The observed CVs have well-known spatial frequencies (frequency pairs in the usual twodimensional case) and contain unknown complex-amplitude errors. The visibilities result from a Fourier integral over the intensity distribution in the image space. For practical purposes, the continuous intensity distribution has to be approximated by a discrete intensity grid. While it is not strictly necessary, it is nevertheless convenient, also for practical purposes, to consider a rectangular equidistant grid of constant-size intensity pixels.

Thus, we start with such grid and intensities *z*_{jl} at grid positions
.
Given an observed frequency pair (*u*_{k},*v*_{k}), the observed CV (complex
amplitude *A*_{k}) is given by

where *f*_{k} is a - generally known - telescope-dependent sensitivity
factor which we shall put,
for convenience, as equal to 1. is a complex error. The *z*_{jl} are
the unknowns to be found from the observations. Thus, we call the righthand
side of Eq. (1) the *visibility model* .

Another sensitivity factor *S*_{k,jl} is included to take into account
the influence of the intensity distribution within the pixel *jl* on the
observations. It is not known observationally but some assumption has to
be made, at least implicitly. For a model, it is useful to assume the same distribution in
all pixels: then *S* depends only on frequency, *S*=*S*_{k}.
For example, *S*_{k}=1 if the intensities were concentrated in the
mid-pixel positions . If, instead, one fourth of the intensity is put into
each of the four pixel corners
, then
.
A non-symmetric intensity distribution within the pixel would give rise to
a complex *S*_{k}.
If each pixel has constant surface brightness
then

which is what we adopt in the following. The reason is that the influence
of the pixel size chosen on the deconvolution result is diminished. In any
case, *S*_{k} is near unity for small frequencies and/or small pixel size
because, then, the exponential factor varies little within each pixel.

Note that each CV is a member
of a DFT of the image but, unless the frequency pairs are given on a
rectangular grid, each CV is a member of a different DFT.
Interchanging the role of the two telescopes giving rise to a certain CV
changes the sign of the frequencies.
Both CVs are, of course, equivalent, and either may be used.
Thus, *A*_{-k} = *A*_{k}^{*} and .

The CVs being a linear superposition of (weighted)
intensities (which are the unknowns),
an LSF is apt. The LSF weights
can be understood in the sense that one expects
to reproduce the corresponding term within an error that can be considered
as a member of an (approximately) Gaussian
distribution with variance . In principle, one can choose the
freely, e.g. according to natural weighting, or to uniform weighting.
It would, however, not be useful to choose them smaller than the actual
measuring errors. In other words, the choice should be
where of course the quadratic error expectation *E* is
no more known than the actual error but can often be reasonably
estimated. In MIM, the equations are normalized to
a *unit error expectation* (0, 1), that is, zero average and unit variance.
Also, the intensity units can be normalized to multiples of the *point
source detection limit*

We keep the errors formally in the equations and minimize
the absolute value of model minus observed CVs,

The result of differentiation in respect to *z*_{nm} is

with *z*^{*} = *z*/*z*_{0}

The system is to be solved in the sense that the residuals
fit approximately into a (0, 1) distribution.
The double sum is, apparently, a discrete convolution (DC) of the intensity
map *z*^{*} with a dirty beam *a*, i.e., we have a straight deconvolution problem, *
not* a Fourier deconvolution problem.

It is convenient to put . Then
.
Each term of the dirty map *b* is the real part of one (weighted) term of a DFT of
the CVs. But the DM as a whole is the real part of such DFT only if the
CVs are given on a rectangular frequency grid (and complemented by zeros where
missing). The same holds for the DB. In this case, *a* and *b* would, except
for the weighting, just be the
DB and DM of the usual procedure. Of course, the pixels of the image *z* would have
to be defined on a corresponding rectangular grid; in particular, the pixel
number has to equal the number of (complemented) CVs, and each Fourier
term has to have equal weight. To the contrary, our
DM and DB, i.e., *b* and *a*, can be considered as a (weighted) *direct
Fourier transform* (see Sramek & Schwab 1986) of the CVs and the corresponding
unit numbers using both signs of the frequencies. Note that, inspite
of its name, this transformation is *not* a Fourier transform in the
usual sense. Unless certain conditions are met, not even a unique back
transformation is possible.

We can thus freely choose a common pixel size for CM, DM, and DB, as well as the number of pixels. Of course, a reasonable pixel size is approximately given by the special problem under consideration. The central beam should not be undersampled while there is little advantage in oversampling it.

The size of the image need not be larger than what is needed to contain all expected intensity. The DB should be large enough to allow a full convolution with the CM in the range of the DM. It should be noted that none of the 3 maps is periodic or assumed to be so. Even if the CVs were given on an rectangular grid, there would be no periodicity.

The overall goodness of the fit can be checked by the sum of the squared
residual visibilities. We find