CHARACTERIZATION OF SINGULAR STRUCTURES IN POLARIMETRIC SAR IMAGES BY WAVELET FRAMES

Institute for Environment and SustainabilityGlobal Environment Monitoring Unit

CHARACTERIZATION OF SINGULAR STRUCTURES IN POLARIMETRIC SAR IMAGES BY WAVELET FRAMES

G. F. De Grandi, P. Bunting, A. Bouvet, T. L. AinsworthEuropean Commission DG Joint

Research Centre21027, Ispra (VA), Italy

e-mail: [email protected]

Naval Research LaboratoryWashington, DC 20375-5351, USA

email: [email protected]

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

SY23 3DB.e-mail: [email protected]

mailto:[email protected]


THEORY - CHARACTERIZING BACKSCATTER DISCONTINUITIES BY A MATHEMATICAL MODEL: THE LIPSCHITZ REGULARITY

Approximation by Taylor polynomials

Cxf n )()(

1

00

00

)(

!1!

)()(

NN

n

nn

xxN

Cxx

n

xfxf

00 , xxx

Upper bound to the approximation error by mth order differentiability

refinement

Pointwise Lipschitz α condition at x0

0

000 )()( xxKxxxaxf

N

n

nn

N N largest integer <= α

x

010 N

Non differentiable functions

00 )()( xxKxfxf

0 Non differentiable but bounded by K e.g. step function

0

Extension to distributions

α >=0 non-integer

Function is uniformly Lip α if its primitive is Lip α+1

Uniform Lip α condition on interval a,b

000 sup,, xx KLipbax

Primitive of Dirac ζ -> Step Function Lip α=0

Dirac ζ -> Lip α= -1


THEORY - FROM LIPSCHITZ REGULARITY TO WAVELET FRAMES

Uniform and pointwise Lipschitz regularity

Trajectory in scale of the wavelet transform maxima

f(x) uniformly Lip α≤1 over a,b

KssuWf ),(

)x(

dx

dx

0dx)x()x(

Wavelet frame which is the derivative of a smoothing function and has 1 non-vanishing moment

j

ssjKxfW 2)(log)(log

22

41}4

3,

2

1,

4

1,{log

2 jjjjjsm

mKxWMAX 22

log)(log

Lip estimator for a pure singularity

K,α

Wavelet modulus maxima at fractional

scales

Multi-voice discrete wavelet transform

Linear fitting

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

Courant Institute NY NY, USA,

Ecole Polytechnique, Paris, France


WAVELET LIPSCHITZ ESTIMATOR: EXAMPLES OF SINGULARITIES

22

22

22

1)(

tt

et

edx

dt

Assumption: wavelet is the derivative of a Gaussian

function with σ=12

)/( 2

2

/)(),(

st

est

tfstWf

Continuous wavelet transform

Step function Lip α=0

Trajectory in scale of wavelet modulus maxima

Wf(x, s) s=20.25, 20.5, 20.75, 22Step function

mKxWMAX 22

log)(log



22

22

22

1)(

tt

et

edx

dt

Assumption: wavelet is the derivative of a Gaussian

function with σ=12

)/( 2

2

/)(),(

st

est

tfstWf

Continuous wavelet transform

6.0)( xxf Cusp Lip α=1


Wf(x, s) s=20.25, 20.5, 20.75, 22Cusp

mKxWMAX 22

log)(log



Heuristic conception of the delta functional as a limit of testing functionsA useful conjecture to extend Lip exponents to singular distributions

Dirac delta functional Lip α= -1


Wf(x, s) s=20.25, 20.5, 20.75, 22Dirac delta functional approximations by testing functions

dtt

tt

)/(

)/()(

1

1

1

1exp

0)(

2

t

t

t

t

Testing function in the space D of infinitely smooth functions with finite support

Approximating function

dt

d

s

t

dt

dstWf s

),(

Wavelet transform through derivatives of the dilated approximating functions

mKxWMAX 22

log)(log


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.

)(1

),( xdx

dfs

s

x

sgf

dx

dssxWf s

Finite approximations to singularities are modeled by means of a smoothing Gaussian

kernel gσ with variance σ2

22

2

2

222

11,,

s

x

s ess

sx

21

22 )(),(

sKssxWf

Wavelet modulus trajectories in scale become non-linear

Non-linear regression for estimating

K, α,σ2


POLARIMETRIC EDGE MODELS

vvvvhvvvhhvv

vvhvhvhvhhhv

vvhhhvhhhhhh

C

C soil C forest

Mixture

Wave Scattering ModelU. Texas at Arlington

C matrix rotation to orientation angle ψ

Fading variable

XPOL powerCOPOL power


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

Lipschitz exponent Swing K Smoothing kernel variance


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

Swing K Cross section at 80 incidence angle Cross section at 600 incidence angle


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.

Swing KLip exponent ~ -1 Incidence angle

00

230

450

HH

VV


EXPERIMENTS: LOCAL LIPSCHITZ PARAMETERS ESTIMATION

Relative swing Swing

Lip exponent

Xpol orientation angle Xpol orientation angle Xpol orientation angle

DLR E-SAR P-band image acquired over Oberpfaffenhofen

Color composite HH, HV, VV

Road between two bare-soil fields





Bare-soil forest edge

Swing Lip exponentSmoothing kernel variance

Relative swing





Point target

Lip exponentSwing K

Relative swing

Smoothing variance


LOCAL LIPSCHITZ PARAMETERS: AN OIL SLICK

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


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

12

221j12

),,(log),,(log-j|),,( W|log=),,K( 12

jj

yxWyxWyxyx

jj

221

522 2log2logj j

imageRGByxyxyx ),,K(),,K(),,K( 321

LIP MAP

imageRGByxyxyx )2,,,Wf()2,,,Wf()2,,,Wf( 5431

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


EXAMPLES OF IMAGE-WIDE LIPSCHITZ PARAMETERS REPRESENTATIONS

K MAP

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.

DLR E-SAR P-band image acquired over

Oberpfaffenhofen Color composite HH, HV, VV

φ = 0°, 23°, 45°



LIP MAP HV

LIP MAP VV

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.

scales 23, 24, 25



LIP MAP COPOL

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).

Lip -1 Lip 1

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



SIR-C C-band image acquired over the English ChannelK MAP

PALSAR 40 days repeat pass interferometric coherence Zotino - Central Siberia

RGB composite HH-HV-Xpol45

LIP MAP HH


EPILOGUE – SOME FOOD FOR THOUGHT

Daniel Barenboim speaking of music and life: Everything is connected

Thanks you for following the connection

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.

http://www.orionbooks.co.uk/MP-43941/Everything-Is-Connected.htm

CHARACTERIZATION OF SINGULAR STRUCTURES IN POLARIMETRIC SAR IMAGES BY WAVELET FRAMES

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