Evaluating the impact of spatio-temporal smoothness constraints on the BOLD hemodynamic response function estimation: an analysis based on Tikhonov regularization.

Recently we have proposed the use of Tikhonov regularization with temporal smoothness constraints to estimate the BOLD fMRI hemodynamic response function (HRF). The temporal smoothness constraint was imposed on the estimates by using second derivative information while the regularization parameter was selected based on the generalized cross-validation function (GCV). Using one-dimensional simulations, we previously found this method to produce reliable estimates of the HRF time course, especially its time to peak (TTP), being at the same time fast and robust to over-sampling in the HRF estimation. Here, we extend the method to include simultaneous temporal and spatial smoothness constraints. This method does not need Gaussian smoothing as a pre-processing step as usually done in fMRI data analysis. We carried out two-dimensional simulations to compare the two methods: Tikhonov regularization with temporal (Tik-GCV-T) and spatio-temporal (Tik-GCV-ST) smoothness constraints on the estimated HRF. We focus our attention on quantifying the influence of the Gaussian data smoothing and the presence of edges on the performance of these techniques. Our results suggest that the spatial smoothing introduced by regularization is less severe than that produced by Gaussian smoothing. This allows more accurate estimates of the response amplitudes while producing similar estimates of the TTP. We illustrate these ideas using real data.