W-kernel support size

Introduction

To reserve sufficient space for the convolution by the w-kernel it is important to know the support in advance. This document describes the derivation of an approximate formula for the w-kernel support size

When the W-term is Nyquist sampled in the image domain, the support in the uv-domain fits exactly inside the grid. Strictly speaking the W-term can not be Nyquist sampled, because its support is infinite. However, the W-term can be considered Nyquist sampled when the phase change from pixel to pixel is smaller then $\pi$. This criterion cuts the tails of the convolution function at approximately the 1% level.

Derivation

Let the phase of the W-term be given as function of image coordinate $l$,

   \varphi(l) = 2\pi w(1-n(l)) = 2\pi w(1-\sqrt{1 - l^2}),

where $w$ is the w-coordinate in wavelengths.

The phase change from pixel to pixel is given by

    \Delta\varphi = \frac{\mathrm{d}\varphi(l)}{\mathrm{d}l}\Delta l,

where $\Delta l = s/N $ and $s$ is the image size in radians and $N$ is the image size in number of pixels.

The derivative is given by

    \frac{\mathrm{d}\varphi(l)}{\mathrm{d}l} = -2\pi w \frac{-l}{\sqrt{1 - l^2}}.

This can be simplified by using the approximation

    \sqrt{1 - l^2} \approx 1

for small $l$. And thus

    \frac{\mathrm{d}\varphi(l)}{\mathrm{d}l} \approx 2\pi w l.

The W-term changes fastest at the edge of the image where $l = s/2$. At the edge

    \Delta \varphi = \pi w s^2/N

Equating $\Delta \varphi$ to $\pi$ leads to the final result $N = w s^2$.

Introducing the shift

When imaging a facet in a tangent plane not for the center of the facet, the W-term will depend on the shift parameter as well.

Let the shift be given by $\delta l$. At the edge of the facet $l' = s/2 + \delta l$.

    \Delta\varphi = 2\pi w l' \Delta l = 2\pi w (s/2 + \delta l) s/N

Equation this to $\pi$ leads to

    N = w (s + 2\delta l)s