3. Method of solution

We consider Eq. (5) as a linear system of equations which is to be solved not exactly but only within the (0, 1) error distributions. The justification for doing so within an LSF is that we do not aim at minimizing the residuals, that is, the remaining error, but rather aim at minimizing the deviations from a model that leaves a set of expected residuals. Also, an exact LSF would ask for less variables than data, i.e., for a sufficiently small image.

Of course, there are many solutions within the error expectance, and this is where constraints come in. The first one is (in case of intensities) non-negativeness (which we take into account during the solving algorithm), the second, more important one is smoothness of the image.

Most smoothing constraints, as maximum entropy or minimum second derivative, are global in the sense that smoothing at one place may affect the solution at other places. Suppression of higher spatial frequencies is also of this kind. Our constraint is local. It is derived from the concept of structural information (Pfleiderer 1989) which is minimized. Here, we only need mention that it is built from squared intensity differences. Its derivative which is used in the minimization process is thus linear in the intensities. Added as a Lagrange condition to the system of Eq. (4), it produces again a linear system with modified coefficients. Instead of solving Eq. (5) under a constraint, i.e., deconvolving the DM with the true DB (the a in Eq. (5)) but allowing for the constraint, we deconvolve the DM without constraint but with a modified DB.

The modification consists of adding a "smoothing beam" s to the coefficients, where is a Lagrange parameter which is not fixed by theory but can be freely chosen. The larger , the more smoothing occurs. The choice is generally a good one. The function s is zero outside the main beam (to the effect that we do not smooth over a larger area), is negative inside the main beam with increasing values towards the center and the sum equalling -1, and is positive =+1 in the center of the main beam (). That is, a₀₀ is replaced by , surrounded by . Thus protrudes from the surrounding beam. This spike gave rise to the name Prussian Helmet type beam.

The smoothing action works as follows: If a point source is somewhere introduced, i.e., the beam is subtracted from the dirty map, then the remaining residuals are too small in the center, and too large in the surroundings. That is, further intensity has to be put into the surrounding area in order to achieve an overall fit.

The exact definition of s is not very critical. The theory leaves it open. The more the negative s is concentrated towards the center, the smaller is the region over which smoothing takes place. The whole procedure has been described in some detail by Pfleiderer (1985, 1988).

Cornwell (1983) used a similar approach in his so-called Smoothness- Stabilized CLEAN. He minimized the sum of squared intensities which is equivalent to putting s₀₀=1 and s=0 everywhere else (i.e., ). He was probably the first to use a Prussian Helmet beam.

The normalization of Eq. (5) with expected residuals within a (0, 1) error distribution, together with , has the advantage that it suffices to determine the z^* to within about . That is, it suffices to determine a diophantic solution. Thus, overfitting below the expected noise level is avoided.

We find the solution by a method comparable to the iterative CLEAN where the maximum of the DM is found and a point source is subtracted from it, the intensity of which is determined by the gain factor. The simplest way to solve our system is: Find the maximum of the DM, subtract the beam corresponding to a point source of intensity 1 (i.e., intensity = point source detection limit z₀), and iterate. Subtract a point source of intensity -1 if the minimum of the DM is sufficiently negative, and if the resulting intensity at this point stays non-negative. We have used (with slight modifications) this rather primitive algorithm because a better method to solve a linear system of equations under the constraints that the solution be diophantic and non-negative is not known to us. Of course, the final smoothing of CLEAN over the main beam is not necessary in our case.