5 Application to XMM simulations

5.1 Introduction

One case in astronomical imagery where the PSF varies across the field of view (FOV) is with space X-ray telescope observations. In order to illustrate our methods we have selected one example from an ESA space mission, the X-ray Multi-Mirror Telescope (XMM). One feature of the X-ray images is that they are in a photon noise regime - practically the incoming photons on the detector are counted one by one and their energy is recorded. The resulting photon event list can be binned into an image by choosing both the pixel size and the energy band. The response of the telescope for incoming light from a point source (the PSF) depends on the position of the source across the FOV and also depends on the energy passband. What makes the task of object detection and reconstruction of the parameters difficult can be summarized as follows:

• few photons spread over a large area;
• object classification problems (AGN/point-like or extended) due to the non-constant PSF, and
• source confusion - large extended objects overlayed with many point-like sources.

5.2 Poisson noise modelling

If a wavelet coefficient w_j(x,y) is due to noise, it can be considered as a realization of the sum $\sum_{k \in K} n_k$ of independent random variables with the same distribution as that of the wavelet function (n_k being the number of events used for the calculation of w_j(x,y)). This allows comparison of the wavelet coefficients of the data with the values which can be taken by the sum of n independent variables.

The distribution of one event in wavelet space is then directly given by the histogram H₁ of the wavelet $\psi$. Since we consider independent events, the distribution of a coefficient w_n (note the changed subscripting for w, for convenience) related to n events is given by n autoconvolutions of H₁:

$$% H_n = H_1 \otimes H_1 \otimes ... \otimes H_1.$$ (10)

For a large number of events, H_n converges to a Gaussian. More details can be found in Starck & Pierre (1998).

5.3 Results

In our XMM example, we created simulated images including most of the telescope effects - PSF blurring, vignetting effect, particle and instrumental background. Our objective was to test the ability of the method to deconvolve images with a space-variant PSF. The image contains a set of point sources at different positions and with different fluxes. An energy band from 0.4 to 4.0 keV was used but this is irrelevant for our main aim. Figure 4, upper left, shows the position of the sources. On each radial line, the flux in the sources is identical, but the PSF becomes larger when the distance from the centre increases and the number of lost photons due to vignetting could well reach 50%. Fluxes run from 10 at left and counterclockwise with logarithmic step to 2000 ( [10,12,15,19,...,1002,1261,1588,1999]). Figure 4, upper right, shows the simulated data. The background is 10^-5 counts/pixel/second, so for 10 ks this corresponds to $\sim 0.1$ counts/pixel. Figure 4, bottom left and right, show respectively the result of the detection by the MVM with and without the PSF. Figure 5 shows the recovery of fluxes for the input sources after correction for the vignetting effect.

Figure 5: XMM model simulation. Results for the detections. The cross-identification searching radius was set to $12\hbox {$^{\prime \prime }$ }$ and the limiting distance from the centre to $14\hbox {$^\prime $ }$. The ratio of the detected counts ( $SCTS({\rm out})$) to the input source counts ( $SCTS({\rm in})$) is shown as a function of $SCTS({\rm in})$ (upper panel) and as a function of the off-axis distance (lower panel). The ratios of 1, 0.5 and 1.5 are indicated with continuous and dashed lines. Also the objects with distance greater than $4\hbox {$^{\prime \prime }$ }$ from their corresponding object are indicated with squares

One example of a "realistic'' image, with extended as well as point-like sources is shown in Fig. 6 and the reconstruction after deconvolution with MVM and PSF in Fig. 7.

Figure 6: Realistic X-ray image. The exposure time is 10 000 s, the point-like sources are distributed according to a $\log N-\log S$ relation (Hasinger et al. 1998), there are 5 extended sources - clusters of galaxies at 5 different redshifts (0.6, 1, 1.5, 1.8 and 2) with King $\beta$-model profiles and temperature 5 keV

Figure 7: Realistic X-ray image. The detected objects with $4\sigma$ significance are shown and the extended objects are indicated by an arrow and the redshift

The recovery of the input $\log N-\log S$ relation for the point-like sources is shown in Fig. 8 together with the distribution and the numbers of missing input objects and false detections.

Figure 8: Realistic X-ray image. The input $\log N-\log S$ relation together with the distribution of the non-detected input sources (blue histogram) and the distribution of the possible false detections (in green). Note however that the extended sources here are counted as false detections

Note that in order to perform the cross-identifications with the input list we take all the input sources with photon counts greater than 13. That is a rather low limit and it is the reason for the large number of missed detections.

More comprehensive analysis and comparison of this method with other methods dedicated to detection of objects in XMM-specific X-ray images can be found in Valtchanov (2000).