, estimating source extent, while the height of
a WT maximum gives a measure of source counts.
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PS reprint
F. Damiani, A. Maggio, G. Micela, S. Sciortino
Istituto ed Osservatorio Astronomico
di Palermo 'G.S.Vaiana', ITALY
The WT, computed at a scale a, is mostly sensitive to structures of
size , estimating source extent, while the height of
a WT maximum gives a measure of source counts.
The WT probability distribution of the background has been computed through extensive simulations, in the low-background limit and analytically in the high-background limit, to set significance thresholds for candidate sources (i.e., local WT maxima). Realistic simulations of PSPC images, containing both pure background and background plus sources, assess the frequency of spurious detections as a function of detection threshold, and the algorithm sensitivity, respectively.
The algorithm may also consistently yield upper limits to the X-ray count rate of undetected objects.
Among the various applications of Wavelet Transforms (WT), we explore here their ability to detect sources in X-ray images. The WT of an image consists of a mixed scale-position representation of the data, which enables studying both the shape (size) and location of imaged objects, unlike Fourier analysis or peak-finding methods (sliding-window detect). Analogous exploration of a wavelet-based X-ray source detection have been already presented by Rosati, Burg, & Giacconi (1994), and by Grebenev et al. (1995); a similar technique ('matched filter') has been employed by Vikhlinin et al. (1995). Our method is described in detail by Damiani et al. (1995).
The multiscale approach of wavelet analysis is very useful to analyze images containing sources of various sizes (either intrinsically, or broadened by the PSF), such as PSPC images, since it takes full advantage of the good spatial resolution near the image center, and detects efficiently off-axis sources and truly extended sources.
The WT of an image is a function 
depending on both position
and a scale parameter a, and is defined as:
where 
is the `generating
wavelet', here chosen of the form (`Mexican Hat'):
The most important properties of the WT in the context of source
detection are: First, a constant function 
, or a function such
that 
have a WT 
. Second, 
depends on the values of 
only in a neighborhood of radius
4-5a around the studied point 
, and third, the WT at a scale
a brings into evidence image structures of size 
.
Our analysis procedure consists of the following main steps:
Although the WT of a flat background is null, there are obviously
fluctuations due to noise. Therefore, in order to discriminate true
sources from background fluctuations, it is essential to know the
probability distribution of the WT of the background. For details on
the derivation of such a distribution see Damiani et al. (1995).
Here, we simply note that this distribution is strongly non-gaussian
when there are few background photons in an area 
, as it often
happens in actual PSPC images.
The reference background used to predict the expected WT fluctuations
in the absence of sources is computed by smoothing the image using a
gaussian filter, whose width is taken equal to the local PSF width
, in order to describe real background modulations
on any length scale. We interpolate the background over ribs, to
produce a ribs-free map. Since this map includes the effect of
sources, we then apply a local median filter to build a final
background map.
The WT of the image is computed at each scale a by direct
integration, with a suitable grid spacing. In order to avoid the
generation of spurious WT maxima near PSPC ribs (since the WT is very
sensitive to discontinuities in the data) we compute the WT of the
count-rate image, not of the photon image, dividing the photon
image by the observation exposure map, pixel by pixel. Since this
operation alters the original (Poisson) photon statistics per bin, we
convert the thresholds for the photon WT to thresholds for the
rate WT using a suitable `effective' exposure time 
,
derived analytically under certain approximations.
Candidate sources are identified as local spatial maxima of the
WT at any scale, provided that the WT amplitude 
is higher
than expected for the local background (at that scale a), to an
assigned probability level.
Sources detected at various scales are cross-identified, to yield the
profile of WT amplitude as a function of scale a, for each source.
If 
is a normalized gaussian (source) of width 
, its WT
has a strong peak for r=0. For a source count rate 
, the WT
amplitude 
at the spatial maximum r=0 is
(Figure 1a). The function 
has a
maximum for 
, which allows us to estimate the
source size 
and rate 
.
After having obtained a first list of sources, a second iteration consists in the elimination of all detected point sources from the computation of the reference background map; this step allows us to detect even weak sources close to much stronger ones, which spuriously raise the local background estimate. All previous steps are then repeated to build a final source list.
The upper limit for an undetected source is computed in the same way as
the rate 
for detection. A point source (assumed gaussian) has
a known width 
: this fixes the shape of the
profile, apart from the normalization 
, which is
the sought count-rate upper limit. Once the local background is known,
the detection threshold at a given level k is simply a function of
scale a. Then, the source would be detected if the 
profile intersects the threshold curve, and undetected if it remains below
it, depending on the value of 
(Figure 1a).
The upper limit to 
is therefore the value that makes the two
curves become tangent to each other.
We have studied in detail the performances of the algorithm with the
help of extensive simulations of PSPC images. Simulations were made of
pure background images, to study the distribution of spurious sources
distribution as a function of both position and significance level, and
of images including sources, to study the source detection efficiency,
and the accuracy of the output quantities. Simulations were made as
realistic as possible, including the presence of ribs and wobbling by
using actual exposure maps, a distribution of source intensities
following the 
relation of Hasinger et al. (1993a),
and a model for the PSF after Hasinger et al. (1993b). These
simulations have been repeated for various exposure times.
Figure 1b shows the agreement between input and detected source counts.
Figure 1: ( a, left): Profiles of 
(dashed) for three
values of source intensity 
(corresponding to a positive
detection, a marginal detection, and a non-detection, respectively), and a
source size 
pixels. These curves are compared with the threshold
curve relative to a background of 1.0 cts/pixel
.
( b, right): Source counts as given by the algorithm vs.
counts of input simulated sources. The straight lines indicate
identity, and factor of two differences.
Figure 1a: PS 13 Kb, Figure 1b: PS 97 Kb
Applications to actual PSPC images of galaxies and stellar clusters also show good performances of the algorithm in a variety of cases, including extended sources and crowded fields.
This work was supported by ASI and MURST.
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