The inversion of the visibilities to make the *Dirty Map* is done
using FFT algorithm which requires that the function be sampled at
regular intervals and the number of samples be power of 2. For the
case of mapping the sky using an aperture synthesis telescope, this
implies that the visibility data be available on a regular 2D grid in
the plane. Thus re-gridding of the data onto a regular grid is
required by potentially interpolating the visibility to the grid
points, since the visibility function is measured at discrete
points which are not assured to be at regular intervals along
the and axis.

Interpolation of is done by multiplying a function and averaging all the measured points which lie within the range of the function with a finite support base, centered at each grid point. The resultant average value is assigned to the corresponding grid point. This operation is equivalent to discrete convolution of with the above mentioned function and then sampling this convolution at the grid points. The convolving function is referred to as the Gridding Convolution Function (GCF). There are other ways of doing this interpolation. However the interpolation in practice is done by convolution since this results into predictable results in the map plane which are easy to visualize. Also using GCF with finite support base results into each grid point getting the value of the local average of the visibilities.

After gridding Eq. 11.2.5 becomes

The effect of gridding the visibilities on the map is to multiply the map with function and since has a finite support base (i.e. is of finite extent), is infinite in extent which result into aliasing in the map plane (the other cause of aliasing could be under-sampling of the -plane). The amplitude of the aliased component from a position in the map is determined by . Ideally therefore, this function should be rectangular function with the width equal to the size of the map and smoothly going to zero immediately outside the map. However from the point of efficiency of the gridding process, this is not possible, and GCF used in practice have a trade-off between the roll-off properties at the edge and flatness within the map.

Since the *Dirty Map* is multiplied by , if is well known,
then effect of convolution by the GCF can be removed by point-wise
division of *Dirty Map* and *Dirty Beam* given by
and for later processing, particularly in
deconvolution of . In practice however, this division is not
carried out by evaluating over the map. Instead, for
efficiency purposes, this function is kept in the computer memory
tabulated with a resolution typically times the size of the
cell in the image.

To take the Fourier transform of using the FFT algorithm one
needs to sample the right hand side of Eq. 11.3.6 by
multiplication with the re-sampling function given by

(11.3.7) |

where . Right hand side of this equation then is the approximation of obtained in practice. As discussed in earlier lecture, FFT generates a periodic function (due to the presence of in the left hand side of Eq. 11.3.8) and represents one period of such a function. To map an angular region of sky of size ( ), using the Nyquist sampling theorem we get and where and is the cell size in the map and and are cell sizes in the -plane.

is usually real and even and is assumed to be separable as . Various GCFs used in practice are listed below. Functions listed below are in 1-dimension and are truncated (set to zero) for where is the size of the grid and is the number of such cells used.

- `Pillbox' function

This amounts to simple averaging of all the -points with in the rectangle defined by Eq. 11.3.9. However since its Fourier transform is with large side lobes, it provides poor alias rejection and is almost never used but is useful for intuitive understanding.

- Truncated exponential function

Typically , and is used and can be expressed in terms of error function.

- Truncated function

For and , this is the normal function expressed in terms of function. As increases, the Fourier transform of this function approaches a step function which is constant over the map and zero outside.

- Sinc exponential function

(11.3.12) - Truncated spheroidal function

Of all the square integrable functions, this is the most optimal in the sense that it has maximum contribution to the normalized area from the part of which is with in the map. This is referred to as the

*energy concentration ratio*expressed as is maximized.