Compressive imaging allows one to sample an image below the Nyquist rate yet still accurately recover it from the measurements by solving an L1 optimization problem. The L1 solvers, however, are iterative and can require significant time to reconstruct the original signal. Intuitively, the reconstruction time can be reduced by reconstructing fewer total pixels. The human eye reduces the total amount of data it processes by having a spatially varying resolution, a method called foveation. In this work, we use foveation to achieve a 4x improvement in L1 compressive sensing reconstruction speed for hyperspectral images and video. Unlike previous works, the presented technique allows the high-resolution region to be placed anywhere in the scene after the subsampled measurements have been acquired, has no moving parts, and is entirely non-adaptive.