#### Interferometric imaging tests for the Large Binocular Telescope

Astron. Astrophys. Suppl. Ser.
Interferometric imaging tests for the Large Binocular Telescope?
S. Correia 0 1
A. Richichi 0 1
0 Send o print requests to: S. Correia
1 Osservatorio Astro sico di Arcetri , Largo E. Fermi 5, I-50125 Firenze , Italy
We present an experiment of interferometric imaging for the Large Binocular Telescope (LBT), conducted at the 1.5 m TIRGO infrared telescope. The raw data were produced by simulating the LBT pupil with a mask on the secondary mirror. Two di erent conditions of D=r0, where D is the single aperture diameter and r0 the Fried parameter, were simulated by the choice of D; eld rotation was simulated by rotating the mask. The data set collected consists of several sequences of short exposure interferograms of one pointlike and one binary star in the J -band, for two di erent D=r0 conditions. We show preliminary results, in particular concerning the fringe contrast loss with integration time. The ability of a Lucy-Richardson-based deconvolution algorithm to reconstruct an object from a set of LBT-like interferometric real data was demonstrated on the binary star γ Leo. The retrieved binary parameter values are compared to catalog values, and a good astrometric agreement is found. Photometric and resolution limitations are also discussed. In a rst part, the reconstruction method and preliminary numerical simulations of LBT image restoration using this method are presented.
instrumentation; methods; data analysis | processing | techniques; telescopes | stars; imaging
1. Introduction
The Large Binocular Telescope (LBT) will consist of two
8.4 m mirrors on a common mount. When the two mirrors
are coherently cophased, this will work as a total baseline
of 22.8 m. In several respects, observations with the LBT
in this con guration will di er from those with more
\conventional" interferometers. For example, LBT will o er a
large eld of view, and will permit true imaging by
simultaneously measuring all the Fourier components
(Angel
et al. 1998)
.
At the same time, the peculiar point spread function
(PSF) and its rotation in the sky due to the alt-azimuthal
mount, will require speci c data acquisition algorithms,
and specialized treatment in the data reduction process.
With these points in mind, we started a project to
perform tests and develop relevant software. The aim was to
investigate the process of image formation and
reconstruction at the LBT, taking into account the characteristics
of the atmosphere, the telescope performance and adding
realistic estimates for the detector read-out noise (RON).
Previously, at least two others groups of authors
have already discussed LBT image reconstruction.
However, only numerical simulations, based on di erent
reconstruction techniques, have been carried out up to
date.
Reinheimer et al. (1997)
have applied the so-called
iterative building block method (bispectral analysis) to
some point-like and extended objects. They presented a
reconstruction method to apply to a speckle utilisation
of LBT, with the known limitation in sensitivity of this
observation mode. They complemented their simulations
with a laboratory experiment. However, this latter was
carried out under very favorable turbulence conditions
(r0 = 2 m) and unspeci ed brightness of the source.
Prior to this
Hege et al. (1995)
had explored the use of
iterative blind deconvolution
(IBD algorithm of Je eries
& Christou 1993)
, on simulated LBT images of an
extended object.
On the contrary, our work is based on real LBT-like
data. In Sect. 2 we present the algorithm of
reconstruction that we have used and modi ed. It is based on the
Lucy-Richardson deconvolution algorithm
(Richardson
1972; Lucy 1974)
, widely used in standard image
restoration methods. Section 3 shows the potential of this
reconstruction technique on some preliminary tests
performed on simulated point-like and extended objects.
In Sect. 4 we present the experiment realized at the
TIRGO observatory. The measurements allowed us to
study some quantitative aspects concerning the process
of image formation under low-order atmospheric
degradation conditions (see Sect. 4.1). Section 4.2 shows the result
of the application of the reconstruction method on
LBTlike data from the TIRGO experiment.
2. The reconstruction method
In this work, we used a Lucy-Richardson (LR) iterative
algorithm adapted for multiple deconvolution, i.e. an
algorithm based on the LR method that allows to retrieve
full angular resolution images from LBT interferograms.
This adaptation was rst developed and tested at the
University of Genova (Bertero & Boccacci, in
preparation) on simulated data made of one frame per orientation
angle.
From this code, we have implemented the possibility
to add the information of more than one frame per
orientation angle, i.e. to do a simultaneous deconvolution
of a set of frames per orientation angle. This approach is
useful when dealing with noisy and atmospheric-degraded
data, in the case of the LBT-like data we have collected.
The method yields the common maximum-likelihood
estimate object from the full data set. Details of this
algorithm are outlined hereafter.
Let's denote f (x; y) the brightness distribution of the
object, and hi(x; y) the PSF corresponding to the ith
baseline position angle of the LBT interferometer. For
each position angle, we observe the interferogram intensity
gi(x; y) de ned by the object image convolution
relationship:
gi(x; y) = hi(x; y) f (x; y) + n(x; y)
where n(x; y) refers to a spatially variant noise process.
The original LR algorithm is a non-linear algorithm
derived from Bayesian considerations, and based on
the knowledge of the PSF. The principle consist in
multiplying the result of each iteration by a correction
factor Ck(x; y), with k denoting the iteration number,
that relates to the remaining tting error. In standard
LR image restoration, i.e. for i = 1, the iterative relation
is the following:
(1)
(2)
(3)
to successive estimates f k(x; y) which are implicitly
positive. Since di erent values of this constant do not lead
to signi cant consequences in term of convergence rate,
f 0(x; y) was xed to unity. Another interesting
characteristic of this algorithm is the conservation of the total
energy. It is also demonstrated
(Shepp & Vardi 1982)
that
this relation leads to the maximum-likelihood object
estimate, under the assumption of a Poisson process in image
formation.
In the multiple deconvolution case, i.e. for i > 1, we
simply sum the contribution of each position angle at each
iteration. The iterative scheme is identical to Eq. (2) only
the correction factor is changed into:
Ck(x; y) =
Because of the normalization of each hi(x; y) to unity,
H = N .
The same approach is used when a set of frames per
orientation angle gi;j (x; y) exists, with j denoting the
position of the frame in the set. Adaptation of the algorithm
to this leaves unchanged Eq. (2) and only modi es the
correction factor as following:
Ck(x; y)=
H i=1 j=1
1 XN XM hi (−x; −y)
gi;j (x; y)
hi(x; y) f k(x; y)
(6)
where M is the number of frames per angle in the set.
Note that the PSF may also be di erent, and therefore
written as hi;j (x; y). But, concerning our data, we were
not able to obtain a good knowledge of the
corresponding PSF for each frame and therefore preferred a common
PSF estimate.
3. Image simulation and reconstruction test
In this section, we present the application of the
reconstruction method of Sect. 2 to numerical simulations of
LBT imaging, taking into account the telescope
performance and adding realistic estimates for the detector. In
a rst step, the construction of simulated interferometric
observations was carried out using an ideally AO
corrected PSF within the whole eld of view. In addition we
considered perfect optics and co-phasing of the two pupils.
The PSF's were therefore modeled as cosine-modulated
Airy functions. In the near future we plan to obtain
a more realistic PSF modeling by taking into account
the attainable level of AO correction. The simulated
interferometric observations, performed in R-band, were
obtained as the convolution of the target with the
f k+1(x; y) = f k(x; y) Ck(x; y)
with
Ck(x; y) = h (−x; −y)
g(x; y)
h(x; y) f k(x; y)
where h (−x; −y) is the conjugate of h(−x; −y). With a
positive constant estimate f 0(x; y), this algorithm leads
Fig. 2. Same as Fig. 1 for a binary star with main component magnitude mR = 27:5 and
mR = 2:5
PSF corresponding to each parallactic angle, adding
sky-background emission. Independent Poisson noise
realizations were then computed, and realistic detector
RON added for each parallactic angle. In all simulations
presented below, we assumed 1000 s integration time
per parallactic angle, 30% e ciency (mirrors + optics +
detector), a sky brightness of 20:80 mag=arcsec2 and a
RON equivalent magnitude of 35:8 mag=pix.
Since the large extension ( 3000 in R-band) of the
foreseen AO corrected LBT eld, it will be theoretically
possible to obtain a su ciently bright reference star in
the eld for PSF calibration. Indeed the average density
of stars with mv 21 is about 0.9 per sq arcmin at 90
galactic latitude, and almost ten times greater at lower
galactic latitudes. Therefore the PSF's used in these
restorations were assumed without noise-contamination.
A few applications carried out on both point-like (binary
star) and extended objects are presented hereafter.
3.1. Reconstructed images of binary star objects
We have applied the reconstruction method to binary
stars of di erent relative magnitude, and outlined
effects of the SNR on both astrometric and photometric
precision of the reconstruction. In the two examples
presented here, we have chosen extreme magnitudes and
magnitude di erences. Note that we have considered 4
equidistant parallactic angles (0 , 45 , 90 and 135 ),
and each star was located on a pixel of the 64 64 pixels
array, which leads, with the sampling of 4 pixels per
fringe adopted, to a 0 0:010 eld of view. The separation
of the binaries were xed to a value of 14.1 pixels,
corresponding to 22.6 mas i.e. about 3 times the di raction
angular resolution limit, and the orientation angle to 45 .
The relative photometry was computed by measuring
the peak pixel values of the reconstructed sources since
the sources were initially located at integer pixel locations.
In the rst case, mR = 1 and mR = 29 for the main
component. This leads to an average peak SNR of only
5.5 for the simulated interferograms. If we do not consider
fringe pattern overlapping between main star and
companion, the peak SNR for this latter is only 2.2 (see Fig. 1).
Concerning the reconstructed object, the binary location
is fully retrieved, while we will comment separately on the
photometric accuracy.
In the second case, mR = 2:5 and mR = 27:5 for
the main component. This leads to an average peak SNR
of 11.3 for the simulated interferograms, and 1.1 for the
companion. As in the rst case, the reconstruction leads
to a fully retrieved \astrometric" position of the original
object. The relative magnitude, after 1000 iterations, of
the reconstructed object is 3.1, a good result when the
di culty of detecting this faint companion in the noise
level is considered (see Fig. 2).
In both cases, we measured a magnitude di erence
larger than the true magnitude di erence. We noticed
however that this discrepancy was signi cantly reduced
in tests with higher SNR. In Fig. 3, we show the variation
of the photometric accuracy with iteration number. The
residuals of the magnitude di erence appear to remain
stable after several hundred of iterations in both cases. Note
also that, even though the algorithm convergence takes
place essentially in < 102 iterations (Fig. 4), the
photometric accuracy seems to continue to improve, but this
fact is only a consequence of performing photometry on
the peak pixel. For the purpose of the present work, we
were satis ed to verify this stability, and did not concern
ourselves with a criterion to stop the iterations. In
practical applications, the algorithm will be stopped according
to the usual considerations on noise.
3.2. Reconstructed image of an extended object
We also tested the ability of the algorithm to reconstruct
images of extended objects from LBT interferograms. The
target used here is an image of the spiral galaxy NGC 1288
rebinned in a 128 128 pixels array that, with the
sampling adopted, corresponds to a eld of view of 00:020 in
R-band. The image was apodized in order to avoid the
e ect of edge discontinuities in the restoration. In this
example, the reconstruction was based on simulated
observations at 6 equidistant parallactic angles (0 , 30 , 60 ,
90 , 120 and 150 ). With the 1000 s integration time
assumed per parallactic angle, the total integration time is
about 1.7 hours. The magnitude of the galaxy was set to
mR = 19, which leads to approximately 2 107 photons per
iterations
100
1000
Fig. 4. Variation of the error metric with iterations number for
the mR = 29 − 30 binary (solid line) and the mR = 27:5 − 30
binary (dashed). Error metric is de ned as the sum, for all
parallactic angles, of the Euclidian distance between the simulated
interferogram and the convolution of the result with the PSF
0″.1
Fig. 5. Numerical simulation of interferometric imaging of an
extended object with LBT. Top left: one of the observed
interferograms of the 0 0:014 eld of view of a mR = 19 galaxy
(parallactic angle = 30 ), obtained after an integration time of
1000 s. Bottom row shows the simulated target, un ltered (left)
and band-pass limited to a 22.65 m perfect circular aperture
(right). Top right: result of the algorithm for 6 interferograms
of 1000 s integration time each after 150 iterations
d
long exposure image and a peak SNR of 80. One can
notice the sharper aspect of the reconstructed galaxy shown
in Fig. 5 with respect to the theoretical di raction-limited
image. This is due to the behaviour of the algorithm in
cases of high SNR. Also, a closer inspection shows that
the nucleus does not appear as smooth as in the original
picture. This is a consequence of the phenomenon of \noise
ampli cation", which basically arises for such a
maximumlikelihood deconvolution algorithm from the di erence of
converging rate between extended objects and point-like
features
(White 1994)
. For our aim, we did not concern
ourselves with this problem, but di erent approaches
concerning the solution of this drawback of the LR algorithm
can be found in the literature
(Lucy 1994; White 1994;
Waniak 1997)
.
4. Experiment at the TIRGO telescope
In early 1998, we started an experiment at the 1.5 m
TIRGO infrared telescope, with a mask simulating the
pupil of the LBT telescope. The main idea was to record
realistic LBT-like data by simulating the level of adaptive
optics (AO) correction expected for LBT by the choice
of the ratio D=r0 (where D is the diameter of one of the
mask apertures projected on the primary and r0 the Fried
parameter), in order to investigate the properties of the
atmospheric parameters and study the process of image
formation and reconstruction (Fig. 6).
4.2. Application of the reconstruction technique to real
data: γ Leo
An observing run, on the nights of 21 and 22 March
1998, permitted to collect a serie of measurements of one
point-like and one binary star under two di erent values of
the ratio D=r0 (Table 1). Actually, it can be noticed that
the reduction in D=r0 obtained with the use of a mask
(i.e., by making D smaller) is not equivalent to that
expected from the use of adaptive optics on the real LBT
(i.e., by making r0 larger). In fact, in our case the mask
holes produce a rescaling of the frequencies in the
turbulence power spectrum. The e ect of correction by adaptive
optics, on the other hand, is not equal at all frequencies.
But, for the aim of this experiment, these two
corrections present enough similarities to be considered equal
in rst approximation. In order to simulate the rotation
of the LBT-like pupil function, i.e. the aperture
synthesis by earth rotation, we recorded object interferograms
at four nearly equidistant mask orientation angles by
rotating the secondary mirror. The data were recorded in
a broad band J lter at the 1.5 m TIRGO infrared
telescope using the ARNICA camera
(Lisi et al. 1996)
which
is equipped with a 256 256 pixels NICMOS 3 detector
and presents a pixel size of 0 0:098. During the run, we made
use of the fast read-out mode of a 32 32 pixels sub-array,
developed for lunar occultations, which allows typical
integration times of around 20 ms. A brief description of this
mode can be found in
Richichi et al. (1996)
. We present
below a summary of some of the data reduction results.
The aperture size of the mask and the seeing conditions
led, for this observation, to D=r0 = 0:84. In the
following, we used mainly a data set of 100 ms integration time
exposures. This data set is composed of a total of 1200
interferograms of the binary γ Leo (Algieba), a sequence of
300 interferograms for each of the four di erent mask
orientation angles (0 , ' 46 , ' 86 , ' 126 ). The object is a
binary star (ADS 7724) composed of a K1III main
component and a G7III companion. According to the Hipparcos
catalogue
(ESA 1997)
and to calibrated V − J colors, the
main component magnitude and the magnitude di erence
are respectively mJ = 0:21 and mJ = 1:39. The angular
separation is 4 0:058, almost two times the di raction limit
of the simulated interferometer. In addition, 300
interferograms of an unresolved bright star (BS 5589, a M4.5III
star with mJ ' 0) were recorded for the rst mask
orientation. These reference data are used for the system
response (optics + atmosphere), the so-called PSF, in the
deconvolution process.
In this run, the total collecting aera was limited by
the xed pixel size of the camera. For this reason, only
0:26% of the mirror area was used, leading for this
integration time to a relatively poor signal-to-noise (S/N)
ratio on the frames. Assuming frozen turbulence driven
by the wind velocity v, the characteristic evolution time
is of the order of B=v, where B is the projection of b on
the primary i.e. here B = 9:5 cm. Taking v = 10 m/s
4.1. Measurement of atmospheric parameters leads to a typical evolution time of the order of 10 ms.
Consequently, the level of atmospheric degradation
conditions (which is evaluated from the D=r0 ratio to a 46%
Strehl ratio) allowed us to assume that the blurring arising
A least-square PSF tting program was developed and ap- from high order atmospheric turbulence of evolution time
plied to our data in order to extract some atmospheric pa- scale inferior to 100 ms remains important. Note that, in
rameters of interest. It allowed us to retrieve the evolution order to minimise variations in seeing conditions during
with time of the fringe contrast, and of the random average data acquisition, all frames of both binary and point-like
optical path di erence between the two apertures (that is stars have been recorded during consecutive periods of the
the di erential piston). This was done for the two values night.
of the ratio D=r0, on a 30 s period, and for an unique base- In the reconstruction, we used 200 interferograms of
line orientation (Fig. 7), and we found consistent results the binary star per orientation angle. A total of 800
interin terms of average contrast and piston root-mean-square ferogram patterns were pre-processed before applying the
(rms) values. Indeed the case of major atmospheric degra- reconstruction method. Pre-processing consisted in
subdation leads to a smaller average contrast: for D=r0 = 0:84 tracting the average sky background, a bilinear
upsamand D=r0 = 0:57 we found respectively an average con- pling from 30 30 pixels to 90 90 pixels and then the
trast of 0.26 and 0.60. Accordingly, we found a piston rms extraction of a 64 64 pixels frame centered on the
photovalue of respectively 0.11 and 0.06 in unit wavelength. center fringe pattern (see Fig. 9). This oversampling
tech
In addition, combining several consecutive short expo- nique allowed a sub-pixel centering of each original 30
sure PSF interferograms, we studied the variation of con- 30 pixels interferogram, i.e. to remove the atmospherically
trast with integration time under the same two conditions induced image motion.
of turbulence degradation (Fig. 8) and de ned a coherence Moreover, in order to obtain an accurate PSF estimate
time tc as the integration time which corresponds to a loss needed for the LR algorithm, a Shift-and-Add (SAA)
proof a certain percentage of fringe contrast. We deduced the cess was computed over the whole PSF data set. Then we
value of tc with 10% loss for our experiment and attempted used the same pre-processing of the object interferograms
to extrapolate the result to the LBT case
(Correia 1998)
. to obtain the PSF estimate (see Fig. 9). Unfortunately,
D/r0 = 0.84
ilsp tihnu 0.0
ia ng
t le
re av- 0.2
n e
e w
iff in(- 0.4
D
- 0.6
0
100
t
s
a 80
r
t
n
o
c
e 60
v
i
t
a
l
e
r
40
%
20
0
D/r0 = 0.57
time (s)
D/r0 = 0.57
20
30
the PSF was only recorded for the rst orientation angle.
The PSF corresponding to other orientation angles were
obtained by computer rotation using a bilinear
interpolation. Note that this rotation was performed on the
oversampled frames, allowing to conserve more information in
the PSF shape. However, it is unlikely that the responses
of the system for each orientation angle are identical to the
rotated responses. Therefore the rotated PSF's obtained
in this way are only an approximation.
The companion is clearly visible in the reconstructed
image of Fig. 10, in spite of numerous limitations in this
data set. Indeed, in addition to the low S/N ratio, and
the approximation of the rotated PSF, the original data
were also slightly under sampled ( 1:8 pixels per fringe
FWHM ).
We further analysed the e ect of the number of
frames on the reduced data. Figures 11 and 12 show
the evolution with this parameter of, respectively, the
reconstructed object, and the pro le of the reconstructed
s
l
e
x
i
p
E
Fig. 11. Reconstruction of the binary γ Leo according to the
number of data frames per orientation angle processed, using
the adapted LR algorithm (1000 iterations). The pixel size is
0 0:098=3 0 0:033. Contour levels are from 5% to 100% in steps
of 5%
1.0
ity 0.8
s
n
e
itn 0.6
d
e
lis 0.4
a
m
r
oN 0.2
0.0
0
5
10
50
200
5
10
arcseconds
15
20
binary along the direction of separation. It can be seen
that the gain obtained in the resulting reconstruction
using multiframes deconvolution is important for a small
number of data frames, and tends to converge rapidly.
The variation of the FWHM of the main component is
consistent with the fact that deconvolving more frames
simultaneously improves the quality of the reconstructed
object, i.e. the sharpness of the reconstruction. In fact,
for a number of 5, 10, 50 and 200 frames per angle
used, the FWHM of the main component obtained is
respectively 2 0:047, 2 0:034, 2 0:031 and 2 0:021. It is interesting
to note that we approached the theoretical di raction
limit for our mask of 1 0:073, but we were unable to reach
it because of the limitations previously mentioned. It is
however important to stress that this 2 0:011 interferometric
resolution does allow us to resolve γ Leo, whereas the
5 0:083 resolution of only one aperture would not permit it.
We noticed moreover that this simultaneous multiframes
deconvolution leads to a better result than deconvolving
simultaneously the sum of the frames for all baseline
orientation.
A data set composed of 50 ms integration time frames
was reduced in the same way and the result obtained
is presented in Fig. 13. This shorter integration time
\freezes" better the atmospheric distorted fringe pattern
and the result is a more detached binary image. Note
that the spurious signal present on the main component
sides probably came from a non perfect centering of each
fringe patterns due to the poor S/N ratio.
We tested the validity of these reconstructed images
by performing the comparison of the retrieved binary
parameter values with those of the Hipparcos catalogue
(included in the CHARA catalogue, Hartkopf et al. 1998)
.
Centroid calculations performed on 5 5 pixel boxes
centered on each maximum intensity pixel of the two star
brightness distributions led to the relative location of the
binary. Two methods were used in order to measure the
magnitude di erence mJ . Firstly, relative aperture
photometry was performed with aperture radii from 3 pixels
to 7 pixels, and we deduced therefore a range of retrieved
mJ . Secondly, we obtained photometry by PSF- tting of
the non-deconvolved frames, using the location retrieved
by means of deconvolution as an input xed parameter.
This PSF- tting is based on a least-square tting routine
identical to that used in Sect. 4.1 and was applied to a
same number of coadded frames for each orientation angle.
Results reported in Table 2 are an average over the four
orientation angles, weighting with the 2 in order to take
into account the error tting of the other free parameters.
From the values reported in Table 2, one can see that we
are able to retrieve both the position angle and the
angular separation of γ Leo, with a better accuracy in the
100 ms case. Concerning the relative magnitude retrieved
by aperture photometry, mJ is larger than the catalogue
value by an amount of 0.77 to 0.58 mag for respectively 3
to 7 pixels aperture radius and 100 ms integration time.
This discrepancy is probably due to the poor S/N ratio
present in the recorded frames, and seems to be a common
characteristic of most of the non-linear image restoration
techniques when applied to strongly noise-contaminated
data
(Lindler et al. 1994; Christou et al. 1998)
. On the
other hand, this interpretation is con rmed by the
simulations performed in Sect. 3. Photometry measurements
obtained by means of PSF- tting lead to more accurate
mJ values and have the advantage to give error
estimates.
5. Conclusion
We have presented a method for interferometric imaging
with the Large Binocular Telescope. We have illustrated
the capability of a Lucy-Richardson based deconvolution
algorithm to reconstruct the object image from a set of
LBT-like interferometric data. We outlined potential
performances on simulated LBT data and robustness of the
algorithm has been tested on real LBT-like data. It is of
great importance, in order to achieve the reconstruction,
to be able to obtain, from a reference star, a su ciently
good estimate of the PSF for each baseline position.
The use of a Lucy-Richardson deconvolution algorithm is
limited by the quality of the PSF calibration. The most
favorable case would be to be able to record simultaneously
object image and PSF. The LBT will provide a wide
coherent eld of view for interferometric imaging, thanks to
innovative technological solutions
(Salinari 1996)
. Therefore
simultaneous measurements of both target and a bright
reference star will be possible and, from this point of view,
the algorithm presented here is of particular interest. The
quality of the PSF calibration will be then limited by the
di erence of wavefront compensation achieved for these
two objects. The future implementation of a AO
correction in the simulation code will allow us to quantify this
limitation.
Acknowledgements. We are indebted to F. Lisi and
G. Marcucci for their help with the experimental set-up.
We thank the director of the TIRGO observatory, Prof.
G. Tofani, for the allocation of observing time. We are grateful
to M. Bertero and P. Boccacci (University of Genova) for
providing the initial version of the reconstruction algorithm,
developed under contract number 16/97 funded by the Italian
Consorzio Nazionale per l'Astronomia e l'Astro sica. We wish
also to thank P. Salinari, M. Carbillet and J.C. Christou
for useful discussions, and A. Marconi and G.P. Tozzi for
assistance during the observations.
Angel J.R.P. , Hill J.M. , Strittmatter P.A. , Salinari P. , Weigelt G. , 1998 , in: Astronomical Interferometry, Proc. SPIE 3352 , Reasenberg R .D. (ed.), p. 881
Christou J.C. , Hege E.K. , Je eries S. , Cheselka M. , 1998 , in: Atmospheric Propagation, Adaptive Systems, and Lidar Techniques for Remote Sensing II, Proc. SPIE 3494 , Davir A. , Kohnle A. , Schreiber U. , Werner C . (eds.), p. 175
Correia S. , 1998 , Infrared interferometric imaging test for the LBT , DEA report , University of Nice Sophia Antipolis (available at http://www.arcetri.astro.it/~correia)
ESA , 1997 , The Hipparcos catalogue , ESA SP-1200
Hartkopf W.I. , McAlister H.A. , Mason B.D. , 1998 , Third catalog of interferometric measurements of binary stars, CHARA (available at http://www.chara.gsu.edu/ DoubleStars/Speckle/intro.html)
Hege E.K. , Angel J.R.P. , Cheselka M. , Lloyd-Hart M. , 1995 , in: Advanced Imaging Technologies and Commercial Applications , Proc. SPIE 2566 , Clark N. , Gonglewski J.D . (eds.), p. 144
Je eries S.M. , Christou J.C. , 1993 , ApJ 415 , 862
Lindler D. , Heap S. , Holbrook J. , et al., 1994 , in: The Restoration of HST Images and Spectra II. Space Telescope Science Institute , Hanish R.J., White R.L . (eds.), p. 286
Lisi F. , Ba a C. , Biloti V. , et al., 1996 , PASP 108 , 364
Lucy L.B. , 1974 , AJ 79 , 745
Lucy L.B. , 1994 , in The Restoration of HST Images and Spectra II. Space Telescope Science Institute , Hanish R.J., White R.L . (eds.), p. 79
Reinheimer T. , Hofmann K .-H., Sch¨oller M. , Weigelt G. , 1997 , A &AS 121 , 191
Richardson W.H. , 1972 , J. Opt. Soc. Am . 62 , 55
Richichi A. , Ba a C. , Calamai G. , Lisi F., 1996 , AJ 112 , 6
Salinari P. , 1996 , in Optical Telescopes of today and Tomorrow , Proc. SPIE 2871 , Ardeberg A .L. (ed.), p. 564
Shepp L.A. , Vardi Y. , 1982 , IEEE Trans . Med . Imaging, Vol. MI - 1, p. 113
Waniak W. , 1997 , A &AS 124 , 197
White R.L. , 1994 , in The Restoration of HST Images and Spectra II. Space Telescope Science Institute , Hanish R.J., White R.L . (eds.), p. 104