Using a convolution is justified by several arguments. First, from Eq.13.1,
. Hence V is already a convolution
of a (nearly Gaussian) function
with the Fourier Transform of I.
Nearby visibilities are not independent. Second, as mentioned above, exact
interpolation not desirable, since original data points are noisy samples of a
smooth function. Third, if the width of the convolution kernel used in gridding is
small compared to
, the convolution added in the gridding process will not
significantly degrade the information. Last, but not least, it is actually possible
to correct for the effects of the convolution gridding.
To demonstrate that, let G be the gridding convolution kernel.
Eq.13.3 becomes
Thus the dirty image and dirty beams are obtained by dividing the Fourier Transform of the gridded data by the Fourier Transform of the gridding function.