** Next:** An Optimal Data Loss Compression Technique for Remote
Surface Multiwavelength Mapping** Up:**
Astrostatistics and Databases** Previous:** Positive Iterative Deconvolution in Comparison to Richardson-Lucy Like Algorithms

Table of Contents -- Index -- PS reprint -- PDF reprint

ASP Conference Series, Vol. 145, 1998

Editors: R. Albrecht, R. N. Hook and H. A. Bushouse

**Jean-Luc Starck and Marguerite Pierre**

CEA/DSM/DAPNIA F-91191 Gif-sur-Yvette cedex

Several statistical models have been used in order to say
whether an X-ray wavelet coefficient *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 hypothesis 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 autoconvolutions.
This method allows one to give a probability that a wavelet coefficient
is due
to noise. No background model is needed, and simulations with different
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 coefficient.

This approach
consists of considering that, if a wavelet coefficient *w*_{j}(*x*,*y*) is due
to the noise, it can be considered
as a realization of the sum 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 coefficients 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

For a large number of events, *H*_{n} converges to a Gaussian.
Knowing the distribution function of *w*_{j}(*x*,*y*), a detection level can
be easily computed in order to define (with a given confidence) whether the
wavelet coefficient is significant or not (i.e not due to the noise).

Significant wavelet coefficients can be grouped into structures (a structure is defined as a set of connected wavelet coefficients 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., ). From a given scale, it is also interesting to count the number structures, and the mean deviation of shape from sphericity.

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 coefficients, and by reconstructing the filtered image
by the inverse wavelet transform. In the case of the *à trous* wavelet
transform algorithm, the reconstruction is obtained by a simple addition
of the wavelet scales and the last smoothed array. The solution *S* is:

where *w*_{j}^{(I)} are the wavelet coefficients of the input data, and *M* is the
multiresolution support (*M*(*j*,*x*,*y*) = 1, the wavelet coefficient 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 Rué, 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 performed
without the last smoothed array for a background free restoration, and
only from a subset of the wavelet coefficients for the restoration of
a set of objects (Bijaoui et Rué 1995).

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
difference 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 co-exist: the central inner 50-100 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
in the cluster central region out to a radius of 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 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 multi-phase plasma and of its inter-connections with the interstellar medium.

Bijaoui, A., & Rué, F., 1995, Signal Processing, 46, 345

Biviano, A., Durret, F., Gerbal, D., Le Fèvre, O., Lobo, C., Mazure, A., & Slezak, E., 1996, A&A311, 95

Bury, P., 1995, Thesis, University of Nice-Sophia Antipolis.

Damiani, F., Maggio, A., Micela, G., Sciortino, S, 1996, in Astronomical Data Analysis Software and Systems V, ASP Conf. Ser., Vol. 101, eds. G. H. Jacoby and J. Barnes (San Francisco, ASP), 143

Escalera, E., Mazure, A., 1992, ApJ, 388, 23

Freeman, P.E., Kashyap, V., Rosner, R., Nichol, R., Holden, B, & Lamb, D.Q., 1996, in Astronomical Data Analysis Software and Systems V, ASP Conf. Ser., Vol. 101, eds. G. H. Jacoby and J. Barnes (San Francisco, ASP), 163

Grebenev, S.A., Forman, W., Jones, C., & Murray, S., 1995, ApJ, 445, 607

Nulsen P. E. J., 1986, MNRAS, 221, 377

Pierre, M., & Starck, J.L., to appear in Astronomy and Astrophysics, (astro-ph/9707302).

Rosati, P., Della Ceca, R., Burg, R., Norman, C., & Giacconi, R., 1995, Apj, 445, L11

Scharf, C.A., Jones, L.R., Ebeling, H., Perlman, E., Malkan, M., & Wegner, G., 1997, ApJ, 477, 79

Slezak, E., Bijaoui, A., & Mars, G., 1990, Astronomy and Astrophysics, 227, 301

Slezak, E., de Lapparent, V. & Bijaoui, A. 1993, ApJ, 409, 517

Slezak, E., Durret, F. & Gerbal, D., 1994, AJ, 108, 1996.

Starck, J.L., & Pierre, M., to appear in Astronomy and Astrophysics, (astro-ph/9707305).

Starck, J.L., Bijaoui, A., & Murtagh, F., 1995, in CVIP: Graphical Models and Image Processing, 57, 5, 420

Vikhlinin, A., Forman, W., and Jones, C., 1996, astro-ph/9610151.

© Copyright 1998 Astronomical Society of the Pacific, 390 Ashton Avenue, San Francisco, California 94112, USA

** Next:** An Optimal Data Loss Compression Technique for Remote
Surface Multiwavelength Mapping** Up:**
Astrostatistics and Databases** Previous:** Positive Iterative Deconvolution in Comparison to Richardson-Lucy Like Algorithms

Table of Contents -- Index -- PS reprint -- PDF reprint