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 do- main 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 LI 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 SLIMMER concerning the following questions. How accurately can the positions of two closely spaced scatterers be estimated? What is the closest distance of two scat- terers 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 approxima- tions. Although we take TomoSAR as the preferred application, the SLIMMER algorithm and our results on SR are generally applicable to sparse spectral estimation, including SR SAR focus- ing 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 1.5-25 for the aforementioned parameter ranges of N and SNR.

[1]  Stefano Tebaldini,et al.  On the Role of Phase Stability in SAR Multibaseline Applications , 2010, IEEE Transactions on Geoscience and Remote Sensing.

[2]  D. Mitchell Wilkes,et al.  The effects of phase on high-resolution frequency estimators , 1993, IEEE Trans. Signal Process..

[3]  David R. Anderson,et al.  Multimodel Inference , 2004 .

[4]  J. Rissanen,et al.  Modeling By Shortest Data Description* , 1978, Autom..

[5]  Gianfranco Fornaro,et al.  Four-Dimensional SAR Imaging for Height Estimation and Monitoring of Single and Double Scatterers , 2009, IEEE Transactions on Geoscience and Remote Sensing.

[6]  David L Donoho,et al.  Compressed sensing , 2006, IEEE Transactions on Information Theory.

[7]  Gianfranco Fornaro,et al.  Three-dimensional multipass SAR focusing: experiments with long-term spaceborne data , 2005, IEEE Transactions on Geoscience and Remote Sensing.

[8]  Joachim H. G. Ender,et al.  On compressive sensing applied to radar , 2010, Signal Process..

[9]  Petre Stoica,et al.  Spectral Analysis of Signals , 2009 .

[10]  G. Schwarz Estimating the Dimension of a Model , 1978 .

[11]  Michael Elad,et al.  From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images , 2009, SIAM Rev..

[12]  Fabrizio Lombardini,et al.  Differential tomography: a new framework for SAR interferometry , 2005, IEEE Transactions on Geoscience and Remote Sensing.

[13]  Gilda Schirinzi,et al.  SAR tomography from sparse samples , 2009, 2009 IEEE International Geoscience and Remote Sensing Symposium.

[14]  Richard G. Baraniuk,et al.  Compressive Sensing , 2008, Computer Vision, A Reference Guide.

[15]  Paco López-Dekker,et al.  A Novel Strategy for Radar Imaging Based on Compressive Sensing , 2010, IEEE Transactions on Geoscience and Remote Sensing.

[16]  Michael Eineder,et al.  High Resolution Interferometric Stacking with TerraSAR-X , 2008, IGARSS 2008 - 2008 IEEE International Geoscience and Remote Sensing Symposium.

[17]  Richard Bamler,et al.  Tomographic SAR Inversion by $L_{1}$ -Norm Regularization—The Compressive Sensing Approach , 2010, IEEE Transactions on Geoscience and Remote Sensing.

[18]  David N. Swingler,et al.  Frequency Estimation for Closely Spaced Sinsoids: Simple Approximations to the Cramer-Rao Lower Bound , 1993, IEEE Trans. Signal Process..

[19]  Alberto Moreira,et al.  Estimation of the Minimum Number of Tracks for SAR Tomography , 2009, IEEE Transactions on Geoscience and Remote Sensing.

[20]  E.J. Candes Compressive Sampling , 2022 .

[21]  Michael A. Saunders,et al.  Atomic Decomposition by Basis Pursuit , 1998, SIAM J. Sci. Comput..

[22]  Mengdao Xing,et al.  Achieving Higher Resolution ISAR Imaging With Limited Pulses via Compressed Sampling , 2009, IEEE Geoscience and Remote Sensing Letters.

[23]  Gilda Schirinzi,et al.  Three-Dimensional SAR Focusing From Multipass Signals Using Compressive Sampling , 2011, IEEE Transactions on Geoscience and Remote Sensing.

[24]  Richard Bamler,et al.  Very High Resolution Spaceborne SAR Tomography in Urban Environment , 2010, IEEE Transactions on Geoscience and Remote Sensing.

[25]  Peyman Milanfar,et al.  Improved spectral analysis of nearby tones using local detectors , 2005, Proceedings. (ICASSP '05). IEEE International Conference on Acoustics, Speech, and Signal Processing, 2005..

[26]  R.G. Baraniuk,et al.  Compressive Sensing [Lecture Notes] , 2007, IEEE Signal Processing Magazine.

[27]  Volker Nannen,et al.  A Short Introduction to Model Selection, Kolmogorov Complexity and Minimum Description Length (MDL) , 2010, ArXiv.

[28]  Richard Bamler,et al.  Let's Do the Time Warp: Multicomponent Nonlinear Motion Estimation in Differential SAR Tomography , 2011, IEEE Geoscience and Remote Sensing Letters.

[29]  S. Gernhardt,et al.  Interferometric Potential of High Resolution Spaceborne SAR , 2009 .