6 Test 3 - Realistic model with only point-like sources

6.1 Input configuration

We simulate an extragalactic field including only point-like sources with fluxes drawn from the $\log N-\log S$ relation (Hasinger et al. 1998, 2001; Giacconi et al. 2000). PSF, vignetting and background models are applied as described in Sect. 2. The aim is to test the detection procedures in more realistic cases where confusion and blending effects are important and not controlled. The raw photon image is shown in Fig. 8 together with its visual representation - the same input configuration for a much larger exposure time and no background, only keeping the objects with counts greater than 10. It displays better the input object sizes, fluxes and positions and it is instructive to compare it to the MR/1 filtered and WAVDETECT images shown on the same figure.

Figure 7: Test 2. WAVDETECT classification based on object size to PSF size ratio - $R_{\rm PSF}$. The identified extended (filled circles) and point-like sources (triangles) are plotted over the results from 10 simulated images with only point-like sources (see Sect. 6), the dashed line marks a ratio of unity

6.2 Cross-identification and positional error

We need to define a searching radius in order to cross-identify the output and the input lists. The input list contains many objects with counts well below the detection limit ( $\log N-\log S$ extends to very faint fluxes) and a lower limit must be chosen. For each detected object, we search for the nearest neighbour inside a circle within the reduced input list.

Figure 8: Test 3. A simulated XMM-Newton extragalactic field with only point-like sources for 10 ks exposure time and the total sensitivity of the three EPIC instruments (upper left), its representation for much larger exposure time and no background (upper right). The MR/1+SE filtering (lower left) and WAVDETECT images both with 10-4 significance threshold (lower right) are shown

 

 

Table 6: One sigma positional error and number of detected objects inside the inner $10\hbox {$^\prime $ }$ from the center of the FOV and more than 100 counts for a 10 ks exposure

Procedure	$\Delta r\hbox{$^{\prime\prime}$ }$	number
EMLDETECT	2.9	13
G+SE	3.5	14
MR/1+SE	3.2	13
WAVDETECT	4.1	12

The positional difference for the brightest detected sources (more than 100 counts) in the inner $10\hbox {$^\prime $ }$ from the center of the FOV is shown in Table 6. The region beyond $10\hbox {$^\prime $ }$ is subject to serious problems caused by the vignetting and PSF blurring, the detected object centroid can be few PSFs widths from the true input identification.

Figure 9: Test 3. Recovery of the input flux (upper panel of each figure). The continuous line is exact match between detected and input counts while the dashed lines are for two times differences. The limit of 50 input counts is shown by a vertical dotted line and the mean and the st.dev. (in brackets) of SCTS(out)/SCTS(in) for the two regions are indicated. The black circles denote objects with detect-input position difference larger than $4\hbox {$^{\prime \prime }$ }$, suggesting blending effects; the grey circles denote objects with more than one neighbour inside the searching radius. In the bottom panels, the corresponding rate and distribution of input counts (continuous histogram), missed input objects (dashed histogram) and possible false detections (dotted histogram) are shown

We therefore adopt the following cross-identification parameters: the input list is constrained to counts greater than 10; a $6\hbox {$^{\prime \prime }$ }$searching radius; we consider only the central $10\hbox {$^\prime $ }$ of the FOV.

6.3 Detection rate and photometric reconstruction

The detection rate and flux reconstruction results are shown in Fig. 9. There are different effects playing a role in the distribution and the numbers of missed and false detections:

(1)   "false'' detections - non-existent objects, or two or more sources blended into a single detected object. The result will be a "false'' detection if the blended objects are not        in the input list (count limit) or the merged object centroid is beyond the searching radius;

(2)   source confusion - in the cross-identification process the nearest neighbour to the detected source is not the true assignment; or as in case (1), when a blend of         objects is wrongly identified by one input source;

(3)   missed detections - depending on the local noise properties, some objects can be missed even if their input counts are above the adopted limiting counts for         cross-identification.

6.4 Discussion

The results in terms of detection rate are similar for all procedures. The best detection rate shows G+SE but at the price of twice as many false detections.

The photometry reconstruction for the sources above 50 counts shows a spread about 10-15% for the WT based methods and $\sim$$25\%$ for EMLDETECT. However, EMLDETECT clearly outperforms the other procedures when we use the same PSF model as the one hard-coded into the programme. This fact shows that using a correct PSF representation has a crucial importance for the ML technique. More discussion about the detection limits, completeness and confusion is left for Sect. 8.