Akbari&anfinsen&doulgeris&eltoft igarss13
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Transcript of Akbari&anfinsen&doulgeris&eltoft igarss13
FACULTY OF SCIENCE AND TECHNOLOGY
DEPARTMENT OF PHYSICS AND TECHNOLOGY
The Hotelling-Lawley Trace Statistic forChange Detection in Polarimetric SAR DataUnder the Complex Wishart Distribution
Vahid AkbariStian N. AnfinsenAnthony P. Doulgeris
IGARSS 2013
Overview
• Propose a new test statistic for change detection
• Polarimetric SAR
• MLC covariance Matrix Data
• Assume scaled complex Wishart distribution
• Hotelling-Lawley trace statistic
• Approximated by a Fisher-Snedecor distribution
• Demonstrated as a CFAR detector
• Compared to Conradsen’s Likelihood Ratio Test statistic
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Theory
• Assume scaled complex Wishart distribution
C ∼ WCd (L,Σ)
• Pixel-wise comparison between two images of C matrices
• “No-change” hyptothesis test
H0 : Σ1 = Σ2
• Hotelling-Lawley trace statistic
τHL = tr(C−11 C2)
• Matrix-variate equivalent to ratio imaging
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Theory
• Two-sided test for both increasing and decreasing values
• Test both tr(C−11 C2) and tr(C−12 C1) at p = α/2
• Null hypothesis sampling distribution approximated by aFisher-Snedecor distribution FS(µ, ξ, ζ)
• FS parameters found by matching first three moments
m(FS)ν (µ, ξ, ζ) = m
(HL)ν (L1, L2, d)
• ENL estimated for each image
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Theory
• Compare to Conradsen’s Likelihood Ratio Test statistic• Also known as the Bartlett distance
τLRT = τB = −2ρ logQ
Q =|C1|L1|C2|L2
∣∣∣L1C1+L2C2
L1+L2
∣∣∣L1+L2
ρ = 1− 2d2 − 1
6d
(1
L1
+1
L2− 1
L1 + L2
)
• Sampling distribution is a sum of three χ-squareddistributions
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Results - Simulated Pauli Images
12-Looks, Σs from real data, 3 changesBrightness increase, decrease and polarimetric change
(a) test image 1 (b) test image 2 (c) log(⌧HL) (d) log(⌧B)
(e) pdf of ⌧HL (f) pdf of ⌧B (g) HL test detection result (h) Bartlett test detection result
Fig. 1. Comparison of HL test and Bartlett test with data simulated from the complex Wishart model.
and the third-order moment is given as
m(HL)3 = E{⌧3
HL}
=L3
a
Q5a � 5Q3
a + 4Qa⇥
d3
✓(Q2
a � 1) +3Qa
Lb+
4
L2b
◆
+ d2
✓3Qa +
3(Q2a + 2)
Lb+
6Qa
L2b
◆
+ d
✓4 +
6Qa
Lb+
2Q2a
L2b
◆�.
(10)
We seek expressions for the parameters of the FS distri-bution in terms of the distribution parameters of the scaledWishart matrices A and B, i.e., L = La = Lb and d. Thesolutions for µ, ⇠ and ⇣ are defined by the equation system
m(FS)⌫ (⇠, ⇣, µ) = m(HL)
⌫ (L, d) , ⌫ = 1, 2, 3 . (11)
An analytic solution for µ is easily found by matching thefirst-order moments. To match the second and third-ordermoments and retrieve the shape parameters, we use minimumdistance optimization [17] and minimize
✏2 =
3X
⌫=2
(m(HL)⌫ � m(FS)
⌫ )2 . (12)
Estimation of the equivalent number of looks is a critical pointin this procedure. To do this accurately and automatically, weuse the unsupervised method by Anfinsen et al. in [18].
3. RESULTS
We have performed a number of test on simulated and realdata, that will all be presented in a future journal paper. Dueto limited space, we here only show an experiment where wehave generated two test images with eight classes that followa scaled complex Wishart distribution with L = 12 and theirindividual ⌃ computed from samples of real data. The testimages represent times t0 and t1. They are shown in Fig. 1(a) and (b) as Pauli images. Here we see that the changesoccuring are: (i) a transition from class 1 to class 8 along avertical strip centered in the upper half of the images; (ii) atransition from class 5 to class 7 in a vertical strip centeredin the lower half of the images; (iii) the transition from class5 to class 1 in the central square. Test statistic images forthe proposed HL test (⌧HL) and the Bartlett test (⌧B) [3] arerespectively shown in (c) and (d) on logarithmic scale to en-hance the contrast. The histograms of ⌧HL and ⌧B follow in(e) and (f), before the change detection results obtained withthresholds set to obtain a CFAR of 1% are shown in (g) and(h).
A clear difference in the behaviour of the test statistics
����
5 / 9
Results - Test statistics
(a) test image 1 (b) test image 2 (c) log(⌧HL) (d) log(⌧B)
(e) pdf of ⌧HL (f) pdf of ⌧B (g) HL test detection result (h) Bartlett test detection result
Fig. 1. Comparison of HL test and Bartlett test with data simulated from the complex Wishart model.
and the third-order moment is given as
m(HL)3 = E{⌧3
HL}
=L3
a
Q5a � 5Q3
a + 4Qa⇥
d3
✓(Q2
a � 1) +3Qa
Lb+
4
L2b
◆
+ d2
✓3Qa +
3(Q2a + 2)
Lb+
6Qa
L2b
◆
+ d
✓4 +
6Qa
Lb+
2Q2a
L2b
◆�.
(10)
We seek expressions for the parameters of the FS distri-bution in terms of the distribution parameters of the scaledWishart matrices A and B, i.e., L = La = Lb and d. Thesolutions for µ, ⇠ and ⇣ are defined by the equation system
m(FS)⌫ (⇠, ⇣, µ) = m(HL)
⌫ (L, d) , ⌫ = 1, 2, 3 . (11)
An analytic solution for µ is easily found by matching thefirst-order moments. To match the second and third-ordermoments and retrieve the shape parameters, we use minimumdistance optimization [17] and minimize
✏2 =
3X
⌫=2
(m(HL)⌫ � m(FS)
⌫ )2 . (12)
Estimation of the equivalent number of looks is a critical pointin this procedure. To do this accurately and automatically, weuse the unsupervised method by Anfinsen et al. in [18].
3. RESULTS
We have performed a number of test on simulated and realdata, that will all be presented in a future journal paper. Dueto limited space, we here only show an experiment where wehave generated two test images with eight classes that followa scaled complex Wishart distribution with L = 12 and theirindividual ⌃ computed from samples of real data. The testimages represent times t0 and t1. They are shown in Fig. 1(a) and (b) as Pauli images. Here we see that the changesoccuring are: (i) a transition from class 1 to class 8 along avertical strip centered in the upper half of the images; (ii) atransition from class 5 to class 7 in a vertical strip centeredin the lower half of the images; (iii) the transition from class5 to class 1 in the central square. Test statistic images forthe proposed HL test (⌧HL) and the Bartlett test (⌧B) [3] arerespectively shown in (c) and (d) on logarithmic scale to en-hance the contrast. The histograms of ⌧HL and ⌧B follow in(e) and (f), before the change detection results obtained withthresholds set to obtain a CFAR of 1% are shown in (g) and(h).
A clear difference in the behaviour of the test statistics
����
6 / 9
Results - Histograms(a) test image 1 (b) test image 2 (c) log(⌧HL) (d) log(⌧B)
(e) pdf of ⌧HL (f) pdf of ⌧B (g) HL test detection result (h) Bartlett test detection result
Fig. 1. Comparison of HL test and Bartlett test with data simulated from the complex Wishart model.
and the third-order moment is given as
m(HL)3 = E{⌧3
HL}
=L3
a
Q5a � 5Q3
a + 4Qa⇥
d3
✓(Q2
a � 1) +3Qa
Lb+
4
L2b
◆
+ d2
✓3Qa +
3(Q2a + 2)
Lb+
6Qa
L2b
◆
+ d
✓4 +
6Qa
Lb+
2Q2a
L2b
◆�.
(10)
We seek expressions for the parameters of the FS distri-bution in terms of the distribution parameters of the scaledWishart matrices A and B, i.e., L = La = Lb and d. Thesolutions for µ, ⇠ and ⇣ are defined by the equation system
m(FS)⌫ (⇠, ⇣, µ) = m(HL)
⌫ (L, d) , ⌫ = 1, 2, 3 . (11)
An analytic solution for µ is easily found by matching thefirst-order moments. To match the second and third-ordermoments and retrieve the shape parameters, we use minimumdistance optimization [17] and minimize
✏2 =
3X
⌫=2
(m(HL)⌫ � m(FS)
⌫ )2 . (12)
Estimation of the equivalent number of looks is a critical pointin this procedure. To do this accurately and automatically, weuse the unsupervised method by Anfinsen et al. in [18].
3. RESULTS
We have performed a number of test on simulated and realdata, that will all be presented in a future journal paper. Dueto limited space, we here only show an experiment where wehave generated two test images with eight classes that followa scaled complex Wishart distribution with L = 12 and theirindividual ⌃ computed from samples of real data. The testimages represent times t0 and t1. They are shown in Fig. 1(a) and (b) as Pauli images. Here we see that the changesoccuring are: (i) a transition from class 1 to class 8 along avertical strip centered in the upper half of the images; (ii) atransition from class 5 to class 7 in a vertical strip centeredin the lower half of the images; (iii) the transition from class5 to class 1 in the central square. Test statistic images forthe proposed HL test (⌧HL) and the Bartlett test (⌧B) [3] arerespectively shown in (c) and (d) on logarithmic scale to en-hance the contrast. The histograms of ⌧HL and ⌧B follow in(e) and (f), before the change detection results obtained withthresholds set to obtain a CFAR of 1% are shown in (g) and(h).
A clear difference in the behaviour of the test statistics
����
7 / 9
Results - CFAR 1% change maps(a) test image 1 (b) test image 2 (c) log(⌧HL) (d) log(⌧B)
(e) pdf of ⌧HL (f) pdf of ⌧B (g) HL test detection result (h) Bartlett test detection result
Fig. 1. Comparison of HL test and Bartlett test with data simulated from the complex Wishart model.
and the third-order moment is given as
m(HL)3 = E{⌧3
HL}
=L3
a
Q5a � 5Q3
a + 4Qa⇥
d3
✓(Q2
a � 1) +3Qa
Lb+
4
L2b
◆
+ d2
✓3Qa +
3(Q2a + 2)
Lb+
6Qa
L2b
◆
+ d
✓4 +
6Qa
Lb+
2Q2a
L2b
◆�.
(10)
We seek expressions for the parameters of the FS distri-bution in terms of the distribution parameters of the scaledWishart matrices A and B, i.e., L = La = Lb and d. Thesolutions for µ, ⇠ and ⇣ are defined by the equation system
m(FS)⌫ (⇠, ⇣, µ) = m(HL)
⌫ (L, d) , ⌫ = 1, 2, 3 . (11)
An analytic solution for µ is easily found by matching thefirst-order moments. To match the second and third-ordermoments and retrieve the shape parameters, we use minimumdistance optimization [17] and minimize
✏2 =
3X
⌫=2
(m(HL)⌫ � m(FS)
⌫ )2 . (12)
Estimation of the equivalent number of looks is a critical pointin this procedure. To do this accurately and automatically, weuse the unsupervised method by Anfinsen et al. in [18].
3. RESULTS
We have performed a number of test on simulated and realdata, that will all be presented in a future journal paper. Dueto limited space, we here only show an experiment where wehave generated two test images with eight classes that followa scaled complex Wishart distribution with L = 12 and theirindividual ⌃ computed from samples of real data. The testimages represent times t0 and t1. They are shown in Fig. 1(a) and (b) as Pauli images. Here we see that the changesoccuring are: (i) a transition from class 1 to class 8 along avertical strip centered in the upper half of the images; (ii) atransition from class 5 to class 7 in a vertical strip centeredin the lower half of the images; (iii) the transition from class5 to class 1 in the central square. Test statistic images forthe proposed HL test (⌧HL) and the Bartlett test (⌧B) [3] arerespectively shown in (c) and (d) on logarithmic scale to en-hance the contrast. The histograms of ⌧HL and ⌧B follow in(e) and (f), before the change detection results obtained withthresholds set to obtain a CFAR of 1% are shown in (g) and(h).
A clear difference in the behaviour of the test statistics
����
8 / 9