Since synthesis telescopes are linear devices, one might expect linear
algebra to be of use in image deconvolution. Andrews &
Hunt (1977) first analyzed image deconvolution problems
in terms of linear algebra. In principle, one can express the
deconvolution problem as a matrix equation 
where S is a
vector of the (unknown) intensity distribution on the sky and D
represents the observed data that constrain S via the measurement
matrix A.
In the image plane, D represents the pixel values in the dirty
image and A the dirty beam pattern that relates values in S and
D. The elements of S would be the strengths of the 
-function
components in the 'CLEAN' representation, for example.
In the u,v plane, D represents the real and imaginary parts of the visibility samples V(u,v), and A contains the sine and cosine terms that represent the Fourier transform relationship between S and D.
If the extent of the source brightness is poorly known then the S
vector can contain many elements. The A matrix is then almost
certainly singular, so there are either no solutions to , or
infinitely many (the ``invisible distribution'' problem). However, if
the source extent is sufficiently small then the A matrix may be
non-singular and a unique solution may be possible. Even if A is
mildly singular, it may be that quite reasonable contraints on the
solution S lead to an effectively unique solution. A serious
practical obstacle to the use of linear algebra in the past has been the
computing problem: since the size of A goes roughly as the square
of the number of pixels, for many solution algorithms, the solution time
goes roughly as the sixth power of the number of pixels. However,
modern workstations have sufficient resources to allow linear
algebra-based deconvolution of images with up to 5000-6000 pixels. Such
algorithms have been investigated by Briggs
(1995).
1996 November 4
10:52:31 EST