Theoretical understanding of `CLEAN' is relatively poor even though
the original algorithm is quite old. Schwarz (1978,
1979) analyzed the Högbom
`CLEAN' algorithm in detail. He notes that in the noise-free case the
least-squares minimization of the difference between observed and
model visibility, which `CLEAN' performs, produces a unique answer if
the number of cells in the model is not greater than the number of
independent visibility measurements contributing to the dirty image
and beam (*cf.* Equations 7 and 8),
counting real and imaginary parts separately. This rule is unaffected
by the distribution of *u*,*v* data so that, in principle,
super-resolution is possible if enough visibility samples are
available. In practice, however, the presence of noise and the use of
the FFT algorithm to calculate the dirty image and beam corrupt our
knowledge of the derivatives of the visibility function upon which
super-resolution is based. Clearly, even if the FFT is not used, the
presence of noise means that independence of the data must be
redefined. Schwarz produced a noise analysis of the least-squares
approach but it involves the inversion of a matrix of side
and so is impractical for large images; furthermore, we are really
interested in `CLEAN', not the more limited least-squares method,
since `CLEAN' will still produce a unique answer in circumstances
where the least-squares method is guaranteed to fail. To date no one
has succeeded in producing a noise analysis of `CLEAN' itself. The
existence of instabilities in `CLEAN' makes such an analysis highly
desirable.

Schwarz also proves three conditions for the convergence of `CLEAN':

- The beam must be symmetric.
- The beam must be positive definite or positive semi-definite. Thus the eigenvalues must be non-negative.
- The dirty image must be in the
*range*of the dirty beam. Roughly speaking, there must be no spatial frequencies present in the dirty image which are not also present in the dirty beam.

All three conditions are obeyed in principle for the dirty image and
beam calculated by Equations 7 and 8 if
the weight function *W* is nowhere negative. In practice, however,
numerical errors, and the gridding and grid-correction process may
create violations of these conditions, so `CLEAN' will eventually
diverge. `CLEAN'ing close to the edge of a dirty image computed by an
FFT is particularly risky.

Most of our understanding of `CLEAN' comes from a combination of guessing how to apply intuition and Schwarz's analysis to real cases, and much practical experience with real and test data. We will now try to summarize the available lore about how `CLEAN' should be used, and how it can fail.

1996 November 4

10:52:31 EST