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 ?

Page 3April 2009

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.

Page 4April 2009

IEEE Journal of selected topics in Signal Processing – October 2008

Page 5April 2009

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

Page 6April 2009

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

Page 7April 2009

High dynamic range - Array beam

Page 8April 2009

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

Page 11April 2009

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

Page 12April 2009

Coordinate system

Correlations are measured only for baselines ( ) ( )i jt t−r r

Page 13April 2009

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− += ∫∫

Page 14April 2009

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− + +

=

− −∫∫

Page 15April 2009

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

Page 16April 2009

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

Page 17April 2009

The non co-planar array case

Page 18April 2009

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

Page 19April 2009

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⎡ ⎤⎢ ⎥⎣ ⎦

Page 20April 2009

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

⎤ ⎡ ⎤⎥ ⎢ ⎥⎥ ⎢ ⎥⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥⎦ ⎣ ⎦

Page 21April 2009

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

Page 22April 2009

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

Page 25April 2009

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

Page 26April 2009

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

Page 27April 2009

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

Page 28April 2009

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

Page 29April 2009

Dirty images

(a) Classical

(b) MVDR

(c) AAR

Page 30April 2009

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)

Page 31April 2009

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

Page 32April 2009

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

Page 33April 2009

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

Page 34April 2009

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

Page 35April 2009

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

Page 36April 2009

Other enhancements

Clark type algorithm using SDP

Cotton-Schwab with off-grid estimation

Page 37April 2009

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!

Page 38April 2009

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

Page 39April 2009

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

Page 40April 2009

Robust beamforming based self calibration

2( ) ( )

1 1

ˆ ˆarg min ( ) ( )K N

n nk k

k n= =

= −∑ ∑ΓΓ a s Γa s

Page 41April 2009

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

Page 42April 2009

Simulated experiments

-East west array with 10 elements and longest baseline 1000

720 covariance matrices- Fully calibrated array

λ.−

Page 43April 2009

Resolution example

Page 44April 2009

Extended sources example

Page 45April 2009

Extended sources example

(a) CLEAN

(b) LS_MVI

Page 46April 2009

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