The document presents a multitexture model for analyzing multilook polarimetric radar data. It proposes modeling the scattering vector using multiple texture variables to represent different polarization components rather than a single texture variable. It derives the probability density function of the sample covariance matrix under this model. Experimental results on Amazon rainforest data show the multitexture model fits the data better than single or dual texture models.

1. A Multitexture Model for Multilook
Polarimetric Radar Data
by
Torbjørn Eltoft, Stian Anfinsen, and
Anthony Doulgeris

2. Outline
• Background and motivation
• Multitexture model
• Mellin type statistics and the log-cumulant diagram
• Experimental studies
• Conclusions
2 7/25/11

3. The scalar product model
The product model has been proposed as a generic model for
generation of non-Gaussian distributions for polarimetric radar
signals.
The scalar product model:
is a multivariate Gaussian speckle variable
is a scalar texture variable
3 7/25/11

4. Scattering mechanisms
Figure from Pottier et al. (2003)
Question: Can the different polarization components be represented
by the same texture variable?
4 7/25/11

5. Multitexture model
Scattering vector:
Multilook sample covariance matrix:
where
5 7/25/11

6. Conditional pdf of sample covariance matrix:
Pdf of sample covariance matrix:
6 7/25/11

7. Reciprocity
Reflection symmetry
where
7 7/25/11

8. Multitexture pdf of C
Assume the texture components thh = tvv
Let qi,j denote entry (i,j) of
Let ci,j denote entry (i,j) of
8 7/25/11

9. 2-D log-cumulant diagram
5
4
3
2 X
κ2
1
K G0
0
W
95%, N=1000
−1
95%, N=100
−2
−6 −4 −2 0 2
κ3
9 7/25/11

10. Mellin kind statistics
Sample covariance matrix in the multitexture model
Mellin type characteristic function:
Statistical independence implies:
10 7/25/11

11. W is Wishart distributed:
T is diagonal:
11 7/25/11

12. Log-moments:
Log-cumulants:
12 7/25/11

13. Sample matrix log-moments:
Relation between log-cumulants & log-moments:
13 7/25/11

14. Asymptotic distribution of sample
matrix log-cumulants:
14 7/25/11

15. Multi -, dual- or scalar texture models
0.6
0.5
0.5 HH VV
HH
VV
0.4 VH
0.4 VH HV
HV
0.3
0.3
κ2
κ2
0.2 0.2
HH test
0.1 HV test 0.1 co−pol test
VH test K G0
VV test x−pol test
0 K G0
0 W
W
−0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 −0.2 −0.15 −0.1 −0.05 0 0.05
κ3 κ3
0.5
HH VV
0.4 VH
HV
0.3
κ2
0.2
0.1
scalar test K G0
0
W
−0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02
κ3
15 7/25/11

16. Study case 1: Amazon Rainforest
550 4
500
1
450
400
350 Spaceborne:
Amazon rainforest;
300 ALOS L-band,
2 No. of looks = 8
250
200
150
100
3
50
16 7/25/11
50 100 150 200 250 300 350

17. Amazon: Box 1 (water)
20
10 µ = 0.000129
0.5 0
HH VV 0 0.5 1 1.5 2 2.5 3
20
0.4 VH
HV 10 µ = 1.48e−05
0.3 0
0 0.5 1 1.5 2 2.5 3
κ2
20
0.2 10
µ = 2.05e−05
co−pol test 0
0.1 0 0.5 1 1.5 2 2.5 3
x−pol test 0 20
scalar test K G
10
0 µ = 7.41e−05
W
0
0 0.5 1 1.5 2 2.5 3
−0.2 −0.15 −0.1 −0.05 0 0.05
κ3
Amazon: Box 2 (forest)
20
10 µ= 0.115
0.7
0
0.6 VV 0 0.5 1 1.5 2 2.5 3
HH 20
VH
0.5 HV 10 µ = 0.0117
0.4 0
0 0.5 1 1.5 2 2.5 3
κ2
20
0.3
10 µ = 0.0116
0.2
co−pol test 0
0 0.5 1 1.5 2 2.5 3
0.1 x−pol test 0 20
scalar test K G
0 10 µ = 0.0946
W
0
−0.4 −0.3 −0.2 −0.1 0 0 0.5 1 1.5 2 2.5 3
κ3
17 7/25/11

18. Amazon: Box 3 (clear-cut area)
20
µ= 0.078
1 10
0
VH 0 0.5 1 1.5 2 2.5 3
0.8 HV 20
VV
HH 10 µ = 0.000467
0.6
0
0 0.5 1 1.5 2 2.5 3
κ2
20
0.4 µ = 0.000493
10
co−pol test 0
0.2 0 0.5 1 1.5 2 2.5 3
x−pol test 0 20
scalar test K G
10 µ= 0.075
0
W
0
0 0.5 1 1.5 2 2.5 3
−0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2
κ3
Amazon: Box 4 (low-intensity water)
20
10 µ = 5.03e−05
0.6 0
HH 0 0.5 1 1.5 2 2.5 3
20
0.5 VV
10 µ = 6.96e−06
0.4
0
0 0.5 1 1.5 2 2.5 3
HV
κ2
0.3 20
VH
10 µ = 7.42e−06
0.2
co−pol test 0
0 0.5 1 1.5 2 2.5 3
0.1 x−pol test 0 20
scalar test K G
0 10 µ = 2.84e−05
W
0
−0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0 0.5 1 1.5 2 2.5 3
κ3
18 7/25/11

19. Study case 2: Nezer Forest
400
350
300
250
3
200 Airborne:
Nezer forest;
150 AIRSAR P-band,
2 No. of looks = 16
100
4 1
50
50 100 150 200 250 300 350 400 450
19 7/25/11

20. Nezer: Box 1 (bare soil)
20
0.1 10 µ = 0.00013
HH 0
0.08 VV 0 0.5 1 1.5 2 2.5 3
20
VH
HV
0.06 10 µ = 1.48e−05
0
κ2
0 0.5 1 1.5 2 2.5 3
0.04 20
co−pol test 10 µ = 2.02e−05
0.02
0 x−pol test
K G scalar test
0
0 0.5 1 1.5 2 2.5 3
0 20
W
10 µ = 7.36e−05
−0.02
−0.01 −0.005 0 0.005 0.01 0
0 0.5 1 1.5 2 2.5 3
κ3
Nezer: Box 2 (old forest)
40
0.16 µ= 0.185
20
0.14 HH
0
0 0.5 1 1.5 2 2.5 3
0.12 VHVV
HV 40
0.1 20 µ = 0.0419
0.08 0
0 0.5 1 1.5 2 2.5 3
κ2
40
0.06
20 µ = 0.0419
0.04 co−pol test
x−pol test 0
0.02
scalar test K G0 0 0.5 1 1.5 2 2.5 3
40
0
W 20 µ = 0.0832
−0.02
0
−0.04 −0.03 −0.02 −0.01 0 0.01 0 0.5 1 1.5 2 2.5 3
κ3
20 7/25/11

21. Nezer: Box 3 (young forest)
40
0.12
20 µ = 0.0572
0.1 0
VH
HV 0 0.5 1 1.5 2 2.5 3
40
0.08
HH
20 µ = 0.00785
0.06 VV
0
0 0.5 1 1.5 2 2.5 3
κ2
40
0.04
20 µ = 0.00785
co−pol test
0.02 x−pol test
scalar test K G0 0
0 0.5 1 1.5 2 2.5 3
40
0
W
20 µ = 0.0384
−0.02
−0.015 −0.01 −0.005 0 0.005 0.01 0
0 0.5 1 1.5 2 2.5 3
κ3
Nezer: Box 4 (class mixture)
40
1.8 20 µ= 0.101
1.6 0
VH
HV 0 0.5 1 1.5 2 2.5 3
1.4 50
µ= 0.029
1.2
1 0
0 0.5 1 1.5 2 2.5 3
HH
κ2
50
0.8 µ= 0.029
0.6 VV
co−pol test 0
0.4 0 0.5 1 1.5 2 2.5 3
x−pol test 20
0.2
scalar test K G0 µ = 0.0624
10
0
W 0
0 0.5 1 1.5 2 2.5 3
−4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0
κ3
21 7/25/11

22. Conclusions
• Introduced a multitexture statistical model for the multi-look
polarimetric sample covariance matrix
• Shown that in the reciprocal, reflection symmetrical case the pdf
can be explicitly formulated as a dual texture model
• Used Mellin kind statistics to develop experimental procedures to
test the nature of the texture variables
• Preliminary experimental studies conclude:
§ The scalar texture model is often valid
§ When multitexture is needed, the dual-model is often sufficient
§ Mixtures often appear as multitexture
22 7/25/11