Random matrix theory enabled
algorithms for sensing and
surveillance
PI: Raj Rao Nadakuditi – University of Michigan
PM: Dr. Jon Sjogren/ Dr. Tristan Nguyen
Thanks to Dr. Rangaswamy (AFRL)
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Prologue
Recognize this sound?
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Modem speed & fundamental theoretical
limits
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Fundamental limit of communication
• 56.6 k is fundamental limit of dial-up modem
• Above 56.6 k not possible
• Independent of sophistication of algorithm or CPU speed
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Role of theory
• Fundamental limits
• What can or cannot be done?
• Algorithm design
• How we can attain these limits?
• Unifying framework
• What is the right way to look at the problem?
• Consequence: New solutions for old applications & enabling technology for new
applications
• My work: Use & develop Random Matrix Theory to understand fundamental theoretical limits
• YIP Focus: Enabling new sensing & surveillance algorithms for AF
• Perspective: Random scalar and vector theory centuries old – matrix analogs new and natural
next step
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Example AF applications with noisy matrices
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Time series collected at a radar sensor array (e.g. aircraft detection)
Hyperspectral images (e.g. Gas detection and tracking)
Video images (e.g. convoy detection and tracking)
“Clutter” ubiquitous for AF applications (e.g. smoke, dust, fog, clouds,)
Goal: Detect, estimate, classify and track “as well as possible” despite the
presence of clutter
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Today’s talk
• Extraction of “signals” from clutter in highly corrupted video frames
• New theoretical limits and algorithms
• New methods for making fog and clouds “transparent”
• New theoretical limits and algorithms
• (Newly developed) Random matrix theory is enabling technology
• Many more AF relevant applications
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Empirical validation setup
• Video sequence
• Stationary background with varying time-varying lighting conditions
• People moving in foreground
• Goal: Separate foreground (= “signal”) from background (= “clutter”)
• AF motivation:
• Moving targets on a radar display
with relatively static clutter or
tracking objects in foreground
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Raw video
• Think of background as clutter – people as moving targets
• Isolating moving targets from background clutter can visually improve
detection (and allow running tracking algorithms on “correct” inputs)
Foreground/Background Separation
Raw Data
Low-rank
Sparse
• Think of background as clutter – people as moving targets
• Isolating moving targets from background clutter can visually improve
detection (and allow running tracking algorithms on “correct” inputs)
Nuclear norm based approach
Low-rank
Sparse
Noisy Video
Small Regularizer
Large Regularizer
New approach vs previous approach
Low-rank
Sparse
Missing Data
New Method
Existing Method
Subspace estimation with clutter
• S is matrix of mean zero clutter with some outliers
• L is the low-rank “signal” matrix
• Question: Quality of subspace estimates?
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Theory: Subspace estimate degradation
• Key insight:
• If σq = O(1) then p as small as O(log m/m), subspace estimates are severely degraded!
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Today’s talk
• Extraction of “signals” from clutter in highly corrupted video frames
• New theoretical limits and algorithms
• New methods for making fog and clouds “transparent”
• New theoretical limits and algorithms
• (Newly developed) Random matrix theory is enabling technology
• Many more AF relevant applications
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Opacity of thick random media
• Exponential decay of intensity
with number of layers
• Fundamental limit?
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Where is the random matrix?
• Scattering matrix (S-matrix) representation:
• Modes in coupling to modes out
• Random scatterer locations leads to random matrix
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The random scattering matrix
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Imp. of scattering matrix distribution
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Simulation results & New Theory during YIP
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500 dielectric cylinders with n = 2.82
Theory using “free probability” - universal bimodal shape = Perfect tx!
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Visualizing “open” eigen-wavefronts
• Numerically rigorous simulations (11th digit accuracy)
• 2D and 3D, range of scatterer properties (geometry, etc)
• Joint work with E. Michielssen (supported by Dr. Nachman)
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Real-world evidence
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Demo
• Place cylinders at random locations
• Generate normal incident wave
• Observe field
• Generate optimal wave
• Eigen-wavefront with max Tx Coeff.
• Observe where light channels
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Demos
• 1 PEC cylinder
• 50 PEC cylinders
• Note:
• NO knowledge about scatterers used
• Backscatter analysis used to construct optimal wavefront
• Rapidly converges in 5-10 measurements
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Convergence of new algorithm
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New view of Tx limits in opaque media
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Today’s talk
• Extraction of “signals” from clutter in highly corrupted video frames
• New theoretical limits and algorithms
• New methods for making fog and clouds “transparent”
• New theoretical limits and algorithms
• (Newly developed) Random matrix theory is enabling technology
• Many more AF relevant applications
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