For many years, the basic goal of sparse aperture design has been to maximize the support of the modulation transfer function (MTF). Golay apertures and related nonredundant arrays are typically used to achieve this objective. Unfortunately, maximizing the support of the MTF has the necessary effect of decreasing the magnitude of the MTF at mid-band spatial frequencies. Fienup has shown that the decreased magnitude of the MTF for nonredundant arrays contributes as much as reduced throughput to the loss of SNR in sparse apertures relative to full aperture systems. This paper considers the use of periodic sparse arrays to improve the mid-band MTF at the cost of reduced spatial frequency coverage. We further consider methods to recover lost spatial frequencies using multispectral and multiframe sampling and decompressive inference. © 2010 SPIE.