Super-resolution power and robustness of compressive sensing for spectral estimation with application to spaceborne tomographic SAR

We address the problem of resolving two closely spaced complex-valued points from N irregular Fourier domain samples. Although this is a generic super-resolution (SR) problem, our target application is SAR tomography (TomoSAR), where typically the number of acquisitions is N = 10-100 and SNR = 0-10 dB. As the TomoSAR algorithm, we introduce Scale-down by L1 norm Minimization, Model selection, and Estimation Reconstruction (SL1MMER), which is a spectral estimation algorithm based on compressive sensing, model order selection, and final maximum likelihood parameter estimation. We investigate the limits of SL1MMER concerning the following questions. How accurately can the positions of two closely spaced scatterers be estimated? What is the closest distance of two scatterers such that they can be separated with a detection rate of 50% by assuming a uniformly distributed phase difference? How many acquisitions N are required for a robust estimation (i.e., for separating two scatterers spaced by one Rayleigh resolution unit with a probability of 90%)? For all of these questions, we provide numerical results, simulations, and analytical approximations. Although we take TomoSAR as the preferred application, the SL1MMER algorithm and our results on SR are generally applicable to sparse spectral estimation, including SR SAR focusing of point-like objects. Our results are approximately applicable to nonlinear least-squares estimation, and hence, although it is derived experimentally, they can be considered as a fundamental bound for SR of spectral estimators. We show that SR factors are in the range of 1.525 for the aforementioned parameter ranges of N and SNR.