Download pdf - Compressed Sensing Solutions for Airborne Low Frequency SAR · Compressed Sensing Image Formation Signi cant improvement in imaging of bright targets! October 9, 2014 WF3 - CS solutions

Compressed Sensing Solutions for Airborne LowFrequency SAR

Mike E. Davies & Shaun I. KellyThe University of Edinburgh

WF3 - Compressive sensing for radar applications

Overview

Motivation/Challenges

I Motivation: Why Airborne LF SAR?I Challenges:

I Notching on transmitI Radio Frequency Interference (RFI)I Phase errorsI Near field imaging

I Solutions (CS based)

October 9, 2014 WF3 - CS solutions for Airborne LF SAR - Mike E. Davies 2

Motivation

Why use VHF/UHF Spectrum?

I FoliagePenetration (FoPEN) Radar

I Ground PenetrationRadar (GPR)

I Scattering is dependent onwavelength.

X-BAND VHF/UHF


Challenges

Issues which effect the VHF/UHF spectrum

I Interference between SARsystems and radio, televisionand communications systems.

I Radio frequencyinterference (RFI)

I Interference Types:

1. SAR systems can interfere with other spectrum users.2. Other users in the spectrum can interfere with SAR system.


Challenges

Notched LFM on Transmit

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LFM Chirp

Notch

Notched LFM

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Challenges

Standard Image Formation with Notching

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Solution Outline

CS-based componentsI Sparse Image Formation

I Fast forward/back projections

I Compressive Autofocus

I RFI suppression


System model

Notched LFM on Transmit

System model (after dechirping and deskewing):

Y = diag (w)h(X) + N

Y ∈ CM′×N′, N ∈ CM×N , w ∈ RM

I X is the scene reflectivities

I Y is the phase history

I h(·) is the system model without notching

I w is a weighting that models the transmit notching

I N is the RFI and additive noise


Sparse Image Formation SAR

First Ingredient: Sparsity

I The signal/image must be sparse or well approximated by a sparsesignal/image (compressible)

Will be considered later.

Second Ingredient: “Good” Measurements

I Measurement equation Y = h(X)

An approximate sub-sampling of the k-space!

Third Ingredient: Reconstruction Algorithm

I Sparse reconstruction algorithm, e.g. constrained `1 min., greedyalgorithms - OMP, IHT, etc.

Many fast algorithms available if there are fast operators available!



Sparsity in Wavelets?

Image Domain Wavelet Domain

SAR images are not significantly compressible in any basis!



Interaction of Reflectors in a Range CellI Random interference: Speckle dominates images due to many

random reflectors in a range cell inducing multiplicative noisein the reconstructed image - not compressible.

I Coherent interference: Coherent reflectors (often targets ofinterest) whose intensity tend to be much larger thanincoherent reflections - compressible in spatial domain.



Compressed Sensing Image Formation

argminXs

‖Xs‖1 s.t. ‖Y − h(Xs)‖F ≤ ε

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Bright Targets

argminXbg

∥∥Y − h(Xbg + Xs

)∥∥F

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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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Fully Sampled Reference Imagex (m)

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Degradation in background speckle!


Fast SAR Operators

Compressed Sensing Image FormationI LF SAR typically has long apertures and large beam width

making the aperture non-linear and the imaging near field.I Efficient iterative reconstruction requires fast

forward/backward operators:

× Direct Forward/Backward Projection - too slow: O(N3)

× Polar Format Algorithm - far field imaging only× Range Migration Algorithm - flat terrain model and linear

apertureX Fast decimation-based Forward/Backward Projection

Algorithms, e.g. [McCorkle et al. ’96],...


Fast SAR Operators

Decimation-in-phase-history

I Recursive splitting of image anddecimating of phase history.

I O(N2 logN

)operations.

I e.g. [McCorkle et al. 1996],

[Wahl et al. 2008].

phase history SAR image

Decimation-in-image

I Recursive splitting of phase historyand decimating of image.

I O(N2 logN

)operations.

I e.g. [Kelly and D. 2014]

phase history SAR image


Fast SAR Operators

Image Formation Times (seconds)

N BP fast BP fast BP PFA(dec.-in-image) (dec.-in-phase-history)

256 18.26 4.75 4.64 0.90512 140.22 18.77 18.74 3.241024 1120.96 77.18 76.98 13.152048 9052.47 318.43 317.36 69.15

I NFFT algorithm with an interpolation kernel length of 24 samples.

I Time on a single core of 2.5 GHz Intel Xeon processor with N2 element imagesand N2 element phase histories.

I log2 N − log2 64 decomposition stages.


Fast SAR Operators

Pixel-wise Relative Errors

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decimation-in-image

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decimation-in-phase-history

I Images formed using fast decimation-in-image and decimation-in-phase-historyBP algorithms with three decomposition stages.

I Pixel-wise/k-space wise relative errors in the fast BP algorithms with respect tothe BP algorithm.


Phase Calibration (Autofocus)

Phase ErrorsI Inaccuracies in the propagation delay estimates introduce

unknown phase errors, φτek :

φτek≈ ω0τek − ατ

2ek

with, τek - delay error at aperture position kω0 - carrier freq. and α - chirp rate.

I Modified SAR observation model with phase errors

Y = h (X) diag{

ejφ}

I If not corrected, phase errors can defocus targets and degradereconstructed image.



Classical Autofocus

Classical (image based) autofocus assumes far field small aperture model

I System model ∼ fullydetermined and separable:

Y = h(X) diag{

ejφ}≈ AXΨB

I A and B ∼ Fourier

I Autofocus ∼ deconvolution

columns of Ψ, due toa quadratic phase error

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I X is recovered from XΨ using classical autofocus methods, e.g.Map Drift (MD) or Phase Gradient Autofocus (PGA)



Undetermined System Model

Y = A′XΨ

I A′ ∈ CN×S is undetermined, e.g. due to notching

Post-Reconstruction AutofocusI Can XΨ be recovered from Y followed by a

post-reconstruction autofocus?I CS Stable Sparse Recovery [Rudelson, Vershynin ’08]:

S ≥ CKΨKX log4(N)

with original sparsity KX and blurring factor KΨ

Reconstruction quality deteriorates as phase errors increases!



Compressive AutofocusI Better Solution: perform joint reconstruction

minimiseX,d

‖X‖1

subject to ‖Y diag {d} − h(X)‖F ≤ σ

d∗ndn = 1, n = 1, . . . ,N .

I Fast Block-relaxation algorithms via majorisation-minimisationexist [Kelly et al 2012/14]

I No far field/small aperture assumptions

I Theoretical guarantees: open problem



Reconstruction performance versus under-sampling ratio

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−→ increasing phase errors −→

‘◦’ oracle reconstruction, ‘�’ compressive auto-focus, ‘×’ sparse image formation with

post-processing autofocus.



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Figure: LF SAR image formations: (a) was formed using the BPalgorithm; (b) was formed using sparse reconstruction (no autofocus);and (c) was formed using Compressive Autofocus.


Radio Frequency Interference

RFI suppression

I Strong interference from AM/FMtransmitters.

I RFI pre-processing suppression methods:

1. Estimate-and-subtract: estimatethe frequencies and phases of theRFI and then abstract.Computationally expensive and

approximation dependent.

2. Linear filter: minimise RFI usinglinear filter, e.g. LMS filter andWiener filter.Can produce large side lobes.

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Dechirping

After dechirping and deskewing:

I narrowband interferes become concentrated in time and

I spectral notches become notches in time.

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After dechirping and deskewing



Filter-based RFI suppression

Linear RFI Filtered Reconstruction:

X = g(H vec(Y))

H = diag([H1, · · · ,HN′ ])

I g(·) is the filtered back-projection algorithm.

I Hn′ are the Wiener filters for each slow-time position, i.e.

Hn′ = I−Qnn′(Qyn′

+ Qnn′)-1 for Qx = E

[xxH]

I Qyn′ are the covariance matrices of the received signal at eachslow-time position

I Qnn′ are the covariance matrices of the RFI at each slow-timeposition



RFI-aware Sparse Image Formation

Incorporate RFI into the Basis Pursuit Denoising:

X = minimiseX

‖X‖1

subject to ‖Y − h(X)‖QN-1 ≤ ε,

where, ‖A‖Q = vec(A)HQ vec(A)

I QN is full covariance matrix of the RFI and additive noise.

I QN is well approximated using a diagonal matrix so the datafidelity term becomes a weighted Frobenius norm.



RFI-aware Sparse Image Formation Implementation

Estimate Noise Covariance:

Estimate QN using ten “dead-time” measurements.Assume elements of N are independent.

Unconstrained Optimisation:

X = minimiseX

‖X‖1 + λ(‖Y − diag (w)h(X)‖QN-1 − ε)

Approximately solved using thirty iterations of a fast iterative shrinkagethresholding algorithm.

Project onto Domain of h(·):

X← g(h(X))



VHF/UHF SAR simulation Parameters

parameter valuecarrier frequency (ω0/2π) 308 MHz

altitude 7000 mnumber of targets 20

chirp bandwidth (αT/π) 324 MHzstand-off distance 7000 m

number of interferes 80IF bandwidth 60 MHz

aperture length 7000 msignal to noise ratio (SNR) 60 dB

scene radius (L) 75 mnumber of aperture samples 300

signal to interference ratio (SIR) -30 dBtransmit notch centre frequencies 175, 330, 389, 416 and 448 MHz

transmit notch bandwidths 15, 7, 13, 20 and 10 MHz



Reconstructed Images

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Filtered back-projection

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Wiener filtered followed byfiltered back-projection

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RFI-aware sparse image formation

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Range compression RFI-awaresparse image formation


Conclusions

ConclusionsI Iterative CS-based algorithms provide a good solution to LF

SAR image formation with notch on transmit

I Compressive Autofocus can be performed simultaneously

I Receiver RFI suppression easily incorporated using a weightedFrobenius norm.

I The proposed technique is superior to previous approaches asit does not suffer from poor range side lobes and it canaccommodate a wide range of RFI.


References

I S. I. Kelly, G. Rilling, M. Davies, and B. Mulgrew, 2011, ”Iterative imageformation using fast (re/back)-projection for spotlight-mode SAR,” in Proc.IEEE Radar Conf. 2011, pp. 835-840.

I S. I. Kelly, C. Du, G. Rilling and M. Davies, 2012, ”Advanced image formationand processing of partial synthetic aperture radar data,” Signal Processing, IET6 (5), pp. 511-520.

I S. I. Kelly, M. Yaghoobi and M. E. Davies, 2012, ”Auto-focus for under-sampledsynthetic aperture radar,” in Sensor Signal Processing for Defence (SSPD 2012)pp. 1-5.

I S. I. Kelly and M. E. Davies, 2013, ”RFI suppression and sparse image formationfor UWB SAR,” Radar Symposium (IRS), 14th International 2, pp. 655-660.

I S. I. Kelly, M. Yaghoobi and M. E. Davies, 2014, ”Sparsity-based Autofocus forUnder-sampled Synthetic Aperture Radar” to appear in IEEE Trans. Aerospaceand Electronic Systems, 2014.

I S. I. Kelly and M. E. Davies, 2014, ”A Fast Decimation-in-imageBack-projection Algorithm for SAR,” in Proc. IEEE Radar Conf. 2014.
