Challenges in advanced imaging with radio interferometric telescopes

S.Bhatnagar

NRAO, Socorro

April 08, 2005

Basic Interferometry

<EiE*j>

Terminology:

Visibility: Vij=<EiE*j>, Ei is the electric field

Baseline: Length of the

projected separation between the antennas (B).

Only relative separation matters for incoherent emission

Co-ordinates: (uij,vij,wij)

uij=ui-uj

An N-element array instantaneously measures N(N-1)/2 baselines (complex values)

Basic Interferometry

In terms of the sky brightness distribution (Io(l,m))

S : uv-sampling function, (l,m): direction in the sky

For l2+m2<<1 or small wij, sky is the 2D Fourier transform of the Visibility function (van Cittert-Zernike Theorem)

Sampling in the Fourier plane

N-element array instantaneously measures O(N2) Fourier components.

As earth rotates the projected baseline changes ==> each baseline measures a new Fourier component

Image plane

The Measurement Eq.: where A is the measurement matrix, V, I and N are the Visibility, Image and noise vectors.

A=FS where F is the Fourier Transform operator and S is a diagonal matrix of weights (wij).

Due to finite Fourier plane sampling, A is singular and in general rectangular

Image reconstruction: Solve for Io

B: The Toeplitz Beam Matrix

ATA=Id: The Dirty Image vector

Imaging

The full Measurement Eq.:

Data Corruptions Sky

Gij are the direction independent corruptions (e.g., multiplicative complex gains, etc.)

Xij are the direction dependent corruptions (e.g., ionospheric/atmospheric effects, Primary beam effects, etc.)

Often Gij, Xij are separable into antenna based quantities as Gij=GiG*j

Data corruption

Calibration: Keeping IM fixed, determine Ji's.

Calibrate the data:

For direction independent corrections:

Incorporate direction dependent corrections in imaging

Imaging: Keeping Jij fixed

Estimate IM such that

Data analysis operations

For full polarization treatment:

V=[VRR,VRL,VLR,VLL]

Given IM, solve for Jis such that

Image using

For single polarization: min:{ }

Calibration

Use of FFT requires re-sampling the Vis. (V(u,v) is not measured on a regular grid)

Convolutional gridding:

Make the Dirty Image as:

Deconvolve the PSF to estimate Io

General structure of deconvolution algorithms:

Major cycle: Compute the model and the residual vis as

Minor cycle: Update the model image

Imaging

General formulation

The Clean algorithm:

Image representation in the pixel basis.

Steepest descent

Minor cycle: Step size: s=max{ }

Major cycle: Update using FFT:

Deconvolution: 2D geometry

Example

IM

AT(V-AIM)

IM

V-AIM

Expensive function evaluation is simplified: Convolution becomes a shift-and-add operation.

Clean (and its variants) ignore inherent pixel level correlation in Io (each pixel is a DOF) – errors are correlated with extended emission.

Search space is assumed to be orthogonal.

Too many DOFs for extended emission (no. of pixels).

Search space constrained by user defined boxes.

Some observations

A is more complicated (not a FT operator)

General approach:

Major cycle involves: and

Use Eij as the GCF to predict the model data (AIM)

Compute VR at high accuracy.

Use an approximation for AT: Use ETij as the GCF

Image plane corrections

Primary Beam Effects

Eij as function of direction is measured a priori

Primary beam effects

Polarized primary beam: Beam squint

For full beam polarimetry (EVLA)

Pointing offset calibration

For mosaicing (EVLA, ALMA)

GCF different for each visibility!

Example

Residual image before and after pointing correction.

IV=PB(IRR-ILL)

Peak ~4%

IV=PBRIRR-PBLILL

Peak ~0.2%

Pixel-to-pixel noise is correlated at the scale of the PSF support (Pixon methods not applicable – they are patented anyway!).

The scale of the emission fundamentally separates the signal (Io) from noise (IN).

Use a pixel-model with finite support

Convolution is no more a shift-n-add operation. Function evaluation becomes very expensive.

Scale-sensitive deconvolution

Search space is non-orthogonal

The total DOFs decrease, but computing cost is a strong function of the number of parameters.

Asp-Clean (Adaptive Scale Pixel model) [A&A, 2004]

Use

Find the local scale by fitting

For emission covering 100,000 pixels, required 1000 Pks

Computing cost prohibitive for more ~100 active parameters.

Sub-spaces

Scale sensitive deconvolution

Example

Fast and accurate forward and backward transform in the presence of image plane effects

Solve for direction dependent effects:

Primary beam effects (pointing, squint, variable beam shape,...) [EVLA, ALMA]

Wide-band imaging: [EVLA]

Need more sophisticated parametrization of Eij [EVLA,ALMA,LWA,SKA...]

Ionospheric/atmospheric calibration [LWA, SKA]

Simple functional representation may be difficult

Challenges

Component based image model

Gridding errors limit the highest dynamic range achievable.

FFT based inversion of the ME may not be usable.

Image sizes and visibility data sets will be large

100s of GB – 1TB of data, 15K X 15K images

10-100 transforms between image and data

Parallelization of computing as well as I/O

Challenges

Scale sensitive deconvolution

Pixel model parameterized by the amplitude, (local) scale, position, frequency, and perhaps polarization.

Total no. of parameters: ~10K

Fast algorithm for parameter estimation

Handle the coupling between pixels (non-orthogonal search space)

Adaptively control on the dimensionality of the search space.

Challenges

Interferometry and Synthesis in Radio Astronomy

Thompson, Moran and Swenson, 2001, John Wiley&Sons

Clean: Hogbom, J.A., 1974, A&AS,15,417

MEM: Cornwell,T.J, &Evans, K.J.,1985,A&A,143,77

Review of various methods: Narayan, R. & Nityananda, R.,1986, ARA&A,24,127

Math. analysis of the Clean Algo.: Schwarz, U.J., 1978, A&A, 65,345

Asp-Clean: Bhatnagar,S., & Cornwell, T.J., 2004,A&A,436,747

W-projection: Cornwell,T.J., Golap,K., & Bhatnagar,S.,2005,A&A,in press

Pointing cal: Bhatnagar,S., Cornwell,T.J.,&Golap, K., 2004, EVLA Memo #84

Ninth Synthesis Imaging Summer School lectures,

http://www.aoc.nrao.edu/events/synthesis/2004

References

Sanjay Bhatnagar

2005-04-08T09:41:56

Sanjay Bhatnagar