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3. Data reduction

3.1. Pre-processing

A typical number of 6000 specklegrams are read from the video tapes for each object (binary star and reference star). For faint objects (mag tex2html_wrap_inline1208 9) this number can reach 15000. Frames are digitized within a square sub-image of the video field. The size of this sub-image is generally 128tex2html_wrap_inline1182128 or 256tex2html_wrap_inline1182256, depending on the star separation and the size of the long-exposure image. The total size of the video field is 384tex2html_wrap_inline1182288. Each video grab is then checked for bad points, null images and loss of video synchronization signal. Images showing these features are deleted.

During the observation of a star, seeing conditions may change and images are classified according to the strength of turbulence before processing. This classification is performed the following way. The square modulus of the Fourier transform of each specklegram is computed. An instantaneous Fried parameter r0 is estimated by fitting the low-frequencies of this energy spectrum with a Kolmogorov spectrum (Fried 1966; Roddier 1981):
equation252
where f is the spatial frequency and tex2html_wrap_inline1128 the wavelength. After each specklegram is associated to a Fried parameter, images are binned into 5 classes according to image quality: class 5 contains the best images (low turbulence, good visual quality), class 1 the worst. Figure 1 (click here) gives an illustration of typical frames in each class. The processing of the data is made separately for each class. The classes we have processed are mainly classes 3 and 4 since the number of images belonging to class 5 was often too low (10-40).

  figure256
Figure 1: Classes of specklegrams illustrated with the star c Her. From left to right: classes 1 to 5. The field of view is tex2html_wrap_inline1226. From a total number of 4999 frames of c Her digitized from the tape, the class repartition was the following. Class 1: 468 images, class 2: 2979 images, class 3: 1350 images, class 4: 191 images and class 5: 11 images. On the best images (class 5), the double star can sometimes be seen directly on the specklegrams

3.2. Separation measurement

The separation tex2html_wrap_inline1164 is derived from the autocorrelation function. The power spectrum of both the double star and the reference star are computed for a given image class. The visibility function is obtained by dividing the two power spectra. An apodisation is then made by multiplying the visibility function by a transfer function of a circular telescope. The Fourier transform of the result gives the autocorrelation of the double star corrected from the turbulence. For a binary star, this function shows a central peak and two lateral ones. The distance between the central peak and the lateral ones gives the star separation. In practice we computed the photocenter of the lateral peaks on a 5tex2html_wrap_inline11825 grid centered on the peak maximum. The error bar is obtained by differentiation of the formula used for photocenter determination, the noise being estimated in a 10tex2html_wrap_inline118210 grid located far from the central and the lateral peaks. The accuracy depends on various parameters such as seeing conditions, magnitude difference, and total magnitude of the binary star and the reference star. The 1994 measurements (oversampled images) were more precise (typical errors 3 mas) than the 1995 ones (typical errors 15 mas).

3.3. Position angle measurement

Absolute position angles (i.e. containing the quadrant information) were computed using the recent technique of cross correlation (Aristidi et al. 1996) which is currently under development at the Département d'Astrophysique of Nice University. This technique consists of computing the ensemble average of the cross-correlation between the specklegrams and their square. In practice the computation is made in the Fourier plane: the cross spectrum between the specklegrams and their square is obtained both for the double star and the reference star. It is a complex quantity whose imaginary part contains the absolute quadrant information. The cross-spectrum of the double star is then divided by those of the reference star. An apodizing function (transfer function of a circular telescope) is applied and a Fourier transform is made. Although the result looks very similar to the autocorrelation (a central peak and two lateral ones), the lateral peaks of the cross-correlation are asymmetric. As presented in Fig. 2 (click here) for several binaries, this asymmetry gives the absolute position of the companion.

  figure267
Figure 2: Gray-level plots of cross-correlations computed on the specklegrams of six binary stars. This function shows a central peak surrounded by two smaller asymmetric ones. The asymmetry of the secondary peaks gives the orientation of the couple, the distance between lateral peaks and the central one provides the angular separation. The look-up table used for this representation is drawn on the right. Scale and orientation are indicated at the top of each picture. The case of tex2html_wrap_inline1140Cyg is interesting: due to the large magnitude difference of the couple (tex2html_wrap_inline1244), only one of the lateral peaks is visible, the second one being in the noise

3.4. Relative photometry measurement

The relative photometry of the system is obtained by using ratios of twofold probability density functions (Carbillet et al. 1996b). The twofold probability density function (PDF) is a function of two random variables of intensity (tex2html_wrap_inline1248 and tex2html_wrap_inline1250), and a space-lag tex2html_wrap_inline1252. Whatever the value of tex2html_wrap_inline1252, the twofold PDF for a point-source (a non-resolved reference star) has a symmetrical structure in tex2html_wrap_inline1248 and tex2html_wrap_inline1250. For tex2html_wrap_inline1252 close to the separation vector tex2html_wrap_inline1262 between the two components, the twofold PDF for a binary star has an arrow-head shape with a trend towards a direction tex2html_wrap_inline1250 of the order of tex2html_wrap_inline1266, where tex2html_wrap_inline1142 is the oriented intensity ratio. The direction tex2html_wrap_inline1266 is tremendously enhanced by dividing the twofold PDF of the binary star, computed for tex2html_wrap_inline1272, by the twofold PDF of the reference star, computed for the same space-lag tex2html_wrap_inline1262. The resulting quantity, the so-called function Q, clearly shows a ridge that simply follows: tex2html_wrap_inline1278. The oriented intensity ratio tex2html_wrap_inline1142 can then be easily evaluated by doing a radial integration of tex2html_wrap_inline1282: tex2html_wrap_inline1284, where tex2html_wrap_inline1126 is the angle measured in polar coordinates in the tex2html_wrap_inline1288 plane. The function Q-QT, where QT is the transposed quantity of Q, can also be computed in order to enhance the relevant ridge. In that case, the radial integration gives both tex2html_wrap_inline1142 (corresponding to the maximum of IQ-QT), and tex2html_wrap_inline1300 (corresponding to the minimum of IQ-QT). The error bars are then derived from a binomial fit of the regions close to the extrema of the radial integration, since a convex function has a quadratic behavior close to its extrema.

An alternative quantity may be used when no reference star is available: the division of the twofold PDF computed for tex2html_wrap_inline1272 by the twofold PDF computed for tex2html_wrap_inline1306.

Figure 3 (click here) shows the functions Q-QT (and their radial integrations) computed for three different type of data.

  figure298
Figure 3: Gray-level plots of function Q-QT, and plots of its radial integration IQ-QT, computed on the specklegrams of three binary stars. The radial integrations show two extrema. The maximum is reached for tex2html_wrap_inline1314, the minimum for tex2html_wrap_inline1316, tex2html_wrap_inline1142 being the intensity ratio of the two binary components. A value of tex2html_wrap_inline1320 means that the spatial lag vector goes from the brightest star to the lowest. The opposite if tex2html_wrap_inline1322. This gives the absolute position angle of the couple

  figure304
Figure 4: Restored images of tex2html_wrap_inline1152 Del (a) and 2 Cam (b). These images were computed from 212 specklegrams (class 5) of tex2html_wrap_inline1152 Del and 1162 specklegrams (class 4) of 2 Cam. The image restoration procedure is described in Sect. 3.5

3.5. Image restoration

We have restored images of tex2html_wrap_inline1152 Del and 2 Cam (Fig. 4 (click here)) using the bispectral method described in Lannes (1988) and Prieur et al. (1991).

The mean bispectrum and power spectrum were computed from the elementary frames of the same class (cf. Sect. 3.1). The phasor of the spectrum was derived from that of the mean bispectrum phasor through a global least-square minimization inversion method as described by Lannes (1988). This resulting phasor associated with the modulus of the spectrum derived from the mean power spectrum leads to an image which was then deconvolved by a Point Spread Function (PSF) obtained by observing a reference star (cf. Sect. 2). The deconvolution method we used (Lannes et al. 1987a, b) preserves the photometry, which allowed us to perform photometric analysis of the restored images. The measurements of the intensity ratios (taking the brightest component as a reference) lead to:

which is in very good agreement with the measurements obtained with the twofold probability density functions described in Sect. 3.4 which are summarized in Table 2 (click here).


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