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Statistical modeling, classification, and sensor management. DARPA-MURI Review 2003. Alfred Hero Univ. Michigan Ann Arbor. Target Search Scenario. HighRes Spot Scan. LowRes Spot Scan. Strip Scan. Sensor Deployment Architecture. Our Research themes:. Sequential Sensor Management.
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Statistical modeling, classification, and sensor management DARPA-MURI Review 2003 Alfred Hero Univ. Michigan Ann Arbor
Target Search Scenario HighRes Spot Scan LowRes Spot Scan Strip Scan
Sensor Deployment Architecture Our Research themes: Sequential Sensor Management Image Reconstruction Adaptive Detection
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
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
Polarimetric Field Modeling and Reconstruction h-pol. incidence • Field Distribution On FDTD Box (2 GHz) v-pol. incidence
MRF empirical histogram Conditional Markov transition histogram …estimated from training data
Causal kNN predictor: Non-Causal MRF model: Causal MRF Field Synthesis
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
Cartoon illustration of density estimator K-Nearest Neighbors Estimator Penalized MRF transition Density Estimator
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
Convergence of Euclidean MST Beardwood, Halton, Hammersley Theorem:
MST Estimator of a-Jensen Affinity Two well separated Classes Two overlapping Classes
MST Estimator of Friedman-Rafsky Affinity Two well separated Classes Two overlapping Classes
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
2D manifold Embedding Sampling distribution Sampling A statistical sample
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)
loglogLinear fit to asymptote LS-Soln: d=13 H=120(bits)_
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?
Sensor 1 Sensor 2 Sensor N Processing unit Final estimator Distributed Estimation and Detection Flat Sensor Aggregation Architecture
Distributed M- Estimation Ambiguityfunction for Cauchy distributed points on a manifold
Global maximum Local maxima A slice of ambiguity function
Key Theoretical Result • The asymptotic distribution of M-estimate is (asymptotically) a Gaussian mixture • Parameters Ref: Blatt&Hero:2003
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.
Estimator 1 Estimator 2 Estimator N M-estimator Aggregation Algorithm Estimation of Gaussian Mixture Parameters (EM) Sample Covariance Analysis Aggregation To Final Estimate
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.
Local/Global Maxima Discrimination Algorithm Bad estimates Bad estimates Inverse FIM Good estimates Empirical covariance
Addition of other Discriminants Value-added due to local acquisition and transmission of likelihood values
Sensor 1 Sensor 3 Sensor 2 Sensor 4 Processing unit Final estimator Distributed Estimation and Detection Hierarchical Sensor Aggregation Architecture Sensor 6 Sensor 5
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
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:
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
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
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
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
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
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)
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
Particle Filtering Illustration Initialize: simulate random samples (particles) from proposal density
Particle Filtering Illustration • Model Update: Propose new particles from existing particles based on drawing samples from the importance density
Particle Filtering Illustration Measurement Update: Reweight particles density according to • Resample the particles if necessary
Multitarget Tracking: Adaptive Proposals • When targets are well separated in measurement space, each target-partition of particle evolves independently. • In this case can use independent partition (IP) updates • When targets become “close” target-partitions become dependent • In this case should use coupled partition (CP)updates • Adaptive strategy: use IP unless CP is deemed necessary IP updating CP updating CP updating