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Astron. Astrophys. 343, 983-989 (1999)

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3. Data analysis and image restoration

3.1. Pre-reduction

The usual corrections were applied: The average dark frames were subtracted. The gain tables of the CCDs were obtained from the average flat fields. The single frames were then divided by their appertaining gain tables (with respect to CCD1 and CCD2 and wavelength position for the latter). Treating single flat field frames as regular data gives information on the noise to be used below for the optimum filters. The images were corrected for the above mentioned displacements and differences of scales. Image motion was taken out by choosing the CCD1 (broad-band) frame with highest contrast and correlating the remaining 139 frames to the former. The same displacements were applied to the corresponding simultaneously taken CCD2 (narrow-band) frames. For further image restoration, i.e. removal of seeing (and telescope aberrations) within isoplanatic patches, the frames were divided into 48 partly overlapping subfields of 64[FORMULA]64 pixel each ([FORMULA]). At the end, these subfields were recombined again to a full image.

3.2. Speckle reconstruction

For both scans analyzed here, the 140 frames from the broad-band channel (CCD1) were restored by speckle techniques, which are well established (cf. de Boer et al. 1992, de Boer & Kneer 1992, de Boer 1996, Wilken et al. 1997). We commonly use the spectral ratio method (von der Lühe 1984) to obtain the amplitude correction factors (in Fourier space) as well as the speckle masking method (Weigelt 1977, Weigelt & Wirnitzer 1983, Lohmann et al. 1983) to determine the phases with high reliability. Since the broad-band channel provides much light with high SNR we obtain reliable reconstructions with only 140 frames.

3.3. Restoration of narrow-band images

Given the reconstructed estimate of the object [FORMULA] seen through the broad-band channel, the observed broad-band images [FORMULA] at time l, and the appertaining narrow-band images [FORMULA], we may perform a further image restoration and estimate the object [FORMULA] at wavelength position [FORMULA] in the narrow-band channel. The [FORMULA] denotes estimates. Since the two channels operate in the same wavelength range we assume that their instantaneous point spread functions [FORMULA] are identical. Presumably, they are "broader" for the narrow-band channel compared to the broad-band channel due to more optical components in the former. Yet this is of no concern here, the assumption underestimates the corrections of the narrow-band images. Their spatial resolution is mainly limited by photon noise.

Denoting the Fourier transforms of the images and of the point spread functions by upper case letters, we obtain in Fourier space the relations, within an isoplanatic patch,

[EQUATION]

and

[EQUATION]

or

[EQUATION]

where the `*' denotes the complex conjugate. Summing over all available l, here 5 at each wavelength position, gives

[EQUATION]

Compared to a summation of Eq. (3), the formulation of Eq. (4) avoids to a large extent divisions by very small numbers at Fourier co-ordinates where a single [FORMULA] becomes 0, i.e. the information is efficiently lost at this instant and wavenumber (cf. also Keller & von der Lühe 1992). Replacing now [FORMULA] by means of Eq. (1) and applying an optimum filter H (Brault & White 1971) yields finally

[EQUATION]

For the construction of the optimum filter we assume that the noise in the broad-band frames is negligible. In the same manner as Löfdahl (1996) we obtain

[EQUATION]

The averages [FORMULA] on the right hand side are obtained by smoothing over 5[FORMULA]5 pixels for the fraction and by taking additionally azimuthal and subframe averages of [FORMULA]. Spikes in H larger than 1.0 are cut to 1.0 and values [FORMULA] are set to zero. The noise term [FORMULA] is calculated from the flat field frames, as mentioned above, and the values in the numerator and denominator on the right hand side are taken from the data themselves.

We demonstrate in Fig. 2 the tremendous improvement of image quality by restoration. Similar results had been shown by Krieg et al. (1998).

[FIGURE] Fig. 2. Improvement of spatial resolution by image restoration. Upper images: simultaneously taken single frames from CCD1 (left ) and CCD2 (right ); lower images: speckle reconstruction (left ) and image in the wing of D2 (right ) after restoration according to Eq. (5). The distance of the tickmarks is 1".

Fig. 3 gives results from one spectral scan. The intensity fluctuations in the wings of Na D2 are very similar to those seen in the speckle reconstruction. But one sees already here that the granular pattern fades away in the images close to the line centre.

[FIGURE] Fig. 3. 20"[FORMULA]15" subfields of restored images from a FPI scan across Na D2. The distance of the tickmarks is 1". The speckle reconstruction from the broad-band channel is shown in the lower right corner. The contrasts are scaled separately in each image to the local maximum and minimum. Starting in the upper left corner, moving to the right, and going down row by row, the wavelength displacements from Na D2 line centre are (in Å): -1.18, -0.78, -0.48, -0.28, -0.08, +0.02 (approximately line centre, 2nd row right image), +0.12, +0.32, +0.52, +0.82, +1.22.

Furthermore, to lower the influence of noise, we combine the data on both sides of Na D2 at approximately the same (absolute) wavelength distance from line centre. This increases the number of observed images from 5 to 10 in the sums of Eq. (5) while at the same time the optimum filter H widens. Dopplershifts are of little concern since we will deal with the damping wings.

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999
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