We demonstrate the deconvolution capabilities of MIM as well as the Gibbs
phenomena by a test square
of 
pixels, with constant
intensity
inside and zero outside (frame size 
pixels).
The dirty map was constructed by
convolving
this model with a given VLBI beam (taken from the data described in sec.6).
The size of the main beam and, thus, of the smoothing beam is
pixels (Fig. 1 (click here)d).
The input data are given
in Figs. 1 (click here)a-f. Note that the maximum height
of the DM is more than 300 times
the height of what a single point
source in Fig. 1 (click here)a would give. That is, in
the center of the DM, only about 0.3% of the intensity come from emission
at that point (Fig. 1 (click here)f).
Figure 2: Results of deconvolution of the square
(inner parts of maps, pixels), and intensity histograms in
units of percent of correct result in square. The correct result would
be a flat square without modulation, and, in the histograms,
1600 times 100% and 0% for all the rest of the pixels).
a,b) MIM. c,d) Richardson-Lucy
50 iterations (with slightly changed
DM and DB). e,f): MIM with automatic edge correction
Results are given in Fig. 2 (click here). The left sides show the inner 40
40
pixels. The right sides give intensity histograms of the whole
map in units of % of the
expected result within the square. The correct result would be
100% and 14784 0%. With full smoothing
(Fig. 2 (click here)a), the resulting
Gibbs phenomenon parallel to the edges amounts to less than 
15%
in this particular case (width of the right spike in Fig. 2 (click here)b).
These high intensities cover the inner 
pixels of the
map (no intensity
below 84%). The other
spikes in Fig. 2 (click here)b at 65% (the edge line of 156 pixels at the inner side
of the circumference of the square which should have 100% in a perfect
deconvolution) and at 30% (the adjacent
outer line of 164 pixels which should be empty in a perfect deconvolution)
stem from the smoothing of the edges. The rest of the frame (14620 pixels)
is essentially empty and contains less than 0.3% of the total intensity.
Comparison with
the Richardson-Lucy method (RL; Lucy 1974;
Hook & Lucy 1992) as contained
in the Nov94
version of MIDAS:
RL needs a non-negative beam. We constructed it by setting negative values
of the beam (Fig. 1 (click here)c) to zero, and used a correspondingly changed DM.
Also, due to the true Fourier transforms used, RL needs
a rather large DM (we used 
pixels) in order to avoid aliasing. In
other words, the deconvolved map must be large enough to be essentially empty.
Then the result (Figs. 2 (click here)c,d) is similar to our fully smoothed result.
That the Gibbs phenomenon can be reduced by automatic edge detection and corresponding correction of the dirty map is shown in Figs. 2 (click here)e+f. As already mentioned, our detection code is, at present, not reliable enough to report on further details.
Figure 3: Point source results:
Inner parts ( pixels) and central
cross sections of deconvolved maps.
a,b) Point source (intensity 30 000) on top
of extended emission (intensity 300). Gibbs correction for the square,
i.e., the remaining Gibbs phenomenon is due to the point source only.
c,d) Same point source without underlying square (isolated point source).
The Gibbs phenomenon has completely vanished. The reason is the
non-negativity constraint in the deconvolution which effectively
suppresses the first negative ring and, thus, all outer positive
rings. Also, the width is definitely
narrower than in a). e,f)
Point source (intensity 300) on top of square (intensity 300).
"Correct'' Gibbs correction but, for the point source, at a position that
is shifted by 1 pixel from the correct one
Another test was to put, in the center of Fig. 1 (click here)a, a point source of height 30000 on top of the square with height 300. That is, the point source has an intensity of 1/16 of the total intensity of the square.
Figure 4: Results on 3C 309.1.
a) DM. b) Difference image of
central source and jet (MIM minus
CLEAN - see text). The image is enlarged, as compared to c) and
d),
by a factor of 3.2. The dark regions (MIM is brighter) are surrounded
by brighter rims (here CLEAN is brighter), indicating the better
resolution of MIM. c) MIM, 0.2 mas pixel
size (map is 
pixels). North is top, east is left. d) Contour plot of same
(levels 
,..,
in units of the central point source)
Figure 3 (click here) gives some MIM results. The circular Gibbs phenomenon due to the point source is quite drastic. The first negative ring reaches zero intensity. Such Gibbs phenomena are strongly suppressed (Figs. 3 (click here)c,d) for a point source that is isolated rather than situated on top of an extended source if, as in most deconvolution routines, the result (= intensities) is forced to be non-negative. Also, the point source appears much narrower in Fig. 3 (click here)d than in Fig. 3 (click here)b.
Interesting is the superposition of a positive and a negative Gibbs phenomenon if the assumed position of the point source where smoothing is suppressed is chosen wrong by one pixel (Figs. 3 (click here)e,3 (click here)f).