Quantitative signal subspace imaging

We will present a quantitative signal subspace imaging method. This imaging method is a generalization of MUSIC that uses both the noise and signal subspaces of the data. The noise subspace provides high spatial resolution while the signal subspace provides quantitative information about the targets. We show that this signal subspace method yields high resolution and quantitative images provided that the signal-to-noise ratio is sufficiently high. The relative balance between the noise and signal subspaces depends on the noise level in the data which is controlled through a user-defined regularization parameter, $\epsilon$. We will first consider the single-frequency array imaging problem and then explain how this method can be generalized for synthetic aperture radar (SAR) data. The first main result is the resolution analysis for this modified and generalized MUSIC method that shows an enhancement in resolution compared to classical imaging by a factor $\sqrt{\epsilon}$. The second main result is the stability analysis of the method to random perturbations of the travel times (for the SAR problem). This analysis shows that the method provides stable reconstructions when $\epsilon$ is chosen to satisfy $\sigma^2 \ll \epsilon < 1$ with $\sigma^2$ denoting the maximum variance of the random perturbations of the travel times. These findings will be illustrated with several numerical simulations.