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Astronomical Data Analysis Software and Systems VII
ASP Conference Series, Vol. 145, 1998
R. Albrecht, R. N. Hook and H. A. Bushouse, e
Ö Copyright 1998 Astronomical Society of the Pacific. All rights reserved.
ds.
Structure Detection in Low Intensity X­Ray Images using
the Wavelet Transform Applied to Galaxy Cluster Cores
Analysis
Jean­Luc Starck and Marguerite Pierre
CEA/DSM/DAPNIA F­91191 Gif­sur­Yvette cedex
Abstract. In the context of assessing and characterizing structures in
X­ray images, we compare di#erent approaches. The intensity level is
often very low and necessitates a special treatment of Poisson statistics.
The method based on wavelet function histogram is shown to be the
most reliable one. Multi­resolution filtering methods based on the wavelet
coe#cients detection are also discussed. Finally, using a set of ROSAT
HRI deep pointings, the presence of small­scale structures in the central
regions of clusters of galaxies is investigated.
1. Wavelet Coe#cient Detection
The ability to detect structures in X­ray images 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 to 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 extending over approximately 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.
More recently the ‘a 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 (w 1 , ..., w n , c n ),
I(x, y) = c n (x, y) +
n
# j=1
w j (x, y) (1)
500

Structure Detection in Low Intensity X­Ray Images 501
Several statistical models have been used in order to say whether an X­ray
wavelet coe#cient w j (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 hypoth­
esis that the local noise is Gaussian. In Slezak et al. (1994), the Anscombe
transform was used to transform an image with Poisson noise into an image
with 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). Simulations have shown
(Starck and Pierre, 1997) that the best filtering approach for images containing
Poisson noise with few events is the method based on histogram autoconvolu­
tions. This method allows one to give a probability that a wavelet coe#cient
is due to noise. No background model is needed, and simulations with di#erent
background levels have shown the reliability and the robustness of the method.
Other noise models in the wavelet space lead to the problem of the significance
of the wavelet coe#cient.
This approach consists of considering that, if a wavelet coe#cient w j (x, y)
is due to the noise, it can be considered as a realization of the sum # k#K n k of
independent random variables with the same distribution as that of the wavelet
function (n k being the number of photons or events used for the calculation of
w j (x, y)). Then we compare the wavelet coe#cients of the data to the values
which can taken by the sum of n independent variables. The distribution of one
event in the wavelet space is directly given by the histogram H 1 of the wavelet #.
Since independent events are considered, the distribution of the random variable
W n (to be associated with a wavelet coe#cient) related to n events is given by
n autoconvolutions of H 1 : H n = H
1# H
1# ...# H 1
For a large number of events, H n converges to a Gaussian. Knowing the dis­
tribution function of w j (x, y), a detection level can be easily computed in order
to define (with a given confidence) whether the wavelet coe#cient is significant
or not (i.e not due to the noise).
Significant wavelet coe#cients can be grouped into structures (a structure
is defined as a set of connected wavelet coe#cients at a given scale), and each
structure can be analyzed independently. Interesting information which can be
easily extracted from an individual structure includes the first and second order
moments, the angle, the perimeter, the surface, and the deviation of shape from
sphericity (i.e., 4# Surface
P erimeter 2 ). From a given scale, it is also interesting to count
the number structures, and the mean deviation of shape from sphericity.
2. Image Filtering
In the previous section, we have shown how to detect significant structures in
the wavelet scales. A simple filtering can be achieved by thresholding the non­
significant wavelet coe#cients, and by reconstructing the filtered image by the
inverse wavelet transform. In the case of the ‘a trous wavelet transform algo­
rithm, the reconstruction is obtained by a simple addition of the wavelet scales
and the last smoothed array. The solution S is:
S(x, y) = c (I)
p (x, y) + # p
j=1 M(j, x, y)w (I)
j (x, y)

502 Starck and Pierre
where w (I)
j are the wavelet coe#cients of the input data, and M is the mul­
tiresolution support (M(j, x, y) = 1, the wavelet coe#cient at scale j and at
position (x, y) is significant). A simple thresholding generally provides poor
results. Artifacts appear around the structures, and the flux is not preserved.
The multiresolution support filtering (see Starck et al (1995)) requires only a
few iterations, and preserves the flux. The use of the adjoint wavelet transform
operator (Bijaoui et Ru’e, 1995) instead of the simple coaddition of the wavelet
scale for the reconstruction suppresses the artifacts which may appear around
objects. Partial restoration can also be considered. Indeed, we may want to
restore an image which is background free, objects which appears between two
given scales, or one object in particular. Then, the restoration must be per­
formed without the last smoothed array for a background free restoration, and
only from a subset of the wavelet coe#cients for the restoration of a set of objects
(Bijaoui et Ru’e 1995).
3. Galaxy Cluster Cores Analysis
Cluster cores are thought to be the place where virialisation first occurs and
thus in this respect, should present an overall smooth distribution of the X­ray
emitting gas. However, in cooling flows (CF) ­ and most probably in the whole
ICM ­ the presence of small scale inhomogeneities is expected as a result of the
development of thermal instability (e.g., Nulsen 1986). (Peculiar emission from
individual galaxies may be also observed, although at the redshifts of interest in
the present paper (# 0.04) ­ and S/N ­ such a positive detection would be most
certainly due to an AGN.) It is thus of prime interest to statistically investigate
at the finest possible resolution, the very center of a representative sample of
clusters, in terms of luminosity, redshift and strength of the cooling flow.
Using a set of ROSAT HRI deep pointings, the shape of cluster cores, their
relation to the rest of the cluster and the presence of small scale structures have
been investigated (Pierre & Starck, 1997). The sample comprises 23 objects up
to z=0.32, 13 of them known to host a cooling flow. Structures are detected and
characterized using the wavelet analysis described in section 1.
We can summarize our findings in the following way:
­ In terms of shape of the smallest central scale, we find no significant di#erence
between, CF and non CF clusters, low and high z clusters.
­ In terms of isophote orientation and centroid shift, two distinct regions appear
and seem to co­exist: the central inner 50­100 kpc and the rest of the cluster.
We find a clear trend for less relaxation with increasing z.
­ In general, very few isolated ``filaments'' or clumps are detected above 3.7# in
the cluster central region out to a radius of # 200 kpc. Peculiar central features
have been found in a few high z clusters.
This study, down to the limiting instrumental resolution, enables us to
isolate ­ in terms of dynamical and physical state ­ central regions down to
a scale comparable to that of the cluster dominant galaxy. However it was
not possible to infer firm connections between central morphologies and cooling
flow rates or redshift. Our results allow us to witness for the first time at the
cluster center, the competition with the relaxation processes which should here

Structure Detection in Low Intensity X­Ray Images 503
be well advanced and local phenomena due to the presence of the cD galaxy.
Forthcoming AXAF and XMM observations at much higher sensitivity, over
a wider spectral range and with a better spatial resolution may considerably
improve our understanding of the multi­phase plasma and of its inter­connections
with the interstellar medium.
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