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

$$I(\xi, \eta) = \int_{-\infty}^\infty \int_{-\infty}^\infty V(u,v)e^{2\pi i(\xi u+\eta v)}\,du\,dv$$

$$= {\mathcal F}^{-1}[V(u,v)].$$ (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

$$I'(\xi, \eta) = \int_{-\infty}^\infty \int_{-\infty}^\infty S(u,v)V(u,v)e^{2\pi i(\xi u+\eta v)}\,du\,dv$$

$$= {\mathcal F}^{-1}[VS],$$ (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,

$${\mathcal F}[f*g]={\mathcal F}[f]{\mathcal F}[g]$$ (3)

into the form

$${\mathcal F}[{\mathcal F}^{-1}[f]*{\mathcal F}^{-1}[g]]=fg,$$ (4)

from which it follows that

$${\mathcal F}^{-1}[f]*{\mathcal F}^{-1}[g]={\mathcal F}^{-1}[fg].$$ (5)

Now note that

$$I = {\mathcal F}^{-1}[V]$$ (6)

is the CLEAN Map, and define the ``Dirty Beam'' as the inverse Fourier Transform of the sampling function,

$$b'\equiv {\mathcal F}^{-1}[S].$$ (7)

The dirty map is then given by

$$I' = {\mathcal F}^{-1}[VS] = {\mathcal F}^{-1}[V]*{\mathcal F}^{-1}[S] = I*b'.$$ (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