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2. Image processing steps of the computer and laboratory experiments

  The instantaneous intensity distribution tex2html_wrap_inline1404 of the LBT interferograms can be described by the incoherent, space-invariant imaging equation
equation255
where o(x) is the two-dimensional object intensity distribution, tex2html_wrap_inline1408 is the point spread function of the atmosphere/interferometer system, x is a two-dimensional space vector. The time dependence (tex2html_wrap_inline1412) is mentioned explicitly to point out the dependence of the psf and the optical transfer function (or uv-coverage) on the instantaneous projection of the interferometer pupil function (as seen from the object) in addition to the time dependence caused by the turbulent atmosphere (speckle boiling).

2.1. Calculation of the average speckle masking bispectrum and aperture synthesis

The first image processing step in speckle masking with normal single-dish telescopes (Weigelt 1977; Weigelt & Wirnitzer 1983; Lohmann et al. 1983) is the calculation of the bispectrum
 equation263
of each interferogram. I denotes the Fourier transform of an interferogram, u and v are two-dimensional coordinate vectors in Fourier space, and the asterix denotes complex conjugation. For single-dish telescopes the time variable tex2html_wrap_inline1422 in equation (2 (click here)) describes exclusively the time dependence caused by the turbulent atmosphere as the pupil function is time-invariant. In speckle imaging experiments with single-dish telescopes the object Fourier transform can be measured at all times at all spatial frequencies up to the diffraction cut-off frequency because of the perfect uv-coverage.

In speckle masking experiments with the LBT the conditions are different from single-dish telescopes because of the diluted pupil function. The instantaneous uv-coverage is not perfect, but it consists of an elongated structure of three circular regions extended up to the diffraction cut-off frequency of a single-dish 22 m telescope (see Fig. 1 (click here)). The dashed circle shows the extension of the uv-coverage of a single-dish 22 m telescope, the shaded areas are the uv-coverage of the LBT at two different times. The uv-coverage changes with time since the astronomical object rotates relative to the LBT pupil function during the night. Therefore, in LBT interferometry, uv-coverage and bispectrum coverage are time-variant.

The Fourier transform tex2html_wrap_inline1436 of the instantaneous LBT interferogram tex2html_wrap_inline1438 recorded at time tex2html_wrap_inline1440 can be described by
equation269
where tex2html_wrap_inline1442 is the optical transfer function of the atmosphere/interferometer system at time tex2html_wrap_inline1444 and O(u) is the object Fourier transform. In LBT interferometry the time dependence of tex2html_wrap_inline1448 is caused (1) by the turbulent atmosphere and, additionally, (2) by the time dependence of the projected LBT interferometer pupil function as seen from the object (time dependence due to earth rotation). Each LBT interferogram yields information about the object structure only within the uv-coverage tex2html_wrap_inline1452 which denotes the extension of the instantaneous transfer function tex2html_wrap_inline1454 (see Fig. 1 (click here)). Therefore, the useful bispectrum tex2html_wrap_inline1456 of each LBT interferogram tex2html_wrap_inline1458 consists of only those factors tex2html_wrap_inline1460, tex2html_wrap_inline1462 and tex2html_wrap_inline1464 with u-, v- and u+v-vectors within the instantaneous uv-coverage tex2html_wrap_inline1474 of the LBT transfer function tex2html_wrap_inline1476, i.e. we define
 eqnarray275
Only bispectrum elements at those bispectrum coordinates (u,v) with tex2html_wrap_inline1480 carry information about the source structure. The remaining bispectrum elements with at least one vector outside the actual uv-coverage tex2html_wrap_inline1484 are zero-valued in the detector and photon noise-free case, however they introduce noise to the average bispectrum (integrated throughout the entire observing time) in the case of detector and photon noise. Therefore, it is important that these bispectrum elements are not used. Fig. 2 (click here) shows examples of vector triples (e.g. tex2html_wrap_inline1486, tex2html_wrap_inline1488, and tex2html_wrap_inline1490) of bispectrum elements with and without information about the source structure at two different orientations of the pupil function relative to the object: for example, the bispectrum element tex2html_wrap_inline1492 yields object information at recording time tex2html_wrap_inline1494, but it introduces noise at recording time tex2html_wrap_inline1496. Therefore, tex2html_wrap_inline1498 has to be derived from measurements at time tex2html_wrap_inline1500, but not from measurements at tex2html_wrap_inline1502.

Due to the rotation of the object relative to the LBT pupil function, the uv-coverage of the LBT transfer function changes its orientation continuously by about 180tex2html_wrap_inline1506 within 12 hours observing time. Therefore, continuous observing of the object during one night creates the uv-coverage of a single-dish 22 m telescope (aperture synthesis by earth rotation). The ensemble average LBT interferometry bispectrum is the sum tex2html_wrap_inline1510 over the aforementioned useful bispectra tex2html_wrap_inline1512 (Eq. 4 (click here)) of the LBT interferograms tex2html_wrap_inline1514 recorded during the total observing time. Therefore, the four-dimensional bispectrum coverage is nearly perfect if observing time is about 12 hours. The bispectrum coverage is not completely filled in the sense that bispectrum elements with the largest u-, v- and u+v-vectors accessible with a 22 m single-dish telescope are not obtained with the diluted LBT aperture (see Fig. 3 (click here)). But nevertheless all spatial frequencies up to the diffraction cut-off frequency of a 22 m single-dish telescope can be measured with the LBT.

2.2. Compensation of photon bias terms

The next image processing step is the compensation of the photon bias terms in the average LBT power spectrum tex2html_wrap_inline1524 (Goodman 1985) and in the average LBT bispectrum tex2html_wrap_inline1526 (Wirnitzer 1985; Pehlemann et al. 1992).

  figure297
Figure 4: Laboratory setup for simulating LBT interferograms

2.3. Compensation of the speckle interferometry transfer function

After compensation of the speckle interferometry transfer function (SITF) (Labeyrie 1970) in the photon bias-compensated ensemble average LBT power spectrum, the object power spectrum tex2html_wrap_inline1528 is obtained. Compensation is performed with the average LBT power spectrum of point source interferograms. In astronomical observations the SITF can be determined if the LBT interferometer is pointed alternately at the program object and a nearby point source or if the object and the point source are observed on different nights with similar seeing. If these techniques are not feasible, a theoretical SITF can be used.

From the object power spectrum tex2html_wrap_inline1530 and the photon bias-compensated average LBT bispectrum tex2html_wrap_inline1532 the object bispectrum tex2html_wrap_inline1534 was obtained in the following way: (1) The modulus
equation310
of the object bispectrum tex2html_wrap_inline1536 is derived from the object power spectrum. (2) The complex object bispectrum tex2html_wrap_inline1538 is obtained by combining tex2html_wrap_inline1540 from the previous step with the phasor
equation318
of the photon bias-compensated average LBT bispectrum. This technique is possible as the phasor of the object bispectrum is identical to the phasor of the average bispectrum (Lohmann et al. 1983).

2.4. Image reconstruction from the object bispectrum by the iterative building block method

  The diffraction-limited object intensity distribution o(x) was reconstructed from the object bispectrum tex2html_wrap_inline1544 by the iterative building block method (Hofmann & Weigelt 1990; Hofmann & Weigelt 1993; Reinheimer et al. 1993). The building block method searches for the diffraction-limited reconstruction which has the best agreement with the measured object bispectrum tex2html_wrap_inline1546.


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