Some Roles of Information Theory in Imaging Problems

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000CollaboratorsFacultyStudents and Post-DocsDonald L. SnyderWilliam H. SmithDaniel R. FuhrmannJeffrey WilliamsonBruce WhitingRichard E. BlahutMichael I. MillerJeffrey H. ShapiroMichael D. DeVoreNatalia A. SchmidMetin OzRyan MurphyJasenka BenacAdam CataldoSushil Anand2

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Research ProgramsARO Center for Imaging Science:JHU, MIT, Washington U., Texas,…http://ciscis.jhu.edu/ONR MURI: Performance Analysis of MultipleSensor Systems in Automatic Target RecognitionKrim (NCSU), Miller (JHU),O’Sullivan (WU),Shapiro and Willsky(MIT)National Cancer Institute of NIHJ. Williamson (PI), D. L. Snyder, J. A. O’Sullivan, OBruce Whiting3

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Outline:• Information Theory• Problems of Interest- Detection- Estimation- Classification• Applications- ATR from SAR- Object-ConstrainedComputerized Tomography- Hyperspectral Imaging• Conclusions4

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Some Roles of Information TheoryIn Imaging ProblemsSome Important Ideas:• Roles depend on problem studied• Key problems are in detection,estimation, and classification• Information is quantifiable• SNR as a measure has limitations• Information theory often providesperformance bounds5

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000How to Measure InformationQUESTIONS:Can we measure the information in an image?Does one sensor provide more information than another?Does resolution measure information?Do more pixels in an image give more information?ANSWER:MAYBEInformation for what?Information relative to what?Ill-defined questions clearly defined problem6

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Measuring InformationInformation for what?• Detection• Estimation• Classification• Image formationInformation relative to what?• Noise only• Clutter plus noise• Natural Scenery7

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Measuring Information: DetectionProblem Statement:H 1 : r(y) = s(y) + n(y), y ∈YH 0 : r(y) = n(y), y ∈YP(1) = pOptimal test statistic: l(r)Threshold: γExample: deterministic signal plus WGN, N 0l(r) = 2 /N 0Performance: P FA (γ), P M (γ)Bounds:-lnP FA (γ) ≥ I 0 (γ) = sup[s γ – m 0 (s) ]-lnP M (γ) ≥ I 1 (γ) = I 0 (γ) - γRate functions, Chernoff Bound8

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Measuring Information: DetectionRate functions:Deterministic signal plus WGNI(x) = (x-SNR)2 /(4 SNR)Poisson signal plus Poisson noiseI(x) = x ln [x / λ] – x + λComplex Gaussian Plus CWGNI(x) = x / λ - ln [x / λ] – 1I 1 (x)I 1 (x)I 0 (x)I 0 (x)9

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Measuring Information: EstimationFisher InformationRegularizationObjective (Likelihood) function: ψ(r,c)Estimate: c i = argmax ψ(r i ,c)Discrepancy Measure: d(c 1 ,c 2 ) ≥ 0Problem is well-posed ifsup [ d(c 1 ,c 2 ) / d(r 1 ,r 2 ) ] < ∞Regularization –Sequence of sets C µ such that restrictionsc ∈C µ make problem well-posed10

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Regularizationargmax ψ(r,c) + α φ(c)Interpretations:Penalty: penalize deviations from a nominalφ(c) = d(c,c 0 )Prior likelihood:α φ(c) = ln p(c)Constrained estimation:φ(c) ≤γα = Lagrange multiplier11

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Regularization and Information TheoryDiscrepancy measures motivated by information theorySquared Errord(p,q) = Σ i < p i – q i , p i – q i >Discrimination (or I-divergence)Id(p,q) = Σ i p i ln [ p i / q i ] - p i + q iItakura-Saito distanced(p,q) = Σ i p i / q i - ln [ p i / q i ] - 1Common decomposition for C µd(f S ,f i ) = d(f S ,φ i ) + d(φ i ,f i )φ i = argmin Cµ d(f S ,f)12

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Regularization Geometry.f SFφ 2...φ 1F 1F 2f 2f 1.13

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Information Theory and ImagingBrown-MITMIT-Harvard MURID. Mumford, , U. Grenander, , S. Mitter, , S. Geman,D. McClure, R. BrockettRelated EffortsA. Yuille, , S.-C. ZhuComplexity RegularizationA. Barron, J. Rissanen, , B. Yu, P. MoulinOSA Annual Meeting and Exhibit 2000M. Neifeld, , E. Clarkson, H. J. Caulfield,G. Barbastathis, , T. SchulzOSA Information Processing DivisionQuantum Theory, MicroscopyInformation Theory and Physics Jaynes, …14

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Complexity Regularizationargmax ψ(r,c) + α φ(c)Choose φ(c) as a measure of complexityComments:• Nonconvex optimization• Different measures of complexity exist• Universality versus problem-specific15

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Information Theory and Imaging• Center for Imaging Science established 1995• Brown MURI on Performance Metrics established 1997• Invited Paper 1998J. A. O’Sullivan, OR. E. Blahut, , and D. L. Snyder,“Information-Theoretic Image Formation,” IEEETransactions on Information Theory, , Oct. 1998.• IEEE 1998 Information Theory Workshop on Detection,Estimation, Classification, and Imaging• IEEE Transactions on Information Theory SpecialIssue on Information-Theoretic Imaging, Aug. 2000• Sessions at ISIT, ICIP, SPIE16

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Scientific Framework forImaging ScienceTarget Recognizer(Supercomputer)SceneUnderstandingScene SynthesizerSourceChannelMultiple SensorsDecoderDatabaseXY = Y 1 , Y 2 , ... Y mX^E( X)p(X ) = e−E(X)ZE(Y | X)p(Y | X) = e−E(Y|X)Z(X)p(X|Y)= p(Y|X)p(X)p(Y)17

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Sensor Fusion and InformationTarget Recognizer(Supercomputer)SceneUnderstandingScene SynthesizerSourceChannelMultiple SensorsDecoderDatabaseXE( X)p(X ) = e−E(X)ZUncertainty of SourceEntropy H(X)Y= Y1,Y2E(Y | X)p(Y | X) =,...Yem−E(Y |X)Z(X)Uncertainty of Source GivenOutputEntropy H(X|Y)^XpY ( | X)p(X)p(X|Y)=pY ( )Mutual InformationI(X;Y)=H(X) - H(X|Y)18

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000IEEE Transactions on Information Theory, , October 1998,Special Issue to commemorate the 50th anniversary ofClaude E. Shannon'sA Mathematical Theory of CommunicationProblem DefinitionOptimality CriterionAlgorithm DevelopmentPerformance Quantification19

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000PhysicalPhenomenaPhysical ScenePhysical SensorsMathematicalAbstractionsScene ModelSensor Modelsc ∈C ⊂ L 2 σ ∈Ο ⊂ L 2ReproductionSpaceC ν ⊂ CCriteriaPerformancel( σ | c), σ − fcmin d σ ,c ( )( )( ) , I σ || fc ( )dc, ( c ˆν)= dc, ( c ˆ*)+ d c ˆ* , ˆ( c ν)20

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Problem DefinitionPhysical Phenomena:Electromagnetic Waves,Physical ObjectsMathematical Abstraction: Function Spaces (L 2 ),Dynamical ModelsSensor Models:Projection Models,StatisticsReproduction Space:Finite DimensionalApproximationOptimality CriterionDeterministic:Stochastic:Least Squares,Minimum DiscriminationMaximum Likelihood,MAP, MMSE21

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Algorithm DevelopmentIterative Algorithms:Random Search:Alternating MinimizationsExpectation-MaximizationSimulated Annealing,Diffusion, Jump-DiffusionPerformance QuantificationDiscrepancy Measures:Performance:Performance Bounds:Squared Error,DiscriminationBias, Variance,Mean DiscriminationCramer-RaoRao,Hilbert-Schmidt22

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Information Theory and AlgorithmsLemma: Given p i ≥ 0, i = 1, 2, …, n, with at least onepositive,ln [ Σ i p i ] = - min Σ i q i ln [q i / p i ] ,where the minimum is over q i that sum to 1.cp(s|c)sp(r|s)rln [ p(r|c) ] = - min Σ s p(s|r,c) ln [p(s|r,c) /{p(r|s)p(s|c)}]Alternating minimization (EM):p(s|r,c) p(s|c)23

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Information Theoretic Algorithms:Alternating MinimizationsPoisson data, minimum discrimination,Lucy-RichardsonShepp, Vardi, Lee, Lucy, Richardson,Miller, Snyder, Schulz, O’SullivanProblem: Given r(y) and h(y|x), minimize I(r||Hc).Comment:• Nonnegativity fundamentally accounted for• Use lemma from above p(x|y)min c min p Ι(pr||hc) =min c min p Σ y Σ x r(y)p(x|y) ln [r(y)p(x|y) / h(y|x)c(x)]- r(y)p(x|y) + h(y|x)c(x)24

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Alternating Minimization AlgorithmsDr(y)÷BackProjectDivide byH 0 (x)c (k) (x)ForwardProject25

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Target Classification ProblemTargetClassifierâ=T72Given a SAR image r,determine a correspondingtarget class â∈ATarget Orientation Estimationa=T72OrientationEstimatorθ=135Given a SAR image r anda target class â∈A,estimate target orientationLikelihood Model Approach:p R|Θ,a (r|θ,a) - Conditional Data Modelp Θ,a (θ,a) - Prior on orientation (known or simply uniform)P(a) - Prior on target class (known or simply uniform)26

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Training and Testing Problem:Function Estimation and ClassificationTraining DataScene and SensorPhysicsFunctionEstimationRaw DataProcessingImageL(r|a,θ)Inferenceâ=T72• Labeled training data: target type and pose• Log-likelihood parameterized by a function:mean image, variance image, etc.27

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Function Estimation and Classification• Functions are estimated from- sample data sets only- physical model datasets only (PRISM, XPATCH, etc.)- combination of these• Training sets are finite• Computational and likelihood models have a finitenumber of parameters• Estimation error• Approximation error• Some regularization is needed28

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Function EstimationLet S and T be two target types,f S and f T the corresponding functionsf S : Θ × D R or R +Spatial domain D:Surface of objectRectangular region in R 2ContinuousPose space Θ:Typically SE(3)Simpler to consider SE(2), SO(2)Θ is continuousAssume a null function f 029

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Function Estimation: SieveFunction space FFamily of subsets F µ , µ > 0- Maximum likelihood estimate of f exists for each µ > 0- ∪ µ > 0 F µ = FUlf GrenanderSpline sieves µ = 1/Q,where Q = number of coefficients.Nonnegative functions, nonnegative expansions.Pierre Moulin, et al.f(λ) ) = Σ k∈Λµa(k) ψ k (λ)) , a(k) >= 030

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Sieve Geometry.f SFφ 2...φ 1F 1F 2f 2f 1.31

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Discrepancy MeasuresEstimation:Relative entropy between parameterized distributionsd(f S ,φ µ ) approximation errorExpected relative entropy for estimated functionsE{d(φ µ , f µ )} estimation errorClassification:Probability of error, rate functionsDecompose into approximation and estimation errorExamine performance as a function of µ, optimize32

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Motivation• 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 details2S1T62 BTR 60D7 ZIL 131 ZSU 23/4Publicly available SAR data from the MSTAR program33

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Approaches: Conditionally GaussianModel each pixel as complex Gaussian plus uncorrelated noise:ri( )1−∏Ki( θ , a)+ N0p rθ , a =eR Θ,A( ( ) + )i π Kiθ , a N02GLRT Classification and MAP Estimation:a ˆ Bayes ()= r argmaxmaxa k( )= argmaxˆ θ r,a HSp ˆ rθ ,a kp ˆ ( rθ k ,a)( )θ kJ. A. O’Sullivan Oand S. Jacobs, IEEE-AES 2000J. A. O’Sullivan, OM. D. DeVore, , V. Kedia, , and M. Miller, IEEE-AES to appear 200034

Information Theory and ImagingApproachJ. A. O’Sullivan. OSA Meeting, 10/25/2000• Select 240 combinations of implementation parameters• Execute algorithms at each parameterization• Scatter plot the performance-complexity pairs• Determine the best achievable performance at any complexityBMP2 Variance Image at 6 SizesZIL131 Variance Image at 6 SizesSix different image sizes from 128x128 to 48x4835

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000System Parameters and ComplexityApproximate α(θ,a) ) and σ 2 (θ,a)) as piecewise constant in θImplementations parameterized by:w - number of constant intervals in θd - width of training intervals in θN 2 - number of pixels in an imageDatabase complexity ≡ log 10 (# floating point values / target type)36

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Performance and ComplexityForty combinations of angular resolution and training interval width.Variance image of aT62 tank1 Window trained over 360°Variance images of a T72 tank72 Windows trained over 10°37

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Problem 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 imagesApproach• Select 240 combinations of implementation parameters• Execute algorithms at each parameterization• Scatter plot the performance-complexity pairs• Determine the best achievable performance at any complexity38

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Approaches: Log-MagnitudeMinimize distance between r dB = 20 log |r| and dB templatesd 2( r dB,µ )2LM= r dB− µ LMMake decisions according to:ˆ a LMˆ θ LM()= r argminmina( r a)= argmin( )( )kd 2 r dB,µ LMθ k, a( )( )θ kd 2 r dB,µ LMθ k,aAlternatively, use a form of normalization:d 2 r dB− r dB,µ LM( θ k,a)− µ LMθ k,a( )( )G. Owirka and L. Novak, SPIE 2230, 1994L. Novak, et al., IEEE-AES, Jan. 199939

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Approaches: Quarter PowerMinimize distance between r QP = |r| 1/2 and quarter power templatesd 2Make decisions according to:ˆa QPˆθ QP(2r QP,µ QP )= r QP− µ QP( )( )( r) = argmin mind 2 r QP,µ QPθ k, aa k( r,a)= argmin d 2 r QP,µ QPθ k,aOr, normalized by vector magnitude:d 2⎜⎛⎝kr QP, µ QP θ k ,ar QPµ QPθ k,a( )( )( )( )⎞⎠S. W. Worrell, et al., SPIE 3070, 1997Discussions with M. Bryant of Wright Laboratory40

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Conditionally Gaussian Results41

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Performance-Complexity LegendForty combinations of number of piecewise constantintervals and training window width42

Information Theory and ImagingLog-Magnitude ResultsJ. A. O’Sullivan. OSA Meeting, 10/25/2000Recognition without normalizationArithmetic mean normalized43

Information Theory and ImagingQuarter Power ResultsJ. A. O’Sullivan. OSA Meeting, 10/25/2000Recognition without normalizationRecognition with normalization44

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Normalized Conditionally GaussianResults45

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Side-byby-Side ResultsComparison in terms of:• Performance achievable at a given complexity• Complexity required to achieve a given performance46

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Target ClassificationResults• Probability of correctclassification: 97.2%2S1 BMP 2 BRDM 2 BTR 60 BTR 70 D7 T62 T 72 ZIL131 ZSU 23 42S1 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 234 0 0 0 0 0 2 0 1 0 271 98.91%47

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Conclusions• General problem in training/testing posed asestimation/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 alternativesand selection of implementation parameters• Analysis ongoing48

MULTI-SPECTRAL ACTIVE-PASSIVESENSOR FUSION: FLIR/LADAR/HRRFLIR (30deg)ShapiroLADAR Range (30 deg)SPIE 98 ,99Advanced TechniquesHRR (30deg)

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Mean Square Error Performance for LaserRadar, FLIR, High-Resolution RadarFLIR MSE BoundLADAR MSE BoundHRR MSE BoundSPIEAeroSense1999 Miller,O’Sullivan,Shapiro50

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Multispectral Sensor Fusion for Ground BasedTargetOrientation Estimation: LADAR, FLIR, HRRFixed Error BoundLADAR/FLIRFixed Error BoundFLIR/HRRFixed Error BoundLADAR/HRRSPIEAeroSense1999Miller,O’Sullivan,ShapiroSullivan,Shapiro51

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000CT Imaging in Presence of HighDensity AttenuatorsBrachytherapy applicatorsAfter-loadingcolpostatsfor radiation oncologyCervical cancer: 50% survival rateDose prediction importantObject-Constrained ComputedTomography (OCCT)52

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Filtered Back ProjectionFBP: inverse Radon transformTruthFBP53

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000OCCT: Model and AlgorithmModel: m(y) = Σ h(y|x) [ c a (x:θ)+ c b (x) ]Known attenuation of applicator: c a (x:θ)Unknown pose: θUnknown body attenuation: c b (x)Known point-spread function: h(y|x)Source-detector pairs – yVoxels in image domain – xCriterion: minimize discrepancy betweenmeasurements and model(θ,c b ) =argmin =I(m||HcHc)54

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000OCCT: Algorithm55

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Iterative Algorithm with KnownApplicator PoseTruthKnown Pose56

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000OCCT IterationsKnown PoseOCCT57

Information Theory and ImagingTruthJ. A. O’Sullivan. OSA Meeting, 10/25/2000KnownStandardFBPOCCT58

Information Theory and ImagingMagnified views aroundbrachytherapy applicatorJ. A. O’Sullivan. OSA Meeting, 10/25/2000FBPTruthOCCT59

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Hyperspectral ImagingWU Teamat Washington UniversityDonald L. SnyderWilliam H. SmithDaniel R. FuhrmannJoseph A. O’SullivanOChrysanthe PrezaMetin OzCollaborationsArmy-TECBoeingNASA-AmesAmesAir ForceMIT, CMUSnyderSmithFuhrmannWorkshop on Hyperspectral ImagingMarch 9, 1999, at Boeing in St. LouisO’Sullivan60

Digital Array Scanning Interferometer (DASI)Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Digital Array Scanning Interferometer (DASI)Snyder, WU Smith, WU61

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Hyperspectral Imaging Likelihood Models()= µ ( y)()= hy− ( x : λ)ideal data : r yµ y ∫∫ sx: ( λ)dxdλhy− ( x : λ)= h a( y − x −∆: λ)+ h ay − x +∆: λ∆ shear vector for Wollaston prismhy− x : λsx: λnonideal (more realistic) data :ry ()= Poisson [ µ ()+ y µ 0() y ]+ Gaussian ydata likelihood :⎧Y ∞1Er| ( scene)=−log⎨∑ ∑ny ()! µ ()+ y µ 0 () y⎩ y =1 ny ( )=1( ) 2( ) wavelength dependent amplitude PSF of DASI( ) scene intensity for incoherent radiation at x, λ()[ ] ny[ ] e−[ ry ( )− ny ( )] 2 2σ 2( ) e − µ ( y)+µ 0 ( y)⎫⎬⎭62

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Shrink-Wrap Algorithm for EndmembersS = ΦAΦ = Endmembers, , K ColumnsA = Pixel Mixture ProportionsSVD, Then Simplex Volume MinimizationFuhrmann, , WU63

Alternating Minimization Algorithmsfor Hyperspectral Imagingα 1CInitialEstimatedCoefficientsSnyder, WU O’Sullivan, WUα 2+PoissonProcessAlternatingMinimizationAlgorithmEstimatedCoefficientsα 3S=ΦAGiven Φ and S, , estimate A.Uses I-Divergence IDiscrepancyMeasure.

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Information Theoretic Imaging:Application to Hyperspectral ImagingProblem DefinitionOptimality CriterionAlgorithm DevelopmentPerformance Quantification• Likelihood Models for Sensors• Likelihood Models for Scenes• Spectrum Estimation andDecomposition• Performance Quantification• Applications to Available Sensors65

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Hyperspectral ImagingOngoing Efforts• Scene Models Including Spatial andWavelength Characteristics• Orientation Dependence of HyperspectralSignatures• Expanded Complementary Efforts• Atmospheric Propagation Models• Performance Bounds• Measures of Added Information66

Information Theory and ImagingJ. A. O’Sullivan. OSA Meeting, 10/25/2000Conclusions• Information Theoretic Imaging• Radar Target RecognitionMSTAR DataTen Class ProblemPerformance Analysis• Hyperspectral ImagingApplying Formal MethodologyData CollectionsSensor Modeling, LikelihoodsBroad, Ongoing Efforts67