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5. Tests

We demonstrate the deconvolution capabilities of MIM as well as the Gibbs phenomena by a test square of tex2html_wrap_inline1095 pixels, with constant intensity inside and zero outside (frame size tex2html_wrap_inline1097 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 tex2html_wrap_inline1109 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, tex2html_wrap_inline1095 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 40tex2html_wrap_inline111340 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
tex2html_wrap_inline1115 100% and 14784 tex2html_wrap_inline1113 0%. With full smoothing (Fig. 2 (click here)a), the resulting Gibbs phenomenon parallel to the edges amounts to less than tex2html_wrap_inline111915% in this particular case (width of the right spike in Fig. 2 (click here)b). These high intensities cover the inner tex2html_wrap_inline1121 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 tex2html_wrap_inline1099 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 (tex2html_wrap_inline1095 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 tex2html_wrap_inline1127 pixels). North is top, east is left. d) Contour plot of same (levels tex2html_wrap_inline1129,.., 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).

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