Introduction

Below is a figure inspired by Dale Gary, which summarises the problem XRAYVISION tries to solve. The top row shows a source map, the point spread function (PSF) or dirty beam and the convolution of two, the dirty map (left to right). The bottom row shows the corresponding visibilities, notice the convolution is replaced by multiplication in frequency space. The problem is given the measured visibilities in f can the original map a be recovered?

(Source code, png, hires.png, pdf)

Theory

Synthesis imaging relies upon describing the amplitude of some quantity on the sky (radio flux or x-ray photon flux) in terms of complex visibilities as:

(1)\[I(l,m) = \int^{\infty}_{-\infty}\int^{\infty}_{-\infty}V(u, v)e^{-2 i \pi(ul+vm}) du dv\]

and the complex visibilties are given by:

(2)\[V(u,v) = \int^{\infty}_{-\infty}\int^{\infty}_{-\infty}I(l, m)e^{2 i \pi(ux+vy}) dl dm\]

In the case where the \(u, v\) plane is fully sampled the amplitude can be retrieved by simple inversion. Any real instrument only nosily samples the \(u, v\) plane. Ignoring noise this sampling function can repented as a series of delta functions and written as

\[S(u,v) = \sum_{i} w_{i} \delta (u-u_{i}) \delta ( v - v_{i})\]

substituting this into (1) we obtain the dirty image

\[I^{D} = \mathscr{F}^{-1} SV\]

applying the convolution theorem

\[I^{D} = B * I\]

where \(B = \mathscr{F}^{-1} S\) is the point spread function (PSF) also known as the dirty beam given by

\[B(l, m) = \sum_{i} e^{-2 i \pi(u_{i}l+v_{i}m)}w_{i}.\]

So the problem is to deconvolve the effects of the PSF or diry beam \(B\) from the dirty image \(I^{D}\) to obtain the true image \(I\).

Implementation

In reality the integrals above must be turned into summations over finite coordinates so (1) can be written as

\[I(l_i, m_j) = \sum_{k=0}^{N} e^{2 \pi i ( l_i u_k + m_i v_k)}\]

where \(x_i\)