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 Müjdat Çetin, Emre Ertin, Clem Karl, Randy Moses, Lee Potter November 3, 2008

2. Adaptive Front-End Signal Processing (R. Moses, lead) • Decision-directed Imaging and Reconstruction • Modeling and Feature Extraction • Statistical Shape Estimation Processing with purpose Problem formulations that admit context & priors • Parametric and Nonparametric • Decision directed • Uncertainty characterization MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

3. q q q q y x Where were we last time? • Sparseness v. sparseness • Sparse apertures • Sparse signal representations • Complexity reduction • Physics-driven basis sets • Use prior information in basis sets • Extract object-level information • Physical optics for model-based imaging • 3D • Sparse apertures MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

4. Long term aims • Methods robust to sensor configuration & sparsity of data • “Submissive sensing” matched to backend management • Works with wide range of configurations • No “my way or the highway” signal processing • E.g. Circular SAR, Multistatic SAR, spatial-spectral diversity • Understanding of performance • Presensing impact of sensing choices for management (e.g. frequency versus geometric diversity) • Understanding performance consequences of sensing choices • Postsensing estimates and uncertainties for fusion • Methods for complex scenes, non-conventional uses, and greedy decision makers  expect more, get more • Target motion • 3D scene structure • Anisotropic behavior MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

5. New work in 2008: Themes • Multiple platforms • Multistatic imaging of movers no bandwidth, no problem • Multipass 3D imaging IFSAR on steroids • Responsiveness to AFRL • Recursive imaging persistent imaging made easy • Challenge problem less filling, tastes great • HSI life beyond radar • Closing the loop • Adapt processing what’d’ya want to know? • Provide utility metrics what is useful to measure! • Choosing hyperparameters • Stein’s unbiased risk balancing models and measurements • ML estimation empirical Bayes • Posterior probabilities • Language for fusion reporting ambiguity MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

6. Draft inputs • Recursive imaging • Wide-angle sparse imaging • Bayesian sparse linear regression (FBMP) • Multibaseline-IFSAR with sparse interpolation • HSI (Hero) • Sparse + hyperparams + mocomp (Cetin) 5 slides 3 minutes • Sparse + magnitude dictionary (Cetin) omit • recursive imaging (Ash) 5 slides 3.24 minutes • 3D wideangle sparse (Austin) 4 slides 2.5 minutes • Sparse + posteriors: FBMP (Potter) ? Slides ?? minutes • Multistatic sparse with sensor placement (Karl) • Multistatic sparse + velocity (Karl) MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

7. Recursive Image Updating for Persistent Synthetic Aperture Radar Surveillance Ash • 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

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

9. Third Order Recursion Ash • 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

10. GOTCHA C-SAR Video Snapshots Ash 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

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

12. “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 Austin Squiggle PSF MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

13. k-space Approach Austin • 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

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

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

16. Sparse Linear Regression Potter “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

17. Detection and Estimation Goals Potter • MMSE estimation • Soft-decision detection 0.16 0.51 0.13 0.09 MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

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

19. 0 Signal Model Potter • 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

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

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

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

23. Update behavior Potter MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

24. In a Nutshell… Potter • 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

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

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

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

28. Hyperparameter Selection Cetin backhoe model conventional image sparsity-based image MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation • Sparsity-based L2-Lp reconstruction • …is nice, but requires the selection of the hyper-parameter λ • Focus of this work: Automatic choice of hyper-parameter balancing data with prior information

29. Approaches and Numerical Methods Cetin MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation • Adaptation of the following parameter choice methods for sparsity-driven SAR imaging: • Stein’s unbiased risk estimator (SURE) • Generalized cross-validation (GCV) • L-curve • Numerical tools for efficient implementation of these methods, including • Randomized trace estimation • Derivative-free optimization through Golden section search • Numerical derivative computation and backtracking line search 28 August 2008 Özge Batu

30. Results: Backhoe, 500 MHz bandwidth Cetin ConventionalGCV≈SURE L-curve MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

31. Joint imaging and model correction Cetin • SAR observation model may not be known perfectly, due to e.g. uncertainties in platform location • This leads to phase errors in observed data • We have extended our sparsity-based imaging framework to optimize over the reflectivities and model parameters simultaneously: model parameters MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

32. Results: Backhoe Data Set Cetin Conventional image Sparsity-based image Without Model Errors Conventional image Proposed Sparsity-based image with model error correction With Model Errors MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

33. New BU signal processing • Multistatic imaging I: • Physical modeling • Sparsity-based reconstruction • Multistatic imaging II: • Understanding performance • Mutual coherence as predictor • Imaging dynamic scenes • Overcomplete dictionary formulation • Recursive assimilation of data MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

34. Multistatic Radar • Sensing Model • Different choices for K(t), rx, tx possible Reflectivity Tx frequency Tx/Rx geometry Transmit Freq B = bistatic angle uB = bistatic bisector wtx = transmitted frequency From Wicks et al MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

35. Many Sensing Options… Case 2: Stationary Tx, Moving Rx, UNB waveform Case 1: Stationary Tx/Rx, Wideband waveform Case 3: Stationary Tx, Moving Rx, Wideband waveform Case 4: Monostatic Tx/Rx, Wideband waveform MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

36. Multistatic Comments • Rich framework to study: • sensor tradeoffs • resource optimization • waveform/sensor planning • Waveform diversity: • UNB  wideband • Many transmitters  few transmitters • Etc… • Need new tools for processing non-conventional datasets MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

37. Reconstruction Formulation • Sparsity-based L2-L1 reconstruction using extension of previous SAR work • Leads to a second order cone program, effectively solved by an interior point method MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

38. Example: UNB Multistatic SAR • UNB (single frequency) • Ntx=10, Nrx = 55 Sparse coverage • Uniform circular coverage • Fourier support (resolution) µ UNB frequency MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

39. Results FBP, cw = 2MHz, SNR = 15dB FBP, cw = 4MHz, SNR = 15dB Extension of FBP Truth LS-L1, cw = 2MHz, SNR = 15dB LS-L1, cw = 4MHz, SNR = 15dB LS-L1 MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

40. Understanding Performance • Want to understand performance consequences of different sensor configurations • Guidance for sensor management • Compressed sensing theory says reconstruction performance related to mutual coherence of configurations • # of measurements needed to reconstruct sparse scene µ (mutual coherence)2 MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

41. Initial work • Compare different monostatic and UNB multistatic radar configurations • Mutual coherence • Measure of diversity of sensing probes MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

42. Different Sampling Strategies Monostatic Multistatic MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

43. Results • Mutual coherence lower for multistatic configuration as number of probes are reduced Monostatic Multistatic MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

44. Results (Cont) • Example reconstruction for Ntx/Nq=10 case • Reconstructions confirm prediction Ground Truth Monostatic Multistatic MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

45. Dynamic Scenes: Moving Targets • Augment model to include velocity • Discrete form of forward model: Static targets at a reference time Phase shift due to motion A depends on unknown scatterer velocity v in pixel p, so nonlinear problem! MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

46. Overcomplete Dictionary Approach • Modify forward operator to include all velocity hypotheses • Pixel reflectively becomes a vector • New overcomplete observation model • A is now fully specified, so observation is linear…but solution f must be very sparse • We know how to do this! MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

47. Overcomplete Problem Solution • Idea: sparest solution should automatically identify correct velocity and scattering • Solution via custom made large-scale interior point method MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

48. Example #1: • Multistatic configuration with Ntx= 10, Nrx = 55 • Dictionary does not contain true velocities Truth CW = 4MHz, OD MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

49. Circular SAR Circular SAR provides information about the 3D location of the scattering centers and their reflectivity as a function of aspect angle • 3D Resolution of Circular SAR is constrained by • Limited Persistence of Reflectors • Sparse Elevation Sampling • Dynamically varying nonuniform spacing in elevation MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation

50. Different Passes Multipass Interferometric SAR • Employ high resolution parametric spectral estimation methods to mitigate the effects of sparse elevation sampling[Xiao and Munson 1998, Gini and Lombardini 2005] • Relative phase of the 2D images formed from different passes provide height information for the scattering centers in each resolution cell • Sum of complex exponentials model MURI: Integrated Fusion, Performance Prediction, and Sensor Management for Automatic Target Exploitation