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

EADS Innovation Works Singapore. Derivation of Separability Measures Based on Central Complex Gaussian and Wishart Distributions. Ken Yoong LEE and Timo Rolf Bretschneider July 2011. Content Overview. Objectives. Bhattacharyya distance. Based on central complex Gaussian distribution.

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

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  1. EADS Innovation Works Singapore Derivation of Separability Measures Based on Central Complex Gaussian and Wishart Distributions Ken Yoong LEE and Timo Rolf BretschneiderJuly 2011

  2. Content Overview • Objectives • Bhattacharyya distance • Based on central complex Gaussian distribution • Based on central complex Wishart distribution • Divergence • Based on central complex Gaussian distribution • Based on central complex Wishart distribution • Experiments • Simulated POLSAR data • NASA/JPL POLSAR data • Summary

  3. Introduction • Objectives • Derivations of Bhattacharyya distance and divergence based on central complex distributions • Use of Bhattacharyya distance and divergence as a separability measure of target classes in POLSAR data ? Oil palm plantation Rubber trees NASA/JPL C-band data of Muda Merbok, Malaysia (PACRIM 2000) Scrub Water (River)

  4. Bhattacharyya Distance • Definition (Kailath, 1967): The Bhattacharyya coefficient is given by wheref (x) and g (x) are pdfs of two populations • Properties (Matusita, 1966): blies between 0 and 1 b = 1 if f(x) = g(x) bis also called affinity as it indicates the closeness between two populations Hellinger distance T. Kailath (1967). The divergence and Bhattacharyya distance measures in signal selection. IEEE Trans. Comm. Tech, 15(1), pp. 22992311. K. Matusita (1966). A distance and related statistics in multivariate analysis. Multivariate Analysis, edited by Krishnaiah, P.R., New York: Academic Press, pp. 187200.

  5. Scattering Vector and Central Complex Gaussian Distribution • Scattering vector z in single-look single-frequency polarimetric synthetic aperture radar data: • Scattering vector z can be assumed to follow p-dimensional central complex Gaussian distribution (Kong etal, 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. 171194.

  6. Bhattacharyya Distance from Central Complex Gaussian Distribution • Theorem 1: The Bhattacharyya distance of two central complex multivariate Gaussian populations with unequal covariance matrices is while the Bhattacharyya coefficient is • Remark 1: The application of Bhattacharyya distance for contrast analysis can be found in Morio etal (2008) J. Morio, P. Réfrégier, 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. 21852196 • Corollary 1: If p = 1, then the Bhattacharyya distance is

  7. Proof of Theorem 1 Now, the Bhattacharyya coefficient is Use of the following integration rules: and Hence, the Bhattacharyya coefficient is while the Bhattacharyya distance is (Q.E.D)

  8. Covariance Matrix and Complex Wishart Distribution • Covariance matrix C in n-look single-frequency polarimetric synthetic aperture radar data: • Hermitian matrix A = nC can be assumed to follow central complex Wishart distribution (Lee etal, 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. 22992311.

  9. Bhattacharyya Distance from Central Complex Wishart Distribution • Theorem 2: The Bhattacharyya distance of two central complex Wishart populations with unequal covariance matrices is while the Bhattacharyya coefficient is • Remark 2: The Bhattacharyya distance is proportional to the Bartlett distance (Kersten etal, 2005) with a constant of 2/n P.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. • Corollary 2: If p = 1, then the Bhattacharyya distance is

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

  11. Divergence • Definition (Jeffreys, 1946; Kullback, 1959): where and the functionsf(x) and g(x) are pdf of two populations I1 or I2 is also known as Kullback-Leibler divergence • Properties: Jis zero if if f(x) = g(x), which implies no divergence between a distribution and itself H. Jeffreys (1946). An invariant form for the prior probability in estimation problems. Proc. Royal Soc. London (Ser. A), 186(1007), pp. 453461. S. Kullback (1959). Information Theory and Statistics, New York: John Wiley.

  12. Divergence • Theorem 3: The divergence of two p-dimensional central complex Gaussian populations with unequal covariance matrices is • Theorem 4: The divergence of two p-dimensional central complex Wishart populations with unequal covariance matrices is • Remark 3: The divergence is proportional to the symmetrized normalized log-likelihood distance (Anfinsen etal, 2007) with a constant of (2n)-1 S.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

  13. Proof of Theorem 4 (1/2) We have Both 1 and 2 can be diagonalized simultaneously (Rao and Rao, 1998, p. 186), i.e. and 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, C.R. Rao and M.B. Rao (1998). Matrix Algebra and Its Applications to Statistics and Econometrics. Singapore: World Scientific A.M. Mathai (1997). Jacobians of Matrix Transformations and Functions of Matrix Argument. Singapore: World Scientific

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

  15. Experiment (1) C-band Oil palm (1350 pixels) NASA/JPL Airborne POLSAR Data - Scene title: MudaMerbok354-1 Rubber (1395 pixels) - Polarisation: Full-pol (HH, HV and VV) - Radar frequency: C and L-band - Acquired date: 19 September 2000 Scrub-grassland (1672 pixels) - Number of looks: 9 Rice paddy (924 pixels) • Pixel spacing: 3.33m (range) 4.63m (azimuth) |SHH|2 |SHV|2 |SVV|2 Simulated POLSAR Data - Simulation based on Lee etal (1994) C-band, 9-look L-band, 9-look - Number of looks: 4 and 9 A B C D • Image size: 400 pixels (column), 150 pixels (row) 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. 22992311.

  16. C-band, 9-look L-band, 9-look Bhattacharyya distance Bhattacharyya distance Window size = 7 Window size = 9 Window size = 7 Window size = 9 Threshold = 0.0305 Correct detection rate = 1 False detection rate = 0.0105 Threshold =0.039 Correct detection rate = 1 False detection rate = 0 Threshold = 0.22 Correct detection rate = 1 False detection rate = 0 Threshold = 0.2701 Correct detection rate = 1 False detection rate = 0 Divergence Divergence Window size = 7 Window size = 9 Window size = 7 Window size = 9 Threshold = 0.25086 Correct detection rate = 1 False detection rate = 0.0099 Threshold = 0.322 Correct detection rate = 1 False detection rate = 0 Threshold = 1.98 Correct detection rate = 1 False detection rate = 0 Threshold = 2.43 Correct detection rate = 1 False detection rate = 0 Euclidean distance Euclidean distance Window size = 7 Window size = 9 Window size = 7 Window size = 9 Threshold = 0.005 Correct detection rate = 1 False detection rate = 0.2316 Threshold = 0.00578 Correct detection rate = 1 False detection rate = 0.0692 Threshold = 0.00049 Correct detection rate = 1 False detection rate = 0.2538 Threshold = 0.00057 Correct detection rate = 1 False detection rate = 0.1764

  17. C-band, 9-look L-band, 9-look Bhattacharyya distance Bhattacharyya distance Window size = 7 Window size = 9 Window size = 7 Window size = 9 Threshold = 0.0725 Correct detection rate = 0.8324 False detection rate = 0 Threshold =0.039 Correct detection rate = 1 False detection rate = 0 Threshold = 0.22 Correct detection rate = 1 False detection rate = 0 Threshold = 0.2701 Correct detection rate = 1 False detection rate = 0 Divergence Divergence Window size = 7 Window size = 9 Window size = 7 Window size = 9 Threshold = 0.61 Correct detection rate = 0.8333 False detection rate = 0 Threshold = 0.322 Correct detection rate = 1 False detection rate = 0 Threshold = 1.98 Correct detection rate = 1 False detection rate = 0 Threshold = 2.43 Correct detection rate = 1 False detection rate = 0 Euclidean distance Euclidean distance Window size = 7 Window size = 9 Window size = 7 Window size = 9 Threshold = 0.0492 Correct detection rate = 0.6262 False detection rate = 0 Threshold = 0.0321 Correct detection rate = 0.7071 False detection rate = 0 Threshold = 0.0068 Correct detection rate = 0.7485 False detection rate = 0 Threshold = 0.0046 Correct detection rate = 0.8524 False detection rate = 0

  18. Experiment (2) Legend Bare soil Beet NASA/JPL Airborne POLSAR Data Forest Grass Lucerne - Scene number: Flevoland-056-1 Peas Potatoes Rapeseed - Polarisation: Full-pol (HH, HV and VV) Stem beans Water Wheat - Radar frequency: L-band |SHH|2 |SHV|2 |SVV|2 • Image size: 1024 pixels (range) 750 pixels (azimuth) • Pixel spacing: 6.662m (range) 12.1m (azimuth) - 4 test regions identified: • Potatoes • Rapeseed • Stem beans • Wheat

  19. Bhattacharyya distance Divergence Euclidean distance

  20. Summary • The 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 divergence • The Bhattacharyya distance is proportional to the Bartlett distance The divergence is proportional to the symmetrized normalized log-likelihood distance • Both the Bhattacharyya distance and the divergence perform consistently in measuring the separability of target classes • The latter is more computationally expensive than the former

  21. Colour palette EADS Innovation Works Singapore, Real-time Embedded Systems, IW-SI-I

  22. Divergence • Corollary 3: If p = 1, then the divergence is • Corollary 4: If p = 1, then the divergence is

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

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

  25. Edge Templates 99 77 Euclidean Distance where aij is the matrix element of 1 bij is the matrix element of 2

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