Compressed Sensing, Sparse Inversion, and Model Mismatch

Abstract

The advent of compressed sensing theory has revolutionized our view of imaging, as it demonstrates that subsampling has manageable consequences for image inversion, provided that the image is sparse in an apriori known dictionary. For imaging problems in spectrum analysis (estimating complex exponential modes), and passive and active radar/sonar (estimating Doppler and angle of arrival), this dictionary is usually taken to be a DFT basis (or frame) constructed for resolution of 2π∕n, with n a window length, array length, or pulse-to-pulse processing length. However, in reality no physical field is sparse in a DFT frame or in any apriori known frame. No matter how finely we grid the parameter space (e.g., frequency, delay, Doppler, and/or wavenumber) the sources may not lie in the center of the grid cells and consequently there is always mismatch between the assumed and the actual frames for sparsity. But what is the sensitivity of compressed sensing to mismatch between the physical model that generated the data and the mathematical model that is assumed in the sparse inversion algorithm? In this chapter, we study this question. The focus is on the canonical problem of DFT inversion for modal analysis.