Post on 24-Feb-2016
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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
Research Centre21027, Ispra (VA), Italy
e-mail: frank.de-grandi@jrc.it
Naval Research LaboratoryWashington, DC 20375-5351, USA
email: ainsworth@nrl.navy.mil.jrc.it
Institute of Geography and Earth SciencesAberystwyth University, Aberystwyth, UK,
SY23 3DB.e-mail: pfb@aber.ac.uk
Institute for Environment and SustainabilityGlobal Environment Monitoring Unit
THEORY - CHARACTERIZING BACKSCATTER DISCONTINUITIES BY A MATHEMATICAL MODEL: THE LIPSCHITZ REGULARITY
Approximation by Taylor polynomials
Cxf n )()(
1
00
00
)(
!1!)()(
NN
n
nn
xxNCxx
nxfxf
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
Institute for Environment and SustainabilityGlobal Environment Monitoring Unit
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(
dxdx
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}43,
21,
41,{log2 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
Institute for Environment and SustainabilityGlobal Environment Monitoring Unit
WAVELET LIPSCHITZ ESTIMATOR: EXAMPLES OF SINGULARITIES
22
22
221)(
tt
etedxdt
Assumption: wavelet is the derivative of a Gaussian
function with σ=12
)/( 2
2/)(),(
st
esttfstWf
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
Institute for Environment and SustainabilityGlobal Environment Monitoring Unit
WAVELET LIPSCHITZ ESTIMATOR: EXAMPLES OF SINGULARITIES
22
22
221)(
tt
etedxdt
Assumption: wavelet is the derivative of a Gaussian
function with σ=12
)/( 2
2/)(),(
st
esttfstWf
Continuous wavelet transform
6.0)( xxf Cusp Lip α=1
Trajectory in scale of wavelet modulus maxima
Wf(x, s) s=20.25, 20.5, 20.75, 22Cusp
mKxWMAX 22
log)(log
Institute for Environment and SustainabilityGlobal Environment Monitoring Unit
WAVELET LIPSCHITZ ESTIMATOR: EXAMPLES OF SINGULARITIESHeuristic conception of the delta functional as a limit of testing functions
A useful conjecture to extend Lip exponents to singular distributions
Dirac delta functional Lip α= -1
Trajectory in scale of wavelet modulus maxima
Wf(x, s) s=20.25, 20.5, 20.75, 22Dirac delta functional approximations by testing functions
dtttt
)/()/()(
1
1
11exp
0)(
2
tt
tt
Testing function in the space D of infinitely smooth functions with finite support
Approximating function
dtd
st
dtdstWf s
),(
Wavelet transform through derivatives of the dilated approximating functions
mKxWMAX 22
log)(log
Institute for Environment and SustainabilityGlobal Environment Monitoring Unit
SMOOTHED SINGULARITIESFunctions 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),( xdxdfs
sx
sgf
dxdssxWf s
Finite approximations to singularities are modeled by means of a smoothing Gaussian
kernel gσ with variance σ2
22
2
2222
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
vvvvhvvvhhvvvvhvhvhvhhhvvvhhhvhhhhhh
C
C soil C forestMixture
Wave Scattering ModelU. Texas at Arlington
C matrix rotation to orientation angle ψ
Fading variable
XPOL powerCOPOL power
Institute for Environment and SustainabilityGlobal Environment Monitoring Unit
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
Institute for Environment and SustainabilityGlobal Environment Monitoring Unit
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
Institute for Environment and SustainabilityGlobal Environment Monitoring Unit
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
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
Institute for Environment and SustainabilityGlobal Environment Monitoring Unit
LOCAL LIPSCHITZ PARAMETERS: AN OIL SLICK
SIR-C C-band image acquired over the English Channel
Institute for Environment and SustainabilityGlobal Environment Monitoring Unit
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
jjyxWyxW
yxyx 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
Institute for Environment and SustainabilityGlobal Environment Monitoring Unit
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°
Institute for Environment and SustainabilityGlobal Environment Monitoring Unit
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
Institute for Environment and SustainabilityGlobal Environment Monitoring Unit
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
Institute for Environment and SustainabilityGlobal Environment Monitoring Unit
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
Institute for Environment and SustainabilityGlobal Environment Monitoring Unit
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.