     Next: Singular value decomposition Up: Deconvolution Tutorial Previous: Comparing `CLEAN' and MEM

# Algebraic deconvolution

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).     Next: Singular value decomposition Up: Deconvolution Tutorial Previous: Comparing `CLEAN' and MEM 1996 November 4
10:52:31 EST