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:
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:
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