1. 1 Easy Does It: User Parameter Free Dense and Sparse Methods for Spectral Estimation Jian Li Department of Electrical and Computer Engineering University of Florida Gainesville, Florida USA

2. 2 Spectral Estimation  The goal of spectral estimation is to determine how power distributes over frequency from a finite number of data samples.  Diverse Applications  For example: synthetic aperture radar (SAR) imaging.  Data-Independent Approaches  FFT, Matched Filter, Delay- and-Sum (DAS)  Poor resolution  High sidelobe levels, especially with missing data. A SAR imaging example using FFT.

3. 3 Data-Adaptive Spectral Estimation  Data-Adaptive Approaches  Examples: APES, Capon  Multiple snapshots needed to form reliable sample covariance matrices – fails for single or few snapshots, irregularly sampled data  High computational complexities High resolution Low sidelobe levels  Recent Development  Iterative Adaptive Approach (IAA)  Applicable to single snapshot scenario  High computational complexities High resolution Low sidelobe levels Dense and accurate WFFT IAA

4. 4 Iterative Adaptive Approach (IAA)  Each iteration of IAA includes two steps (user parameter free): Estimate coefficients: Update covariance matrix estimate

5. 5 IAA-R (IAA with Regularization)  Noise effect taken into account explicitly:  Still user parameter free!

6. 6 Active Sensing Example  Active sensing (radar, sonar, etc.)  Received signal decomposition:

7. Range-Doppler Imaging Matched Filter Initialization

8. Movies Are Nice Local Quadratic Convergence of IAA Proven.

9. Radar GMTI Example 9 Terrain map The goal of ground moving target indication (GMTI) is to detect slow moving targets in the stationary background. yellow or green dots: moving vehicles

10. STAP 10 AntennaElements Pulses slowtime 1 M N 1 MN samples for fixed range bin Range bins fasttime (J. Ward ’ 94) STAP: space-time adaptive processing Datacube:

11. Adaptive Processing 11 Space-Time Adaptive Processor (Guerci et al. ’ 06)

12. 12 Angle-Doppler Imaging in STAP dB IAA DAS Clutter power distribution over angle-Doppler for a fixed range

13. 13 Target angle: 195 A total of 200 targets with constant power Average SCNR over range: -18.94 dB Ground truth denoted by x o Simulated Ground Truth Target Detection for Fixed Angle

14. 14 Range-Doppler Images Ideal (total knowledge) Prior (wrong knowledge) IAA dB GLC (partial knowledge)

15. 15 ROC Curves Median CFAR algorithm applied to target detection GLC detector  Automatic diagonal loading  Sample Number N = 20 Prior detector  Wrong prior knowledge of the clutter-and-noise covariance matrix

16. 16 KASSPER DataSet Main-beam width: 5 target angles: 190 - 200 (3-D target detection) A total of 246 targets with varying power Slow-moving targets and/or weak targets present o oo azimuth =

17. 17 ROC Curves (KASSPER Data) Median CFAR algorithm applied for target detection

18. 18 Sparse Approaches  Related work:   is replaced by to yield a convex optimization problem.  LASSO: The least absolute shrinkage and selection operator.  BP: Basis pursuit, very similar to LASSO  FOCUSS: Focal underdetermined system solution  SBL: Sparse Bayesian learning  L1-SVD: L1 – singular value decomposition, similar to BP  CoSaMP: Compressive Sampling Matching Pursuit  Most existing algorithms require  Large computation times  User parameters  Hard to decide  Performance sensitive to choice of user parameter Minimize such that is satisfied.

19. 19 Kragh et al. Approach  Kragh et al. uses optimization transfer technique to obtain an iterative procedure:   A recent paper on SAR imaging states: “ ’’ This is FOCUSS.

20. 20 SLIM  Sparse Learning via Iterative Minimization (SLIM) Solves the User Parameter Problem! (Tan, Roberts, Li, and Stoica, 2010)  SLIM Assumes the Following Hierarchical Bayesian Model:  SLIM is a MAP Approach:

21. 21 SLIM Iterations  SLIM Iterates the Following Steps (Starting with DAS): Given q, SLIM is User Parameter Free – Easy to Use!

22. Regularized Minimization in SLIM 22 Cyclic approach with majorization minimization employed to minimize cost function. Conjugate gradient + FFT can be used for efficient implementation of SLIM. For fixed noise variance (i.e., making it a user parameter), SLIM becomes FOCUSS/Kragh et al. Approach.

23. 23 FFT for GOTCHA

24. 24 SLIM for GOTCHA

25. 25 SLIM for GOTCHA

26. 26 IAA (Dense) vs. SLIM (Sparse)  IAA is dense; SLIM is sparse.  IAA is more accurate; SLIM tends to bias downward.  IAA has high resolution; SLIM has higher resolution.  IAA’s fast implementation is trickier, especially for non- uniformly sampled data; SLIM is faster and its fast implementation is more straightforward.

27. 27 Concluding Remarks  We need to devise dense and sparse methods that are user parameter free – easy to use in practice,  And accurate,  And with high resolution,  And computationally efficient.

28. 28 Thank you!