Using Invariant Theory to Obtain Unknown Size, Shape ... · • ima_presentation_v3.ppt ... 16 Three-dimensional signal separation yields range ... Rigid Body Kinematics

Using Invariant Theory to Obtain Unknown Size, Shape, Motion, and Three-Dimensional Images from

Single Aperture Synthetic Aperture Radar

October 2005Mark Stuff

This talk contains results which have been contributed to by many persons:

Martin Biancalana (GDAIS)Joseph Gabarino (ALTARUM)Jason Hunt (GDAIS)Susan Wei (GDAIS)Gregory Arnold (AFRL)Vincent Velton (AFRL)Michael Woodroofe (U of Mich)Robert Keener (U of Mich)Pedro Sanchez (Eastern Mich U)

Significant portions of the work reported here were supported by the United States Air Force under contract F33615-02-C-1177

SN-05-0378AFRL/WS Approved

Security and Policy Review Worksheet

Reviewed 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.

2

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 radarpath

Moving Object:SAR surface determined

by radar path, object motion.

3

Previous Data From INS on military truck

u Real targets rapidly move through three dimensional, angularly complex, trajectories.

u Signal phase changes measurably for displacements of fractions of the wavelength. uAny simplistic motion model rapidly looses fidelity as the observation time increases

degrees

degrees

Target fixed view of the aircraft trajectory

Earth fixed view of the aircraft trajectory

Ten seconds of angular inertial data from a truck on a nominally straight flat road

4

3D Image Surface

5

Implications of complex, random angular motions in three dimensions:

u“Auto-focus” techniques are not enough: uncompensated target rotations limit SAR/ISAR image quality, no matter how well the range translations are compensated.

u 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.

u 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.

u Finite parametric models for the motion rapidly loose fidelity: who can predict the bumps on the road, or the jitters of the driver?

6

Hope for a New Way to Understand Radar Signals

u 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

u 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

uMost 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

7

3DMAGI System Diagram

SignalPreparationSystem

InputFromRadar

Signal AnalysisSystem

GeometricAnalysisSystem

ImageFormationSystem

Shape&

Motion Images&

ImageProducts

3 Dimensional Motion And Geometric Information (3DMAGI) System

Ranges toScattering

centers

8

Extracting 3D Information

u Radar collapses the three dimensional world into a one dimensional signal.u Coherent collapse confounds the target signals and often annihilates them (destructive interference).u 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

u Prying these signals apart is the key to extracting 3D information from a moving object

Range-compressed phase history

9

Radar pulses are only one dimensional

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

Range è

Time è

Time è

Synthetic range profiles from 15 constant amplitude scattering centers moving a rigid body

Time è

Small angular changes alter the microwave radar response via constructive and destructive interference.

10

3D View of Synthetic

11

Target Rotation Enables Three-Dimensional Signal Separation

uSimultaneous complex valued time, range, velocity analysis allows us to de-interfere range and time coincident scattering mechanism responses.

uDifferential phase extraction allows us to track scattering mechanism ranges with sub-wavelength accuracy.

Range èRange è Range èVelocity

è

Time è

Time è

Time è

12

Tracks 2 and 3 in Time Slices

time→

←range velocity→

13

Dynamic progamming method automates optimal track extraction

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

14

Residuals and models

range →

←tim

e

Pulses 401-600

Signal = Fitted + Residual

Iteration 0 Iteration 1 Iteration 2 Iteration 3

Original Range Compressed Signal

15

Signal Analysis

Signal History Range Compressed Image Translation Compensated Vehicle Signature

Captured signal energy displayed in red.

←tim

e

←tim

e

frequency → range → range →

←cr

oss

rang

e

16

Three-dimensional signal separation yields range histories with sub-wavelength precision

35 feet

3 feet

17



InputFromRadar




Shape&

Motion Images&

ImageProducts


Ranges toScattering

centers

18

Rigid Body Kinematics

Consider a configuration of Q points (landmarks) on a moving rigid body:Coordinates at time t

Rotation matrix at time t

Coordinates at time zero

Translation vector at time t

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

19

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


Fixed 3D coordinates

20

Special Cases of the Geometric Inverse Problem

u The ranges are observed data (or are estimated from observed data).u If the motions are known, then solving for the coordinates of the configuration leads to a linear regression problem.u If the coordinates are known, then solving for the motions (illumination directions and line of sight translation) leads toanother linear regression problem.

u 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

21

Translation Invariance by Centering

Centering the range measurements (at each time sample) eliminates the translation function.

Centered far field ranges at time t


Centered 3D configuration matrix

22

Rotation Invariance

23

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

24

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 > 4, we need another condition to identify the invariants.

25

The Uniqueness Theorem

26

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

27

Proof Sketch for the Uniqueness Theorem

Definitions, centered quadratic form lemma

Quadratic rank 6 condition; linear relation of ranges to illumination directions

Projection matrix properties

Definition of the projection matrix

28

Estimation Strategy

29

Example

uEight points on a rigid bodyuAngular motions giving rise to an eight component vector time series

noise

30

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

31

Noise Study for the Estimated Squared Difference Matrix, ψ

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

32

Noise study for the estimated Geometric invariants matrix, Ω

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

33

What Are These Invariants?

34

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.

35

Scattering center locations extracted from radar signals

Show movie here

36

Constructing a Model = A Choice of a Coordinate Frame

Low noise

High noise

Estimated point coordinates for three realizations of noise (blue, green, red)

37



InputFromRadar




Shape&

Motion Images&

ImageProducts


Ranges toScattering

centers

38

National Ground Intelligence Center Data

T72 with reactive armorin compact range on 9/25/01

T72 Model M1(without reactive armor)

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

39

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

40

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 uniquevehicles or modifications

41

3D Image Surface

42

Moving target signal history formatted in three dimensions

•Signal history extracted from NGIC t72 data

•TEL motion data quantized to NGIC illumination angles

•Dragnet II Collection

•China Lake

•6 seconds collection time

Fz

?Fy ?

43

Recorded Vehicle Motion Example: Illumination Directions Curve

•Original and quantized aspect-elevation angles

•TEL

•Dragnet II Collection

•China Lake

•6 seconds collection time

44

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.

45

Iterative 3D Clean Process for Moving Target Signals

Copy for each input point

Locations of input data samples

Locations of ouput data samples

Copy for each output point

4 MB

1 MBCalculate displacements

250 GB

250 GB

Prepend constant term

250 GB

Convert to squared distance, invert

250 GBDuplicate

250 GB

332 GBDuplicate 4 times

83 GBMultiply

332 GB

Moore-Penrose Inverse

332 GB

Extract First Term Weights for each input point,

one set for each output point

332 GB

83 GB

Create Weights





4 MB

1 MBCalculate displacements

250 GB

250 GB

Prepend constant term

250 GB

Convert to squared distance, invert

250 GBDuplicate

250 GB

332 GBDuplicate 4 times

83 GBMultiply

332 GB

Moore-Penrose Inverse

332 GB

Extract First Term Weights for each input point,

one set for each output point

332 GB

83 GB

Create Weights

Embed in 3D volume

3D Fourier transform

Locate maximum in 3D image

Synthesize 2D moving scattering center signals

Regress remaining target signals on scattering center signals

Initial complete moving target signal history

Weights for each input point, one set for each output point

83 GB





4 MB

1 MBConvert to squared distance, invert

250 GB

250 GB

Multiply, convert to complex values

166 GB

Complex valued residual from previous iteration


2 MB Multiply and sum

83 GB

166 GB

1 MB

To embedding, point selection, and regression

Extrapolate Signal Values

Weights for each input point, one set for each output point

83 GB





4 MB

1 MBConvert to squared distance, invert

250 GB

250 GB

Multiply, convert to complex values

166 GB

Complex valued residual from previous iteration


2 MB Multiply and sum

83 GB

166 GB

1 MB

To embedding, point selection, and regression

Extrapolate Signal Values

Scattering center locations and amplitudes

46

Iterative Experiments / “Clean” methods

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.

Iteration 0 Iteration 1 Residual Modeled Signal 1AutoscaledAutoscaledAutoscaled

47

Three-Dimensional Point Response

48


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.

Iteration 0 Iteration 49 Residual Iteration 248 ResidualAutoscaled Autoscaled Autoscaled

49


Full 3D data set 6 Seconds Moving target subset

248 estimated points

50

Summary

u Moving targets impose 3D information on SAR radar returnsu This extra information confuses any process that does not take it specifically into accountu To take advantage of the 3d information, the motion of the target is needed

• This cannot be predicted, so must be measured

u With the motion information the proper relationships between the target and the data can be determinedu From this data 3D images are possible and are useful for exploiting moving targetsu More research must be done to improve the methods for creating the 3D image products

Documents

Using Invariant Theory to Obtain Unknown Size, Shape ... · • ima_presentation_v3.ppt ... 16 Three-dimensional signal separation yields range ... Rigid Body Kinematics