A Sparse Signal Reconstruction Perspective for Source Localization with Sensor Arrays

1. A Sparse Signal Reconstruction Perspective for Source Localization with Sensor Arrays Alan Willsky Contributors: Dmitry Malioutov, Müjdat Çetin SensorWeb MURI Review Meeting December 2, 2005

2. Source Localization Problem • Source localization based on passive sensor measurements • Our approach: View the problem as one of imaging a “source density” over the field of regard • Ill-posed inverse problem (overcomplete basis representation) • Favor sparsefields with concentrated densities

3. Contributions/Highlights • Source localization framework using lp-norm-based sparsity constraints • Efficient techniques for synergistic use of all data and efficient algorithms for numerical solution • Theoretical results: • Justification based on properties of the data and human goals • Solution of combinatorial optimization problems by computationally feasible algorithms! • Extensive performance analysis on simulated data • Self-calibration for sensor location uncertainties • Experimental results on ARL’s acoustic data

4. Data fidelity Regularizing sparsity constraint Source Localization Framework • Cost functional (notional): • Observation model: Noise Sensor measurements Array manifold matrix Unknown “source density” • Role of the regularizing constraint : • Preservation of strong features (source densities) • Preference of sparse source density field • Can resolve closely-spaced radiating sources

5. Data Processing and Optimization Algorithms • SVD-based approach for processing and using multiple time or frequency snapshots efficiently and synergistically • Two numerical optimization algorithms: • One based on half-quadratic regularization • One based on second-order cone programming and interior point algorithms • Fast multiresolution approach for iteratively refining the search around likely source locations

6. Theoretical Results onlp regularization • Observations: • Preferring the optimally sparse solution would involvel0-norms • But that requires solving combinatorial optimization problems • lp-norm-based techniques have been empirically observed to yield solutions that look sparse • Question: Can we ever get the optimally sparse solution using lp–norms? • Interestingly, the answer, as we have found out, is YES! • Provided that the actual spatial spectrum is sparse enough • This provides a rigorous characterization of the lp–sparsity link • As a result, we can solve a combinatorial optimization problem by tractable algorithms!

7. Performance Analysis on Simulated Data • Narrow-band and wide-band signals • Far-field and near-field sources • Incoherent and coherent (due to multipath) sources • Linear, circular, cross, rectangular arrays • Wide range of SNRs • Wide range of the number of snapshots • We will show only a subset of this analysis

8. Narrowband, uncorrelated sources – high SNR • DOAs: 65, 70 • SNR = 10 dB • Far-field • 200 time samples • Uniform linear array with 8 sensors

9. Narrowband, uncorrelated sources – low SNR • Far-field • 200 time samples • Uniform linear array with 8 sensors • DOAs: 65, 70 • SNR = 0 dB

10. Narrowband, correlated sources • Far-field • 200 time samples • Uniform linear array with 8 sensors • DOAs: 63, 73 • SNR = 20 dB

11. Robustness to limitations in data quantity Single time-sample processing • Uncorrelated sources • Uniform linear array with 8 sensors • DOAs: 43, 73 • SNR = 20 dB

12. Resolving many sources • 7 uncorrelated sources • Uniform linear array with 8 sensors

13. Estimator Variance and the CRB • Correlated sources • Uniform linear array with 8 sensors • DOAs: 43, 73 • Each point on curve average of 50 trials

14. MUSIC Capon’s method (MVDR) Proposed Multi-band example – low SNR Underlying true spectrum

15. Extension to Self-calibration • What if we don’t know the sensor locations exactly? • Extended our framework to include optimization over sensor locations • Setup for experiments: • Far-field case • Narrowband signals • Linear array with 15 sensors • Two uncorrelated sources • DOAs: 45, 75 • SNR = 30 dB • Sensor locations perturbed with a standard deviation of 1/3 of the nominal sensor spacing

16. Moderate calibration errors can be compensated up to intrinsic ambiguities Self-calibration Example

17. Validation on real data provided by ARL ARL Field Experiment Setup • Six acoustic sensor arrays in oval loop. • Each has a circular array configuration with seven microphones. • Tanks and trucks travel on oval loop or on nearby asphalt road.

18. Sensor Configuration

19. Results on single-vehicle data

20. Results on multiple-vehicle data - I

21. Results on multiple-vehicle data - I • A temporal slice – (when the vehicles are closest)

22. Results on multiple-vehicle data – II (limited observations)

23. Results on multiple-vehicle data – III (limited bandwidth)

24. Results on multiple-vehicle data – IV (low SNR)

25. Summary • Sparse signal reconstruction framework & algorithms for source localization with passive sensor arrays • Theoretical analysis justifying the formulation • Extensive performance analysis • Superior source localization performance (Superresolution, Reduced artifacts) • Robustness to resource limitations (SNR, Observation time, Available aperture) • Self-calibration capability • Fruitful interactions with ARL • Validation on ARL field data • Adaptation to signal structures of interest to the Army (including multiband harmonic sources) • Collaboration with Dr. Brian Sadler & successful application of this framework to estimation of sparse communication channels