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