In blood-oxygenation-level-dependent functional magnetic resonance imaging (fMRI), current methods typically acquire ∼ 500,000 imaging voxels at each time point, and then use computer algorithms to reduce this data to the coefficients of a few hundred parcels or networks. This suggests that the amount of relevant information present in the fMRI signal is relatively small, and presents an opportunity to greatly improve the speed and signal to noise ratio (SNR) of the fMRI process. In this work, a theoretical framework is presented for calculating the coefficients of functional networks directly from highly undersampled fMRI data. Using predefined functional parcellations or networks and a compact k-space trajectory that samples data at optimal spatial scales, the problem of estimating network coefficients is reformulated to allow for direct least squares estimation, without Fourier encoding. By simulation, this approach is shown to allow for acceleration of the imaging process under ideal circumstances by nearly three orders of magnitude.