Due to the higher sensitivity to bold oxygenation level dependent (BOLD) signals and the ability to distinguish neurally related signals, multi-echo fMRI (ME-fMRI) has been a hot topic of research recently. Weighted combination in ME-fMRI is critical for the quality of the combined signal, which can be used for the analysis of brain functions. However, a convincing weighting method has never been put forward in previously published research. In this work, we optimized the weighting methods based on the Rayleigh quotient when using temporal signal-noise-ratio (tSNR) and multi-echo temporal signal-noise-ratio (metSNR) as metrics. These two metrics strongly represent the contrast-noise-ratio (CNR) which has been widely used to estimate fMRI quality. In our case of steady-state fMRI, with the optimal weighting methods, both tSNR and metSNR of the combined signal were improved significantly as compared to other widely-used weighting methods. In the meantime, we obtained negative weights that were not used in the combination for previous research, which drove us to find the causes and re-define the normalization as 'L1 Normalization'. We also found the distribution characteristics of negative weights in terms of brain region and TE. The occurrence and features of negative weights were explained by the correlation between echoes' data and the change of the multipliers (e.g. the mean value). In addition, we found that the metSNR performs more robustly than tSNR when comparing non-optimal methods to optimal methods and explained their variations based on the ratio from a mathematical viewpoint.

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.