1. Ken Yoong LEE and Timo Rolf Bretschneider July 2011
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

2. Content Overview Objectives Bhattacharyya distance Divergence
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
where f (x) and g (x) are pdfs of two populations
Properties (Matusita, 1966):
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 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 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. 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 et al (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 (Q.E.D) 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 = 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. 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 et al, 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
Corollary 2: If p = 1, then the Bhattacharyya distance is

10. Complex multivariate gamma function
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): Properties:
where
and
the functions f(x) and g(x) are pdf of two populations
I1 or I2 is also known as Kullback-Leibler divergence
Properties:
J is 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 et al, 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. Scrub-grassland (1672 pixels)
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 et al (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. Bhattacharyya distance Bhattacharyya distance
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 =
Correct detection rate = 1
False detection rate =
Threshold =0.039
Correct detection rate = 1
False detection rate = 0
Threshold = 0.22
Correct detection rate = 1
False detection rate = 0
Threshold =
Correct detection rate = 1
False detection rate = 0
Divergence
Divergence
Window size = 7
Window size = 9
Window size = 7
Window size = 9
Threshold =
Correct detection rate = 1
False detection rate =
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 =
Threshold =
Correct detection rate = 1
False detection rate =
Threshold =
Correct detection rate = 1
False detection rate =
Threshold =
Correct detection rate = 1
False detection rate =

17. Bhattacharyya distance Bhattacharyya distance
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 =
Correct detection rate =
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 =
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 =
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 =
Correct detection rate =
False detection rate = 0
Threshold =
Correct detection rate =
False detection rate = 0
Threshold =
Correct detection rate =
False detection rate = 0
Threshold =
Correct detection rate =
False detection rate = 0

18. Experiment (2) NASA/JPL Airborne POLSAR Data
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
Bare soil
Beet
Forest
Grass
Lucerne
Peas
Potatoes
Rape seed
Stem beans
Water
Wheat A
3.3128
0.3427
0.0729
1.6564
1.2287
0.3328
0.0367
0.9576
0.1945
3.9961
0.5486 B
0.9504
0.4672
1.4361
0.1634
0.3037
0.4835
1.0943
0.1134
1.1076
1.5209
0.2938 C
2.3620
0.1330
0.3780
0.7952
0.3979
0.2733
0.2308
0.5466
0.1077
3.0136
0.2739 D
1.8596
0.1281
0.6487
0.5593
0.4071
0.0800
0.4151
0.1769
0.4402
2.4981
0.0023
Divergence
Bare soil
Beet
Forest
Grass
Lucerne
Peas
Potatoes
Rape seed
Stem beans
Water
Wheat A
152.49
3.1396
0.6014
27.334
17.195
3.1701
0.3010
12.236
1.7010
321.74
5.9335 B
12.656
4.7709
24.141
1.4397
2.8383
4.8502
15.480
0.9934
19.374
28.349
2.7727 C
77.822
1.1663
3.718
9.6602
3.7275
2.5210
2.0898
6.2563
0.9131
171.68
2.5318 D
37.129
1.0853
7.460
6.1186
4.6332
0.6897
4.2561
1.5880
4.6488
78.349
0.0186
Euclidean distance
Bare soil
Beet
Forest
Grass
Lucerne
Peas
Potatoes
Rape seed
Stem beans
Water
Wheat A
0.0236
0.0165
0.0114
0.0234
0.0106
0.0095
0.0147
0.0243
0.0163 B
0.0036
0.0048
0.0037
0.0046
0.0122
0.0145
0.0041
0.0056 C
0.0092
0.0055
0.0084
0.0075
0.0126
0.0087
0.0081
0.0072
0.0097
0.0085 D
0.0099
0.0050
0.0110
0.0104
0.0065
0.0082
0.0035
0.0125
0.0105
0.0011

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. EADS Innovation Works Singapore, Real-time Embedded Systems, IW-SI-I
26 April, 2017
Colour palette 21

22. Divergence Corollary 3: 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 Euclidean Distance 99 77 where
aij is the matrix element of 1
bij is the matrix element of 2