Applied Mathematics in Defense Applications
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Applied Mathematics in Defense ApplicationsAndrea Bertozzi
Department of MathematicsUniversity of California, Los Angeles
Topics• Multisensor fusion• Image and human event fusion for statistical density estimation• Automated boundary tracking for autonomous robots and high dimensional
imagery• Collaborative searching through swarming• Diffuse interface methods in imaging• Segmentation with corners• Imaging through turbulence• Direct sparse deblurring• Geographic profiling• Crime hotspots• Gang violence data• Predicting crime
Data Fusion – Multiple sensors
• Pan Sharpening – panchromatic (greyscale) higher spatial resolution, multiband – lower spatial resolution – IKONOS and QUICKBIRD satellite
• Hyperspectral sharpening – panchromatic obtained separately (may not be perfect match) – hyperspectral can have hundreds of bands – contain material information
• Human Event data – events in space and time fused with geographical data (e.g. residential burgalaries)
• Point sensor data – mobile sensor data
Data Fusion and Segmentation– Multiple sensors
• Pan Sharpening – panchromatic (greyscale) higher spatial resolution, multiband – lower spatial resolution – IKONOS and QUICKBIRD satellite
• Hyperspectral sharpening – panchromatic obtained separately (may not be perfect match) – hyperspectral can have hundreds of bands – contain material information
• Human Event data – events in space and time fused with geographical data (e.g. residential burgalaries)
• Point sensor data – mobile sensor data
Panchromatic signal is not a linear combination of isolated bands
Recent pansharpening techniques
• IHS• Brovey• PCA • Wavelet Fusion • First variational approach: ’A Variational
Model for P+XS Image Fusion’, Ballester, Casselles, Igual, Verdera, 2006
Intensity Hue Saturation Results
Assumes panchromatic is a linear combination of spectral bands.
Variational Wavelet PansharpeningMichael Moeller, Todd Wittman, ALB, preprint
Wavelet matching – data must be registered to dyadic scaling (pansharpening)
Full VWP variational problem
Alternate VWP – avoids switching from wavelet to physical space
Numerical results
Hyperspectral data fusion
Michael Moeller, Todd Wittman, and Andrea L. Bertozzi, A Variational Approach to Hyperspectral Image Fusion, Proc. SPIE Conference on Algorithms and Technologies for Multispectral, Hyperspectral, and Ultraspectral Imagery XV. Orlando, Florida. April 2009.
Spatial detail inherited from master image – spectral detail from AVIRIS data
Spectral preservation - examples
Spectral angle is preserved
George Mohler, Andrea Bertozzi, Tom Goldstein, Stan OsherFast TV regularization for 2D Maximum penalized likelihood estimation,
preprint 2009
• Method for estimating non-smooth probability densities• Important for estimating threat level based on event data and
other intel.• TV based regularization allows for best estimation of densities
with spatial discontinuities.• Computationally challenging in multi-D• Challenge solved using Split Bregman L1 minimization
technique.• Tested using V-fold Cross Validation with large 2D data sets.• Tested on data from LAPD for residential burglaries.
Maximum Penalized Likelihood Estimationbasic problem
• Estimate probability density u(x) from point data x1, x2, x3, etc.
• General approach for regularizer R(u).
• For discontinuous densities, choose R = TV
Example from San Fernando ValleyData courtesy of LAPD
• Point process data for residential burglaries• No residences in area in middle
Actual data TV method (new) kernel estimation (old)
TV method is much closer to real problem, does not bleed threat level into region where threat is not active. Can be fused with other types of data, such as spatial visual, infrared, LIDAR etc as long as one has a model to incorporate this into the problem.
Density Estimation for Sparse DataLaura Smith, Matthew Keegan, Todd Wittman, ALB UCLA
• Point data of individual events that come from a background source
• Examples – human event activity – burglaries – what is the probability of event as a function of space?
• Data is sparse – want to fuse with other information e.g. overhead imagery
Improving Density Estimation by Incorporating Spatial Information, preprint 2009.
Overhead imagery vs. human events
San Fernando Valley Burglaries
Orange County Coastline
Experimental Validation of Cooperative Environmental Boundary
Tracking with On-board Sensors
A. Joshi, T. Ashley, Y. Huang, and A. L. Bertozzi, Experimental validation of cooperative environmental boundary tracking with on-board sensors, American Control Conference, St. Louis, MO, June 2009, pp. 2630-2635.
Control Algorithms for Boundary Tracking
• UUV-gas bang-bang type steering controller• Time-corrected algorithm• Robotic path planning – • Hsieh et al Amer. Contrl. Conf. 2005• Jin and ALB, IEEE CDC 2008• Joshi et al Amer. Contrl. Conf. 2009
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Bang-bang type steering Control Law
• UUV-gas algorithm:
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Time-corrected Steering Control Law
• Time-corrected algorithm:
• Includes time difference between crossing points on boundary,
• Reduces to the bang-bang type controller when,
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Decision Algorithm CUSUM Filter
• Upper : indicates teal tape
• Lower : indicates black tape
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Single Vehicle Implementation• A Kalman inspired pre-filter was used to weakly damp
the noisy signal.
• Essentially a simple proportional model with the empirical gain factor from Kalman filtering a previous data set.
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convergent Kalman gain (this is expected in view of the fairly constant, though high, noise co-variance of the testbed)
Single Vehicle Implementation
• The sharp features when on the black tape cause the decision algorithm to slip-up occasionally
• Pre-filtering reduces these errors considerably
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Single Vehicle Implementation
Vehicle speed 0.3m/s Left raw data -> CUSUM Right raw data -> prefilter -> CUSUM
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Cooperative boundary tracking
• The global control law held admirably
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The time difference between data points is 24 s
A. Chen, T. Wittman, A. Tartakovsky, and A. L. Bertozzi Image segmentation through efficient boundary sampling, in SAMPTA '09, Marseille, May 18-22, 2009.
W. Liu, M.B. Short, Y. E. Taima, and A. L. Bertozzi,
Multiscale Collaborative Searching Through Swarming, preprint.
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• Sensor data processing– Kalman filter to reduce noise– CUSUM filter to check threshold
• Movement control of agents– Models for search phase, target locating phase– Target location estimated
• Performance– Average time to locate a target– Average error in estimate of location– Number of false registers
• Scaling properties– Estimate for swarm size, measured by diameter– Optimal swarm diameter (analytical approximation, upper bound)
Collaborative Searching through SwarmingW. Liu, M.B. Short, Y. E. Taima, and A. L. Bertozzi
Introduction 2
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• Overview of the algorithm
Collaborative Searching through SwarmingW. Liu, M.B. Short, Y. E. Taima, and A. L. Bertozzi
Movement Control of Agents
• Search phase (with Levy flight)
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• Location estimation
• Target locating phase
Collaborative Searching through SwarmingW. Liu, M.B. Short, Y. E. Taima, and A. L. Bertozzi
Sensor data processing
• Agent sensor reading
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• Kalman filter to reduce noise
• CUSUM filter to detect threshold
Collaborative Searching through SwarmingW. Liu, M.B. Short, Y. E. Taima, and A. L. Bertozzi
Performance
• Simulations of a 20 by 20 dimensionless board, 32 agents, target sensing radius 1.0, 200 trials
• Divide-and-conquer and whole region search strategies
• Average time to locate a target and average location estimate error measured
• Larger swarms are more accurate, multiple smaller ones more efficient
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Collaborative Searching through SwarmingW. Liu, M.B. Short, Y. E. Taima, and A. L. Bertozzi
Scaling Properties
• Swarm diameter D scales with inter-agent distance
• For 25% of agents to sense before deciding to locate,
• Optimal D maximizes separation between the center of the swarm and the target location
• The average time to locate a target is
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Diffuse interface methods
Ginzburg-Landau functionalTotal variation
Cahn-Hilliard Inpainting
Bertozzi, Esedoglu, Gillette, IEEE Trans. Image Proc. 2007, SIAM MMS 2007Patent pending. Transitioned to NGA for road inpainting. Transitioned to InQtel for document exploitation.Continue edges in the same direction – higher order method for local inpainting.Fast method using convexity splitting and FFT
H-1 gradient flow for diffuse TVL2 fidelity with known data
Wavelet Allen-Cahn Image Processing
• Dobrosotskaya, Bertozzi, IEEE Trans. Image Proc. 2008, Preprint subm. IFB.• Transitioned to NGA for road inpainting. Transitioned to InQtel for document exploitation.• Nonlocal wavelet basis replaces Fourier basis in classical diffuse interface method.• Analysis theory in Besov spaces. • Gamma convergence to anisotropic TV. H-1 gradient flow for diffuse TV
L2 fidelity with known data
Convex Splitting SchemesSchoenlieb and Bertozzi, submitted
Basic idea:
Art is to choose Ec to give an implicit problem that is easy to solve- e.g. Ec is H1 semi norm – can be solved using FFT- in wavelet case Ec is wavelet Laplace operator
Contraints on Ec and Ee so that splitting is unconditionally stable
Proof of convergence of splitting schemes for various higher order inpainting methods.
Segmentation with Corners
Image Snakes (KWT ‘88)
Droske & Bertozzi – geometric corner snakes2009
Chan-Vese 2001
CV with corners 2009
Idea – segmentation requires a regularizationIt is analogous to denoising. CV, Snakes reduce length of curve.Removes corners as well as noise.Instead regularize with the “curve” analogy of TV – nonlinear penalization of curvature-based functional.Low Curvature Image Simplifiers (Tumblin & Turk SIGGRAPH 2000, Bert. And Greer CPAM 2004)Extend to curve evolution using either Lagrangian framework or Level sets.
Marc Droske and Wenhua Gao
Higher-order feature-preserving geometric regularization, SIAM J. Img. Sci. 2010.
Imaging through turbulence
Images taken at China Lake – courtesy of Alan Vannevel and Gary HewerWhen you image at a kilometer anisplanatic effects are relevant – we need betterdeblurring and deconvolution methods.Often imformation is known about the image – the difficulty is to extract features.
morning afternoon
Direct Sparse DeblurringLou, Bertozzi, Soatto, submitted 2009
Blurry data ROF deblurring Our method
Uses training data
Dictionary based
Inverse problem not solved
Fit data to blurred dictionary then directly unblur
Solves problem of amplifying noise with solution of inverse problem
Geographic ProfilingGeorge Mohler and Martin Short
Estimation of probabilities
2004 San Fernando Valley Data
2007 Los Angeles Burglary Data
M. B. Short, M. R. D'Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi, and L. Chayes, A statistical model of criminal behavior, M3AS: Mathematical Models and Methods in Applied Sciences, special issue on Traffic, Crowds, and Swarms, volume 18, Supp., pages 1249-1267, 2008.
G. Mohler, M. Short, P. Brantingham, F. Schoenberg, and G. Tita, Self-exciting point process modeling of crime, submitted.
Gang violence DataMike Egesdal, Chris Fathauer, Kym Louie, and Jeremy Neuman, Statistical Modeling of Gang Violence in Los Angeles, to appear in SIURO online.
Gang violence Data
Burglary 2005 Robbery 2007
Courtesy of George Mohler
AcknowledgmentsONR grant N000141010221, Information Fusion of Human Activity, Social Networks, and Geography Using Fast Compressive Sensing, co PIs Stan Osher, Jeff Brantingham, George Tita (UCI).
ONR grant N000140810363 Geometry Based Image Analysis and Understanding, 1/08-12/10.
ARO grant (STIR) W911NS-09-1-0559 Mathematical modeling of insurgent activities as compared to urban street crime, 10/09-6/10.
ARO MURI grant 50363-MA-MUR Spatio-temporal event pattern recognition subcontract from Brown University, Boris Rozovsky, PI.
The Department of Defense.
NSF grant DMS-0914856 Algorithms for Threat Detection (ATD): adaptive sensing and sensor fusion for real time chemical and biological threats, (jointly funded by DTRA).
NSF grant DMS-0601395 Research Training Group in Applied Differential Equations and Scientific Computing, 6/06-5/11 (supports undergraduate research).
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