Dirty Map -- from Eric Weisstein's World of Physics

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Dirty Map

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

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 , is measured. It is given by
where S(u,v) is the sampling function and V(u,v) is the observed visibility function. Let denote convolution Eric Weisstein's World of Math and rearrange the convolution theorem, Eric Weisstein's World of Math

into the form

from which it follows that

Now note that

is the CLEAN map, and define the "dirty beam" as the inverse Fourier transform Eric Weisstein's World of Math of the sampling function,

The dirty map is then given by

In order to deconvolve the desired CLEAN map I from the measured dirty map and the known dirty beam , the CLEAN algorithm is often used.

CLEAN Algorithm, CLEAN Map, Dirty Beam