Functional magnetic resonance imaging brain activation directly from k-space

In functional magnetic resonance imaging (fMRI), the process of determining statistically significant brain activation is commonly performed in terms of voxel time series measurements after image reconstruction and magnitude-only time series formation. The image reconstruction and statistical activation processes are treated separately. In this manuscript, a framework is developed so that statistical analysis is performed in terms of the original, prereconstruction, complex-valued k-space measurements. First, the relationship between complex-valued (Fourier) encoded k-space measurements and complex-valued image measurements from (Fourier) reconstructed images is reviewed. Second, the voxel time series measurements are written in terms of the original spatiotemporal k-space measurements utilizing this k-space and image relationship. Finally, voxelwise fMRI activation can be determined in image space in terms of the original k-space measurements. Additionally, the spatiotemporal covariance between reconstructed complex-valued voxel time series can be written in terms of the spatiotemporal covariance between complex-valued k-space measurements. This allows one to utilize the originally measured data in its more natural, acquired state rather than in a transformed state. The effects of modeling preprocessing in k-space on voxel activation and correlation can then be examined.