CHARACTERIZATION OF SINGULAR STRUCTURES IN POLARIMETRIC SAR IMAGES BY WAVELET FRAMES<br />G. F. De Grandi, P. Bunting, A. ...
THEORY - CHARACTERIZING BACKSCATTER DISCONTINUITIES BY A MATHEMATICAL MODEL: THE LIPSCHITZ REGULARITY<br />Approximation b...
THEORY - FROM LIPSCHITZ REGULARITY TO WAVELET FRAMES<br />Trajectory in scale of the wavelet transform maxima<br />Uniform...
WAVELET LIPSCHITZ  ESTIMATOR: EXAMPLES OF SINGULARITIES<br />Assumption: wavelet is the derivative of a Gaussian function ...
WAVELET LIPSCHITZ  ESTIMATOR: EXAMPLES OF SINGULARITIES<br />Assumption: wavelet is the derivative of a Gaussian function ...
WAVELET LIPSCHITZ  ESTIMATOR: EXAMPLES OF SINGULARITIES<br />Heuristic conception of the delta functional as a limit of te...
SMOOTHED SINGULARITIES<br />Functions with singularities (e.g. the step function and the delta functional) are mathematica...
POLARIMETRIC EDGE MODELS<br />Wave Scattering Model<br />U. Texas at Arlington<br />C matrix rotation to orientation angle...
EDGE MODELS: FOREST boundary<br />Lip parameters dependence on incidence angle θ (80-600) and xpol orientation angle ψ (00...
EDGE MODELS: EFFECT OF TERRAIN AZIMUTH TILT<br />Terrain slope in the along-track direction  influences the target reflect...
DIELECTRIC DIHEDRAL SCATTERING<br />Dielectric dihedral model based on compounded Fresnel coefficients  with εra= εrb=25<b...
EXPERIMENTS: LOCAL LIPSCHITZ PARAMETERS ESTIMATION<br />Road between two bare-soil fields<br />DLR E-SAR P-band image acqu...
EXPERIMENTS: LOCAL LIPSCHITZ PARAMETERS ESTIMATION<br />Bare-soil forest edge<br />Smoothing kernel variance<br />Relative...
EXPERIMENTS: LOCAL LIPSCHITZ PARAMETERS ESTIMATION<br />Point target<br />Swing K<br />Relative swing<br />DLR E-SAR P-ban...
LOCAL LIPSCHITZ PARAMETERS: AN OIL SLICK<br />SIR-C C-band image acquired over the English Channel<br />
APPROXIMATIONS OF THE LIPSCHITZ PARAMETERS IN THE IMAGE SPACE-POLARIZATION DOMAIN<br />Estimation of the K parameter (swin...
EXAMPLES OF IMAGE-WIDE LIPSCHITZ PARAMETERS REPRESENTATIONS<br />DLR E-SAR P-band image acquired over Oberpfaffenhofen <br...
EXAMPLES OF IMAGE-WIDE LIPSCHITZ PARAMETERS REPRESENTATIONS<br />LIP MAP HV<br />scales 23, 24, 25<br />White features cor...
EXAMPLES OF IMAGE-WIDE LIPSCHITZ PARAMETERS REPRESENTATIONS<br />Yellow-red features (Lip >0 discontinuities) correspond t...
EXAMPLES OF IMAGE-WIDE LIPSCHITZ PARAMETERS REPRESENTATIONS<br />PALSAR 40 days repeat pass interferometric coherence<br /...
EPILOGUE – SOME FOOD FOR THOUGHT<br />We have traced a connection leading from the abstract theory of function regularity,...
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CHARACTERIZATION OF SINGULAR STRUCTURES IN POLARIMETRIC SAR IMAGES BY WAVELET FRAMES

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CHARACTERIZATION OF SINGULAR STRUCTURES IN POLARIMETRIC SAR IMAGES BY WAVELET FRAMES

  1. 1. CHARACTERIZATION OF SINGULAR STRUCTURES IN POLARIMETRIC SAR IMAGES BY WAVELET FRAMES<br />G. F. De Grandi, P. Bunting, A. Bouvet, T. L. Ainsworth<br />European Commission DG Joint Research Centre<br />21027, Ispra (VA), Italy<br />e-mail: frank.de-grandi@jrc.it<br />Naval Research Laboratory<br />Washington, DC 20375-5351, USA<br />email: ainsworth@nrl.navy.mil.jrc.it<br />Institute of Geography and Earth Sciences<br />Aberystwyth University, Aberystwyth, UK, SY23 3DB.<br />e-mail: pfb@aber.ac.uk<br />
  2. 2. THEORY - CHARACTERIZING BACKSCATTER DISCONTINUITIES BY A MATHEMATICAL MODEL: THE LIPSCHITZ REGULARITY<br />Approximation by Taylor polynomials<br />PointwiseLipschitzα condition at x0<br />α >=0 non-integer<br />Upper bound to the approximation error by mth order differentiability<br />refinement<br />N largest integer <= α<br />Uniform Lip α condition on interval a,b<br />Non differentiable functions<br />Extension to distributions<br />Function is uniformly Lip α if its primitive is Lip α+1<br />Primitive of Dirac ζ-> Step Function Lip α=0<br />Non differentiable but bounded by K e.g. step function<br /> Dirac ζ -> Lip α= -1<br />
  3. 3. THEORY - FROM LIPSCHITZ REGULARITY TO WAVELET FRAMES<br />Trajectory in scale of the wavelet transform maxima<br />Uniform and pointwise Lipschitz regularity<br />Wavelet frame which is the derivative of a smoothing function and has 1 non-vanishing moment<br />f(x) uniformly Lip α≤1 over a,b <br />Multi-voice discrete wavelet transform<br />Lip estimator for a pure singularity<br />K,α<br />Wavelet modulus maxima at fractional scales <br />Linear fitting <br />S. Mallat, W.L. Hwang, S. Zhong,<br /> Courant Institute NY NY, USA, <br />Ecole Polytechnique, Paris, France<br />
  4. 4. WAVELET LIPSCHITZ ESTIMATOR: EXAMPLES OF SINGULARITIES<br />Assumption: wavelet is the derivative of a Gaussian function with σ=1<br />Continuous wavelet transform<br />Step function Lip α=0<br />Trajectory in scale of wavelet modulus maxima<br />Step function<br />Wf(x, s) s=20.25, 20.5, 20.75, 22<br />
  5. 5. WAVELET LIPSCHITZ ESTIMATOR: EXAMPLES OF SINGULARITIES<br />Assumption: wavelet is the derivative of a Gaussian function with σ=1<br />Cusp Lip α=1<br />Continuous wavelet transform<br />Trajectory in scale of wavelet modulus maxima<br />Cusp<br />Wf(x, s) s=20.25, 20.5, 20.75, 22<br />
  6. 6. WAVELET LIPSCHITZ ESTIMATOR: EXAMPLES OF SINGULARITIES<br />Heuristic conception of the delta functional as a limit of testing functions<br />A useful conjecture to extend Lip exponents to singular distributions<br />Wavelet transform through derivatives of the dilated approximating functions<br />Approximating function<br />Testing function in the space D of infinitely smooth functions with finite support<br />Dirac delta functional Lip α= -1<br />Trajectory in scale of wavelet modulus maxima<br />Dirac delta functional approximations by testing functions<br />Wf(x, s) s=20.25, 20.5, 20.75, 22<br />
  7. 7. SMOOTHED SINGULARITIES<br />Functions with singularities (e.g. the step function and the delta functional) are mathematical idealizations. Due to the sensor’s finite resolution we need in reality to consider smoothed singularities.<br />Wavelet modulus trajectories in scale become non-linear<br />Finite approximations to singularities are modeled by means of a smoothing Gaussian kernel gσ with variance σ2<br />Non-linear regression for estimating<br />K, α,σ2<br />
  8. 8. POLARIMETRIC EDGE MODELS<br />Wave Scattering Model<br />U. Texas at Arlington<br />C matrix rotation to orientation angle ψ<br />Fading variable<br />XPOL power<br />COPOL power<br />Mixture<br />C soil<br />C forest<br />
  9. 9. EDGE MODELS: FOREST boundary<br />Lip parameters dependence on incidence angle θ (80-600) and xpol orientation angle ψ (00-900)<br />UTA model simulations for grassland and dense coniferous forest (35 cm DBH) at L-band<br />Swing K<br />Smoothing kernel variance<br />Lipschitz exponent<br />
  10. 10. EDGE MODELS: EFFECT OF TERRAIN AZIMUTH TILT<br />Terrain slope in the along-track direction influences the target reflection symmetry and as a consequence the copol to crosspol correlation terms of the covariance matrix <br />The xpol Lip signatures mirror this effect by a shift of the maximum from 450 which is notably relevant at steep incidence angles<br />Cross section at 80 incidence angle<br />Swing K<br />Cross section at 600 incidence angle<br />
  11. 11. DIELECTRIC DIHEDRAL SCATTERING<br />Dielectric dihedral model based on compounded Fresnel coefficients with εra= εrb=25<br />The copol Lip signatures mirror the dependence on angle of incidence due to the π shift between the copol terms of the scattering matrix. <br />VV<br />HH<br />00<br />Swing K<br />Lip exponent ~ -1<br />Incidence angle<br />230<br />450<br />
  12. 12. EXPERIMENTS: LOCAL LIPSCHITZ PARAMETERS ESTIMATION<br />Road between two bare-soil fields<br />DLR E-SAR P-band image acquired over Oberpfaffenhofen<br />Color composite HH, HV, VV<br />Relative swing<br /> Swing<br /> Lip exponent<br />Xpol orientation angle<br />Xpol orientation angle<br />Xpol orientation angle<br />
  13. 13. EXPERIMENTS: LOCAL LIPSCHITZ PARAMETERS ESTIMATION<br />Bare-soil forest edge<br />Smoothing kernel variance<br />Relative swing<br /> Swing<br /> Lip exponent<br />DLR E-SAR P-band image acquired over Oberpfaffenhofen<br />Color composite HH, HV, VV<br />
  14. 14. EXPERIMENTS: LOCAL LIPSCHITZ PARAMETERS ESTIMATION<br />Point target<br />Swing K<br />Relative swing<br />DLR E-SAR P-band image acquired over Oberpfaffenhofen<br />Color composite HH, HV, VV<br />Lip exponent<br />Smoothing variance<br />
  15. 15. LOCAL LIPSCHITZ PARAMETERS: AN OIL SLICK<br />SIR-C C-band image acquired over the English Channel<br />
  16. 16. APPROXIMATIONS OF THE LIPSCHITZ PARAMETERS IN THE IMAGE SPACE-POLARIZATION DOMAIN<br />Estimation of the K parameter (swing) for each pixel (x,y) in the image using wavelet modulus trajectories from scale 22 to 25 and three polarizations (cross-polarisation at orientation φ = 0°, 23°, 45°)<br />K MAP<br />Approximation of the Lip exponent α for each pixel (x,y) in the image at one polarization (e.g. HH, HV, VV) by combining in a RGB image the wavelet modulus at scales 23, 24, 25<br />LIP MAP<br />
  17. 17. EXAMPLES OF IMAGE-WIDE LIPSCHITZ PARAMETERS REPRESENTATIONS<br />DLR E-SAR P-band image acquired over Oberpfaffenhofen <br />Color composite HH, HV, VV<br />K MAP<br />φ = 0°, 23°, 45°<br />The red dots correspond to stronger swing at HV. These discontinuities appear mainly in the forested areas, and correspond to intensity variation from volume scattering. <br />The blue dots are stronger discontinuities at φ=450, and correspond mainly to man-made targets.<br />
  18. 18. EXAMPLES OF IMAGE-WIDE LIPSCHITZ PARAMETERS REPRESENTATIONS<br />LIP MAP HV<br />scales 23, 24, 25<br />White features correspond to Lip 0 discontinuities e.g. edges (no wavelet maxima decay).<br />Red spots correspond to Lip -1 targets e.g. point targets (decreasing wavelet maxima with scale).<br />Positive Lip discontinuities Lip > 0 are marked with colors tending to blue.<br />LIP MAP VV<br />
  19. 19. EXAMPLES OF IMAGE-WIDE LIPSCHITZ PARAMETERS REPRESENTATIONS<br />Yellow-red features (Lip >0 discontinuities) correspond to edges surrounding surfactant features (oil-slick). Also neighborhoods of point targets (ships) appear as Lip>0 because the estimator is not limited to the local maxima.<br />Black spots (Lip -1 discontinuities) correspond t o the center of strong point targets (ships).<br />SIR-C C-band image acquired over the English Channel<br />LIP MAP COPOL<br />Lip -1<br />Lip 1<br />
  20. 20. EXAMPLES OF IMAGE-WIDE LIPSCHITZ PARAMETERS REPRESENTATIONS<br />PALSAR 40 days repeat pass interferometric coherence<br /> Zotino - Central Siberia<br />RGB composite HH-HV-Xpol45<br />SIR-C C-band image acquired over the English Channel<br />K MAP<br />LIP MAP HH<br />
  21. 21. EPILOGUE – SOME FOOD FOR THOUGHT<br />We have traced a connection leading from the abstract theory of function regularity, through singular distributions, wavelet frames, up to the characterization of discontinuities in a natural or man-made target, as seen by a polarimetric radar.<br />This connection has opened up an interesting field of investigation. <br />Whether practical fall-outs will follow remains to be assessed.<br />Daniel Barenboim speaking of music and life:<br /> Everything is connected<br />Thanks you for following the connection<br />

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