Array signal processing for radio-astronomy imaging and ... · Array signal processing for...
Transcript of Array signal processing for radio-astronomy imaging and ... · Array signal processing for...
Page 1April 2009
Array signal processing for radio-astronomy imagingand future radio telescopes
Amir LeshemSchool of Engineering Faculty of EEMCSBar-Ilan University Delft University of Technology
Joint work with Alle-Jan van der Veen, Chen Ben-David, Ronny Levanda, A. Boonstra
Some data and simulations were provided by S. van der Toll, S. Wijnholds, Huib Intema, Huub Rottgering and T.
Clarke
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What is parametric imaging ?
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ReferencesC. Ben David and A. Leshem. “Parametric high resolution techniques for radio astronomical image formation”. IEEE JSTSP October 2008.
A. Leshem and A-J. van der Veen. “Radio-astronomical imaging in the presence of strong radio interference”. IEEE trans. on Information Theory. Special issue on information theoretic imaging, no. 5, pages 1730-1747. August 2000.
A. Leshem, A-J. van der Veen and A. J. Boonstra. “Multichannel interference mitigation techniques in radio-astronomy”. The Astrophysical Journal supplements. Volume 131, 355-373, November 2000.
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IEEE Journal of selected topics in Signal Processing – October 2008
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Presentation overview
The matrix measurement equation– Planar array– Non co-planar array– Polarization– Calibration*– Spatially varying calibration*
Some ideas related to parametric imaging– LS-MVI Acceleration and hardware implementation– Joint imaging and calibration through SDP– L1 optimization*
ExamplesConclusions
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High dynamic range1 dimensional images – for simplicity
Dirty images only!
Theoretical dynamic range within the image 1,000,000:1
Weakest source is only 10x noise RMSE!
Sources are extended off the grid
One source is moved through FOV
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High dynamic range - Array beam
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High dynamic range - sources
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High dynamic range - Dirty images
Three 1-D dirty images:Classical, AAR and AAR with 3% RMSE calibration error on
each array element in each direction
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Imaging – The correlation process
1Signals are stacked into vectors ( ) ( ),..., ( )
Location of antenna at time is ( )Baseline between antennas , is ( ) ( )
T
p
i
i j
t x t x t
i t ti j t t
⎡ ⎤= ⎣ ⎦
−
x
rr r
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Imaging - the correlation process
( )
10 /* *
,0
, ,
1( ) ( ) ( ) ( )
corresponds to the visibility ( ), ( ) at frequency 2
ss T
k k k s k sn
i k j ki j
E t t t nT t nTN
V t t
ω ω ω ω ω
ωωπ
=
⎡ ⎤= = + +⎣ ⎦
⎡ ⎤⎣ ⎦
∑k
k
R x x x x
R r r
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Coordinate system
Correlations are measured only for baselines ( ) ( )i jt t−r r
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Imaging equation – Planar array
( )
•
•
•
•
We limit ourselves to single frequency.
( , ) is the "known" amplitude response
of the antennas.
( , ) is the intensity at location ( , ).
, is the visibility function.
f
f
A l m
I l m l m
V u v
( ) ( )22, ( , ) ( , ) j ul vmf fV u v A l m I l m e dldmπ− += ∫∫
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Imaging equation: Non-coplanar array
( )( )22
2 2
, ,
1 ( , , ) ( , )1
f
j ul vm wnf
V u v w
A l m n I l m e dldml m
π− + +
=
− −∫∫
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The discrete measurement equation
( )
1
( ), ( ) ( )Tl
dj
ijk i k j k ll
V V t t I e− −
=
⎡ ⎤= =⎣ ⎦ ∑ i js r rr r s
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Matrix formulation of the imaging equation
( ) ( ) ( ) ( )
( ) ( )
− − −−
= =
≡ = =
⎡ ⎤⎢ ⎥⎡ ⎤= =⎣ ⎦ ⎢ ⎥⎢ ⎥⎦
•
⎣
•
∑ ∑,1 1
)
1
Recall
In "our" notation this translates to
=
where(
,..., ,( )
T TTi j ji
d dj jj
k ijki j
Hk k k
k k k d
d
V I e e I e
I
I
s r r s rs r
1
R s s
R A BA
s 0A a s a s B
0 s
( )−
−
⎡ ⎤⎢ ⎥
= ⎢ ⎥⎢ ⎥⎣ ⎦
1( )
( )
Tk
Tp k
j t
k
j t
e
e
s r
s r
a s
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The non co-planar array case
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Matrix formulation of the imaging equation
( )
•
•
= ( + ) ( )
( ) is a self-calibration slowly time varying pxp matrix
( ) is the noise covariance matrix. Typically AWGN but when interference exists it dom
H Hk k k k k k
k
k
t t t
t
t
nn
nn
R Γ A BA Γ R
Γ
R
inates
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Polarimetric imaging
When imaging polarized sources each element receives the two polarizations.The steering vector at epoch k is now replaced by two orthogonal vectors describing circularly left and right polarization co
, ,
, 1 , 2
mponents( ), ( )
For all , we have ( ) ( )For a quasi mono-chromatic source the polarization can be described by the parameters
k k
k k⊥L R
1 2 L R
a s a ss s a s a s
11 12
21 22
which are the coherence between right and left circular polarization components.
b bb b⎡ ⎤⎢ ⎥⎣ ⎦
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Polarimetric imaging
11 12 21 22
11
12
21
22
The sources cross polarization coefficients , , , are connected to the Stokes parameters I,Q,U,V through the matrices
b 1 0 0 1b 0 1 0
b 0 1 0b 1 0 0 1
b b b b
jj
⎡ ⎤ ⎡⎢ ⎥ ⎢⎢ ⎥ ⎢=⎢ ⎥ ⎢ −⎢ ⎥ −⎣⎣ ⎦
11 12 21 22
This implies that it will be sufficient to deconvolve the parameters , , , and then recover the Stokesparameter
IQUV
b b b b
⎤ ⎡ ⎤⎥ ⎢ ⎥⎥ ⎢ ⎥⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥⎦ ⎣ ⎦
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Polarimetric imaging
, 1 , 1 , ,
1 111 121 121 22
11 12
21 22
The matrix form of the (calibrated) measurement equation now becomes (assuming N point sources)
( ), ( ),..., ( ), ( )k k L k k N k N
N N
N N
k
s s
b bb b
b bb b
⎡ ⎤= ⎣ ⎦⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
=
R L R
k k
A a s a a s a
B
R A BA , k=1,...,KH σ 2+ I
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Self calibration with polarized sources
( )( ) ( )( ) ( )
,1 ,
11 12, ,,
21 22, ,
, k=1,...,K
diag ,...,
H Hk k k k
k k k p
k i k ik i
k i k i
σ
γ γ
γ γ
2= +
=
⎡ ⎤= ⎢ ⎥⎢ ⎥⎣ ⎦
kR Γ A BA Γ I
Γ Γ Γ
Γ
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Deconvolution
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Deconvolution in the (u,v) plain
Clark proposed to estimate several CLEAN component in a single dirty image and subtract them
Cotton-Schwab proposed to perform the subtraction in the (u,v) domain. This allows for non grid points
– We will always consider the Cotton-Schwab approach– Perform non-grid estimation whenever possible– We will discuss extension of the Clark approach
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Deconvolution
=
=
=
=
=
∑
∑
D1
'D
1
'D
I ( ) ( ) ( )
C
I ( ) ( ) ( )
ˆ ( ) arg min I ( )
lassical Fourier imaging
Super-resolution MVDR based imaging
Pseudo-spectrum
MVDR criterio
such tha
n
t
KHk k
k
KHk k
k
k
k
k
w
s a s R a s
s w s R w s
w s s
( ) ( ) ( )
( ) ( )
β β−−
−=
=
= =
= ∑'D
1
( ) ( ) 1
1ˆ ( ) where
1 I (
So
)
lution
Hk k
k k k k k Hk k k
K
Hk k k k
11
1
w s a s
w s R a sa s R a s
sa s R a s
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The adapted angular response
We would like to maintain the Gaussian receiver noise isotropicTo that end we add to the MVDR equations the requirement 1
Still we want a solution of the form (k
=w
w
2
''D 2
1
) ( )We obtain that
( ) ( ) ( )
and the angular spectrum becomes( ) ( ) I ( )( ) ( )
k k
k kHk k k
HKk k kH
k k k k
α −
−
−
−
−=
=
=
=∑
1
1
1
s R a s
R a swa s R a s
a s R a ssa s R a s
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Enhancing the image using independence
1
Since the covariance matrices at different epochs represent indepndent observations it is reasonable to assume that the overall covariance of the data is given by
=
K
RR
R
( )
( )
1
MVDR
1
Choosing a vector for yields ,..., where
In this case the MVDR dirty image becomes:
1 I ( )
TT TK
k k k
KHk k k
k
−
−
=
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
⎡ ⎤= ⎣ ⎦=
=
∑
1
1
w R w w w
w R a s
sa s R a ( )s
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The AAR dirty image
( ) ( )
( ) ( )1
AAR2
1
I ( )
KHk k k
kK
Hk k k
k
−
=
−
=
=∑
∑
1a s R a ss
a s R a s
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Dirty images
(a) Classical
(b) MVDR
(c) AAR
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Bandwidth synthesis
( ) ( )
( ) ( )
1, , ,
1AAR
2, , ,
1
I ( )
KHk k k
kK
Hk k k
k
ω ω ωω
ω ω ωω
−
=
−
=
=∑ ∑
∑ ∑
a s R a ss
a s R a s
•Extension to bandwidth synthesis is different than the classical imaging due to non-linear structure
•Spectral index of sources can be included as extra parameter (not shown below)
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Classical beamforming based CLEAN
( ) ( ) σ
α
γα
=
=
=
= +
=
= =
= −
∑
∑
∑
1
2
1
D1
D D
max ( ) ( )
I ( ) ( ) ( )
ˆ ˆˆarg max I ( ) max I ( )
ˆ ( ) ( )
KHk k
kN
Hk n k n k n
nK
Hk k
k
Hk k k k
b s s
ks
k
s
w s R w s
R a a I
s a s R a s
s s s
R R a s a s
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MVI CLEAN imaging
α
γα
= =
= −
ˆ ˆˆarg max I ( ) max I ( )
ˆ ( ) ( )AAR AAR
Hk k k k
ss s s
R R a s a s
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Least Squares power estimator to remove bias
2K) ( ) ( ) ( )
k=1 F
arg min ( ) ( )
Independently used in the context of multi-resolution CLEAN by Bhatnagar & Cornwell, A&A, 2004
n n Hkα
λ α( = −∑ n nR a s a s
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The LS-MVI algorithm
1(0)
( ) (n)AAR
) ( )
Measure ,...,
Set Until convergence criterionSource location estimator
arg max I ( )
Power estimator
arg min
K
k k
n nkα
γ
λ α(
= = 0.1
=
= −
n
s
R R
R R
s s
R2K
( ) ( )
k=1 F( )
( 1) ( ) ) ( ) ( )
( ) ( )
Update
( ) ( )
H
nk
n n n Hk k γλ+ (= −
∑ n n
n n
a s a s
R
R R a s a s
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Enforcing the non-negative definite constraint
( ) ( )
All source powers are non-negativeand residual covariance is also non-negative
( ) ( )
Solution is simple: Solve the unconstrained proble
Hk σ α
α
2− − ≥≥ 0
n nR I a s a s 0
m and use bisection on finding sequentially over k that is good for all k
αα
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Other enhancements
Clark type algorithm using SDP
Cotton-Schwab with off-grid estimation
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Complexity analysis
2 2
2
Main bottleneck: Recomputing the dirty image at each stage Complexity of naive implemntation is M K per iteration where M is the number of pixels in the image, p number of antennas
We can reduce th
p
2
22
is to M K per iteration Still high compared to gridded CLEAN with 2M log operations forrecomputing the dirty image, but managable.These techniques are easy to implement using parallel processors!De
pM
dicated accelerators might be a viable option!
Can use coarse grid and use local optimization to obtain accurate location estimate! We do not need intermediate images just accurate parameter estimates!
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Robust beamforming and self calibration
( ) ( )
Assume that ( ) is not completely known.( ) is the nominal value. is a positive definite matrix describing the uncertainty
at epoch
( ) ( ) ( ) ( ) 1
The robust beamforming problem is
k
k
Hk k k k k
k
− − ≤
k
a sa sC
a s a s C a s a s
[ ]
( ) ( )
11 ,...,
given by
ˆ ˆ( ),..., ( ) arg max
subject to
( ) ( ) 1
KK
Hk k k
Hk k k
ρρ ρ
σ ρρ
,
2
, =
− − ≥≥ 0
− − ≤
a a
k k
a s a s
R I a a 0
a a s C a a s
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Equivalent SDP formulation
[ ]
( )( )
11 ,...,
ˆ ˆ( ),..., ( ) arg min
subject to
( )
( ) 1
KK
k kHk
k kH
k
τρ τ
στ
ρ τ
,
2
, =
⎡ ⎤− ≥≥⎢ ⎥
⎣ ⎦⎡ − ⎤
≥⎢ ⎥−⎢ ⎥⎣ ⎦
=1/
a a
k
k
a s a s
R I 0 a0
a
C a a s0
a a s
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Robust beamforming based self calibration
2( ) ( )
1 1
ˆ ˆarg min ( ) ( )K N
n nk k
k n= =
= −∑ ∑ΓΓ a s Γa s
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ExtensionsSimplified Maximum likelihood (Leshem and van der Veen 2000)
Maximum likelihood using EM (Lanterman)
Robust beamforming techniques, e.g., diagonal loading
Generalized LS-MVI using parametric source templates (templets: shapelets, curvelets, ridgelets, wavelets…)
Global optimization techniques: L1 and others
Performance analysis
Sensitivity to modeling assumptions
Resolution limits
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Simulated experiments
-East west array with 10 elements and longest baseline 1000
720 covariance matrices- Fully calibrated array
λ.−
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Resolution example
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Extended sources example
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Extended sources example
(a) CLEAN
(b) LS_MVI
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ConclusionsExciting science can be done with future instruments
Science strongly depends on technology
Great challenges: – Calibration is extremely challenging !
– Image processing can take us beyond the equipment !!
– Real time adaptive interference mitigation is crucial
– Architectures and algorithms for extremely large arraysComm – 2-4 sensors
Radar - thousands of sensors
Radio astronomy – 10,000-100,000,000 elements