Figure 2 (click here) (left) shows a simulated image of a galaxy cluster. Two point sources are superimposed (on the left of the cluster), a cooling flow is at the center, a substructure on its left, and a group of galaxies at the top. From this image, a "noisy'' images has been created (Fig. 2 (click here) (right)). The mean background level is equal to 0.1 events per pixel. This corresponds typically to X-ray cluster observations. In the noisy image, the maximum value is equal to 23 events. The background is not very relevant. The problem in this kind of images is the small number of photons per object. It is very difficult to extract any information from them.
Figure 2: Left, simulated image. The central luminosity is equal to 12, and the two first isophots are at 1 and 2.62. Right, same image with a Poisson noise
Figure 3: Top left and right, convolution of the noisy image with a Gaussian with a standard deviation
equal to 3 and 5 respectively. Bottom left, filtered image using a sigma clipping on each wavelet scale, and a 10 sigma detection. Bottom right, filtered image using an hypothesis of local Gaussian noise, and a 10 sigma detection
Figure 4: Results of the filtering using the method
based on the histogram autoconvolutions. Left, image obtained with a confidence level equal to 1e-3 (which is equivalent to a 3.09 sigma detection for
the case of Gaussian noise), and right, image obtained with a confidence level equal to 10-4 (
Gaussian equivalence)
Figure 3 (click here), top left and top right, shows the filtering of the image by convolving the noisy image by a Gaussian, with a standard deviation equal to 3 and 5 respectively. Using the Anscombe transform, we were unable to obtain an image with a reasonable quality. It seems that this transform should only be used in the condition defined in Murtagh et al. (1995), i.e. with a minimum number of photons equal to 30 per pixel. In the case of very low photons count, the results are very poor.
Figure 3 (click here) bottom left shows the result after a filtering using a sigma clipping on each wavelet scale, and a ten sigma detection. Figure 3 (click here) bottom right shows the filtering using an hypothesis of local Gaussian noise, and a ten sigma detection. For both, even at a detection level of ten sigma, the filtered image presents residual noise.
Figure 4 (click here) shows the results of the filtering using the method based on the histogram autoconvolutions with two different confidence levels. Figure 4 (click here) left corresponds to a confidence interval of 10-3 (which is equivalent to a 3.09 sigma detection for the case of Gaussian noise), and Fig. 4 (click here) right, with a confidence level equal to 10-4 (3.72 Gaussian equivalence). Even if the two point sources could not have been distinguished by eye in the noisy image, they have been detected and correctly restored.
Figure 5: Filtering of the simulated image with different background levels.
From left to right and top to bottom, the background level was respectively
equal to 0.1, 0.5, 1, 2 counts per pixel
Figure 6: Left, multiresolution support of the simulated image
(see Fig. 2).
Right, multiresolution support of the same simulated field, but all
objects contains less flux. The maximum of the noisy image is equal to 7
Figure 5 (click here) shows the result of the filtering with
different background levels. The detections in the wavelet scale were
done using 
. From left to right and top to bottom, the background level was respectively
equal to 0.1, 0.5, 1, 2 counts per pixel. If the background level
is high, there is more noise, and we see that the second source disappears
when the background level increases, which is normal behavior.
The best filtering is clearly obtained using the method based on wavelet transform and the histogram autoconvolutions. For other methods which use the wavelet transform, we did not use Monte Carlo simulations and the exact level for signal detection is difficult to find. Furthermore, the level is certainly not the same for the whole scale. For this reason, a simple Gaussian filtering seems to be better.