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Astron. Astrophys. 359, 907-931 (2000)

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3. Data reduction

3.1. Source extraction

Once the plates/films are digitized, the next step consists of identifying all point sources in these frames. The source extraction is performed on each frame using SExtractor (Bertin & Arnouts 1996), a software dedicated to the automatic analysis of astronomical images using a multi-threshold algorithm allowing good object deblending. The detection of the stars is done at a 3-[FORMULA] level above the background. This software, which can deal with huge amount of data (up to 60,000 [FORMULA] 60,000 pixels) is not suited for very crowded field like the centers of the globular clusters. Since the radial surface density is so much unreliable towards the very crowded parts of these globular clusters, we just ignore it in all crowded inner areas. From the catalogues in B and R filters, produced by SExtractor for each cluster, we construct a color B and color index [FORMULA] catalogue with the instrumental magnitudes. We do not calibrate our data, except in the case of NGC 5139, since we need only relative magnitudes and colors for the purpose of establishing cluster membership. The magnitude error from the photographic plate is found to be up to 0.2 mag for the faintest stars. Typically, we get, for each field, a total number of stars from [FORMULA] up to [FORMULA] for the richest fields. We do not apply any crowding correction to our stellar counts, first, because crowding is nearly constant and weak in the outer areas (the only ones we consider) surrounding of the clusters (see also Grillmair et al. 1995), and, second, because crowding is completely dominated by the observational biases for the overdensities.

3.2. Star/Galaxy separation

In case of low fore- and background densities towards a considered globular cluster, i.e., for GCs located at high Galactic latitudes, we perform a star/galaxy separation by using the method of star/galaxy magnitude vs. log(star/galaxy area) - which is shown to work well down to the 18 instrumental magnitude, as displayed in Fig. 1. The galaxies have a lower surface brightness than the stars and in the magnitude vs. log(area) plane, the two classes of objects follow different loci as clearly shown on Fig. 1. The star sequence is fitted by a fifth or sixth order polynomial and the objects more than 3 [FORMULA] fainter than the fit are considered as galaxies, through an iterative process which stops as soon as no new galaxies are detected (typically after about 10 iterations).

[FIGURE] Fig. 1. Star/galaxy magnitude vs. log(star/galaxy area) diagram for the field around NGC 288. At large areas and bright magnitudes, the upper sequence is made of stars, while the lower sequence is made of galaxies. The lowest line represents the 3-[FORMULA] limit below the continuous line, fitting the star sequence.

We detect some background clusters of galaxies from the overdensities at high spatial frequencies, using the Wavelet Transform (WT, cf. Slezak et al. 1994 and hereafter). As a check, the galaxy catalogue obtained with this method is shown to correlate very well, with the Abell cluster catalog (Abell et al. 1989). We present in Fig. 2 the detection of clusters of galaxies in a wide field observed with the plate SRC678: the use of the WT allows a clear detection of about 50 clusters and substructures of clusters of galaxies at the typical scale of these structures. We point out that we refer to Abell cluster without any distinction between north and south Abell clusters. We emphasize that the star catalogues produced from such galaxy/star separation is polluted by galaxies at faint magnitudes, as visible in Fig. 2 because of confusion at faint magnitude for the detection. Our problem being the genuine detection of star overdensities, we look at the correlation between the galaxy clusters detected and the star overdensities, in order to disentangle the confusion, assuming that the faint magnitude galaxies follow the distribution of the bright galaxies detected. This assumption is justified by comparing some fields heavily polluted by galaxies with the so-called stellar overdensities (see, e.g., Fig. 25 hereafter). In order to remove the bad classification on the globular cluster region because of the crowding, we set the surface density of this area ([FORMULA]) to zero.

[FIGURE] Fig. 2. Overdensities of the galaxies (plate SRC678) using the spatial details down to the plane 2 of the WT. North is up, East to the left. The Abell clusters in this field are labeled. The contours levels represent 3-[FORMULA] to 8-[FORMULA] detections. The voids near the Eastern corners are due to artifacts from this plate.

3.3. Star selection

Following the method of Grillmair et al. (1995) we perform, for each considered globular cluster, a star selection from the color-magnitude diagram (CMD) in which cluster stars and field stars exhibit different colors. In this way we can differentiate present and past cluster members from the fore- and background field stars by identifying in the CMD the area occupied by cluster stars. The envelope of this area is empirically chosen so as to optimize the ratio of cluster stars to field stars in the relatively sparsely populated outer regions of each cluster.

This search for a mask is done by subdividing the color-magnitude plane into a 50 [FORMULA] 50 array in which individual sub-areas are about 0.1 mag wide in color and 0.15 mag high in B mag depending slightly on the cluster. Assuming that the color-magnitude distribution of the field stars is constant across the plate, a color-magnitude sequence for each cluster can be estimated from:

[EQUATION]

in the notation of Grillmair et al. (1995) where [FORMULA] and [FORMULA] refer to the number of stars with color index i and the instrumental magnitude index j counted within the central region of the cluster and in an annulus outside the cluster, respectively. The factor g is the ratio of the area of the cluster annulus to the field annulus. We compute the "signal-to-noise" ratio for each color-magnitude sub-area:

[EQUATION]

Given the magnitude resolution of about 0.2 mag, for both plates and films, we perform a wavelet transform (WT, cf. Slezak et al. 1994) of the s(i,j) function on 5 planes, with [FORMULA]. We remove the planes 0 and 1 to obtain the S/N function

[EQUATION]

with a magnitude resolution of at most 0.2 mag, see below for more details on the WT.

Grillmair et al. (1995) estimated a CMD mask for the selection of stars by computing a cumulative function from the S/N function for each cluster. This cumulative function reaches a maximum for a sub-area of the color-magnitude plane. Then by selecting all sub-areas with S/N values higher than this maximum, it is possible to construct the mask. In the present case, we depart slightly from Grillmair et al. (1995) procedure by selecting, for some fields, a subset of the mask, with a higher S/N value relative to the background stars (see Fig. 3 and Fig. 4). It must be a compromise between the S/N ratio and the number of stars selected, in order to get a sufficient spatial resolution which is lowered by Poissonian noise for small star counts. Given the S/N chosen, we have been able to eliminate from 50% up to 99% of the field stars. We show in Fig. 4 the CMD selected for the less contaminated fields.

[FIGURE] Fig. 3. Upper right panel: color-magnitude diagram of stars in the cluster NGC 2298 (r [FORMULA]) using instrumental magnitude. Upper left panel: color-magnitude diagram of stars in the cluster field (for clarity only 10% of the total stars is shown). Lower panel: Signal/Noise [FORMULA] distribution (see text) in the color-magnitude diagram (CMD) built from the NGC 2298 plates. The resolution used with the WT takes into account the error on the magnitude ([FORMULA] 0.2 mag). Values greater than zero (dark isopleths) indicate greater contributions from the cluster stars than from the background stars (see Equ.2).

[FIGURE] Fig. 4. Color magnitude diagrams (left panel), using instrumental magnitude, of stars in the fields of 8 clusters with the area selected from the highest contrast between the cluster and the field. Note that the range can be different for each cluster. The right panel shows the S/N distribution [FORMULA] which gives the best contrast for the selection in the CMD space (see Equ.3). For the constrast a darker color means a greater contribution from the cluster stars. In the case of NGC 288, there is no smoothing. Note that each CMD has been scaled for matching the [FORMULA] map.

On the CMD-selected star-count map [FORMULA], we fit a background map [FORMULA], following Grillmair et al. (1995), by masking the GC (1 to 2 [FORMULA]) and using a blanking value inside, equal to the mean between 1.5 and 2.5 [FORMULA] to get a smooth background. We fit, on a [FORMULA] binned grid, a low-order bivariate polynomial surface [FORMULA], mainly first- or second-order surface, to avoid to erase some local variation:

[EQUATION]

We subtract this background from the CMD-selected map to get a surface-density map [FORMULA] of the overdensities that we can attribute to the tidal extension of the GC:

[EQUATION]

after having analyzed the potential observational biases that could create the fluctuations in the star-counting analysis.

3.4. Wavelet analysis

The wavelet transform is a powerful signal processing technique which provides a decomposition of the signal into elementary local contribution labeled by a scale parameter (Grossman & Morlet 1985). They are the scalar products with a family of shifted and dilated functions of constant shape called wavelets. The data are unfolded in a space-scale representation which is invariant with respect to dilation of the signal. Such an analysis is particularly suited to study signals which exhibit space-scale discontinuities and/or hierarchical features. Its ability to detect structures at particular scales has already been used in several astrophysical problems (Gill & Henriksen 1990; Slezak et al. 1994; Cambrézy, L. 1999; Chereul et al. 1999)

3.4.1. "A trous" algorithm

We perform on the raw tidal map [FORMULA] a wavelet analysis using the "à trous" algorithm (see Bijaoui 1991). It allows to get a discrete wavelet decomposition within a reasonable CPU time. The kernel function [FORMULA] for the convolution is a [FORMULA] spline function. The wavelet decomposition [FORMULA] is obtained from the following steps:

[EQUATION]

The last plane, called Last Smoothed Plane (LSP), is the residuals of the last convolution and not a wavelet plane, but afterwards we will abusively speak of wavelet plane for all these planes. Each plane [FORMULA] represents the details of the image at the scale i. We divide each image in [FORMULA] bins, a process which changes the spatial resolution of each cluster according to the different sizes of the fields: typically we get star-count maps of 3´ to 16´ resolution. The spatial resolution [FORMULA] for each wavelet-rebuilt cluster field can be computed, in arcmin, from the following relation: [FORMULA], with [FORMULA] being the total field size in arcmin. It is so far possible to do a filtering of each plane to get only the relevant wavelet component. One problem is to find the noise level for each plane. We know that the raw tidal map is blurred by the Poissonian noise of the background objects; this is especially true at low galactic latitudes. We could perform an Anscombe transformation (Murtagh et al. 1995) to transform the Poissonian noise into a Gaussian noise on each scale. Actually we choose to perform Monte-Carlo simulations, because of the varying Poissonian noise through the field, in order to follow easily the spatial variation of the rms noise at each scale. The contours for the surface density are computed to be above 3 [FORMULA] level, with [FORMULA] being the rms fluctuation of the selected wavelet coefficients computed in an area avoiding the central cluster.

3.4.2. Filtering of high varying density background noise

In case of strong gradient density of the galactic background, the noise is varying with the location in the field. To filter properly this noise, we perform N Poissonian simulations from the fitted background star counts [FORMULA] and we take the WT of the N realizations and perform statistics on each pixel for the whole set of wavelet planes,

[EQUATION]

Practically we have taken N=100. Then we fit by a low-order bivariate polynomial surface a rms noise map ([FORMULA]), for each wavelet plane, to obtain an estimate of the rms fluctuation on the N realizations. This allows a good estimate of the rms noise without the need of performing a great number of simulations, which are CPU time-consuming because of the WT:

[EQUATION]

We filter each component of the raw map [FORMULA], above a given threshold [FORMULA], using the rms noise [FORMULA] map on each wavelet scale i to get the filtered wavelet planes [FORMULA] of our image:

[EQUATION]

In this study the coefficients are filtered at the 3-sigma level. We show the case of the globular cluster NGC 5139, located at low galactic latitude, for which we present the raw surface density map and the filtered map at different resolution (see Fig. 5 and Fig. 6). We remind that a wavelet plane i has a typical resolution of [FORMULA] pixels, which is the Gaussian-equivalent resolution of the wavelet function at the scale i. We have to point out that our filtered solution is not the optimum solution (cf. Starck et al. 1997, e.g. for 1-D optimum solution) since it is only a selection of significant coefficients from the raw map. The "à trous" algorithm implemented permits a WT transform with a rate of about 2 kPixel/s/plane on a DecAlpha500 workstation 1

[FIGURE] Fig. 5. Filtered image of color-selected star-count overdensities (Log) in NGC 5139 using the Wavelet Transform (WT) (upper panel) to be compared with the raw star-count (lower panel). The upper panel displays the full resolution of [FORMULA] using the whole set of wavelet planes.

[FIGURE] Fig. 6. Different resolutions of the tidal tail extensions towards NGC 5139 using the WT filtered planes. The spatial resolutions are 3.2´, 6.4´, 12.9´, 25.8´, 51.6´, and 103.2´ from the upper-left panel to the lower-right panel.

The final tidal map [FORMULA] is built as follows:

[EQUATION]

where [FORMULA] is the filtered WT in case of strong background noise. The lower and upper indexes [FORMULA] and [FORMULA] constrain the resolution of the rebuilt map by filtering the higher and/or the lower space scale wavelet planes. For the adopted binning, we find that the map with the planes 3 to 7 (LSP) gives the best compromise between the spatial resolution and the Poissonian noise of the star-counting after the filtering of the background noise. It provides, in most cases, a higher spatial resolution than in Grillmair et al. (1995), because the wavelet decomposition extract the energy only at the useful scales. We have to point out nevertheless that the wavelet basis used here is not orthogonal, mixing up slightly the scale energy on different planes, but this does not affect our rebuilt map.

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© European Southern Observatory (ESO) 2000

Online publication: July 13, 2000
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