     Next: The `CLEAN' algorithm Up: Solutions of the convolution equation Previous: The ``principal solution'' and ``invisible distributions''

## Problems with the principal solution

If the visibility data V(u,v) are obtained on a regular grid, then the principal solution can easily be computed: one simply chooses the weight function W in Equation 7 to correct the bias in weight due to any vagaries of sampling. For each grid point, the visibility samples are summed with the appropriate weights, and the total weight is normalized to unity. Using such uniform weighting, the principal solution is the same as the dirty image: the convolution of the true brightness distribution with the dirty beam.

For most synthesis arrays now used in radio astronomy, the dirty beam has sidelobes in the range 1% to 10%. Sidelobes represent unavoidable ambiguity about the true distribution of emission in the dirty image. There are two ways to resolve this ambiguity:

1. make further observations to improve the sampling,
2. incorporate a priori information, e.g. about the extent of the emission on the sky, about positivity where appropriate, or about an upper bound to the degree of polarization, etc.

For example, consider uniformly weighted observations of a point source: the dirty image is the dirty beam centered on the point source position. Without a priori information we cannot distinguish whether the source is truly a point, or is shaped like the dirty beam. Of course we know that the Stokes parameter I must be positive, and that most radio sources are much better localized than dirty beams (they certainly do not have sidelobe patterns extending to infinity). A further unsatisfactory aspect of the principal solution, besides its implausibility, is that it changes (sometimes drastically) as more visibility data are added. A more stable estimator is obviously desirable.

The key to ``successful'' deconvolution is to make good use of such a priori information as positivity and the extent (support) of the radio source to steer the choice of invisible distributions to add while constructing the image. This process can also be thought of as specifying plausible boundary conditions while solving the deconvolution equation.

Image deconvolution in radio astronomy has so far been dominated by two algorithms, `CLEAN' and the Maximum Entropy Method (MEM), which solve the convolution equation by placing strikingly different constraints on the invisible distributions. This tutorial focuses on these dominant approaches, and on a third that has recently shown particular promise for VLBI imaging: direct algebraic solution of the convolution equation.     Next: The `CLEAN' algorithm Up: Solutions of the convolution equation Previous: The ``principal solution'' and ``invisible distributions'' 1996 November 4
10:52:31 EST