two-point statistic of polarimetric sar data two-point statistic of polarimetric sar data

G. F. De Grandi1, R.M. Lucas2, A. Bouvet1

European Commission Joint Research Centre21027, Ispra (VA), Italy

e-mail: frank.de-grandi@jrc.ec.europa.eu

Two-point statistic of polarimetric SAR data provided by a wavelet frame

The importance of being scaling…(suggested by Oscar Wilde)

Institute of Geography and Earth SciencesAberystwyth University, Aberystwyth, UK,

SY23 3DB.e-mail: rml@aber.ac.uk

Bits of History

Problem Statement

vvhvhh SSS 2

Two-point statistic provided by wavelet

coefficients

Structure Function

2)()()( xfxff

Wavelet Variance

)()(2 scalefxw

Spatial random field (SRF)

Tvvhvhh SSS 2

),(),( * yxSSyxP hvhvhv

Wavelet frame transform

What happens when the polarization basis is changed?

vvhvhh SSS 2

Wavelet variance of the crospolar and

copolar power in the rotated polarization

basis

),()(2 fxw

Tvvhvhh SSS 2

),,(),,( '*2

'2 yxyxPxpol

T' 2cos1

2sin

),,(),,( '*1

'1 yxyxPcopol

Wavelet variance of PolSAR power in a rotated basis: a model for a WS stationary process

vvhvhh SSS 2

Hj Fourier transform of the wavelet dilated at scale j

jH

ACF of wavelet coefficients computed in the frequency domain from the

power spectrum G of the input process 21 jinout HGR

outG inG

0Rw out2i Wavelet variance at cross-polarized

state with orientation ψ

Wavelet transformPower

spectrum inPower

spectrum out

Power spectrum of the crosspolar power in the rotated basis

vvhvhh SSS 2 VVHVHH S,S2,S

HHVVHVSScScS

21

Rotation to a linear basis with orientation ψ

Cross-polarized component in the new basis ψ

ii

iab XaP )(

**2 Re)( jijij

iiii

iab SSccSScP

VVHVHHi SSSS ,,

2cos21c

2sin2

22c

Power spectrum of the input process

j

jji

iiin XaXaG *)(

jii XXj

ijiX

iiin

GaaGaG,

2)(

**2 Re)( jijij

iiii

iab SSccSScP

ii

iab XaP )(

Power spectrum in the rotated basis is a linear combination of the power spectra (auto-correlation) and cross-spectra (cross correlation) between dyads of the vector in the H,V basis

Copol-xpol correlation

Power-Correlation

HV2

HH2, VV2

A numerical model for the wavelet variance of a correlated K-distributed stationary clutter

vvhvhh SSS 2

The Wavelet Scaling Polarimetric Signature (WASPS)

),()(2 fxw

Correlated K-distributed clutter, C. Oliver

From theory to practice

vvhvhh SSS 2

Multi-voice wavelet frame

transform

),()(2 sfxWs

VVHVHH SSSfP ,2,

Power synthesis over a range / azimuth transect in a linear

basis with orientation ψ

Supervised wavelet statistics analysis

Single look complex slant range polarimetric data

),()(

)(22

4

sfxW

xW

s

s

Wavelet variance

Wavelet kurtosis (flatness factor)

ALOS PALSAR: Hawai’i island Papau Seamount

vvhvhh SSS 2

Flat sea surface

Rayleigh term

Correlation length

White noise

Data delivered by JAXA ALOS PI program

ALOS PALSAR: Hawai’i island Papau Seamount

vvhvhh SSS 2

Sea surface features

1

3

2

Periodicity

Unbounded

Correlation length

Data delivered by JAXA ALOS PI program

ALOS PALSAR: Hawai’i island Papau Seamount

vvhvhh SSS 2

Sea surface features

1

3

2

Maximum shift

Symmetric signature

Local Maxima

Data delivered by JAXA ALOS PI program

A View from Fourier Kingdom

vvhvhh SSS 2

Spectral Characteristics in the H-V Basis

Sea surface featuresVV VV* HH VV* HH HH*

HH HV* HV VV*

HV HV* Xspec(HH HH*,HH HV*)

Xspec(HV HV*,HH HV*)

Xspec(HV HV*,HH HH*)

Flat sea surface Wind waves

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%

Cross Power Spectrum (Power, Power)(HH HH*,HV HV*): 77%

FLATX WIND

X

*

/

HVHV

XXXWINDFLATWIND

Xspec(Power, Crosscor)

Xspec(Power, Power)

Going to Higher Resolution

DLR Tandem-X dual-pol data

Data provided by DLR AO-2010 VEGE0330

Swamp

Lowland

Mosaic

Clear-cut

Lulonga River – Basankuso - DRC

Going to Higher Resolution

DLR Tandem-X InSAR Coherence

Data provided by DLR AO-2010InSAR processing by SARMAP

Swamp

Lowland

Mosaic

Clearcut

CONCLUSIONS

I will not be the same without the jungleCiao, Leb Wohl, Goodbye, Sayonara