The ability of detecting structures in X-ray image of celestial objects
is crucial, but the task is highly complicated due to
the low photon flux, typically from 0.1 to a few photons
per pixel. Point sources
detection can be done by fitting the Point Spread Function,
but this method does not allow extended sources detection.
One way of detecting extended features in a image is to convolve it
by a Gaussian. This increases the signal to noise ratio, but at the same
time, the resolution is degraded. The VTP method (Scharf et al.
1997) allows detection of extended objects, but it is not
adapted for the detection of substructures. Furthermore, in some cases, an
extended object can be detected as a set of point sources (Scharf
et al. 1997). The wavelet transform (WT) has been introduced
(Slezak et al. 1990) and presents considerable advantages
compared to traditional methods. The key point is that the wavelet
transform is able to discriminate structures as a function of scale, and
thus is well suited to detect small scale structures embedded within larger
scale features. Hence, WT has been used for clusters and subclusters
analysis (Slezak et al. 1994; Grebenev et al.
1995; Rosati et al. 1995; Biviano
et al. 1996), and has also allowed the discovery of a long,
linear filamentary feature extended over approximatily 1 Mpc from the Coma
cluster toward NGC 4911 (Vikhlinin et al. 1996). In the
first analyses of images by the wavelet transform, the Mexican hat was used.
The method simply consists in applying the correlation product between the
image I and the wavelet function:
Where a is the scale parameter. By varying a, we obtain a
set of images, each one corresponding to the wavelet coefficients
of the data at a given scale. The wavelet function corresponding to the
Mexican hat is
More recently the à trous wavelet
transform algorithm has been used because it allows an
easy reconstruction (Slezak et al. 1994;
Vikhlinin et al. 1996).
By this algorithm, an image I(x,y)
can be decomposed into a set (w1,..., wn, cn),
Several statistical models have been used in order to say if a X-ray wavelet coefficient wj(x,y) is significant, i.e. not due to the noise. In Viklinin et al. (1996), the detection level at a given scale is obtained by an hypothesis that the local noise follows a Gaussian noise. In Slezak et al. (1994), the Anscombe transform was used in order to transform an image with a Poisson noise into an image with a Gaussian noise. Other approaches have also been proposed using k sigma clipping on the wavelet scales (Bijaoui & Giudicelli 1991), simulations (Slezak et al. 1990; Escalera & Mazure 1992, Grebenev et al. 1995), a background estimation (Damiani et al. 1996; Freeman et al. 1996), or the histogram of the wavelet function (Slezak et al. 1993; Bury 1995).
We discuss and compare in this paper the different methods for signal detection using the à trous wavelet transform algorithm and present how X-ray images can be restored even in the case of very low photon flux.