# Advanced Imaging Methods for Long-Baseline Optical Interferometry

**ABSTRACT** We address the data processing methods needed for imaging with a long baseline optical interferometer. We first describe parametric reconstruction approaches and adopt a general formulation of nonparametric image reconstruction as the solution of a constrained optimization problem. Within this framework, we present two recent reconstruction methods, Mira and Wisard, representative of the two generic approaches for dealing with the missing phase information. Mira is based on an implicit approach and a direct optimization of a Bayesian criterion while Wisard adopts a self-calibration approach and an alternate minimization scheme inspired from radio-astronomy. Both methods can handle various regularization criteria. We review commonly used regularization terms and introduce an original quadratic regularization called ldquosoft support constraintrdquo that favors the object compactness. It yields images of quality comparable to nonquadratic regularizations on the synthetic data we have processed. We then perform image reconstructions, both parametric and nonparametric, on astronomical data from the IOTA interferometer, and discuss the respective roles of parametric and nonparametric approaches for optical interferometric imaging.

**0**Bookmarks

**·**

**80**Views

- [Show abstract] [Hide abstract]

**ABSTRACT:**With the advent of infrared long-baseline interferometers with more than two telescopes, both the size and the completeness of interferometric data sets have significantly increased, allowing images based on models with no a priori assumptions to be reconstructed. Our main objective is to analyze the multiple parameters of the image reconstruction process with particular attention to the regularization term and the study of their behavior in different situations. The secondary goal is to derive practical rules for the users. Using the Multi-aperture image Reconstruction Algorithm (MiRA), we performed multiple systematic tests, analyzing 11 regularization terms commonly used. The tests are made on different astrophysical objects, different (u,v) plane coverages and several signal-to-noise ratios to determine the minimal configuration needed to reconstruct an image. We establish a methodology and we introduce the mean-square errors (MSE) to discuss the results. From the ~24000 simulations performed for the benchmarking of image reconstruction with MiRA, we are able to classify the different regularizations in the context of the observations. We find typical values of the regularization weight. A minimal (u,v) coverage is required to reconstruct an acceptable image, whereas no limits are found for the studied values of the signal-to-noise ratio. We also show that super-resolution can be achieved with increasing performance with the (u,v) coverage filling. Using image reconstruction with a sufficient (u,v) coverage is shown to be reliable. The choice of the main parameters of the reconstruction is tightly constrained. We recommend that efforts to develop interferometric infrastructures should first concentrate on the number of telescopes to combine, and secondly on improving the accuracy and sensitivity of the arrays.Astronomy and Astrophysics 06/2011; 533. · 5.08 Impact Factor - SourceAvailable from: Sridharan RengaswamyFabien Malbet, William Cotton, Gilles Duvert, Peter Lawson, Andrea Chiavassa, John Young, Fabien Baron, David Buscher, Sridharan Rengaswamy, Brian Kloppenborg, Martin Vannier, Laurent Mugnier[Show abstract] [Hide abstract]

**ABSTRACT:**We present the results of the fourth Optical/IR Interferometry Imaging Beauty Contest. The contest consists of blind imaging of test data sets derived from model sources and distributed in the OI-FITS format. The test data consists of spectral data sets on an object "observed" in the infrared with spectral resolution. There were 4 different algorithms competing this time: BSMEM the Bispectrum Maximum Entropy Method by Young, Baron & Buscher; RPR the Recursive Phase Reconstruction by Rengaswamy; SQUEEZE a Markov Chain Monte Carlo algorithm by Baron, Monnier & Kloppenborg; and, WISARD the Weak-phase Interferometric Sample Alternating Reconstruction Device by Vannier & Mugnier. The contest model image, the data delivered to the contestants and the rules are described as well as the results of the image reconstruction obtained by each method. These results are discussed as well as the strengths and limitations of each algorithm. Comment: To be published in SPIE 2010 "Optical and infrared interferometry II"Proc SPIE 07/2010; -
##### Article: Principles of Stellar Interferometry

Principles of Stellar Interferometry: , Astronomy and Astrophysics Library. ISBN 978-3-642-15027-2. Springer-Verlag Berlin Heidelberg, 2011. 01/2011;

Page 1

IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 2, NO. 5, OCTOBER 2008 767

Advanced Imaging Methods for Long-Baseline

Optical Interferometry

Guy Le Besnerais, Sylvestre Lacour, Laurent M. Mugnier, Eric Thiébaut, Guy Perrin, and Serge Meimon

Abstract—We address the data processing methods needed for

imaging with a long baseline optical interferometer. We first de-

scribe parametric reconstruction approaches and adopt a general

formulationofnonparametricimagereconstructionasthesolution

of a constrained optimization problem. Within this framework,

we present two recent reconstruction methods, MIRA and WISARD,

representative of the two generic approaches for dealing with the

missing phase information. MIRA is based on an implicit approach

and a direct optimization of a Bayesian criterion while WISARD

adopts a self-calibration approach and an alternate minimization

scheme inspired from radio-astronomy. Both methods can handle

variousregularizationcriteria.Wereviewcommonlyusedregular-

ization terms and introduce an original quadratic regularization

called “soft support constraint” that favors the object compact-

ness. It yields images of quality comparable to nonquadratic regu-

larizations on the synthetic data we have processed. We then per-

form image reconstructions, both parametric and nonparametric,

on astronomical data from the IOTA interferometer, and discuss

the respective roles of parametric and nonparametric approaches

for optical interferometric imaging.

Index Terms—Fourier synthesis, image reconstruction, optical

interferometry, phase closure.

I. INTRODUCTION

T

today’s technology limits diameters to 10 m or so for ground

based telescopes and to a few meters for space telescopes. Op-

tical interferometry (OI) allows one to surpass the resulting res-

olution limitation, currently by a few factors of ten, and in the

next decade by a factor 100.

Interferometershaveallowedbreakthroughsinstellarphysics

with the first measurements of diameter and more generally of

fundamental stellar parameters, see recent reviews [1], [2]. Star

pulsations have been detected allowing to understand both the

physicsofstarsandthewaytheyreleasematterintheinterstellar

medium. Also, the measurement of the pulsation of Cepheid

HE ultimate resolution of an individual telescope is lim-

ited by its diameter. Because of size and mass constraints,

Manuscript received February 01, 2008; revised August 06, 2008. The as-

sociate editor coordinating the review of this manuscript and approving it for

publication was Dr. Julian Christou.

G. Le Besnerais, L. M. Mugnier, and S. Meimon are with ONERA, 29,

92122 Châtillon Cedex, France (e-mail: lebesner@onera.fr; mugnier@onera.fr;

meimon@onera.fr).

S. Lacour and G. Perrin are with the LESIA, Observatoire de Paris, 92190

Meudon, France (e-mail: sylvestre.lacour@obspm.fr; guy.perrin@obspm.fr).

E. Thiébaut is with CRAL, École Normale Supérieure de Lyon, 46, allée

d’Italie, 69364 Lyon cedex 07, France (e-mail: thiebaut@obs.univ-lyon1.fr).

Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JSTSP.2008.2005353

stars has allowed astronomers to establish an accurate distance

scale in the universe. Many subjects have been addressed, a

spectacular one being the shape of fast rotating stars, which are

now clearly known to be oblate by sometimes a huge amount

[3], [4]. With the advent of large telescopes and adaptive optics,

more distant sources, beyond our own galaxy are now acces-

sible. An important result has been the direct study of the dusty

torus around super-massive black holes in the center of these

galaxies, which is the corner stone of the unified theory to ex-

plain the active galactic nuclei phenomenon [5]–[7]. With the

success of large interferometers, and especially of the European

VLTI, interferometry is now used as a regular astrophysical tool

by nonexpert astronomers and many more results are to be ex-

pected with the steadily increasing amount of published mate-

rial.

OI consists in taking the electromagnetic fields received at

each of the apertures of an array (elementary telescopes or

mirror segments) and making them interfere. For each pair of

apertures, the data contain high-resolution information at an

angular spatial frequency proportional to the vector separating

the apertures projected onto the plane of the sky, or baseline.

With baselines of several hundred meters, this spatial frequency

can be much larger than the cut-off frequency of the individual

apertures.

Longbaselineinterferometers,

line-to-aperture ratio is quite large, usually provide a discrete

set of spatial frequencies of the object brightness distribution,

from which an image can be reconstructed by means of Fourier

synthesis techniques. For the time being, interferometers able

to provide direct images are not common: the Large Binocular

Telescope (LBT) , cf. lbto.org/, will be the first of this kind

with a baseline of the same order as the diameter of the two

individual apertures. Recent, comprehensive reviews of OI and

its history can be found for instance in [2], [8].

This paper addresses optical interferometry imaging (OII),

i.e., the data processing methods needed for imaging sources

with today’s long baseline optical interferometers. Many re-

construction methods for OII are inspired from techniques

developed for radio interferometry, as can be seen in the

methods which were compared in the recent Interferometry

ImagingBeautyContests:IBC’04[9]andIBC’06[10].Another

body of work is the set of parametric reconstruction (a.k.a.

model-fitting) methods. This latter class of methods is bound

to remain a reference, partly because in interferometry, optical

data will long remain much more sparse than radio data. In

some instances, e.g. with the very extended object of IBC’06

[10], OII is very difficult even with relatively large data set, and

thus often relies on the information provided by a parametric

forwhichthebase-

1932-4553/$25.00 © 2008 IEEE

Authorized licensed use limited to: IEEE Xplore. Downloaded on January 6, 2009 at 11:43 from IEEE Xplore. Restrictions apply.

Page 2

768IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 2, NO. 5, OCTOBER 2008

reconstruction. The latter is used at least as a guidance for

judging the (nonparametric) image reconstruction, and often

as a constraint for the support of the observed object, although

this process is not always explicit.

We adopt a general formulation of nonparametric image

reconstruction as the solution of a constrained optimization

problem. Within this framework, methods may differ by many

aspects, notably: the approximation of the data statistics, the

type of regularization, the optimization strategy and the explicit

or implicit accounting of missing phase information.

We presenttworecent

methods, representative of the two generic approaches for

dealing with the missing phase information. These nonpara-

metric reconstruction methods are evaluated on synthetic and

on astronomical data. The synthetic data allow us to study the

influence of several types of prior knowledge. In particular, we

show that contrarily to what is generally believed, appropriate

quadratic regularizations are able to perform frequency inter-

polation and are suitable for the problem at hand if the object

is compact: we propose a separable quadratic regularization

which favors the object compactness and yields images of

quality comparable to nonquadratic regularizations.

On the astronomical data we demonstrate the operational

imaging capabilities of these methods; for these data, which

may be considered representative of today’s optical long-base-

line interferometers, we show that the parametric approach

remains a choice of reference for OII. Finally, we discuss the

possible associations of both kinds of reconstruction methods.

The paper is organized as follows: Section II presents the

instrumental process so as to define the observation model.

Section III adresses the two main categories of prior informa-

tion used for the reconstruction of the observed astronomical

object: parametric models on the one hand, and regularization

terms for nonparametric reconstruction methods on the other

hand. Section V presents results on real data. Discussion and

concluding remarks are gathered in Section VI.

nonparametric reconstruction

II. OBSERVATION MODEL OF LONG-BASELINE

OPTICAL INTERFEROMETRY

Let us consider a monochromatic source of wavelength

with a limited angular extension. Its brightness distribution can

then be represented by,

portion of the plane of the sky around the mean direction of ob-

servation.

An intuitive way of representing data formation in a

long-baseline interferometer is Young’s double hole experi-

ment, in which the aperture of each telescope is modeled by

a (small) hole letting through the light coming from an object

located at a great distance [11], [12]. At each observation time

, each pair of telescopes yields a fringe pattern with a

spatial frequency of

, where the baseline

the vector linking telescopes

normal to the mean direction of observation. The coherence of

the electromagnetic fields at each aperture is measured by the

visibility or contrast

and the position of the fringes,

which are often grouped together in a complex visibility

with a small

is

andprojected onto the plane

. In an ideal experiment, the

Van Cittert-Zernike theorem [11], [13] states that the coherence

function (hence the complex visibility) is the Fourier transform

(FT) of the flux-normalized object

frequency

. Let us introduce notations for Fourier

quantities

at spatial

(1)

(2)

(3)

In ground based interferometry, interferometric data are

corrugated by the atmospheric turbulence. Inhomogeneities in

the air temperature and humidity of the atmosphere generate

inhomogeneities in the refractive index of the air, which perturb

the propagation of light waves through the atmosphere. These

perturbationsleadtospaceandtimevariationsoftheinputpupil

phase

, which can be modeled by a Gaussian spatio-temporal

random process [14]–[16]. The spatial behavior of this process

is generally described by the Fried’s diameter

smaller

, the stronger the turbulence. Typically, its value is

about 15–20 cm for

evolution time

of the turbulent phase is given by the ratio

ofto the velocity dispersion of turbulence layers in the

atmosphere

[14]: a typical value is a few milliseconds

at

. In the sequel, short exposure (respectively,

long exposure) refers to data acquired with an integration time

shorter (respectively, markedly longer) than

[17]. The

at good sites. The typical

.

A. Short Exposure System Response

Forapertures ofdiameter

coherence due to the turbulence perturbations reduces the vis-

ibility of the fringes. This can be counterbalanced if the wave-

fronts are corrected by adaptive optics [18] (AO), at a rate faster

than

, before the beams are made to interfere. In the sequel,

we assume that each aperture is indeed either small enough or

corrected by AO. Note, however, that it is possible to operate in

the multi-speckle mode [19].

IntheYoung’sholesanalogymentionedabove,theremaining

effect of turbulence on interferometric measurements is to add

a phase shift (or piston)

going through it. The interference between two apertures

are thus outof phasebya random “differential piston”

, whose typical evolution time is of the order of

depends on the baseline [20].

A short exposure observation finally writes

notablylargerthan,thelossof

at each apertureto the wave

and

and

(4)

(5)

When a complete interferometer array of

used, i.e., one in which all the possible two-telescope baselines

can be formed simultaneously, there are

visibilityphasemeasurements(5)foreachinstant .Theseequa-

tions can be put in matrix form

telescopes is

(6)

where the baseline operator

mally defined in Appendix A.

of dimensionsis for-

Authorized licensed use limited to: IEEE Xplore. Downloaded on January 6, 2009 at 11:43 from IEEE Xplore. Restrictions apply.

Page 3

LE BESNERAIS et al.: ADVANCED IMAGING METHODS FOR LONG-BASELINE OI769

Fig.1. FrequencycoveragesobtainedwiththeIOTAinterferometeron?Cygni

(observing run of May 2006). For a given baseline ? ? ?

channel of mean wavelength ?, the measured frequency is ? ? ? ? ?????

? ? ?

?????.

??? and a given spectral

?

Note that the baseline

pends on time. Indeed, the aperture configuration as seen from

the object changes as the Earth rotates. It is thus possible to use

“Earth rotation synthesis”, a technique that consists, when the

source emissiondoesnotvaryintime, inrepeatingthemeasure-

ments in the course of a night of observation to increase the fre-

quencycoverageoftheinterferometer.Atypicalfrequencycov-

erage obtained with the IOTA interferometer (see Section V-A)

is presented in Fig. 1. This Fourier coverage can be formally

represented by a short-exposure transfer function

between aperturesandde-

(7)

where

all observation instants and used pairs of telescope.

The complex gains

originate from various causes. Some of them can be estimated

using observations of a calibrator (i.e., a star unresolved by the

interferometer, or whose diameter is precisely known, located

near the object of interest and with similar spectral type) and

compensated for. In the following, we consider that the

are pre-calibrated, i.e.,

Equation (7) and Fig. 1 provide a first insight on the data pro-

cessing problem at hand. It is a Fourier synthesis problem, i.e.,

it consists in reconstructing an object from a sparse subset of its

Fourier coefficients. As shown by Fig. 1, interferometry gives

access to very high frequency coefficients, but the number of

data is verylimited(afewhundreds). Measuringthese datawith

a sufficient signal-to-noise ratio (SNR) is quite delicate. Indeed,

in a short exposure, the differential pistons are expressed by

random displacements of the fringes without attenuation of the

contrast. But in long exposure measurements, averaging these

displacements leads to a dramatic visibility loss: a specific av-

eraging process must be used, as described in the next section.

denote the Dirac function and summations extend over

account for visibility losses that

.

B. Long Exposure Data

1) Principle: As mentioned above, the main obstacle to

long exposure data measurement is the differential pistons

which affect the phase of the visibility. On the one hand,

averaging the modulus of the visibility is possible; on the other

hand, some phase information can be obtained by carrying out

phase closure [21] before the averaging. The principle is to

sum short-exposure visibility phase data

measured on a triangle of telescopes

From (5), one can check that turbulent pistons are canceled out

in the closure phase defined by

and

.

(8)

To form this type of expression it is necessary to measure three

visibilityphasessimultaneously,andthustouseanarrayofthree

telescopes or more.

In the case of a complete interferometer array of

scopes, the set of closure phases that can be formed is generated

by, for instance, the

,

phases measured on the triangles of telescopes including tele-

scope

. There are

closure phases. In what follows, the vector grouping together

these independent closure phases will be noted

sure operator

is defined such that

tele-

, i.e., the closure

of these independent

and a clo-

The second equation is a matrix version of (8): the closure op-

erator cancels the differential pistons, a property that can be

written

, with the baseline operator introduced in

(6) and Appendix A. It can be shown [22] that this equation im-

pliesthattheclosureoperatorhasa kernelofdimension

given by

,

(9)

where

The closure phase measurement thus does not contain all the

phase information. This classical result can also be obtained by

counting up the phase unknowns for each instant of measure

. There are unknown object visibility phases

and

independent measured phase clo-

sures, which gives

missing phase data. In other words,

optical interferometry through turbulence is a Fourier synthesis

problem with partial phase information. As is well known, the

moreaperturesinthearray,thesmallertheproportionofmissing

phase information.

2) Data Reduction and Averaging: In practice, the basic ob-

servables of optical interferometry are then sets of three simul-

taneous fringe patterns obtained on a triangle of telescopes. The

outputofthepre-processingstage(seeforinstance[23]forade-

scription of the pre-processing with IOTA data) are as follows.

• Power spectra

is obtained by removing the first column from.

:

(10)

Authorized licensed use limited to: IEEE Xplore. Downloaded on January 6, 2009 at 11:43 from IEEE Xplore. Restrictions apply.

Page 4

770 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 2, NO. 5, OCTOBER 2008

• Bispectra

, , defined by

(11)

Notation

instant . The integration time

spatial frequency to be considered constant during the integra-

tiondespitetherotationoftheEarth.Italsoimpactsthestandard

deviation of the residual noises on the measurement. Equation

(10) and (11) are biased estimates of the object spectrum and

bispectrum, see [24], [25] for the expressions of these bias.

The phases of the bispectra

expresses the averaging in a time interval around

must be short enough for the

constitute unbiased long-exposure closure phase estimators.

3) Observation Model and Data Likelihood: Using notation

and concatenating all instantaneous

measurements in vectors denoted by bold letters, the long-ex-

posure observation model writes

(12)

Noisetermsareusuallyonlycharacterizedbyestimatedsecond-

order statistics, hence they are modeled as Gaussian processes:

,

matrices

and are generally assumed to be diagonal

(as, for instance in the OIFITS data exchange format [26]), al-

though correlations are for instance produced by the use of the

same reference stars in the calibration process [27].

Note that the observation model (12) corresponds to a

minimal dataset for a complete

eter. In practice, the data may contain closures without the

corresponding power spectra, or bispectra amplitudes. These

supplementary data are not processed the same way by all the

data reconstruction methods, in particular by the MIRA and

WISARD algorithms described in Section IV.

The neg-log-likelihood derived from this observation model

writes

. Covariance

-telescope interferom-

(13)

The notation

visibility residuals at time

denotes the statistics of the squared

In the sequel, we shall use the term “likelihood” to denote the

various goodness-to-fit terms such as (13) derived from the dis-

tribution of the data.

The closure term

is usually also a

closures residuals, but in order to account for phase wrapping

and to avoid excessive nonlinearity, the term

the measured phase closures can also be chosen as a weighted

quadratic distance between the complex phasors

over phase

related to

(14)

where

closure

is the model of the measured phase

.

III. OBJECT MODELS

Imaging amounts to finding a flux-normalized positive func-

tion

defined over the support

way is to minimize the likelihood (13). Three problems are then

encountered.

1) Under-determination: because of the noise, the object

which minimizes the likelihood is not necessarily the good

solution: actually, several objects are compatible with the

data. This is a usual situation in statistical estimation,

which is here emphasized by the small number of mea-

sured Fourier coefficients, the noise level and the missing

phase information.

2) Nonconvexity: the phase indetermination leads to a non

convex1and often multi-modal data likelihood.

3) Non-Gaussian likelihood: phase and modulus measure-

ments with Gaussian noise leads to a non-Gaussian

likelihood in

. In other words, even if all the visibility

phases were measured instead of just the closure phases,

the data likelihood would still be nonconvex. We shall

come back to this point in Section IV-B.

To deal with under-determination, one is led to assume some

further prior knowledge on the object. In this section we re-

view two approaches: parametric modeling and regularized re-

construction.

which fits the data (12). One

A. Parametric Models

1) Introduction: The object is sought by minimization

of (13) using a parametric form

oftenexhibitsfurthernonlinearities,butasthe

number of parameters is very limited (typically

global minimization is achievable. The minimal value of the

criterion gives an information on whether the chosen model is

appropriate to describe the brightness distribution of the object.

Additionally, the second derivative of

minimum allows the estimation of error bars.

For years, interferometric data were very sparse, essentially

because the number of telescopes in interferometers was quite

small. Most interferometers were two-telescope arrays and in

few cases three telescopes were available. The only way to in-

terpret the data was then to use parametric models with a very

small number of parameters, typically two or three. Among the

most used models, let us mention the uniform disk to measure

stellar diameters, and binary system models.

When objects are as simple as individual or binary regular

stars, such simple models can be used beforehand to prepare

the observations and anticipate likely visibility values. This

is very useful to conduct “baseline bootstrapping”, a process

which consists in observing a visibility of very low SNR using

a triangle of telescopes with two other baselines having a higher

SNR. Simple parametric models are also used to compute the

expected visibility of reference stars in order to calibrate the

. The resulting criterion

around its

1Convexity is a desirable property of a criterion when a minimization process

is conducted, which can furnish sufficient conditions for convergence of iter-

ative local optimization techniques toward a global minimum. A well-known

reference is the book by R.T. Rockafellar [28].

Authorized licensed use limited to: IEEE Xplore. Downloaded on January 6, 2009 at 11:43 from IEEE Xplore. Restrictions apply.

Page 5

LE BESNERAIS et al.: ADVANCED IMAGING METHODS FOR LONG-BASELINE OI771

response of the interferometer and overcome residual visibility

losses, such as those due to polarization effects.

Now that current interferometers yield richer data, more so-

phisticated models can be used. It is outside the scope of this

paper do describe the large number of parametric models which

are used nowadays, however we present in the following sub-

sections two trends in parametric modeling.

2) Fitting of a Geometrical Model: Parametric inversion can

be used to derive the geometrical structure of the brightness dis-

tribution of the object. An example among others is the deriva-

tion of the brightness profile

darkening is an optical depth effect, which results in a drop of

the effectivetemperature (and hence intensity) towards the edge

of the stellar disk. Numerous types of limb-darkening models

exist in the literature. To cite only two, one can use a power law

[29] as follows:

of a limb-darkened disk. Limb

(15)

or a quadratic law [30]

(16)

where , the cosine of the azimuth of a surface element of the

star, is equal to

from the star center, and

is the angular diameter of the pho-

tosphere. The parametric fit is actually done on complex visi-

bilities. In the Fourier domain, the power law limb darkening

model yields [31]

, being the angular distance

(17)

where the parameters are

frequency and

The quadratic law model yields

, is the radial spatial

the Euler gamma function.

(18)

where the parameters are

are the first and second-order Bessel functions, respectively,

and

,; and

(19)

3) Physical Parameter Determination: An interesting possi-

bility offered by parametric inversion is to directly adjust phys-

ical parameters of the objects. An example can be found on a

study about the star

Cep from FLUOR interferometric obser-

vation [32]. Data was fitted with an analytical expression of the

brightness distribution that includes a temperature for the pho-

tosphere and a radiative transfer model of the molecular layer.

The model used is radial, and writes

(20)

for

and

(21)

areotherwise,wherethe parameters

, withand

the diameters of the star and the molecular layer respectively,

and

the opacity of the molecular layer as a function of the

wavelength.

is the Planck function.

This model illustrates how to obtain a direct estimation of the

temperatures of the star and of the molecular layer from inter-

ferometric data. Interestingly, this type of model allows an ex-

ploitation of multi-wavelength observations that takes into ac-

count the chromaticity of the astronomical object.

B. Regularized Reconstruction

1) Introduction: In this framework, the sought object distri-

bution

is represented by its projection onto a basis of func-

tions, often defined as a shifted separable kernel basis

(22)

where dimensions

chosen so as to span the object support

Shannon–Nyquist condition with respect to the experimental

frequency coverage. Kernels

functions, sometimes wavelets or prolate spheroidal functions

[33], [34]. The estimation aims at finding the coefficients

, and sampling steps, are

and to satisfy the

are often box functions or sinc

so as to fit the data.

This approach is sometimes loosely called a non parametric

approach because the parametrization (22) is here not to put

further constraints on the solution, but only to allow its numer-

ical computation. To tackle the under-determination the data

likelihood (13) is combined with a regularization term in a

criterion of the form

(23)

Theregularizationterm

theobject(smoothness,positivity,compactness,etc.).Theregu-

larization parameter

allows to tune the regularization strength

or, equivalently, to select a data term level set

Of course, the choice of

depends on the noise level. The com-

pound criterion (23) can be derived within a Bayesian para-

digm: the data model (12) is translated into a data likelihood

(13) which is combined with a prior distribution on the object

by Bayes’ rule to form the a posteriori distribution. Maximiza-

tion of the posterior distribution is equivalent to minimization

of the likelihood, i.e., of a regularized criterion such as (23).

Most regularization terms penalize the discrepancy between

the current solution and some a priori object

the null object. In the OII context of very sparse and ambiguous

enforcesthedesiredpropertiesof

.

, be it simply

Authorized licensed use limited to: IEEE Xplore. Downloaded on January 6, 2009 at 11:43 from IEEE Xplore. Restrictions apply.

Page 6

772 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 2, NO. 5, OCTOBER 2008

datasets, the use of a meaningful prior object can be an efficient

way to orient the reconstruction and to improve the results. In

IBC’06 for instance, an a priori object was finally provided to

the participants: see [10, Fig. 3].

previous observations of the source with other instruments, or

derived from the fit of a parametric model to the interferometric

data at hand—see Section V-C below.

In the following, we briefly discuss the choice of the regular-

ization terms and introduce an original regularization criterion

that can be used on compact objects to enforce a “soft support”

constraint.

2) Quadratic Regularization: Quadratic regularization has

been applied to Fourier synthesis and OII by A. Lannes et al.

[34]–[36]. For relatively smooth objects, one can use a corre-

lated quadratic criterion expressed in the Fourier domain, with

a parametric model of the object’s power spectrum

model was proposed for deconvolution of AO corrected images

in [37]

can be obtained from

. Such a

(24)

This model, which relies on a prior object and three “hyper-pa-

rameters”

has been used in various image reconstruc-

tion problems, including OII [9], [38]. Parameter

frequency which is typically the inverse of the diameter of the

object’s support and avoids divergence at the origin,

terizes the decrease rate of the object’s energy in the Fourier

domain, and

plays the role of the inverse of hyperparameter

of (23) and can replace this parameter. As already mentioned,

anadvantageofquadraticcriteriaisthatitispossibletoestimate

thehyper-parameters,bymaximumlikelihoodforexample[39].

A simple and efficient quadratic regularization is a separable

quadratic distance to the prior object

show that the general expression of such regularization terms

undertheOII-specificconstraintsofunitsumandpositivity[see

(40)] is

is a cutoff

charac-

. In Appendix B, we

(25)

where the a priori object

and normalized to unity.

Intheabsenceofameaningfulobjecttobeusedfor

width

of the observed source is usually more or less known,

sowehavefoundthatareasonableaprioriobjectisanisotropic

one such as the Lorentzian model

. Such a prior object can then be seen as enforcing a

loose support constraint.

3) Edge-Preserving Regularization: For extended objects

with sharp edges such as planets, a quadratic criterion tends

to over-smooth the edges and introduce spurious oscillations,

or ringing, in their neighborhood. A solution is thus to use an

edge-preserving criterion such as the so-called quadratic-linear,

or

criterion, which are quadratic for weak gradients

of the object and linear for the stronger ones. The quadratic (or

) part ensures good noise smoothing and the linear (or

part cancels edge penalization. Here we present an isotropic

is chosen to be strictly positive

,the

)

version [40] of the criterion proposed by Rey [41] in the context

of robust estimation and used by Brette and Idier in image

restoration [42]

(26)

(27)

(28)

with

ferences in the two spatial directions. The two parameters to be

adjustedareascaleparameter andathresholdparameter .Pa-

rameter

plays the same role as the conventional regularization

parameter

andcanreplaceit,with

values of

each term of (26) reads

.

4) Spike-Preserving or Entropic Regularization: For objects

composedofbrightpointsonafairlyextendedandsmoothback-

ground, such as are often found in astronomy, a useful regular-

ization tools is entropy. Here, we adopt the pragmatic point of

view of Narayan and Nityananda [43] and consider that entropy

is essentially a separable criterion

and the gradient approximations by finite dif-

;indeed,forsmall

(29)

where each pixel is drawn toward a prior value

cording to a nonquadratic potential

Classical examples of “entropic potential” are the Shannon en-

tropy

ac-

also termed neg-entropy.

and the Burg entropy

but many other non quadratic

potentials can be used, as shown in [43]. The major interest of

the nonlinearity of entropic potentials is that they help to inter-

polate holes in the frequency coverage. Side effects are empha-

sizing spikes and smoothing low level structures. As it result in

ripples suppression in the flat background and enhanced spa-

tial resolution near sharp structures, this behavior may be con-

sidered as beneficial in the context of interferometric imaging

though it also introduces substantial biases. Note that the inter-

ferometric imaging method BSMEM [44], winner of the IBC’s

2004 and 2006 [9], [10], or the VLBMEM method [9] are based

upon entropic regularization with potential

Here we propose an entropic-like criterion which re-employs

the potential

of (28) in a “white

same tools as in Appendix B, it can be shown that the general

form of a white

prior under the OI-specific constraints

of unit sum and positivity is

.

” prior. Using the

(30)

where

first derivative.

A interesting refinement of such priors is to model the ob-

served object with the combination of a correlated

ization for the extended component of the object and a white

regularization for the bright spots [45].

is a function such as (28), anddenotes its

regular-

Authorized licensed use limited to: IEEE Xplore. Downloaded on January 6, 2009 at 11:43 from IEEE Xplore. Restrictions apply.

Page 7

LE BESNERAIS et al.: ADVANCED IMAGING METHODS FOR LONG-BASELINE OI773

IV. ALGORITHMS FOR REGULARIZED IMAGING

The regularized criterion (23) is not strictly convex and actu-

ally often multi-modal, because of missing phase information.

Therefore, the solution to the OII problem cannot be simply de-

fined as the minimizer of (23). Actually, it should be defined as

the point where the minimization algorithm stops. Most OII al-

gorithms (BSMEM, MIRA, WISARD) are iterative local descent

algorithmswiththeexceptionofMACIM[10]whichusessimu-

latedannealingtosearchfortheobject’ssupport.Inthissection,

we present two iterative algorithms designed for OII: MIRA and

WISARD. Both are at the state of the art: WISARD ranked second

in IBC’04, while MIRA ranked second in IBC’06 and very re-

cently won IBC’08. They are able, like MACIM, to handle var-

ious prior terms—while BSMEM is dedicated to entropic regu-

larization. They differ, however, in their treatment of the phase

problem. There are essentially two approaches of this problem:

explicitalgorithmsuseasetofphasevariablesandproceedwith

a joint minimization overthese variables and the object , while

implicit approaches search for a minimum of (23) with respect

to ,oftenwith someheuristicsin ordertoavoidgetting trapped

in local minima. Explicit algorithms include VLBMEM [9],

WIPE [46], [47] and WISARD. BSMEM, MACIM and MIRA are

implicit algorithms. In this respect, MIRA and WISARD are rep-

resentative of the two main streams of current OII algorithms.

A. Direct Minimization (MIRA)

The MIRA [48], [49] method (MIRA standsfor Multi-aperture

Image Reconstruction Algorithm) seeks for the image by min-

imizing directly criterion (23). MIRA accounts for power spec-

trum and closure phase data via penalties defined in (13) and

(14). MIRA implicitly accounts for missing phase information,

as it only searches for the object

tempt to explicitly solve degeneracies, it can be used to restore

an image (of course with at least a 180 orientation ambiguity)

from the power spectrum only, i.e., without any phase informa-

tion, see examples in [49], [50].

To minimize the criterion, the optimization engine is

VMLMB [51], a limited memory variable metric algorithm

which accounts for parameter bounds. This last feature is used

to enforce positivity of the solution. Only the value of the cost

function and its gradient are needed by VMLMB. Normaliza-

tion of the solution is obtained by a change of variables, i.e.,

the image brightness distribution becomes

where

are the internal variables seen by the optimizer with

the constraints that

, . Thus,

positive. The gradient is modified as follows:

. Since MIRA does not at-

,

is both normalized and

(31)

To avoid getting trapped into a local minimum of the data

penalty

which is multi-modal, MIRA starts the mini-

mization with a purposely too high regularization level. After

convergence, the reconstruction is restarted from the previous

solution with a smaller regularization level (e.g. the value of

is divided by two). These iterations are repeated until the

chosen value of

is reached. This procedure mimics the more

clever strategy proposed by Skilling & Bryan [52] and which is

implemented in MemSys the optimization engine of BSMEM.

B. A Self Calibration Approach (WISARD)

The self calibration approach developed in [22], [38], [53]

relies on an explicit modeling of the missing phase information

and allows one to obtain a convex intermediate image recon-

struction criterion. It is inspired by self-calibration algorithms

in radio-interferometry [54], but uses a more precise approxi-

mationoftheobservationmodelthanfirstattempts suchas[47].

This approach consists in jointly finding the object

-dimensional phase vector

components in the closure operator kernel of (9). It starts from a

generalized inverse solution to the phase closure equation (12),

using the operator

left to (12) and (9), the missing phase components are made

explicit

and an

, corresponding to phase

. By applyingon the

(32)

Itisthustemptingtodefineapseudo-equationofvisibilityphase

measurement by identifying the term

a noise affecting the visibility phase [55]

of (32) with

(33)

Unfortunately, as matrix

rigorously possible and one is led to associate an ad hoc covari-

ance matrix

with the term

fit the statistical behavior of the closures. Recently, [22], [38]

have discussed possible choices for

Finding a suitable approximation for the covariance of the

amplitude measurements (12), cf. [22] and [38], gives a “my-

opic” measurement model, i.e., one that depends on the un-

knowns

and

is singular, this identification is not

so as to approximately

.

(34)

with Gaussian noise terms on the modulus and on the ampli-

tude. Still, the resulting likelihood is not quadratic with respect

to , because a Gaussian additive noise in phase and modulus

is not equivalent to an additive complex Gaussian noise on the

visibility. This is the problem of converting measurements from

polarcoordinatestoCartesianones,whichhaslongbeenknown

in the field of radar [56] and was identified only very recently in

optical interferometry [57]. The myopic model of (34) is thus

further approximated by a complex additive Gaussian model

such as

(35)

The mean value and covariance matrix of the additive complex

noise term

can be chosen so that the corresponding

data likelihood criterion is convex quadratic w.r.t. the complex

Authorized licensed use limited to: IEEE Xplore. Downloaded on January 6, 2009 at 11:43 from IEEE Xplore. Restrictions apply.

Page 8

774IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 2, NO. 5, OCTOBER 2008

Fig. 2. Synthetic object from IBC’04. Left: original full resolution true object. Right: true object convolved by the PSF of a 132-m perfect telescope.

while remaining close to the

real nonconvex likelihood [22], [57]. Finally, using (33)

(36)

where

instantaneous frequency coverage at time and

ponent-wisemultiplicationofvectors.Asclearlyshownby(36),

the resulting model is now linear in

This last step leads to a data fitting term

quadratic in the real and imaginary parts of the residuals—see

[22] and [38] for a complete expression. As discussed in

Section III, this data term is then combined with a convex

regularization term, so as to obtain a composite criterion

is the discrete time FT matrix corresponding to the

denotes com-

for a fixed.

that is

(37)

Let us emphasize the interesting properties of

hand

. On one

is convex in ; on the other hand,

is separable over measure-

ment instants , which allows handling the phase step by several

parallel low-dimensional optimizations.

The WISARD algorithm makes use of these properties and

minimizes

alternately in

the current . The structure of WISARD is the following: after a

first step that casts the true data model into the myopic model

(34), a second step “convexifies” the obtained model w.r.t. , to

obtainthe modelof(36). Aftertheselection oftheguess and the

prior, WISARD performs the alternating minimization.

For the moment, this approach is less versatile than a di-

rect all-purpose minimization method such as MIRA: WISARD

cannot cope with missing phase closure information or take into

account bispectrum moduli. Indeed, as the pseudo-likelihood

associated to model (36) is derived, data that do no fit this re-

casting stage are not taken into account. Extending WISARD to

make it more versatile in the above-mentioned sense deserves

some future work.

for the currentand infor

C. Results on Synthetic Data

This section presents results of nonparametric reconstruction

methods on the synthetic interferometric data that were pro-

Fig. 3. Frequency coverage from the IBC’04.

ducedbyC.Hummelforthe2004InternationalImagingBeauty

Contest(IBC’04)organizedbyP.LawsonfortheIAU[9].These

data simulate the observation of the synthetic object shown in

Fig. 2 with the NPOI 6-telescope interferometer. The corre-

sponding frequency coverage, shown in Fig. 2, contains 195

square visibility moduli and 130 closure phases. The resolution

of the interferometric configuration, as given by the ratio of the

minimum wavelength over the maximum baseline, is 0.9 mas.

In Fig. 2 right, we present the image that a 132-m perfect

telescope would provide of the object. The cutoff frequency of

such an instrument would be twice the maximum value of the

frequency coverage used to produce the synthetic dataset (see

Fig. 3). It is therefore relevant to compare the reconstructions

with this image.

Various results of MIRA with quadratic regularizations are

presented in Fig. 4. The top image is essentially a “dirty recon-

struction”: it uses a separable quadratic penalty with a very low

value of regularization parameter

tained in the same setting, but with a positivity constraint. The

improvement is dramatic, as both the object support and its low

resolution features are recovered. An interpretation is that the

positivity plays the role of a floating support constraint, which

. The middle image is ob-

Authorized licensed use limited to: IEEE Xplore. Downloaded on January 6, 2009 at 11:43 from IEEE Xplore. Restrictions apply.

Page 9

LE BESNERAIS et al.: ADVANCED IMAGING METHODS FOR LONG-BASELINE OI 775

Fig. 4. Results on IBC’04 with Mira. Top: “dirty” reconstruction (see text).

Middle:positivityconstraint.Bottom:softsupportquadraticregularization(25),

with prior object Lorentzian of 5 mas FWHM and 0.1 mas pixel size. All recon-

structions have 256 ? 256 pixels.

favors smooth spectra and interpolates the missing spatial fre-

quencies. The bottom image uses the soft support quadratic reg-

ularizationof(25),withaLorentzianof5masFWHMasaprior

object and a positivity constraint. This regularization, although

quadratic, leads to a very good reconstruction, with a central

peak clearly separated from the asymmetric shell.

Fig. 5 presents two WISARD reconstructions. The left one

is obtained with the same soft support quadratic regularization

than the MIRA reconstruction Fig. 4, bottom. Although MIRA

and WISARD are based on different criteria and follow different

paths during the optimization, the reconstructions are visually

very close. With such a “rich” 6-telescope dataset the missing

phase information (33%) is reasonable and the differences be-

tween reconstructions, when they are present, originate mainly

from the choice of different regularization terms. As an ex-

ample, a reconstruction based on the white

serving prior of (30) with a constant prior object is shown in

Fig. 5, right. This last reconstruction presents finer details than

quadratic ones, possibly even finer than the smoothed object of

Fig. 2, at the price of some artefacts on the asymmetric shell.

However, the validity of these details is difficult to assess.

As a conclusion, the proposed soft support quadratic regular-

ization yields images of quality comparable to those obtained

with spike-preserving priors. Contrarily to what is generally

believed (see for instance Narayan and Nityananda [43]), spe-

cial-purpose quadratic separable regularizations are perfectly

suitable for image reconstruction by Fourier synthesis as soon

as the object is compact and positivity constraints are active.

spike-pre-

V. PROCESSING REAL DATA

A. The Infrared Optical Telescope Array (IOTA)

The IOTA interferometer, operated from 1993 to 2006 (cf.,

tdc-www.harvard.edu/iota/) on Mt Hopkins (Arizona, USA),

had variable baseline lengths and thus gave access to a broad

frequency coverage. It operated with three 45 cm siderostats

that could be located at different stations on each arm of an

L-shaped array (the NE arm is 35 m long, the SE arm 15 m).

The maximum nonprojected baseline length was 38 m, and the

minimum one 5 m. It used fiber optics for spatial filtering, and

an integrated optics beam combiner called IONIC [58]. It was

decommissioned in July 2006.

B. The Dataset

The dataset presented here correspond to observations of the

star

Cygni. Of the class of the Mira variables,

an evolved star whose extended atmosphere is puffed up by

the strong radiation pressure induced by fusion of metals (here,

metals means chemical elements heavier than Helium) in its

core. This late stage of evolution is appropriate for interfero-

metric imaging since the large stellar radius can be resolved by

optical interferometers. Moreover, these stars are usually bright

in the infrared, allowing robust fringe tracking.

Cygni was observed during a six-night run in May 2006.

Night-time is used for observing and daytime is used to move

and configure the telescopes. The log of the interferometer con-

figurations is presented in Table I.

The reduced dataset is plotted in the two panels of Fig. 6.

Cygni was observed over the whole H Band

, and fringes were dispersed in order to obtain spectral

information. In this paper, we shall not address the chromaticity

of the object. Therefore, we use the diverse wavelengths only

as a way to increase the Fourier coverage. The frequency plane

coverage was previously presented in Fig. 1. The visibilities are

Cygni is

Authorized licensed use limited to: IEEE Xplore. Downloaded on January 6, 2009 at 11:43 from IEEE Xplore. Restrictions apply.

Page 10

776IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 2, NO. 5, OCTOBER 2008

Fig. 5. Results on IBC’04 with Wisard. Left: 120 ? 120 pixels reconstruction with 0.125 mas pixel size using a soft support quadratic regularization (25), with

prior object Lorentzian of 2.5 mas FWHM. Right: 60 ? 60 pixels reconstruction with 0.25 mas pixel size based on a white ? ? ? regularization (30) with a

constant prior object and parameters ? ? ?, ? ? ?????.

TABLE I

? CYGNI OBSERVING LOG

Configuration refers to the location in meters of telescopes A, B, C on the

NE, SE, and NE arms, respectively. Position “0” corresponds to the arms’

intersection.

presented in the upper panel of Fig. 6 as a function of the base-

line length in wavelength units. The closure phases are plotted

on the bottom panel. Due to the difficulty to represent these

phases as a function of a physical parameter, we simply present

them as a function of the observation data-point number. The

vertical lines indicate a change of interferometer configuration.

A closure phase equals to zero or

symmetric object. Thus, a preliminary inspection of the clo-

sure phases can show the presence of asymmetries. The higher

the frequency, the more apparent the asymmetry is. This makes

sense to an astronomer because photospheric inhomogeneities

are likely to be present at a smaller scale than the size of the

photosphere. In the case of

Cygni, the photosphere’s size es-

timate is 21.3 mas, to be compared with the resolution of the

interferometer, slightly less than 5 mas.

corresponds to a centro-

C. Image Reconstruction

In Fig. 7, we present three reconstructed images, obtained

using different methods and priors.

The first image corresponds to a parametric inversion of the

data using all the available spectral channels merged together.

Thejustificationforsuchapolychromaticprocessingofthedata

is ongoing work, however first results confirm a variation of the

angular diameter of less than 1 mas in the H band (S. Lacour,

private communication, 2008).

As stated, the quality of the reconstruction will depend

heavily on the correctness of the model of the object. For-

tunately, Mira variables are not completely unknown, and

Fig. 6. IOTAdataset on ? Cyg. Top panel: Visibility square? ? ?

of the baseline length. Bottom panel: closures phases ? ? ?

the observation number. Labels of the type Axx-Bxx-Cxx correspond to the

telescope configurations (see Table I).

as a function

as a function of

previous astronomical observations tell us that the star is ex-

pected to possess: i) a large limb-darkened photosphere; ii)

important asymmetries of theform of photospheric “hot-spots”;

and iii) a close, warm, molecular layer surrounding the photo-

sphere at around one stellar radius of the photosphere.

Authorized licensed use limited to: IEEE Xplore. Downloaded on January 6, 2009 at 11:43 from IEEE Xplore. Restrictions apply.

Page 11

LE BESNERAIS et al.: ADVANCED IMAGING METHODS FOR LONG-BASELINE OI777

Fig. 7. Reconstructed images of the star ? Cygni, right: contour plot with

levels 10%, 20%, ???, 90% of the maximum. From top to bottom: parametric

reconstruction, WISARD with white ? ? ?

quadratic penalization towards a prior parametric solution. Details on the

different methods are given in Section V-C. Note the colossal size of the star:

at the distance of ? Cygni (170 pc; [59]), 5 mas correspond roughly to the

distance Sun-Earth (1 astronomical unit).

prior, MIRA with a separable

This simplistic theoretical model (i.e., a limb darkened disk,

a spot and a spherical thin layer) is converted to a geometrical

parametricmodel,whichisadjustedtothedatathroughthemin-

imizationof

of(13).Theimagepresentedintheleft

panel of Fig. 7 corresponds to the geometrical model with best

fit parameters.

These parameters give direct information on the structure of

the object, and error bars can be estimated: the star diameter is

21.49

0.11masandthehotspotcontrastis1.70

that the requirements of a parametric reconstruction, in terms

of frequency coverage, are much less stringent than that of a

nonparametric one. Thus, parametric inversion can also be used

with each spectral channel separately, to determine the spectral

energy distribution of the surrounding atmospheric layer.

The second image was produced using the WISARD software

described in Section IV-B with a white

(26). The last image was reconstructed using MIRA, see

Section IV-A, and more importantly, using a prior solution

in the white quadratic setting of (25). The prior solution is a

0.04%.Note

prior—see

limb-darkened disk whose parameters are determined by model

fitting on the visibilities.

D. Discussion

Fig. 7 shows that, for the sparse data at hand, the more strin-

gent the prior, the more convincing the reconstruction looks to

an astronomer.

Moreprecisely,thewhite

not allow to distinguish more than a resolved photosphere and

the fact that some asymmetry is present. The form of the re-

constructed photosphere and its surrounding can be questioned

whencomparedtowhatisexpectedfromthetheory.Besides,on

thepresentedreconstructions,MIRAwasusedwithamuchmore

informative prior and is in good agreement with the parametric

reconstruction. This image is however interesting because the

reconstruction is notably different from a simple disk, and adds

an asymmetry—a “hotspot”—on the surface. The presence of

an asymmetry could be foreseen by looking at the raw closure

phases (right panel of Fig. 6). The fact that this asymmetry ap-

pears similarly—in terms of flux and position—on the para-

metricandonthenonparametricimagereconstructionsisacon-

vincing argument to validate both images.

Note that, on the MIRA reconstruction, an emission sur-

rounding the photosphere is present, but its reality is difficult to

assert on the reconstructed image. Hence, it should be pointed

out that neither of the nonparametric reconstructions exhibits

the molecular layer which is revealed by the parametric recon-

struction.

priorusedbyWISARDdoes

VI. CONCLUSIONS

In recent years, long baseline optical interferometers with

better capabilities have become available. Routine observations

with three or more telescope interferometers have become a re-

ality.Althoughquitesparsewithrespecttoradioarrays,thespa-

tial frequency coverage allows one to study more complex ob-

jectsandtoreconstructimages.Inthispaper,wehavedescribed,

besides the parametric reconstruction approach, two nonpara-

metricimagereconstructionmethods,MIRAandWISARD.MIRA

is based on the direct optimization of a Bayesian criterion while

WISARD adopts a self-calibration approach inspired from radio-

astronomy. As such, these two methods are representative of

the two families of state-of-the-art nonparametric reconstruc-

tion methods [9], [10].

On rich-enough data, which are currently available only from

simulations, both methods demonstrate a valuable and compa-

rable capability for imaging complex objects. On such data,

the differences between reconstructions originate mostly from

the choice of different regularization terms. We have reviewed

common regularization criteria and proposed an original regu-

larization criterion that can be used on compact objects to en-

force a “soft support” constraint. This criterion, although it is

quadratic, yields images of quality comparable to that obtained

with spike-preserving priors on the IBC’04 dataset.

We have demonstrated the operational imaging capabilities

of these methods on a IOTA dataset of

these data, which may be considered representative of today’s

optical long-baseline interferometers, we have shown that the

parametric approach remains a choice of reference for OII.

Cygni. However, for

Authorized licensed use limited to: IEEE Xplore. Downloaded on January 6, 2009 at 11:43 from IEEE Xplore. Restrictions apply.

Page 12

778IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 2, NO. 5, OCTOBER 2008

The experience gathered while trying to extract the most in-

formation from real-world data, both in the work described here

andelsewhere[31],suggeststhattheoptimalprocessingofmea-

surements from present-day interferometers should make use

of both approaches in an alternate fashion as described below.

With a sparse frequency coverage, a parametric reconstruction

is useful to obtain ab initio a first estimate for the observed ob-

ject. A parametric reconstruction will not reveal any unguessed

feature, but it can be used to guide nonparametric reconstruc-

tions as an initial guess or as a prior object for instance. Then

thereconstructedimagesareveryusefultounderstandthestruc-

ture of a complex object since they are most often the very first

insight one gets about the source at this angular resolution.

Thefidelityofnonparametricreconstructionsremainslimited

in a photometricsense and cantherefore seldom be used toinfer

astrophysical parameters. In fine, parametric models remain the

choice of reference for estimating astrophysical parameters re-

lated to the very physics of the objects of interest.

It is therefore very likely that even in the yet to come imaging

era of optical interferometry, i.e., when much larger optical

interferometric arrays become operational, the parametric

approach will remain a useful tool for astrophysical modeling,

even though it will no longer be necessary to initialize the

imaging process.

APPENDIX A

THE CLOSURE AND BASELINE OPERATORS

AND

Let be the number of telescopes of a complete interfero-

metric array. We have the following definitions:

(38)

for

inverse

. It is easy to see that

of , defined by

.

. The generalized

, is such that

APPENDIX B

QUADRATIC REGULARIZATION TOWARDS

A PRIOR OBJECT IN OI

A general expression for a quadratic separable regularization

is given by

(39)

where

The default solution

function in the absence of data and subject to the constraints

(normalization and nonnegativity)

, ,otherwisethecriterionisdegenerated.

is obtained by minimizing the cost

(40)

where

ment that all inequality constraints are inactive at the solution,

the Lagrangian for the constrained problem can be written as

means,. Assuming for the mo-

(41)

Minimizing

with respect to only readily yields

. The optimal Lagrange

multiplier

is identified by requiring the normalization of

and, finally, the default solution is

(42)

which is normalized and strictly positive since

ditionallyvalidatesourhypothesisthattheinequalityconstraints

were all inactive at the solution. Combining (39) and (42) yields

the expression of the quadratic, separable, loose support regu-

larization term of (25).

. This ad-

ACKNOWLEDGMENT

The authors would like to thank all the people who con-

tributedtotheexistenceandsuccessoftheIOTAinterferometer.

They also thank the anonymous reviewers for their numerous

suggestions, which resulted in a great improvement of the

paper’s quality.

REFERENCES

[1] A. Quirrenbach, “Optical interferometry,” Annu. Rev. Astron. and As-

trophys., 2001.

[2] J. D. Monnier, “Optical interferometry in astronomy,” Rep. Progr.

Phys., vol. 66, pp. 789–857, May 2003.

[3] J. D. Monnier et al., “Imaging the surface of Altair,” Science, vol. 317,

p. 342, Jul. 2007.

[4] A. D. da Souza et al., “The spinning-top be star achernar from VLTI-

VINCI,” Astron. Astrophys., vol. 407, no. 3, pp. L47–L50, 2003.

[5] W. Jaffe et al., “The central dusty torus in the active nucleus of NGC

1068,” Nature, vol. 429, pp. 47–49, 2004.

[6] A. Poncelet, G. Perrin, and H. Sol, “A new analysis of the nucleus of

NGC 1068 with MIDI observations,” Astron. Astrophys., vol. 450, pp.

483–494, 2006.

[7] K. R. W. Tristram et al., “Resolving the complex structure of the dust

torus in the active nucleus of the circinus galaxy,” Astron. Astrophys.,

vol. 474, pp. 837–850, 2007.

[8] P. R. Lawson, “Notes on the history of stellar interferometry,” in Prin-

ciplesofLongBaselineStellarInterferometry,CourseNotesfrom1999

MichelsonSummerSchool,P.R.Lawson,Ed.

JPL, 2000, pp. 325–32, no. 00-009.

[9] P. R. Lawson et al., “An interferometric imaging beauty contest,” in

NewFrontiersinStellarInterferometry,Proc.SPIEConf.,W.A.Traub,

Ed.Bellingham, WA: SPIE, 2004, vol. 5491, pp. 886–899.

[10] P.R.Lawsonetal.,“The2006interferometryimagingbeautycontest,”

in Advances in Stellar Interferometry, J. D. Monnier, M. SchÖller, and

W. C. Danchi, Eds.Bellingham, WA: SPIE, 2006, vol. 6268, p. 59.

[11] J. W. Goodman, Statistical Optics.

[12] M. Born and E. Wolf, Principles of Optics, 6th ed.

amon, 1993.

[13] J.-M. Mariotti, “Introduction to Fourier optics and coherence,” in

Diffraction-Limited Imaging With Very Large Telescopes, ser. NATO

ASI Series C, D. M. Alloin and J.-M. Mariotti, Eds.

Kluwer, 1989, vol. 274, pp. 3–31.

[14] F. Roddier, “The effects of atmospherical turbulence in optical as-

tronomy,” in Progress in Optics, E. Wolf, Ed.

Netherlands: North Holland, 1981, vol. XIX, pp. 281–376.

Pasadena,CA:NASA-

New York: Wiley, 1985.

New York: Perg-

Norwell, MA:

Amsterdam, The

Authorized licensed use limited to: IEEE Xplore. Downloaded on January 6, 2009 at 11:43 from IEEE Xplore. Restrictions apply.

Page 13

LE BESNERAIS et al.: ADVANCED IMAGING METHODS FOR LONG-BASELINE OI 779

[15] F. Roddier, J. M. Gilli, and G. Lund, “On the origin of speckle boiling

and its effects in stellar speckle interferometry,” J. Opt., vol. 13, no. 5,

pp. 263–271, 1982.

[16] J.-M. Conan, G. Rousset, and P.-Y. Madec, “Wave-front temporal

spectra in high-resolution imaging through turbulence,” J. Opt. Soc.

Amer. A, vol. 12, no. 12, pp. 1559–157, Jul. 1995.

[17] D. L. Fried, “Statistics of a geometric representation of wavefront Dis-

tortion,” J. Opt. Soc. Amer., vol. 55, no. 11, pp. 1427–143, 1965.

[18] F. Roddier, Ed., Adaptive Optics in Astronomy.

Cambridge Univ. Press, 1999.

[19] D. Mourard, I. Bosc, A. Labeyrie, L. Koechlin, and S. Saha, “The ro-

tating envelope of the hot star Gamma Cassiopeiae resolved by optical

interferometry,” Nature, vol. 342, pp. 520–522, Nov. 1989.

[20] W. J. Tango and R. Q. Twiss, “Michelson stellar interferometry,”

in Progr. Opt. (A81-13109 03-74) .

North-Holland Publishing, 1980, vol. 17 , pp. 239–277.

[21] R. C. Jennison, “A phase sensitive interferometer technique for the

measurement of the fourier transforms of spatial brightness distribu-

tion of small angular extent,” Monthly Notices Roy. Astron. Soc., vol.

118, pp. 276–284, 1958.

[22] S. Meimon, L. M. Mugnier, and G. Le Besnerais, “A self-calibration

approach for optical long baseline interferometry,” J. Opt. Soc. Amer.

A, accepted for publication.

[23] J. D. Monnier et al., “First results with the IOTA3 imaging interferom-

eter:Thespectrocopicbinaries?virginisandWR140,”apJL,vol.602,

no. 1, pp. L57–L60, Feb. 2004.

[24] J. C. Dainty and A. H. Greenaway, “Estimation of spatial power

spectra in speckle interferometry,” J. Opt. Soc. Amer., vol. 69, no. 5,

pp. 786–79, May 1979.

[25] B. Wirnitzer, “Bispectral analysis at low light levels and astronomical

speckle masking,” J. Opt. Soc. Amer. A, vol. 2, no. 1, pp. 14–2, Jan.

1985.

[26] T. Pauls et al., “A data exchange standard for optical (visible/ir) in-

terferometry,” in New Frontiers in Stellar Interferometry, Proc. SPIE

Conf.. Bellingham, WA: SPIE, 2004, vol. 5491.

[27] G. Perrin, “The calibration of interferometric visibilities obtained with

single-mode optical interferometers. Computation of error bars and

correlations,” Advanc. Appl. Prob., vol. 400, pp. 1173–1181, Mar.

2003.

[28] R. T. Rockafellar, Convex Analysis.

Press, 1996.

[29] D. Hestroffer, “Centre to limb darkening of stars. New model and ap-

plication to stellar interferometry,” A&A, vol. 327, pp. 199–206, Nov.

1997.

[30] A. Manduca, R. A. Bell, and B. Gustafsson, “Limb darkening co-

efficients for late-type giant model atmospheres,” A&A, vol. 61, pp.

809–813, Dec. 1977.

[31] S.Lacouretal.,“Thelimb-darkenedArcturus;ImagingwiththeIOTA/

IONIC interferometer,” ArXiv E-Prints, vol. 804, Apr. 2008.

[32] G. Perrin et al., “Study of molecular layers in the atmosphere of the

supergiant star ? Cep by interferometry in the K band,” A&A, vol. 436,

pp. 317–324, Jun. 2005.

[33] F. R. Schwab, “Optimal gridding of visibility data in radio interfer-

ometry,” in Measurement and Processing for Indirect Imaging, J. A.

Roberts,Ed. Cambridge,U.K.: CambridgeUniv.Press,1984,p.333.

[34] A. Lannes, E. Anterrieu, and K. Bouyoucef, “Fourier interpolation

and reconstruction via Shannon-type techniques. I regularization

principle,” J. Mod. Opt., vol. 41, no. 8, pp. 1537–1574, 1994.

[35] A. Lannes, S. Roques, and Casanove, “Stabilized reconstruction in

signal and image processing: Part i: Partial deconvolution and spectral

extrapolation with limited field,” J. Mod. Opt., vol. 34, pp. 161–226,

1987.

[36] A. Lannes, E. Anterrieu, and P. Maréchal, “Clean and wipe,” Astron.

and Astrophys. Suppl., vol. 123, pp. 183–198, May 1997.

[37] J.-M. Conan, L. M. Mugnier, T. Fusco, V. Michau, and G. Rousset,

“Myopic deconvolution of adaptive optics images using object and

point spread function power spectra,” Appl. Opt., vol. 37, no. 21, pp.

4614–4622, Jul. 1998.

[38] S. Meimon, “Reconstruction d’images astronomiques en inter-

férométrie optique,” Ph.D. dissertation, Université Paris Sud, Paris,

France, 2005.

[39] A. Blanc, L. M. Mugnier, and J. Idier, “Marginal estimation of aber-

rations and image restoration by use of phase diversity,” J. Opt. Soc.

Amer. A, vol. 20, no. 6, pp. 1035–1045, 2003.

Cambridge, U.K.:

Amsterdam, The Netherlands:

Princeton, NJ: Princeton Univ.

[40] L. M. Mugnier, C. Robert, J.-M. Conan, V. Michau, and S. Salem,

“Myopic deconvolution from wavefront sensing,” J. Opt. Soc. Amer.

A, vol. 18, pp. 862–872, Apr. 2001.

[41] W. J. Rey, Introduction to Robust and Quasi-Robust Statistical

Methods. Berlin, Germany: Springer-Verlag, 1983.

[42] S. Brette and J. Idier, “Optimized single site update algorithms for

image deblurring,” in Proc. IEEE ICIP, Lausanne, Switzerland, Sep.

1996, pp. 65–68.

[43] R. Narayan and R. Nityananda, “Maximum entropy image restoration

in astronomy,” Ann. Rev. Astron.Astrophys.,vol. 24, pp. 127–170, Sep.

1986.

[44] D. F. Buscher, “Direct maximum-entropy image reconstruction from

the bispectrum,” in IAU Symp. 158: Very High Angular Resolution

Imaging, J. G. Robertson and W. J. Tango, Eds., 1994, p. 91, –+.

[45] J.-F. Giovannelli and A. Coulais, “Positive deconvolution for superim-

posedextendedsourceandpointsources,”Astron.Astrophys.,vol. 439,

pp. 401–412, 2005.

[46] L. Delage, F. Reynaud, E. Thiebaut, K. Bouyoucef, P. Marechal, and

A. Lannes, “Présentation d’un démonstrateur de synthèse d’ouverture

utilisant des liaisons par fibres optiques,” in Actes du ?? colloque

GRETSI, Grenoble, France, Sep. 1997, pp. 829–832.

[47] A. Lannes, “Weak-phase imaging in optical interferometry,” J. Opt.

Soc. Amer. A, vol. 15, no. 4, pp. 811–82, Apr. 1998.

[48] E. Thiébaut, P. J. V. Garcia, and R. Foy, “Imaging with Amber/VLTI:

The case of microjets,” Astrophys. Space. Sci., vol. 286, pp. 171–176,

2003.

[49] E.Thiébaut,“Mira:Aneffectiveimagingalgorithmforopticalinterfer-

ometry,”presentedattheAstronomicalTelescopesandInstrumentation

SPIE Conf., 2008, paper no. 7013-53, vol. 7013.

[50] E. Thiébaut, “Reconstruction d’image en interférométrie optique,” in

XXIe Colloque GRETSI, Traitement du signal et des images, Troyes,

France, 2007.

[51] E. Thiébaut, “Optimization issues in blind deconvolution algorithms,”

in Astronomical Data Analysis II, Proc. SPIE Conf., J.-L. Starck and

F. D. Murtagh, Eds. Bellingham, WA: SPIE, 2002, vol. 4847, pp.

174–183.

[52] J. Skilling and R. K. Bryan, “Maximum entropy image reconstruc-

tion: General algorithm,” Monthly Not. Roy. Astron. Soc., vol. 211, pp.

111–124, 1984.

[53] S. Meimon, L. M. Mugnier, and G. Le Besnerais, “Reconstruction

method for weak-phase optical interferometry,” Opt. Lett., vol. 30, no.

14, pp. 1809–1811, Jul. 2005.

[54] T. J. Cornwell and P. N. Wilkinson, “A new method for making maps

with unstable radio interferometers,” Monthly Notices Roy. Astron.

Soc., vol. 196, pp. 1067–1086, 1981.

[55] A. Lannes, “Integer ambiguity resolution in phase closure imaging,” J.

Opt. Soc. Amer. J. A, vol. 18, pp. 1046–1055, May 2001.

[56] Y. Bar-Shalom and X.-R. Li, Multitarget-Multisensor Tracking: Prin-

ciples and Techniques.Storrs, CT: Univ. Connecticut, 1995.

[57] S. Meimon, L. M. Mugnier, and G. Le Besnerais, “A convex approxi-

mation of the likelihood in optical interferometry,” J. Opt. Soc. Amer.

A, Nov. 2005.

[58] J.-P. Berger et al., “An integrated-optics 3-way beam combiner for

IOTA,” in Interferometry for Optical Astronomy II—Proc. SPIE, W.

A. Traub, Ed., Feb. 2003, vol. 4838, pp. 1099–1106.

[59] F. Van Leeuwen, Hipparcos, The New Reduction of the Raw Data.

New York: Springer, 2007.

GuyLeBesneraiswasborninParis,France,in1967.

HegraduatedfromtheÉcoleNationaleSupérieurede

Techniques Avancées in 1989 and received the Ph.D.

degree in physics from the Université de Paris-Sud,

Orsay, France, in 1993.

Since 1994, he has been with the Office National

d’Études et Recherches Aérospatiales, Châtillon,

France. His main interests are in the fields of image

reconstruction and spatio-temporal processing of

image sequences. He made various contributions in

optical interferometry, super-resolution, optical-flow

estimation, and 3-D reconstruction from aerial image sequence.

Authorized licensed use limited to: IEEE Xplore. Downloaded on January 6, 2009 at 11:43 from IEEE Xplore. Restrictions apply.

Page 14

780IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 2, NO. 5, OCTOBER 2008

Sylvestre Lacour received the Ph.D. degree in astro-

physics in 2007, specializing in astronomical instru-

mentation, mainly involving interferometry and high

angular resolution.

He is a tenured Assistant Researcher at CNRS.

He is working within the National Institute for

Earth Sciences and Astronomy, Meudon, France.

His astrophysical interests involve the interstellar

medium, evolved stars, and extrasolar planets.

Laurent M. Mugnier graduated from Ecole Poly-

technique, France, in 1988. He received the Ph.D.

degree in 1992 from Ecole Nationale Supérieure des

Télécommunications (ENST), France, for his work

onthedigitalreconstructionofincoherent-lightholo-

grams.

In 1994 he joined ONERA, where he is currently a

Senior Research Scientist in the field of inverse prob-

lemsandhigh-resolutionopticalimaging.Hiscurrent

research interests include image reconstruction and

wavefront-sensing, in particular for adaptive-optics

corrected imaging through turbulence, for retinal imaging, for Earth observa-

tion and for optical interferometry in astronomy. His publications include five

contributions to reference books and 30 papers in peer-reviewed international

journals.

Eric Thiébaut was born in Béziers, France, in 1966.

He graduated from the École Normale Supérieure in

1987 and received the Ph.D. degree in astrophysics

from the Université Pierre and Marie Curie de Paris

VII, Paris, France, in 1994.

Since 1995, he has been an Astronomer at the

Centre de Recherche Astrophysique de Lyon,

France. His main interests are in the fields of signal

processing and image reconstruction. He has made

various contributions in blind deconvolution, optical

interferometry, and optimal detection.

Guy Perrin was born in Saint-Etienne, France, in

1968. He graduated from École Polytechnique in

1992 and received the Ph.D. degree in astrophysics

from Université Paris Diderot in 1996.

He has been an Astronomer with Observatoire de

Parissince1999.Hisresearchtopicsfocusbothonin-

strumentaltechniques for high angular resolution ob-

servations and on the use of interferometers and tele-

scopes equipped with adaptive optics to study point-

like objects such as evolved stars, active galactic nu-

clei, and the Galactic Center.

SergeMeimonwasborninParis,France,in1978.He

graduatedfromÉcoleCentraledeNantesin2002and

received the Ph.D. degree in physics from Université

Paris-Sud Orsay in 2005 for his work on image-re-

construction methods in optical interferometry.

Since then, he has been with the Office National

d’Études et Recherches Aérospatiales, Châtillon,

France. He has made various contributions in the

field of optical high-angular resolution.

Authorized licensed use limited to: IEEE Xplore. Downloaded on January 6, 2009 at 11:43 from IEEE Xplore. Restrictions apply.

#### View other sources

#### Hide other sources

- Available from free.fr
- Available from Laurent M Mugnier · Jun 2, 2014