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UDRC Signal Processing in a Networked Battlespace
Compressive Sensing Applications andDemonstrations:
Synthetic Aperture Radar
Shaun I. KellyThe University of Edinburgh
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UDRC Signal Processing in a Networked Battlespace
Outline
1 SAR Basics
2 Compressed Sensing SAR
3 Other Applications of Sparsity in SAR
4 Compressed Sensing SAR: Example Code
2
UDRC Signal Processing in a Networked Battlespace
Outline
1 SAR Basics
2 Compressed Sensing SAR
3 Other Applications of Sparsity in SAR
4 Compressed Sensing SAR: Example Code
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UDRC Signal Processing in a Networked Battlespace
SAR Data Acquisition
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UDRC Signal Processing in a Networked Battlespace
SAR Data Acquisition
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UDRC Signal Processing in a Networked Battlespace
SAR Data Acquisition
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UDRC Signal Processing in a Networked Battlespace
SAR Data Acquisition
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UDRC Signal Processing in a Networked Battlespace
SAR Data Acquisition
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UDRC Signal Processing in a Networked Battlespace
SAR Data Acquisition
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UDRC Signal Processing in a Networked Battlespace
SAR Data Acquisition
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UDRC Signal Processing in a Networked Battlespace
SAR Data Acquisition
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UDRC Signal Processing in a Networked Battlespace
SAR Data Acquisition
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UDRC Signal Processing in a Networked Battlespace
SAR Data Acquisition
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UDRC Signal Processing in a Networked Battlespace
SAR Data Acquisition
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UDRC Signal Processing in a Networked Battlespace
SAR Data Acquisition
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UDRC Signal Processing in a Networked Battlespace
SAR Data Acquisition
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UDRC Signal Processing in a Networked Battlespace
SAR Data Acquisition
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UDRC Signal Processing in a Networked Battlespace
SAR Data Acquisition
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UDRC Signal Processing in a Networked Battlespace
SAR Data Acquisition
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UDRC Signal Processing in a Networked Battlespace
SAR Data Acquisition
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UDRC Signal Processing in a Networked Battlespace
SAR Data Acquisition
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UDRC Signal Processing in a Networked Battlespace
SAR Data Acquisition
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UDRC Signal Processing in a Networked Battlespace
SAR Data Acquisition
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UDRC Signal Processing in a Networked Battlespace
SAR Data Acquisition
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UDRC Signal Processing in a Networked Battlespace
SAR Data Acquisition
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UDRC Signal Processing in a Networked Battlespace
SAR Data Acquisition
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UDRC Signal Processing in a Networked Battlespace
SAR Data Acquisition
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UDRC Signal Processing in a Networked Battlespace
SAR Data Acquisition
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UDRC Signal Processing in a Networked Battlespace
SAR Data Acquisition
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UDRC Signal Processing in a Networked Battlespace
SAR Data Acquisition
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UDRC Signal Processing in a Networked Battlespace
SAR System Model
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UDRC Signal Processing in a Networked Battlespace
SAR System Model
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UDRC Signal Processing in a Networked Battlespace
SAR System Model — Spatial Fourier Domain
kx
ky
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UDRC Signal Processing in a Networked Battlespace
SAR System Model — Spatial Fourier Domain
kx
ky
kx
ky
kx
ky
kx
ky
kx
ky
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UDRC Signal Processing in a Networked Battlespace
SAR System Model — Spatial Fourier Domain
kx
ky
kx
ky
kx
ky
kx
ky
kx
ky
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UDRC Signal Processing in a Networked Battlespace
SAR System Model — Spatial Fourier Domain
kx
ky
kx
ky
kx
ky
kx
ky
kx
ky
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UDRC Signal Processing in a Networked Battlespace
SAR System Model — Spatial Fourier Domain
kx
ky
kx
ky
kx
ky
kx
ky
kx
ky
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UDRC Signal Processing in a Networked Battlespace
SAR System Model — Spatial Fourier Domain
kx
ky
kx
ky
kx
ky
kx
ky
kx
ky
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UDRC Signal Processing in a Networked Battlespace
SAR system model — Spatial Fourier Domain
kx
ky
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UDRC Signal Processing in a Networked Battlespace
Standard Image Formation
Approximate 2-D Matched Filter
1 Back-projection Algorithm
2 Polar Format Algorithm
3 Range Migration Algorithm
4 Range/Doppler Algorithm
Example SAR Image
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UDRC Signal Processing in a Networked Battlespace
Outline
1 SAR Basics
2 Compressed Sensing SAR
3 Other Applications of Sparsity in SAR
4 Compressed Sensing SAR: Example Code
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UDRC Signal Processing in a Networked Battlespace
Why is Compressed Sensing Interesting for SAR?
Data Reduction
If the phase history could be sampled at less than the Nyquist rate itwould lead to improved storage and transmission capabilities.
Sensor Constraints
Missing radar echoes: interrupted pulses allow other tasks to beperformed (multi-function radar)
Missing frequency bands: the radar band may contain frequencies wherethere is interference or transmission is not allowed
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UDRC Signal Processing in a Networked Battlespace
Aperture Undersampling
Undersampling of K-space
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UDRC Signal Processing in a Networked Battlespace
Aperture Undersampling
Standard Image Formation
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UDRC Signal Processing in a Networked Battlespace
Compressed Sensing of a SAR Phase History
First Ingredient: SparsityThe signal/image x ∈ CN must be sparse or well approximated by a sparse signal/image (compressible):‖x‖0 = K � N
Will be considered later.
Second Ingredient: “Good” MeasurementsMeasurement equation y = SFx(+e), with F the DFT matrix and S ∈ {0, 1}mq×n a subset selection
SHS is the projector on the selected subset, PSF = (SF)H (SF) is the “point spread function”
(more precisely PSF = F−1diag(SHS) and PSFx = PSF ∗ x)
An approximate sub-sampling of the k-space!
Third Ingredient: Reconstruction AlgorithmOptimisation algorithm. ex: constrained `1 minimisation
Greedy algorithm. ex: Orthogonal Matching Pursuit (OMP), CoSaMP, Iterative Hard Thresholding (IHT)
Many fast algorithms available if there are fast operators available!
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UDRC Signal Processing in a Networked Battlespace
Sparsity of SAR Image
Sparsity in Wavelets?
Image Domain Wavelet Domain
SAR images are not significantly compressible in any basis!
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UDRC Signal Processing in a Networked Battlespace
SAR Image Statistics
Interaction of Reflectors in a Range Cell
Random interference: Speckle dominates images due to manyrandom reflectors in a range cell inducing multiplicative noisein the reconstructed image - not compressible.
Coherent interference: Coherent reflectors (often targets ofinterest) whose intensity tend to be much larger thanincoherent reflections - compressible in spatial domain.
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UDRC Signal Processing in a Networked Battlespace
Aperture Undersampling
Compressed Sensing Image Formation
arg. min.xs
‖xs‖1 s.t. ‖y − SFxs‖2 ≤ ε
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Bright Targets
arg. min.xbg
‚‚y − SF`xbg + xs
´‚‚2
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Background Speckle
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UDRC Signal Processing in a Networked Battlespace
Aperture Undersampling
Compressed Sensing Image Formation
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Standard Image Formationx (m)
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Compressed Sensing Image Formation
Significant improvement in imaging of bright targets!
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UDRC Signal Processing in a Networked Battlespace
Aperture Undersampling
Compressed Sensing Image Formation
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Reference Standard Image Formationx (m)
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Compressed Sensing Image Formation
Degradation in background speckle!
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UDRC Signal Processing in a Networked Battlespace
Issues in Compressed Sensing SAR
Topics to Consider:
Fast operators
“Sparsity-parameter” selection
“Off-the-grid” targets
Deterministic undersampling
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UDRC Signal Processing in a Networked Battlespace
Outline
1 SAR Basics
2 Compressed Sensing SAR
3 Other Applications of Sparsity in SAR
4 Compressed Sensing SAR: Example Code
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UDRC Signal Processing in a Networked Battlespace
Phase Calibration (Autofocus)
Motion and propagationerrors produce unknownphase errors.
Each aperture position hasa phase error.
Without a sparsityassumption the problemill-posed.
kx
ky
eθ0
eθ1
eθN
...
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UDRC Signal Processing in a Networked Battlespace
Ground Moving Target Indication
Targets may have asignificant velocity.
Target velocities can causetarget displacement and/orblurring.
Without a sparsityassumption on the numberof moving targets, manyantennas and a largeamount of apertureoversampling is required.
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UDRC Signal Processing in a Networked Battlespace
3-D Imaging
Multi-pass/2-D apertureis required.
Meeting the Nyquistsampling raterequirement for a 2-Daperture is typically notpossible.
Scattering occurs atpropagation mediumboundaries so 3-Dimages are sparse.
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UDRC Signal Processing in a Networked Battlespace
Outline
1 SAR Basics
2 Compressed Sensing SAR
3 Other Applications of Sparsity in SAR
4 Compressed Sensing SAR: Example Code
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UDRC Signal Processing in a Networked Battlespace
Compressed Sensing SAR: Example Code
simulation cs sar.m
Main function ofexample code.
Range/Doppleralgorithm foroperators.
Random apertureundersampling.
simple sar simulator.m
Mono-staticstrip-map SAR.
Returns are delayedand scaled versionsof a reference chirpsignal.
Dechirp-on-receive.
fista.m
Implementation ofthe Fast IterativeShrinkage-ThresholdingAlgorithm (Beck etal. 2009).
Optimal first ordermethod.
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UDRC Signal Processing in a Networked Battlespace
Compressed Sensing SAR: Example Code
Results
Reference StandardImage Formation
StandardImage Formation
Compressed SensingImage Formation
Things to Experiment WithUse a different solver.
Change the λ parameter.
Change the amount ofundersampling.
Use different operators.
Use a more complicated SARsimulator.
Add in clutter.
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UDRC Signal Processing in a Networked Battlespace
Thank you for your attention!Questions?
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