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  1. 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. Bouvet1European Commission Joint Research Centre Institute of Geography and Earth Sciences 21027, Ispra (VA), Italy Aberystwyth University, Aberystwyth, UK, e-mail: SY23 3DB. e-mail:
  2. 2. Bits of History
  3. 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. 4. What happens when the polarization basis is changed? T sin 2 Shh 2Shv Svv T 1 cos 2 Shh 2Shv Svvxpol * 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. 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 computedRout 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. 6. Power spectrum of the crosspolar power in the rotated basisS 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. 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 aPower 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. 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. 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 4Power 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) sP f S HH , 2 S HV , SVV Wavelet kurtosis (flatness factor)
  10. 10. ALOS PALSAR: Hawai’i island Papau Seamount Flat sea surface Data delivered by JAXA ALOS PI program Correlation WhiteShh 2Shv Svv length noise Rayleigh term
  11. 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 21
  12. 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. 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 powerPower 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. 14. Going to Higher Resolution DLR Tandem-X dual-pol data Lulonga River – Basankuso - DRC Data provided by DLR AO-2010 VEGE0330 Swamp Mosaic LowlandClear-cut
  15. 15. Going to Higher Resolution DLR Tandem-X InSAR Coherence Data provided by DLR AO-2010 InSAR processing by SARMAP Swamp Mosaic LowlandClearcut
  16. 16. CONCLUSIONSI will not be the same without the jungle Ciao, Leb Wohl, Goodbye, Sayonara