Graph Slepian Framework for Guided Filtering With Application to Neuroimaging

Objective: Graph signal processing (GSP) has enabled new approaches for jointly analyzing graphs and graph signals. Various classical operations, such as the Fourier transform and filtering, have been extended to this setting, along with more advanced constructs including Slepian functions. The latter provides a basis for bandlimited graph signals that are maximally concentrated in a given subgraph. Here, we propose a novel approach that introduces complex values to encode several subgraphs, enabling a richer analysis of how graph signals are expressed.

Methods: The motivating application from neuroscience is to jointly analyze brain graphs obtained from diffusion-weighted magnetic resonance imaging (MRI), with brain graph signals from functional MRI. The brain activity measured by the latter is constrained by the underlying brain graph. Complex-valued graph Slepians constructed with prior knowledge from well-known task-positive and -negative functional networks can then reflect how activity is dynamically reorganizing.

Results: The feasibility of the approach is demonstrated using synthetic data first, and then applied to data from the Human Connectome Project, revealing patterns of brain network interactions. Results are currently limited to two subgraphs, but future work will explore more extensive graph configurations.

Conclusion: Slepian functions offer new ways to decode graph signals lying on top of a graph structure.

Significance: This confirms that the proposed method provides a new representation for studying brain activity constrained by the brain's structural connectivity.