Lecture - Fundamental Concepts

Sampling

As previously mentioned, the visibilities are only measured at discrete locations $(u_\text{l}, v_\text{l})$. Therefore, the measured visibilities are given by the multiplication of the true visibility function $V$ and a sampling function $S$. Hence, the dirty image $B^\text{D}$ is then given by the convolution of their Fourier transforms:

$\displaystyle B^\text{D} = \mathcal{FT}[V^\text{S}] = \mathcal{FT}[V \cdot S] = \mathcal{FT}[V] \star \star \mathcal{FT}[S] \text{,}$

in which $\mathcal{FT}[x]$ denotes the Fourier transform of $x$ and the double-star $\star \star$ indicates a two-dimensional convolution.

Weighting

Furthermore, as also previously mentioned, the visibilities can be weighted by multiplying a weighting function $W$ to the visibilities $V^\text{S}$:

$\displaystyle V^\text{W} = W \cdot V^\text{S} = W \cdot V \cdot S \text{.}$

This weighting function $W$ consists of three individual factors:

• $R_\text{l}$ accounts for different telescope properties within an array (e.g. different $A_\text{eff}$, $T_\text{sys}$, $\Delta \nu$ and $\tau_\text{a}$),
• $T_\text{l}$ is a taper that controls the beam shape,
• $D_\text{l}$ weights the density of the measured visibilities.

The $D_\text{l}$-factor affects the angular resolution and sensitivity of the array. On the one hand side, the highest angular resolution can be achieved by weighting the visibilities as if they had been measured uniformly over the entire $(u,v)$-plane. Therefore, this weighting scheme is called uniform weighting. Since the density of the measured visibilities is higher at the center of the $(u,v)$-plane, the visibilities in the outer part are over-weighted leading to the highest angular resolution. On the other hand, the highest sensitivity is achieved if all measured visibilities are weighted by identical weights. This weighting scheme is called natural weighting. The main properties of both schemes are summarized in the following:

• natural weighting:
  • $D_\text{l} = 1$
  • broader synthesized beam
  • highest sensitivity

• uniform weighting:
  • $D_\text{l} = \frac{1}{n(\text{l})}$, in which $n(\text{l})$ is the number of visibilities occurring within an area of constant size around the weighted visibility
  • highest resolution
  • lowest sensitivity

With this weighting function $W$, the dirty image $B^\text{D}$ is given by

$\displaystyle B^\text{D} = \mathcal{FT}[V^\text{W}] = \mathcal{FT}[W \cdot V^\text{S}] = \mathcal{FT}[W] \star \star \mathcal{FT}[V^\text{S}] = \mathcal{FT}[W] \star \star (\mathcal{FT}[V] \star \star \mathcal{FT}[S]) \text{.}$

Gridding

To use the time advantage of the fast Fourier transform (FFT) algorithm, the visibilities must be interpolated onto a regular grid of size $2^{\text{M}_\text{V}}\times 2^{\text{N}_\text{V}}\,(\text{e.g.}\,256\times 256,\,512\times 512,\, 1024 \times 1024,\,...)$, resulting in an image of size $2^{\text{M}_\text{B}}\times 2^{\text{N}_\text{B}}$. This interpolation, also called gridding, is done at first by convolving the weighted discrete visibility $V^\text{W}$ with an appropriate function $C$ to obtain a continuous visibility distribution. This continuous visibility distribution is then resampled at points of the regular grid with spacings $\Delta u$ and $\Delta v$ by multiplying a two-dimensional Shah-function $G$, given by

Fig. 2.32 Illustration of the gridding process. The real $(u,v)$-tracks measured by the IRAM interferometer at Plateau de Bure in France (black dots) have to be interpolated onto a regular grid of $2^N\times2^N$ pixels (red dots). Taken from: U. Klein

$\displaystyle G(u,v) = \sum_{\text{j} = -\infty}^{\infty} \sum_{\text{k} = -\infty}^{\infty} \delta^2\left(j-\frac{u}{\Delta u}, k-\frac{v}{\Delta v}\right) \text{.}$

After these modifications, the visibility $V^\text{G}$ is given by

$\displaystyle V^\text{G} = G \cdot (C \star \star V^\text{W}) = G \cdot [C \star \star (W \cdot V^\text{S})] = G \cdot [C \star \star (W \cdot V \cdot S)]$

and the dirty image $\tilde{B^\text{D}}$ reads

$\displaystyle \tilde{B^\text{D}} = \mathcal{FT}[G] \star \star (\mathcal{FT}[C] \cdot \mathcal{FT}[V^\text{W}]) =  \mathcal{FT}[G] \star \star \{ \mathcal{FT}[C] \cdot [\mathcal{FT}[W] \star \star (\mathcal{FT}[V] \star \star \mathcal{FT}[S])]\} \text{.}$ 

This gridding process is illustrated in the Fig. 2.29, in which real $(u,v)$-tracks measured by the IRAM interferometer at Plateau de Bure in France are shown as black dotted ellipses. Furthermore, a regular grid, onto which the visibilities have been interpolated, is shown as red dots.

Since the sampling intervals $\Delta u$ and $\Delta v$ in the $(u,v)$-plane are inversely proportional to the sampling intervals $\Delta \xi$ and $\Delta \eta$ in the image plane ( $\Delta u^{-1} = M\Delta \xi$ and $\Delta v^{-1} = N\Delta \eta$ for a grid of size $M\times N$), the maximum map size in one domain is given by the minimum sampling interval in the other. Therefore, it is important to choose appropriate sampling intervals $\Delta u$ and $\Delta v$. If $\Delta u$ and $\Delta v$ are chosen to be too large for example, this will result in artefacts in the image plane produced by reflections of structures from the map edges, which is called aliasing.

The most effective way to deal with this aliasing is to use a convolution function $C$ for which the Fourier transform in the image plane $\mathcal{FT}[C]$ decreases rapidly at the image edges and is nearly constant over the image. Therefore, the simplest choice for the convolution function $C$ is a rectangular function, but with this choice the aliasing would be strongest. A better choice would be a sinc function, however a Gaussian-sinc function (product of a Gaussian and a sinc function) leads to the best suppression of aliasing.

Finally, the gridding modifications must be corrected after the Fourier transform by dividing the dirty image $\tilde{B^\text{D}}(\xi_\text{m},\eta_\text{m})$ by the inverse Fourier transform of the gridding convolution function $C(u,v)$. Therefore, the so-called grid-corrected image $\tilde{B^\text{D}_\text{C}}(\xi_\text{m},\eta_\text{m})$ is given by

$\displaystyle \tilde{B^\text{D}_\text{C}}(\xi_\text{m},\eta_\text{m}) = \frac{\tilde{B^\text{D}}(\xi_\text{m},\eta_\text{m})}{\mathcal{FT}[C](u,v)} \text{.}$