A fundamental bound for super-resolution - With application to 3D SAR imaging

Resolution is a crucial aspect for urban imaging where structures are in the same spatial scale as the resolution of the imaging instrument. This is particularly true for 3D SAR imaging, also referred to as SAR Tomography (TomoSAR). We address the problem of super-resolution (SR), i.e. the ability to resolve two closely spaced complex-valued points from N irregular Fourier domain samples. Our target application is TomoSAR where the typical number of acquisitions N = 10...100 and the SNR = 0...10dB. As the TomoSAR algorithm we introduce "Scale-down by L1 norm Minimization, Model selection, and Estimation Reconstruction" (SL1MMER), 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: 1) How accurately can the positions of two closely spaced scatterers be estimated? 2) What is the closest separable distance of two scatterers? 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, establish a fundamental bound for SR of spectral estimators and imaging. We show that SR factors are in the range of 1.5 to 25 for the aforementioned parameter ranges of N and SNR.