compressive sensing applications and demonstrations ... · compressive sensing applications and...

UDRC Signal Processing in a Networked Battlespace

Compressive Sensing Applications andDemonstrations:

Synthetic Aperture Radar

Shaun I. KellyThe University of Edinburgh

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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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Outline

1 SAR Basics




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SAR Data Acquisition

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SAR System Model

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SAR System Model — Spatial Fourier Domain

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SAR System Model — Spatial Fourier Domain

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SAR system model — Spatial Fourier Domain

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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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Outline

1 SAR Basics




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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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Aperture Undersampling

Undersampling of K-space

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Standard Image Formation

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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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Sparsity of SAR Image

Sparsity in Wavelets?

Image Domain Wavelet Domain

SAR images are not significantly compressible in any basis!

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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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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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Standard Image Formationx (m)

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Significant improvement in imaging of bright targets!

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Reference Standard Image Formationx (m)

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Degradation in background speckle!

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Issues in Compressed Sensing SAR

Topics to Consider:

Fast operators

“Sparsity-parameter” selection

“Off-the-grid” targets

Deterministic undersampling

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Outline

1 SAR Basics




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Phase Calibration (Autofocus)

Motion and propagationerrors produce unknownphase errors.

Each aperture position hasa phase error.

Without a sparsityassumption the problemill-posed.

kx

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eθ0

eθ1

eθN

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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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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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Outline

1 SAR Basics




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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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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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Thank you for your attention!Questions?

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