UC San Diego

This dissertation considers the problems of sparse signal recovery (SSR) and nuisance regression in functional MRI (fMRI). The first part of the dissertation introduces a Bayesian framework to recover sparse non-negative solutions in under-determined systems of linear equations. A novel class of probability density functions named Rectified Gaussian Scale Mixtures (R-GSM) is proposed to model the sparse non-negative solution of interest. A Bayesian inference algorithm called Rectified Sparse Bayesian Learning (R-SBL) is developed, which robustly recovers the solution in numerous experimental settings and outperforms the state-of-the-art SSR approaches by a large margin.

The rest of the dissertation investigates the effects of nuisance regression in fMRI. Chapter 3 proposes a mathematical framework to investigate the effects of global signal regression (GSR). GSR is a widely used nuisance removal approach in resting-state fMRI, however its use has been controversial since it introduces artifactual anti-correlations between pairs of fMRI signals. The proposed framework shows that the main effects of GSR can be well-approximated as a temporal down-weighting or temporal censoring process, in which the data from time points with relatively large GS magnitudes are greatly attenuated (or censored) while data from time points with relatively small GS magnitudes are largely retained. The censoring approximation reveals that the anti-correlated networks are intrinsic to the brain's functional organization and are not simply an artifact of GSR.

In Chapters 4 and 5, the effects of nuisance terms on the relationship between pairs of fMRI signals both before and after nuisance regression are investigated. It is shown that geometric norms of various nuisance regressors can significantly influence the correlation-based functional connectivity (FC) estimates in both static FC and dynamic FC studies. It is demonstrated that nuisance regression is largely ineffective in removing the significant correlations observed between FC estimates and nuisance norms. Consequently, a mathematical bound is derived on the difference between correlation coefficients before and after nuisance regression. This bound restricts the removal of nuisance norm effects from FC estimates.