Global Signal Removal (GSR) as graph spatial filtering.

Global Signal Removal (GSR) is a widely applied step in functional magnetic resonance imaging (fMRI) preprocessing. Although GSR conventionally denotes 'Global Signal Regression,' we use 'Global Signal Removal' to encompass a broader family of spatial filtering operations. GSR in general remains controversial due to concerns about introducing spurious anticorrelations and removing neurally meaningful signals. In this paper, we provide a precise geometric characterization by formalizing GSR as graph spatial filtering. We demonstrate that the most common form of GSR, Regression-GSR, equates to a rank-1 deflation of the covariance matrix (i.e. functional connectivity) by the degree vector. Empirically, the degree vector is dominated by the first principal component of the functional connectivity matrix (correlation = 0.88 ± 0.12 in resting-state HCP data), making Regression-GSR an approximation to first eigenmode removal. This view of GSR as a spatial projection framework allows us to develop a family of GSR variants, each expressible in a unified spatial filter: Naive-GSR removes the uniform vector, PCA-GSR precisely removes the first eigenvector, and SC-GSR, a new variant we introduce that removes the first harmonic of the structural connectivity matrix. A key distinction emerges: while Naive, PCA, and SC-GSR are orthogonal projections, Regression-GSR is an oblique projection that computes regional weights proportional to the degree vector but removes a spatially uniform signal. All GSR variants induce numerical singularity in the covariance matrix, but they differ in their effects on task-state separability, which we examine empirically. In summary, we reframe GSR as a family of graph spatial filters that enable interpretability of its effects, with systematically varying effects on network connectivity across variants.