This document presents a method for characterizing singular structures in polarimetric synthetic aperture radar (SAR) images using wavelet frames. It introduces the Lipschitz regularity model for characterizing discontinuities mathematically. Wavelet frames can estimate Lipschitz parameters like the exponent and swing from the scale trajectory of wavelet transform maxima. Examples show Lipschitz parameter estimation for edges, point targets, and oil slicks. Maps of the swing and Lipschitz exponent parameters across full SAR images are demonstrated to identify different target types and discontinuities.

1. CHARACTERIZATION OF SINGULAR STRUCTURES IN POLARIMETRIC SAR IMAGES BY WAVELET FRAMES G. F. De Grandi, P. Bunting, A. Bouvet, T. L. Ainsworth European Commission DG Joint Research Centre 21027, Ispra (VA), Italy e-mail: frank.de-grandi@jrc.it Naval Research Laboratory Washington, DC 20375-5351, USA email: ainsworth@nrl.navy.mil.jrc.it Institute of Geography and Earth Sciences Aberystwyth University, Aberystwyth, UK, SY23 3DB. e-mail: pfb@aber.ac.uk

2. THEORY - CHARACTERIZING BACKSCATTER DISCONTINUITIES BY A MATHEMATICAL MODEL: THE LIPSCHITZ REGULARITY Approximation by Taylor polynomials PointwiseLipschitzα condition at x0 α >=0 non-integer Upper bound to the approximation error by mth order differentiability refinement N largest integer <= α Uniform Lip α condition on interval a,b Non differentiable functions Extension to distributions Function is uniformly Lip α if its primitive is Lip α+1 Primitive of Dirac ζ-> Step Function Lip α=0 Non differentiable but bounded by K e.g. step function Dirac ζ -> Lip α= -1

3. THEORY - FROM LIPSCHITZ REGULARITY TO WAVELET FRAMES Trajectory in scale of the wavelet transform maxima Uniform and pointwise Lipschitz regularity Wavelet frame which is the derivative of a smoothing function and has 1 non-vanishing moment f(x) uniformly Lip α≤1 over a,b Multi-voice discrete wavelet transform Lip estimator for a pure singularity K,α Wavelet modulus maxima at fractional scales Linear fitting S. Mallat, W.L. Hwang, S. Zhong, Courant Institute NY NY, USA, Ecole Polytechnique, Paris, France

4. WAVELET LIPSCHITZ ESTIMATOR: EXAMPLES OF SINGULARITIES Assumption: wavelet is the derivative of a Gaussian function with σ=1 Continuous wavelet transform Step function Lip α=0 Trajectory in scale of wavelet modulus maxima Step function Wf(x, s) s=20.25, 20.5, 20.75, 22

5. WAVELET LIPSCHITZ ESTIMATOR: EXAMPLES OF SINGULARITIES Assumption: wavelet is the derivative of a Gaussian function with σ=1 Cusp Lip α=1 Continuous wavelet transform Trajectory in scale of wavelet modulus maxima Cusp Wf(x, s) s=20.25, 20.5, 20.75, 22

6. WAVELET LIPSCHITZ ESTIMATOR: EXAMPLES OF SINGULARITIES Heuristic conception of the delta functional as a limit of testing functions A useful conjecture to extend Lip exponents to singular distributions Wavelet transform through derivatives of the dilated approximating functions Approximating function Testing function in the space D of infinitely smooth functions with finite support Dirac delta functional Lip α= -1 Trajectory in scale of wavelet modulus maxima Dirac delta functional approximations by testing functions Wf(x, s) s=20.25, 20.5, 20.75, 22

7. SMOOTHED SINGULARITIES 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. Wavelet modulus trajectories in scale become non-linear Finite approximations to singularities are modeled by means of a smoothing Gaussian kernel gσ with variance σ2 Non-linear regression for estimating K, α,σ2

8. POLARIMETRIC EDGE MODELS Wave Scattering Model U. Texas at Arlington C matrix rotation to orientation angle ψ Fading variable XPOL power COPOL power Mixture C soil C forest

9. EDGE MODELS: FOREST boundary Lip parameters dependence on incidence angle θ (80-600) and xpol orientation angle ψ (00-900) UTA model simulations for grassland and dense coniferous forest (35 cm DBH) at L-band Swing K Smoothing kernel variance Lipschitz exponent

10. EDGE MODELS: EFFECT OF TERRAIN AZIMUTH TILT 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 The xpol Lip signatures mirror this effect by a shift of the maximum from 450 which is notably relevant at steep incidence angles Cross section at 80 incidence angle Swing K Cross section at 600 incidence angle

11. DIELECTRIC DIHEDRAL SCATTERING Dielectric dihedral model based on compounded Fresnel coefficients with εra= εrb=25 The copol Lip signatures mirror the dependence on angle of incidence due to the π shift between the copol terms of the scattering matrix. VV HH 00 Swing K Lip exponent ~ -1 Incidence angle 230 450

12. EXPERIMENTS: LOCAL LIPSCHITZ PARAMETERS ESTIMATION Road between two bare-soil fields DLR E-SAR P-band image acquired over Oberpfaffenhofen Color composite HH, HV, VV Relative swing Swing Lip exponent Xpol orientation angle Xpol orientation angle Xpol orientation angle

13. EXPERIMENTS: LOCAL LIPSCHITZ PARAMETERS ESTIMATION Bare-soil forest edge Smoothing kernel variance Relative swing Swing Lip exponent DLR E-SAR P-band image acquired over Oberpfaffenhofen Color composite HH, HV, VV

14. EXPERIMENTS: LOCAL LIPSCHITZ PARAMETERS ESTIMATION Point target Swing K Relative swing DLR E-SAR P-band image acquired over Oberpfaffenhofen Color composite HH, HV, VV Lip exponent Smoothing variance

15. LOCAL LIPSCHITZ PARAMETERS: AN OIL SLICK SIR-C C-band image acquired over the English Channel

16. APPROXIMATIONS OF THE LIPSCHITZ PARAMETERS IN THE IMAGE SPACE-POLARIZATION DOMAIN 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°) K MAP 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 LIP MAP

17. EXAMPLES OF IMAGE-WIDE LIPSCHITZ PARAMETERS REPRESENTATIONS DLR E-SAR P-band image acquired over Oberpfaffenhofen Color composite HH, HV, VV K MAP φ = 0°, 23°, 45° 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. The blue dots are stronger discontinuities at φ=450, and correspond mainly to man-made targets.

18. EXAMPLES OF IMAGE-WIDE LIPSCHITZ PARAMETERS REPRESENTATIONS LIP MAP HV scales 23, 24, 25 White features correspond to Lip 0 discontinuities e.g. edges (no wavelet maxima decay). Red spots correspond to Lip -1 targets e.g. point targets (decreasing wavelet maxima with scale). Positive Lip discontinuities Lip > 0 are marked with colors tending to blue. LIP MAP VV

19. EXAMPLES OF IMAGE-WIDE LIPSCHITZ PARAMETERS REPRESENTATIONS 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. Black spots (Lip -1 discontinuities) correspond t o the center of strong point targets (ships). SIR-C C-band image acquired over the English Channel LIP MAP COPOL Lip -1 Lip 1

20. EXAMPLES OF IMAGE-WIDE LIPSCHITZ PARAMETERS REPRESENTATIONS PALSAR 40 days repeat pass interferometric coherence Zotino - Central Siberia RGB composite HH-HV-Xpol45 SIR-C C-band image acquired over the English Channel K MAP LIP MAP HH

21. EPILOGUE – SOME FOOD FOR THOUGHT 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. This connection has opened up an interesting field of investigation. Whether practical fall-outs will follow remains to be assessed. Daniel Barenboim speaking of music and life: Everything is connected Thanks you for following the connection