Graph Partitioning and Distance Correlation Approaches to Brain Parcellation

Abstract

Functional brain parcellation is the task of dividing the brain into regions that reflect the statistical dependencies of their activation patterns as measured by functional MRI techniques. We adopt the graph partitioning approach to brain parcellation wherein we assign voxels in the fMRI data set to vertices of a weighted, undirected graph and connect them to their spatially adjacent neighbors. We introduce the use of a recently developed statistic for measuring linear and non-linear statistical dependency called distance correlation as the weights of edges adjoining neighboring voxels. Based on this idea, we pose a new graph cut-type problem called Maximize Average Within Edge (MAWE), wherein the objective is to maximize the sum for each component, of the average weight of all edges with both endpoints in the component. We explore a number of methods, both original and borrowed, for approximately solving MAWE subject to constraints on parcel smoothness and size balance. These methods include a family of heuristic graph-growing algorithms, spectral methods, methods based on symmetric nonnegative and binary matrix factorization, and mixed integer programming formulations. We conclude with an empirical analysis of these proposed partitioning methods on 6 fMRI scans of patients with autism spectrum disorder and 6 control subjects.