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. 256Ã256.
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