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
 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
Page 2
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
Scrub
Water (River)
Page 3
NASA/JPL C-band data of
Muda Merbok, Malaysia
(PACRIM 2000)
Bhattacharyya Distance
• Definition (Kailath, 1967):
B  log b
The Bhattacharyya coefficient is given by
b
f x  g x  dx
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 h  1  b
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.
Page 4
Scattering Vector and Central Complex Gaussian
Distribution
• Scattering vector z in single-look single-frequency polarimetric
synthetic aperture radar data:
S HH  HH   i( HH )
z  S HV    HV   i( HV ) 
S VV   VV   i(VV ) 
• Scattering vector z can be assumed to follow p-dimensional central
complex Gaussian distribution (Kong et al, 1987):
pz  
1
π Σ
p

exp  z T Σ 1z

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.
Page 5
Bhattacharyya Distance from Central Complex
Gaussian Distribution
• Theorem 1: The Bhattacharyya distance of two central complex
multivariate Gaussian populations with unequal covariance matrices is
B  log
1
2
Σ1  12 Σ2  12 log Σ1  12 log Σ2
while the Bhattacharyya coefficient is b 
Σ1
1
2
12
Σ2
12
Σ1  12 Σ 2
• 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


 
 
B  log 12 σ12  12 σ22  12 log 12  12 log 22
Page 6
Proof of Theorem 1
Now, the Bhattacharyya coefficient is
bπ
p
Σ1
1 2
Σ2

1 2


1 * T 1
1 T 1
exp

z
Σ
z

z Σ2 z dz
1

2
2

Use of the following integration rules:



 exp  ax  bx dx 

2
 b2 

exp  
a
 4a 
 exp  ax  dx 

and
2


a
Hence, the Bhattacharyya coefficient is

p
b
12
12
 p Σ1 Σ2 
1
 Σ 12 Σ 12
2
 1
1
Σ  12 Σ2  12 Σ1  12 Σ2
2 1

Σ1 Σ2
while the Bhattacharyya distance is
B  log 12 Σ1  12 Σ2  12 log Σ1  12 log Σ2
7
(Q.E.D)
Covariance Matrix and Complex Wishart Distribution
• Covariance matrix C in n-look single-frequency polarimetric
synthetic aperture radar data:

 S HH S HH


C   S HV S HH


S
S
 VV HH

S HH S HV

S HV S HV

S VV S HV


S HH S VV


S HV S VV 


S VV S VV 

• Hermitian matrix A = n C can be assumed to follow central complex
Wishart distribution (Lee et al, 1994):
pA  
A
n p
p n  Σ
n
 
exp  tr Σ 1A

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.
Page 8
Bhattacharyya Distance from Central Complex
Wishart Distribution
• Theorem 2: The Bhattacharyya distance of two central complex Wishart
populations with unequal covariance matrices is
B  n log
1
2
Σ1  12 Σ2  n2 log Σ1  n2 log Σ2
while the Bhattacharyya coefficient is b 
Σ1
1
2
n2
Σ2
n2
Σ1  Σ 2
1
2
n
• 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. 519-527.
• Corollary 2: If p = 1, then the Bhattacharyya distance is


 
 
B  n log 12 σ12  12 σ22  n2 log 12  n2 log 22
Page 9
Proof of Theorem 2
Now, the Bhattacharyya coefficient is
b  p-1n Σ1
n 2
Σ2
n 2
A
A
The Jacobian of B to A: B 
1
2
Σ11

1
2
Σ 21
n-p

1
2
 
exp - tr
1
2
 
Σ11  12 Σ21 A dA
Σ11  12 Σ 21
 A
12
1
2
Σ11  12 Σ 21
b
n
12
is
p
(Mathai, 1997, Th. 3.5, p. 183)
Complex multivariate
gamma function
Hence, the Bhattacharyya coefficient is
p-1

n 2 1
Σ1Σ 2
2
Σ11

1
2
Σ21
n
B
n-p
exp - trB  dB
B
Σ1Σ 2

1
2
Σ11

n 2
1
2
Σ 21
n

Σ1
n
Σ1
n
Σ1Σ 2
1
2
Σ11

n 2
1
2
Σ 21
n
Σ2
n
Σ2
n

Σ1
1
2
n 2
Σ2
n 2
Σ1  Σ 2
1
2
n
A.M. Mathai (1997). Jacobians of Matrix Transformations and Functions of Matrix Argument. Singapore: World Scientific
10
(Q.E.D)
Divergence
• Definition (Jeffreys, 1946; Kullback, 1959):
J  I1  I 2
where
I1   f x  log
f x 
dx and
g x 
I 2   g x  log
g x 
dx
f x 
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.
Page 11
Divergence
• Theorem 3: The divergence of two p-dimensional central complex
Gaussian populations with unequal covariance matrices is

 

J  tr Σ11Σ 2  tr Σ 21Σ1  2 p
• Theorem 4: The divergence of two p-dimensional central complex
Wishart populations with unequal covariance matrices is




J  n tr Σ11Σ 2  n tr Σ 21Σ1  2np
• 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
Page 12
Proof of Theorem 4 (1/2)
n  Σ1 A exp - trΣ11A 
n
n p
g A   p1 n  Σ 2 A
exp - trΣ 21A 
We have f A  
p1
n
n p
Both 1 and 2 can be diagonalized simultaneously (Rao and Rao, 1998,
p. 186), i.e.
C Σ1C  I and C Σ 2C  D
where C is nonsingular matrix; I is identity matrix; D is diagonal matrix
containing real eigenvalues 1,…, p of Σ11Σ 2
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,
f W   p1 n  W
g W   p1 n  D
n p
n
exp - trW 
W
n p
 
exp - tr D1W

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
13
Proof of Theorem 4 (2/2)
I1  n log D  p1 n  trW W
W


 p1 n  tr D1W W
W
 p1
 p1 n D
n
exp  trWdW
W  TT
exp  trW dW
W
n
n

1
p
 n log D  np 
I 2  n log D
n p
n p
n D  trD
n
 trW W
n p
 p 2 n  2 j 1 

dW  2  t jj
dT


 j 1

p

exp  trD WdW
exp  trD WdW
VD
1
W
n p
W W
n p
1
1
1 2
W
dV  D
 n log D  np  n1    n p
WD 1 2
p
dW
Finally, the divergence is
n
n
J  n1    n p 

 2np  n tr Σ11Σ 2  n tr Σ 21Σ1  2np
1
p
(Q.E.D)

14



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)
- Number of looks: 4 and 9
C-band, 9-look
L-band, 9-look
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.
Page 15
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
Page 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 = 0.0725
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
Page 17
Experiment (2)
Legend
Bare soil
Beet
Forest
Grass
Lucerne
Peas
Potatoes
Rapeseed
Stem beans
Water
Wheat
NASA/JPL Airborne POLSAR Data
- Scene number: Flevoland-056-1
- Polarisation: Full-pol (HH, HV and VV)
- Radar frequency: L-band
- Image size: 1024 pixels (range)
750 pixels (azimuth)
- Pixel spacing: 6.662m (range)
12.1m (azimuth)
- 4 test regions identified:
A. Potatoes
B. Rapeseed
C. Stem beans
D. Wheat
Page 18
|SHH|2
|SHV|2
|SVV|2
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
0.0234
0.0106
0.0095
0.0165
0.0147
0.0243
0.0163
B
0.0036
0.0048
0.0147
0.0037
0.0046
0.0122
0.0122
0.0048
0.0145
0.0041
0.0056
C
0.0092
0.0055
0.0106
0.0084
0.0075
0.0126
0.0087
0.0081
0.0072
0.0097
0.0085
D
0.0099
0.0050
0.0110
0.0099
0.0104
0.0065
0.0082
0.0035
0.0125
0.0105
0.0011
Page 19
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 loglikelihood 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
Page 20
EADS Innovation Works Singapore, Real-time Embedded Systems, IW-SI-I
Colour palette
Page 21
28 July, 2017
Divergence
• Corollary 3: If p = 1, then the divergence is
σ12 σ22
J  2  2 -2
σ2 σ1
• Corollary 4: If p = 1, then the divergence is
 σ12 σ 22 
J  n  2  2 - 2 
 σ 2 σ1

Page 22
Proof of Theorem 3 (1/2)
We have f z   π  p Σ1
1

exp - z Σ11z

and g z   π  p Σ 2
1

exp - zΣ 21z

Both 1 and 2 can be diagonalized simultaneously (Rao and Rao, 1998,
p. 186), i.e.
C Σ1C  I and C Σ 2C  D
where C is nonsingular matrix; I is identity matrix; D is diagonal matrix
containing real eigenvalues 1,…, p of Σ11Σ 2
Let w = C*z, the Jacobian of the transformation from w to z is |C*C|
(Mathai, 1997)

Hence, f w   π  pexp - ww


p
 1
and g w   π D exp - w D w
-1
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
23

Proof of Theorem 3 (2/2)
1
I1  log D  p

 log D  p 



1
w
w
exp

w
w
d
w


p







 1

w
D
w
exp

w
w dw


1
1

1
p
1
I 2  log D  p
 D



1
 1
 1
w
D
w
exp

w
D
w
d
w


p

D





 1
w
w
exp

w
D w dw


 log D  p  1     p
Finally, the divergence is
J  I1  I 2  1     p 

 
1
1

 2p
1
p

 tr Σ11Σ 2  tr Σ 21Σ1  2 p
24
(Q.E.D)
Edge Templates
99
77
Euclidean Distance
p
p
d   aij  bij
2
j 1 i 1
where
aij is the matrix element of 1
bij is the matrix element of 2
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