Zero-spacing Problem

Since visibility and the brightness distribution are related via a Fourier transform, $V(0,0)$ measures the total flux from the sky. However, since the difference between the antenna positions is always finite, $V(0,0)$ is never measured by an interferometer. For a point source, it is easy to estimate this value by extrapolation from the smallest $u$ and $v$ for which a measurement exist, since $V$ as a function of baseline length is constant. However for an extended source, this value remains unknown and extrapolation is difficult.

For the purpose of understanding the effect of missing zero-spacings, we can multiply the visibility in Eq. 11.3.6 by a rectangular function which is 0 around $(u,v)=(0,0)$ and 1 elsewhere. In the map domain then, the Dirty Map gets convolved with the Fourier transform of this function, which has a central negative lobe. As a result, extended sources will appear to be surrounded by negative brightness in the map which cannot be removed by any processing. This can only be removed by either estimating the zero-spacing flux while restoring $I$ from $I^d$ or $V$, or by supplying the zero-spacing flux as an external input to the mapping/deconvolution programs. The Maximum Entropy class of image restoration algorithms attempt to estimate the zero-spacing flux, while the CLEAN class of image restoration algorithms needs to be supplied this number externally. Both these will be discussed in the later lectures.