13.1.3 Gridding Process

The goal of gridding is to resample the visibilities on a regular grid for subsequent use of the FFT. At each grid point, gridding involves some sort of interpolation from the values of the nearest visibilities. The visibilities being affected by noise, the interpolating function needs not fit exactly the original data points. Although any interpolation scheme could a priori be used, such as smoothing spline functions, it is customary to use a convolution technique to perform the gridding.

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 $\hat{B}$, 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

$$I_w^g \ensuremath{\rightleftharpoons} G \ensuremath{\ast\!\ast} (S \times W \times V)$$ (13.6)

  

$$I_w^g \ensuremath{\rightleftharpoons} G \ensuremath{\ast\!\ast} (S \times W \times V) I_w^g = \hat{G} \times (\widehat{S W} \ensuremath{\ast\!\ast}\hat{V}) = \hat{G} \times I_w$$ (13.7)

$$D_w^g \ensuremath{\rightleftharpoons} G \ensuremath{\ast\!\ast} (S \times W) D_w^g = \hat{G} \times \widehat{S W}$$ (13.8)

$$\frac{I_w^g}{\hat{G}} = \frac{D_w^g}{\hat{G}} \ensuremath{\ast\!\ast} (B I)$$ (13.9)

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.