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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. Institute of Geography and Earth Sciences

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slide1

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

Institute of Geography and Earth Sciences

Aberystwyth University, Aberystwyth, UK, SY23 3DB.

e-mail: pfb@aber.ac.uk

Naval Research Laboratory

Washington, DC 20375-5351, USA

email: ainsworth@nrl.navy.mil.jrc.it

slide2

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

slide3

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

Wavelet modulus maxima at fractional scales

Lip estimator for a pure singularity

K,α

Linear fitting

S. Mallat, W.L. Hwang, S. Zhong,

Courant Institute NY NY, USA,

Ecole Polytechnique, Paris, France

slide4

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

slide5

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

slide6

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

slide7

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

slide8

POLARIMETRIC EDGE MODELS

Wave Scattering Model

U. Texas at Arlington

C matrix rotation to orientation angle ψ

XPOL power

COPOL power

Fading variable

Mixture

C soil

C forest

slide9

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

slide10

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

slide11

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

slide12

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

slide13

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

slide14

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

slide15

LOCAL LIPSCHITZ PARAMETERS: AN OIL SLICK

SIR-C C-band image acquired over the English Channel

slide16

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

slide17

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.

slide18

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

slide19

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

slide20

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

slide21

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