All smoothing deconvolutions show Gibbs phenomena ("ringing") near steep slopes, as point sources or edges, which are smeared out over the smoothing area. The data are not fitted well near such places. E.g., a smoothed point source has too little intensity in the center, and too much around. This overshooting of intensity in the vicinity of the point source is compensated by undershooting farther out, and this again by overshooting still farther out, etc.
In order to reduce such ringing, it is necessary to deconvolve the corresponding steep features with reduced smoothing. One possibility is Two Channel Restoration (Gratl 1998) in which steep features, especially point sources, are recognized and their convolution with the DB subtracted from the DM, followed by the deconvolution proper which can include smoothing because the steep features have been removed. The method permits subpixel accuracy and can be used with any deconvolution method.
MIM offers another possibility because it uses two different beams, a and s. Smoothing can be switched off locally by putting locally (Pfleiderer 1985). We at present prefer to change the DM such that the code will consider edges and point sources formally as being smooth. Subpixel accuracy is not possible.
The dirty map DM (errors neglected) is a convolution of the unknown
true map TM with the dirty beam a,
which is equivalent to
By adding the true map convolved with the smoothing beam to the dirty map, one can deconvolve with the smoothness-modified beam and still recover the edges. It follows from that is large only near edges.
On the other hand, the clean map CM of the smoothed deconvolution obeys
where r is the residual map. Subtraction from Eq. (10) gives
where TM-CM also is large only near edges. The right side being known, we can use Eq. (13) - not to properly solve it; it is too ill-conditioned - but to estimate the maxima of TM-CM and thus TM, or the edge height, near the edges. The corresponding estimate of , calculated only near suspected edges, is added to r, and the deconvolution with the modified beam is continued.
Figure 1: (a,b): top; e,f) bottom): Deconvolution of a square ( pixels in a frame). a) The model. b) uv coverage of the beam. c) Dirty beam (4 times larger: pixels). d) Central cross section through beam. e) Dirty map 128) = model a) convolved with DB c). f) Central cross section through DM e). Also, central cross section through model a), enlarged by a factor of 100 and shifted by 50 units. Finally, cross section through edge correction for DM, again multiplied by a factor 100
If the solution is known, as in the test examples of the next section, a perfect compensation is possible. We have done this in Figs. 3 (click here)a and 3 (click here)e for the square in order to show the Gibbs ringing of the point sources more clearly. The changes need not be large. For instance, the data of the DM (Fig. 1 (click here)e) have to be changed only near the edges of Fig. 1 (click here)a but nowhere by more than 0.3% (Fig. 1 (click here)f). Note that these changes of the DM are enlarged, in Fig. 1 (click here)f, by a factor 100.
In practice, one does not know the edges or point sources and can, thus, only estimate positions and heights of these from preliminary results of a smoothed deconvolution. The criterion for successful steepening is the decrease of the Gibbs phenomenon. We compensate for point sources interactively (example: the point source in Sect. 6). Edges are more difficult. We do not yet have a reliable code that finds all edges correctly and does not invent spurious edges. In simple cases, as the square of Sect. 5, our automatic code works sufficiently well (example: Fig. 2 (click here)e).