The instantaneous intensity distribution of the LBT interferograms
can be described by the incoherent, space-invariant imaging equation
where o(x) is the two-dimensional object intensity distribution,
is the
point spread function of the atmosphere/interferometer system,
x is a two-dimensional space vector.
The time dependence (
) 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).
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
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 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 of the instantaneous LBT interferogram
recorded at time
can be described by
where is the optical transfer function of the
atmosphere/interferometer system at time
and O(u) is the object
Fourier transform. In LBT interferometry the time dependence of
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
which denotes the extension of the instantaneous transfer function
(see Fig. 1 (click here)). Therefore, the useful bispectrum
of each LBT interferogram
consists of only
those factors
,
and
with u-, v- and
u+v-vectors within the instantaneous uv-coverage
of the LBT
transfer function
, i.e. we define
Only bispectrum elements at those bispectrum coordinates (u,v) with
carry information about the source structure. The remaining bispectrum elements
with at least one vector outside the actual uv-coverage
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.
,
, and
)
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
yields object information at
recording time
, but it introduces noise at recording time
. Therefore,
has to be derived from measurements at
time
, but not from measurements at
.
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 180 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
over the aforementioned useful bispectra
(Eq. 4 (click here)) of the LBT interferograms
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.
The next image processing step is the compensation of the photon bias terms
in the average LBT power spectrum
(Goodman 1985) and in the average LBT bispectrum
(Wirnitzer
1985; Pehlemann et al. 1992).
Figure 4: Laboratory setup for simulating LBT interferograms
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 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 and the photon bias-compensated
average LBT bispectrum
the object
bispectrum
was obtained in the following way: (1) The modulus
of the object bispectrum
is derived from the object power spectrum. (2) The complex object bispectrum
is obtained by combining
from the
previous step with the phasor
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).