From the van Cittert-Zernicke theorem, the relationship between observed
visibility function and source brightness in
synthesis imaging is given by
But the visibility function is sampled only at discrete points (finite sampling),
so only an approximation to , called the ``dirty map'' and denoted , is measured. It is given by
where is the sampling function and is the observed
visibility function. Let denote Convolution and rearrange the Convolution Theorem,
|
(3) |
into the form
|
(4) |
from which it follows that
|
(5) |
Now note that
|
(6) |
is the CLEAN Map, and define the ``Dirty Beam'' as the inverse Fourier Transform of the sampling function,
|
(7) |
The dirty map is then given by
|
(8) |
In order to deconvolve the desired CLEAN Map from the measured dirty map and the known Dirty Beam
, the CLEAN Algorithm is often used.
See also CLEAN Algorithm, CLEAN Map, Dirty Beam
© 1996-9 Eric W. Weisstein
1999-05-24