Consider a situation in which some spatial frequencies that are present in the source model are not sampled by the data. The fit of the model to the data is unaffected by changing the amplitudes of the sinusoids corresponding to these frequencies.

The dirty beam filters out these un-sampled spatial frequencies: if *Z*
is an intensity distribution containing only such frequencies, then
*B***Z* = 0. Thus, if *I* is a solution of the convolution equation,
then so is where is any number. As usual, the
existence of homogeneous solutions implies the non-uniqueness of any
solution, *in the absence of boundary conditions.*

In interferometry, the solution in which all the un-sampled spatial
frequencies have zero amplitude is called the ``principal'' solution.
It is useful to think of the homogeneous solutions, or ``invisible
distributions'' (Bracewell & Roberts 1954),
as originating via two main shortcomings of our *u*,*v* coverage:

- the coverage extends only out to a finite spatial-frequency limit, and
- there are holes in the coverage.

Invisible distributions of the first sort correspond to finer detail
than can be resolved. We deal with these by accepting a
finite-resolution image as our final product. The most vexing
problems of image construction come from finding *plausible*
invisible distributions of the second sort to merge with the principal
solution. To see why we need to do this, consider the
shortcomings of the principal solution.

1996 November 4

10:52:31 EST