Analyzing Dynamical Brain Functional Connectivity as Trajectories on Space of Covariance Matrices

Abstract

Human brain functional connectivity (FC) is often measured as the similarity of functional MRI responses across brain regions when a brain is either resting or performing a task. This paper aims to statistically analyze the dynamic nature of FC by representing the collective time-series data, over a set of brain regions, as a trajectory on the space of covariance matrices, or symmetric-positive definite matrices (SPDMs). We use a recently developed metric on the space of SPDMs for quantifying differences across FC observations, and for clustering and classification of FC trajectories. To facilitate large scale and high-dimensional data analysis, we propose a novel, metricbased dimensionality reduction technique to reduce data from large SPDMs to small SPDMs. We illustrate this comprehensive framework using data from the Human Connectome Project (HCP) database for multiple subjects and tasks, with task classification rates that match or outperform state-of-the-art techniques.

Fractional Sobolev metrics on spaces of immersed curves

“…More recently, curves with values in a manifold have emerged as a topic of interest in shape analysis as well. Examples include the study of trajectories on the earth [34,35], of computer animations [18], or of brain connectivity data [19]. Here the brain connectivity of a patient over time is represented as a path in the space of positive, symmetric matrices.…”

Supervised deep learning of elastic SRV distances on the shape space of curves

“…To demonstrate our network's generalization capabilities, we trained it on one type of data and tested it on a different dataset, e.g., we trained on synthetic data but tested on CPC precipitation data. While this leads to a slight increase in prediction error, the network still outperforms DP on both measures by a large margin, see Table III Datasets: We used data from the MPEG-7 4 and Swedish leaf datasets 5 , which contain images of objects whose boundaries were extracted and treated as 2D curves, discretized with n = 100 points, see Fig. 3.…”

“…Motivated by applications from computer vision to bioinformatics, the field of elastic shape analysis deals with problems where one needs to analyze the variability of geometric objects [18], [1], [19], [13], [5]. In this article, we address the computation of elastic geodesic distances between geometric curves in one and higher dimensions.…”

“…Source: https://dabi.temple.edu/external/shape/MPEG7/dataset.html5 Source: https://www.cvl.isy.liu.se/en/research/datasets/swedish-leaf/…”

$k$-means on Positive Definite Matrices, and an Application to Clustering in Radar Image Sequences

“…For example, SPD matrices correspond bijectively to mean centered Gaussian distributions, and are used to model Brownian motion in Diffusion Tensor Imaging (DTI), where they are referred to as tensors [1]. The finite-lag autocovariance matrices of time-series are SPD, and have been used in compression based clustering [2], for analysing dynamical brain functional connectivity [3], and in our application (Section III). Many more examples are mentioned in [1], [4].…”