Application of iterative blind deconvolution to the reconstruction of LBT LINC-NIRVANA images

Abstract

Context. The paper is about methods for multiple image deconvolution and their application to the reconstruction of the images acquired by the Fizeau interferometer, denoted LINC-NIRVANA, under development for the Large Binocular Telescope (LBT). The multiple images of the same target are obtained with different orientations of the baseline. Aims. To propose and develop a blind method for dealing with cases where no knowledge or very poor knowledge of the point spread functions (PSF) is available. Methods. The approach is an iterative one where object and PSFs are alternately updated using deconvolution methods related to the standard Richardson-Lucy method. It is basically an extension, to the multiple image case, of iterative blind deconvolution methods proposed in the case of a single image. Results. The method is applied to simulated LBT LINC-NIRVANA images and its limitations are investigated. The algorithm has been implemented in the module BLI of the software package AIRY (Astronomical Image Reconstruction in interferometrY), available under request. The preliminary results we have obtained are promising but an extensive simulation program is still necessary for a full understanding of the applicability of the method in the practice of the reconstruction of LINC-NIRVANA images.

Full text

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A&A 452, 727–734 (2006)
DOI: 10.1051/0004-6361:20054481
c ? ESO 2006
Astronomy
&
Astrophysics
Application of iterative blind deconvolution to the reconstruction
of LBT LINC-NIRVANA images
G. Desiderá1, B. Anconelli1, M. Bertero1, P. Boccacci1, and M. Carbillet2
1DISI, Università di Genova, via Dodecaneso 35, 16146 Genova, Italy
e-mail: bertero@disi.unige.it
2Laboratoire Universitaire d’Astrophysique de Nice, UMR 6525, Parc Valrose, 06108 Nice Cedex 02, France
Received 7 November 2005 / Accepted 11 February 2006
ABSTRACT
Context. The paper is about methods for multiple image deconvolution and their application to the reconstruction of the images
acquired by the Fizeau interferometer, denoted LINC-NIRVANA, under development for the Large Binocular Telescope (LBT). The
multiple images of the same target are obtained with different orientations of the baseline.
Aims. To propose and develop a blind method for dealing with cases where no knowledge or very poor knowledge of the point spread
functions (PSF) is available.
Methods. The approach is an iterative one where object and PSFs are alternately updated using deconvolution methods related to the
standard Richardson-Lucy method. It is basically an extension, to the multiple image case, of iterative blind deconvolution methods
proposed in the case of a single image.
Results. Themethod isappliedtosimulatedLBTLINC-NIRVANAimagesand itslimitationsareinvestigated. Thealgorithm hasbeen
implemented in the module BLI of the software package AIRY (Astronomical Image Reconstruction in interferometrY), available
under request.
The preliminary results we have obtained are promising but an extensive simulation program is still necessary for a full understanding
of the applicability of the method in the practice of the reconstruction of LINC-NIRVANA images.
Key words. methods: numerical – techniques: image processing
1. Introduction
In previous papers (Bertero & Boccacci 2000a,b; Correia
et al. 2002; Carbillet et al. 2002; Anconelli et al. 2005a)
we developed methods and software for the deconvolution
of multiple interferometric images of the same astronom-
ical target. Moreover, our group has produced the soft-
ware package AIRY (see http://dirac.disi.unige.it),
which can be used in conjunction with the software pack-
age CAOS (Code for Adaptive Optics Systems; Carbillet
et al. 2005), within the common CAOS “problem solving
environment” (see http://www.arcetri.astro.it/caos or
http://www-astro.unice.fr/caos). This tool can be ap-
plied to Fizeau interferometers such as LINC-NIRVANA (Lbt
INterferometric Camera and Near-InfraRed/Visible Adaptive
iNterferometer for Astronomy), the German-Italian beam com-
biner for LBT (in the following LINC-NIRVANA will be de-
noted by LN). We recall that LBT consists of two 8.4m mirrors
on a common mount, with a spacing of 14.4m between their
centres, so that a maximum baseline of 22.8m will be available.
The first primary mirror has been installed in March 2004, and
“first light” has been achieved on October 12, 2005. The second
primary mirror has been aluminized on board in January 2006,
so that LBT is now a true binocular telescope with two coated
primary mirrors.
The methods we have implemented and validated are based
on the extension of the well-known Richardson-Lucy method
(RLM) to the problem of multiple image deconvolution.Indeed,
LN will provide different images of the same astronomical
target, corresponding to different orientations of the baseline. In
order to get a unique high-resolution reconstruction of the tar-
get, one needs the knowledge of the interferometric PSFs corre-
spondingto the differentimages. These PSFs cannot be modeled
and computed since they depend on the configuration of the at-
mosphere during the exposures; therefore they must be obtained
from the images of reference stars. These images are corrupted
by several factors such as noise and backgrounds (we assume
calibrated images, already corrected for effects such as bad pix-
els, flat field, etc.); moreover, as a consequence of non-uniform
adaptive optics (AO) correction, they may not correspond to the
PSFs in the region of the scientific object. In other words, one is
faced with the problemof improvingthe quality of the measured
PSFs. Sometimes the knowledge may be so poor that the PSFs,
in practice, are not known.
In the case of single-image deconvolution, methods of blind
deconvolution (BD) have been developed for extracting both the
object and the PSF from the recorded image. These methods can
be used also when a poor knowledge of the PSF is available, as
discussed above. In such a case the term myopic deconvolution
is sometimes used.
BD methodscan be dividedinto two classes. In the first class
BD is attemptedby minimizinganerrorfunctiondesignedto op-
timize both the reconstructed object and the PSF, with the addi-
tion of suitable constraints (non-negativity, bounded domain in
the angular or u,v variables, etc.). The underlyingobject and the
PSF are obtained simultaneously. An example of these methods
is that proposedbyJefferies &Christou(1993).Thesecondclass
consistsofmethodsthatrestoretheobjectandthePSFseparately
Article published by EDP Sciences and available at http://www.edpsciences.org/aa or http://dx.doi.org/10.1051/0004-6361:20054481

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728G. Desiderá et al.: Application of iterative blind deconvolution to the reconstruction of LBT LINC-NIRVANA images
in an iterative form: within each iteration, which we will call a
cycle, either the object or the PSF is kept fixed while the other
is updated, using one of the usual deconvolution techniques (re-
member that the convolution product is symmetric with respect
to the exchangeof object and PSF). Thereforethe output of each
cycle updates both the object and the PSF, as provided by the
previous one. These methods are referred to as iterative blind
deconvolution (IBD).
IBD was first proposed with the Wiener filter as deconvolu-
tion method (Ayers & Dainty 1988). However, most of the IBD
methods are based on RLM (Holmes 1992; Tsumuraya et al.
1994; Fish etal.1995;Biggs &Andrews1998).Itis obviousthat
one can use different deconvolution methods for the object and
the PSF. For instance, an IBD approach based on the projected
Landweber method, with different constraints for the object and
the PSF, is proposed in Bertero et al. (1998).
In this paper we extend IBD to the case of multiple-image
deconvolution and we apply the method to the reconstruction of
LN images. In Sect. 2 we describe the general structure of IBD
in terms of cycles: if p is the number of detected images, each
cycle provides an update both of the object and of the p PSFs
and is described in terms of an Object box and of p PSF boxes.
As suggested by the names, the Object box updates the object
while the PSFs boxes update the p PSFs. We describe the in-
put/output relationship between the boxes and the initialization
of the cycles. In Sect. 3 we give the algorithms implemented
within the two boxes. In the Object box we use the method
denoted ordered subset expectation maximization (OSEM), in-
troduced by Hudson & Larkin (1994) in emission tomography,
and extended to LBT imaging in our previous papers (Bertero &
Boccacci 2000a; Anconelli et al. 2005a); in the p PSF boxes we
use single image RLM, with additional constraints on the PSFs.
In Sect. 4 we describe the module BLI of the software package
AIRY, implementing the IBD method we have developed, and,
finally, in Sect. 5 we give the results of our numerical experi-
ments. In Sect. 6 we discuss the limitations of the method, as
derived from our experience.
2. The structure of the method
We use bold letters for denoting N × N arrays, whose pixels are
indexed by a multi-index n = {n1,n2}. Moreover, we assume to
have p images acquired with LN, corresponding to p different
orientations of the baseline and denoted by g1,g2,...,gp. Then,
according to the model proposed by Snyder et al. (1993) for
images acquired with a CCD camera, the value of one of these
images at pixel n is given by:
gj(n) = gobj,j(n) + gback,j(n) + rj(n),
where: gobj,j(n) is the number of photoelectrons due to radiation
from the object; gback,j(n) is the number of photoelectrons due
to external and internal background, dark current, etc.; rj(n) is
the read-out noise due to the amplifier. The first two terms are
realizations of independentPoisson processes (photonnoise), so
that their sum is also a Poisson process; its expected value is
given by:
E{gobj,j(n) + gback,j(n)} = (Kj∗ f)(n) + bj(n),
where: Kjis the point spread function (PSF), corresponding to
the jth orientation of the baseline; f is the object array; bjis
the expected value of the (constant) background. We assume, as
usual, that the PSFs are normalized to unit volume:
?
(1)
(2)
n
Kj(n) = 1.
(3)
Fig.1. Pictorial representation of a blind cycle.
The data of the problem are the p detected images and back-
grounds, {gj, bj} (j = 1,..., p), and the goal is an estimate of the
unknown PSFs Kj, (j = 1,..., p), and object f.
Fortheconvenienceofthereader,inFig.1wegiveapictorial
representation of one IBD cycle, which consists of one Object
box, followed by p PSF boxes, one for each PSF. We introduce
an index k characterizing the IBD cycles. If {K(k−1)
f(k−1)} is the output of the cycle k − 1 (or the initial estimate in
the case k = 1), then the functions of the boxes in the cycle k are
the following.
j
, j = 1,..., p;
– Object box: this box performs a multiple-image deconvolu-
tion. The input consists of {gj, bj}, (j = 1,..., p) and of the
outputs of the previous cycle, {K(k−1)
while the output will be an update f(k)of the object.
– PSF boxes: each box performsa single-image deconvolution
for estimating one of the PSFs. For a given j, the input of
the jth PSF box consists again of {gj, bj} and, in addition, of
the estimate of the jth PSF, K(k−1)
cycle, as well as of the estimate of the object, f(k), provided
by the previous Object box. The output will be an update of
the jth PSF, K(k)
j
, j = 1,..., p; f(k−1)},
j
, provided by the previous
j.
Since the input data {gj, bj}, (j = 1,..., p) are common to all
boxes, in Fig. 1 they are indicated as inputs of all IBD cycles.
Moreover, it is clear that the initialization of IBD is just the ini-
tialization of the first Object box which, in Fig. 1, is indicated by
{K(0)
The deconvolution algorithms implemented in the different
boxes, and derived from OSEM and RLM, will be described in
detail in the next Section. Therefore we conclude by discussing
possible initializations of the first IBD cycle. The initialization
requires initial estimates both of the object and of the PSFs.
A possible choice of f(0)is provided by the standard initial-
ization of the RLM algorithm, namely a constant array. For the
PSFs the following choices can be considered:
j, f(0)}.
– if the object consists mainly of stars, then, as suggested by
Biggs & Andrews (1998), the initial PSFs can be obtained

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G. Desiderá et al.: Application of iterative blind deconvolution to the reconstruction of LBT LINC-NIRVANA images729
from the autocorrelations of the observed images and are
defined as follows:
Rj= gj? gj= gT
Hj= Rj− min{Rj} + ?(max{Rj} − min{Rj}),
K(0)
j
= Hj/(
n
j∗ gj,
(4)
?
Hj(n)),
where ? denotes the autocorrelation and gT
The effect of the background is limited by subtracting the
minimumvalueof theautocorrelation;then a small amount?
(for instance of the order of 10−2) of the dynamic range of
the autocorrelation is re-added, to prevent pixels with zero
values(werecall thatzerovaluescannotbechangedbymul-
tiplicative algorithms such as RLM or OSEM);
– in the case of diffuseobjects, the initial PSFs can be obtained
from the autocorrelations of the ideal PSFs. Also in this case
the autocorrelations are used, instead of the ideal PSFs, to
prevent pixels with zero values;
– measured PSFs, extracted from the images of one or more
detected stars; in such a case the method can be called
myopic deconvolution.
j(n) = gj(−n).
Our numerical simulations have confirmed that the best results
are provided by the first choice in the case of point objects, and
by the second choice in the case of diffuse objects.
Itis obviousthatIBD mustbestoppedafterkmaxcycles;then,
the correspondingestimates ofthePSFs couldbe usedforastan-
dard multiple image deconvolution to improve the estimate of
the object.
3. Deconvolution algorithms
We describe separately the algorithms implemented in the
Object box and in the PSF boxes, by assuming that these boxes
are working inside the cycle k, for k = 2,...,kmax; therefore
{K(k−1)
input of the cycle k.
j
, j = 1,..., p; f(k−1)} is the output of the cycle k−1 and the
3.1. Object box
Several deconvolution algorithms can be implemented and used
in this box. However, in our numerical simulations, we mainly
used OSEM.
We recall that the expectation maximization (EM) method,
introduced by Shepp & Vardi (1982) in emission tomography,
is an iterative method for the maximization of the likelihoood
function, derived from the assumption of images dominated by
photonnoise;inthe caseofimagedeconvolutionthis methodco-
incideswithRLM. ThemethodOSEM,introducedbyHudson&
Larkin (1994), is an accelerated version of the EM algorithm for
tomography,hence with data consisting of a set of projections of
the same object. In OSEM the projections are grouped into sub-
sets carrying different pieces of information and one EM itera-
tion is replaced by a cycle over the subsets in a given order; the
acceleration factor depends on the number and ordering of the
subsets. In the LBT problem the projections are replaced by the
images corresponding to different orientations of the baseline.
Since their number is small, it is not necessary to group these
images into subsets and also the order is not relevant. Therefore
one iteration of multiple image RLM is replaced by a cycle over
the p images, each step being a single image iteration.
Since the index k is used for characterizing the blind cy-
cles (external iterations), a different index, let us say l, will be
used for the iterations of OSEM internal to the Object box.
Accordingly, the result of the lth iteration of OSEM inside the
cycle k will be denoted by f(k,l). We use the version of OSEM
described in Anconelli et al. (2005a), which includes also a con-
straint on the flux of the reconstructed image.
Preprocessing step
– Compute the flux constant c defined by:
p ?
Reconstruction step
– Initialize the algorithm with f(k,0)= f(k−1), the output of the
cycle k − 1 (k = 2,...,kmax).
– For l = 0,..., lobj− 1, given f(k,l), set j = (l + 1) mod p and
compute:
⎛⎜⎜⎜⎜⎜⎜⎝[K(k−1)
n
– Set:
f(k,l+1)=
˜ c(k,l+1)˜f(k,l+1).
In these equations, as usual, quotients and products of arrays are
defined pixel by pixel. Moreover lobjis the number of OSEM
iterations performed inside the Object box. The output of this
box is just the array f(k)= f(k,lobj), that is part of the output of
cycle k.
c =1
p
j=1
?
n
?
gj(n) − bj(n)
?
.
(5)
˜f(k,l+1)= f(k,l)
j
]T∗
gj
K(k−1)
j
∗ f(k,l)+ bj
⎞⎟⎟⎟⎟⎟⎟⎠,
(6)
˜ c(k,l+1)=
?
˜f(k,l+1)(n).
c
(7)
3.2. PSF boxes
Inside the cycle k, the input of the jth PSF box (j = 1,..., p) is
given by {K(k−1)
means of RLM, that is equivalent to EM for single image decon-
volution. Therefore we have internal iterations characterized by
an index denoted again as l. The maximum number of iterations
performed inside each PSF box is denoted by lpsf.
Since RLM can perform an out-of-band extrapolation, it is
necessarytoreducethiseffect.ThereforewecombineeachRLM
iteration with a suitable low-pass filter, related to the band of the
jth image. The choice of this filter, denoted by Fj, is discussed
at the end of this subsection. However, the filter can induce neg-
ative values in the filtered PSF and these value must be removed.
A simple clipping is not recommended, because zero values are
preservedbythe multiplicativestructureof thealgorithm.We in-
troduce an operation,denotedby POS, which consists in replac-
ing the negative value in one pixel by the smallest positive value
in a suitable neighborhood of that pixel. Finally, the normaliza-
tion of the PSF to unit volume is performed. In conclusion, the
algorithm including all these constraints is the following.
– Initialize the algorithm with K(k,0)
cycle k − 1.
– For l = 0,...,lpsf− 1, given K(k,l)
⎛⎜⎜⎜⎜⎜⎜⎝[f(k)]T∗
F,jj
,
˜K(k,l+1)
P,j
= POS
F,j
.
j
, f(k)}. The updating of the PSF is obtained by
j
= K(k−1)
j
, the result of the
j
, compute:
˜K(k,l+1)
j
= K(k,l)
j
gj
f(k)∗ K(k,l)
j
+ bj
⎞⎟⎟⎟⎟⎟⎟⎠,
(8)
˜K(k,l+1)
= Fj∗˜K(k,l+1)
?˜K(k,l+1)
?

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730G. Desiderá et al.: Application of iterative blind deconvolution to the reconstruction of LBT LINC-NIRVANA images
– Set:
K(k,l+1)
j
=˜K(k,l+1)
P,j
/
?
n
˜K(k,l+1)
P,j
(n).
(9)
The output of the jth PSF box is just K(k)
of the jth PSF provided by the cycle k.
About the choice of the low-pass filter, we observe that, for
any given telescope, the band B of the instrument, namely the
domain of the {u,v}-plane where the modulus transfer function
(MTF) of the telescope is different from zero, is known, since
it is determined by the pupil of the instrument. In the particular
case of LN, which is the main subject of this paper, B consists
of the union of three discs, which are replicas of the band of a
8.4m mirror. More precisely, we have different bands Bjcorre-
sponding to the different orientations of the baseline. Then, with
reference to the LN-case, given Bj, the most simple choice of
the low-pass filter consists in taking Fjas the inverse Fourier
transform of the mask of Bj, namely the array Mjwhich is one
over Bjand zero outside:
ˆFj= Mj.
j
= K(k,lpsf)
j
, the update
(10)
However, as a consequence of the discontinuity at the boundary
of the band, this filter array contains strong side lobes of the
central peak, producing strong artifacts.
In order to circumvent this difficulty one can proceed in the
followingway.First, defineaneffectivebandBeff,jas thedomain
in the {u,v}-planewherethe MTF is greaterthansomethreshold-
ingvalueσ (i.e.|ˆKj(n)| > σ); next,convolvethemaskofthis do-
main with some suitable smoothing function such as a Gaussian
or a Butterworth function.
4. The module BLI of AIRY
On the basis of the ideas presented in the previous Sections,
we have developed a module BLI (BLInd deconvolution) of the
software package AIRY. As already mentioned, a description of
AIRY is given in Correia et al. (2002).
The input of the module BLI is a structure containing the
images and additionalinformation(estimate of the backgrounds,
pixel size, etc.), while the output consists of two structures: one
containing the reconstructed object and the other containing the
reconstructed PSFs. In the case of a simulation study, the data
are generated inside an AIRY project, while in the case of data
reduction application they must be uploaded.
The user has first to create, in the worksheet of CAOS
(Correia et al. 2002), an AIRY project linking together a set of
modules includingBLI. In the worksheet he has to set the global
number of iterations which, in this application, coincides with
the maximum number of cycles, kmax, as defined in Sect. 2. In
Fig. 2 we show the graphical user interface (GUI) of this module
which allows the user to set the parameters of a blind deconvo-
lution application, as described in the previous sections.
Aboutthe choiceof the PSFs we observethat the first item in
the enumeration of Sect. 2 concerns PSFs which are computed
by the module while the second and third item must be intended
as PSFs providedby the user. Moreover,other parameters which
must be selected by the user are: the maximum number of iter-
ations inside the Object box, lobj, denoted as Object iterations
in the GUI; the maximum number of iterations inside the PSFs
boxes, lpsf, denoted as PSFs iterations in the GUI. The choice of
these parameters will be discussed in the next section.
The choice of the filter is an important issue and can be done
through a specific GUI (see Fig. 3), which allows to create the
Fig.2. GUI of the module BLI of AIRY.
Fig.3. GUI for the choice of the filter.
different types of filters described in Sect. 3 or to upload a user
defined filter.

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G. Desiderá et al.: Application of iterative blind deconvolution to the reconstruction of LBT LINC-NIRVANA images731
5. Numerical experiments
IBD contains several parameters which must be selected by the
user,suchas thenumberofiterationswithinthe ObjectandPSFs
boxes,lobjandlpsfrespectively,andthetotalnumberofcycles(or
global iterations) kmax. Following Biggs & Andrews (1998), we
will call symmetric IBD the case lobj= lpsfand asymmetric IBD
the case lobj? lpsf.
Other important degrees of freedom are the initialization of
the IBD cycles and the choice of the filter in the PSFs boxes.
About the last point, after a number of numerical experiments,
we concluded that, in the case of LN, the best choice is provided
by a superposition, clipped to one, of three circular Butterworth
filters of order 10; the three filters are centered on the three discs
forming the band, and have a width slightly smaller than the ra-
dius of the discs.
In our numerical experiments we consider objects repre-
sented by 256×256arrays and we assume that they are observed
in K-band by means of the LN detector; hence the pixel size is
5.12 mas. Moreover we assume that each observation of a given
object provides a set of three images obtained with different ori-
entations of the baseline, corresponding to relative parallactic
angles of 0◦,60◦and 120◦. Therefore, for simulating one obser-
vation we need a set of three PSFs. Since we are interested to in-
vestigatethe dependenceoftheresults onthe Strehlratio (SR) of
thePSFs,weconsiderthreedifferentsets,withdifferentSR,each
consisting of three PSFs with the orientations of the baseline in-
dicated above. The first is a set of ideal interferometric PSFs of
LN,thesecondandthirdonearesetsofAO-correctedPSFs,with
SR of about 70% and 26%, respectively. These PSFs have been
obtained by means of the software package CAOS, according to
a model already described in previous papers (Anconelli et al.
2004, 2005a).
For a given observation of a given object, the three im-
ages are generated first by convolving the object with the se-
lected set of PSFs. Next, sky background emission in K band
(12.5 mag/arcsec2) is added to the results, which are also per-
turbed with Poisson and read-out Gaussian noise (10 e−rms).
Moreover the following parameters are used: a total efficiency
of 25%, an integration time of 20 min and a telescope surface
of 104 m2.
With these observational parameters, the average number of
background photons per pixel is about 1.4 × 104; moreover the
total numberof object photons is about 5.4×109for an object of
magnitude 10, and 2.1 × 109for an object of magnitude 11. We
remark that the assumed read-out noise, that corresponds to the
expected value for the LN detector, is negligible with respect to
background and photon noise.
In the following subsections we describe the results pro-
vided by the implementedIBD methodfor two differentkinds of
objects: binary systems and diffuse objects.
5.1. Binary systems
We consider three different binary systems, with different angu-
lar separations: 22 mas, 36 mas and 195 mas. The first is just
at the limit of the angular resolution of LN in K-band (20 mas),
while the two othersare well resolved.Moreoverwe assume that
the two stars have magnitude 10 and 11 in K-band, so that, by
taking into account background and photon noise, the average
SNR per pixel is about 320.
In the following the three binaries will be denoted as
Binary 1, 2 and 3, respectively. For each of them we consider
three sets of images, corresponding to the three sets of PSFs
Fig.4. Reconstruction of the binary with angular separation of 36 mas,
using exact PSFs(inverse crime). Behaviour of the flux of the two com-
ponents of the binary as a function of the number of OSEM iterations:
ideal PSFs (dotted line); AO-corrected PSFs with SR = 70% (dashed
line) and SR = 26% (dash-dotted line). The horizontal full lines give
the exact values of the fluxes.
indicated above. Hence, each set simulates an observation with
a given AO-correction.
The best accuracy achievable in the reconstruction of the bi-
naries is estimated by an inverse crime experiment, namely the
set of PSFs used for generating the images is also used for re-
constructing the binaries by means of the OSEM method im-
plemented in AIRY. The accuracy is tested by computing the
fluxes of the two components at each iteration, using a square
of 3 × 3 pixel centered on each one of the two stars. The results
in the case of Binary 2 are shown in Fig. 4. We use a maximum
number of 100 iterations and in all cases the accuracy is satis-
factory, even if the fluxes are a bit underestimated in the case
of Binary 1 and of the PSFs with the smallest SR. Probably one
needs a larger number of iterations in this case. Indeed, the main
difference between the three cases consists in the convergence
rate: convergenceis faster for higher values of SR and for wider
angular separation of the two stars.
Since we are considering point objects, we first use a sym-
metric IBD with lobj= lpsf= 1, as suggested by Tsumurayaet al.
(1994) and Biggs & Andrews (1998). In all cases the initializa-
tion of the PSFs is provided by the autocorrelations of the im-
ages.InthecaseofidealPSFs theresultsarequitegood.Therate
of convergence depends on the angular separation of the binary
but one can conclude that, for all binaries, after about 50 IBD
cycles, the fluxes of the two stars, as well as the PSFs, are re-
constructed with a few percent error. The correct value of ∆m is
reached after about 25 iterations (errors of the order of 1%).
In the case of AO-corrected PSFs, results are acceptable for
Binary 3. For both sets of PSFs, the fluxes of the two stars are
underestimated (approximately one half of the correct value)
but the difference of magnitude is correctly reproduced (er-
ror smaller than 1% after about 30 IBD cycles). Finally the
AO-corrected PSFs are reconstructed with good accuracy. In
Fig.5weshowtheresultwehaveobtainedinthecaseSR=26%,
for the PSF with an orientation angle of 0◦. Important details of
the AO-corrected PSF are recovered; the reconstruction error is
of the order of 15%.
The results are not so good for the two other binaries. Both
the fluxes and the difference of magnitude are underestimated.
In the attemptof circumventingthis difficultywe considerasym-
metric IBD with lobj> lpsf. In the case SR = 70% we do not find
an improvement with respect to the symmetric case. Therefore,
for both binaries, the fluxes and the ∆m are underestimated with
an error of about 20%. On the other hand, in the case SR = 26%
the best resultsare obtainedwiththe asymmetry{18 : 1},namely
18 OSEM iterations in the Object box and 1 iteration in the PSF

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732 G. Desiderá et al.: Application of iterative blind deconvolution to the reconstruction of LBT LINC-NIRVANA images
Fig.5. Upper panels: theAO-correctedPSFwithSR=26% and 0◦(left)
and the corresponding image of the binary with angular separation of
195 mas (right). Lower panels: autocorrelation of the previous image
(left) and reconstruction of the PSF provided by IBD (right).
boxes. The fluxes are underestimated with errors ranging be-
tween 30 and 40%, but the ∆m is overestimated with an error
of 20% for Binary 1 and an error of 10% for Binary 2.
It mustbeobservedthat thelackofaccuracydiscussedabove
concerns photometry; astrometry is quite good in all cases. The
reconstructed binaries consist mainly of two bright pixels, cor-
responding to the correct positions of the two stars, and of a
number of weaker scattered pixels (which can be visualized by
a representation of the reconstruction in logarithmic scale), ac-
counting for the under-estimation of the flux of the binary. A
way for improving photometry could consist in estimating the
positions of the two stars and the PSFs by means of IBD and
then estimate the magnitudes by a least-square fitting of the de-
tected images, using the procedure proposed in Anconelli et al.
(2005b).
5.2. Diffuse objects
We first consider a diffuse object already used in previous simu-
lations (Correia et al. 2002), and precisely the image of a young
stellar object (YSO) derived from HST near-infrared observa-
tion of IRAS 04302+2247. The integrated magnitude of the ob-
ject is set to 11 in K-band, so that the average SNR per pixel
is about 150 (again only background and photon noise is taken
into account). Moreover, three sets of LN images are produced
by means of the three sets of PSFs mentioned above.
Also in this case we first perform an inverse crime ex-
periment in order to estimate the best accuracy which can be
achieved by means of the different sets of PSFs. At each OSEM
iteration the accuracy is evaluated by computingthe relative rms
error, defined as the Euclidean norm of the difference between
the iterate and the true object, divided by the Euclidean norm of
the object. We call this quantity the reconstruction error, and in
Fig. 6 we plot the results of the first 4000 OSEM iterations for
the three sets of PSFs. In all cases the convergence is very slow.
Indeed, in the case of ideal PSFs, the minimum reconstruction
Fig.6. Behaviour of the reconstruction error as a function of the num-
ber of OSEM iterations in the case of exact PSFs (inverse crime). The
full line corresponds to ideal PSFs, the dotted and dashed lines to
AO-corrected PSFs, respectively with SR = 70% and 26%.
error is about 3.7%, and is reached after 7000 iterations, while
in the cases SR = 70% and SR = 26%, the minimum reconstruc-
tion error is about 4% and 4.8%, respectively, and is reached
after 9000 and 20000 iterations. Therefore, if the quality of the
PSFs decreases, both the minimum reconstruction error and the
number of iterations increase, even if the quality of the recon-
struction is not significantly degraded.
We also checked how good is the data fitting provided by
the reconstructed objects. To this purpose, we compute at each
iteration the Euclidean norm of gj− b − Kj∗ f(k), divided by
the Euclidean norm of gj− b, a quantity that can be called rela-
tive discrepancy. In all cases we find that this quantity decreases
monotonically, as a function of the number of iterations, reach-
ing a value of 0.2% in the case of ideal PSFs, and values of
0.62% and 0.65% in the two other cases. In all cases, data fitting
is quite good.
The slow convergence in the inverse crime experiment in-
dicates the need of a large number of IBD cycles. Since we
are working with an extended object, we use asymmetric IBD
with lobj < lpsf, as suggested by Biggs & Andrews (1998), and
lobj= 1, to reduce the number of free parameters. Moreover, the
PSFs boxes are initialized with the autocorrelations of the ideal
PSFs, as proposed in Sect. 2.
In the case of ideal PSFs and AO-corrected PSFs with
SR = 70%, the asimmetry {1 : 7} is optimal, since an increase
of the asymmetry does not produce an improvement of conver-
gence. In the ideal case, after 500 IBD cycles, the reconstruc-
tion error of the object is about 8% (against 3.7% in the case
of inverse crime) while the reconstruction error of the PSFs is
about 24%. In the case SR = 70%, after 1000 IBD cycles, the
reconstruction error of the object is about 13.5% (against 4.0%)
while the reconstruction error of the PSFs depends on the PSF
andisrangingbetween22and24%.InFig.7weshowtheresults
obtained in this case at the end of the IBD cycles. The recon-
struction of the object is quite satisfactory also from the visual
pointofviewandtheimprovementinthePSF, withrespecttothe
initial guess provided by the autocorrelation of the ideal PSF, is
evident.On theotherhand,in thecase SR = 26% results are very
poor: the error in the reconstruction of the object is of the order
of 30% (against 4.8%), while the error in the reconstruction of
the PSFs is ranging between 40 and 50%.
The failure of IBD in the case of PSFs with low SR may be
due to the difficulty of reconstructing an object such as that of
Fig. 7, which is characterized by sharp edges in its central part.
Thereforeweconsidereda smootherobject,namelytheimageof
the galaxyNGC 1288,as extractedfromaHST image.Againthe
integrated magnitude in K band is set to 11 and three LN images

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G. Desiderá et al.: Application of iterative blind deconvolution to the reconstruction of LBT LINC-NIRVANA images733
Fig.7. Upper panels: the object (left) and the AO-corrected PSF with
SR = 70% and 0◦(right). Middle panels: the image corresponding to
the previous PSF (left) and the autocorrelation of the ideal PSF with 0◦
(right). Lower panels: the reconstruction of the object provided by IBD
(left) and the reconstruction of the PSF with 0◦(right).
are generated by convolving the object with the three PSFs with
SR = 26%. The same observational parameters of the previous
simulations are used.
The inverse crime experiment produces an error of 8% after
3000 iterations. Therefore a large number of IBD cycles must
be expected. We find that, in this case, the optimal asymmetry
is {1 : 3}; the best reconstruction is obtained with 2000 IBD cy-
cles and the reconstruction error is of the order of 15%, even
if the reconstruction error of the PSFs is still quite large, of the
order of 40%. In Fig. 8 we show the original object, the simu-
lated LN image with 0◦, the result provided by the inverse crime
and that provided by IBD. The result looks satisfactory, even if
someartifacts duetothe“hexagonal”structureofthecoveringin
the u,v plane are evident. We recall that we consider only three
equi-spacedorientationsofthebaseline;therefore,theseartifacts
could be reduced by a bit larger number of observations.
6. Concluding remarks
In this paper we describe the implementation of IBD in the
case of multiple image deconvolutionand its application to LBT
LINC-NIRVANA interferometer. The method has been inserted
in a module of the software package AIRY, version 3.0, which
can be downloaded from http://dirac.disi.unige.it or
Fig.8. Upper panels: the object (left) and the image corresponding to
the PSF with 0◦and SR = 26%. Lower panels: the inverse-crime recon-
struction (left) and the IBD reconstruction (right).
from http://www-astro.unice.fr/caos. The results ob-
tained with a limited number of numerical experiments indicate
that it is possible to obtain satisfactory results both in the case of
stellar objects and in the case of diffuse objects if the SR of the
PSFs is sufficiently high. However, it is obvious that the use of
IBD in the practice of the reconstruction of the LN images will
require a large amount of simulations to optimize the parame-
ters involved in the method for different classes of objects and
different observation conditions.
In this first implementation of IBD the methods used in
the Object and PSFs boxes are respectively OSEM and RLM.
The convergence of these methods is notoriously slow and, as
a consequence, a very large number of IBD cycles (global iter-
ations) is typically required before reaching reliable estimates.
Therefore it is crucial to investigate the applicability of accel-
eration methods such as those proposed by Biggs & Andrews
(1997, 1998). These accelerationmethods must be applied to the
global iterations and not to the internal iterations of the Object
and PSFs boxes.
We also stress that our software is very flexible, so that other
methods, in addition to OSEM and RLM, can be easily inserted
in the different boxes. For instance, reduction of noise propa-
gation and artifacts can be obtained by means of Bayesian ap-
proaches, leading to the regularization of functionals such as
the least-squares functional or the Csiszàr I-divergence. It is
worth mentioning that iterative methods for the minimization of
these functional can be derived from the so-called split gradient
method (SGM), proposed by Lanteri et al. (2001, 2002) in the
caseofsingleimagedeconvolution,andextendedtomultipleim-
age deconvolution by Anconelli et al. (2004). Lastly, reduction
of boundary effects can be obtained by implementing methods
recently proposed by Bertero & Boccacci (2005) and Anconelli
et al. (2006) respectively for single image and multiple image
deconvolution.

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734G. Desiderá et al.: Application of iterative blind deconvolution to the reconstruction of LBT LINC-NIRVANA images
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