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, *a*_{00} 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 *s*_{00}=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 *z*_{0}), 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.