Gridding and Interpolation

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 $uv$ 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 $V(u,v)$ is measured at discrete points $(u,v)$ which are not assured to be at regular intervals along the $u$ and $v$ axis.

Interpolation of $V$ 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 $V$ 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

$$(V.S.W)*C \rightleftharpoons (I*DB).c,$$ (11.3.6)

where $C$ represents the GCF and $c \rightleftharpoons C$.

The effect of gridding the visibilities on the map is to multiply the map with function $c$ and since $C$ has a finite support base (i.e. is of finite extent), $c$ is infinite in extent which result into aliasing in the map plane (the other cause of aliasing could be under-sampling of the $uv$-plane). The amplitude of the aliased component from a position $(l,m)$ in the map is determined by $c(l,m)$. 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 $c$, if $c$ 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 $\bar I^d=I^d/c$ and $\bar{DB}=DB/c$ for later processing, particularly in deconvolution of $I^d$. In practice however, this division is not carried out by evaluating $c(l,m)$ over the map. Instead, for efficiency purposes, this function is kept in the computer memory tabulated with a resolution typically $1/100$ times the size of the cell in the image.

To take the Fourier transform of $(V.S.W)*C$ using the FFT algorithm one needs to sample the right hand side of Eq. 11.3.6 by multiplication with the re-sampling function $R$ given by

$$R(u,v)=\sum_{j=-\infty}^\infty \sum_{k=-\infty}^\infty \delta(j-u/\Delta u, k-v/\Delta v),$$ (11.3.7)

where $\Delta u$ and $\Delta v$ are the cell size in the $uv$-domain. Eq. 11.3.6 then becomes

$$R.((V.S.W)*C) \rightleftharpoons r*((I*DB).c),$$ (11.3.8)

where $R \rightleftharpoons r$. Right hand side of this equation then is the approximation of $I^d$ obtained in practice. As discussed in earlier lecture, FFT generates a periodic function (due to the presence of $R$ in the left hand side of Eq. 11.3.8) and $I^d$ represents one period of such a function. To map an angular region of sky of size ( $N_l \Delta l, N_m\Delta m$), using the Nyquist sampling theorem we get $N_l\Delta u = 1/\Delta l$ and $N_m\Delta v = 1/\Delta m$ where $\Delta l$ and $\Delta m$ is the cell size in the map and $\Delta u$ and $\Delta v$ are cell sizes in the $uv$-plane.

$C$ is usually real and even and is assumed to be separable as $C(u,v)=C_1(u)C_2(v)$. Various GCFs used in practice are listed below. Functions listed below are in 1-dimension and are truncated (set to zero) for $\vert u\vert \ge m\Delta u/2$ where $\Delta u$ is the size of the grid and $m$ is the number of such cells used.

1. `Pillbox' function

   
   

   $$C(u)=\left\{\matrix{1, ~\vert u\vert < m\Delta u/2\cr 0, ~otherwise\cr } \right\}.$$ (11.3.9)

   
   

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

2. Truncated exponential function

   
   

   $$C(u)=e^{{-\vert u\vert^\alpha \over {w\Delta u}}}.$$ (11.3.10)

   
   

   Typically $m=6$, $w=1$ and $\alpha=2$ is used and $c$ can be expressed in terms of error function.

3. Truncated $sinc$ function

   
   

   $$C(u)=sinc\left({u \over w\Delta u}\right).$$ (11.3.11)

   
   

   For $m=6$ and $w=1$, this is the normal $sinc$ function expressed in terms of $sin$ function. As $m$ increases, the Fourier transform of this function approaches a step function which is constant over the map and zero outside.

4. Sinc exponential function

   
   

   $$C(u)=e^{{-\vert u\vert^\alpha \over {w_1\Delta u}}} sinc\left({u \over w_2\Delta u}\right).$$ (11.3.12)

   
   
   For $m=6$, $w_1=2.52$, $w_2=1.55$, $\alpha=2$, the above equation reduces to multiplication of gaussian with the exponential function. This optimizes between the flat response of exponential within the map and suppression of the side-lobes due the presence of the gaussian.

5. Truncated spheroidal function

   
   

   $$C(u)=\vert 1-\eta^2(u)\vert^\alpha \phi_{\alpha0}(\pi m/2.\eta(u)),$$ (11.3.13)

   
   
   where $\phi_{\alpha0}$ is the 0-order spheroidal function, $\eta(u)=2u/m\Delta u$ and $\alpha > -1$.

   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 $c(l)$ which is with in the map. This is referred to as the energy concentration ratio expressed as ${\int_{map}\vert c(l)\vert^2 d l \over {\int\limits_{-\infty}^\infty\vert c(l)\vert^2 d l}}$ is maximized.