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A CLEAN­based method for mosaic
deconvolution
By F. Gue t hy, S. Gu i l l o t e a uy AND F. V i a l l e f o ndz
e­mail: gueth@iram.fr
y Institut de Radio Astronomie Millim'etrique, 300, rue de la Piscine, F­38406 St Martin
d'H`eres, FRANCE
z DEMIRM, Observatoire de Paris, 61, avenue de l'Observatoire, F­75014 Paris, FRANCE
Mosaicing may be used in aperture synthesis to map large fields of view. So far, only MEM
techniques have been used to deconvolve mosaic images (Cornwell (1988)). A CLEAN­based
method has been developed, in which the components are located in a modified expression. This
allows a better utilization of the information and consequent noise reduction in the overlapping
regions.
Simulations show that this method gives correct clean maps and recovers most of the flux of
the sources. The introduction of the short­spacing visibilities in the data set is strongly required.
Their absence actually introduces artificial lack of structures on the corresponding scale in the
mosaic images. The formation of ``stripes'' in clean maps may also occur, but this phenomenon
can be significantly reduced by using the Steer--Dewdney--Ito algorithm (Steer, Dewdney & Ito
(1984)) to identify the CLEAN components. Typical IRAM interferometer pointing errors do
not have a significant effect on the reconstructed images.
1. Introduction
The field of view of aperture synthesis observations is limited by the size of the primary
beams of the antennas. Whereas this is not critical in centimeter interferometry (field
of view ? 10 0 ), it becomes a very strong constraint in the millimeter range. In the case
of the IRAM millimeter interferometer of Plateau de Bure (Hautes Alpes, France), the
15­m dishes limit the field of view to about 50 arcsec at – ¸ 3 mm. To map larger
fields, one has to perform a mosaic, in which several overlapping fields are observed in
a sequence which guarantees some homogeneity in terms of uv coverage and noise level
(see, for example, Cornwell (1994)).
One of the most important step in the analysis of interferometric data is the decon­
volution of the image, necessary to reduce the effects of the very strong sidelobes of the
observed ``dirty beam'' and thus to obtain a map which allows a reliable interpretation.
Two methods are used for the deconvolution of radio synthesis images: the CLEAN
algorithm (H¨ogbom (1974), Clark (1980)) and the Maximum Entropy Method (see, for
example, Narayan & Nityananda (1986)).
So far, only MEM techniques have been used to deconvolve mosaics (Cornwell (1988)).
We present here a CLEAN­based method for the mosaic deconvolution.
2. Mosaic reconstruction
2.1. Single field observation
The maps produced by an interferometer are given by the following measurement equa­
tion:
F = D \Lambda (B \Theta S) + N (2.1)
1

2 F. Gueth et al.: Mosaic deconvolution
where F is the map, D the dirty beam, B the primary beam, S the sky brightness
distribution, N the noise distribution, and \Lambda indicates a convolution product.
The CLEAN algorithm attempts to replace directly D by a clean gaussian beam. Note
that it is possible to correct for the primary beam attenuation after the deconvolution
by a simple division.
2.2. Mosaics
To build a mosaic, one has to make the sum of the different observed fields and to
correct for the primary beam attenuation, which is inhomogeneous over the whole region
covered by the observations. For optimum results from the signal to noise point of view,
this operation is done by a weighted sum:
J =
P
i
(B i \Theta oe \Gamma2
i \Theta F i )
P
i
(B 2
i
\Theta oe \Gamma2
i
)
=
P
i
(B i \Theta (D i \Lambda (B i \Theta S) +N i ) \Theta oe \Gamma2
i
)
P
i
(B 2
i
\Theta oe \Gamma2
i
)
(2.2)
where the subscript i corresponds to the field number i, and oe i is the noise level: oe i =
rms (N i ). J is here directly homogeneous to a sky brightness distribution.
The mosaic reconstruction has to be done before the deconvolution, because of the
non­linearity of the CLEAN or MEM algorithms. The joint deconvolution actually gives
better results than a mosaic of individually deconvolved fields, because the adjacent
pointings reinforce each other in the estimation of the missing spacings (see Ekers &
Rots (1979), Cornwell (1988)), and of course because of the improvement in the signal­
to­noise ratio due to the redundancy of the observations.
But the equation 2.2 is not a simple convolution equation, and does not allow to
directly and properly replace the dirty beam by a clean beam. This is the reason why a
specific CLEAN method had to be developed for the mosaic deconvolution.
3. Mosaic deconvolution
3.1. Primary beam truncation
In practice, we reconstruct the mosaic in a slightly different way. To avoid contaminating
field centers by (small, but possibly significant) errors coming from poor knowledge of
the far­off center beam pattern from nearby fields, we use the following expression:
J =
P
i
(B t
i
\Theta oe \Gamma2
i
\Theta F i )
P
i
(B t
i
2
\Theta oe \Gamma2
i
)
(3.3)
where B t
i
is a truncated version of the primary beam, down to some arbitrary level,
typically 10 to 30 %.
3.2. A CLEAN based algorithm
The noise level in J is not constant:
N =
q P
i
(B t
i
2
\Theta oe \Gamma2
i
)
P
i
(B t
i
2 \Theta oe \Gamma2
i
)
= 1
q P
i
(B t
i
2
\Theta oe \Gamma2
i
)
: (3.4)
At the edges of the mosaic, where the B i are low, the noise is thus very high. This is a
basic problem in mosaic deconvolution: if we intend to apply CLEAN on J , the risk of
taking a noise peak as a CLEAN component is too important. Instead, we construct the

F. Gueth et al.: Mosaic deconvolution 3
Figure 1. Left: Model of a sky brightness distribution. Right: Simulation of a ten fields mosaic
observed with the IRAM Plateau de Bure interferometer. The separation between the adjacent
pointings is half the primary beam. The clean beam is 3:5 \Theta 2:9 arcsec 2 .
following expression:
H = J
N
=
P
i
(B t
i \Theta F i \Theta oe \Gamma2
i
)
q P
i
(B t
i
2
\Theta oe \Gamma2
i
)
=
P
i
(B t
i \Theta (D i \Lambda (B i \Theta S) + N i ) \Theta oe \Gamma2
i
)
q P
i
(B t
i
2
\Theta oe \Gamma2
i
)
: (3.5)
In the limit of identical noise oe on all fields, one obtains:
H = 1
oe
P
i
(B t
i
\Theta (D i \Lambda (B i \Theta S) + N i ))
q P
i
B t
i
2
: (3.6)
H is essentially the signal to noise ratio at each point. Thus, locating CLEAN components
from H is a safer procedure than from J .
The proposed algorithm is then, for each iteration k:
ffl locate the CLEAN component position (l k ; m k ) from the maximum of H (or oeH)
and find the corresponding value j k (l k ; m k ) of J .
ffl correct J , i.e. subtract from J the contribution of a supposed point source at
(l k ; m k ):
J k+1 = J k \Gamma
P
i
B t
i \Theta (D i \Lambda (g \Theta j k \Theta B i (l k ; m k )))
P
i
B t
i
2 (3.7)
where g is the loop gain. H k+1 can be calculated in the same way.
Note that it is possible to use the different enhancements of CLEAN (e.g. Clark
or Steer­Dewdney­Ito variants) in the process, the basic idea being always to find the
position of the components on H and to correct J . However, the Multi Resolution
Clean (Wakker & Schwarz (1988)) seems not to be easily adaptable, since it requires a
convolution equation.
3.3. Simulations
To check and test this algorithm, various simulations have been performed. The flow­
chart of these simulations is the following:

4 F. Gueth et al.: Mosaic deconvolution
Figure 2. Left: Model of a sky brightness distribution. Middle: Simulation of a ten field mosaic
observed with the PdBI. Right: The same with the short spacing information: the continuous
structure is reconstructed.
ffl start with a model of the sky brightness distribution;
ffl multiply by the primary beam of the antennas, centered at the positions of the
observed fields;
ffl calculate the Fourier transform (i.e. the visibility) of each field;
ffl sample each visibility in a way corresponding to a real interferometric observation;
ffl reconstruct the mosaic and deconvolve, as with real observations.
Various effects, such as pointing errors or simply a general noise, can be added at the
different steps.
Figure 1 shows one of the model and the corresponding map obtained with a mosaic of
ten fields. The algorithm reconstructs correctly the images, and recovers most of the flux
of the sources (¸ 65 % in this example). These results are better than those obtained
in the same conditions with MEM techniques, which seem to need a better uv coverage
than the one available with the Plateau de Bure interferometer.
4. Problems in image reconstruction
4.1. Short spacings
The problem of the short­spacing information is well­known in aperture synthesis: due
to the physical size of the dishes, the shortest baselines cannot be observed with an
interferometer. This usually limits the maximal size of a structure it is possible to map
in a single­field observation. But is appears to be even more important in the case of
mosaics, where the absence of the short spacings introduces artificial lack of structure
on intermediate scale. Figure 2 shows, on a particular example, how a continuous large
source can be split into several structures, which are only reconstruction artefacts.
The correct way to get rid of this problem is to observe the same object with a single­
dish antenna that is larger than the shortest baseline available, or with a smaller inter­
ferometer, and to combine these observations with the interferometric data. The third

F. Gueth et al.: Mosaic deconvolution 5
Figure 3. Left: Mosaic deconvolved with the Clark algorithm, and a very high gain loop, so as
to more clearly show the stripes formation. Right: The same simulation, deconvolved with the
SDI­Clean and the same parameters.
image of Figure 2 corresponds to such a simulation: the continuous structure is now
reconstructed in a correct way.
4.2. Stripes formation
Another phenomenon that occurs in CLEANed maps is the so­called ``stripes formation''
(Schwarz (1984)). This can however be significantly reduced by using the Steer, Dewdney
& Ito (1984) enhancement of CLEAN. Figure 3 shows the improvements in the image
quality when using SDI­Clean instead of the Clark algorithm.
4.3. Pointing errors
Simulations including pointing errors have also been performed, each field of the mosaic
being shifted randomly. This global error on the position of the whole field maximizes
the effect. Rms shifts of 1/10 of the primary beam (i.e. the typical pointing error at
Plateau de Bure) do not change significantly the images obtained after deconvolution.
Important modifications of the reconstructed structures occur with pointing errors of 1=5
of the primary beam.
5. Conclusion
A new CLEAN­based algorithm for mosaic deconvolution has been developed. Sim­
ulations of Plateau de Bure observations and image reconstruction show the efficiency
of the method, which recovers in a correct way both the structure and the flux of the
sources. It actually seems to give better results than MEM techniques in the case of the
``sparse'' uv coverages obtained with the Plateau de Bure interferometer.
The most important artefacts in the final maps are due to the missing short spacing
information. Its introduction in the data set (single­dish observation) is then strongly
required, especially for observations of very large and continuous structures.

6 F. Gueth et al.: Mosaic deconvolution
This deconvolution method will allow now the use of the Plateau de Bure interferometer
for mapping large objects (e.g. galaxies, molecular outflows).
REFERENCES
Clark B., 1980, A&A, 89, 377.
Cornwell T., 1988, A&A, 202, 316.
Cornwell T., 1994, in Astronomy with millimeter and submillimeter wave interferometry, eds
Ishiguro M. & Welch W.J., p.96.
Ekers R. & Rots A., 1979, in Image formation from coherence functions in astronomy, ed. van
Schooneveld C., p.61.
H¨ogbom J., 1974, A&AS, 15, 417.
Narayan R. & Nityananda R., 1986, ARA&A, 24, 127.
Schwarz U., 1984, in Indirect imaging ed J.A. Roberts p.255.
Steer D., Dewdney P. & Ito M., 1984, A&A, 137, 159.
Wakker B. & Schwarz U., 1988, A&A, 200, 312.