Institute for Environment and Sustainability Global Environment Monitoring Unit 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: [email protected]Naval Research Laboratory Washington, DC 20375-5351, USA email: [email protected]Institute of Geography and Earth Sciences Aberystwyth University, Aberystwyth, UK, SY23 3DB. e-mail: [email protected]
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CHARACTERIZATION OF SINGULAR STRUCTURES IN POLARIMETRIC SAR IMAGES BY WAVELET FRAMES
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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
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
Institute for Environment and SustainabilityGlobal Environment Monitoring Unit
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
Institute for Environment and SustainabilityGlobal Environment Monitoring Unit
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
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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
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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
Institute for Environment and SustainabilityGlobal Environment Monitoring Unit
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
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EXPERIMENTS: LOCAL LIPSCHITZ PARAMETERS ESTIMATION
DLR E-SAR P-band image acquired over Oberpfaffenhofen
Color composite HH, HV, VV
Road between two bare-soil fields
Institute for Environment and SustainabilityGlobal Environment Monitoring Unit
EXPERIMENTS: LOCAL LIPSCHITZ PARAMETERS ESTIMATION
DLR E-SAR P-band image acquired over Oberpfaffenhofen
Color composite HH, HV, VV
Bare-soil forest edge
Swing Lip exponentSmoothing kernel variance
Relative swing
Institute for Environment and SustainabilityGlobal Environment Monitoring Unit
EXPERIMENTS: LOCAL LIPSCHITZ PARAMETERS ESTIMATION
DLR E-SAR P-band image acquired over Oberpfaffenhofen
Color composite HH, HV, VV
Point target
Lip exponentSwing K
Relative swing
Smoothing variance
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LOCAL LIPSCHITZ PARAMETERS: AN OIL SLICK
SIR-C C-band image acquired over the English Channel
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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
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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°
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EXAMPLES OF IMAGE-WIDE LIPSCHITZ PARAMETERS REPRESENTATIONS
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
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EXAMPLES OF IMAGE-WIDE LIPSCHITZ PARAMETERS REPRESENTATIONS
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
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EXAMPLES OF IMAGE-WIDE LIPSCHITZ PARAMETERS REPRESENTATIONS
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
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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.