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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:

(1)  displaymath886

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 tex2html_wrap_inline797 -functions whose strengths are proportional to the intensity. The model can be thought of as a `bed of nails' with strengths tex2html_wrap_inline799 , where tex2html_wrap_inline801 and tex2html_wrap_inline803 are the element separations on a grid in two orthogonal sky coordinates. The visibility tex2html_wrap_inline805 predicted by this model is given by

(2)  displaymath888

For simplicity we notate the discrete model tex2html_wrap_inline799 as tex2html_wrap_inline809 . Assuming reasonably uniform sampling of a region of the u,v plane, one can expect to estimate source features with widths ranging from tex2html_wrap_inline815 up to tex2html_wrap_inline817 . The grid spacings, tex2html_wrap_inline801 and tex2html_wrap_inline803 , and the number of pixels on each axis, tex2html_wrap_inline823 and tex2html_wrap_inline825 , must be chosen so that all these scales can be represented. In terms of the range of u,v points sampled, the requirements are:

  1. tex2html_wrap_inline831 ,
  2. tex2html_wrap_inline833 ,
  3. tex2html_wrap_inline835 , and
  4. tex2html_wrap_inline837 .

The model has tex2html_wrap_inline839 free parameters: the flux densities tex2html_wrap_inline809 in each cell. The measurements constrain the model such that at the sampled u,v points

(3)  displaymath890

where tex2html_wrap_inline847 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

(4)  displaymath892

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

(5)  displaymath894

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

(6)  displaymath896


(7)  displaymath898


(8)  displaymath900

tex2html_wrap_inline861 in Equation 6 is the noise image obtained by replacing V in Equation 7 by tex2html_wrap_inline847 . Note that the tex2html_wrap_inline867 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 tex2html_wrap_inline809 , when convolved with the point spread function tex2html_wrap_inline867 (also known as the dirty beam) corresponding to the sampled and weighted u,v coverage, should yield tex2html_wrap_inline877 (known as the dirty image).

The weight function W(u,v) can be chosen to favor certain aspects of the data. For example, setting tex2html_wrap_inline881 to the reciprocal of the variance of the error in tex2html_wrap_inline883 optimizes the signal-to-noise ratio in the final image. Setting tex2html_wrap_inline881 to the reciprocal of some approximation of the local density of samples minimizes the sidelobe level.

next up previous contents external
Next: Solutions of the convolution equation Up: Deconvolution Tutorial Previous: Purpose

1996 November 4
10:52:31 EST