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Cornwell, Holdaway, and Uson (1994) proposed a novel mosaicing algorithm for the upcoming millimeter array (MMA): generate the mosaic of the dirty images and a single approximate point-spread function (PSF), and then proceed with any conventional single field deconvolution algorithm. For the MMA's high-quality Fourier-plane coverage and similar PSF's for all fields in the mosaic, this approach was not limited by the differences in the approximate PSF and each field's actual PSF until the possible image dynamic range exceed a few hundred to one.
AIPS++ takes this approach to mosaicing a step further: perform an incremental deconvolution of the residuals with the approximate PSF, with an exact subtraction of the cumulative model brightness distribution at the end of each incremental ``major cycle'' (similar in concept to the major cycles of the Clark CLEAN).
If all of the fields are observed with many short snapshots over and over again (this is the usual way to make a mosaic observation) then each field will have similar Fourier coverage and hence similar synthesized beams. An approximate PSF can be created which is a fairly good match to the actual PSF of each of the fields. Also, if the sky-coverage of the observed fields is Nyquist or better, then the approximate, shift-invariant PSF will be a reasonable match to the actual PSF of sources at various locations across the mosaic. The residual visibilities from each field can be transformed and mosaiced to make a single residual mosaic image. This mosaic image can be deconvolved with the deconvolution method of your choice; for example, with Clark CLEAN, Multiscale CLEAN, maximum entropy, or maximum emptiness.
The deconvolution algorithm cannot deconvolve arbitrarily deeply, because at some level the discrepancies between our approximate shift-invariant PSF and the true PSF at any location in the image will become apparent, and we will start ``cleaning'' error flux. Hence, we need to stop deconvolving when we have gotten down to the level of these PSF discrepancies. At this point, we take the part of the model brightness distribution we have just deconvolved and calculate model visibilities (using the measurement equation) and subtract them from the (corrected) data visibilities. To the extent that the primary beam and sky pointing are exact, the visibility subtraction is also exact. The residual visibilities can then be re-mosaiced, but the peak residual is at a much lower level. The process of deconvolving with the approximate, shift-invariant PSF then continues, and another increment to the model brightness distribution is formed, removed from the remaining residual visibilities, and added to the cumulative model brightness distribution. Borrowing from the Clark CLEAN's terminology, we call each cycle of incremental deconvolution and exact visibility subtraction a ``major cycle''.