derivation of separability measures based on central complex gaussian and wishart distributions

Derivation of Separability Measures Based on Central Complex Gaussian and Wishart Distributions

Ken Yoong LEE and Timo Rolf Bretschneider July 2011EADS Innovation Works Singapore

EADS Innovation Works Singapore, Real-time Embedded Systems, IW-SI-I

Page *Content OverviewObjectivesDivergenceExperimentsSimulated POLSAR dataNASA/JPL POLSAR dataBased on central complex Gaussian distributionBased on central complex Wishart distributionBhattacharyya distance SummaryBased on central complex Gaussian distributionBased on central complex Wishart distribution

Page *IntroductionObjectivesRubber treesOil palm plantationWater (River)Scrub?NASA/JPL C-band data of Muda Merbok, Malaysia (PACRIM 2000)Use of Bhattacharyya distance and divergence as a separability measure of target classes in POLSAR dataDerivations of Bhattacharyya distance and divergence based on central complex distributions

Definition (Kailath, 1967):T. Kailath (1967). The divergence and Bhattacharyya distance measures in signal selection. IEEE Trans. Comm. Tech, 15(1), pp. 22992311.The Bhattacharyya coefficient is given bywhere f (x) and g (x) are pdfs of two populationsProperties (Matusita, 1966):K. Matusita (1966). A distance and related statistics in multivariate analysis. Multivariate Analysis, edited by Krishnaiah, P.R., New York: Academic Press, pp. 187200.b lies between 0 and 1 b = 1 if f(x) = g(x) b is also called affinity as it indicates the closeness between two populationsBhattacharyya DistancePage *Hellinger distance

Page *Scattering Vector and Central Complex Gaussian DistributionScattering vector z can be assumed to follow p-dimensional central complex Gaussian distribution (Kong et al, 1987):J. A. Kong, A. A. Swartz, H. A. Yueh, L. M. Novak, and R. T. Shin (1987). Identification of terrain cover using the optimum polarimetric classifier. JEWA, 2(2) pp. 171194.Scattering vector z in single-look single-frequency polarimetric synthetic aperture radar data:

Page *Bhattacharyya Distance from Central Complex Gaussian DistributionTheorem 1: The Bhattacharyya distance of two central complex multivariate Gaussian populations with unequal covariance matrices isCorollary 1: If p = 1, then the Bhattacharyya distance iswhile the Bhattacharyya coefficient isRemark 1: The application of Bhattacharyya distance for contrast analysis can be found in Morio et al (2008)J. Morio, P. Rfrgier, F. Goudail, P.C. Dubois-Fernandez, and X. Dupuis (2008). Information theory-based approach for contrast analysis in polarimetric and/or interferometric SAR images. IEEE Trans. GRS, 46, pp. 21852196

*Use of the following integration rules:andHence, the Bhattacharyya coefficient is while the Bhattacharyya distance is (Q.E.D)Now, the Bhattacharyya coefficient isProof of Theorem 1

Page *Covariance Matrix and Complex Wishart DistributionHermitian matrix A = n C can be assumed to follow central complex Wishart distribution (Lee et al, 1994):J. S. Lee, M. R. Grunes, and R. Kwok (1994). Classification of multi-look polarimetric SAR imagery based on complex Wishart distribution. IJRS, 15(11), pp. 22992311.Covariance matrix C in n-look single-frequency polarimetric synthetic aperture radar data:

Page *Bhattacharyya Distance from Central Complex Wishart DistributionTheorem 2: The Bhattacharyya distance of two central complex Wishart populations with unequal covariance matrices isCorollary 2: If p = 1, then the Bhattacharyya distance iswhile the Bhattacharyya coefficient isRemark 2: The Bhattacharyya distance is proportional to the Bartlett distance (Kersten et al, 2005) with a constant of 2/nP.R. Kersten, J.-S. Lee, and T.L. Ainsworth (2005). Unsupervised classification of polarimetric synthetic aperture radar images using fuzzy clustering and EM clustering. IEEE GRS, 43(3), pp. 519-527.

*The Jacobian of B to A: is Complex multivariate gamma functionHence, the Bhattacharyya coefficient is (Q.E.D)Now, the Bhattacharyya coefficient isProof of Theorem 2A.M. Mathai (1997). Jacobians of Matrix Transformations and Functions of Matrix Argument. Singapore: World Scientific(Mathai, 1997, Th. 3.5, p. 183)

Definition (Jeffreys, 1946; Kullback, 1959):H. Jeffreys (1946). An invariant form for the prior probability in estimation problems. Proc. Royal Soc. London (Ser. A), 186(1007), pp. 453461.wherethe functions f(x) and g(x) are pdf of two populationsProperties:S. Kullback (1959). Information Theory and Statistics, New York: John Wiley.J is zero if if f(x) = g(x), which implies no divergence between a distribution and itselfDivergencePage *andI1 or I2 is also known as Kullback-Leibler divergence

Page *DivergenceTheorem 3: The divergence of two p-dimensional central complex Gaussian populations with unequal covariance matrices isTheorem 4: The divergence of two p-dimensional central complex Wishart populations with unequal covariance matrices isRemark 3: The divergence is proportional to the symmetrized normalized log-likelihood distance (Anfinsen et al, 2007) with a constant of (2n)-1S.N. Anfinsen, R. Jensen, and T. Eltolf (2007). Spectral clustering of polarimetric SAR data with Wishart-derived distance measures. Proc. POLinSAR 2007, Available at earth.esa.int/workshops/polinsar2007/papers/140_anfinsen.pdf

*Both 1 and 2 can be diagonalized simultaneously (Rao and Rao, 1998, p. 186), i.e. andWe haveProof of Theorem 4 (1/2)where C is nonsingular matrix; I is identity matrix; D is diagonal matrix containing real eigenvalues 1,, p of Let W = C*AC, the Jacobian of the transformation from W to A is |C*C|p (Mathai, 1997, Theorem 3.5, p. 183) Hence,A.M. Mathai (1997). Jacobians of Matrix Transformations and Functions of Matrix Argument. Singapore: World ScientificC.R. Rao and M.B. Rao (1998). Matrix Algebra and Its Applications to Statistics and Econometrics. Singapore: World Scientific

*Proof of Theorem 4 (2/2)Finally, the divergence is(Q.E.D)

Page *Experiment (1)Scrub-grassland (1672 pixels)Oil palm (1350 pixels)Rubber(1395 pixels)Rice paddy (924 pixels)NASA/JPL Airborne POLSAR Data- Radar frequency: C and L-band- Polarisation: Full-pol (HH, HV and VV) Pixel spacing: 3.33m (range)4.63m (azimuth)- Scene title: MudaMerbok354-1|SHH|2|SHV|2|SVV|2C-band Image size: 400 pixels (column), 150 pixels (row)Simulated POLSAR Data- Number of looks: 9A B C D C-band, 9-lookL-band, 9-look- Number of looks: 4 and 9J. S. Lee, M. R. Grunes, and R. Kwok (1994). Classification of multi-look polarimetric SAR imagery based on complex Wishart distribution. IJRS, 15(11), pp. 22992311.- Simulation based on Lee et al (1994)- Acquired date: 19 September 2000

Page *Bhattacharyya distanceWindow size = 7DivergenceEuclidean distanceC-band, 9-lookL-band, 9-lookWindow size = 9Window size = 7Window size = 9Window size = 7Window size = 9Window size = 7Window size = 9Window size = 7Window size = 9Window size = 7Window size = 9Threshold =0.039 Correct detection rate = 1False detection rate = 0Threshold = 0.0305Correct detection rate = 1False detection rate = 0.0105Threshold = 0.322Correct detection rate = 1False detection rate = 0Threshold = 0.25086Correct detection rate = 1False detection rate = 0.0099Threshold = 0.00578Correct detection rate = 1False detection rate = 0.0692Threshold = 0.005Correct detection rate = 1False detection rate = 0.2316Threshold = 0.2701Correct detection rate = 1False detection rate = 0Threshold = 0.22Correct detection rate = 1False detection rate = 0Threshold = 2.43Correct detection rate = 1False detection rate = 0Threshold = 1.98Correct detection rate = 1False detection rate = 0Threshold = 0.00057Correct detection rate = 1False detection rate = 0.1764Threshold = 0.00049Correct detection rate = 1False detection rate = 0.2538Bhattacharyya distanceDivergenceEuclidean distance

Page *Bhattacharyya distanceWindow size = 7DivergenceEuclidean distanceC-band, 9-lookL-band, 9-lookWindow size = 9Window size = 7Window size = 9Window size = 7Window size = 9Window size = 7Window size = 9Window size = 7Window size = 9Window size = 7Window size = 9Threshold =0.039 Correct detection rate = 1False detection rate = 0Threshold = 0.0725Correct detection rate = 0.8324False detection rate = 0Threshold = 0.322Correct detection rate = 1False detection rate = 0Threshold = 0.61Correct detection rate = 0.8333False detection rate = 0Threshold = 0.0321Correct detection rate = 0.7071False detection rate = 0Threshold = 0.0492Correct detection rate = 0.6262False detection rate = 0Threshold = 0.2701Correct detection rate = 1False detection rate = 0Threshold = 0.22Correct detection rate = 1False detection rate = 0Threshold = 2.43Correct detection rate = 1False detection rate = 0Threshold = 1.98Correct detection rate = 1False detection rate = 0Threshold = 0.0046 Correct detection rate = 0.8524False detection rate = 0Threshold = 0.0068Correct detection rate = 0.7485False detection rate = 0Bhattacharyya distanceDivergenceEuclidean distance

Page *NASA/JPL Airborne POLSAR Data- Radar frequency: L-band- Polarisation: Full-pol (HH, HV and VV) Pixel spacing: 6.662m (range)12.1m (azimuth)- Scene number: Flevoland-056-1Bare soilBeetForestGrassLucernePotatoesStem beansWaterWheatPeasRapeseedLegend|SHH|2|SHV|2|SVV|2 Image size: 1024 pixels (range)750 pixels (azimuth)Experiment (2)PotatoesRapeseedStem beansWheat- 4 test regions identified:

Page *Bhattacharyya distanceDivergenceEuclidean distance

Bare soilBeetForestGrassLucernePeasPotatoesRape seedStem beansWaterWheatA3.31280.34270.07291.65641.22870.33280.03670.95760.19453.99610.5486B0.95040.46721.43610.16340.30370.48351.09430.11341.10761.52090.2938C2.36200.13300.37800.79520.39790.27330.23080.54660.10773.01360.2739D1.85960.12810.64870.55930.40710.08000.41510.17690.44022.49810.0023



Page *SummaryThe Bhattacharyya distances for complex Gaussian and Wishart distributions differ only in term of the number of degrees of freedom (number of looks in POLSAR data)Same observation for the divergenceThe Bhattacharyya distance is proportional to the Bartlett distanceThe divergence is proportional to the symmetrized normalized log-likelihood distanceBoth the Bhattacharyya distance and the divergence perform consistently in measuring the separability of target classesThe latter is more computationally expensive than the former

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Page *DivergenceCorollary 4: If p = 1, then the divergence isCorollary 3: If p = 1, then the divergence is

*Both 1 and 2 can be diagonalized simultaneously (Rao and Rao, 1998, p. 186), i.e. andWe haveProof of Theorem 3 (1/2)where C is nonsingular matrix; I is identity matrix; D is diagonal matrix containing real eigenvalues 1,, p of andLet w = C*z, the Jacobian of the transformation from w to z is |C*C| (Mathai, 1997) Hence,andA.M. Mathai (1997). Jacobians of Matrix Transformations and Functions of Matrix Argument. Singapore: World ScientificC.R. Rao and M.B. Rao (1998). Matrix Algebra and Its Applications to Statistics and Econometrics. Singapore: World Scientific

*Proof of Theorem 3 (2/2)Finally, the divergence is(Q.E.D)

Page *Edge Templates7799Euclidean Distancewherebij is the matrix element of 2aij is the matrix element of 1

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derivation of separability measures based on central complex gaussian and wishart distributions

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