Correlated gaussian designs for compressive imaging

Statistical correlations among wavelet transform coefficients of images are commonly represented using graphical models. But in linear inverse problems like compressed sensing, the sensing matrix linearly mixes up these dependencies, making recovery of the transform coefficients difficult. Past work has involved using greedy methods to recover images in a compressed sensing framework. Recently, message passing and group lasso based methods have been shown to perform at least as well as traditional approaches. Group lasso based methods are especially viable, since they provide the guarantees that come along with solving a convex program. Standard sensing matrices are well-suited to the recovery of unstructured sparse signals, but the sparsity patterns of natural images are highly structured. In this paper, we look to exploit the intra-group dependencies among coefficients to design sensing matrices that are better matched to image structure than conventional compressed sensing matrices. We show that the new sensing matrices based on structural prior knowledge yield considerably better results compared to standard sensing matrices.