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
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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
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4. Scattering mechanisms
Figure from Pottier et al. (2003)
Question: Can the different polarization components be represented
by the same texture variable?
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5. Multitexture model
Scattering vector:
Multilook sample covariance matrix:
where
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6. Conditional pdf of sample covariance matrix:
Pdf of sample covariance matrix:
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7. Reciprocity
Reflection symmetry
where
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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
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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
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10. Mellin kind statistics
Sample covariance matrix in the multitexture model
Mellin type characteristic function:
Statistical independence implies:
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11. W is Wishart distributed:
T is diagonal:
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12. Log-moments:
Log-cumulants:
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13. Sample matrix log-moments:
Relation between log-cumulants & log-moments:
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14. Asymptotic distribution of sample
matrix log-cumulants:
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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
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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
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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
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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
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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
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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
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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
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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
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