Astron. Astrophys. 330, 57-62 (1998)

3. Image combining/deconvolution

The frames were combined in two ways. First, the standard reduction and image combination techniques implemented in the IRAF package were used in order to average the frames. The sigma-clipping algorithm was used for bad pixel rejection. This leads to two deep J and images over a field of 1. The total exposure times were 5040 sec in J and 8100 sec in . The resulting detection limit is 22 in J and 20 in K (3 , integrated over the whole object). Fig. 1 presents the field in the J band.

Fig. 1. A field of one arc-minute around HE 1104-1805. The detection limit on this deep J band image is 22. West is to the top, North to the left. The PSF stars are labeled. No obvious galaxy overdensity near the QSO pair can be seen; only galaxies G1 and/or G2 might be involved in the lensing potential.

3.1. Image deconvolution

In order to study the immediate environment of HE 1104-1805, we used the new MCS deconvolution algorithm described in full detail by Magain, Courbin & Sohy (1997).

Deconvolution of an image by the total observed Point-Spread-Function (PSF) leads to the so-called "deconvolution artifacts" or "ringing effect" around the point sources. This results from the deconvolution algorithm attempting to recover spatial frequencies higher than the Nyquist frequency, thus violating the sampling theorem. Instead, the MCS algorithm uses a narrower PSF which ensures that the deconvolved image will not violate the sampling theorem. Additionally, the MCS algorithm takes advantage of important prior knowledge: in the deconvolved images, all the point sources have the same (known) shape. This allows us to decompose the deconvolved image into a sum of analytical point sources plus a diffuse background which is smoothed to the final resolution chosen by the user. Most of the deconvolution artifacts are thus avoided. This is of particular interest when one wishes, like in the present study, to discover faint objects embedded in the seeing disks of much brighter point sources.

If the deconvolution of a single image already yields very good results (e.g. Courbin et al. 1997), the simultaneous deconvolution of numerous dithered exposures is even more efficient (e.g. Courbin & Claeskens, 1997). In particular, the MCS code allows the pixel size of the deconvolved image to be as small as desired. This over-sampling possibility, already applicable to the deconvolution of a single frame, is of considerable interest when dealing with the spatial information contained in many dithered frames.

Another advantage of the MCS algorithm is that the PSF can vary from frame to frame. For example, one can combine good quality images with trailed or even defocused images or, in a more reasonable way, frames of differing image quality and signal-to-noise ratios. The resulting frame is an optimal combination of the whole data set, with improved resolution and sampling.

The seeing in the original IRAC2b images was of the order of 0:006 for the very best frames and up to 1:002 for the worse ones, in both J and . We adopted for the deconvolution a sampling step of 0:001381, two times smaller than the 002762 which still samples well the resulting image (2 original pixel size. This allows us to reach a final resolution of 0: pixels Full-Width-Half-Maximum (FWHM)).

Since the signal-to-noise of individual images is very low and since we had to reject very numerous bad pixels, we first combined the images in groups of nine. Thus, we obtained 6 intermediate images in J and 12 images in . A PSF was derived for each of these images. In , only "Star 1" is bright enough, - i.e. comparable to the QSO's luminosity - to compute an accurate PSF (See Fig. 1 for the labeling of the stars used). In J, "Star 1" shows extended luminosity so we used both "Star 2" and "Star 3". The resulting total exposure time of the co-added images is different from the one of the images combined using IRAF and the standard methods. We rejected more frames with bad pixels falling right on the object or the PSF stars. On the other hand, we included in the different stacks more images with bad seeing. Thus, the total exposure time of the deconvolved images is 6480s in both J and .

The program requires initial estimates for the positions and intensities of the point sources in the field. This was done by choosing the central pixel of each QSO image. During deconvolution, the centres of the point sources are forced to be the same in all the images, only an image translation (no rotation) being allowed. The data are never aligned or rebinned; only the deconvolved model (on which the highest spatial frequencies are modeled analytically) is transformed. The intensities of the point sources can be allowed to be different in each image so that even variable objects may be considered in the deconvolution.

The shape of IRAC2b PSF shows significant variations across the field. In J, the variation is still acceptable, mainly because we used "Star 2" and "Star 3", which are closer to HE 1104-1805 than "Star 1", which is used for the PSF computation in the band. It is possible in our algorithm to let the PSF depart slightly from its original shape, during the deconvolution process. It is in fact re-determined directly from the point sources in the field that is deconvolved. In the present case of simultaneous deconvolution, the correction on the PSFs is well constrained by the numerous images considered. The quality of the PSF correction is even better if numerous stars are present in the field. With 2 point sources and 12 images in , it has been possible to correct rather well the PSFs of the 12 images. The deconvolution is first performed with variable PSFs, and then repeated with the corrected PSFs fixed.

The background component of the deconvolution is smoothed on the length scale of the final resolution. The weight attributed to the smoothing (see Magain, Courbin & Sohy, 1997 for more detail) is chosen so that the residual map between each data frame and the model image (reconvolved with the PSF) in units of the photon noise, has the correct statistical distribution, i.e. is equal to unity all over the field. In other words, we chose the smoothing term by inspecting the local residual maps. This ensures that the deconvolved image is compatible with the whole data set in any region of any of the data frames.

The deconvolution consists of a minimization between the deconvolved model image and the whole data set, using an algorithm derived from the conjugate gradient method. Again, the residual maps are used as a quality check of the result. We stop the iteration process only when the residual maps show the correct statistical distribution all over the field so that we avoid local over or under-fitting.

The program produces the following outputs: a deconvolved image, the centre of the point sources, the shifts between the images, the intensities of the point sources for each of the individual frames and an image of the deconvolved galaxy, free of any contamination by the QSOs.

3.2. Results

Fig. 2 displays the result of the deconvolution for the J band images. Six images were used to obtain this result. The spatial resolution is 0:002762, comparable to the resolution reached by the HST in the IR domain. We chose the same final resolution for the simultaneous deconvolution of 12 images. The lensing galaxy is clearly detected and displayed in Fig. 2. It is also seen in , were it is in fact brighter.

Fig. 2. Left: A J band image of HE 1104-1805 obtained with 2.2m ESO/MPI telescope. The field is 8.8 and the total exposure time is 5040 sec. Right: Simultaneous deconvolution of the 6 intermediate images (see text). The adopted pixel size is 0:001381. The FWHM of the point sources on the deconvolved image is 0:002762.

The images were deconvolved several times, with different initial guesses as to the position and the intensity of the QSO pair. In Table 1 the relative positions of the QSOs are tabulated. The errors correspond to the dispersion in the different deconvolutions (1 error bars).

Table 1. Summary of the astrometry (in the same orientation as Fig. 1 and Fig. 2) and photometry for HE 1104-1805 and the lensing galaxy. The 1 error bars are also indicated.

The photometry of the QSO images is also given. The 1 errors correspond to the dispersion in the peak intensities in each of the images considered in the simultaneous deconvolution (6 in J, 12 in ).

The position of the lensing galaxy was determined on the deconvolved background image by both Gaussian fitting and first order moment calculation. The results were averaged together and taken as the position of the lensing galaxy. We estimate the 1 error on the galaxy position to be about 0:0008 in both bands. The values given in Table 1 are the average of the positions in J and 0005. The angular separation between the lensing galaxy and QSO A is 1.14 and have an estimated error of 0: 0.06 and the distance between the two QSO images is 3.14 0.04.

We derived the magnitude of the lensing galaxy by aperture photometry on the deconvolved background image to avoid contamination by the QSO's light. A diaphragm of 0:009 diameter was used. Due to the too low signal-to-noise ratio in the lensing galaxy, we could not determine its shape parameters.

Fig. 3 shows the position of the galaxy, relative to the QSO images. A slight misalignment between the lens and the line joining QSO A and QSO B can be seen. It is larger than our error bars and is apparent in both J and . In addition, the PSF's shape does not show any significant variations across the deconvolved field (only 8.8) so that any geometric distortion can be ruled out. The observed misalignment seems therefore real.

Fig. 3. Geometry of HE 1104-1805. Note the slight misalignment between the 2 QSOs and the lensing galaxy.

No obvious galaxy overdensity is detected in the immediate surrounds of the QSO, although the detection limit of 22 in J and 20 in K would have allowed us to see any rich cluster up to . The two nearest objects to the double QSO are galaxies and (see Fig. 1). Table 2 gives their position relative to QSO A and their photometry, both derived on the "un-deconvolved" image since they are outside the field used for the deconvolution.

Table 2. Astrometry and photometry of galaxies G1 and G2, relative to QSO A. The astrometry is given in the same orientation as Fig. 1 and Fig. 2. These values were derived from the "un-deconvolved" images.

© European Southern Observatory (ESO) 1998

Online publication: January 8, 1998
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