This document discusses using wavelet transforms to analyze two-point statistics of polarimetric synthetic aperture radar (PolSAR) data. It introduces wavelet variance and kurtosis as metrics that can be applied to PolSAR data transformed using a wavelet frame. It then provides an example of applying this analysis to ALOS PALSAR data over Hawaii's Papau Seamount to characterize sea surface features.
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More info at http://summerschool.ssa.org.ua
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More info at http://summerschool.ssa.org.ua
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Talk given at "Quantum Information, Computation and Logic" workshop at Perimeter Institute in 2005.
Based on the preprint http://arxiv.org/abs/quant-ph/0509193
Talk was recorded and is viewable online at http://pirsa.org/05070102/
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The derivative of a composition of functions is the product of the derivatives of those functions. This rule is important because compositions are so powerful.
Nondeterministic testing of Sequential Quantum Logic Propositions on a Quant...Matthew Leifer
Talk given at "Quantum Information, Computation and Logic" workshop at Perimeter Institute in 2005.
Based on the preprint http://arxiv.org/abs/quant-ph/0509193
Talk was recorded and is viewable online at http://pirsa.org/05070102/
I now regard these ideas as flawed, but they may be revisited in future work.
Second Order Perturbations During Inflation Beyond Slow-rollIan Huston
This is a talk I gave at the University of Sussex in June 2011. It outlines the newly released numerical code Pyflation and the results published in arXiv:1103.0912.
Presentation at the 15th World Conference on Earthquake Engineering (2012) of paper no. 620: Craifaleanu, I. G., Borcia, I. S., "Damage Index vs. Instrumental Intensity: A Comparison of Two Different Approaches in Seismic Damage Assessment"
Research Inventy : International Journal of Engineering and Scienceresearchinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
The International Journal of Engineering and Science (IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
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TWO-POINT STATISTIC OF POLARIMETRIC SAR DATA TWO-POINT STATISTIC OF POLARIMETRIC SAR DATA
1. Two-point statistic of polarimetric SAR data provided
by a wavelet frame
The importance of being scaling…(suggested by Oscar Wilde)
G. F. De Grandi1, R.M. Lucas2, A. Bouvet1
European Commission Joint Research Centre Institute of Geography and Earth Sciences
21027, Ispra (VA), Italy Aberystwyth University, Aberystwyth, UK,
e-mail: frank.de-grandi@jrc.ec.europa.eu SY23 3DB.
e-mail: rml@aber.ac.uk
3. Problem Statement
T Two-point statistic
Shh 2Shv Svv Wavelet frame provided by wavelet
transform coefficients
Shv *
Phv ( Shh y )2Shv SvvShv ( x , y )
x,
Wavelet
Variance
w2 ( x) f ( scale )
Structure Function
Spatial random field
2
(SRF) f( ) f ( x) f (x )
4. What happens when the polarization basis is changed?
T ' sin 2
Shh 2Shv Svv T
1 cos 2
Shh 2Shv Svv
xpol ' '*
P ( x, y, ) 2 2 ( x, y, )
copol ' '*
P ( x, y, ) 1 1 ( x, y, )
2 Wavelet variance of
w ( x) f( , ) the crospolar and
copolar power in the
rotated polarization
basis
5. Wavelet variance of PolSAR power in a rotated basis: a
model for a WS stationary process
Wavelet Power
Power
transform spectrum out
spectrum in
Shh 2Shv Svv
G Hj Gout
in
Hj Fourier transform of the wavelet dilated at scale j
1 2 ACF of wavelet coefficients computed
Rout Gin Hj in the frequency domain from the
power spectrum G of the input process
Wavelet variance at cross-polarized
w i2 R out 0 state with orientation ψ
6. Power spectrum of the crosspolar power in the rotated basis
S HH , 2S HV , S VV S c1 Sc1
HV
c22 SVV 2 S HH
cos
Shh 2Shv Svv 2
Rotation to a linear basis with
orientation ψ
c2 in the new basis ψ2
sin
Cross-polarized component
2
2 * *
Pab ( ) c Si S
i i ci c j Re Si S j
i ij
Si S HH , S HV , SVV
Pab ( ) ai X i
i
7. HV2
Copol-xpol
Power spectrum of the input process
correlation2 HH , VV2
Power-
Correlation
2 * *
Pab ( ) c Si S
i i ci c j Re S i S j
i ij
Pab ( ) ai X i
i
*
G ( ) a X a
Power spectrum in the rotated basis is a linear combination jof the
in i i j X
i j
power spectra (auto-correlation) and cross-spectra (cross
correlation) between dyads of the vector in the H,V basis
2
G ( )
in
aG i Xi
aaG i j Xi ,X j
i ij
8. A numerical model for the wavelet variance of a
correlated K-distributed stationary clutter
The Wavelet Scaling Polarimetric
Signature (WASPS)
w2 ( x) f( , )
Shh 2Shv Svv
Correlated K-distributed
clutter, C. Oliver
9. From theory to practice
Supervised wavelet statistics analysis
Single look complex slant range
polarimetric data
Shh 2Shv Svv
2
W ( x) s f ( s, )
Wavelet variance
Multi-voice
4
Power synthesis over a range /
wavelet frame
transform
W ( x)
s
azimuth transect in a linear
2
f ( s, )
basis with orientation ψ 2
W ( x)
s
P f S HH , 2 S HV , SVV
Wavelet kurtosis
(flatness factor)
10. ALOS PALSAR: Hawai’i island Papau Seamount
Flat sea surface
Data delivered by JAXA
ALOS PI program
Correlation White
Shh 2Shv Svv length noise
Rayleigh
term
11. ALOS PALSAR: Hawai’i island Papau Seamount
3
Sea surface features
Data delivered by JAXA
ALOS PI program
Shh 2Shv Svv Unbounded
Periodicity
Correlation
length
2
1
12. Data delivered by JAXA
ALOS PI program ALOS PALSAR: Hawai’i island Papau Seamount
Local
Maxima Sea surface features
3
Maximum
shift
Shh 2Shv Svv
Symmetric
signature
2
1
13. A View from Fourier Kingdom
Spectral Characteristics in the H-V Basis
Sea surface features
VV VV* HH VV* HH HH*
HH HV* HV VV*
HV HV*
Shh 2Shv Svv Xspec(HH HH*,HH HV*)
Xspec(Power, Crosscor)
Xspec(HV HV*,HH HH*)
Xspec(HV HV*,HH HV*)
Xspec(Power, Power)
Flat sea surface X FLAT
Wind waves X WIND
Relative differences in energy normalized by HV power
Power Spectrum copol-xpol correlation (HH HV*): 91%
Cross Power Spectrum (Power, Cross) (HH HH*,HV HH*): 95%
X WIND
X /X
FLAT WIND
*
Cross Power Spectrum (Power, Power)(HH HH*,HV HV*): 77% HV HV
14. Going to Higher Resolution
DLR Tandem-X dual-pol data
Lulonga River – Basankuso - DRC
Data provided by DLR AO-2010 VEGE0330
Swamp
Mosaic
Lowland
Clear-cut
15. Going to Higher Resolution
DLR Tandem-X InSAR Coherence
Data provided by DLR AO-2010
InSAR processing by SARMAP
Swamp
Mosaic
Lowland
Clearcut