Cosmic String Enhancement in AS BP and BP+ Reconstruction What

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Prior Information : string signal follows GGD in wavelet space. • GGD scale and .... y = ΦΨα + n,. 0. 0.05. 0.1. 0.15. 0.2. 0.25. 100 random Gaussians. 256×256.
Compressed sensing imaging techniques for aperture synthesis by radio interferometry L.

1,2 Jacques ,

Y.

1,3 Wiaux ,

G.

1 Puy

and P.

1 Vandergheynst

1

Institute of Electrical Engineering, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland 2 Communications and Remote Sensing Laboratory, Université catholique de Louvain (UCL), B-1348 Louvain-la-Neuve, Belgium 3 Centre for Particle Physics and Phenomenology, Université catholique de Louvain (UCL), B-1348 Louvain-la-Neuve, Belgium

What is Aperture Synthesis ?

Radio sources

BP and

+ BP

Cosmic String Enhancement in AS

Reconstruction

Compressed Sensing Model : Fourier Acquisition e p co

ts c A

a s a

r e g lar

y = ΦI = SF I,

es l e t

m measurements

!ω !⊥

B

E1 (!ω , t)

Iω! (x, y)

⊥ ˆ ! Iω! (B )

Iˆω! (B ) =

!

baseline

Fourier transform

Time correlation ! " N 2



using N telescopes,



and baselines undergo Earth rotation !



!⊥

Context similar to Magnetic Resonance Imaging MRI



We may use Basis Pursuit [2] : (BP)

or, if positive image (additional prior)

αest = arg min !u!1 s.t. y = ΦΨu, Ψu ! 0 (BP ) +

u



or, noisy version :



Solver : Proximal Methods and Douglas-Rachford Splitting [3]

Each telescope pair = one elliptic path

y = ΦΨα + n, ni ∼ N (0, σ 2 ), y = ΦΨu → "y − ΦΨu"2 ! !

0

100 random Gaussians 256×256

4 - 20m

Telescope illumination

4

6

8

String signal

10

12

14

16

(its gradient here)

Very low SNR for string signal



String signals not yet observed but simulated [4]



Prior Information : string signal follows GGD in wavelet space



(i.e. low string tension)

: -30 dB !

scale and shape parameters deduced in steerable wavelets [5] ! ! sj πj (uw ) ∼ exp !uw /ρbj ! , with w = {j, θk , p$i }

Reconstruction : Statistical BP DeNoise (with some sj < 1 !)

arg min !u!S s.t. !W y − W ΦΨu!2 ! !, u

Other configurations : • Very Large Array VLA • Square Kilometer Array SKA

“Visibility”

2



BP+, m = 10%, 13.44 dB

CLEAN, m = 10%, 11.84 dB

1500

Laboratory to test cosmological models

(acknowledgements to M. J. Fadili)

u

1000



• GGD

Simulations : random interferometer, Ψ = Dirac, 1.8°×1.8°

500

Gaussian Noise

u



Arcminute Microkelvin Imager AMI

v

−500

αest = arg min !u!1 s.t. y = ΦΨu

possible pairing (visibilities)

Example :

+

m×N 2



∗ !E1 E2 "∆t

String tension ρ

M = ST S

S∈R

Van Cittert Zernike Theorem :

! B

with I = Ψα sparse in Ψ

Planar approximation

E2 (!ω , t)

=

Cosmic Microwave Background (CMB) signal

Visibility Selection

Sensing matrix

y ∈ Cm



with !u!S =

! w

(SBP! )

sj

|uw /ρbj | , ε2 = 99th percentile χ2(2m)

W = whitening of the Gaussian Noise (known spectrum),

Problem : In matrix notations, find I ∈ R

d = BI, Dirty map

R

N 2 ×N 2

Circulant matrix, i.e. convolution

N2

0

0.05

0.1

0.15

0.2

50

with B = F M F 2

N ×N

2

Diagonal matrix, i.e. the visibility mask

0

0.05

Fourier basis, ˆ = Fu i.e. u



CLEAN is a (γ damped) Matching Pursuit in the Dictionary B Other methods : Multi-scale CLEAN, MR CLEAN, MEM, ...

• •

0.2

0.25

0

0.05

0.1

0.15

0.2

0.25

Solver : re-weighted !1with SPGL1 toolbox

=

4.5

m/N 2

SBPε, m = 20%, 4.3 dB

40

BP+

30

CLEAN BP

SBP BP CLEAN

4 3.5

20

3 2.5

BP SNR controlled by the compressibility of I

10

2

4

6

8

(reconstruction gradient)

0

5

10

Conclusion : •

0.15

Average over 30 simulations

 Assume I sparse in space, i.e. in the canonical (Dirac) basis.



0.1

Coverage



R

0.25

from

SNR (dB)



(Högbom, 1974 [1]) :

SNR (dB)

CLEAN Mathematical Model



CS is a flexible framework for image reconstruction from radio-interferometric data through convex optimization. The inclusion of prior knowledge on the signal under scrutiny improve the quality of signal reconstruction. In progress : control of the actual visibility coverage, inclusion of TV sparsity term, mosaicking.

15 20 Coverage (%)

25

30

References :

10

Average over 30 simulations

2

(comparison between gradients)

1.5

5

10

15 20 Coverage (%)

25

30

References :

[4] Fraisse A. A. et al, 2008, Phys. Rev. D, 78, 043535 [1] Högbom J. A., 1974, Astron. Astrophys. Suppl., 15, 417 [2] Candès E. J., Tao T., 2006, IEEE Trans. Inform. Theory, 52, 489 [5] Hammond D. K., Wiaux Y., Vandergheynst P., arXiv:0811.1267v1 [3] Combettes, P.L.; Pesquet, J.-C., 2007, IEEE JSTSP, 1(4), 564