Sparse Reconstruction and Feature Extraction

1. Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation Sparse Reconstruction and Feature Extraction MURI Review Meeting Lee Potter, Müjdat Çetin, Emre Ertin, Clem Karl, Randy Moses September 14, 2007

2. Draft inputs: OSU signal processing • Recursive imaging • Wide-angle sparse imaging • Bayesian sparse linear regression (FBMP) MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

3. Recursive Image Updating for Persistent Synthetic Aperture Radar Surveillance • Persistent SAR • SAR video • Imagery on demand • Variable aperture integration • Insight: recursive imaging spreads computation over time and avoids block processing memory load MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

4. Convolution Backprojection • Range profile by filtering and backprojecting • Window wj controls crossrange sidelobes • J N2 computations per image • Recursive formulation: MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

5. Third Order Recursion • Can choose ai coefficients to emulate many common apodization windows (e.g. Hamming). MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

6. GOTCHA C-SAR Video Snapshots Block-Processing Hamming Az Window Recursive Processing Third Order GOTCHA: fc=9.6 GHz, 640 MHz BW, 45 elev, 3 azimuth MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

7. Flop-free change in effective aperture Recursive Processing 3°Azimuth Window Recursive Processing 25°Azimuth Window MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

8. “Squiggle” Path Dataset Air Force Research Laboratory construction backhoe challenge dataset Data collected over a very sparse “squiggle” flight path Radar: fc: 10 GHz, BW 6 GHz Polarization: HH, VV, VH k-space Az » [65.5°, 114.5°] El » [17.5°, 42.5°] Wide-Angle Sparse 3D Synthetic Aperture Radar Squiggle PSF MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

9. k-space Approach • Insights • Sparsity of strong reflectors • Low persistence: uncorrelated subimages • Approach • Form narrow-angle sub-images by l1-penalized least-squares inversion • Noncoherently combine subaperture images MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

10. Results 1.29% data of benchmark MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

11. Animation MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

12. Sparse Linear Regression: FBMP “Are you guys still working on As + n ?” Thomas Kailath, c. 1988 “The thing that hath been, it is that which shall be; and that which is done is that which shall be done: and there is no new thing under the sun.” Ecclesiastes 1:9, c. BC 250 “There is nothing new under the sun but there are lots of old things we don't know.” Ambrose Bierce, The Devil's Dictionary, US author & satirist (1842 - 1914) “Neurosis is the inability to tolerate ambiguity.” Sigmund Freud (1856 – 1939) [graphic courtesy of Rich Baraniuk] MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

13. Desiderata • Allow arbitrary correlation of columns in design matrix • Ambiguity in variable selection • Report ambiguity • Compute posterior probabilities for variable sets & posterior on vbls • Minimize estimation error • MMSE estimation of variables • Compute with low complexity • Keep order of complexity of Orthogonal Matched Pursuits • Use domain knowledge, if available • Flexible family of priors with known hyperparameters, or • ML estimation of hyperparameters • Admit complex-valued data • Band-pass signals in radar, spectroscopy and communications • Provide non-asymptotic bounds on variable detection • Characterize performance of the MAP detection of variables MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

14. Applications: Radar Radar: Wide Area-Coverage, all weather, day/night, persistent illumination High resolution imaging of objects and tracking of moving targets MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

15. In a Nutshell… • Bayesian signal model • Effective tree search for high-probability set • Fast update of posterior • Generalized EM for unknown hyperparameters • Return MAP solution, MMSE solution and list of candidate solutions with relative probabilities MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

16. 0 Signal Model • Variables drawn from a Gaussian mixture with point mass at origin • Multinomial for mixing indicator • Simplest illustration: • Procedures implemented for complex-valued case and arbitrary Gaussian mixtures MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

17. Model Posteriors MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

18. Posterior, p(x|y) Typical realization: s_true ranks fourth in p(s | y) MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

19. NMSE 9 dB MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

20. Sparsity MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

21. Runtime “Fast” but no free lunch MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

22. MAP detection bounds • Perfect detection is very low probability • Sufficient condition for detection of a coefficient with probability 1-δ • Necessary condition for detection of a coefficient with probability 1-δ • Sufficient condition for detection of 1-ε energy with probability 1-δ MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation