Because synthesis arrays sample the (u,v) plane at discrete locations,
there is incomplete knowledge about the Fourier transform of the source
intensity distribution. The measured u,v) data can be thought of as
the true distribution, V(u,v), in the u,v) plane multipled by some
sampling function, S(u,v). The convolution theorem states that the
Fourier transform of the sampled distribution (the dirty map, 
) is
equal to the convolution of the Fourier transform of the true source
u,v) distribution (the true image, I) and the Fourier transform of
the sampling function (the dirty beam, 
):
where 
indicates convolution, and 
indicates the
Fourier transform. Deconvolution algorithms attempt to account for the
unsampled regions of the u,v) plane. If it was fully sampled, there would be
no sidelobes, since the sampling function would be a constant, and the Fourier
transform of a constant is a delta function; a perfect beam. Thus,
deconvolution tries to remove the side lobes of the dirty beam that are present
in the image. It is important to realize that in doing so, the algorithm is
guessing at what the visibilities are in the unsampled part of the u,v) plane.
The solution to the convolution equation is not unique, and the problem of
image reconstruction is reduced to that of choosing a plausible image from the
set of possible solutions.
You should be extremely cautious when deconvolving images formed from a small number of snapshots. In these cases, there will be large areas of the u,v) plane that are unsampled, because of the poor instantaneous u,v) coverage of the ATCA. If the source is complicated, the deconvolution algorithm may go badly wrong in its guess of what the source really looks like in the gaps. The best way to make a decent image of an object is to observe it, not allow a deconvolution algorithm to guess what it looks like.
There are two techniques used commonly in radio atronomy; CLEAN and maximum entropy (MEM). CLEAN is rarely used outside of radio astronomy, but MEM is more far reaching. For detailed discussion on the `pros' and `cons' of these algorithms, see the NRAO imaging work shops and references therein. Much blood has been spilt over their relative merits in the last decade or so.
It is probably fair to say that in general, CLEAN is easier to drive than MEM, although use of MEM can result in reduced processing times for large problems. However, if you need to use MX as your imaging tool, there is no choice; CLEAN is what you get. A dirty image made with HORUS, UVMAP, or MX can can be deconvolved with one of HBCLN, APCLN, SDCLN (CLEAN), or VTESS\ (MEM). VTESS can also be used for mosaicing, although I will not discuss this further.