Statistical modeling, classification, and sensor management

1. Statistical modeling, classification, and sensor management DARPA-MURI Review 2003 Alfred Hero Univ. Michigan Ann Arbor

2. Target Search Scenario HighRes Spot Scan LowRes Spot Scan Strip Scan

3. Sensor Deployment Architecture Our Research themes: Sequential Sensor Management Image Reconstruction Adaptive Detection

4. Research Loci • Image modeling and reconstruction • Markov random field (MRF) polarimetric models (Hory&Blatt) • 3D Imaging with uncalibrated sensor nets (Rangarajan&Patwari) • Adaptive detection and classification • Pattern matching and modeling (Costa) • Distributed detection and classification (Blatt&Patwari) • Sequential sensor management • Myopic information-driven approaches (Kreucher) • Non-myopic approaches (Kruecher&Blatt) Common theme: adaptive robust non-parametric methods

5. Detection: Target or Clutter Alone?

6. Detection: Target or Clutter Alone?

7. Target Returns Not Additive or Gaussian • 1cm x 1cm x 1mm plate at 1m from ground • Plate under forest canopy (10 deciduous trees) • 2GHz SAR illumination • Aggregate of three look angles (azimuth=35,45,55, elev=180) SNR=0dB SNR=6dB

8. Polarimetric Field Modeling and Reconstruction h-pol. incidence • Field Distribution On FDTD Box (2 GHz) v-pol. incidence

9. MRF empirical histogram Conditional Markov transition histogram …estimated from training data

10. Causal kNN predictor: Non-Causal MRF model: Causal MRF Field Synthesis

11. Example: K-NN MRF Extrapolation

12. Non-parametric MRF density estimator • General penalized MRF transition density estimate • y is observed data • parameter b enforces smoothness • function g(f) captures data-fidelity • g(f)=|f|^2: standard L2 quadratic regularization • g(f)=|f|: L1 regularization for denoising • w(x): smoothing within and across neighborhoods

13. Cartoon illustration of density estimator K-Nearest Neighbors Estimator Penalized MRF transition Density Estimator

14. Visual Validation of MRF Model g(f)=|f|

15. MRF Transition Density Comparisons

16. Target Modeling and Classification • Pattern matching in high dimensions • Standard techniques (histogram, density estimation) fail due to curse of dimensionality • Entropic graphs recover inter-distribution distance directly • Robustification to outliers through graph pruning • Manifold learning and model reduction • Standard techniques (LLE, MDS, LE, HE) rely on local linear fits and provide no means of getting at sample density • Our geodesic entropic graph methods fit the manifold globally • Computational complexity is only n log n

17. A Planar Sample and its Euclidean MST

18. Convergence of Euclidean MST Beardwood, Halton, Hammersley Theorem:

19. Pattern Matching

20. MST Estimator of a-Jensen Affinity Two well separated Classes Two overlapping Classes

21. MST Estimator of Friedman-Rafsky Affinity Two well separated Classes Two overlapping Classes

22. Target model reduction • 128x128 images of three land vehicles over 360 deg azimuth at 0 deg elevation • The 3(360)=1080 images evolve on a lower dimensional imbedded manifold in R^(16384) Courtesy of Center for Imaging Science, JHU

23. Target-Image Manifold

24. 2D manifold Embedding Sampling distribution Sampling A statistical sample

25. Geodesic Entropic Graph Manifold Learning and Pattern Matching Algorithm • Construct geodesic edge matrix (ISOMAP,C-ISOMAP) • Build entropic graph over geodesic edge matrix • MST: consistent estimator of manifold dimension and process alpha-entropy • MST-Jensen: consistent estimator of Jensen difference between labeled vectors • Use bootstrap resampling and LS fitting to extract rate of convergence (intrinsic dimension) and convergence factor (entropy)

26. Illustration for 3 land Vehicles

27. loglogLinear fit to asymptote LS-Soln: d=13 H=120(bits)_

28. Distributed Multisensor Estimation and Detection • Distributed M-estimation (Blatt) • Ambiguity function is often multimodal: local and global M • Distributed measurements make local M more difficult • We develop method to discriminate between local/global M • Use unsupervised clustering and Fisher information matching • Distributed change detection (Patwari) • Bandwidth and computation constraints • Multilayer vs flat store-detect-forward architecture • We study perfromance loss due to bandwidth constraints • How much information should be sent to what layers?

29. Sensor 1 Sensor 2 Sensor N Processing unit Final estimator Distributed Estimation and Detection Flat Sensor Aggregation Architecture

30. Distributed M- Estimation Ambiguityfunction for Cauchy distributed points on a manifold

31. Global maximum Local maxima A slice of ambiguity function

32. Key Theoretical Result • The asymptotic distribution of M-estimate is (asymptotically) a Gaussian mixture • Parameters Ref: Blatt&Hero:2003

33. Validation of Key Result – QQ-plots M-estimates are clustered into two groups. Each group is centered according to the analytical mean and normalized according to the analytical variance.

34. Estimator 1 Estimator 2 Estimator N M-estimator Aggregation Algorithm Estimation of Gaussian Mixture Parameters (EM) Sample Covariance Analysis Aggregation To Final Estimate

35. Illustration Model: • 200 Sensors • 100 snapshots per sensor • Snapshots are 1D Gaussian 2-mixture • Known covariance • Unknown means • Sensors generate i.i.d. M-estimates of means and forward to central processor Global maximum Local maximum Ambiguity function.

36. Local/Global Maxima Discrimination Algorithm Bad estimates Bad estimates Inverse FIM Good estimates Empirical covariance

37. Addition of other Discriminants Value-added due to local acquisition and transmission of likelihood values

38. Sensor 1 Sensor 3 Sensor 2 Sensor 4 Processing unit Final estimator Distributed Estimation and Detection Hierarchical Sensor Aggregation Architecture Sensor 6 Sensor 5

39. 1 Optimal 7-Sensor 2 3 4 5 6 7 r = 0.30 r= 0.10 1 Legend 2 3 Flat r = 0.03 4 5 6 7 Hier. w/o Feedback Hier. w/ Feedback Optimal 1-Sensor Detection: Flat vs Hierarchical Architecture Optimal 7-Sensor • ‘Flat’ [Rago, Willett, et al] • Hierarchical, w/ and w/o Feedback • Each sensor is limited with identical r • At low PF, Hierarchical outperforms Flat r = 0.30 r = 0.10 Legend Flat r = 0.03 Hier. w/o Feedback Hier. w/ Feedback Optimal 1-Sensor

40. Sequential Adaptive Sensor Management • Sequential: only one sensor deployed at a time • Adaptive: next sensor selection based on present and past measurements • Multi-modality: sensor modes can be switched at each time • Detection/Classification/Tracking: task is to minimize decision error • Centralized decisionmaking: sensor has access to entire set of previous measurements Single-target state vector:

41. Sequential Adaptive Sensor Management • Myopic information-based strategies (Kruecher) • Multi-target tracking capabilities • Fully Bayesian approach • Non-linear particle filtering with adaptive partitioning • Renyi-alpha divergence criterion • Non-Myopic strategies (Blatt&Kreucher) • MDP value function approximations and rollout methods • Bayesian path averaging • Reinforcement feedback and learning

42. Sensor scheduling objective function • Prospective value of deploying sensor s at time t: Sensor agility Prediction Retrospective value of deploying sensor s Available measurements at time t-1

43. Information-based Value Function • Incremental information gained from data collected from using sensor s. Can be measured by divergence • Requires posterior distributions of future target state X given future Z and given present Z, resp., • Main issues for evaluation of E[D(s,t)|Z] • Computation complexity • Robustness to model mismatch • Decisionmaking relevance

44. Value Function : Alpha Divergence • Properties of Renyi divergence • Simpler and more stably implementable than KL (Kreucher&etal:TSP03) • Parameter alpha can be adapted to non-Gaussian posteriors • More robust to mis-specified models than KL (Kreucher&etal:TSP03) • Related directly to decision error probability via Sanov (Hero&etal:SPM02) • Information theoretic interpretation

45. Relevance of alpha-D to Decision Error • Consider testing hypotheses • Sanov’s theorem: optimal decision rule has error • Implication: nearly-optimal decision rule for H1 is if can generate good estimate of alpha-D

46. Multi-Target Bayesian Filtering • Joint multiple target posterior density (JMPD) jointly represents all target states (Kastela) • Update eqns must generally be approximated Model Update (Prediction using prior kinematic model) Measurement Update(Bayes Rule)

47. Particle Filter (Metropolis) Approximation • Propose (draw) a set of particles based on some importance (proposal) density q chosen to be as close to the posterior as possible • Weight the particles using the principle of importance sampling • Resample particles using above density to avoid degeneracy time t time t-1

48. Particle Filtering Illustration Initialize: simulate random samples (particles) from proposal density

49. Particle Filtering Illustration • Model Update: Propose new particles from existing particles based on drawing samples from the importance density

50. Particle Filtering Illustration Measurement Update: Reweight particles density according to • Resample the particles if necessary