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Independent component analysis (ICA) has proved to be a powerful method for exploratory analysis of functional magnetic resonance imaging (fMRI) data. It has been used to uncover unexpected activations in fMRI data derived from brain activation. ICA has been used to characterize other sources of variability in the fMRI signal besides task-related activity, as well as challenging some of the assumptions inherent in other fMRI analysis methods. As a data-driven fMRI analysis technique, the philosophy of ICA is often in disagreement with hypothesis-driven methods. By exploiting the fact that much of fMRI data has deterministic spatial-temporal structure, a scheme employing ICA denoising and least squares (LS) estimation of the evoked hemodynamic response (HDR) is proposed. Simulations suggest that the method is more robust to different noise models compared to naive application of LS. The result is a considerably increased level of significance of activation for a given voxel but still qualitatively similar spatial distribution of activations over all voxels. This suggests that the proposed method has the potential to substantially reduce total scanning time requirements to achieve the same level of statistically significant activation.