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School of Electrical and Computer Engineering

A Mathematical Theoryof Automatic Target Recognition

Aaron D. Lanterman

(lanterma@ece.gatech.edu)

What Makes ATR “Harder” than Factoring Large Numbers?

- Factoring large numbers may be NP-hard, but...
- At least it’s easy to precisely specify what the problem is!
- Not so easy in ATR
- Subject to controversy

Can You Build an Airplane Without a Theory of Aerodynamics?

- Sure! Without aerodynamic theory, you can do this...

- …but with a theory, you can do this!

Can You Build an Communication Systems w/out Information Theory?

- Sure! Without Information Theory, you can do this…

- …but with Information Theory, you can do this!

Steam Engines and Thermodynamics

- Dick Blahut likens the situation to steam engines coming before the science of thermodynamics
- First steam engines build by entrepreneurs and “inventors”
- Thomas Savery: 17th and 18th centuries
- Necessity the mother of invention!
- Thermodynamics didn’t begin to crystallize until mid 19th century… but with it, you eventually get

Shannon’s Lightning Bolt

- 1948: Claude Shannon’s “A Mathematical Theory of Communication” (1948)
- Later renamed “The Mathematical Theory of Communication”
- Found fundamental limits on what is possible, i.e. channel capacity

- Before Shannon, your boss might ask you to do the impossible, and fire you if you failed to do it!
- Your boss cannot fire your for failing to exceed channel capacity!
- You can tell your boss you need a better channel

Theory and Technology

- Advances in theory are not enough;

also need the technology

- Aerodynamic theory alone won’t get you a B-2;

need advances in materials, manufacturing

- Information theory along won’t get you cell phones;

need fast DSP chips, good batteries, even more theory (i.e. coding theory)

- Theory tells you what’s possible, but sometimes only hints at how to get there
- Quantum computing folks: does this sound familiar?

Multiple Sensors

Database

(Statistical Estimation-Theoretic)

Info-Theoretic View of ATRTarget Recognizer

Scene

Understanding

Channel

Decoder

Source

Performance

Bounds

Optimality Criteria

Miss, false alarm rate

Confusion matrices

Bias, Variance, M.S.E.

Hypothesis testing (LRT, GLRT)

ML, Bayes, Neyman Pearson

Estimation

ML, MAP, M.M.S.E., Bayes

Chernoff

Stein’s Lemma

Cramer-Rao

CIS/MIM

What Makes ATR “Harder” than Designing a Cell Phone?

- The space of X for real-world scenes is extremely complicated
- You don’t get to pick p(x)
- Likelihood p(y|x) is difficult to formulate
- The “channel” is often deliberately hostile
- Targets hiding in clutter
- Using decoys and camouflage
- Radars can be subject to jamming

Variability in Complex Scenes

- Geometric variability
- Position
- Orientation
- Articulation
- “Fingerprint”
- Environmental variability
- Thermal variability in infrared
- Illumination variability in visual
- Complexity variability
- Number of objects not known

Ulf Grenander

- Student of Cramér (yes, that Cramér)
- PhD on statistical inference in function spaces (1950)
- “Toeplitz Forms and their Applications” (with Szegö)
- Fundamental work on spectral estimation (1958)
- “Probabilities on Algebraic Structures” (1968)
- “Tutorial on Pattern Theory” - unpublished manuscript
- Inspired classic paper by Geman & Geman (1983)

General Pattern Theory

- Generalize standard probability, statistics, and shape theory
- Put probability measures on complex structures
- Biological structures
- Mitochondria
- Amoebas
- Brains
- Hippocampus
- Natural language
- Real-world scenes of interest in ATR

The 90’s GPT Renaissance

- Made possible by increases in computer power
- Michael Miller (Washington Univ., now at JHU) did a sabbatical with Grenander
- Fields Medalist David Mumford moves from Harvard to Brown; shifts from algebraic geometry to pattern theory

Composite Parameter Spaces

- Naturally handles obscuration
- Don’t know how many targets are in the scene in advance

- Move away from thinking of detection, location, recognition, etc. as separate problems

Applying the Grenander Program (1)

- Take a Bayesian approach
- Many ATR algorithms seek features that are invariant to pose (position and orientation)
- Grenander’s Pattern Theory treatspose as nuisance variable in the ATR problem, and deals with it head on
- Co-estimate pose, or integrate it out
- At a given viewing angle, Target A at one orientation may look much like Target B at a different orientation
- “…the nuisance parameter of orientation estimation plays a fundamental role in determining the bound on recognition” - Grenander, Miller, & Srivastava

U. Grenander, M.I. Miller, and A. Srivastava, “Hilbert-Schmidt Lower Bounds for Estimators on Matrix Lie Groups for ATR,” IEEE Trans. PAMI, Vol. 20, No. 2, Aug. 1998, pp. 790-802.

Applying the Grenander Program (2)

- Develop statistical likelihood
- Data fusion is natural
- At first, use as much of the data as possible
- Be wary of preprocessing: edge extraction, segmentation etc.
- Processing can never add information
- Data processing inequality from information theory

- If you need to extract features, i.e. for real-time computational tractability, try to avoid as much loss of information as possible

- Estimation bounds on continuous parameters
- Cramér-Rao bounds for continuous pose parameters
- Hilbert-Schmidt metrics for orientation parameters
- Bounds on detection/recognition probabilities
- Stein’s Lemma, Chernoff bounds
- Asymptotic analysis to approximate probabilities of error
- Performance in a binary test is dominated by a term exponential in a distance measure between a “true” and an “alternate” target
- Adjust pose of “alternate” target to get closest match to “true” target as seen by the sensor system
- Secondary term involving CRB on nuisance parameters
- Links pose estimation and recognition performance

Anuj

Srivastava

U. Grenander, A. Srivastava, and M.I. Miller, “Asymptotic Performance Analysis of Bayesian Target Recognition,” IEEE Trans. Info. Theory, Vol. 46, No. 4, July 2000, pp. 1658-1665.

Reading One of DARPA’s BAAs…

- DARPA’s E3D program seeks:
- “efficient techniques for rapidly exploiting 3-D sensor data to precisely locate and recognize targets.”
- BAA full of demands (hopes?) for different stages of the program, such as:
- “The Target Acquisition and Recognition technology areas will develop techniques to locate and recognize articulating, reconfigurable targets under partial obscuration conditions, with an identification probability of 0.85%, a target rejection rate less than 5%, and a processing time of 3 minutes per target or less”

…Leads Us to Wondering

- If such a milestone is not reached,

is that the fault of the algorithm or the sensor?

- How does the DARPA Program Manager know who to fire?
- Without a theory, the DARPA PM may fire someone who was asked to “exceed channel capacity,” i.e. given an impossible task
- What performance from a particular sensor is necessary to achieve a certain level of ATR performance,

independent of the question of what algorithm is used?

- CCD loglikelihood of Snyder et. al

where

- Cascade with

- Sensor fusion natural; just add loglikelihoods

- Write posterior in Gibbs form:

- Fix number of targets and target types
- Simulate Langevin diffusion:

- Distribution of

- Computed desired statistics from the samples
- Generalizes to non-Euclidean groups like rotations
- Gradient computation
- Numeric approximations
- Easy and fast on modern 3-D graphics hardware

- Gibbs style
- Sample from a restricted part of the posterior
- Metropolis-Hastings style
- Draw a “proposal” from a “proposal density”
- Accept (or reject) the proposal with a certain probability

Simulations from PRISM: Discretizes target surface using regions from CAD template and internal heat transfer model

Average Static State

Average Dynamic State

CIS/MIM

Can’t Hide from Thermal Variations

Profile 8 Profile 45 Profile 75 Profile 140

Performance Variations

Due To Thermodynamic Variability

Performance Loss Due To

Inaccurate Thermodynamic Information

Cooper, Miller SPIE 97

CIS/MIM

Model radiance as scalar random field on surface

Compute empirical mean & covariance from database of 2000 radiance profiles

Karhunen-Loeve expansion using eigenfunctions of covariance on surface - “Eigentanks”

Add expansion coefficients to parameter space

Fortunately, able to estimate directly given pose

Principle Component Representation of Thermal StateMatt Cooper

(now with Xerox)

A younger, much

thinner Aaron

Lanterman

SPIE 97 Cooper, Grenander, Miller, Srivastava

CIS/MIM

Meteorological Variation

Operational Variation

Remember, we’re

showing 2-D views of

full 3-D surfaces

Composite Mode of Variation

SPIE 97 Cooper, Grenander,

Miller, Srivastava

CIS/MIM

Joint MAP Est. of Pose and Thermal Signature

Real NVESD M60 data (courtesy James Ratches)

Initial

Estimate

Final

Estimate

CIS/MIM

SPIE 98 Cooper and Miller

Information Bound

Ladar/IR Sensor FusionTom Green

Joe Kostakis

Jeff Shapiro

FLIR

(intensity)

LADAR

(range)

CIS/MIM

LADAR/FLIR Hannon Curve

15 degrees error

LADAR/FLIR Hannon Curve

9 degrees error

SPIE 98 Advanced Techniques ATR III

Kostakis, Cooper, Green, Miller,

OSullivan, Shapiro Snyder

CIS/MIM

Target Models

Panzer IILight Tank

Sturmgeschultz IIISelf-Propelled Gun

Semovente M41 Self-Propelled Gun

M48 A3 Main Battle Tank

Hull Length: 4.81 mWidth: 2.28 mHeight: 2.15 m

Hull Length: 6.77 mWidth: 2.95 mHeight: 2.16 m

Hull Length: 5.205 mWidth: 2.2 mHeight: 2.15 m

Hull Length: 6.419 mWidth: 3.63 mHeight: 3.086 m

(Info and Top Row of Images from 3-D Ladar Challenge Problem Slides by Jacobs Sverdrup)

CR-Bound on Orientation

Position assumed known

We take a performance hit!

Strum

Position unknown, must be

co-estimated

Semo

Interesting knee at 0.2 meters

M48 vs. Others

M48 and Panzer have dissimilar signatures; most easily distinguished

M48 and Semo have similar signatures; most easily confused

Semovente vs. Others

At higher resolutions,

Semo and M48 have most dissimilar signatures; most easily distinguished

(perhaps there are nice features which only become apparent at higher resolutions?)

At lower resolutions,

Semo and Panzer have most dissimilar signatures; most easily distinguished

Semoand Sturm have similar signatures; most easily confused

Michael

DeVore

Joseph

O’Sullivan

- • MSTAR Data Set
- Conditionally Gaussian model for pixel values with variances trained from data
- • Likelihood based classification
- • Target orientation unknown and uniformly distributed over 360° of azimuth
- • Joint orientation estimation and target classification
- • Train on 17° depression angle
- • Test on 15° depression angle

T72

BMP 2

Variance Images

SAR Images

CIS/MIM

• Results using 72 variance images per target of 10° each, and using 80 x 80 pixel sub-images to reduce background clutter

• Probability of correct classification: 98%

• Average orientation error: < 10°

Orientation

MSE effects ID!

CIS/MIM

Supported by ARO Center for Imaging Science DAAH 04-95-1-04-94 and ONR MURI N00014-98-1-06-06

Where Should Clutter Go? (1)

A “forward model,” i.e. a “scene simulator”

non-Gaussian minimax entropy texture models by Song Chun Zhu

- A forest might go well in the “noise” part…

Where Should Clutter Go? (2)

- …but downtown Baghdad will not “whiten”
- Structured clutter is the most vexing
- May need to go in here, and directly manipulate the clutter

…or a bit of each

- Where to draw the line?

Acknowledgments

- Much of the work described here was funded by the ARO Center for Imaging Science
- Also ONR (William Miceli) and AFOSR (Jon Sjogren)
- Slides with CIS/MIM tag were adapted from slides provided by Michael Miller

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