Robust imaging of localized scatterers using the singular value decomposition and ℓ1 minimization

We consider narrow band, active array imaging of localized scatterers in a homogeneous medium with and without additive noise. We consider both single and multiple illuminations and study ℓ1 minimization-based imaging methods. We show that for large arrays, with array diameter comparable to range, and when scatterers are sparse and well separated, ℓ1 minimization using a single illumination and without additive noise can recover the location and reflectivity of the scatterers exactly. For multiple illuminations, we introduce a hybrid method which combines the singular value decomposition and ℓ1 minimization. This method can be used when the essential singular vectors of the array response matrix are available. We show that with this hybrid method we can recover the location and reflectivity of the scatterers exactly when there is no noise in the data. Numerical simulations indicate that the hybrid method is, in addition, robust to noise in the data. We also compare the ℓ1 minimization-based methods with others including Kirchhoff migration, ℓ2 minimization and multiple signal classification. (paper)