Sensitivity considerations in compressed sensing

In [1]-[4], we considered the question of basis mismatch in compressive sensing. Our motivation was to study the effect of mismatch between the mathematical basis (or frame) in which a signal was assumed to be sparse and the physical basis in which the signal was actually sparse. We were motivated by the problem of inverting a complex space-time radar image for the field of complex scatterers that produced the image. In this case there is no apriori known basis in which the image is actually sparse, as radar scatterers do not usually agree to place their ranges and Dopplers on any apriori agreed sampling grid. The consequence is that sparsity in the physical basis is not maintained in the mathematical basis, and a sparse inversion in the mathematical basis or frame does not match up with an inversion for the field in the physical basis. In [1]-[3], this effect was quantified with theorem statements about sensitivity to basis mismatch and with numerical examples for inverting time series records for their sparse set of damped complex exponential modes. These inversions were compared unfavorably to inversions using fancy linear prediction. In [4] and this paper, we continue these investigations by comparing the performance of sparse inversions of sparse images, using apriori selected frames that are mismatched to the physical basis, and by computing the Fisher information matrix for compressions of images that are sparse in a physical basis.