subject.

ABSTRACT

Neuronal cell bodies mostly reside in the cerebral cortex. The study of this thin and highly convoluted surface is essential for understanding how the brain works. The analysis of surface data is, however, challenging due to the high variability of the cortical geometry. This paper presents a novel approach for learning and exploiting surface data directly across surface domains. Current approaches rely on geometrical simplifications, such as spherical inflations, a popular but costly process. For instance, the widely used FreeSurfer takes about 3 hours to parcellate brain surfaces on a standard machine. Direct learning of surface data via graph convolutions would provide a new family of fast algorithms for processing brain surfaces. However, the current limitation of existing state-of-the-art approaches is their inability to compare surface data across different surface domains. Surface bases are indeed incompatible between brain geometries. This paper leverages recent advances in spectral graph matching to transfer surface data across aligned spectral domains. This novel approach enables a direct learning of surface data across compatible surface bases. It exploits spectral filters over intrinsic representations of surface neighborhoods. We illustrate the benefits of this approach with an application to brain parcellation. We validate the algorithm over 101 manually labeled brain surfaces. The results show a significant improvement in labeling accuracy over recent Euclidean approaches, while gaining a drastic speed improvement over conventional methods.

Motivation & Objective

• Address the challenge of learning surface data across highly variable, non-uniform cortical geometries in neuroimaging.
• Overcome the limitation of existing graph convolutional methods that cannot compare data across different surface domains due to incompatible Laplacian eigenbases.
• Enable direct learning of cortical surface data using spectral filters over intrinsic neighborhood representations.
• Achieve high accuracy and speed in cortical parcellation by transferring spectral coordinates across aligned spectral domains.
• Demonstrate that spectral-based graph convolutions outperform Euclidean-based approaches in both accuracy and computational efficiency.

Proposed method

• Use spectral graph matching to align Laplacian eigenbases across different brain surface geometries, enabling cross-domain comparison of surface data.
• Represent cortical surfaces as graphs and embed them into a spectral manifold using the first few eigenvectors of the graph Laplacian.
• Apply graph convolutional networks (GCNs) with spectral filters that operate on intrinsic neighborhood structures in the spectral domain.
• Construct weighted adjacency matrices based on spectral coordinates to preserve local geometric patterns during convolution operations.
• Train the GCN using a supervised learning setup on manually labeled cortical parcellations, optimizing for Dice score.
• Apply Markov random field (MRF) regularization with Potts model to refine boundary smoothness post-prediction.

Experimental results

Research questions

• RQ1 Can spectral graph convolutions be effectively applied to cortical surface data across different geometries, despite incompatible Laplacian eigenbases?
• RQ2 Does learning in the spectral domain improve parcellation accuracy compared to conventional Euclidean-based graph convolutional methods?
• RQ3 To what extent does incorporating neighborhood information in the spectral domain enhance performance over pointwise spectral learning?
• RQ4 Can the proposed method achieve state-of-the-art parcellation accuracy while drastically reducing computation time compared to traditional methods like FreeSurfer?
• RQ5 How stable is the performance of the spectral approach across diverse cortical surfaces, as measured by variance in Dice scores?

Key findings

• The proposed spectral graph convolution framework achieves a mean Dice score of 82.2% (±5.3%) across 32 cortical parcels, significantly outperforming the Euclidean baseline (59.0%).
• The method reduces processing time from approximately 3 hours (FreeSurfer) to just 18 seconds per subject, representing an order-of-magnitude speedup.
• Incorporating neighborhood information in the spectral domain improves performance to 82.2% Dice score, compared to 75.6% when using only pointwise spectral information.
• The standard deviation of Dice scores across parcels is 5.3% for the spectral method, which is substantially lower than FreeSurfer’s 10.3% and the Euclidean approach’s 20.5%, indicating greater stability.
• MRF regularization further improves the spectral method’s accuracy to 85.4% and reduces the Hausdorff distance from 12.3mm to 10.6mm, demonstrating enhanced boundary quality.
• The method achieves performance on par with FreeSurfer (83.2% Dice in MindBoggle dataset) while offering a major advantage in computational efficiency.