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CS and GLRT in SAR Tomography
Compressive Sensing and Generalized LikelihoodRatio Test in SAR Tomography
G. Fornaro1,2, A. Pauciullo1, D. Reale1, M. Weiss3, A. Budillon2, G. Schirinzi2
1 - Institute for Electromagnetic Sensing of the Environment (IREA)
National Research Council (CNR), Naples, Italy
2 - University of Napoli “Parthenope”
Department of Engineering, Naples, Italy
3- Fraunhofer FHR
Wachtberg, Germany.
CS and GLRT in SAR Tomography 2
Outline
� Synthetic Aperture Radar (SAR) Tomography and motivation for super-resolution
� Compressive Sensing in Tomo-SAR
� Scatterer detection problem Tomo-SAR . Generalized Likelihood Ratio Test (GLRT) schemes: the GLRT with cancellation (GLRT-C) and the support based GLRT (sup-GLRT) approaches
� Application to real data: CS, GLRT-C vs sup-GLRT and sup-GLR vs CS
� Conclusions and future works
CS and GLRT in SAR Tomography 3
SAR Tomography for full 3D Imaging
� SAR Interferometry and Differential SAR Interferometry has important applicationsin Digital Elevation Model (DEM) reconstruction and monitoring of deformation.
� SAR Tomography extends interferometric approaches for application to complexscenarios.
� By synthesizing an antenna also in the slant height direction (orthogonal tothe line of sight) it is possible to analyze the vertical structure of the scatteringthus extending SAR imaging form 2D (azimuth-slant range) to 3D (azimuth-slantrange-slant height).
CS and GLRT in SAR Tomography 4
3D SAR Imaging
( )∫−
−=max
max
2
s
s
sjn dsesx nπξγ ( ) Nnrbnn ,....12 == λξ
r
s FOURIER INVERSION FROM IRREGULAR SAMPLES:
� BeamForming (BF)
� Regularized inversion (SVD)
� Adaptive Beamforming (Capon)
� Compressive sensing (CS)1,2
signal to the n-th antenna
backscattering distribution
along the slant height
N acquisitions with spatial (orthogonal) baseline distribution Nbb ......1
( )Brs 2λ=∆
RAYLEIGH
RESOLUTION
minmax bbB −=
1. A. Budillon, A. Evangelista, G. Schirinzi. Three-Dimensional SAR Focusing From Multipass
Signals Using Compressive Sampling. IEEE Trans. Geosci. Remote Sens., 49 (1):488-499,
2011
2. X. X. Zhu, and R. Bamler. Tomographic SAR Inversion by L1 -Norm Regularization—The
Compressive Sensing Approach. IEEE Trans. Geosci. Remote Sens., 48(10):3839-3846, 2010.
CS and GLRT in SAR Tomography
INVERSION
5
[ ]∫ −= − dssjsNx nn πξγ 2exp)(21
Beamforming – Matched filter
( ) xaxAγHH ˆˆmms =⇒= γ
[ ]Nxx ....1T =x N measurements
Aγγx =
=−− LNN sjsj
eeN πζπζ 22 1
111
DISCRETIZATION
[ ]LaaA ...1= sensing matrix
{ }Ll ssss ,..,,...,1∈ L (bins): L≥N typically L>>N
[ ] Ne lN sjl
πξ2T ...1−=a
steering vector: response or “firm” of a scatterer at a given height
CS and GLRT in SAR Tomography 6
4D SAR Imaging (Differential SAR Tomography)
( )( )
∫−
−−=max
max
,4
2
s
s
tsdjsj
n dseesxn
n λπ
πξγ
Signal to the n-th antenna
Deformation term
r
s
N acquisitions with spatial baseline distribution and temporal distributionNbb ......1 Ntt ......1
( )∫ ∫− −
−−=max
max
max
max
22,
s
s
v
v
vjsjn dvdsevsax nn πηπξ
γ
( )rbt nnnn λξλη 22 ==
( )( )∫
−
−−=
max
max
2,
4
,
v
v
vjtsdj
dvevsae nn πηλ
π
λη nn t2=
CS and GLRT in SAR Tomography 7
Compressive Sensing
Result: if γγγγ is a sparse signal (can be well represented as few non zero contributions) then,
under the condition that a sufficient number of samples are acquired, the unknown can be
well reconstructed via the compressive sensing technique.
[ ] γaaAγx L...1== [ ] Ne lN sj
l
πζ2T ...1−=a
Hypothesis: the number K of non-zero entries of γγγγ ( )
is sufficiently smaller than L,N (K<<N<<L)
K=0
γ Aγx =
1γ
2γ
ε<−=20
tosubjectminargˆ Aγxγγ
γ
L >> N i.e. few measurements (N) and many unknown (L)
NP Complete problem, i.e. requires the check of all L!/K!/(L-K)! combinations
{ }12
minargˆ γAγxγγ
δ+−=
Under “some” mild hypothesis the solution above is equal to that of one of the following (equivalent) convex problems:
Basis Pursuit De-Noising (BPDN)
ε<−=21
tosubjectminargˆ Aγxγγγ
CS and GLRT in SAR Tomography 8
Scatterer Detection Problem
Key elements: the False Alarm Probability (FAP) and the Detection Probability (DP).
waax
wax
wx
++=
+=
=
2211
11
:
:
:
γγ
γ
2
1
0
H
H
H21; γγ are the scatterers’ reflectivities
Generalized Likelihood Ratio Test (GLRT) is required.
The maximization at the numerator selects the highest peak of the beamforming and normalizes to the data vector norm.
( ) ( ) ( ) ( )T
H
H
0
1
2
2H
2
2H
2
2HH
2
2H
22
2H
min
1
maxmaxmaxmax
<>−====
⊥
x
xPx
x
Pxx
x
xpapax
x
xpa
xa
xpappppp
( ) HH1HaaIaaaaIP −=−=
−⊥
w is the additive noise
� A. De Maio, G. Fornaro, A. Pauciullo, Detection of Single Scatterers in Multidimensional SAR Imaging, IEEE Trans. Geosci. Remote
Sens.,Vol. 47, No. 7, pp. 2284-2297, Jul. 2009
Letting p to collect the unknown parameters (height, height/velocity, …) for a
Gaussian model the discrimination between the first two hypotheses is achieved as:
CS and GLRT in SAR Tomography 9
Sequential GLRT with cancellation (GLRT-C)
2
2
1H
2
2
1H
22
2
1H
11
1
1
min
1)(
max
)(max
)(x
xPx
x
xpa
xa
xpa
xp
p
p
⊥
−===L
Estimation from the data of the direction of the first scatterer and evaluation of the coherence L1(x)
Projection of the data in the complement orthogonal to the subspace spanned by the estimated direction
Estimation from the projected data of the direction of the second scatterer and evaluation of the coherence L2(x)
)ˆ(ˆ
ˆˆˆ
11
H111
paa
aaIP
=
−=⊥
2
c
2
2c2
2
c2H
c2
2
)(
)(max)(
2 xpa
xpax
p=L
Advantages: requires only one dimensional maximizations (computational time)
Disadvantages: no super-resolution
212c
1c
ˆ
ˆ
aPa
xPx
⊥
⊥
=
=
� A. Pauciullo, D. Reale, A. De Maio, G. Fornaro, Detection of Double Scatterers in SAR Tomography, IEEE Trans. Geosci.
Remote Sens., Vol. 50, No. 9, pp. 3567-3586, Sept. 2012
CS and GLRT in SAR Tomography
Scheme of the GLRT-C detector
22 )( TL >x 11 )( TL >xFalse False
True True
x0
H
2H
1H
The thresholds T1 and T2 are set based on the fixed values of PFA, on the first
and on the second scatterers
xpap
)(max 1H
1
1p̂
CS and GLRT in SAR Tomography 11
sup-GLRT: an advance over the GLRT-C
2
1H
2
21H
,
2
)(min
),(min
1
1
21
xpPx
xppPx
p
pp
⊥
⊥
−=gML estimation
Advantages: allows super-resolution (i.e. detection of targets below the Rayleigh resolution)
Disadvantages: computationally demanding
xx
xppPxpp
H
2
21H
,
1
),(min
1 21
⊥
−=g
11
0
0
TgH
H
<>
22
1
2
TgH
H
<>
2
21H
,
2
21H
,),(maxarg),(minarg
2121
xppPxxppPxpppp
=⊥
waax
wax
wx
++=
+=
=
2211
11
:
:
:
γγ
γ
2
1
0
H
H
H
2
1H
2
1H )(maxarg)(minarg
11
xpPxxpPxpp
=⊥
sup-GLRT
11 Tg >
� A. Budillon, G. Schirinzi, GLRT Based on Support Estimation for Multiple Scatterers Detection in SAR Tomography, IEEE
Journ. Select. Topic. Appl. Earth Observ. Remote Sens., Vol. 9, No. 3, pp. 1086-1094, March 2016
CS and GLRT in SAR Tomography 12
• 25 TerraSAR-X Spotlight acquisitions over the city of Las Vegas USA (from 2008. 02. 02 to 2009. 04. 06)
• Imaging Mode: HS (High Resolution Spotlight)
• Orbit Direction: Ascending
• Beam Identification: spot_042
• Orbit Number: 3522
• Incidence Angle: 35.8°
• Look Direction: Right
• Azimuth resolution: ∼ 1.1 meters
• Slant Range resolution: ∼ 0.6 meters
• Polarisation Mode: Single
• Polarisation: VV
The TERRASAR-X dataset over Las Vegas
CS and GLRT in SAR Tomography 13
Rayleigh resolution: 40m (slant height) 27m (vertical)
Acquisition distribution of the Las Vegas dataset
CS and GLRT in SAR Tomography
CS and GLRT in SAR Tomography 15
Application to high resolution data (The Mirage Hotel)
CS and GLRT in SAR Tomography 16
CS Detected Single Scatterers
CS and GLRT in SAR Tomography 17
CS Detected Double Scatterers (lower)
CS and GLRT in SAR Tomography 18
CS Detected Double Scatterers (higher)
CS and GLRT in SAR Tomography 19
CS Detected Double Scatterers: height difference
CS and GLRT in SAR Tomography 20
sup-GLRT Detected Single Scatterers
CS and GLRT in SAR Tomography 21
GLRT-C Detected Single Scatterers
CS and GLRT in SAR Tomography 22
sup-GLRT Detected Double Scatterers(lower)
CS and GLRT in SAR Tomography 23
GLRT-C Detected Double Scatterers(lower)
CS and GLRT in SAR Tomography 24
Sup-GLRT Detected Double Scatterers(higher)
CS and GLRT in SAR Tomography 25
GLRT-C Detected Double Scatterers(higher)
CS and GLRT in SAR Tomography 26
sup-GLRT Detected Double Scatterers: height difference
CS and GLRT in SAR Tomography 27
GLRT-C Detected Double Scatterers: height difference
CS and GLRT in SAR Tomography 28
Topographic Difference Between CS and sup-GLRT
Histogram of the difference between the height of double scatterers estimated by CS and sup-GLRT (mask of points detected by sup-GLRT)
CS and GLRT in SAR Tomography 29
Deformation Mean Velocity Difference between CS and sup-GLRT
Histogram of the difference between the deformation mean velocity of double scatterers estimated by CS and sup-GLRT (mask of points detected by sup-GLRT)
CS and GLRT in SAR Tomography 30
CS Post Detection Single Scatterers
CS and GLRT in SAR Tomography 31
Sup-GLRT Single Scatterers
CS and GLRT in SAR Tomography 32
CS Post Detected Double Scatterers(lower)
CS and GLRT in SAR Tomography 33
Sup-GLRT Double Scatterers (lower)
CS and GLRT in SAR Tomography 34
CS Post Detected Double Scatterers(higher)
CS and GLRT in SAR Tomography 35
Sup-GLRT Double Scatterers (higher)
CS and GLRT in SAR Tomography 36
Conclusions and future works
SAR Tomography allows implementing a radar scanner from the space to reconstruct 3D point clouds and monitor deformations.
Next generation VHR sensors (COSMO-SkyMED II Generation, HRWS) will allow further improving this technology for application to urban area and critical infrastructure monitoring.
Super-resolution in SAR tomography allows achieving improvements in the generation of 3D point clouds.
An open issue is the coupling between the reliability of the reconstruction and the computational performances. To this end a key point seems to be associated with the “assimilation” of proper detection schemes within computationally efficient L1 methods.
THANK YOU DANKE