IGARSS 2011 Esteban Aguilera Compressed Sensing for Polarimetric SAR Tomography E. Aguilera, M....
Transcript of IGARSS 2011 Esteban Aguilera Compressed Sensing for Polarimetric SAR Tomography E. Aguilera, M....
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IGARSS 2011Esteban Aguilera
Compressed Sensing forPolarimetric SAR Tomography
E. Aguilera, M. Nannini and A. Reigber
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IGARSS 2011Esteban Aguilera
1. Polarimetric SAR tomography
2. Compressive sensing of single signals
3. Multiple signals compressive sensing: Exploiting correlations
4. Compressive sensing for volumetric scatterers
5. Conclusions
Overview
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IGARSS 2011Esteban Aguilera
azimuthground range
M parallel tracks for 3D imaging
Tomographic SAR data acquisition
Side-looking illumination at L-Band
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IGARSS 2011Esteban Aguilera
The tomographic data stack
Our dataset is a stack of M two-dimensional SAR images per polarimetric channel
M images
azimuthrange
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IGARSS 2011Esteban Aguilera
The tomographic data stack
Projections of the reflectivity in the elevation direction are encoded in M pixels (complex valued)
azimuthrange
1
2
M
b
bB
b
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IGARSS 2011Esteban Aguilera
The tomographic signal model: B = AX
11,1 1,2 1,3 1,1
22,1 2,2 2,3 2,2
33,1 3,2 3,3 3,
,1 ,2 ,3 ,
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
N
N
N
MM M M M N N
xa r a r a r a rb
xa r a r a r a rb
xa r a r a r a r
ba r a r a r a r x
,4
,( )i jj r
i ja r e
height
B : measurementsA : steering matrixX : unknown reflectivity
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IGARSS 2011Esteban Aguilera
What’s the problem?
High resolution and low ambiguity require a large number of tracks:
1. Expensive and time consuming
2. Sometimes infeasible
3. Long temporal baselines affect reconstruction
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IGARSS 2011Esteban Aguilera
Where does this work fit?
Beamforming (SAR tomography):
1. Beamforming (Reigber, Nannini, Frey)
2. Adaptive beamforming (Lombardini, Guillaso)
3. Covariance matrix decomposition (Tebaldini)
Physical Models (SAR interferometry):1. PolInSAR (Cloude, Papathanassiou)2. PCT (Cloude)
Compressed sensing (SAR tomography)1. Single signal approach (Zhu, Budillon)2. Multiple signal/channel approach
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IGARSS 2011Esteban Aguilera
Elevation profile reconstruction
A
B AX
AMxN : steering matrix
XN : unknown reflectivityBM : stack of pixels
height
gnd. rangeazimuth
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IGARSS 2011Esteban Aguilera
The compressive sensing approach
We look for the sparsest solution that matches the measurements
minX 1
X
2AX B subject to
Convex optimization problem
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IGARSS 2011Esteban Aguilera
How many tracks?
In theory:
take
measurements
frequencies selected at random
In practice:
we can use our knowledge about the signal and sample less:
low frequency components seem to do the job!
0 log( )M C S N
2M S
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IGARSS 2011Esteban Aguilera
CS for vegetation mapping ?
The elevation profile can be approximated by a summation of sparse profiles
Different to conventional models (non-sparse). And probably a bad one…
elevation
amplitude
= + + … +
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IGARSS 2011Esteban Aguilera
Tomographic E-SAR Campaign
Testsite: Dornstetten, GermanyHorizontal baselines: ~ 20mVertical baselines: ~ 0mAltitude above ground: ~ 3800m# of baselines: 23
3,5 m
2 corner reflectors in layover and ground
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IGARSS 2011Esteban Aguilera
CAPON using 23 tracks (13x13 window) = ground truth
40 m
2 corner reflectors in layover
Canopy and groundGround
40 m
Single Channel Compressive Sensing
CS using only 5 tracks
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IGARSS 2011Esteban Aguilera
Normalized intensity – 40 m
Beamforming (23 passes, 3x3)
SSCS (5 passes, 3x3)
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IGARSS 2011Esteban Aguilera
Multiple Signal Compressive Sensing
Assumption: adjacent azimuth-range positions are likely to have targets at about the same elevation
1 1 1
2 2 2...
M M M
b c d
b c d
b c d
L columns
azimuthrange
range
azimuthM images
GHH
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IGARSS 2011Esteban Aguilera
Polarimetric correlations
We can further exploit correlations between polarimetric channels
G
3L columns
GHH GHV GVV
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IGARSS 2011Esteban Aguilera
Elevation profile reconstruction
A
G AY
AMxN : steering matrixYNx3L : unknown reflectivities
HH HV VV Mx3L : stacks of pixelsG
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IGARSS 2011Esteban Aguilera
YNx3L : unknown reflectivity
Y
minY
2AY G subject to
2,1Y
Elevation profile reconstruction
We look for a matrix with the least number of non-zero rows that matches the measurements
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IGARSS 2011Esteban Aguilera
Mixed-norm minimization
minY
2AY G subject to
0
Number of columns in Y (window size + polarizations)
Probability of recovery failure
(Eldar and Rauhut, 2010)
2,1Y
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IGARSS 2011Esteban Aguilera
SSCS (saturated) MSCS (span saturated)
MSCS (polar) MSCS (span)
Layover recovery with CS
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IGARSS 2011Esteban Aguilera
Beamforming (23 passes, 3x3)
SSCS (5 passes, 3x3)
MSCS (5 passes, 3x3)
MSCS (pre-denoised) (5 passes, 3x3)
Layover recovery with CS
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IGARSS 2011Esteban Aguilera
Volumetric Imaging
Single signal CS (5 tracks)
Multiple signal CS (5 tracks)
40 m
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IGARSS 2011Esteban Aguilera
Volumetric Imaging
Single signal CS (5 tracks)
Multiple signal CS (5 tracks)
40 m
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IGARSS 2011Esteban Aguilera
Volumetric Imaging
Polarimetric Capon beamforming (5 tracks)
Multiple signal CS (5 tracks)
40 m
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IGARSS 2011Esteban Aguilera
Towards a “realistic” sparse vegetation model
elevation
amplitude
Canopy and ground component
Possible sparse description in wavelet domain!
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IGARSS 2011Esteban Aguilera
Sparsity in the wavelet domain
Daubechies wavelet example: 4 vanishing moments 3 levels of decomposition
groundcanopy ground
canopy
0.5
1
0
0.5
1
0
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IGARSS 2011Esteban Aguilera
Elevation profile reconstruction
minY 1
WY
( )AY D Gs.t.
Additional regularization
1
L1 norm of wavelet expansion
(W: transform matrix)
synthetic aperture
2,1Y
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IGARSS 2011Esteban Aguilera
Volumetric Imaging in Wavelet Domain
Fourier beamforming using 23 tracks (23x23 window)
Wavelet-based CS (5 tracks)
40 m
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IGARSS 2011Esteban Aguilera
Volumetric Imaging in Wavelet Domain
Fourier beamforming using 23 tracks (23x23 window)
Wavelet-based CS (5 tracks)
40 m
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IGARSS 2011Esteban Aguilera
Conclusions
Single signal CS:
1. High resolution with reduced number of tracks2. Recovers complex reflectivities but polarimetry problematic3. Model mismatch is not catastrophic (CS theory)4. It’s time-consuming (Convex optimization)
Multiple signal CS:
1. Polarimetric extension of CS2. Higher probability of reconstruction, less noise3. More robust for distributed targets4. Vegetation reconstruction in the wavelet domain
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IGARSS 2011Esteban Aguilera
Convex optimization solvers
CVX (Disciplined Convex Programming): http://cvxr.com/cvx/
SEDUMI: http://sedumi.ie.lehigh.edu/