4 Source detection

Once all data have been calibrated, the final raster map R(x,y) and its associated rms map $R_\sigma(x,y)$ can be created. If several rasters of the same field are available, they can be co-added in order to improve the signal to noise ratio. The noise in R(x,y) (i.e. $R_\sigma(x,y)$) is non-homogeneously distributed over the map, first because some pixels have been masked (short glitches) and second because some areas of the field (particularly the border of the mosaic) present low redundancy (few readouts per sky position). For this reason the noise around the border of the image can be relatively high with respect to the noise toward the image center. Therefore, if we made the simple hypothesis of uniform noise (for instance Gaussian noise of standard deviation $\sigma$), it would lead to a large amount of false detections on the border. The correct solution is to use the $R_\sigma(x,y)$ map. In order to detect faint sources on the final image, we can use the multi-scale vision model (MVM) in two-dimensions. This time we use the "à trous'' algorithm because the linearity of the wavelet transform allows us to derive a robust modelling of the noise in wavelet space (using the rms map $R_\sigma(x,y)$), which is impossible using the MMT. Moreover, in this case the artefacts around sources are negligible since we have no strong sources.

For each wavelet coefficient w_j(x,y) of R, the exact standard deviation $\sigma_j(x,y)$ has to be calculated from the root mean square map $R_\sigma(x,y)$.

A wavelet coefficient w_j(x,y) is obtained by the correlation product between the image R and a function g_j:

$$w_j(x,y) = \sum_k \sum_l R(x,y) g_j(x+k,y+l)$$ (8)

then we have:

$$\sigma_j^2(x,y) = \sum_k \sum_l R_{\sigma}^2(x,y) g_j^2(x+k,y+l).$$ (9)

In the case of the "à trous'' algorithm, the coefficients g_j(x,y) are not known exactly, but they can be computed by taking the wavelet transform of a Dirac ($w^{\delta}$, in our notation):

$$g_j(x,y) = w_j^{\delta} (x,y).$$ (10)

Then the map $\sigma_j^2$ is calculated by correlating the square of the wavelet scale j of $w^{\delta}$ by $R^2_\sigma(x,y)$. A wavelet coefficient is significant if:

$$\mid w_j(x,y) \mid \gt N_\sigma \sigma_j(x,y).$$ (11)

$N_\sigma$ is a parameter fixing the confidence level (generally taken equal to 3). Once this step is performed, the object selection and their reconstructions can be done as described in Bijaoui & Rué (1995). One can therefore produce a map containing only the reconstructed objects, i.e. the sources (galaxies, stars) that we were looking for. This image can be used for comparison at other wavelengths.

Finally, the outputs of PRETI are numerous and contain all information at all scales divided into several cubes of data and images in FITS format, but the most commonly used outputs are the following:

1.

the final image and its associated rms image.

2.

the image of the reconstructed objects

3.

the list of objects with their position, flux and flux error (assuming no transient error), the sigma level of the detection and the scale at which the object was detected (size of the object).

The confidence level associated with the faint sources detected with PRETI cannot be directly understood as a usual signal-to-noise ratio typical of a Gaussian noise. In fact, because of the presence of residual glitches, the detection level is not fixed by the overall rms but by the level at which false detections begin to appear. One can check the robustness of the source detection by using a high detection level and comparing the source list with the brightest sources of the corresponding optical image. However, to tackle adequately the faint source detection, the only solution is to tune the method with simulations.

A field where this faint source detection method is widely applied is the study of source number counts in galaxy surveys, as for example the ISOCAM survey in the Hubble Deep Field region (Aussel et al. 1999). We will explain in the next section how simulations allows us to determine the accuracy of the so called $\log N - \log S$ diagrams, which show the number N of sources as a function of the emitted flux S.

  

Figure:9 Simulation of the ISO-HDF mosaic (I): (left) this image contains only simulated photon noise plus readout noise, i.e. Gaussian noise, (right) image of the ISO-HDF mosaic without any noise and with simulated sources according to a distribution without evolution (Franceschini 1997). Sources ranges from $0.1~\mu$Jy to 1 mJy

  

Figure 10: Simulation of the ISO-HDF mosaic (II): (left) sum of the two previous images, i.e. simulated sources plus readout and photon noise, (right) image of the ISO-HDF mosaic simulated from a staring observation, i.e. all sources of noise are present but no source at all is present (this image can be used to estimate the number of false detections due to glitches, since it does not contain any real sources)