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FACULTY OF SCIENCE AND TECHNOLOGY
DEPARTMENT OF PHYSICS AND TECHNOLOGY
The Hotelling-Lawley Trace Statistic for
Change Detection in Polarimetric SAR Data
Under the Complex Wishart Distribution
Vahid Akbari
Stian N. Anfinsen
Anthony 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
1 / 9
Theory
• Assume scaled complex Wishart distribution
C ∼ WC
d (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−1
1 C2)
• Matrix-variate equivalent to ratio imaging
2 / 9
Theory
• Two-sided test for both increasing and decreasing values
• Test both tr(C−1
1 C2) and tr(C−1
2 C1) at p = α/2
• Null hypothesis sampling distribution approximated by a
Fisher-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
3 / 9
Theory
• Compare to Conradsen’s Likelihood Ratio Test statistic
• Also known as the Bartlett distance
τLRT = τB = −2ρ log Q
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-squared
distributions
4 / 9
Results - Simulated Pauli Images
12-Looks, Σs from real data, 3 changes
Brightness increase, decrease and polarimetric change
(a) test image 1 (b) test image 2
5 / 9
Results - Test statistics
(c) log(⌧HL) (d) log(⌧B)
6 / 9
Results - Histograms
(a) test image 1 (b) test image 2
(e) pdf of ⌧HL (f) pdf of ⌧B (g) HL
Fig. 1. Comparison of HL test and Bartlett test with data sim7 / 9
Results - CFAR 1% change maps
(c) log(⌧HL) (d) log(⌧B)
(g) HL test detection result (h) Bartlett test detection result
with data simulated from the complex Wishart model.8 / 9
Conclusions
• Clearly different behaviour
• Detection accuracy: HL 97.9%, Bartlett 90.6%
• Error rate: HL 1.02%, Bartlett 1.40%
• Measured 1% FAR: HL 0.9%, Bartlett 1%
• Submitted to TGRS, in review
- H0 histogram fitting
- ROC curves show HL is superior
- Results for real images
9 / 9

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Akbari&anfinsen&doulgeris&eltoft igarss13

  • 1. FACULTY OF SCIENCE AND TECHNOLOGY DEPARTMENT OF PHYSICS AND TECHNOLOGY The Hotelling-Lawley Trace Statistic for Change Detection in Polarimetric SAR Data Under the Complex Wishart Distribution Vahid Akbari Stian N. Anfinsen Anthony P. Doulgeris IGARSS 2013
  • 2. 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 1 / 9
  • 3. Theory • Assume scaled complex Wishart distribution C ∼ WC d (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−1 1 C2) • Matrix-variate equivalent to ratio imaging 2 / 9
  • 4. Theory • Two-sided test for both increasing and decreasing values • Test both tr(C−1 1 C2) and tr(C−1 2 C1) at p = α/2 • Null hypothesis sampling distribution approximated by a Fisher-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 3 / 9
  • 5. Theory • Compare to Conradsen’s Likelihood Ratio Test statistic • Also known as the Bartlett distance τLRT = τB = −2ρ log Q 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-squared distributions 4 / 9
  • 6. Results - Simulated Pauli Images 12-Looks, Σs from real data, 3 changes Brightness increase, decrease and polarimetric change (a) test image 1 (b) test image 2 5 / 9
  • 7. Results - Test statistics (c) log(⌧HL) (d) log(⌧B) 6 / 9
  • 8. Results - Histograms (a) test image 1 (b) test image 2 (e) pdf of ⌧HL (f) pdf of ⌧B (g) HL Fig. 1. Comparison of HL test and Bartlett test with data sim7 / 9
  • 9. Results - CFAR 1% change maps (c) log(⌧HL) (d) log(⌧B) (g) HL test detection result (h) Bartlett test detection result with data simulated from the complex Wishart model.8 / 9
  • 10. Conclusions • Clearly different behaviour • Detection accuracy: HL 97.9%, Bartlett 90.6% • Error rate: HL 1.02%, Bartlett 1.40% • Measured 1% FAR: HL 0.9%, Bartlett 1% • Submitted to TGRS, in review - H0 histogram fitting - ROC curves show HL is superior - Results for real images 9 / 9