IGARSS 2011 Esteban Aguilera Compressed Sensing for Polarimetric SAR Tomography E. Aguilera, M. Nannini and A. Reigber.
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IGARSS 2011Esteban Aguilera
Compressed Sensing forPolarimetric SAR Tomography
E. Aguilera, M. Nannini and A. Reigber
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
IGARSS 2011Esteban Aguilera
azimuthground range
M parallel tracks for 3D imaging
Tomographic SAR data acquisition
Side-looking illumination at L-Band
IGARSS 2011Esteban Aguilera
The tomographic data stack
Our dataset is a stack of M two-dimensional SAR images per polarimetric channel
M images
azimuthrange
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
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
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
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
IGARSS 2011Esteban Aguilera
Elevation profile reconstruction
A
B AX
AMxN : steering matrix
XN : unknown reflectivityBM : stack of pixels
height
gnd. rangeazimuth
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
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
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
= + + … +
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
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
IGARSS 2011Esteban Aguilera
Normalized intensity – 40 m
Beamforming (23 passes, 3x3)
SSCS (5 passes, 3x3)
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
IGARSS 2011Esteban Aguilera
Polarimetric correlations
We can further exploit correlations between polarimetric channels
G
3L columns
GHH GHV GVV
IGARSS 2011Esteban Aguilera
Elevation profile reconstruction
A
G AY
AMxN : steering matrixYNx3L : unknown reflectivities
HH HV VV Mx3L : stacks of pixelsG
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
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
IGARSS 2011Esteban Aguilera
SSCS (saturated) MSCS (span saturated)
MSCS (polar) MSCS (span)
Layover recovery with CS
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
IGARSS 2011Esteban Aguilera
Volumetric Imaging
Single signal CS (5 tracks)
Multiple signal CS (5 tracks)
40 m
IGARSS 2011Esteban Aguilera
Volumetric Imaging
Single signal CS (5 tracks)
Multiple signal CS (5 tracks)
40 m
IGARSS 2011Esteban Aguilera
Volumetric Imaging
Polarimetric Capon beamforming (5 tracks)
Multiple signal CS (5 tracks)
40 m
IGARSS 2011Esteban Aguilera
Towards a “realistic” sparse vegetation model
elevation
amplitude
Canopy and ground component
Possible sparse description in wavelet domain!
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
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
IGARSS 2011Esteban Aguilera
Volumetric Imaging in Wavelet Domain
Fourier beamforming using 23 tracks (23x23 window)
Wavelet-based CS (5 tracks)
40 m
IGARSS 2011Esteban Aguilera
Volumetric Imaging in Wavelet Domain
Fourier beamforming using 23 tracks (23x23 window)
Wavelet-based CS (5 tracks)
40 m
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
IGARSS 2011Esteban Aguilera
Convex optimization solvers
CVX (Disciplined Convex Programming): http://cvxr.com/cvx/
SEDUMI: http://sedumi.ie.lehigh.edu/
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