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Dirty Map

From the van Cittert-Zernicke theorem, the relationship between observed visibility function $V(u,v)$ and source brightness $I(\xi,\eta)$ in synthesis imaging is given by

$\displaystyle I(\xi, \eta)$ $\textstyle =$ $\displaystyle \int_{-\infty}^\infty \int_{-\infty}^\infty V(u,v)e^{2\pi i(\xi u+\eta v)}\,du\,dv$  
  $\textstyle =$ $\displaystyle {\mathcal F}^{-1}[V(u,v)].$ (1)

But the visibility function is sampled only at discrete points $S(u,v)$ (finite sampling), so only an approximation to $I$, called the ``dirty map'' and denoted $I'$, is measured. It is given by
$\displaystyle I'(\xi, \eta)$ $\textstyle =$ $\displaystyle \int_{-\infty}^\infty \int_{-\infty}^\infty S(u,v)V(u,v)e^{2\pi i(\xi u+\eta v)}\,du\,dv$  
  $\textstyle =$ $\displaystyle {\mathcal F}^{-1}[VS],$ (2)

where $S(u,v)$ is the sampling function and $V(u,v)$ is the observed visibility function. Let $*$ denote Convolution and rearrange the Convolution Theorem,
{\mathcal F}[f*g]={\mathcal F}[f]{\mathcal F}[g]
\end{displaymath} (3)

into the form
{\mathcal F}[{\mathcal F}^{-1}[f]*{\mathcal F}^{-1}[g]]=fg,
\end{displaymath} (4)

from which it follows that
{\mathcal F}^{-1}[f]*{\mathcal F}^{-1}[g]={\mathcal F}^{-1}[fg].
\end{displaymath} (5)

Now note that
I = {\mathcal F}^{-1}[V]
\end{displaymath} (6)

is the CLEAN Map, and define the ``Dirty Beam'' as the inverse Fourier Transform of the sampling function,
b'\equiv {\mathcal F}^{-1}[S].
\end{displaymath} (7)

The dirty map is then given by
I' = {\mathcal F}^{-1}[VS] = {\mathcal F}^{-1}[V]*{\mathcal F}^{-1}[S] = I*b'.
\end{displaymath} (8)

In order to deconvolve the desired CLEAN Map $I$ from the measured dirty map $I'$ and the known Dirty Beam $b'$, the CLEAN Algorithm is often used.

See also CLEAN Algorithm, CLEAN Map, Dirty Beam

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© 1996-9 Eric W. Weisstein