15.1 Fourier Transform

The first step in imaging is to evaluate the dirty image, using Fourier Transform. Several techniques are available.

15.1.1 Direct Fourier Transform

The simplest approach would be to directly compute $\sin$ and $\cos$ functions in Eq.15.4 for all combinations of visibilities and pixels in the image. This is straightforward, but slow. For typical data set from the VLA, which contain up to $10^5$ visibilities per hour and usually require large images ( $1024 \times 1024$ pixels), the computation time can be prohibitive. On the other hand, the IRAM Plateau de Bure interferometer produces about $10^4$ visibilities per synthesis, and only require small images ( $128 \times 128$). The Direct Fourier Transform approach could actually be efficient on vector computers for spectral line data from Plateau de Bure interferometer, because the $\sin$ and $\cos$ functions needs to be evaluated only once for all channels. Moreover, the method is well suited to real-time display, since the dirty image can be easily updated for each new visibility.

15.1.2 Fast Fourier Transform

In practice, everybody uses the Fast Fourier Transform because of its definite speed advantage. The drawback of the methods is the need to regrid the visibilities (which are measured at arbitrary points in the $(u,v)$ plane) on a regular grid to be able to perform a 2-D FFT. This gridding process will introduce some distortion in the dirty image and dirty beams, which should be corrected afterwards. Moreover, the gridded visibilities are sampled on a finite ensemble. As discussed in more details below, this sampling introduces aliasing of the dirty image (and beam) in the map plane.

15.1.3 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.15.1, $V = \widehat{BI} = \hat{B} \ensuremath{\ast\!\ast}\hat{I}$. Hence $V$ is already a convolution of a (nearly Gaussian) function $\hat{B}$ with the Fourier Transform of $I$. Nearby visibilities are not independent. Second, as mentioned above, exact interpolation is 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.15.3 becomes

$$I_w^g \ensuremath{\rightleftharpoons}G \ensuremath{\ast\!\ast}(S \times W \times V)$$ (15.6)

We thus have for the image $I$ the following relations

$$I_w^g = \hat{G} \times (\widehat{S W} \ensuremath{\ast\!\ast}\hat{V}) = \hat{G} \times I_w$$ (15.7)

and for the dirty beams

$$D_w^g \ensuremath{\rightleftharpoons}G \ensuremath{\ast\!\ast}(S \times W)   D_w^g = \hat{G} \times \widehat{S W}$$ (15.8)

from which we derive the relation

$$\frac{I_w^g}{\hat{G}} = \frac{D_w^g}{\hat{G}} \ensuremath{\ast\!\ast}(B I)$$ (15.9)

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.