Improving the resolution of magnetic resonance imaging (MRI), or, alternatively, reducing the acquisition time, can be quite beneficial for many applications. The main motivation of this work is the assumption that any information that is a priori available on the target image could be used to achieve this goal. In order to demonstrate this approach, we present a novel partial acquisition strategy and reconstruction algorithm, suitable for the special case of detection of pseudoperiodic patterns. Pseudoperiodic patterns are frequently encountered in the cerebral cortex due to its columnar functional organization (best exemplified by orientation columns and ocular dominance columns of the visual cortex). We present a new MRI research methodology, in which we seek an activity pattern, and a pattern-specific experiment is devised to detect it. Such specialized experiments extend the limits of conventional MRI experiments by substantially reducing the scan time. Using the fact that pseudoperiodic patterns are localized in the Fourier domain, we present an optimality criterion for partial acquisition of the MR signal and a strategy for obtaining the optimal discrete Fourier transform (DFT) coefficients. A by-product of this strategy is an optimal linear extrapolation estimate. We also present a nonlinear spectral extrapolation algorithm, based on projections onto convex sets (POCSs), used to perform the actual reconstruction. The proposed strategy was tested and analyzed on simulated signals and in MRI phantom experiments.