An interferometric array samples the complex visibility function
*V*(*u*,*v*) of the sky at points in the *u*,*v* plane. Under
approximations that are valid for a sufficiently small sources in an
otherwise empty sky, the visibility function *V*(*u*,*v*) is related to
the angular distribution of the source intensity *I*(*l*,*m*) (multiplied
by the primary beam of the array elements) through a two-dimensional
Fourier transform:

where *S* denotes integration over the whole sky.

Practical arrays provide only a finite number of noisy samples of the
visibility function *V*(*u*,*v*), so *I*(*l*,*m*) cannot be recovered directly.
Instead, *I*(*l*,*m*) must be estimated either from a model with a finite
number of parameters, or from a non-parametric approach.

For radio astronomical imaging, a convenient (and sometimes realistic) model of the source intensity is a 2-d grid of -functions whose strengths are proportional to the intensity. The model can be thought of as a `bed of nails' with strengths , where and are the element separations on a grid in two orthogonal sky coordinates. The visibility predicted by this model is given by

For simplicity we notate the discrete model as . Assuming reasonably uniform
sampling of a region of the *u*,*v* plane, one can expect to estimate
source features with widths ranging from up to
. The grid spacings, and ,
and the number of pixels on each axis, and , must be chosen
so that all these scales can be represented. In terms of the range of
*u*,*v* points sampled, the requirements are:

- ,
- ,
- , and
- .

The model has free parameters: the
flux densities in each cell. The measurements
constrain the model such that at the sampled *u*,*v* points

where is a complex, normally-distributed random error due to
receiver noise, and *r* indexes the samples.

At points in the *u*,*v* plane where no sample was taken, the transform
of the model can have any value without conflicting with the data. One
can think of Equation 3 as a multiplicative relation

where *W*(*u*,*v*) is a weighted sampling function which is non-zero only
where we have samples in the *u*,*v* plane,

By the convolution theorem, this corresponds to a convolution relation in the image plane:

where

and

in Equation 6 is the noise image obtained by
replacing *V* in Equation 7 by . Note that the
given by Equation 8 is the point spread
function (beam) that is synthesized after all weighting has been
applied (and after gridding and grid correction if an FFT was used; to
keep the notation concise, the gridding and grid correction are not
explicitly included). The Hermitian nature of the visibility has been
used in this rearrangement.

Equation 6 represents the constraint that the model
, when convolved with the point spread function
(also known as the *dirty beam*) corresponding to the
sampled and weighted *u*,*v* coverage, should yield (known
as the *dirty image*).

The weight function *W*(*u*,*v*) can be chosen to favor certain aspects of
the data. For example, setting to the reciprocal of the
variance of the error in optimizes the signal-to-noise ratio in
the final image. Setting to the reciprocal of some
approximation of the local density of samples minimizes the sidelobe
level.

1996 November 4

10:52:31 EST