Parallel CLEAN: beyond the frequency domain
I. M. Stewart (formerly of JBCA, University of Manchester, UK; now Dept of Astronomy, University of Cape Town, SA)
The CLEAN algorithm1,2, despite its problematic convergence properties and its need for heuristic intervention and tuning, is the most commonly used means ofattenuating sampling artifacts in images made through radio interferometry. Subsequent modifications by Conway3 and Sault and Wieringa4 adapted CLEAN toobservations made using a wide fractional bandwidth, which are not well handled by the original algorithm. Here a generalized version of the Sault-Wieringa
algorithm is applied to other cases where standard CLEAN also fails, including time-variable5 and off-pixel-centre sources6. A 1-dimensional version of the latter
technique can also be applied to extract non-integer frequency values from periodograms.

The parallel CLEAN algorithm4:

Generalized parallel CLEAN: dirty image as a sum of convolutions.
I0

I1

I2

**

*=+
B0

Dirty image D+…

B1

B2

Ri = D * Bi(* here represents correlation)
Aij = Bi * BjConstruct M such that mij = Aij(0,0)
for k=1, Niter:R = ΣRi
Find rmax, the location of max(R)c'i = λ Rj(rmax)
Ri = Ri – Σj c'j Aij(r-rmax)(λ usually <<1)c = M-1c'
Save the vector clean component c

5 sources, all ~1 Jy, varying in both flux and spectral index by ~x10.

Time-varying sources: the wide-band case

Input to the simulation (average flux;
convolved with restoring beam)

Dirty image

Högbom clean: 2000 cycles, λ=0.1

Parallel clean: 2000 cycles, λ=0.1
Nf=3, Nt=6

Expand the sky brightness map
I(r,ν,t) in basis functions:Nf −1N t −1

I (r,ν , t ) = ∑∑ Ap,q (r ) Fp (ν )Tq (t )p=0 q =0

Each FpTq generates a ‘dirty beam’,giving N = Nf × Nt beams in total.

Gibbs

Just as Högbom CLEAN models I(r)as sum of scalar clean components,
so Sault-Wieringa CLEAN models
Ap,q(r) as a sum of vector clean
components cp,q.
With a proper choice of time basis
function, this expansion in the image
plane allows one to avoid Gibbs
ringing either at the boundaries or at
gaps in the time sequence of thedata.

All brightness scales -0.02 to +0.02 Jy

A single ~1 Jy, variable source surrounded by faint diffuse emission.

Time-varying sources:
narrow-band observation,
unresolved source

Since light curves are expected to be
more rapidly varying than spectra, Nt
and therefore the total number ofbeams N may need to be large toobtain good modelling of the sources.
But the algorithm (see top right)
requires O(N2) images to be stored.
This can be reduced to O(N) if thebeams are orthogonalized (eg via
Gram-Schmidt) before commencing
cleaning. Clean components can berecovered afterwards via a matrix
inversion.

If time-variation is restricted to asingle point source in a narrow-band
observation, perfect decomposition
can be achieved with just 2 beams:
the usual dirty beam for one, and abeam constructed from the entire light
curve of the data for the other. Eachpixel of the dirty image must be acombination of these two.

Input to the simulation (average flux;
convolved with restoring beam)

Brightness scale -0.005 to +0.005 Jy
Högbom clean: 5000 cycles, λ=0.01

Brightness scale -0.05 to +0.05 Jy

Sub-pixel cleaning (1D example).
The ‘beams’:
B0 = F{S(x)/2}
B1 = F{πix/X S(x)}Fourier transforming gives
F{y}=δ(ω-n)*Aeiφ(B0+fB1)+F{Sy'}
A, f and φ are easily
recovered after SaultWieringa processing.
Higher orders  greater accuracy.

Data: y(x) = S(x)·[Acos(2π[n+f]x/X + φ) + y'(x)];n is integer, f in [-0.5,0.5].

Spectrum7 before cleaning

xSpectrum after cleaningxDirty image

Brightness scale -0.05 to +0.05 Jy
Parallel clean: 5000 cycles, λ=0.01

Brightness scale -0.005 to +0.005 Jy

1: Högbom J A 1974, Astron. Astrophys. Suppl. 15, 417
2: Schwab F R 1984, Ap. J. 89, 1076
3: Conway J E, Cornwell T J & Wilkinson P N 1990, MNRAS 246, 490
4: Sault R & Wieringa M H 1994, A&A Suppl. Ser. 108, 585
5: Stewart I M, Fenech D M, Muxlow T B M & Rushton A 2009, inpreparation
6: Stewart I M 2009, in preparation
7: Schröder A C, private communication.