Information-Theoretic Imaging Hyperspectral …jao/Talks/InvitedTalks/cmu...Information-Theoretic...

InformationInformation--Theoretic Imaging Theoretic Imaging with Applications inwith Applications in

HyperspectralHyperspectral Imaging and Imaging and TransmissionTransmission TomographyTomography

Joseph A. OJoseph A. O’’SullivanSullivan

Electronic Systems and Signals Research Laboratory

Department of Electrical Engineering

Washington University

[email protected]

http://essrl.wustl.edu/~jao

Supported by: ONR, ARO, NIHONR, ARO, NIH

J. A. O’Sullivan. CMU Seminar, 03/13/2002Information-Theoretic Imaging

2

CollaboratorsCollaborators

Michael D. DeVore

Natalia A. Schmid

Metin Oz

Ryan Murphy

Jasenka Benac

Adam Cataldo

Donald L. Snyder

William H. Smith

Daniel R. Fuhrmann

Jeffrey F. Williamson

Bruce R. Whiting

Richard E. Blahut

Michael I. Miller

Chrysanthe Preza

David G. Politte

James G. Blaine

Faculty Students and Post-Docs


3

Outline:Outline:

•• Information Theory and ImagingInformation Theory and Imaging

•• HyperspectralHyperspectral ImagingImaging

-- Information Value DecompositionInformation Value Decomposition

•• TransmissionTransmission TomographyTomography

-- Estimate in Exponential FamilyEstimate in Exponential Family

-- ObjectObject--ConstrainedConstrained

ComputerizedComputerized TomographyTomography

•• Alternating Minimization AlgorithmsAlternating Minimization Algorithms

•• Applications: HSI, CTApplications: HSI, CT

•• ConclusionsConclusions


4

Some Roles of Information TheorySome Roles of Information TheoryIn Imaging ProblemsIn Imaging Problems

Some Important Ideas:Some Important Ideas:

•• Roles depend on problem studiedRoles depend on problem studied

•• Key problems are in detection, Key problems are in detection,

estimation, and classificationestimation, and classification

•• Information is quantifiableInformation is quantifiable

•• SNR as a measure has limitationsSNR as a measure has limitations

•• Information theory often providesInformation theory often provides

performance boundsperformance bounds


5

How to Measure InformationHow to Measure Information

Information for what?Information for what?

Information relative to what?Information relative to what?

IllIll--defined questions defined questions clearly defined problemclearly defined problem

Can we measure the information in an image?Can we measure the information in an image?

Does one sensor provide more information than another?Does one sensor provide more information than another?

Does resolution measure information?Does resolution measure information?

Do more pixels in an image give more information?Do more pixels in an image give more information?

QUESTIONS:QUESTIONS:

ANSWERANSWER::

MAYBEMAYBE


6

Measuring InformationMeasuring Information

Information for what?Information for what?

•• DetectionDetection

•• EstimationEstimation

•• ClassificationClassification

•• Image formationImage formation

Information relative to what?Information relative to what?

•• Noise onlyNoise only

•• Clutter plus noiseClutter plus noise

•• Natural SceneryNatural Scenery


7

Information Theory and ImagingInformation Theory and Imaging

•• Center for Imaging Science established 1995Center for Imaging Science established 1995

•• Brown MURI on Performance Metrics established 1997Brown MURI on Performance Metrics established 1997

•• Invited Paper 1998Invited Paper 1998

J. A. OJ. A. O’’Sullivan, R. E. Sullivan, R. E. BlahutBlahut, and D. L. Snyder,, and D. L. Snyder,

““InformationInformation--Theoretic Image Formation,Theoretic Image Formation,”” IEEEIEEE

Transactions on Information TheoryTransactions on Information Theory, Oct. 1998., Oct. 1998.

•• IEEE 1998 Information Theory Workshop on Detection,IEEE 1998 Information Theory Workshop on Detection,

Estimation, Classification, and ImagingEstimation, Classification, and Imaging

•• IEEE Transactions on Information TheoryIEEE Transactions on Information Theory SpecialSpecial

Issue on InformationIssue on Information--Theoretic Imaging, Aug. 2000Theoretic Imaging, Aug. 2000

•• Complexity Regularization, Large Deviations, Complexity Regularization, Large Deviations, ……


8

IEEE Transactions on Information TheoryIEEE Transactions on Information Theory, October 1998, , October 1998,

Special Issue to commemorate the 50th anniversary of Special Issue to commemorate the 50th anniversary of

Claude E. Shannon's Claude E. Shannon's

A Mathematical Theory of CommunicationA Mathematical Theory of Communication

Problem DefinitionProblem Definition

Optimality CriterionOptimality Criterion

Algorithm DevelopmentAlgorithm Development

Performance QuantificationPerformance Quantification


9

HyperspectralHyperspectral ImagingImaging

at Washington Universityat Washington University

Donald L. SnyderDonald L. Snyder

William H. SmithWilliam H. Smith

Daniel R. Daniel R. FuhrmannFuhrmann

Joseph A. OJoseph A. O’’SullivanSullivan

Chrysanthe PrezaChrysanthe Preza

WU TeamWU Team

SnyderSnyder SmithSmith FuhrmannFuhrmann OO’’SullivanSullivan

Digital Array Scanning Digital Array Scanning

InterferometerInterferometer

(DASI)(DASI)


10

HyperspectralHyperspectral ImagingImaging• Scene Cube Data Cube

• “Drink from a fire hose”

• Filter wheel, interferometer,

tunable FPAs

• Modeling and processing:

- data models

- optimal algorithms

- efficient algorithms


11

Hyperspectral Imaging Likelihood ModelsHyperspectral Imaging Likelihood Models

ideal data : r y y

y h y x : s x : dxd

h y x : ha y x : ha y x :2

shear vector for Wollaston prism

h y x : wavelength dependent amplitude PSF of DASI

s x : scene intensity for incoherent radiation at x,

nonideal (more realistic) data :

r y Poisson y 0 y Gaussian y

data likelihood :

E r | scene log1

n y !y 0 y

n ye

y 0 ye

r y n y2

2 2

n y 1y 1

Y


12

Idealized Data ModelIdealized Data Model

•• Data spectrum for each pixel Data spectrum for each pixel ssjj

•• Linear combination of constituent spectraLinear combination of constituent spectra

•• Problem: Estimate constituents and proportions Problem: Estimate constituents and proportions subject to subject to nonnegativitynonnegativity;; positivitypositivity ofof SS assumedassumed

•• Ambiguity if Ambiguity if

•• Comments: Radiometric Calibration; Comments: Radiometric Calibration; Constraints FundamentalConstraints Fundamental

K

k

kjkj as1

AS


13

Idealized Problem Statement:Idealized Problem Statement:MaximumMaximum--LikelihoodLikelihood Minimum IMinimum I--divergencedivergence

•• Poisson distributed data Poisson distributed data loglikelihoodloglikelihood functionfunction

•• Maximization over Maximization over and A equivalent to and A equivalent to minimization of Iminimization of I--divergencedivergence

AS

I

i

J

j

K

k

kjik

K

k

kjikij aasASl1 1 11

ln)|(

I

i

J

j

K

k kjikijK

k kjik

ijij as

a

ssASI

1 11

1

ln)||(

•• Information Value Decomposition ProblemInformation Value Decomposition Problem


14

Markov ApproximationsMarkov Approximations•• XX andand YY RVRV’’s on finite sets, s on finite sets, p(x,y)p(x,y) unknownunknown

•• Data:Data: NN i.i.d. pairs {i.i.d. pairs {XXii,Y,Yii}}

•• Unconstrained ML Estimate of Unconstrained ML Estimate of p(x,y)p(x,y)

•• Lower rank Markov approximation Lower rank Markov approximation XX MM YYMM in a set of cardinality in a set of cardinality KK

•• Factor analysis, contingency tables, economicsFactor analysis, contingency tables, economics

•• Problem: Approximation of one matrix by another of Problem: Approximation of one matrix by another of lower ranklower rank

•• C.C. EckartEckart and G. Young, and G. Young, PsychometrikaPsychometrika, vol. 1, pp. 211, vol. 1, pp. 211--218 1936.218 1936.

•• SVDSVD IVDIVD

N

yxnS

),(

AS

)||(min ASIA


15

CT Imaging in Presence of High CT Imaging in Presence of High Density AttenuatorsDensity Attenuators

BrachytherapyBrachytherapy applicatorsapplicators

AfterAfter--loadingloading colpostatscolpostats

for radiation oncologyfor radiation oncology

Cervical cancer: 50% survival rateCervical cancer: 50% survival rate

Dose prediction importantDose prediction important

ObjectObject--Constrained Computed Constrained Computed

TomographyTomography (OCCT)(OCCT)


16

Filtered Back ProjectionFiltered Back Projection

Truth FBP

FBP: inverse Radon transform


17

TransmissionTransmission TomographyTomography

•• SourceSource--detector pairs indexed by detector pairs indexed by yy; pixels indexed by ; pixels indexed by xx

•• DataData d(y)d(y) Poisson, means Poisson, means g(y:g(y: ),), loglikelihoodloglikelihood functionfunction

•• MeanMean unattenuatedunattenuated countscounts II00, mean background , mean background

•• Attenuation function Attenuation function (x,E)(x,E),, EE indexes energiesindexes energies

•• Maximize over Maximize over oror ccii;; equivalently minimize Iequivalently minimize I--divergencedivergence

•• Comment: pose search Comment: pose search c(x) = cc(x) = caa(x:(x: ) +) + ccbb(x)(x)

l (d : ¹ ) =X

y2 Y

[d(y) ln g(y : ¹ ) ¡ g(y : ¹ )]

¹ (x; E ) =P I

i = 1 ¹ i (E )ci (x)

g(y : ¹ ) =P

E I 0(y; E) exp [¡P

x h(yjx)¹ (x; E )] + ¯(y)


18

Alternating Minimization AlgorithmsAlternating Minimization Algorithms

•• Define problem as Define problem as minminqq (q)(q)

•• DeriveDerive VariationalVariational Representation:Representation: (q) = (q) = minminpp J(p,q)J(p,q)

•• JJ is an auxiliary function is an auxiliary function pp is in auxiliary set is in auxiliary set PP

•• Result: double minimization Result: double minimization minminqq minminpp J(p,q)J(p,q)

•• Alternating minimization algorithmAlternating minimization algorithm

),(

),(

)1()1(

)()1(

minarg

minarg

qpJq

qpJp

l

Qq

l

l

Pp

l

Comments: Guaranteed Monotonicity; J selected carefully


19

Alternating Minimization Algorithms:Alternating Minimization Algorithms:II--Divergence, Linear, Exponential FamiliesDivergence, Linear, Exponential Families

•• Special Case of Interest:Special Case of Interest: JJ is Iis I--divergencedivergence

•• Families of Interest:Families of Interest:Linear Family Linear Family L(A,b) = {p: L(A,b) = {p: ApAp = b}= b}Exponential FamilyExponential Family E(E( ,B) = {q: ,B) = {q: qqii == ii exp[exp[ jj bbijij jj]}]}

)||(

)||(

)1()1(

)()1(

minarg

minarg

qpIq

qpIp

l

Eq

l

l

Lp

l

CsiszCsiszáárr andand TusnTusnáádydy;; DempsterDempster, Laird, Rubin;, Laird, Rubin; BlahutBlahut;;

Richardson; Lucy;Richardson; Lucy; VardiVardi,, SheppShepp, and Kaufman; Cover;, and Kaufman; Cover;

Miller and Snyder; O'SullivanMiller and Snyder; O'Sullivan


20

Alternating Minimization ExampleAlternating Minimization Example

•• Linear family: pLinear family: p11 + 2 p+ 2 p22 = 2= 2

•• Exponential family: qExponential family: q11 = exp (= exp (vv), q), q22 = exp (= exp (--vv))

0 1 2 3 4 50

1

2

3

4

5

p1, q

1

p 2,

q2

0 1 2 3 4 50

1

2

3

4

5

p1, q

1

p 2,

q2

0.6 0.8 1 1.2 1.40.2

0.4

0.6

0.8

1

1.2

p1, q

1

p2,

q 2

0.6 0.8 1 1.2 1.40.2

0.4

0.6

0.8

1

1.2

p1, q

1

p2,

q 2

)||(minmin qpILpEq


21

Information GeometryInformation Geometry

•• II--divergence is nonnegative, convex in pair divergence is nonnegative, convex in pair (p,q)(p,q)

•• Generalization of relative entropy, Generalization of relative entropy, example of fexample of f--divergencedivergence

•• First triangle equality: First triangle equality: pp inin LL

•• Second triangle equality:Second triangle equality: qq inin EE

)||()||()||()||( *** minarg qpIppIqpIqpIp

Lp

)||()||()||()||( *** minarg qqIqpIqpIqpIq

Eq


22

VariationalVariational RepresentationsRepresentations

•• Convex decomposition lemma. Let Convex decomposition lemma. Let ff be convex. Thenbe convex. Then

•• Special Case:Special Case: ff isis lnln

•• Basis for EM; see also De Basis for EM; see also De PierroPierro, Lange, , Lange, FesslerFessler

i

ii

i i

ii

rii

rr

xfrxf

0,1

)()(1

i

i

i i

ii

Ppi

i

ppP

q

ppq

1:

lnminln


23

ShrinkShrink--Wrap Algorithm for Wrap Algorithm for EndmembersEndmembers

FuhrmannFuhrmann, WU, WU

SS == AA

== EndmembersEndmembers, K Columns, K Columns

AA = Pixel Mixture Proportions= Pixel Mixture Proportions

SVD, Then Simplex Volume MinimizationSVD, Then Simplex Volume Minimization

SS== AA

GivenGiven andand SS, estimate , estimate AA..

Uses IUses I--Divergence Discrepancy Divergence Discrepancy

Measure.Measure.

Alternating Minimization AlgorithmsAlternating Minimization Algorithms

for Hyperspectral Imagingfor Hyperspectral Imaging

++PoissonPoisson

ProcessProcess

CC

AlternatingAlternating

MinimizationMinimization

AlgorithmAlgorithm

InitialInitial

EstimatedEstimated

CoefficientsCoefficients

EstimatedEstimated

CoefficientsCoefficients

Snyder, WUSnyder, WU OO’’Sullivan, WUSullivan, WU


25

Information Value DecompositionInformation Value Decomposition

Applied toApplied to HyperspectralHyperspectral DataData

• Downloaded spectra from USGS website

• 470 Spectral components

• Randomly generated A with 2000 columns

• Ran IVD on result


26

Information Value DecompositionInformation Value DecompositionApplied to Applied to HyperspectralHyperspectral DataData


27

Information Theoretic Imaging:Information Theoretic Imaging:

Application to Hyperspectral ImagingApplication to Hyperspectral Imaging


Optimality CriterionOptimality Criterion



•• Likelihood Models for SensorsLikelihood Models for Sensors

•• Likelihood Models for ScenesLikelihood Models for Scenes

•• Spectrum Estimation andSpectrum Estimation andDecompositionDecomposition

•• Performance QuantificationPerformance Quantification

•• Applications to Available SensorsApplications to Available Sensors


28

HyperspectralHyperspectral ImagingImagingOngoing EffortsOngoing Efforts

•• Scene Models Including Spatial and Scene Models Including Spatial and Wavelength CharacteristicsWavelength Characteristics

•• Sensor Models Including Sensor Models Including ApodizationApodization

•• Orientation Dependence ofOrientation Dependence of HyperspectralHyperspectralSignaturesSignatures

•• Expanded Complementary EffortsExpanded Complementary Efforts

•• Atmospheric Propagation ModelsAtmospheric Propagation Models

•• Performance BoundsPerformance Bounds

•• Measures of Added InformationMeasures of Added Information


29

CT Minimum ICT Minimum I--Divergence FormulationDivergence Formulation

L(d) = f p(y; E ) ¸ 0 :X

E

p(y; E) = d(y)g

I (dkg(y : c)) = minp2 L (d)

I (pkq)

l(d : ¹ ) =X

y2 Y

[d(y) ln g(y : ¹ ) ¡ g(y : ¹ )]

g(y : ¹ ) =P

E I 0(y; E) exp [¡P

x h(yjx)¹ (x; E )] + ¯(y)

Maximize loglikelihood function over minimize I-divergence

Use Convex Decomposition Lemma

g is a marginal over q


30

New Alternating Minimization AlgorithmNew Alternating Minimization Algorithmfor Transmission for Transmission TomographyTomography

minq2 E( I 0;H ¢¹ )

minp2 L (d)

I (pkq)

q(k ) (y; E ) = I 0(y; E ) exp ¡X

x 2 X

X

i

¹ i (E )h(yjx)c(k )i (x)

#

p(k ) (y; E) = q(k ) (y; E)d(y)

PE 0q(k ) (y; E 0)

~b(k )i (x) =

X

y2 Y

X

E

¹ i (E )h(yjx)p(k ) (y; E )

b(k )i (x) =

X

y2 Y

X

E

¹ i (E )h(yjx)q(k ) (y; E )


31

New Alternating Minimization AlgorithmNew Alternating Minimization Algorithmfor Transmissionfor Transmission TomographyTomography

c(k+ 1)i (x) = c

(k )i (x) ¡

1

Zi (x)ln

Ã~b

(k )i (x)

b(k )i (x)

!

Interpretation: Compare predicted data to measured data

via ratio of backprojections

Update estimate using a normalization constant

Comments: Choice for constants; monotonic convergence;

Linear convergence; Constraints easily incorporated


32

Iterative Algorithm with Known Iterative Algorithm with Known Applicator PoseApplicator Pose

Truth Known Pose


33

OCCT IterationsOCCT Iterations

Known Pose

OCCT


34

TruthTruth Known

Standard

FBP

OCCT


35

Magnified views around Magnified views around

brachytherapybrachytherapy applicatorapplicator

Truth OCCT

FBP


36

ConclusionsConclusions•• InformationInformation--Theoretic ImagingTheoretic Imaging

•• Alternating Minimization AlgorithmsAlternating Minimization Algorithms

•• HyperspectralHyperspectral ImagingImaging

-- Applying Formal MethodologyApplying Formal Methodology

-- Data CollectionsData Collections

-- Sensor Modeling, Sensor Modeling, LikelihoodsLikelihoods

-- Broad, Ongoing EffortsBroad, Ongoing Efforts

•• CT ImagingCT Imaging

-- New Image Formation AlgorithmNew Image Formation Algorithm

-- Simulations and Scanner DataSimulations and Scanner Data

-- Modeling Issues: Physics CrucialModeling Issues: Physics Crucial


37


38

PhysicalPhysical

PhenomenaPhenomena

MathematicalMathematical

AbstractionsAbstractions

ReproductionReproduction

SpaceSpace

CriteriaCriteria

PerformancePerformance

Physical ScenePhysical Scene Physical SensorsPhysical Sensors

Scene ModelScene Model Sensor ModelsSensor Models

c C L2 L2

C C

l | c , f c , I || f c

min d ,c

d c, c d c, c*

d c*, c


39


Physical Phenomena:Physical Phenomena: Electromagnetic Waves, Electromagnetic Waves, Physical ObjectsPhysical Objects

Mathematical Abstraction:Mathematical Abstraction: Function Spaces (LFunction Spaces (L22),),Dynamical ModelsDynamical Models

Sensor Models:Sensor Models: Projection Models, Projection Models, StatisticsStatistics

Reproduction Space:Reproduction Space: Finite Dimensional Finite Dimensional ApproximationApproximation

Optimality CriterionOptimality CriterionDeterministic:Deterministic: Least Squares, Least Squares,

Minimum DiscriminationMinimum Discrimination

Stochastic:Stochastic: Maximum Likelihood, Maximum Likelihood, MAP, MMSEMAP, MMSE


40


Iterative Algorithms:Iterative Algorithms: Alternating MinimizationsAlternating MinimizationsExpectationExpectation--MaximizationMaximization

Random Search:Random Search: Simulated Annealing, Simulated Annealing, Diffusion, JumpDiffusion, Jump--DiffusionDiffusion


Discrepancy Measures:Discrepancy Measures: Squared Error, Squared Error, DiscriminationDiscrimination

Performance:Performance: Bias, Variance, Bias, Variance, Mean DiscriminationMean Discrimination

Performance Bounds:Performance Bounds: CramerCramer--RaoRao,,HilbertHilbert--SchmidtSchmidt


41

Target Orientation EstimationTarget Orientation Estimation

Likelihood Model Approach:Likelihood Model Approach:ppRR|| ,,aa((rr|| ,,aa)) -- Conditional Data ModelConditional Data Model

pp ,,aa(( ,,aa)) -- Prior on orientation (known or simply uniform)Prior on orientation (known or simply uniform)

P(P(aa)) -- Prior on target class (known or simply uniform)Prior on target class (known or simply uniform)

TargetTarget

ClassifierClassifierââ=T72=T72

OrientationOrientation

EstimatorEstimator=135=135

aa=T72=T72

Given a SAR image Given a SAR image rr,,

determine a corresponding determine a corresponding

target class target class ââ AA

Given a SAR image Given a SAR image rr andand

a target class a target class ââ AA,,

estimate target orientationestimate target orientation

Target Classification ProblemTarget Classification Problem


42

Training and Testing Problem:Training and Testing Problem:Function Estimation and ClassificationFunction Estimation and Classification

Function

Estimation

L(r|a, ) Inferenceââ=T72=T72

Scene and Sensor

Physics

Training Data

Raw DataProcessing

Image

• Labeled training data: target type and pose

• Log-likelihood parameterized by a function:

mean image, variance image, etc.


43

Function Estimation and ClassificationFunction Estimation and Classification

•• Functions are estimated fromFunctions are estimated from-- sample data sets onlysample data sets only-- physical model datasets only (PRISM, XPATCH, etc.)physical model datasets only (PRISM, XPATCH, etc.)-- combination of thesecombination of these

•• Training sets are finiteTraining sets are finite

•• Computational and likelihood models have a finite Computational and likelihood models have a finite number of parametersnumber of parameters

•• Estimation errorEstimation error

•• Approximation errorApproximation error

•• Some regularization is neededSome regularization is needed


44

Function EstimationFunction Estimation

LetLet SS andand TT be two target types,be two target types,

ffSS andand ffTT the corresponding functionsthe corresponding functions

fS : D R or R+

Spatial domain Spatial domain DD::

Surface of objectSurface of object

Rectangular region in RRectangular region in R22

ContinuousContinuous

Pose space Pose space ::

Typically SE(3)Typically SE(3)

Simpler to consider SE(2), SO(2)Simpler to consider SE(2), SO(2)

is continuousis continuous

Assume a null function fAssume a null function f00


45

Function Estimation: SieveFunction Estimation: Sieve

Function space Function space FF

Family of subsets Family of subsets FF

-- Maximum likelihood estimate of Maximum likelihood estimate of ff exists for each exists for each

-- FF FF

UlfUlf GrenanderGrenander

SplineSpline sievessieves = 1/= 1/Q,Q, wherewhere QQ = number of coefficients.= number of coefficients.

Nonnegative functions, nonnegative expansions.Nonnegative functions, nonnegative expansions.

Pierre Moulin, et al.Pierre Moulin, et al.

ff(( ) = ) = kk a(k)a(k) kk(( ) , a(k) >= 0) , a(k) >= 0


46

Sieve GeometrySieve Geometry

.

.

.

.

.fS

F

F1

F2

f2

f1

12


47

Discrepancy MeasuresDiscrepancy Measures

Estimation:

Relative entropy between parameterized distributions

d(fS, approximation error

Expected relative entropy for estimated functions

E{d( f )} estimation error

Classification:

Probability of error, rate functions

Decompose into approximation and estimation error

Examine performance as a function of , optimize


48

MotivationMotivation

• Many reported approaches to ATR from SAR

• Performance and database complexity are interrelated

• We seek to provide a framework for comparison that:

- Allows direct comparison under identical conditions

- Removes dependency on implementation details

2S12S1 T62T62 BTR 60BTR 60 D7D7 ZIL 131ZIL 131 ZSU 23/4ZSU 23/4

Publicly available SAR data from the MSTAR program


49

Approaches: Conditionally Approaches: Conditionally GaussianGaussian

J. A. OJ. A. O’’Sullivan and S. Jacobs, IEEESullivan and S. Jacobs, IEEE--AES 2000AES 2000

Model each pixel as complex Model each pixel as complex GaussianGaussian plus uncorrelated noise:plus uncorrelated noise:

i

NaK

r

i

Ai

i

eNaK

ap 0

2

,

0

,,

1,r

R

aBayes r argmaxa

maxk

p r k ,a

ˆHS r,a argmax

k

p r k ,a

GLRT Classification and MAP Estimation:GLRT Classification and MAP Estimation:

J. A. OJ. A. O’’Sullivan, M. D. Sullivan, M. D. DeVoreDeVore, V. , V. KediaKedia, and M. Miller, IEEE, and M. Miller, IEEE--AES to appear 2000AES to appear 2000


50

ApproachApproach

• Select 240 combinations of implementation parameters

• Execute algorithms at each parameterization

• Scatter plot the performance-complexity pairs

• Determine the best achievable performance at any complexity

BMP2 Variance Image at 6 Sizes

ZIL131 Variance Image at 6 Sizes

Six different image sizes from 128x128 to 48x48


51

System Parameters and ComplexitySystem Parameters and ComplexityApproximateApproximate (( ,,aa) and ) and 22(( ,,aa) as piecewise constant in ) as piecewise constant in

Implementations parameterized by:Implementations parameterized by:

ww -- number of constant intervals in number of constant intervals in

dd -- width of training intervals in width of training intervals in

NN22 -- number of pixels in an imagenumber of pixels in an image

Database complexity log10(# floating point values / target type)


52

Performance and ComplexityPerformance and Complexity

Forty combinations of angular resolution and training interval width.

Variance image of aT62 tank

1 Window trained over 360°

Variance images of a T72 tank

72 Windows trained over 10°


53

Problem StatementProblem Statement• Directly compare conditionally Gaussian, log-magnitude MSE,

and quarter power MSE ATR Algorithms

- identical training and testing data

- identical spatial and orientation windows

• Plot performance vs. complexity

- probability of classification error

- orientation estimation error

- log-database size as complexity

• Use 10 class MSTAR SAR images

ApproachApproach• Select 240 combinations of implementation parameters

• Execute algorithms at each parameterization

• Scatter plot the performance-complexity pairs

• Determine the best achievable performance at any complexity


54

Approaches: LogApproaches: Log--MagnitudeMagnitude

Minimize distance between rdB = 20 log |r| and dB templates

d2

rdB, LM rdB LM

2

Make decisions according to:

aLM r argmina

mink

d2

rdB, LM k, a

ˆLM r a argmin

k

d2 rdB, LM k ,a

Alternatively, use a form of normalization:

d2

rdB rdB, LM k,a LM k ,a

G. Owirka and L. Novak, SPIE 2230, 1994

L. Novak, et al., IEEE-AES, Jan. 1999


55

Approaches: Quarter PowerApproaches: Quarter Power

d2

rQP, QP rQP QP

2

Minimize distance between rQP = |r|1/2 and quarter power templates

Make decisions according to:

aQP r argmina

mink

d2

rQP, QP k, a

ˆQP r,a argmin

k

d2 rQP, QP k,a

d2 rQP

rQP

,QP k ,a

QP k ,a

Or, normalized by vector magnitude:

S. W. Worrell, et al., SPIE 3070, 1997

Discussions with M. Bryant of Wright Laboratory


56

ConditionallyConditionally GaussianGaussian ResultsResults


57

PerformancePerformance--Complexity LegendComplexity Legend

Forty combinations of number of piecewise constant

intervals and training window width


58

LogLog--Magnitude ResultsMagnitude Results

Recognition without normalization Arithmetic mean normalized


59

Quarter Power ResultsQuarter Power Results

Recognition without normalization Recognition with normalization


60

Normalized Conditionally Normalized Conditionally GaussianGaussianResultsResults


61

SideSide--byby--Side ResultsSide ResultsComparison in terms of:

• Performance achievable at a given complexity

• Complexity required to achieve a given performance


62

2S1 BMP 2 BRDM 2 BTR 60 BTR 70 D7 T62 T 72 ZIL131 ZSU 23 4

2S1 262 0 0 0 0 0 4 8 0 0 95.62%

BMP 2 0 581 0 0 0 0 0 6 0 0 98.98%

BRDM 2 5 3 227 1 0 14 3 5 4 1 86.31%

BTR 60 1 0 0 193 0 0 0 0 0 1 98.97%

BTR 70 4 5 0 0 184 0 0 3 0 0 93.88%

D7 2 0 0 0 0 271 1 0 0 0 98.91%

T 62 1 0 0 0 0 0 259 11 2 0 94.87%

T 72 0 0 0 0 0 0 0 582 0 0 100%

ZIL131 0 0 0 0 0 0 2 0 272 0 99.27%

ZSU 23 4 0 0 0 0 0 2 0 1 0 271 98.91%

•• Probability of correctProbability of correct

classification: 97.2%classification: 97.2%

Target ClassificationTarget Classification

ResultsResults


63

• General problem in training/testing posed as

estimation/classification

• Method of sieves (polynomial splines chosen)

• Comprehensive performance-complexity study for

- Ten class MSTAR problem

- Conditionally Gaussian model

- Log-magnitude MSE

- Quarter power MSE

• Provided a framework for direct comparison of alternatives

and selection of implementation parameters

• Analysis ongoing

ConclusionsConclusions

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