15.3 Convolution and Aliasing

The combination of Gridding and Sampling produces the $uv$ data set

$$V_m = \frac{1}{\Delta u \Delta v} \ensuremath{\mathrm{III}}(\frac{u}{\D... ...rac{v}{\Delta v}) \times (G \ensuremath{\ast\!\ast}(S \times W \times V)) (u,v)$$ (15.13)

$$= \ensuremath{\mathrm{III}}\times (G \ensuremath{\ast\!\ast}(S \times W \times V)) / (\Delta u \Delta v)$$ (15.14)

which analogous with Eq.15.6

The Fourier Transform of this $uv$ data set is

$$\widehat{V_m} = \ensuremath{\mathrm{III}}(x\Delta u, y\Delta v) \ensuremath{\ast\!\ast}(\hat{G} \times (\widehat{SW} \ensuremath{\ast\!\ast}\hat{V} ))$$ (15.15)

$$= \ensuremath{\mathrm{III}}\ensuremath{\ast\!\ast}(\hat{G} \times (\widehat{SW} \ensuremath{\ast\!\ast}(BI)))$$ (15.16)

$\widehat{V_m}$ is thus the sky brightness multiplied by the primary beam (BI), convolved by the the dirty beam $\widehat{SW}$, then multiplied by the Fourier transform of the gridding function $\hat{G}$ and periodically replicated (by the convolution with the Shah function).

Accordingly, aliasing of $\hat{G}$ in the map domain will thus occur. Note at this stage that, providing aliasing of $\hat{G}$ remains negligible, an exact convolution equation is preserved

$$\frac{\widehat{V_m}}{\hat{G}} = \widehat{SW} \ensuremath{\ast\!\ast}BI$$ (15.17)

The gridding function will thus have to be selected to minimize aliasing of $\hat{G}$. This criterion will depend on the image fidelity required. Obviously, if the data is very noisy, aliasing of the $\hat{G}$ can be completely negligible.

Furthermore, the weighting function $W$ is usually smooth, while the gridding function $G$ is a relatively sharp function (since it ensures the re-gridding by convolution from nearby data points). Thus, to first order $G \ensuremath{\ast\!\ast}W = W$, and we could rewrite Eq.15.14 as

$$V_m = \ensuremath{\mathrm{III}}\times W \times (G \ensuremath{\ast\!\ast}(S \times V)) / (\Delta u \Delta v)$$ (15.18)

Hence, the weighting can be performed after the gridding. The choice of weighting before or after gridding is essentially based on computational speed or algorithmic simplicity.

Let us focus on the choice of the gridding function. The gridding function will be selected according to the following principles:

1. small support, typically one or two cells wide ($\Delta u$).
2. small aliasing.
3. fast computation.

Points 1 and 2 are contradictory, since a small support for $G$ implies a large extent of $\hat{G}$. Some compromise is required. For simplicity, gridding functions are usually selected among those with separable variables:

$$G(u,v) = G_1(u) G_1(v)$$

although this could break the rotation invariance.

The simplest gridding function is the Rectangular function

$$G(u) = \frac{1}{\Delta u} \Pi (\frac{u}{\Delta u})$$ (15.19)

$$\hat{G}(x) = \frac{sin(\pi \Delta u x)}{\pi \Delta u x}$$ (15.20)

where $\Pi$ is the unit rectangle function. Obviously, aliasing will be important, since the sinc function falls off very slowly.

A better choice could be the Gaussian function

$$G(u) = \frac{1}{\alpha \Delta u \sqrt{\pi}} e^{-(u/\alpha \Delta u)^2}$$ (15.21)

$$\hat{G}(x) = e^{-(\pi \alpha x \Delta u)^2}$$ (15.22)

By proper selection of $\alpha$ (not too small, not too large), a compromise between computation speed (better for small $\alpha$) and aliasing (better for large $\alpha$) can be found. $\alpha = 2\sqrt{ln(4)} \simeq 0.750$ is a standard choice.

However, a Gaussian still has fairly significant wings. $\hat{G}$ should ideally be a rectangular function (1 inside the map, 0 outside). $G$ would be a sinc function, but this falls off too slowly, and would require a lot of computations in the gridding. Moreover, the (unavoidable) truncation of $G$ would destroy the sharp edges of $\hat{G}$ anyhow. Hence the idea to use an apodized version of the sinc function, the Gaussian-Sinc function

$$G(u) = \frac{\sin{\pi u/(\alpha \Delta u)}}{\pi u} e^{-(u/(\beta \Delta u))^2}$$ (15.23)

$$\hat{G}(x) = \Pi(\alpha x \Delta u) \ast (\sqrt{\pi} \beta \Delta u e^{-(\pi \beta x \Delta u)^2})$$ (15.24)

It provides good performance for $\alpha = 1.55$ and $\beta = 2.52$.

The empirical approaches mentioned above do not guarantee any optimal choice of the gridding function. A completely different approach is based on the desired properties of the gridding function. We actually want $\hat{G}$ to fall off as quickly as possible, but $G$ to be support limited. Mathematically, this defines a class of functions known as Spheroidal functions. Spheroidal functions are solutions of differential equations, and cannot be expressed analytically. In practice, this is not a severe limitation since numerical representations can be obtained by tabulating the gridding function values. Given the limited numerical accuracy of the computations, the tabulation does not require a prohibitively fine sampling of the gridding function, and is quite practical both in term of memory usage and computation speed. Tabulated values are used in the task UV_MAP.

Note that the finite accuracy of the computation may ultimately limit the image dynamic range.