Radio Aperture synthesis as a practical example of sparse ... · My work (Part I): Planetary radio...

CEA - Irfu 2nd INFIERI - Session 8 - July 23rd 2014

Julien Girard

Radio Aperture synthesis as a practical example of sparse signal reconstruction

CEA Saclay - Cosmostat

J.-L. Starck, J. Bobin, H. Garsden, S. Corbel, C. Tasse, A. Woiselle


My work (Part I): Planetary radio emissions

May 2013 - PhD at Observatoire de Paris at the LESIA - pôle plasma

● Jupiter




● Jupiter- Auroral cyclotron emission ( λ ~ Dm )

f < 40 MHz

f

t





f < 40 MHz

f

t



- Synchrotron emission ( λ ~ cm-dm-m )f = 100 MHz - 10 GHz

e-!

Girard et al., SF2A 2012 and A&A in prep. 2014



f < 40 MHz

f

t




e-!


R

11/11/2012 VLA 1.4 GHz!



f < 40 MHz

f

t



● Saturn


e-!


R

11/11/2012 VLA 1.4 GHz!



f < 40 MHz

f

t



● Saturn- Detection of Lightning f = broadband


e-!


R

11/11/2012 VLA 1.4 GHz!



f < 40 MHz

f

t



● Saturn

- Measurements of the water abundance ( λ ~ dm-m )f = 100-200 MHz

- Detection of Lightning f = broadband


e-!


R

11/11/2012 VLA 1.4 GHz!



f < 40 MHz

f

t



● Exoplanetary systems

● Saturn




e-!


R

11/11/2012 VLA 1.4 GHz!



f < 40 MHz

f

t




● Saturn



- Detection and characterization in radio( λ ~ dm-m? )


e-!


R

11/11/2012 VLA 1.4 GHz!



f < 40 MHz

f

t




● Saturn



- Detection and characterization in radio( λ ~ dm-m? )


e-!


R

11/11/2012 VLA 1.4 GHz!


Mostly using LOFAR


Outline

● Basics/recalls of Radio antennas & interferometry

● Imaging interferometric data

● Sparse reconstruction of interferometric data


Antennas basics


Antennas basics

● Measures electric fields


Antennas basics



Antennas basics


● Measures wave polarization


Antennas basics



● Sensitivity depends on effective collecting area, quality of receiver...

● Angular resolution depends on aperture size (+ diffraction)


Antennas basics






Antennas basics





Point source

Point Spread Function


Antennas basics





Point source



Antennas basics





Point source



Antennas basics





Point source



Antennas basics

Visible'λ'='500'nm'

D'='15'cm' 0,8’’%

D'='50'cm' 0,25’’%D'=1'm'

(Meudon)' 0,12’’%

D='8'm'(Subaru)'

0,015’’%%(15%mas)%

Infrared'λ'='10'µm'

Radio'HF'λ'='10'mm'

Radio'BF'λ'='1'm'

D=%3%m% D=%3%km% D=%300%km%%

D=%10%m% %D=%10%km% D=%1000%km%

D=%21%m% D=%21%km% D=%2100%km%

D=%168%m% D=%168%km% D=%16800%km%





Point source


Green Bank Transit Telescope (100 m)


November 15th, 1988Green Bank Transit Telescope (100 m)




Arrays of antennas I

�✓ / �

B12


Arecibo


�✓ / �

B12



�✓ / �

B12



�✓ / �

B12

→ Antenna arrays



�✓ / �

B12

→ Antenna arrays

GMRT (Pune, Inde)30 paraboles de 45 mBase max: 25 kmλ~1m, fmin = 153 MHzA ~50000 m2

Westerbork(ASTRON, Pays-Bas)14 paraboles de 6mBase max: 2.7 km λ~10cm – 1mA ~400 m2

VLA (NRAO, Nouveau Mexique)

27 paraboles de 25 mBase max: 36 km

λ~1cm – 1m, fmin = 74 MHzA ~14000 m2





λ~1cm – 1m, fmin = 74 MHzA ~14000 m2

B12

�✓ / �

B12

GMRT VLA



�✓ / �

B12

→ Antenna arrays





λ~1cm – 1m, fmin = 74 MHzA ~14000 m2





λ~1cm – 1m, fmin = 74 MHzA ~14000 m2

B12

�✓ / �

B12

GMRT VLA






λ~1cm – 1m, fmin = 74 MHzA ~14000 m2


SMA (USA – Taïwan)‏Hawaïi8 antennes de 6 mBase max: 0.5 km λ~0.5mm

Plateau de Bure (IRAM, ‏France)6 antennes de 15m Base max: ~1 kmλ~1mm


ALMA (Atacama Large Millimeter Array)

• Chili: 5000m d’altitude• 50 antennes de 12m• f = [30-900GHz]• λ= [1 cm-0.3mm]• S = 5600m²• lignes de base ⇒ 14km• résolution ⇒ 0.007” @ 0.4mm (750 GHz)

⇒ Spectro-imagerie de très haute résolution dans le mm/sub-mm

ALMA (Atacama Large Millimeter Array)

• Chili: 5000m d’altitude• 50 antennes de 12m• f = [30-900GHz]• λ= [1 cm-0.3mm]• S = 5600m²• lignes de base ⇒ 14km• résolution ⇒ 0.007” @ 0.4mm (750 GHz)

⇒ Spectro-imagerie de très haute résolution dans le mm/sub-mm

ALMA (ESO/NRAO/NAOJ, Chile)

66 Antennas: - 54 of 12 m - 12 of 7 m

Base max: 16 km

λ ~ 100 µm - 1 mm



The pastEffelsberg dish (diam. 100 m)


The future:



The future:



Arrays of antennas II

Parabolas!Directive antennas!

VLA!

mechanically steerable!




VLA!





VLA!


Electronically steerable!

Dipolar, wide FOV antenna!

~ isotropic antennas!

zenith!

Element beam pattern!!




VLA!





zenith!


N antenna compound!Beam pattern!




VLA!


•  Interferometer: Pair to pair correlation of antenna signals!•  Phased array: Phased sum of antenna signals!

Arrays!




zenith!


N antenna compound!Beam pattern!


LOFAR: the LOw Frequency ARray

● Giant digital & multi-purpose radio telescope distributed across Europe ● Radio interferometer composed of ∼48 phased arrays (stations) ● Working bands: LBA 30-80 MHz & HBA 120-240 MHz ● Improved angular (arcsec), temporal (µs), spectral (kHz) resolutions ● High sensitivity (~mJy) 1 Jy = 10-26 W.m-2.Hz-1 NL Station!


F = 50 MHz -10-30 GHzλ ~ 6 m ~ 1.2 cmCollecting area = ~1 km2

Base max = >3000 km


My work (Part II) : instrumentation at low frequencies (1/2)

Nançay

3 Gbits/s

● Development of the « LOFAR Super Station » in Nançay = NenuFAR

- Major extension of LOFAR (10-90 MHz) - New stand-alone instrument in Nançay (96 →1824 antennas)

Very high instantaneous sensitivity (70-80% de LOFAR, ~2x LOFAR core)


My work (Part II) : instrumentation at low frequencies (1/2)

Nançay

3 Gbits/s

● Development of the « LOFAR Super Station » in Nançay = NenuFAR

- Major extension of LOFAR (10-90 MHz) - New stand-alone instrument in Nançay (96 →1824 antennas)

Very high instantaneous sensitivity (70-80% de LOFAR, ~2x LOFAR core)

~400m


My work (Part II) : instrumentation at low frequencies (2/2)● EM simulation & Multi-scale optimization, tests, commissionning

- elementary antenna




x (m)

- topology (and phasing strategy) of sub-arrays

y (m

)




x (m)

- topology (and phasing strategy) of sub-arrays

y (m

)

• Thèse, 2013; Girard et al., CRAS, 2012; Zarka, Girard et al., SF2A, 2012; Girard et al., en rév. 2014

• Collab. ASTRON/NL, IRA/Kharkov, Subatech/Nantes, NRAO/USA

- global distribution (and cabling) of sub-arrays

% d

u m

axim

um

% d

u m

axim

um

Déc

linai

son (

°)

Déc

linai

son (

°)

Ascension droite (°) Ascension droite (°)

% d

u m

axim

um

% d

u m

axim

um

Déc

linai

son (

°)

Déc

linai

son (

°)

Ascension droite (°) Ascension droite (°)

couverture (u,v)

radiale azimuthale


Two-element interferometer

1 baseline b (t,ν) + 1 direction s =

1 spatial frequency of the sky brightness

through the measurement of the fringe contrast

or fringe « visibility»

complex visibility Vν


Multi-element interferometer I

Nant = 4

Nant = 3

Nant = 2

PSF of the interferometer


Multi-element interferometer II

N

N(N � 1)

2

antennas/telescopes

independent baselines

1 projected baseline = 1 sample in the Fourier « u,v » plane

VLA

Lm



N

N(N � 1)

2

antennas/telescopes



VLA

Lm

u

v(u,v) plane

sampling



N

N(N � 1)

2

antennas/telescopes



VLA

Lm

u

v(u,v) plane

sampling



N

N(N � 1)

2

antennas/telescopes



VLA

Lm

u

v(u,v) plane

sampling


Outline





Sampling with a Multi-element interferometer

Illustra8on of the PSFFourier (u,v) coverage


Inversion 1/2

Stolen from D. Wilner presentation


Inversion 1/2



Inversion 1/2



Inversion 1/2



Inversion 1/2



Inversion 1/2



Inversion 2/2Fourier domain

Snapshot (u,v) coverage

discontinuous sampling of the (Fourier) (u,v) plane





Image domain

Reconstructed image = « true » sky * PSF =

~FT-‐1

Dirty image





Image domain


~FT-‐1

Dirty image





}Usually: ● Poor Fourier sampling● Not a true FT relation● Simplifying hypotheses don’t hold

insufficient samples, redundancy

non-coplanar interferometer

small field approximation

difficult inversion problem

+ all other Direction Dependent Effects (DDE) (Beam pattern, ionosphere...)

Image domain


~FT-‐1

Dirty image

CEA - Irfu

Animation ionosphere


A classical deconvolution method

Stolen from D. Wilner presentation held at NRAO 14th synthesis imaging workshop



CLEAN

● Iterative PSF subtraction from the dirty map

● Optimal on point sources




CLEAN




Basic Algorithminitialize: i) residual map = dirty map ii) Clean Component list = 0



CLEAN




Basic Algorithminitialize: i) residual map = dirty map ii) Clean Component list = 0

1. identify the highest peak in the residual map as a point source

2. subtract a fraction of this peak from the residual map using a scaled dirty beam

s(l,m) x gain

3. add this point source location and amplitude to the Clean Component list

4. goto step 1 (an iteration) unless stopping criterion reached


CLEAN RUNNING



CLEAN RUNNING



CLEAN RUNNING



CLEAN RUNNING



CLEAN RUNNING



CLEAN RUNNING



Many others deconvolution algorithms

- Multi-scale CLEAN (Cornwell, 2008)

CLEAN (Högbom, 1974) has known many evolutions

- Clark CLEAN (Clark, 1980)

- Cotton-Schwab CLEAN (Schwab, 1984)

- SDI (Steer, Dewdney, Ito, 1984)

- Multi-resolution CLEAN (Wakker & Schwarz, 1988)

- Maximum Entropy Method (MEM, Skilling & Gull, 1984)

- Multi-scale Multifrequency synthesis (Rau, 2011)

- Multi-frequency CLEAN (Sault & Wieringa, 1994)

- Wavelet CLEAN (Starck & Bijaoui, 1994)

...


Outline




* E. Candès and T. Tao, “Near Optimal Signal Recovery From Random Projections: Universal Encoding Strategies? “, IEEE Trans. on Information Theory, 52, pp 5406-5425, 2006. * D. Donoho, “Compressed Sensing”, IEEE Trans. on Information Theory, 52(4), pp. 1289-1306, April 2006. * E. Candès, J. Romberg and T. Tao, “Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information”, IEEE Trans. on Information Theory, 52(2) pp. 489 - 509, Feb. 2006.

“Signals with exactly K components different from zero can be recovered perfectly from ~K log N incoherent measurements”

A non linear sampling theorem

Compressed Sensing




Compressed Sensing




- Underdetermined system - Sparsity of x - Incoherence ( random)

Assumption on the signal x:

Compressed Sensing




Reconstruction based on non-linear algorithms

- Underdetermined system - Sparsity of x - Incoherence ( random)

Assumption on the signal x:

Compressed Sensing


General idea

slides by Redouane LGUENSAT


General idea



General idea



Sparsity



Sparsity



Sparsity



Sparsity



Sparsity



Proximal methods



Proximal methods



Proximal methods



Sparse representation with wavelets

Most xm are small

Sparsity =

Small number of essential xm =

WT


Wavelet transform (DWT)


Curvelet transform

J.L. Starck, E. Candès and D. Donoho, 2003


Wavelet vs. Curvelet : sparse representation

Using Wavelets Using Curvelets

J.L. Starck, E. Candès and D. Donoho, 2003


Wavelet vs. Direct : sparse representation

J. Rapin PhD thesis

Original image



J. Rapin PhD thesis

Original image

5% of the highest coeffDirect space



J. Rapin PhD thesis

Original image

5% of the highest coeffWavelet space




J. Rapin PhD thesis

Original image

5% of the highest coeffWavelet space


Approximate image


For more information on Compressed Sensing / Sparsity

Check-out http://ada7.cosmostat.org Presentations,Tutorials...

http://ada7.cosmostat.org

Radio-Interferometry Image Reconstruction

==> See (McEwen et al, 2011; Wenger et al, 2010; Wiaux et al, 2009; Cornwell et al, 2009; Suskimo, 2009; Feng et al, 2011; Garsden, Starck and Corbel, 2013).

Measurement System

FOURIER

H

X

Y = HX + N

Compressed Sensing Theory and Radio-Interferometry {

Visibilities

Sky

VLA

Radio interferometry & Compressed Sensing

[McEwen et al, 2011; Wenger et al, 2010; Wiaux et al, 2009; Cornwell et al, 2009; Suskimo, 2009; Feng et al, 2011; Garsden, Starck and Corbel, 2013]

Visibilities

Sky

VLA

Sparsee.g. Wavelets Tr.



Visibilities

Sky

VLA

Measurement matrix (Fourier + Sampling)

H

FOURIER

{Sparsee.g. Wavelets Tr.



Y = HX + N

Visibilities

Sky

VLA


H

FOURIER




Y = HX + N

Visibilities

Sky

min��p

p subject to �Y �H��2 � ⇥

VLA


H

FOURIER





Sparse Recovery: Example

Test Image

FFT

Apply mask + Noise to FFT Sampling/Sensing

Sparse Recovery

Starting imageInverse FFT Dirty Map

True Sky Mask

Sparse reconstruction


Compressed Sensing & LOFAR

How does CS work on LOFAR data ? !How good is the photometry ? !How well does it work on extended sources ? !How good is the reconstructed image resolution ?


LOFAR Specific Compressed Sensing Imaging



HLOFAR operator much more complicated than simple FT § Visibilities are in 3-D. Need W-Projection § Rotation of the Earth, changing orientations -> time and direction

dependent effects (DDE). Need A-projection. § Points in (U,V) space sparsely populated and non-equispaced.



HLOFAR operator much more complicated than simple FT § Visibilities are in 3-D. Need W-Projection § Rotation of the Earth, changing orientations -> time and direction

dependent effects (DDE). Need A-projection. § Points in (U,V) space sparsely populated and non-equispaced.

- Use directly the HLOFAR implementation in the LOFAR pipeline developed by C. Tasse - Chose wavelets (undecimated isotropic wavelets) for sparsifying the solution. - Use minimization software developed/used at Saclay (FISTA).

Strategy:

Experiment #1: Photometry

Right Ascension (J2000)

Decli

natio

n (J2

000)

+50°

Jy/beam

+52°

+54°

+56°

+48°13h50m00s14h0m00s10m00s20m00s30m00s

9000

7500

6000

4500

3000

1500

0

-1500

Dirty map

Random flux densities

[1-10000] Jy

Large field of view

8°x8° centered at zenith

10x10 grid of point sources

Widefield imaging

- CLEAN

- Sparse reconstruction

Simulated dataset

➢ recover flux densities from model images

==> Sparse recovery provides similar results to CLEAN

0 2000 4000 6000 8000 10000Input Flux density(Jy)

0

2000

4000

6000

8000

10000

12000O

utpu

tFl

uxde

nsity

(Jy)

Point source reconstructionCLEANSparse Rec.

0 2000 4000 6000 8000 10000Input Flux density(Jy)

10− 1100101102103

Abso

lute

Erro

r(Jy

)

Experiment #1: Photometry

Experiment #2: Angular separation

- Imaging with CLEAN and Sparse recovery

- Filled with simulated data

* Source angular separation = from 10’’ to 5’

* Two point sources of 1 Jy at zenith

* Injected noise corresponding to SNR = 2.7, 8.9, 16 and 2000 (noiseless)

- Simulated LOFAR dataset

* Core stations only (N=24)

➢ restricts artificially the resolution to ~2-3 arcminutes

* Radial cut in the Fourier (u,v) plane at Ruv=1.6 kλ

* ΔT=1h - ΔF=195 KHz - F=150 MHz

CLEAN Sparse reconstruction

5’


CLEAN Sparse reconstruction

5’


0

5

10

15

Jy/Beam

δθ=1’ δθ=2’ δθ=3’ δθ=4’

Sparse recovery

CLEAN CLEAN beam = 3.2’x2.5’

● Sparse Recovery resolution improved by at least 2 compared the CLEAN beam.

● Recovered « sub-beam » sources have correct fluxes (~2% error) & positions

Noiseless dataExperiment #2: Angular separation

● On noisy data ➢ (rough) measurement of the source separability angle.

Rayleigh criterion

23% drop

Separated sources when decrease > 23%

Effective source separability vs. SNR

Angu

lar s

epar

atio

n (°)

SNR

CLEANSparse reconstruction

==> Sparse reconstruction: angular separation improved by 2 for SNR > 10, and converges to CLEAN resolution at low SNR regimes.


● VLA 21-cm image of W50 + empty simulated LOFAR dataset

● Set to an arbitrary flux scale and converted to visibilities (AWimager)

VLA @ 21 cm

Experiment #3: Extended source

Model image



VLA @ 21 cm

(u,v) coverage

u

v

FFT +

(u,v) Sampling


Model image



VLA @ 21 cm

(u,v) coverage

u

v

FFT +

(u,v) Sampling

Dirty image


Model image

CLEAN Multiscale CLEAN Sparse Reconstruction

RMS error = 3.50

RMS error = 3.28

RMS error = 0.76

Experiment #3: Extended sourceR

econ

stru

cted

Erro

r im

age

● Using CLEAN, Multiscale CLEAN and Sparse reconstruction

RMS error = 3.50

RMS error = 3.28

RMS error = 0.76

Experiment #3: Extended sourceR

econ

stru

cted

Erro

r im

age

CLEAN Multiscale CLEAN Sparse Reconstruction

● Using CLEAN, Multiscale CLEAN and Sparse reconstruction

Dec

linat

ion

CLEAN

Total Flux density = 9393 JyTotal Flux density = 9393 JyExperiment #4: Real data Cygnus A

F = 151 MHz - ΔF = 195 kHzΔT = 6 Hr36 LOFAR Stations

(dataset courtesy of John Mckean)

● Threshold = 0.5 mJy

● Pixel = 1’‘ size = 512 x 512

● Weighting = super uniform

Right Ascension

Restored imageTotal Flux density = 9393 Jy

ResidualsResidual std-dev = 2,65 Jy/beam

Right Ascension

Dec

linat

ion

Multi-Scale CLEAN


Residual std-dev = 0,26 Jy/beam

Restored image

ResidualsTotal Flux density = 10553 Jy



● Pixel = 1’‘ size = 512 x 512


● Scales = [0, 5, 10, 15, 20] pixels

Cygnus A

Right Ascension

Dec

linat

ion

Multi-Scale CLEAN



Restored image




● Pixel = 1’‘ size = 512 x 512


● Scales = [0, 5, 10, 15, 20] pixels

Cygnus A

Right Ascension

Dec

linat

ion


Sparse Reconstruction


● Pixel = 1’‘ size = 512 x 512




Restored image


● Scales = 7 wavelets scales

● Minimization algorithm: FISTA Fast Iterative Shrinkage-Thresholding Algorithm

Cygnus A

Right Ascension

Dec

linat

ion


Sparse Reconstruction


● Pixel = 1’‘ size = 512 x 512




Restored image


● Scales = 7 wavelets scales

● Minimization algorithm: FISTA Fast Iterative Shrinkage-Thresholding Algorithm

Cygnus A

+ 40° 43m00s

30s

44m00s

30s

45m00sDe

c(J2

000)

19h59m24s27s30s33s

RA (J2000)0

25

50

75

100

125

150

175

200

225

250

Jy/be

am

Colorscale: reconstructed 512512 image of Cygnus A at 151 MHz (with resolution 2.8” and a pixel size of 1”). Contours levels are![1,2,3,4,5,6,9,13,17,21,25,30,35,37,40] Jy/Beam from a 327.5 MHz Cyg A VLA image (Project AK570) at 2.5” angular resolution and a pixel size of 0.5”. Most of the recovered features in the CS image correspond to real structures observed at higher frequencies.


Conclusions

ü Sparse recovery is a totally new imaging method for LOFAR and other modern interferometers. !

ü Experimental results are good § Photometry: similar to CLEAN on point sources. § ResoluNon: improved by a factor 2 for SNR > 10. § Extended objects reconstrucNon much beSer than CLEAN and MulNscale CLEAN. § Improved image quality (RMS beSer by factor 5 compared to CLEAN) !

ü Will conNnue to develop (CLEAN has had 40 years) !

ü Paper § H. Garsden, J-‐L. Starck, S. Corbel et al., "Compressed sensing imaging

reconstrucNon for the LOFAR Radio Telescope", Proceedings of SPIE Vol. 8833 (2013)

§ Journal Paper submiSed to A&A (arXiv: 1406.7242)

VLA

Multi-channel Sparse reconstruction

VLA


Visibilities

…

…

VLA


Visibilities

…

…


H

FOURIER

{

VLA


Visibilities

…

…


H

FOURIER

{Snapshot imaging: - Snapshot = Bad (u,v) coverage = bad PSF ⟶ deconvolution problem - Noise: limited by noise in a single timeshot

VLA


Visibilities

…

…


H

FOURIER

{Snapshot imaging: - Snapshot = Bad (u,v) coverage = bad PSF ⟶ deconvolution problem - Noise: limited by noise in a single timeshot

Solution? - Project the signal in a dictionary where temporal signals are sparse - Peal all extragalactic non-variable radio sources - Performing 3D FFTs of the visibilities (as in T-awimager, Tasse et al.)

VLA


Visibilities

…

…


H

FOURIER

{Image cube

xy

Snapshot imaging: - Snapshot = Bad (u,v) coverage = bad PSF ⟶ deconvolution problem - Noise: limited by noise in a single timeshot

Solution? - Project the signal in a dictionary where temporal signals are sparse - Peal all extragalactic non-variable radio sources - Performing 3D FFTs of the visibilities (as in T-awimager, Tasse et al.)


⌧xy


⌧xy

Fast

Slow

Slice 1

(Courtesy of C. Tasse)


⌧xy

Fast

Slow

Slice 1


y

x

Slice 21


⌧xy

Fast

Slow

Slice 1


Slice 43

x

y

y

x

Slice 21


Bibliography

Sparse Image and signal processing

Kraus - Radio astronomy

Bracewell - the Fourier Transform and its applications

Taylor et al. Synthesis Imaging in Radio

astronomy II

+ tons of articles...

Scale 1 Scale 2 Scale 3 Scale 4 Scale 5

h h h h h

WT

ISOTROPIC UNDECIMATED WAVELET TRANSFORM

€

I(k, l) = cJ ,k,l + w j,k,lj=1

J∑

The STARLET Transform Isotropic Undecimated Wavelet Transform (a trous algorithm)

€

ϕ = B3 − spline, 12ψ(x

2) =

12ϕ( x

2) −ϕ(x)

h = [1,4,6,4,1]/16, g =δ - h, ˜ h = ˜ g =δ


Deconvolution

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Radio Aperture synthesis as a practical example of sparse ... · My work (Part I): Planetary radio...

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