The combination of Gridding and Sampling produces the uv data set
The Fourier Transform of this uv data set is
Accordingly, aliasing of in the map domain will thus occur.
Note at this stage that, providing aliasing of
remains negligible, an exact convolution equation is preserved
![]() |
(13.17) |
The gridding function will thus have to be selected to minimize
aliasing of . This criterion will depend on the image fidelity
required. Obviously, if the data is very noisy, aliasing of the
can be completely negligible.
Furthermore, the weighting function W is usually smooth, while the gridding
function G is a relatively sharp function (since it ensures the re-gridding by
convolution from nearby data points). Thus, to first order
,
and we could rewrite Eq.13.14 as
Let us focus on the choice of the gridding function. The gridding function will be selected according to the following principles:
Points 1 and 2 are contradictory, since a small support for Gimplies a large extent of . Some compromise is required.
For simplicity, gridding functions are usually selected among those
with separable variables:
The simplest gridding function is the Rectangular function
G(u) | = | ![]() |
(13.19) |
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= | ![]() |
(13.20) |
A better choice could be the Gaussian function
G(u) | = | ![]() |
(13.21) |
![]() |
= | ![]() |
(13.22) |
However, a Gaussian still has fairly significant wings. should ideally be
a rectangular function (1 inside the map, 0 outside). G would be a sinc
function, but this falls off too slowly, and would require a lot of computations in
the gridding. Moreover, the (unavoidable) truncation of G would destroy the sharp
edges of
anyhow. Hence the idea to use an apodized version of the sinc function, the Gaussian-Sinc function
G(u) | = | ![]() |
(13.23) |
![]() |
= | ![]() |
(13.24) |
The empirical approaches mentioned above do not guarantee any optimal choice of the
gridding function. A completely different approach is based on the desired
properties of the gridding function. We actually want to fall off as
quickly as possible, but G to be support limited. Mathematically, this defines a
class of functions known as Spheroidal functions. Spheroidal functions are
solutions of differential equations, and cannot be expressed analytically. In
practice, this is not a severe limitation since numerical representations can be
obtained by tabulating the gridding function values. Given the limited numerical
accuracy of the computations, the tabulation does not require a prohibitively fine
sampling of the gridding function, and is quite practical both in term of memory
usage and computation speed. Tabulated values are used in the task UV_MAP.
Note that the finite accuracy of the computation may ultimately limit the image dynamic range.