compressed sensing for polarimetric sar tomography e. aguilera, m. nannini and a. reigber
DESCRIPTION
Compressed Sensing for Polarimetric SAR Tomography E. Aguilera, M. Nannini and A. Reigber. Polarimetric SAR tomography Compressive sensing of single signals Multiple signals compressive sensing: Exploiting correlations Compressive sensing for volumetric scatterers Conclusions. Overview. - PowerPoint PPT PresentationTRANSCRIPT
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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 tomography2. Compressive sensing of single signals3. Multiple signals compressive sensing: Exploiting
correlations4. Compressive sensing for volumetric scatterers5. 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
bb
B
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 consuming2. Sometimes infeasible3. 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 matrixXN : 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 profilesDifferent to conventional models (non-sparse). And probably a bad one…
elevation
amplitude
= + + … +
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IGARSS 2011Esteban Aguilera
Tomographic E-SAR CampaignTestsite: 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 db c d
b c d
L columnsazimuthrange
rangeazimuth
M 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 ImagingSingle signal CS (5 tracks)
Multiple signal CS (5 tracks)
40 m
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IGARSS 2011Esteban Aguilera
Volumetric ImagingSingle signal CS (5 tracks)
Multiple signal CS (5 tracks)
40 m
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IGARSS 2011Esteban Aguilera
Volumetric ImagingPolarimetric 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 DomainFourier beamforming using 23 tracks (23x23 window)
Wavelet-based CS (5 tracks)
40 m
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IGARSS 2011Esteban Aguilera
Volumetric Imaging in Wavelet DomainFourier 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/