Using Invariant Theory to Obtain Unknown Size, Shape, Motion, and Three-Dimensional Images from Single Aperture Synthetic Aperture Radar October 2005 Mark Stuff

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Using Invariant Theory to Obtain Unknown Size,
Shape, Motion, and Three-Dimensional Images from
Single Aperture Synthetic Aperture Radar
October 2005Mark Stuff
SN-05-0378AFRL/WS Approved Security and Policy
Review WorksheetReviewed File  
ima_presentation_v3.ppt SN-05-0378  Your
document (Presentation/Brief), Using Invariant
Theory to Obtain Unknown Size, Shape, Motion, and
Three-Dimensional Images from Single Aperture
Synthetic Aperture Radar was cleared by AFRL/WS
on 11-OCT-05 as Document Number AFRL/WS-05-2360.
Significant portions of the work reported here
were supported by the United States Air Force
under contract F33615-02-C-1177
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2D Imaging Simplified
Cone
Plane
Surface
Turn Table SAR cone determined by table
center and radar position
Stationary Object SAR plane determined by
scene center and radar path
Moving Object SAR surface determined by radar
path, object motion.
3
Previous Data From INS on military truck

• Real targets rapidly move through three
  dimensional, angularly complex, trajectories.
• Signal phase changes measurably for
  displacements of fractions of the wavelength.
• Any simplistic motion model rapidly looses
  fidelity as the observation time increases

degrees
Target fixed view of the aircraft trajectory
Earth fixed view of the aircraft trajectory
degrees
Ten seconds of angular inertial data from a truck
on a nominally straight flat road
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3D Image Surface
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Implications of complex, random angular motions
in three dimensions

• Auto-focus techniques are not enough
  uncompensated target rotations limit SAR/ISAR
  image quality, no matter how well the range
  translations are compensated.
• Radar data contains three dimensional target
  size and shape information No two dimensional
  image can contain all the target information
  present in the radar data.
• Sensor flight geometry does not determine
  viewing geometry side views are as common as top
  views mixed up, mangled, some of each views are
  what you really get.
• Finite parametric models for the motion rapidly
  loose fidelity who can predict the bumps on the
  road, or the jitters of the driver?

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Hope for a New Way to Understand Radar Signals

• Moving targets overload conventional processors
  with information
• System design often ignores or destroys much of
  this information
• Result is smeared, offset images, in 2 dimensions
• The 3DMAGI system exploits the additional
  information
• No restricting assumptions on the complexity of
  the motions
• No prior knowledge of the target type needed
  build models on the fly, track objects which have
  never been seen before
• Developed to work with single aperture systems
  extends to multi-aperture systems
• Requires understanding of geometric theory and
  unconventional signal processing
• Most of the work remains to be done Progress has
  been made by adapting the work to perceived
  immediate needs
• Focus moving targets
• Track moving targets

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3DMAGI System Diagram
3 Dimensional Motion And Geometric Information
(3DMAGI) System
Signal Preparation System
Input From Radar
Ranges to Scattering centers
Signal Analysis System
Shape Motion
Images Image Products
Geometric Analysis System
Image Formation System
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Extracting 3D Information

• Radar collapses the three dimensional world into
  a one dimensional signal.
• Coherent collapse confounds the target signals
  and often annihilates them (destructive
  interference).
• For movers, the radar signal provides more
  information in the form of diverse angular views
  
• Radar signals change rapidly because the signal
  phases change rapidly.
• The rich target information is encoded in those
  phase changes
• Prying these signals apart is the key to
  extracting 3D information from a moving object

Range-compressed phase history
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Radar pulses are only one dimensional

• Multiple scattering centers frequently
  contribute signals to the same range bin
• Combinations of simple stable scattering
  mechanisms give rise to complicated unstable
  interference patterns.

Time ?
Time ?
Time ?
Synthetic range profiles from 15 constant
amplitude scattering centers moving a rigid body
Range ?
Small angular changes alter the microwave radar
response via constructive and destructive
interference.
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3D View of Synthetic
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Target Rotation Enables Three-Dimensional Signal
Separation

• Simultaneous complex valued time, range, velocity
  analysis allows us to de-interfere range and time
  coincident scattering mechanism responses.
• Differential phase extraction allows us to track
  scattering mechanism ranges with sub-wavelength
  accuracy.

Time ?
Time ?
Time ?
Velocity ?
Range ?
Range ?
Range ?
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Tracks 2 and 3 in Time Slices
time?
?range
velocity?
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Dynamic progamming method automates optimal track
extraction

• The globally optimal path is guaranteed for each
  pass
• Several locally optimal paths per pass, in
  practice
• Optimizing over exponentially many possible
  trajectories in linear computational complexity
• Decisions depend on the integrated scores over
  the entire dwell this enables success at lower
  signal to noise ratios
• Natural tendency to find scattering mechanisms
  well spread around on the target
• Natural opportunity to enforce continuous
  differentiability
• Any local optimality score can be used phase
  information can be exploited

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Residuals and models
Pulses 401-600
range ?
Iteration 0
Iteration 1
Iteration 2
Iteration 3
? time
?
Original Range Compressed Signal
Signal Fitted Residual
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Signal Analysis
frequency ?
range ?
range ?
? time
? time
? cross range
Signal History
Range Compressed Image
Translation Compensated Vehicle Signature
Captured signal energy displayed in red.
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Three-dimensional signal separation yields range
histories with sub-wavelength precision
?
35 feet
?
3 feet
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3DMAGI System Diagram
3 Dimensional Motion And Geometric Information
(3DMAGI) System
Signal Preparation System
Input From Radar
Ranges to Scattering centers
Signal Analysis System
Shape Motion
Images Image Products
Geometric Analysis System
Image Formation System
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Rigid Body Kinematics
Consider a configuration of Q points (landmarks)
on a moving rigid body
What if we can only observe one of the three
coordinates?
Line of sight translation at time t
Far field range at time t
Illumination direction at time t
Fixed 3D coordinates
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Far Field Range Data for Multiple Landmarks
Observing just one coordinate approximates the
ranging sensor situation for remote
objects. Spherical wave-fronts become planar
wave-fronts Ignore wave-front curvature on the
scale of the rigid body
Line of sight translation at time t
Far field ranges at time t
Illumination direction at time t
Fixed 3D coordinates
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Special Cases of the Geometric Inverse Problem

• The ranges are observed data (or are estimated
  from observed data).
• If the motions are known, then solving for the
  coordinates of the configuration leads to a
  linear regression problem.
• If the coordinates are known, then solving for
  the motions (illumination directions and line of
  sight translation) leads to another linear
  regression problem.
• What if the motions and coordinates are both
  unknown?
• A nonlinear estimation problem
• Non-trivial uniqueness question (well-posedness,
  identifiability)
• The number of unknowns grows with the number of
  observations

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Translation Invariance by Centering
Centering the range measurements (at each time
sample) eliminates the translation function.
Centered far field ranges at time t
Illumination direction at time t
Centered 3D configuration matrix
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Rotation Invariance
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Proof Sketch for the Existence Theorem
Concept Just solve for the illumination
direction vectors and use the normalization
constraint to eliminate them.
Uses the non-co-planarity assumption
Uses the normalization constraint
Uses a lemma concerning centered quadratic forms
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How We Want to Use the Invariant Equation
Try to create an over-determined set of equations
for the invariants
Problems Matrix rank never exceeds 6 (S has rank
3, so centered range histories have rank 3, so
quadratic combinations of range histories span,
at most, 6 dimensions), Motion must escape any
elliptic cone to reach rank 6, For Q gt 4, we need
another condition to identify the invariants.
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The Uniqueness Theorem
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Definitions for Discrete Data Sets The
Projection Matrix
Matrix of centered ranges
Q by Q Covariance matrix of centered ranges
Q by Q projection matrix, determined by the
leading eigenvectors
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Proof Sketch for the Uniqueness Theorem
Definitions, centered quadratic form lemma
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Estimation Strategy
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Example

• Eight points on a rigid body
• Angular motions giving rise to an eight component
  vector time series

noise
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Noise Study for the Estimated Projection Matrix, P
Mean and standard deviation of SSE for the
estimated projection matrix P
SSE for the estimated projection matrix P as
function of input noise level for 100 realizations
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Noise Study for the Estimated Squared Difference
Matrix, ?
Mean and standard deviation of MSE for the
estimated squared differences matrix psi
MSE for the estimated squared differences matrix
psi as function of input noise level for 100
realizations
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Noise study for the estimated Geometric
invariants matrix, ?
Mean and standard deviation of SSE for the
estimated invariant matrix Omega
SSE for the estimated matrix of geometric
invariants, Omega, as function of input noise
level for 100 realizations
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What Are These Invariants?
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Constructing a Model A Choice of a Coordinate
Frame
The configuration S is determined up to Euclidean
isometry. The orientation (and reflection
symmetry) remains unknown. But the size and
shape of S is determined.
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Scattering center locations extracted from radar
signals
Show movie here
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Constructing a Model A Choice of a Coordinate
Frame
Estimated point coordinates for three
realizations of noise (blue, green, red)
Low noise
High noise
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3DMAGI System Diagram
3 Dimensional Motion And Geometric Information
(3DMAGI) System
Signal Preparation System
Input From Radar
Ranges to Scattering centers
Signal Analysis System
Shape Motion
Images Image Products
Geometric Analysis System
Image Formation System
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National Ground Intelligence Center Data
T72 with reactive armorin compact range on
9/25/01

• T72 with reactive armor in the compact range on
  9/25/01.
• Replaced with T72 Model M1 for compact range
  collections from 3/12/02 to 3/26/02.
• Although the reactive armor should not change
  the 3D-MAGI result, it adds complication to the
  goal of quantifying 3D-MAGI performance.

T72 Model M1(without reactive armor)
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Three orthographic views
Multiple orthogonal views clarify relation
between vehicle parts and signature elements,
leading to unambiguous interpretations

• NGIC 3D data
• T72
• 10 degree by 10 degree aperture

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Masked Range Image From Dense 10 Degree Data

• NGIC 3D data
• T 72
• 10 deg by 10 deg angle-angle aperture

Aid in characterizing new or unique vehicles or
modifications
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3D Image Surface
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Moving target signal history formatted in three
dimensions
Fz ?

• Signal history extracted from NGIC t72 data
• TEL motion data quantized to NGIC illumination
  angles
• Dragnet II Collection
• China Lake
• 6 seconds collection time

Fy ?
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Recorded Vehicle Motion Example Illumination
Directions Curve

• Original and quantized aspect-elevation angles
• TEL
• Dragnet II Collection
• China Lake
• 6 seconds collection time

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Image From Extrapolation to Neighboring 3D Grid
Locations
Illumination direction view of 3D image from
extrapolation to two sided nearest neighbor grid
locations using a local linear, inverse squared
distance weighted, model with all the moving
target signal data.
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Iterative 3D Clean Process for Moving Target
Signals
Initial complete moving target signal history
Scattering center locations and amplitudes
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Iterative Experiments / Clean methods
Autoscaled
Autoscaled
Autoscaled
Iteration 0
Iteration 1 Residual
Modeled Signal 1
Illumination direction view of 3D image from
extrapolation to two sided nearest neighbor grid
locations using a local linear, inverse squared
distance weighted, model with all of the moving
target signal data.
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Three-Dimensional Point Response
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Iterative Experiments / Clean methods
Autoscaled
Autoscaled
Autoscaled
Iteration 0
Iteration 49 Residual
Iteration 248 Residual
Illumination direction view of 3D image from
extrapolation to two sided nearest neighbor grid
locations using a local linear, inverse squared
distance weighted, model with all of the moving
target signal data.
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Iterative Experiments / Clean methods
Full 3D data set
6 Seconds Moving target subset 248 estimated
points
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Summary

• Moving targets impose 3D information on SAR
  radar returns
• This extra information confuses any process that
  does not take it specifically into account
• To take advantage of the 3d information, the
  motion of the target is needed
• This cannot be predicted, so must be measured
• With the motion information the proper
  relationships between the target and the data can
  be determined
• From this data 3D images are possible and are
  useful for exploiting moving targets
• More research must be done to improve the
  methods for creating the 3D image products