[Advances in Imaging and Electron Physics] Volume 141 || Phase Diversity: A Technique for Wave-Front Sensing and for Diffraction-Limited Imaging

$Page 1: [Advances in Imaging and Electron Physics] Volume 141 || Phase Diversity: A Technique for Wave-Front Sensing and for Diffraction-Limited Imaging$
ADVANCES IN IMAGING AND ELECTRON PHYSICS, VOL. 141

Phase Diversity: A Technique for Wave-Front Sensing and forDiffraction-Limited Imaging

LAURENT M. MUGNIERa, AMANDINE BLANCa, ANDJÉRÔME IDIERb

aOffice National d’Etudes et de Recherches Aérospatiales, Department d’Optique Théorique etAppliquée, 92322 Châtillon cedex, France

bInstitut de Recherche en Communication et Cybernétique de Nantes, Analyse et Décision enTraitement du Signal et de l’Image, 1 rue de la Noe, BP 92101, 44321 Nantes cedex 3, France

I. Introduction and Problem Statement . . . . . . . . . . . . . . . 3A. Context . . . . . . . . . . . . . . . . . . . . . . 3B. Image Formation . . . . . . . . . . . . . . . . . . . . 4

1. PSF of a Telescope . . . . . . . . . . . . . . . . . . 52. Origin of PSF Degradations: Intrinsic Aberrations . . . . . . . . . 53. Origin of PSF Degradations: Atmospheric Turbulence . . . . . . . . 64. Parameterization of the Phase . . . . . . . . . . . . . . . 85. Discrete Image Model . . . . . . . . . . . . . . . . . 10

C. Basics of Phase Diversity . . . . . . . . . . . . . . . . . 111. Uniqueness of the Phase Estimate . . . . . . . . . . . . . . 112. Inverse Problems at Hand . . . . . . . . . . . . . . . . 11

II. Applications of Phase Diversity . . . . . . . . . . . . . . . . 12A. Quasi-Static Aberration Correction of Optical Telescopes . . . . . . . . 13

1. Monolithic-Aperture Telescope Calibration . . . . . . . . . . . 132. Cophasing of Multiaperture Telescopes . . . . . . . . . . . . 14

B. Diffraction-Limited Imaging Through Turbulence . . . . . . . . . . 141. A posteriori Correction . . . . . . . . . . . . . . . . . 152. Real-Time Wave-Front Correction . . . . . . . . . . . . . . 16

III. Phase Estimation Methods . . . . . . . . . . . . . . . . . . 16A. Joint Estimator . . . . . . . . . . . . . . . . . . . . 17

1. Joint Criterion . . . . . . . . . . . . . . . . . . . 172. Circulant Approximation and Expression in Fourier Domain . . . . . . 203. Tuning of the Hyperparameters . . . . . . . . . . . . . . 21

B. Marginal Estimator . . . . . . . . . . . . . . . . . . . 221. Expression of R−1

I. . . . . . . . . . . . . . . . . . 23

2. Determinant of RI . . . . . . . . . . . . . . . . . . 233. Marginal Criterion . . . . . . . . . . . . . . . . . . 244. Relationship Between the Joint and the Marginal Criteria . . . . . . . 245. Expression in the Fourier Domain . . . . . . . . . . . . . . 256. Unsupervised Estimation of the Hyperparameters . . . . . . . . . 25

1ISSN 1076-5670/05 Copyright 2006, Elsevier Inc.DOI: 10.1016/S1076-5670(05)41001-0 All rights reserved.

2 MUGNIER ET AL.

C. Extended Objects . . . . . . . . . . . . . . . . . . . 261. Apodization . . . . . . . . . . . . . . . . . . . . 262. Guard Band . . . . . . . . . . . . . . . . . . . . 27

IV. Properties of the Phase Estimation Methods . . . . . . . . . . . . . 28A. Image Simulation . . . . . . . . . . . . . . . . . . . 28B. Asymptotic Properties of the Two Estimators for Known Hyperparameters . . . 29C. Joint Estimation: Influence of the Hyperparameters . . . . . . . . . . 32D. Marginal Estimation: Unsupervised Estimation . . . . . . . . . . . 34E. Performance Comparison . . . . . . . . . . . . . . . . . 35F. Conclusion . . . . . . . . . . . . . . . . . . . . . 36

V. Restoration of the Object . . . . . . . . . . . . . . . . . . 36A. With the Joint Method . . . . . . . . . . . . . . . . . . 37B. With the Marginal Method . . . . . . . . . . . . . . . . . 38

1. Principle . . . . . . . . . . . . . . . . . . . . . 382. Results . . . . . . . . . . . . . . . . . . . . . . 393. Influence of the Hyperparameters . . . . . . . . . . . . . . 39

C. With a “Hybrid” Method . . . . . . . . . . . . . . . . . 401. Principle . . . . . . . . . . . . . . . . . . . . . 402. The Three Steps . . . . . . . . . . . . . . . . . . . 403. Results . . . . . . . . . . . . . . . . . . . . . . 41

D. Conclusion . . . . . . . . . . . . . . . . . . . . . 42VI. Optimization Methods . . . . . . . . . . . . . . . . . . . 42

A. Projection-Based Methods . . . . . . . . . . . . . . . . . 42B. Line-Search Methods . . . . . . . . . . . . . . . . . . 44

1. Strategies of Search Direction . . . . . . . . . . . . . . . 442. Step Size Rules . . . . . . . . . . . . . . . . . . . 45

C. Trust-Region Methods . . . . . . . . . . . . . . . . . . 45VII. Application of Phase Diversity to an Operational System: Calibration of NAOS-CONICA 46

A. Practical Implementation of Phase Diversity . . . . . . . . . . . . 461. Choice of the Defocus Distance . . . . . . . . . . . . . . 462. Image Centering . . . . . . . . . . . . . . . . . . . 473. Spectral Bandwidth . . . . . . . . . . . . . . . . . . 47

B. Calibration of NAOS and CONICA Static Aberrations . . . . . . . . . 481. The Instrument . . . . . . . . . . . . . . . . . . . 482. Calibration of CONICA Stand-Alone . . . . . . . . . . . . . 493. Calibration of the NAOS Dichroics . . . . . . . . . . . . . 514. Closed-Loop Compensation . . . . . . . . . . . . . . . 52

C. Conclusion . . . . . . . . . . . . . . . . . . . . . 53VIII. Emerging Methods: Measurement of Large Aberrations . . . . . . . . . 53

A. Problem Statement . . . . . . . . . . . . . . . . . . . 53B. Large Aberration Estimation Methods . . . . . . . . . . . . . 55

1. Estimation of the Unwrapped Phase . . . . . . . . . . . . . 552. Estimation of the Wrapped Phase (Then Unwrapping) . . . . . . . . 56

C. Simulation Results . . . . . . . . . . . . . . . . . . . 591. Choice of an Error Metric . . . . . . . . . . . . . . . . 592. Results . . . . . . . . . . . . . . . . . . . . . . 59

IX. Emerging Applications: Cophasing of Multiaperture Telescopes . . . . . . . 63A. Background . . . . . . . . . . . . . . . . . . . . . 63B. Experimental Results on an Extended Scene . . . . . . . . . . . . 65

A TECHNIQUE FOR WAVE-FRONT SENSING 3

C. Conclusion . . . . . . . . . . . . . . . . . . . . . 68References . . . . . . . . . . . . . . . . . . . . . . 68

I. INTRODUCTION AND PROBLEM STATEMENT

A. Context

The theoretical angular resolution of an optical imaging instrument such as atelescope is given by the ratio of the imaging wavelength λ over the aperturediameter D of the instrument.

For a real-world instrument, optical aberrations often prevent this so-calleddiffraction-limit resolution λ/D from being achieved. These aberrations mayarise both from the instrument itself and from the propagation medium of thelight. When observing Space from the ground, the aberrations are predom-inantly due to atmospheric turbulence: inhomogeneities of air temperatureinduce inhomogeneities of refraction index.

The aberrations can be compensated either during the image acquisition byreal-time techniques or a posteriori (i.e., by postprocessing). Adaptive optics(AO) is a technique to compensate in real time for turbulence-induced aber-rations (Roddier, 1999). Most of these techniques require the measurement ofthe aberrations, also called wave-front, by a wave-front sensor (WFS).

A large number of WFSs exist today; these are thoroughly reviewed in(Rousset, 1999) and can be classified into two families: focal-plane sensorsand pupil-plane sensors. Today’s AO systems use either Shack–HartmannWFSs (Shack and Plack, 1971) or curvature WFSs (Roddier, 1988), whichboth divert part of the incoming light by means of a (dichroic) beam-splitterinto some auxiliary optics and belong to the second family. For AO they bothhave the appealing property that they work with broadband light (because theyare well described by geometrical optics) and that the relationship between theunknown wave-front and the recorded data is linear, so that it can be invertedin real time.

The focal-plane family of sensors was born from the very natural idea thatan image of a given object contains information not only about the object,but also about the wave-front. A focal-plane sensor thus requires little or nooptics other than the imaging sensor; it is also the only way to be sensitiveto all aberrations down to the focal plane. The first practical method forwave-front sensing from focal-plane data was proposed by Gerchberg andSaxton (1972) in the electron microscopy context and later rediscovered byGonsalves (1976). If the pupil (or aperture) function of the imaging system isknown, this method only requires one focal-plane image of a point sourcein order to estimate the aberrations, which are coded in the phase of the

4 MUGNIER ET AL.

pupil transmittance. It finds the aberrations that are the most compatibleones with the known constraints in the pupil plane (known aperture) andin the focal plane (measured image). The original implementation usesprojections: it works by imposing the known constraints on the wave’scomplex amplitude alternatively in the two domains until convergence. Theconnection between this projection-based algorithm and the minimization ofa least-square functional of the unknown aberrations was later made by Fienup(1982). This so-called phase-retrieval method has two major limitations. First,it only works with a point source. Second, there is generally a sign ambiguityin the recovered phase (i.e., the solution is not unique), as detailed below.

Gonsalves (1982) showed that by using a second image with an additionalknown phase variation with respect to the first image (such as defocus), itis possible to estimate the unknown phase even when the object is extendedand unknown. The presence of this second image additionally removes theabove-mentioned sign ambiguity of the solution. This technique is referred toas “phase diversity” by analogy with a technique used in wireless telecom-munications. The idea of using two images of the same object with a knownfinite relative defocus in order to determine phase information from intensitymeasurements can actually be traced back to the work of Misell (1973), againin the electron microscopy context.

This contribution attempts to provide a (necessarily incomplete) survey ofthe phase diversity technique, with an emphasis on its wave-front sensing ca-pabilities. In most of this text, we consider a single-aperture1 optical imaginginstrument working with spatially incoherent light, such as a telescope. Theremainder of this section provides an introduction to the image formation forsuch an instrument, reviews the sources of image degradation, and states theinverse (estimation) problem to be solved.

B. Image Formation

Image formation is well described by the scalar theory of diffraction, pre-sented in detail in reference books (Goodman, 1968; Born and Wolf, 1993).It can be modeled by a convolution of the observed object by the instrument’spoint spread function (PSF), at least within the so-called isoplanatic patch ofthe instrument. At visible wavelengths, this patch is typically of the order of1 degree when considering only the aberrations of the telescope itself and ofthe order of a few arc-seconds (1 arcsec = 1/3600◦) for a telescope observingSpace through atmospheric turbulence.

1 As opposed to multiple-aperture instruments such as imaging interferometers, see Section IX.


1. PSF of a Telescope

The PSF of a telescope or of the “telescope + atmosphere” system at animaging wavelength λ is the square modulus of the (inverse) Fourier transformof the complex amplitude ψ , where ψ = P exp(jϕ) is the electromagneticfield in the instrument pupil (or aperture) when the observed object is a pointsource:

hopt(x, y) = ∣∣FT−1(P(λu, λv) ejϕ(λu,λv))∣∣2

(x, y). (1)

In this expression, the Fourier transform models the transformation of theelectromagnetic field between infinity and the focal plane, the square modulusis due to the quadratic detection (i.e., the detection of the field’s intensity),and x and y are angles on the sky, in radians (rd). For a perfect telescope andwithout turbulence, P is constant within the pupil and ϕ is zero; for a realtelescope, the variations of ψ are due to the aberrations of the telescope itselfand to those introduced by turbulence.

From this point forward in the text, we shall write P(u, v) = P(λu, λv)

and φ(u, v) = ϕ(λu, λv) to deal with dimensionless quantities.In the following, we assume that P is simply the indicatrix of the pupil,

(i.e., that the intensity variations in the pupil are negligible). This assumptionis generally valid in astronomical imaging and is called the near fieldapproximation (Roddier, 1981). With this assumption, the PSF is completelydescribed by the pupil phase φ.

Eq. (1) indicates that the optical transfer function (OTF) hopt is the auto-correlation of P exp(jφ):

hopt(u, v) = (P exp(jφ) ⊗ P exp(jφ)

)(u, v), (2)

where the correlation of two complex-valued functions f1 and f2 is defined byf1⊗f2(x) �

∫f �

1 (t)f2(t+x) dt . In the absence of aberrations (i.e., if φ = 0),the OTF is thus the autocorrelation of P . Consequently, it has a cutoff spatialfrequency of D/λ rd−1, where D is the pupil diameter, and is strictly zerobeyond. In a real system for Space observation from the ground, turbulence-induced aberrations, if uncorrected, lead to a cutoff frequency much smallerthan D/λ.

2. Origin of PSF Degradations: Intrinsic Aberrations

Some aberrations are intrinsic to the instrument; they originate in imperfec-tions in the design, fabrication and assembly, as well as in the environmentof the instrument (e.g., thermomechanical stresses and vibrations for a spacetelescope).

Optical telescopes for Space observation from the ground or for Earthobservation from Space are usually instruments of very good optical quality,

6 MUGNIER ET AL.

so that the overall amplitude of these aberrations is small (notably less thanthe imaging wavelength λ). Their spatial spectrum is related to their origins:

• Some aberrations originate in the optical design; they are fixed and of lowspatial frequencies.

• Some aberrations are due to the fabrication process (polishing); they arealso fixed but, conversely, of high spatial frequencies.

• Some aberrations are due to misalignments, either because of an imperfectalignment during the integration or because of thermomechanical driftsduring operation. Such aberrations are slowly varying and of low spatialfrequencies.

• Finally, some aberrations may occur at the location where the opticalcomponents are supported. These too are slowly varying but may be ofvariable spatial frequencies.

In summary, the salient features of intrinsic aberrations are that they areslowly varying and usually of small overall amplitude.

3. Origin of PSF Degradations: Atmospheric Turbulence

Inhomogeneities of the temperature of atmospheric air induce inhomo-geneities of the air refraction index, which perturb the propagation of lightwaves through the atmosphere. We shall assume that these random indexfluctuations follow the Kolmogorov law: their probability density function isGaussian, with zero mean and a power spectral density (PSD) proportionalto |ν|11/3, where ν is the three-dimensional (3D) spatial frequency (Roddier,1981). This assumption is usually valid, at least for the spatial frequencies oftoday’s single-aperture telescopes.

Astronomical observation from the ground In the case of astronomicalobservations from the ground, the light wave coming from a point source isplanar at the entrance of the atmosphere. By integration of the index fluctua-tions’ PSD along the optical path and within the near field approximation, itis possible to derive the spatial statistics of the phase in the telescope’s pupil.This phase is Gaussian as it results from the sum of all index perturbationsfrom the high atmosphere down to the ground (Roddier, 1981). Its PSDdepends on only one parameter denoted by r0 and reads (Noll, 1976):

Sφ(f ) = 0.023 r−5/30 f −11/3, (3)

where f is the modulus of the 2D spatial frequency in the pupil and r0,called Fried diameter (Fried, 1965), is the key parameter that quantifies theturbulence’s strength. It is all the smaller as the turbulence is stronger; it istypically 10 cm in the visible in a relatively good site.


The typical evolution time τ of the turbulent phase in the pupil is given bythe ratio of its characteristic scale r0 over an average wind speed, which is,more precisely, the standard deviation of the wind speed modulus distribution(Roddier et al., 1982):

τ = r0/Δv. (4)

For r0 ≈ 10 cm and Δv ≈ 10 m · s−1, τ ≈ 10−2 sec, images corresponding toan integration time that is notably longer than this time will be referred to as“long exposure”; those of shorter exposure time will be referred to as “shortexposure.” For a comprehensive exposition on the temporal statistics of theturbulent phase, see Conan et al. (1995).

The long-exposure turbulent OTF (without AO correction) is the product ofthe telescope’s OTF without atmosphere T by an atmosphere transfer functionB of cutoff frequency r0/λ (Roddier, 1981):

hopt(f ) �⟨h

optt (f )

⟩ = T (f )B(f ),

where B(f ) = exp{−3.44(λf/r0)5/3} and 〈·〉 denotes a temporal average

on an arbitrarily long time. Because the cutoff frequency of T is D/λ, thisequation shows that the phenomenon that limits the resolution depends onthe ratio D/r0: if D < r0, the instrument is diffraction limited, that is, itsresolution is given by its diameter (if its intrinsic aberrations are reasonable),whereas if D � r0, the long-exposure resolution of the instrument is limitedby turbulence and is not better than that of a telescope of diameter r0.

As noted by Labeyrie (1970), when the exposure time is short enough tofreeze the atmospheric turbulence [typically shorter than 10 ms, cf. Eq. (4)],some high-frequency information is preserved in the images, in the form ofspeckles, whose typical size is λ/D and whose position is random. This isillustrated in Figure 1. If a number of these (uncorrected) short-exposureimages are processed jointly in a more clever way than a simple average, it isthus possible to restore a high-resolution image of the observed object.

Earth observation from Space In the case of Earth observation fromSpace, a light wave coming from a point source on the ground is sphericaland not planar as in astronomical observation. Because it is spherical, sucha wave intersects less and thus interacts less with the lower layers of theatmosphere, which are the ones where the turbulence is strongest. It can beshown theoretically (Fried, 1966a) that the lower layers contribute less tothe overall turbulence strength r0, whose value is typically of a few tens ofmeters (see, e.g., Blanc, 2002). As a consequence, turbulence is not a limitingfactor in the case of a space telescope observing the Earth.

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FIGURE 1. Simulated short-exposure (left) and long-exposure (right) images of a star throughturbulence. The turbulence strength is D/r0 = 10; the sampling rate respects the Nyquist-Shannoncriterion.

4. Parameterization of the Phase

The pupil phase φ can often be described parsimoniously when expanded on amodal basis. The Zernike polynomials (Noll, 1976) form an orthonormal basison a disk and thus make a convenient basis for the expansion of the phase onthe circular pupil of a telescope. Each of these polynomials is the product of atrigonometric function of the polar angle θ with a polynomial function of theradius r:

Zi(r) = Rmn (r)Θm

n (θ), (5)

where the trigonometric function reads:

Θmn (θ) =

⎧⎪⎨⎪⎩

√n + 1 if m = 0,√2(n + 1) cos(mθ) if m = 0 and i even,√2(n + 1) sin(mθ) if m = 0 and i odd

(6)

and the polynomial function reads:

Rmn (r) =

(n−m)/2∑s=0

(−1)s(n − s)!s![(n + m)/2 − s]![(n − m)/2 − s]!r

n−2s . (7)

Parameter n in Eqs. (6) and (7) is called the radial degree of the correspondingZernike polynomial, and parameter m is called the azimuthal degree; m andn have the same parity and are such that 0 ≤ m ≤ n. The first Zernikepolynomials are represented in Figure 2. Several additional properties makethis Zernike basis very commonly used:


FIGURE 2. First Zernike polynomials.

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• The first Zernike polynomials correspond to the well-known low-orderoptical aberrations: Z4 is defocus, Z5 and Z6 are astigmatism, Z7 and Z8are coma, and Z11 is spherical aberration.

• Their ordering by increasing radial degree corresponds to an order ofincreasing spatial frequency.

The expansion of the phase φ on this basis reads:

φ(r) =∞∑

k=1

ak Zk(r), (8)

where r denotes the spatial coordinates in the pupil, normalized to a unitradius. For a multiple-aperture instrument, such an expansion can be usedon each of the apertures. The first term is called piston (Z1) and codes for theaverage optical path difference of a given aperture; the two following terms(Z2 and Z3) are tip/tilt and code for the position of the aperture’s PSF. Fora single-aperture telescope, the sum in Eq. (8) is usually started with k = 4,which corresponds to a centered PSF. Additionally, in practice, the sum isnecessarily limited to a finite number kmax, which depends on the problem athand. It is of the order of 10 or a few tens when estimating the aberrations ofa space telescope and of the order of 100 or a few hundreds when observingSpace from the ground.

5. Discrete Image Model

The image is recorded by a detector such as a charge-coupled device (CCD)camera, which integrates the flux on a grid of pixels. This can be convenientlymodeled as the convolution by a detector PSF hdet followed by a samplingoperation. The global PSF of the instrument is thus:

h = hdet � hopt. (9)

Due to the inevitable noise of the recording process (photon noise and detectornoises), the recorded image reads:

i = [h � o]x + n, (10)

where [·]x denotes the sampling operation.This model is generally approximated by a discrete convolution with the

sampled version of the (unknown) object o and written in matrix form:

i = h � o + n = Ho + n, (11)

where H is the matrix representing the discrete convolution by the sampledversion h of h, and where i is the vector obtained by stacking together thecolumns of the corresponding image. Similarly, o is the vector obtained bystacking together the columns of the sampled object.


C. Basics of Phase Diversity

1. Uniqueness of the Phase Estimate

We mentioned in Section I.A that phase retrieval from a single imagegenerally faces non-uniqueness of the solution, even if the object is known.This is due to the relationship between the OTF and the pupil phase [Eq. (2)],as shown below.

For any complex-valued function f , a simple change of variables in theintegration shows that the function f ′ defined as f ′(t) � f �(−t) and f haveidentical autocorrelations: f ⊗ f = f ′ ⊗ f ′.

Let f (t) = P(t) ejφ(t), one obtains f ′(t) = P(−t) e−jφ(−t) as P

is real-valued; thus, if P is centrosymmetrical (i.e., even) for any phaseφ(t), the phase defined by φ′(t) = −φ(−t) yields the same OTF, thatis, hopt(φ′) = hopt(φ) (as noted by Gonsalves, 1976), and thus the sameimage. This result can be cast into a somewhat more informative form: ifthe phase is decomposed (uniquely) into its even and odd components, that is,φ(t) = φeven(t) + φodd(t), then φ′(t) = −φeven(t) + φodd(t). In other words,there is an indetermination on the sign of the even part of the phase (Blanc,2002).

Recording a second image of the same object with the instrument sufferingfrom the same unknown phase plus a known additional even one removes thisindetermination and adds enough information to retrieve both the phase andthe possibly unknown object. More quantitative results on the uniqueness ofthe phase estimate can be found in Idier et al. (2005). Let φd be this “diversity”phase (often defocus), the two images read:

i1 = [hdet � hopt(φ)

]� o + n1, (12)

i2 = [hdet � hopt(φ + φd)

]� o + n2. (13)

2. Inverse Problems at Hand

The phase-diversity technique can be used in two different contexts. (1) Onecan be interested in imaging a remote object, for instance in solar astronomyor Space surveillance. (2) One can be interested in measuring the aberrationsof an imaging system, either to correct the latter in real time or to restorea posteriori the images it takes. These two problems are obviously veryrelated, but they are not identical.

In particular, when interested in imaging a remote object through unknownaberrations, the aforementioned sign ambiguity on the phase can be tolerated,provided the object is recovered satisfactorily. Indeed, multiframe “blind”deconvolution from short-exposure turbulent images has been successfullydemonstrated (Schulz, 1993; Thiébaut and Conan, 1995), where blind means

12 MUGNIER ET AL.

deconvolution without a dedicated WFS but with the use of the strongconstraints that each PSF is fully described by a pupil phase [cf. Eq. (1)] andthat the unknown object is identical in all images. Yet, it can be advantageousto record WFS data simultaneously with the images, in particular becauseblind deconvolution is usually impaired by the presence of local minima inthe criterion to minimize. The WFS can be a focal-plane WFS consisting of adiverse image for each recorded image or a pupil-plane WFS such as a Shack–Hartmann (Fontanella, 1985; Primot et al., 1988; Mugnier et al., 2001).

When interested in estimating wave-fronts, the presence of (at least) twoimages is necessary to avoid a sign indetermination in the phase, as shown inSubsection I.C.1.

In both problematics, the basis of the inversion consists in estimating thephase2 and the object that are consistent with the measurements, given therecorded images. The most “natural” solution, which is the one used byGonsalves (1982) originally, is based on the minimization of the followingleast-square criterion3:

J (o,φ) = ∥∥i1 − H(φ)o∥∥2 + ∥∥i2 − H(φ + φd)o

∥∥2 (14)

as a function of (o,φ).The remainder of this contribution is organized as follows: Section II re-

views the domains of application of phase diversity. Then, Sections III and IVreview the wave-front estimation methods associated with this technique andtheir properties, while Section V examines the possible object estimation (i.e.,image restoration) methods. Section VI gives some background on the variousminimization algorithms that have been used for phase diversity. Section VIIillustrates the use of phase diversity on experimental data for wave-frontsensing. Finally, Sections VIII and IX highlight two fields of phase-diversitywave-front sensing that have witnessed noteworthy advances. Section VIIIreviews the methods used to estimate the large-amplitude aberrations faced inimaging through turbulence and proposes a novel approach for this difficultproblem. Section IX reviews the developments of phase diversity for a recentapplication: the phasing (also called cophasing) of multiaperture telescopes.

II. APPLICATIONS OF PHASE DIVERSITY

The concept of phase diversity has been first proposed by Gonsalves in 1982as a WFS for adaptive optics. Since 1990, this method has been successfullyused in several applications, including astronomy, space observation, and

2 The phases in the case of a sequence of image pairs.3 For simplicity, this criterion is stated for the case of two images per phase screen and one single

phase screen; it is readily generalizable to more than two images and several phase screens.


Earth observation. Phase diversity has the particularity of providing theestimation of the unaberrated object as well as the aberrations responsible forthe blurring. This method directly uses image data for the estimation of theaberrations. It is thus sensitive to all aberrations degrading the quality of theimaging telescope, contrarily to WFSs such as the Shack–Hartman, which usea dedicated light path and thus are affected by noncommon-path aberrations.Furthermore, the optical hardware of this technique is simple. These are someof the reasons why phase diversity is becoming a widespread method both tocompensate quasi-static optical aberrations and to obtain diffraction-limitedimaging through turbulence.

A. Quasi-Static Aberration Correction of Optical Telescopes

Imperfections of an optical telescope can originate from design, fabricationof the optical system (e.g., polishing errors), misalignments (integration andlaunch), and thermomechanical stresses. These aberrations correspond todifferent ranges in term of spatial frequencies, but all are slowly changing.Phase diversity using the image data from the science camera obviates theneed for important auxiliary optics and thus is a strong candidate for thecalibration of telescopes.

1. Monolithic-Aperture Telescope Calibration

Space-based telescopes In the case of imaging space or Earth from space,the images are only perturbed by the imperfections of the optical system.

A first practical application of the phase-diversity technique has beenthe determination of the Hubble Space Telescope aberrations (Roddier andRoddier, 1991, 1993; Fienup et al., 1993). In this case, the observed objectwas known (an unresolved star), so only the aberrations had to be estimated,resulting in a much easier problem, referred to phase-diverse phase retrieval(Ellerbroek et al., 1997).

Using the ability of phase diversity to also work with extended objects(including those extending beyond the field of view), studies have been madefor the calibration of telescopes imaging the Earth (Blanc et al., 2003b). Theimplementation of the real-time correction of static optical aberrations hasalso successfully been demonstrated (Kendrick et al., 1998).

Ground-based telescopes Phase diversity has also been used to calibratespace-imaging systems on Earth. The images obtained from Earth are mostlydegraded by the deleterious effects of the atmospheric turbulence but also bythe static aberrations of the system. Aberrations induced by the atmosphere

14 MUGNIER ET AL.

are zero mean and quickly changing, unlike aberrations due to imperfectionsof the system.

The calibration of the whole optical system, from the entrance pupil ofa telescope to the focal plane, can be done by averaging a large numberof aberration estimates corresponding to a series of short-exposure pairs ofimages of an astronomical object (Acton et al., 1996; Baba and Mutoh,2001). How these estimates are obtained is explained in Subsection II.B.1.The calibration through the atmosphere has also been done, by Lee et al.(1997b), in the case where the optical instrument contains an AO system.In the latter reference, the diversity introduced in the images is unusual: noadditional defocused image is required, and successive changes to the AOintroduce the diversity.

If only the AO and the camera (and not the telescope itself) are to becalibrated, the most effective procedure is to install an internal point sourceat the entrance of the AO system. The calibration of the noncommon-pathaberrations of the European very large telescope (VLT) AO system (calledNAOS) and its camera (called CONICA) has been recently done this way; seeBlanc et al. (2003a), Hartung et al. (2003), and Section VII for details. Phasediversity is also a practical tool for calibrating deformable mirrors (Löfdahlet al., 2000).

2. Cophasing of Multiaperture Telescopes

The resolution of a telescope is ultimately limited by its aperture diameter.The latter is limited by current technology to approximately 10 m for ground-based telescopes and to a few meters for space-based telescopes because ofvolume and mass considerations.

Multiaperture telescopes (interferometers) have the potential to removethese limitations. In order to reach the diffraction-limited resolution, all sub-apertures must be precisely phased with respect to one another. This so-calledcophasing of the interferometer can be performed by use of a phase diversitysensor. Indeed, for a multiaperture telescope as well as a single-aperture one,the image is the result of interferences between all aperture points. Thus,there is information in the image (whether focused or defocused) about themisalignments between subapertures, which are the specific aberrations ofinterferometry and can be described on each subaperture by the first threeZernike polynomials, called piston and tip-tilt [see Eq. (8) and Figure 2].Section IX is dedicated to this relatively recent application of phase diversity.

B. Diffraction-Limited Imaging Through Turbulence

Phase diversity can be used to correct the phase errors due to atmosphericturbulence in two ways: it can be used as an a posteriori correction technique


(image restoration) or as a real-time WFS for adaptive optics. Note that theability of phase diversity to recover both wave-front phase and amplitudehas been demonstrated on simulated (Gonsalves, 1997) and experimental data(Jefferies et al., 2002).

1. A posteriori Correction

For this application, the object is the parameter of interest. Image restorationby means of phase diversity can either correct all the aberrations degradingan imaging system without an AO, or it can be used after AO correction, as asecond step, to correct for the residual aberrations.

A special processing approach has been proposed to use phase diversity forimaging through the atmosphere, called phase-diverse speckle (a techniquethat blends the speckle imaging and the phase diversity concepts). Severalshort-exposure pairs of phase diversity data (in and out of focus) are collected.This method has been applied for imaging through turbulence without AO, inparticular for imaging satellites (Seldin et al., 1997; Thelen et al., 1999b) andfor imaging the Sun.

Additionally, when the object being imaged through turbulence is very ex-tended, the PSF is no longer space invariant in the field of view. The problemof correcting for turbulence-induced blur becomes thus more complicated.Phase diversity can accommodate for space-variant blur. Two methods havebeen investigated for solving this problem: correcting separately subfields thatare smaller than the isoplanatic patch, which is the field of view in whichthe PSF can be considered as space invariant (Löfdahl and Scharmer, 1994;Seldin and Paxman, 1994; Paxman et al., 1996) or using a tomographic phasereconstruction (Gonsalves, 1994; Acton et al., 1996; Thelen et al., 1999a,2000; Paxman et al., 1994, 1998). With subfielding, a series of overlappingsubframe reconstructions is combined to provide the entire corrected fieldof view. In the other more sophisticated approach, the volumic nature ofturbulence is taken into account by reconstructing the phase in several screenslocated at different altitudes.

Postcorrection by means of phase diversity is also useful for AO-correctedtelescopes. A first reason is the existence of noncommon-path aberrations,either unseen because of being outside the AO loop or corrected by the AOloop while not in the science path. A second reason is that the AO correction isalways partial (Roggemann, 1991; Conan, Madec and Rousset, 1994; Conan,1994). Phase-diverse techniques have been successfully demonstrated forpostcorrection of binary stars in Seldin et al. (1996b), of satellites in Seldinet al. (1996a), and of the Sun in Löfdahl and Scharmer (2002).

16 MUGNIER ET AL.

2. Real-Time Wave-Front Correction

The correction of atmospheric turbulence can be done in real time by usingAO systems. Phase diversity is potentially a good candidate for use as areal-time AO WFS for a number of reasons: it is very simple optically; itis also easy to calibrate; and it directly relies on the image so it corrects allaberrations degrading the images (no noncommon-path aberration). However,the computational time required on today’s computers to obtain estimates ofthe wave-front with phase diversity is, for the moment, considerable comparedto the evolution time of the turbulence (a few milliseconds) so that current AOsystems generally use other (pupil-plane) sensors.

Demonstrations of real-time correction have been obtained for very fewcorrected aberrations by Gates et al. (1994) and Kendrick et al. (1994a, 1998).Efforts in making phase-diversity estimation faster have thus been made: firstby proposing better numerical algorithms in Vogel et al. (1998), Löfdahl et al.(1998a), then by modifying the error metric used to estimate the aberrationsand object from the data (Kendrick et al., 1994b; Scharmer et al., 1999;Löfdahl and Scharmer, 2000). Phase-diversity sensors depend on an imagingmodel [Eq. (10)] involving convolutions that are usually implemented usingfast Fourier transforms (FFTs) and thus computationally demanding. Thegoal of new metrics is to reduce the number of computed FFTs. The use ofthese metrics for real-time correction has been demonstrated only for a fewaberrations; even these new methods involve rapidly increasing computingtime as the number of aberrations increases.

Another difficulty of the phase-diversity WFS for this application is thatit exhibits phase wrapping when the peak-to-valley phase variation is higherthan 2π , which is often the case for turbulence-induced dynamic aberrationsbefore closing the loop of the AO system. This is due to the fact that this sensoris sensitive only to the phase modulo 2π , as can be seen from Eq. (1). Recentworks provide some methods to alleviate this problem—see Section VIII fordetails.

III. PHASE ESTIMATION METHODS

The main problem to address in the phase-diversity framework is to estimatethe unknown quantities (the object o and/or the aberrated phase φ) from thedata (focused and defocused images). The choice of a relevant estimator isthus essential. This section presents the conventional phase estimator in thephase-diversity literature. More precisely, it focuses on the estimation of theaberrated phase from a focal image i1 and an additional defocused one i2obtained from a single-aperture telescope. The estimation methods presentedhere can be easily generalized to more than one phase screen (i.e., to the


phase-diverse speckle context). The estimation of the object is discussedin Section V and the specificity of the estimation from segmented-aperturetelescope in Section IX.

A. Joint Estimator

1. Joint Criterion

The conventional processing scheme found in the literature is based on thejoint estimation of the aberrations and of the observed object (Paxman et al.,1992). The Bayesian interpretation of such an approach is that it consists incomputing the joint maximum a posteriori (JMAP) estimator:

(o, φ)MAP = arg maxo,φ

p(i1, i2, o,φ; θ)

= arg maxo,φ

p(i1|o,φ; θn)p(i2|o,φ; θn)p(o; θo)p(φ; θφ),(15)

where p(i1, i2, o,φ; θ) is the joint probability density function of the data(i1, i2), the object o, and the aberrations φ. It may also depend on a set ofhyperparameters θ = (θn, θo, θφ). The likelihood of the data ik is denotedby p(ik|o,φ; θ); p(o; θo) and p(φ; θφ) are the a priori probability densityfunctions of o and φ.

The majority of the estimation structures used in the phase-diversity litera-ture can be rewritten as Eq. (15) even if they were not originally introduced ina Bayesian framework. Gonsalves (1982) proposed to use a joint least-squareapproach for the estimation of aberrations parameters. A maximum likelihoodestimation of the unknowns was later presented in Paxman et al. (1992) underGaussian and Poisson noise models. The Gonsalves least-square approachis equivalent to the joint maximum likelihood (JML) approach presented inPaxman et al. (1992) under the Gaussian noise model. The JML, in turn,is obtained by setting p(o; θo) = p(φ; θφ) = 1 in the JMAP approach[Eq. (15)]. Bucci et al. (1999) introduce regularization under a deterministicapproach. A first stochastic interpretation of joint estimation is presented inVogel et al. (1998) and in Thelen et al. (1999b). By introducing regularizationon the aberrations (p(φ; θφ) = 1), they propose a so-called generalizedmaximum likelihood (GML) estimation.4 The use of statistical informationon both the object and the aberrations leads to the JMAP estimator (Vogelet al., 1998; Blanc et al., 2003b) of Eq. (15).

4 The denomination GML comes from the statistics literature and refers to a JML criterion that ispenalized by a regularization term on some of the unknowns only.

18 MUGNIER ET AL.

Noise It is assumed here, for simplicity, that the noise is stationary whiteGaussian with the same variance σ 2 for each image. The case of differentvariances has been presented in Löfdahl and Scharmer (1994). Hence, thelikelihood p(ik|o,φ; θ) reads

p(ik|o,φ; θ)

= 1

(2πσ 2)N2/2

exp

(− 1

2σ 2(ik − Hko)t (ik − Hko)

), k = {1, 2},(16)

where N2 is the number of pixels in the image and the hyperparametervector θn reduces to σ 2. This stationary white Gaussian model is a reasonableapproximation for a bright and extended object (Earth or solar observations).

Object prior probability distribution Various methods have been pro-posed to introduce regularization on the object in the phase-diversity litera-ture. Some authors (Löfdahl and Scharmer, 1994; Lee et al., 1997a) use alow-pass filter. Terminating the iterations of the criterion before convergenceis also a (somewhat ad hoc) regularization strategy (Seldin and Paxman, 1994;Thelen et al., 1999b). A quadratic regularization model has been proposed(Vogel et al., 1998; Bucci et al., 1999). We choose the latter method, which iseasily interpretable in a Bayesian framework, as a Gaussian prior probabilitydistribution for the object. The general expression for such a prior is:

p(o; θo) = 1

(2π)N2/2 det(Ro)1/2

exp

(−1

2(o − om)tR−1

o (o − om)

), (17)

where om is the mean object and Ro its covariance matrix.

Phase prior probability distribution Concerning the aberrations, implicitregularization is achieved by expanding the phase on a finite linear combina-tion of basis functions. Usually the aberrated phase is expanded on a finite setof Zernike polynomials (see Noll, 1976, and Section I.B.4):

φ(r) =kmax∑k=4

akZk(r). (18)

Note that coefficients a1−3 have not been introduced as mentioned in Sec-tion I.B.4: the piston coefficient a1 is the average phase and has no influenceon the point spread function, and the tilt coefficients a2−3 introduce a shift inthe image that is of no importance for extended objects. In the following, wenote a = (a4, . . . , akmax)

t , the {kmax − 3}-dimensional vector gathering theaberration coefficients to be estimated. Additionally, in the case of imagingthrough turbulence, a statistical prior on the turbulent phase is available


according to the Kolmogorov model (Thelen et al., 1999b). It leads to aGaussian prior probability distribution for the aberrations, with a zero meanand a covariance matrix Ra given by Noll (1976):

p(φ(a); θφ

) = 1

(2π)(kmax−3)/2 det(Ra)1/2exp

(−1

2atR−1

a a

). (19)

The a priori information on the aberrations a is the covariance matrix Ra ,so that in this case, θφ = Ra . Note that in the particular case wherethe aberrations are only intrinsic (see Subsection I.B.2) and high-frequency(polishing) errors are negligible, a few Zernike coefficients are enough todescribe all the aberrations, regularization due to the truncated expansion ofthe phase is sufficient, and the a priori probability density function p(φ(a); θ)

can be omitted (leading to a GML estimator).

Criterion Under the above Gaussianity assumptions, we have:

p(i1, i2, o, a; θ)

= 1

(2π)N2/2 σN2 exp

[− 1

2σ 2(i1 − H1o)t (i1 − H1o)

]

× 1

(2π)N2/2 σN2 exp

[− 1

2σ 2(i2 − H2o)t (i2 − H2o)

]

× 1

(2π)N2/2 det(Ro)1/2

exp

[−1

2(o − om)tR−1

o (o − om)

]

× 1

(2π)(kmax−3)/2 det(Ra)1/2exp

(−1

2atR−1

a a

). (20)

The JMAP approach amounts to maximizing p(i1, i2, o, a; θ), which isequivalent to minimizing the criterion:

LJMAP(o, a, θ)

= − ln p(i1, i2, o, a; θ)

= N2 ln σ 2 + 1

2ln det(Ro) + 1

2ln det(Ra)

+ 1

2σ 2(i1 − H1o)t (i1 − H1o) + 1

2σ 2(i2 − H2o)t (i2 − H2o)

+ 1

2(o − om)tR−1

o (o − om) + 1

2atR−1

a a + A, (21)

where A is a constant.

20 MUGNIER ET AL.

Expression of o Canceling the derivative of LJMAP with respect to theobject gives (Gonsalves, 1982; Paxman et al., 1992) a closed-form expressionfor the object o(a, θ) that minimizes the criterion for given (a, θ):

o(a, θ) = R(Ht

1i1 + Ht2i2 + σ 2R−1

o om

), (22)

where R = (H t1H1+Ht

2H2+σ 2R−1o )−1. Substituting o(a, θ) into the criterion

of Eq. (21) yields a “new” criterion that does not explicitly depend on theobject:

L′JMAP(a, θ)

= LJMAP(o(a, θ), a, θ

)= N2 ln σ 2 + 1

2ln det(Ro) + 1

2ln det(Ra)

+ 1

2σ 2

(it

1i1 + it2i2

)

− 1

2σ 2

(it

1H1 + it2H2 + σ 2ot

mR−1o

)R

(Ht

1i1 + Ht2i2 + σ 2R−1

o om

)

+ 1

2atR−1

a a + A. (23)

The dimension of the parameter space over which the minimization ofthis new criterion is performed is dramatically reduced compared to theminimization of the criterion LJMAP(o, a, θ) since the N2 object parametershave been eliminated (Paxman et al., 1992). Note that there is no such closed-form expression of o with a Poisson noise model.

2. Circulant Approximation and Expression in Fourier Domain

H1 and H2 correspond to convolution operators, thus they are Toeplitz-block-Toeplitz (TBT) matrices. If (o − om) can be assumed stationary, thecovariance matrix Ro is also TBT. Such matrices can be approximated bycirculant-block-circulant matrices, with the approximation corresponding toa periodization (Hunt, 1973). Under this assumption, the covariance matrixRo and the convolution matrices H1 and H2 are diagonalized by the discreteFourier transform (DFT). We can write

Ro = F−1diag[So]F,

H1 = F−1diag[h1]F,

H2 = F−1diag[h2]F,

where F is the 2D DFT matrix, diag[x] denotes a diagonal matrix having x onits diagonal, tilde (˜) denotes the 2D DFT, and So is the object power spectral


density model. Thus the criterion LJMAP and the closed-form expressiono(a, θ) can be written in the discrete Fourier domain, leading to a fastercomputation:

LJMAP(o, a, θ) = N2 ln σ 2 + 1

2

∑v

ln So(v) + 1

2ln det(Ra)

+∑

v

1

2σ 2|ı1 − h1o|2 +

∑v

1

2σ 2|ı2 − h2o|2 (24)

+∑

v

|o − om|22So(v)

+ 1

2atR−1

a a + A

and ˆo(a, θ , v) = h∗1(a, v)ı1 + h∗

2(a, v)ı2 + σ 2om(v)So(v)

|h1(a, v)|2 + |h2(a, v)|2 + σ 2

So(v)

, (25)

where v is the spatial frequency. The expression in the Fourier space of thecriterion L′

JMAP(a, θ) is:

L′JMAP(a, θ)

= LJMAP(o(a, θ), a, θ

)= N2 ln σ 2 + 1

2

∑v

ln So(v)

+ 1

2

∑v

|ı1(v)h2(a, v) − ı2(v)h1(a, v)|2σ 2

(|h1(a, v)|2 + |h2(a, v)|2 + σ 2

So(v)

)

+ 1

2

∑v

|h1(a, v)om(v) − ı1(v)|2 + |h2(a, v)om(v) − ı2(v)|2So(v)

(|h1(a, v)|2 + |h2(a, v)|2 + σ 2

So(v)

)

+ 1

2ln det(Ra) + 1

2atR−1

a a + A. (26)

Note that the objective function first proposed by Gonsalves (1982) is onlycomposed of the first and third terms of criterion (26). L′

JMAP(a, θ) mustbe minimized with respect to the aberrations a. There is no closed-formexpression for a, so the minimization is done using an iterative method (seeSection VI for a description of minimization methods). Additionally, beforeminimizing the criterion, the value of the regularization parameters must bechosen.

3. Tuning of the Hyperparameters

Noise The noise model requires the tuning of the variance of the noise inthe image θn = {σ 2}. It can be estimated using the total flux in the image andthe previously calibrated electronic noise level of the camera.

22 MUGNIER ET AL.

Aberrations In the case of atmospheric turbulence, the a priori informationon the aberrations a is the covariance matrix Ra . Noll (1976) has shown thatthis matrix is completely defined by the ratio D/r0, where D is the diameterof the telescope and r0 is the Fried diameter (Fried, 1966b). The value of thelatter can be obtained by a seeing monitor, for example.

Object We choose the following model for So:

So(v) � E[∣∣o(v) − om(v)

∣∣2] = k/[v

po + vp

] − ∣∣om(v)∣∣2

, (27)

where E stands for the mathematical expectation. This heuristic modeland similar ones have been quite widely used (Kattnig and Primot, 1997;Conan et al., 1998). This model introduces four hyperparameters θo =(k, vo, p, om). The tuning of the hyperparameters of the object θo is noteasy; their optimum values depend on the structure of the object. Thus, theymust be estimated for each object. Unfortunately, in a joint estimation ofthe object and the aberrations, these hyperparameters θo cannot be jointlyestimated with o and a. Indeed, the criterion of Eq. (21) degenerates when,for example, one seeks θo together with o and a. In particular, for the pair{θo = (k = 0, vo, p, om), o = om}, which does not depend on the data,the criterion tends to minus infinity. So before minimizing LJMAP, thesehyperparameters must be chosen empirically by the user or estimated usinga sounder statistical device.

B. Marginal Estimator

In this method, the aberrations a and the hyperparameters (θn, θo) linked tothe noise and the object are first estimated. Then if the parameter of interest isthe object, it can be restored, in a second step, using the estimated aberrationsand hyperparameters (see Section V for a detailed explanation).

The marginal estimator restores the sole aberrations by integrating theobject out of the problem.5 It is a maximum a posteriori (MAP) estimatorfor a, obtained by integrating the joint probability density function:

aMAP = arg maxa

p(i1, i2, a; θ) = arg maxa

∫p(i1, i2, o, a; θ) do

= arg maxa

∫p(i1|a, o; θ)p(i2|a, o; θ)p(a; θ)p(o; θ) do. (28)

Let I = (i1i2)t denote the vector that concatenates the data. As a linear

combination of jointly Gaussian variables (o and n), I is a Gaussian vector.

5 In the vocabulary of probabilities, to integrate out (i.e., to marginalize) a quantity means tocompute a marginal probability law by summing over all possible values of the quantity.


Maximizing p(i1, i2, a; θ) = p(I , a; θ) is thus equivalent to minimizing thefollowing criterion:

LMAP(a, θ) = 1

2ln det(RI ) + 1

2(I − mI )

tR−1I (I − mI )

+ 1

2ln det(Ra) + 1

2atR−1

a a + B, (29)

where B is a constant, mI = (H1omH2om)t , and RI � E[II t ] − E[I ]E[I ]tis the covariance matrix of I .

1. Expression of R−1I

The expression of R−1I is obtained by the block matrix inversion lemma

(Gantmacher, 1966):

R−1I =

(Q11 Q12Q21 Q22

)(30)

with

Q11 = [(H1RoH

t1 + σ 2Id

) − H1RoHt2

(H2RoH

t2 + σ 2Id

)−1H2RoH

t1

]−1,

Q12 = −Q11(H1RoH

t2

)(H2RoH

t2 + σ 2Id

)−1,

Q21 = −(H2RoH

t2 + σ 2Id

)−1(H2RoH

t1

)Q11,

Q22 = [(H2RoH

t2 + σ 2Id

) − H2RoHt1

(H1RoH

t1 + σ 2Id

)−1H1RoH

t2

]−1.

(31)

2. Determinant of RI

Let Δ be a matrix that reads

Δ =[A B

C D

]

in a block form. Its determinant is given by det(Δ) = det(A) det(D −CA−1B). Using this formula, it is easy to calculate the determinant of RI :

det(RI ) = det(H1RoH

t1 + σ 2Id

)× det

[H2RoH

t2 + σ 2Id

− H2RoHt1

(H1RoH

t1 + σ 2Id

)−1H1RoH

t2

]. (32)

24 MUGNIER ET AL.

3. Marginal Criterion

Substituting the definition of mI and the expression of R−1I of Eq. (30) into

(I − mI )tR−1

I (I − mI ) yields the following expression:

(I − mI )tR−1

I (I − mI )

= (i1 − H1om)tQ11(i1 − H1om) + (i1 − H1om)tQ12(i2 − H2om)

+ (i2 − H2om)tQ21(i1 − H1om) + (i2 − H2om)tQ22(i2 − H2om).

Basic algebraic manipulations yield the following expression for the marginalcriterion:

LMAP(a, θ)

= 1

2ln det(RI ) + 1

2ln det(Ra)

+ 1

2σ 2

(it

1i1 + it2i2

)

− 1

2σ 2

(it

1H1 + it2H2 + σ 2ot

mR−1o

)R

(Ht

1i1 + Ht2i2 + σ 2R−1

o om

)

+ 1

2atR−1

a a + B, (33)

where R = (H t1H1 + Ht

2H2 + σ 2R−1o )−1.

4. Relationship Between the Joint and the Marginal Criteria

The comparison of the expression of the criterion LMAP [Eq. (33)] and ofthe criterion LJMAP [Eq. (23)] shows that the two criteria are related by thefollowing relationship:

LMAP(a, θ) = 1

2ln det(RI ) − N2 ln σ 2

− 1

2ln det Ro + L′

JMAP(a, θ) + C, (34)

where C is a constant. If we focus only on the terms depending on the phase(i.e., suppose that the hyperparameters are known), relationship (34) can besummarized by (Goussard et al., 1990):

LMAP(a) = 1

2ln det(RI ) + LJMAP(a) + C′, (35)

where C′ is a constant. Thus, the difference between the marginal and the jointestimator consists of a single additional term dependent on the phase, whichis ln det(RI ). Although the two estimators differ only by one term, it is shownin Section IV that their properties differ considerably.


5. Expression in the Fourier Domain

In practice, the marginal estimator is computed in the Fourier domain. Usingthe circulant approximations (see Subsection III.A.2) for H1, H2, and Ro andnoting that ln det(Ro) = ∑

v ln So(v), the term ln det(RI ) [Eq. (32)] can beexpressed as follows:

ln det(RI ) =∑

v

ln So(v) + N2 ln σ 2

+∑

v

ln

(∣∣h1(a, v)∣∣2 + ∣∣h2(a, v)

∣∣2 + σ 2

So(v)

). (36)

Combining Eqs. (26), (34), and (36) gives the marginal estimator in theFourier domain:

LMAP(a, θ)

=∑

v


+∑

v

ln

(∣∣h1(a, v)∣∣2 + ∣∣h2(a, v)

∣∣2 + σ 2

So(v)

)

+ 1

2

∑v

|ı1(v)h2(a, v) − ı2(v)h1(a, v)|2σ 2

(|h1(a, v)|2 + |h2(a, v)|2 + σ 2

So(v)

)

+ 1

2

∑v

|h1(a, v)om(v) − ı1(v)|2 + |h2(a, v)om(v) − ı2(v)|2So(v)

(|h1(a, v)|2 + |h2(a, v)|2 + σ 2

So(v)

)

+ 1

2ln det(Ra) + 1

2atR−1

a a + B. (37)

Let us see, now, how the hyperparameters can be estimated.

6. Unsupervised Estimation of the Hyperparameters

For the marginal estimator, the estimation of θo = (k, vo, p, om) and θn = σ 2

can be jointly tackled with the aberrations, according to:

(a, θo, θn) = arg maxa,θo,θn

p(i1, i2, a; θ). (38)

The criterion LMAP(a) of Eq. (29) becomes LMAP(a, σ 2, k, vo, p, om, θa).It must be minimized with respect to the aberrations a and the five hyper-parameters (σ 2, k, vo, p, om). If we adopt the change of variable μ =σ 2/k, the cancellation of the derivative of the criterion with respect to k

26 MUGNIER ET AL.

gives a closed-form expression k(a, μ, vo, p, om, θa), which minimizes thecriterion for given values of the other parameters. Injecting k into LMAP yieldsLMAP(a, μ, vo, p, om, θa). There is no closed-form expression for μ, vo, p,and om but it is easy to calculate the analytical expression of the gradients ofthe criterion with respect to these hyperparameters and then to use numericalmethods for the minimization of the criterion.

C. Extended Objects

The problem of edge effects must be addressed in order to process extendedscenes (Earth or solar observations). This problem is due to the fact that thejoint and the marginal criteria are expressed in the Fourier domain thanks toan approximation: the convolutions are made using FFTs. This introduces aperiodization that produces severe wraparound effects on extended objects. Tosolve this problem, two solutions have been proposed in the phase-diversityliterature: the apodization technique (Löfdahl and Scharmer, 1994) and theguard-band technique (Seldin and Paxman, 1994).

1. Apodization

To reduce the computing time required to minimize the joint and the marginalcriteria, they are computed in Fourier space and thus do not take intoaccount the effects of boundaries of the images. To reduce the edges effectswhile still computing o(a) with FFTs, the images can be apodized. Theentire apodization of the data by a Hanning window has been first used byPaxman and Crippen (1990) but led to poor results. Löfdahl and Scharmer(1994) suggested to apodize only the edges of the images by a modifiedHanning window (an example of a 1D Hanning window and of a modifiedHanning window are shown in Figure 3 for comparison). In this technique,the summation in the joint criterion expression [Eq. (24), line 2] is computedin the image space instead of the Fourier space (according to Parseval’stheorem). It allows keeping in the summation only data that have not beenapodized (i.e., the ones for which the apodization function is unity). Notethat this method can be easily adapted to the marginal criterion. This type ofapodization has been used already in speckle techniques (Von der Lühe, 1993)and works well with phase-diversity data. The advantage of this technique isthat it provides fast computation of the criterion. Its disadvantage, apart fromthe fact that it is approximate, is that a part of the data is apodized and is notused in the estimation.


FIGURE 3. A 1D Hanning window (dotted line) and a 1D modified Hanning window (solidline) for comparison.

2. Guard Band

Another way of tackling the edge effects has been proposed for the joint esti-mator by Seldin and Paxman (1994) and is given below for both estimators.

Joint estimator The technique consists first in acknowledging the fact thatobject pixels beyond the field of view of the image do influence the data dueto the convolution operator involved in the image formation [see Eq. (10)] andestimating the object value on the guard-band pixels (i.e., pixels beyond theeffective field of view) as well. The criterion is minimized numerically withrespect to the aberrations and the object (no fast solution for computing o).The guard-band width depends on the severity of the aberrations (i.e., on theeffective PSF support width). Second, in practice, the object and the PSF 2Darrays are immersed in arrays of size larger than the sum of their support inorder to be able to compute h � o exactly by means of FFTs.

Marginal estimator To apply the guard-band technique to the marginalestimator LMAP(a, θ), a new algorithm is used, called the “alternating”marginal estimator and noted Lalt

MAP(o, a, θ). The relationship between thejoint estimator and the marginal one [see Eq. (34)] can be summarized byLMAP(a, θ) = L′

JMAP(a, θ) + ε(a, θ). The alternating marginal criterion isthen defined by Lalt

MAP(o, a, θ) = LJMAP(o, a, θ) + ε(a, θ) and:

arg mino,a,θ

LaltMAP(o, a, θ) = arg min

a,θ

{arg min

oLalt

MAP(o, a, θ)}

= arg mina,θ

{arg min

o

[LJMAP(o, a, θ)

] + ε(a, θ)}

28 MUGNIER ET AL.

= arg mina,θ

[L′

JMAP(a, θ) + ε(a, θ)]

= arg mina,θ

LMAP(a, θ). (39)

The minimization of LaltMAP(o, a, θ) with respect to o, a, and θ is therefore

equivalent to the minimization of LMAP(a, θ) with respect to the sole a and θ .The guard band can then be applied to the criterion Lalt

MAP(o, a, θ).In the guard-band technique, the measured data are unperturbed but the

disadvantage of this method is the extensive computation time (due to theiterative estimation of the object).

IV. PROPERTIES OF THE PHASE ESTIMATION METHODS

This section studies the properties of the two phase estimation methodspresented in the previous section, by means of simulations. Their asymptoticproperties, the influence of the hyperparameters on the quality of the estimatedphase, and finally their performance are compared.

A. Image Simulation

The simulations have been obtained in the following way: our object isan Earth view. The aberrations are due to the imperfections of the opticalsystem. The phase is a linear combination of the first 21 Zernike polynomials,with coefficients listed in Table 1; the estimated phase is expanded on thesame polynomials. The defocus amplitude for the second observation planeis 2π radians, peak to valley. The simulated images are monochromatic andare sampled at the Shannon rate. They have been obtained by convolutionbetween the PSF and the object, computed in the Fourier domain using FFTs.The result is corrupted by a stationary white Gaussian noise (Figure 4). Theimages generated in this way are periodic. This is an artificial situation, under

TABLE 1VALUES OF THE COEFFICIENTS USED FOR SIMULATIONS

Coefficient a4 a5 a6 a7 a8 a9 a10 a11 a12

Value (rd) −0.2 0.3 −0.45 0.4 0.3 −0.25 0.35 0.2 0.1

Coefficient a13 a14 a15 a16 a17 a18 a19 a20 a21

Value (rd) 0.05 −0.05 0.05 0.02 0.01 −0.01 −0.02 0.01 0.01


(a) (b)

(c) (d)

FIGURE 4. Aberrated phase (a) of RMS value λ/7 and true object (b) used for the simulation.Simulated focused (c) and defocused (d) images.

which Ro is truly circulant block circulant. The fact that the images areperiodic allows estimation of the phase without the additional computing costof the guard-band technique.

B. Asymptotic Properties of the Two Estimators for Known Hyperparameters

For the time being, we consider that the hyperparameters are the “true” ones—we fit the PSD of the object using the true object, and we assume that σ 2 is

30 MUGNIER ET AL.

known (note that the mean object is set to zero). Additionally, as in all thefollowing, no regularization on the aberrations is introduced, save the factthat only the Zernike coefficients a4 to a21 are estimated. Consequently, themarginal estimation (which was based on a MAP approach) now correspondsto a maximum likelihood (ML) estimation. Similarly, the JMAP approachcorresponds to a GML estimation because the phase is not regularized (seeSubsection III.A.1). The minimized criteria will then be denoted LML andLGML, respectively. Figure 5 shows the bias, standard deviation, and rootmean square error (RMSE) on the phase provided by the joint method (left)and the marginal method (right) as a function of the noise level for three imagesizes (128 × 128, 64 × 64, and 32 × 32 pixels). These three quantities aredefined as:

• The empirical bias, b = ∑21k=4[〈ak〉 − atrue

k ]• The empirical standard deviation, σ = [∑21

k=4〈(ak − 〈ak〉)2〉]1/2

• The empirical RMSE, e = (b2 + σ 2)1/2

The empirical average is done on 50 different noise realizations. Furthermore,for the image sizes of 32 × 32 and 64 × 64 pixels, the quantities are averagedon all the subimages of 32 × 32 pixels (respectively 64 × 64) contained in theimage of 128 × 128 pixels. For joint estimation, the bias increases with thenoise level. Furthermore, processing a larger number of data is not favorable interms of bias. On the contrary, the standard deviation of the phase estimate is adecreasing function of the image size. Finally, the RMSE, which is dominatedby the bias term, does not decrease as the number of data increases. Thispathological behavior meets several statistical studies (Champagnat and Idier,1995; Little and Rubin, 1983): the estimate does not converge toward the truevalue as the size of the data set tends to infinity. An intuitive explanation ofthis phenomenon is that, if a larger image is used to estimate the aberrations,the size of the object, which is jointly reconstructed, increases also, so thatthe ratio of the number of unknowns to the number of data does not tendtoward zero. On the contrary, for marginal estimation, the ratio of unknownsto data tends toward zero because the number of unknowns stays the sameregardless of the size of the data set. In this case, the bias, standard deviation,and RMSE of the phase decrease when the number of data increases (seeFigure 5, right). Indeed, under broad conditions, the marginal estimator isexpected to converge, since it is a true ML estimator (Lehmann, 1983; DeCarvalho and Slock, 1997).

The curve of the joint estimation standard deviation presents an irregularityfor the noise level of 14% and an image size of 128 × 128 pixels. Thissurprising result can be interpreted by looking at the different phase estimatesobtained in this condition, which are shown Figure 6. The minimization ofthe joint criterion leads to two different sets of aberration coefficients. This


FIGURE 5. Bias, standard deviation, and RMSE of phase estimates as a function of noise levelgiven in percent (it is the ratio between the noise standard deviation and the mean flux per pixel). Leftfigures are for the joint estimator, right figures for the marginal one. The solid, dashed, and dottedlines, respectively, correspond to images of dimensions 128 × 128, 64 × 64, and 32 × 32 pixels. Allestimates were obtained as empirical averages on 50 independent noise realizations.

explains why the standard deviation reaches a much larger value for thissimulation condition. It also shows that the joint criterion presents localminima. Furthermore, we have empirically checked that it presents localminima regardless of the size of the data set, whereas such local minimaare not seen with the marginal criterion (see Figure 5, right). Indeed, the

32 MUGNIER ET AL.

FIGURE 6. The different aberration estimates obtained with the joint method for the image sizeof 128 × 128 pixels and a noise level of 14%. Dashed line: true aberrations; solid line: aberrationestimates.

marginal criterion tends to be asymptotically more and more regular. Thelatter observation is in agreement with the asymptotic Gaussianity of thelikelihood, which is expected under suitable statistical conditions.

C. Joint Estimation: Influence of the Hyperparameters

An important problem for the estimation of the aberrations (and the object)is the tuning of the hyperparameters. For the joint estimator, we have pointedout that they must be adjusted by hand. Particularly important is the globalhyperparameter, which we shall denote by μ and is the one that quantifiesthe trade-off between goodness of fit to the data and to the prior.6 Let usstudy its influence on the joint method. Figure 7 shows the RMSE on thephase estimates and on the object estimate as a function of the value of thishyperparameter (its true value is μ = 1). The RMSE on the object is definedas [∑r〈(o(r) − otrue(r))2〉]1/2/[∑r o(r)2]1/2. We see that the best value ofthis hyperparameter (i.e., the one that gives the lower error on the estimate)is not the same for the object and for the phase. It means that the objectand the phase cannot be jointly optimally restored. Note that the optimalhyperparameter value for the object coincides with the true value μ = 1.If the parameter of interest is the phase, the object must be underregularized

6 For the Gaussian prior used in this work, tuning μ is equivalent to tuning a scale factor in theobject PSD So.


(a)

(b)

FIGURE 7. Plots of RMSE for joint phase estimates (dashed line—see right vertical axis) andjoint object estimate (solid line—see left vertical axis) as a function of the value of the hyperparameterμ for an image size of 32 × 32 pixels. Figure (a) depicts a noise level of 14%; Figure (b) is 4%.

to have a better estimation of the aberrations. The behavior of RMSE on thephase strongly depends on the noise level. For a high noise level (14% here),there is an optimal basin. However, for lower noise levels (4% and below), anyvalue under 1, including a near null regularization (“near null regularization”means that the parameter μ is not equal to zero but to a small arbitraryconstant [10−16 in our case] to avoid numerical problems due to computerprecision), is almost optimal with respect to estimation of the aberrations,even though the jointly estimated object becomes of the poorest quality. Thisobservation sheds some light on the fact that, when the parameters of interestare the aberrations and when the noise level is low, estimation without objectregularization can be successfully used as the literature testifies (Thelen et al.,1999b; Meynadier et al., 1999; Seldin and Paxman, 2000; Carrara et al.,

34 MUGNIER ET AL.

FIGURE 8. Performance of the joint estimator. RMSE of the phase estimates as a function ofnoise level for a near null regularization, for three image sizes.

2000). This empirical observation has also led us to study the asymptoticbehavior of the joint estimator with near null regularization. Figure 8 showsthe results. In this case, when the number of data increases, the RMSE onthe aberrations estimates decreases. Although the number of data samplesover the number of unknowns is the same as that of the estimation with thetrue hyperparameters, the estimator behaves as if the object were not beingestimated. This surprising behavior of joint aberration estimates when theregularization parameter μ vanishes has been recently explained in Idier et al.(2005); this study has shown that the GML is a consistent phase estimator(i.e., it converges toward the true value as the number of data increases).

D. Marginal Estimation: Unsupervised Estimation

For the marginal estimator, we must show that the unsupervised estima-tion of the hyperparameters (i.e., when the hyperparameters are estimatedjointly with the aberrations) yields good aberration estimates. To this end,we compare the quality of the aberrations reconstruction obtained eitherby minimizing LML(a) with the true hyperparameters or by minimizingLML(a, μ, vo, p), for several image sizes. As shown in Figure 9, for low noiselevels, the unsupervised restoration is very good (the maximum difference isless than 5%). For 128 × 128 pixels, it is quite good (the maximum differenceis less than 15%) for any noise level. Only for 32 × 32 pixels and highnoise levels is the reconstruction seriously degraded because of the lack ofinformation contained in the noisy data.


FIGURE 9. Performance of the marginal estimator with the true hyperparameters (plus signs)and with an unsupervised estimation (diamonds), as measured by the RMSE of the phase estimates asa function of the noise level.

E. Performance Comparison

We present the performance comparison of the joint and the marginal estima-tors for phase estimation. In order to compare these estimators in a realisticway, we use the joint estimator with a near null regularization (which providesgood results for the estimation of the aberrations as seen in Subsection IV.C)and the unsupervised marginal estimator, described in Subsection III.B.6. Wecompare the RMSE of the phase estimates as a function of noise level for twoimage sizes (32 × 32 pixels and 128 × 128 pixels). The results for the twoestimators are plotted in Figure 10. We can see two different domains: whenthe signal to noise ratio (SNR) is high (noise level < 5%), the two estimatorsgive approximately the same results. At lower SNR (5% < noise level <

20%), marginal estimation is significantly better. Note that this result hasbeen checked on experimental data. The performance comparison dependson the studied object. A comprehensive comparison of the two methods hasbeen done in Blanc (2002). This study has shown that the marginal methodleads to better phase estimates for high noise levels. For low noise levels, thejoint estimator performs sometimes slightly better than the marginal one; thesignificance of the difference between the two phase estimates depends onthe observed object. These performance differences between several observedobjects are probably due to the Gaussian hypothesis on the object used in themarginal approach—see Subsection III.B.

36 MUGNIER ET AL.

FIGURE 10. RMSE of the aberrations estimates for the unsupervised marginal estimator(diamonds) and for the joint one with near null regularization (plus signs), as a function of the noiselevel.

F. Conclusion

The two latter sections have presented two methods for estimating the aberra-tions. The conventional estimator found in the literature, which is interpretableas a joint maximum a posteriori approach, is based on a joint estimation ofthe aberrated phase and the observed object. It has been shown, by means ofsimulations, that it has poor asymptotic properties unless it is underregularizedand that it does not allow an optimal joint estimation of the object and theaberrated phase. The joint estimator without an object regularization cannevertheless be successfully used to estimate the aberrations, thanks to thefact that the GML is a consistent phase estimator. The marginal estimator,which estimates the sole phase by maximum a posteriori, is obtained byintegrating the observed object out of the problem. This reduces drastically thenumber of unknowns, allows the unsupervised estimation of the regularizationparameters, and provides better asymptotic properties than the joint approach.Finally, the comparison of the quality of the phase restoration has shown thatthe marginal method leads to better phase estimates for high noise levels andthat the two estimators yield quite similar performance (sometimes slightlybetter for the joint one depending on the observed object) for low noise levels.

V. RESTORATION OF THE OBJECT

When phase diversity is used as a WFS, the estimation of the aberrations isthe unique goal. On the contrary, in the case of image restoration, the object is


(a) (b) (c)

FIGURE 11. Joint estimation: (a) True object. (b) and (c) Object restored by the joint method(with near null regularization used). The noise level is equal to 1% for (b). In this case, the RMSE isequal to 115%. Restoration (c) depicts a noise level of 14%; the RMSE is 1500%.

the parameter of interest. This section shows how the object can be estimatedby the joint and by the marginal approaches.

A. With the Joint Method

In the case of the joint method, the object is estimated jointly with theaberrations by JMAP. Section III.A.3 has shown that the hyperparametersof the object must be adjusted empirically by the user, which is not easy,especially with an extended object. Furthermore, we have seen in Subsec-tion III.A.3 that the joint method does not allow an optimally joint estimationof the object and of the aberrations. In particular, a near null regularization isfavorable for a good estimation of the aberrations. Let us now see the objectobtained with no regularization. Figure 11 shows the results of the restorationof the object for two noise levels (1% and 14%) and an image size of 64× 64 pixels (the first image is the true object). The image simulations areobtained in the same conditions as in Section IV.A. As shown, when thereis no regularization on the object, at the minimum of the joint criterion,the object estimate is completely buried in noise. In order to obtain a goodobject estimate, the object estimation needs to be regularized, which hasbeen done either by incorporating an explicit regularization term into thecriterion (see Section III.A.1) or by interrupting the iterative minimizationbefore the convergence (Strand, 1974; Thelen et al., 1999b). Yet, as mentionedabove, this is prejudicial to the phase estimation and thus does not lead to thebest object estimate. Additionally, it leads to the problem of hyperparametertuning.

38 MUGNIER ET AL.

B. With the Marginal Method

1. Principle

The marginal estimator has been obtained by integrating the observed objectout of the problem. It is based on a MAP approach and restores the soleaberrations. The previous section has shown that this estimator has goodasymptotic properties and allows the unsupervised estimation of the noisevariance and of the regularization parameters of the object. Furthermore, themarginal method provides a simple way to restore the object. The idea isto calculate o once the aberrations amarg and the hyperparameters θmarg areestimated by the marginal method. In particular, this estimation can be doneby MAP:

oMAP(amarg, θmarg) = arg maxo

f (o|i1, i2, amarg; θmarg)

= arg maxo

f (i1, i2, amarg|o; θmarg)f (o; θmarg). (40)

Maximizing f (i1, i2, amarg|o; θmarg)f (o; θmarg) is thus equivalent to mini-mizing the following criterion:

LMAP(o) ∝ ‖i1 − H1(amarg)o‖2

σ 2+ ‖i2 − H2(amarg)o‖2

σ 2

+ (o − om)tR−1o (o − om). (41)

The cancellation of the derivative of this criterion with respect to o gives thefollowing closed-form expression:

oMAP(amarg, θmarg) = R−1(Ht1i1 + Ht

2i2 + σ 2R−1o om

), (42)

where R = (H t1H1+Ht

2H2+σ 2R−1o ). The object oMAP is the bi-frame Wiener

filter associated with the aberration and hyperparameter estimates amarg andθmarg. Note that the restoration is given by a Wiener filter as with the jointestimator. But the two methods differ in their tuning of the regularizationparameters and in their estimation of the aberrations.

In the case of a compact object, the criterion LMAP(o) and the closed-form expression of oMAP can be written in the Fourier domain. For objectsmore extended than the image, the computation of the criterion requires theintroduction of a guard band and the minimization is done iteratively (seeSection III.C.2 for details).

To summarize, the object is restored in two steps: first the phase and thehyperparameters linked to the noise and the object (θn, θo) are estimated.Then the object is restored by a MAP approach. This approach benefits from


(a) (b) (c)

FIGURE 12. Marginal estimation. (a) True object. (b) and (c) Object restored with the marginalmethod. The noise level is equal to 1% for (b). In this case, the RMSE is equal to 22%. Restoration (c)is for a noise level of 14%; the RMSE is 38%.

the good properties of this estimator: oMAP(amarg, θmarg) converges toward

oMAP(atrue, θ true) as the size of the data increases.Additionally, although the restoration of the object is formally done in a

second step, in practice, it is in fact performed at each computation of themarginal criterion [see Eqs. (34) and (25)], so it is available at convergence.

2. Results

Figure 12 presents the restoration of the object obtained by this approach(same conditions as for the joint restoration). The RMSE on the object isequal to 22% for a noise level of 1% and 38% for a noise level of 14%. Thequality of the object estimates provided by this method is satisfactory. Thus,this method provides a robust and easy (although indirect) MAP estimation ofthe object.

3. Influence of the Hyperparameters

Section IV.C has shown that the JMAP estimation of the object and the phasedoes not allow optimal reconstruction of the two quantities. We now study theinfluence of the global hyperparameter on the RMSE of marginal estimates.By contrast with the joint estimator, Figure 13 indicates that here there isa unique optimal hyperparameter for both the object and the aberrations.We have drawn the curves only for the 4% noise level because the generalbehavior of this estimator is the same for all noise levels. The marginal methodis thus able to optimally restore the aberrations and the object.

40 MUGNIER ET AL.

FIGURE 13. Plots of the RMSE for the marginal phase estimate (dashed line; see right verticalaxis) and for the marginal object estimate (solid line; see left vertical axis) as a function of the valueof the hyperparameter μ, for an image size of 32 × 32 pixels, and a noise level of 4%.

C. With a “Hybrid” Method

1. Principle

For low noise levels, the joint method with no regularization on the objectprovides a very degraded object estimate but a good restoration of the aber-rations (sometimes even better than the marginal method—see Section IV.E).Additionally, the estimation of the aberrations by the joint method requiresless computation time than the restoration by the marginal method becausethe latter requires hyperparameter estimation. We thus propose to use the jointmethod to estimate the aberrations, then to estimate the sole hyperparametersfor these aberrations by the marginal method. This step is fast since onlyfew parameters remain to be estimated. At this point, the conditions arethe same as in the case of the marginal estimation: the aberrations and thehyperparameters are known. Then we estimate the object by a MAP approach.This method is called the “hybrid” method and consists in the three stepsdetailed below.

2. The Three Steps

• Estimation of the aberrationsThe estimate of the aberrations is obtained jointly with the object by a GMLapproach, that is, without a regularization of the object (see Section III.A.1and IV.B):

(o, a)GML = arg maxo,a

f (i1, i2|a; o, θ)f (a; θ). (43)

From this estimation, the sole aberrations aGML are kept.


• Estimation of the hyperparametersThe estimation of the hyperparameters relative to the noise and to the objectis done by a ML approach. The joint probability density function of theobject o, the images i1 and i2, and the aberrations a is marginalized withrespect to the object o, as done for the marginal estimator (except that herethe aberrations a are fixed to their values estimated by GML):

θ = arg maxθ

∫f (i1, i2, o, aGML; θ) do

= arg maxθ

f (i1, i2, aGML; θ). (44)

The expression of the associated criterion, as well as its implementationand computation, are identical to the ones used in Section III.B.6 for themarginal estimation of the hyperparameters and the aberrations. This step isvery fast due to the fact that the minimization is done with fixed aberrationsand only the (three) hyperparameters are estimated.

• Estimation of the MAP objectFrom the aberrations aGML and the hyperparameters θ estimated in the twoprevious steps, the object is restored by a MAP approach:

oMAP(aGML, θ ) = arg maxo

f (i1, i2, aGML|o; θ )f (o; θ ). (45)

This step is identical to the restoration of the object from the marginalestimator (see the previous subsection) except that here this step must beexplicitly computed.

3. Results

We present the results of the object estimation obtained by the hybrid methodin Figure 14 (the simulation conditions are always the same). The RMSEon the object is 22% for a noise level of 1% and 41% for a noise level of14%. The object estimates obtained by the joint method with no regularization(Section V.A) do not bear comparison with these estimates: here the qualityof the restoration is close to the one provided by the marginal method. Forthe noise level of 1%, the hybrid method and the marginal one yield nearlyidentical object estimates. For the lower SNR, the marginal method performsslightly better. This is not surprising: for high noise level, the quality of theaberration estimates obtained by the marginal method is better than the onegiven by the joint method (see Section IV.E).

42 MUGNIER ET AL.

(a) (b) (c)

FIGURE 14. Hybrid estimation: (a) True object. (b) and (c) Restored object with the marginalmethod. The noise level is equal to 1% for (b). In this case, the RMSE is equal to 22%. Restoration (c)is for a noise level of 14%; the RMSE is 41%.

D. Conclusion

We have shown that the joint method does not provide good object estimates.For high noise levels, the marginal method provides a simple and robust wayto estimate the object. For lower noise levels, we propose to use the goodphase estimate obtained by the joint method, in a novel hybrid method, leadingto a fast and satisfactory estimation of the object.

VI. OPTIMIZATION METHODS

A. Projection-Based Methods

As mentioned in Section I, phase-diversity techniques originate from theproblem of phase retrieval, where the observed object o is a point source andonly one focused image i is available. In such conditions, it is clear fromEqs. (1) and (11) that the observed image reads i = |FT−1(P ejφ)|2 + n.Neglecting the noise component and assuming that P is known (as theindicatrix of the pupil), the issue of recovering the pupil phase φ can berestated in a more basic way: how to recover the phase φ of a function ofknown modulus P , given the modulus ρ = i1/2 of its Fourier transform.

The most common approach for solving this problem is to alternatelyenforce the known moduli in the two domains, which leads to the Gerchberg–Saxton algorithm (Gerchberg and Saxton, 1972; see also Gerchberg, 1974;Papoulis, 1975; Fienup, 1982):

1. Given a current value of φ, compute g0 = FT−1(P ejφ).2. Replace the modulus of g0 by ρ, that is, compute g = ρ g0/|g0|.


3. Compute G = FT(g).4. Take the phase of G as the new current value of φ, and go back to Step 1.

The enforcement of the constraints can be mathematically interpreted asprojections onto two subsets S1 and S2 of the set of phase vectors φ. Atconvergence, the aim is to find a phase vector that belongs to S1 ∩ S2.

Variants of the same alternating procedure are also encountered, for in-stance, to tackle the problem where P is an unknown nonnegative function ofknown support (Fienup, 1982; Bauschke et al., 2002).

In the simplest phase-diversity problem, a couple of images of a pointsource are available according to Eqs. (12) and (13). Adapted versions of theGerchberg–Saxton algorithm have been proposed to cope with this situation(Misell, 1973; Baba and Mutoh, 2001).

In the general phase-diversity problem, the object is unknown, and theGerchberg–Saxton algorithm does not easily generalize (see Baba and Mutoh,1994, for an attempt). Instead, optimization procedures are more commonlyfound in the literature of phase diversity, either to minimize the plain least-square criterion of Eq. (14) or a penalized version of it [e.g., Eq. (21)].In the authors’ mind, least-square minimization techniques should not beconsidered as stopgap solutions when the Gerchberg–Saxton algorithm is notreadily usable. On the contrary, for several reasons, adopting the optimizationframework should rather be seen as highly recommendable to solve phase-diversity (or even phase retrieval) problems:

• The Gerchberg–Saxton algorithm will not converge if S1 ∩ S2 is an emptyset (i.e., if the data set is not feasible). This difficulty is shared by nearlyall projection techniques. Obviously, the nonfeasible case is far fromacademic. It is rather expected to correspond to practical situations, asfar as noisy measurements and approximated models must be handled. InCensor et al. (1983) and in Byrne (1997), modified projection techniquesare introduced to cope with nonfeasible, linear problems. Convergence isthen shown toward a least-square solution.

• In Fienup (1982), the Gerchberg–Saxton algorithm is tested against aleast-square approach based on a gradient-search method, on a phaseretrieval problem. The former approach is reported to converge very slowlycompared to the latter.

Finally, it is the authors’ view that much clarity is gained when anobjective function (such as a least-square criterion) is explicitly defined,first and foremost. Only then, an appropriate algorithm is to be selectedamong different families of optimization schemes, one of which is based onsuccessive projections. In the remaining part of this section, two other familiesare introduced: line-search methods and trust-region schemes. For the sake ofsimplicity, we mainly restrict our attention to the minimization of L = L′

JMAP

44 MUGNIER ET AL.

defined by Eq. (23) [or indifferently to L = LMAP defined by Eq. (33)], as afunction of aberration parameters a.

B. Line-Search Methods

Line-search minimization is by far the most commonly adopted frameworkin the context of phase diversity (see, e.g., Gonsalves, 1982; Fienup, 1982;Paxman et al., 1992; Blanc et al., 2003b).

Each iteration of a line-search method is twofold: first, a search directionp� is computed. Then, a stepsize α� is determined, which corresponds to howfar to move along p�. The resulting iteration reads:

a�+1 = a� + α�p�.

Important particular cases are obtained according to specific choices of thesearch direction.

1. Strategies of Search Direction

A steepest descent method is obtained when p� is chosen opposite to thecriterion gradient g� = ∇L(α�). Such a method is computationally simplebut rather slow to converge.

Conjugate gradient methods form a very useful extension of steepestdescent, since they usually converge much faster, for almost the same lowcomputational requirement. They were originally designed to minimize con-vex quadratic functions, but they also provide an efficient and popularapproach to general optimization problems. According to conjugate gradientmethods, each direction p� is built as a linear combination of −g� and theprevious direction p�−1:

p� = −g� + β�p�−1,

where β� is chosen to generate mutually conjugated search directions whenthe criterion is convex quadratic. For instance, the Fletcher–Reeves choice forthe conjugate directions corresponds to

β� = gt� g�

gt�−1g�−1

.

In the phase-diversity context, conjugate gradient methods are proposed inGonsalves (1982), Fienup (1982), Paxman et al. (1992), and Blanc et al.(2003b).

p� = −H−1� g� yields Newton’s method when H� is the Hessian ∇2L(α�).

The asymptotic convergence of Newton’s method is fast, but its behavior far


from the solution is hardly predictable, especially when L is not a convexcriterion. Moreover, it is more computationally demanding than the previousmethods, since each search direction p� is the solution of a linear system.

Quasi-Newton methods correspond to

p� = −B−1� g�, (46)

where B� is “well-chosen.” Typically, matrix B� results from a trade-off: (1) itis close to H� for large values of �; (2) it is a symmetric and structurallypositive definite, even for the first iterations (contrarily to the Hessian H�);(3) it is more easily invertible than H�. The most popular quasi-Newtonalgorithm is the BFGS method (for Broyden–Fletcher–Goldfarb–Shanno),where B−1

� is maintained and updated, rather than matrix B� itself. For large-scale problems, it is still too demanding to handle matrices B−1

� . Limited-memory BFGS directly approximates B−1

� g�, using information gatheredfrom the m earlier iterations, where m is usually a small number. For moredetails on BFGS and on limited-memory BFGS, see Nocedal and Wright(1999). Vogel et al. (1998) reports that the performance of BFGS appliedto the minimization of L = L′

JMAP is disappointing; Vogel (2000) resorts tolimited-memory BFGS.

2. Step Size Rules

Choosing a clever strategy of search direction is not a sufficient condition toobtain an efficient minimization algorithm. Modern analyzes of convergencecombine both the search direction and the choice of the step size α�. Thepresumably “ideal” step size rule α� = arg minα L(a� + α�p�) usually hasno closed-form expression. We are thus led to employ so-called inexact stepsize strategies (i.e., inexact line search strategies [Moré and Thuente, 1994]).To find even a local minimizer of f (α) = L(a� + α�p�) needs a certainamount of iterations of a univariate minimization method, depending onthe required precision. In this respect, a key result is brought by sufficientdecrease conditions such as Wolfe’s (Nocedal and Wright, 1999): inexact stepsize rules that satisfy such conditions, in conjunction with appropriate searchdirections, form the ingredients of algorithms that are both implementable andconverging.

C. Trust-Region Methods

Trust-region methods are based on a model function M� whose behavior nearthe current point a� is similar to that of the actual criterion L. The modelM� is usually a chosen convex quadratic. Then, the trust-region approachcorresponds to the following strategy:

46 MUGNIER ET AL.

1. Solve

pr�+1 = arg min

pM�(a� + p), (47)

where a� +p is restricted to belong to a trust region. The latter is typicallychosen as a ball of radius, say r .

2. If the candidate solution pr�+1 does not produce a sufficient decrease

L(a�) − L(a� + pr�+1), then the trust region is considered as too large:

the radius r is decreased, and the minimization step is resolved, as manytimes as necessary.

In practice, it may be computationally expensive to solve Eq. (47). Actually,it suffices to compute inexact solutions, provided that a condition of sufficientdecrease still holds.

The Levenberg–Marquardt method is often presented as a method of choiceto solve nonlinear least-square problems (Press et al., 1992). It is based onupdates of the form of Eq. (46), with B� = J t

� J� + λ�I , where:

• J t� J� is a positive semi-definite approximation of the Hessian.

• I is the identity matrix.• λ� is a parameter that must be adjusted along the iterations to ensure the

sufficient decrease of L.

The Levenberg–Marquardt method can be viewed as a pioneering trust-region method, where the trust-region is a ball of radius λ�, and M�(p) =‖J�p + g�‖2 (Nocedal and Wright, 1999).

In the context of phase diversity, Löfdahl and Scharmer (1994) propose asimplified form of the Levenberg–Marquardt method, where λ� = 0. Sucha choice does not ensure the theoretical convergence of {a�} for arbitraryinitial points. Vogel et al. (1998) also resort to a variant of the Levenberg–Marquardt method, and Luke et al. (2000) apply limited memory BFGS withtrust regions.

VII. APPLICATION OF PHASE DIVERSITY TO AN OPERATIONAL

SYSTEM: CALIBRATION OF NAOS-CONICA

A. Practical Implementation of Phase Diversity

1. Choice of the Defocus Distance

An appropriate choice of the defocus distance d is essential to optimizethe performance of the phase-diversity sensor. If it is too small, the imagesare almost identical (not enough diversity), which brings back the problems


associated with phase retrieval. If it is too large, the defocused image containsvirtually no information.

The RMS defocus coefficient ad4 [such that φd(r) = ad

4 Z4(r), see Eq. (13)]depends on d , on the wavelength λ, the telescope diameter D, and the focallength F through:

ad4 = πd

8√

3λ(F/D)2(in radian), (48)

φd(r) is quadratic, minimum for r = 0 and maximum for r = 1. Thecorresponding peak-to-valley optical path difference � is equal to

� = λ

2π× 2

√3ad

4 =√

3λad4

π= d

8(F/D)2. (49)

Some studies (Lee et al., 1999; Meynadier et al., 1999) have shown thatchoosing d such that � is approximately equal to λ provides accurate results.In fact, the “optimal” defocus distance depends on the object structure, on theamplitude of the aberrations, and on the SNR of the images. In practice, alarge domain around this value (typically, � ≈ λ ± λ/2) still provides goodresults. This result has been first obtained experimentally (Meynadier et al.,1999). It has been later confirmed in a simplified theoretical framework, usingthe expression of the Cramer–Rao lower bound for the variance of unbiasedestimators (Lee et al., 1999; Prasad, 2004).

2. Image Centering

The estimation of aberrations from experimental data requires the determi-nation of two parameters in addition to those describing the aberrated phase(Löfdahl and Scharmer, 1994). These parameters correspond to the relativealignment in x and y between the focused and the defocused images. Theycan be described by tip-tilt coefficients a2 and a3 in the Zernike basis and canbe estimated in the same way as those corresponding to the wave-front, exceptthat they relate only to the defocused image. This estimation of the alignmentparameters eliminates the need for subpixel alignment of the images, butthe images still need to be first recentered to an accuracy of about one ortwo pixels (e.g., by cross-correlation) because phases larger than ±π aredifficult to estimate by phase diversity and often induce local minima (seeSection VIII).

3. Spectral Bandwidth

The phase-diversity concept, as described so far, is a monochromatic WFS.Nevertheless, Meynadier et al. (1999) have shown that the use of broadband

48 MUGNIER ET AL.

filters does not significantly degrade the accuracy as long as �λ/λ is rea-sonable (typically �λ/λ ≤ 0.15). Note that the concept can be extendedto polychromatic images through the appropriate change in the data model(Seldin and Paxman, 2000).

B. Calibration of NAOS and CONICA Static Aberrations

1. The Instrument

The European Very Large Telescope (VLT) instrument NAOS-CONICA iscomposed of the AO (Roddier, 1999) system NAOS (Rousset et al., 1998)and of the high-resolution camera CONICA (Lenzen et al., 1998). It isaimed at providing very high-quality images on one (UT4-Yepun) of the 8-mtelescopes of the European Cerro Paranal observatory. Figure 15 presents thesimplified outline of such an AO instrument. It consists of two different opticalpaths separated by a beam-splitter: the imaging path and the AO path (havingin common the so-called “common path”). The imaging path is composed ofthe different filters and objectives of CONICA. The AO path consists of theWFS that estimates the optical phase delays of the real-time computer, whichapplies the appropriate corrections via the deformable mirror (DM). In orderto reach the ultimate performance of the instrument, it has been necessaryto calibrate the remaining aberrations induced by its optical components.Defects of the wave-front originating from any component within the AO loop(common and AO path) are seen by the AO WFS and thus corrected. Thisis not the case for a degradation of image quality induced by components

FIGURE 15. Simplified outline of the VLT instrument NAOS-CONICA.


outside the AO loop, in the imaging path (i.e., the beam-splitters and filtersand objectives of the camera). Phase diversity has been used to calibratethese unseen aberrations. More details about the calibration procedure canbe found in Blanc et al. (2003a) and Hartung et al. (2003). Because of thehuge number of observation modes of NAOS-CONICA, the calibration hasbeen split into several parts: NAOS dichroics (i.e., the beam-splitters) andCONICA (i.e., the different filters and objectives). The goal is to be able toassign the aberration contributions to the various optical components. Twodifferent procedures have been used to obtain the phase-diversity data (i.e.,the focused and defocused images). The first procedure has provided thecalibration of the optical path of CONICA, the second one, the aberrationsof NAOS dichroics.

2. Calibration of CONICA Stand-Alone

Phase diversity setup The estimation of CONICA stand-alone aberrations(objectives and filters) is obtained through the use of pinholes located atdifferently defocused positions on a wheel in the camera entrance focalplane. Figure 16 depicts the setup for the corresponding measurements.CONICA has as many as 40 possible filters and 7 different camera objectives.A calibration point source is slid in front of CONICA. The telescope pupilis simulated by a cold pupil placed inside CONICA. After rotating the wheelthat holds the pinholes, the focused and defocused images are recorded andthen the phase-diversity estimation is performed.

Example of CONICA aberration estimation The true value of the aber-rations is of course not available in the real world. A practical way to beconfident in the correctness of the estimated aberrations is to compare therecorded images to the ones reconstructed with these estimated aberrations.

Figure 17 shows an example of the comparison between experimentalimages and the reconstructed ones. Note that, here, the observed object is

FIGURE 16. Calibration of CONICA stand-alone: use of pinholes in the entrance focal plane.

50 MUGNIER ET AL.

FIGURE 17. Comparison between measured images and PSFs reconstructed from the estimatedaberrations. Left: focused plane; right: defocused plane (a log scale is adopted for each image).

close to a Dirac function. The phase estimation is obtained using the jointmethod with no regularization on the object. The phase estimate is expandedon the first 15 Zernike polynomials. The Strehl ratio (SR)7 computed from theaberrations’ estimate is equal to 87%. It compares nicely to the SR directlycomputed on the focal plane image, which is equal to 85%.

Calibration of the filters and objectives The aberrations estimated fromCONICA phase diversity data correspond to the contributions of a filter and acamera objective. Thanks to the fact that the camera objectives are achromaticand that there are many different filters with very different aberrations, thecontributions of the filters and of the objectives can be separated. First, thetotal aberrations for a given camera and a given filter i (1 ≤ i ≤ n) aremeasured. This operation is repeated for all the n filters. The camera objectivecontribution aCam is then estimated by taking the median value of these totalaberrations aCtot,filti for all different filters:

aCam = median(aCtot,filt1, aCtot,filt2, . . . , aCtot,filtn). (50)

Finally, filter aberrations are obtained as the difference between the measuredaberrations and the estimated camera aberrations:

afilti = aCtot,filti − aCam. (51)

Figure 18 shows the calibration results in the J–H band, for a specific cameraobjective (C50S) and several filters. The solid line corresponds to the medianrepresenting the camera objective aberrations.

7 The Strehl ratio is a common way to describe the quality of the point spread function. It is given bythe ratio of the measured to the theoretical, diffraction-limited, peak intensity in the image of a pointsource.


FIGURE 18. CONICA’s aberrations measured by several filters in the J and H bands withcamera objective C50S. The solid lines indicate the median value representing the aberrations of thecamera objective.

3. Calibration of the NAOS Dichroics

Phase-diversity setup The estimation of the NAOS dichroic aberrations isobtained through the use of the AO system. Note that NAOS has five differentdichroics. A focused image of a fiber source, located in the entrance focalplane of NAOS, is recorded in closed-loop in order to avoid the common-path aberrations from the optical train between the source and the dichroic.Then a known defocus is introduced on the DM with the AO loop still closedto record the defocused image. This approach gives the sum of aNCtot, theNAOS dichroic aberrations and the CONICA aberrations.

Calibration of NAOS dichroic aberrations The contribution of the NAOSdichroics adichro can be determined by subtracting the total CONICA in-strument aberrations aCtot (provided by the previous calibration of CONICAstand-alone) from the overall NAOS-CONICA instrument aberrations aNCtot:

adichro = aNCtot − aCtot. (52)

52 MUGNIER ET AL.

4. Closed-Loop Compensation

Thanks to the individual calibration of the different optical components,the correction coefficients are known for any possible configuration of theinstrument. The precompensation of these static aberrations can be done byusing NAOS, which is able to introduce known static aberrations on the DM inclosed loop. In this manner, the DM takes the shape needed for compensationof the static optical aberrations. To demonstrate the final gain in opticalquality, we compare the originally acquired images without correction forstatic aberrations with the images obtained after closed-loop compensation.Figure 19 shows two examples of closed-loop compensations in J-band and inK-band. We achieve a striking correction in J-band, visible with the naked eye

FIGURE 19. Comparison of the PSFs before and after closed-loop compensation. The SR ofeach PSF is indicated in percent.


between the images before and after correction. In K-band the noncorrectedimage is already very close to the diffraction limit, and the improvement isbarely visible. However, the computation of the SR (see Figure 19) showsthat even in K-band the performed correction is still significant.

C. Conclusion

These calibrations have shown that phase diversity is a simple and powerfulapproach to improve the overall optical performance of an AO system. Thisis of a great interest for future very high SR systems, in which an accurateestimation and correction of aberrations is essential to achieve the ultimateperformance and to reach the scientific goals (for instance, regarding exo-planet detection).

VIII. EMERGING METHODS: MEASUREMENT OF LARGE

ABERRATIONS

A. Problem Statement

Large-amplitude aberrations refer to the fact that variations higher than 2π

can be encountered. In such a situation, the data used in phase diversity (i.e.,focused and defocused images) do not contain the full information on theaberrated pupil phase. In fact, Eq. (1), recalled below, shows that the PSF isrelated to the pupil phase φ by a nonlinear relationship such that the phaseappears in a complex exponential:

hopt(x, y) = ∣∣FT−1(P(λu, λv) ejϕ(λu,λv))∣∣2

(x, y).

Any 2π -variation of any phase point in the pupil leaves the PSF unchangedand thus the image unchanged. Hence the phase-diversity data only containinformation on the wrapped phase φ[2π ]. The sole data (whatever theirnumber) do not allow discrimination between the true continuous phase andall the equivalent phases. Figure 20 shows a 1D example of two phasesthat yield the same image. In the joint criterion as in the marginal one,the term corresponding to the likelihood presents an infinite number ofequivalent local minima. Figure 21 illustrates this in a 1D representation:the unwrapped true phase corresponds to one of these minima, but if onlyusing the data, it is not differentiated from the others. Note that, unfortunately,the defocused image, which removes the indetermination of the sign of theeven part of the pupil phase (see Subsection I.C.1), has no effect on thisindetermination. In practice, this indetermination appears for any strongly

54 MUGNIER ET AL.

FIGURE 20. The pupil phase in solid line and the one in dashed line yield the same PSF.

FIGURE 21. Symbolic 1D representation of the likelihood term.

degraded system (i.e., for phase amplitudes greater than 2π ). It is the case,for example, for astronomical observations from the ground where images arestrongly degraded by the atmospheric turbulence. This indetermination doesnot affect a posteriori correction. Indeed, for this application, the object is theparameter of interest; thus it does not bear the indetermination provided thatthe wrapped phase φ[2π ] is correctly estimated. By contrast, real-time wave-front correction requires the estimation of the unwrapped phase8 and thus the2π -indetermination must be removed. Regularity criteria are usually invokedto perform phase unwrapping. However, the latter is a difficult task, both fromthe informational viewpoint (only poor information is available) and from thecomputational viewpoint (the criterion to optimize is highly multimodal).

8 Especially if the scientific imaging is not performed at the same wavelength as the wave-frontsensing or if it is polychromatic.


B. Large Aberration Estimation Methods

In the phase-diversity literature, two different ways have been proposed toestimate large aberrations. The first one is based on a single-step estimationof the unwrapped phase; the second one is composed by two steps: first, thewrapped phase is estimated, then, if necessary, the phase is unwrapped.

1. Estimation of the Unwrapped Phase

This method consists in simultaneously estimating and unwrapping the phase.It requires introducing information in the estimation process in addition tothose contained in the data, in order to remove the 2π -indetermination andto construct a criterion that presents (if possible) a unique global minimumcorresponding to the unwrapped phase. This information can be brought byan appropriate choice of the basis function for the expansion of the phaseand/or by the use of an explicit regularization term on the phase parameters.In the phase-diversity literature, two different basis have been used to expandthe phase.

Zernike polynomials In all previous sections and as often done in thephase-diversity literature, the phase has been expanded on the Zernike modalbasis (see Subsection I.B.4). A parameterization of the phase on an infinitenumber of Zernike polynomials leaves the likelihood criterion unchanged(Figure 21). But in practice, the phase is expanded on a finite number ofpolynomials. In this case, the minima of the criterion are no more equivalentand a unique global minimum corresponds to the unwrapped phase. Evenif this decomposition seems to remove the ambiguity, in practice all theminima are very close to each other (see Figure 22), and it is almostimpossible to reach the global minimum (corresponding to the unwrappedphase). In order to better differentiate between the global minimum and theseothers, it is essential to add more information on the phase by introducing

FIGURE 22. 1D representation of the likelihood term for the pupil phase expanded on atruncated Zernike decomposition.

56 MUGNIER ET AL.

a regularization term in the criterion. For imaging through turbulence, onecan use the statistical knowledge on the atmospheric turbulence and introducethe a priori probability distribution presented in Subsection III.A.1, Eq. (19).This simultaneous estimation and unwrapping of the phase has been chosenby Thelen et al. (1999b).

Delta functions Another possible basis for the phase expansion is the deltafunctions: the phase is estimated point by point in the pupil (i.e., at eachsampled point in the pupil plane). By contrast with the truncated Zernikebasis, this expansion of the phase leaves the indetermination unchanged, i.e.,the likelihood term keeps the shape of Figure 21. Thus, in order to reach theunwrapped phase solution, one must introduce a priori information on thephase to “distort” the criterion and remove the equivalence between all 2π -variation phase estimates. The solution proposed by Jefferies et al. (2002) is toparameterize the phase by a function convolved with a smoothing kernel. Thisis a mean to impose smoothness on the estimated phase, and it has the sameunwanted effect as using a truncated Zernike basis: it distorts the criterion intosomething like Figure 22.

To summarize, for the Zernike modal regularization approach as well asfor the point-by-point one, the regularized criterion presents many localminima. The global minimum corresponding to the searched unwrappedphase is unique but very close to the other local solutions. The success ofthe minimization of such criteria by the local minimization method (seeSection VI) is very uncertain and strongly depends on the starting point ofthe iterative minimization (which should be very close to the unwrappedsolution). Thus, the simultaneous estimation and unwrapping of the phaseis very difficult. This is due to the fact that each process (estimation andunwrapping) is, by itself, a difficult problem.

2. Estimation of the Wrapped Phase (Then Unwrapping)

An alternative approach is to keep the problem separated in two steps: first,estimate the wrapped phase φ[2π ], then, if necessary, unwrap the phase toobtain φ. This approach is guided by, on one hand, the above-mentioneddifficulties to solve the two problems simultaneously, and, on the other hand,by noting that for some applications, the estimation of the sole wrapped phaseφ[2π ] is enough (as for postcorrection). In the phase-diversity literature,Löfdahl et al. (1998b) and Baba and Mutoh (2001) have chosen this two-step method. Their application was the estimation of static aberrations of atelescope by observing an unresolved point source (a star) through turbulence.We have also developed a phase estimation method based on this two-stepapproach. It is briefly presented in the following. More details can be foundin Blanc (2002).


Estimation of φ[2π ]: Construction of a regularized criterion In thisapproach, we first acknowledge the fact that it is almost impossible to find theglobal minimum of a criterion that has many almost equivalent minima. Thus,to estimate φ[2π ] without the minimization problems associated with localminima, all minima corresponding to a 2π -variation of the phase must remainstrictly equivalent: the criterion has to keep the form of Figure 21. Concerningthe likelihood term, it requires the choice of an appropriate basis for the phaseexpansion. As we indicate in Subsection VIII.B.1, the delta functions keep theequivalence. Hence, the phase is expanded on this basis. The implementationof the point-by-point phase estimation leads to a very large parameter space(compared to a truncated Zernike basis) and thus requires the use of an explicitregularization term on the phase. In order to keep the equivalence between alllocal minima, this term must also remain the indetermination.

• Choice of the phase regularization functionThe goal of the a priori information on the phase is to ensure a goodsmoothing of the small gradients (i.e., corresponding to noise) but to beinsensitive to the large gradients (corresponding to 2π -variations). Wechoose the following expression for the phase regularization function:

Jregul(φ) =∑

(l,m)∈S

[∣∣ej (φl−1,m−φl,m) − ej (φl,m−φl+1,m)∣∣2

+ ∣∣ej (φl,m−1−φl,m) − ej (φl,m−φl,m+1)∣∣2]

. (53)

The summation is done on all the pixels within the pupil (S is the pupilsupport). Furthermore, we impose a strict support constraint, that is, allterms | . . . |2 that contain, at least, a pixel out of the pupil support aresuppressed.

To ensure that the regularization function is insensitive to any 2π -variation of the phase, the a priori information has been imposed on thephasors (i.e., ejφ). Note that, even if the regularization term involves thephasors, the estimated parameters are still the phase values on pixels. Inorder to keep all the indeterminations of the global criterion, the regular-ization function has been constructed in such a way that it is insensitive, asthe data are, to a global piston and to tip-tilt (see Sections I.B.4 and III.A.1).

A first-order expansion of Eq. (53) provides insight on the effect of theregularization term on the phase. For small phase differences, Eq. (53) isapproximately given by:

Jregul(φ) �∑

(l,m)∈S

[|φl−1,m − 2φl,m + φl+1,m|2

+ |φl,m−1 − 2φl,m + φl,m+1|2].

58 MUGNIER ET AL.

It corresponds to a quadratic penalization on the second-order differencesof the phase values.

• MAP estimationWe propose a MAP approach for the point-by-point phase estimation:

φMAP = arg minφ

JMAP(φ). (54)

The expression of φMAP is obtained by using Eq. (37), where the regular-ization term on the Zernike coefficients atR−1

a a is replaced by Eq. (53).Eq. (37) becomes:

LMAP(φ, θ)

=∑

v


+∑

v

ln

(∣∣h1(φ, v)∣∣2 + ∣∣h2(φ, v)

∣∣2 + σ 2

So(v)

)

+ 1

2

∑v

|ı1(v)h2(φ, v) − ı2(v)h1(φ, v)|2σ 2

(|h1(φ, v)|2 + |h2(φ, v)|2 + σ 2

So(v)

)

+ 1

2

∑v

|h1(φ, v)om(v) − ı1(v)|2 + |h2(φ, v)om(v) − ı2(v)|2So(v)

(|h1(φ, v)|2 + |h2(φ, v)|2 + σ 2

So(v)

)+ γ

∑(l,m)∈S

[∣∣ej (φl−1,m−φl,m) − ej (φl,m−φl+1,m)∣∣2

+ ∣∣ej (φl,m−1−φl,m) − ej (φl,m−φl,m+1)∣∣2]

, (55)

where γ is the hyperparameter that quantifies the trade-off between good-ness of fit to the data and to the prior. There is no closed-form expressionfor φ that minimizes the criterion LMAP(φ, θ), so the minimization has tobe done using an iterative method.

• Tuning of the hyperparameters— Noise and object: As in Subsection III.B.6, the hyperparameters linked

to the noise and the object are estimated jointly with the phase parame-ters. We recall that:

(φ, θo, θb) = arg minφ,θo,θb

JMAP(φ, θo, θb, θφ). (56)

— Phase: The value of γ has to be adjusted by hand. In fact, γ cannot bejointly estimated with φ (the solution γ = 0 corresponding to a nullregularization minimizes the criterion taken as a function of γ for anygiven value of the phase).


Phase-unwrapping The preceding estimation procedure produces an esti-mate of the wrapped phase. For applications requiring the phase itself, it isnecessary, in a second step, to calculate the phase from the wrapped phase. Inthis second step, a phase-unwrapping method is used. The phase-unwrappingproblem is found in a variety of applications and several phase-unwrappingmethods have been proposed in the literature (see, e.g., Ghiglia and Pritt,1998). We will not develop these methods here. In the following, we willfocus on the wrapped-phase estimation, which is specific to phase diversitycontrary to the phase-unwrapping problem.

C. Simulation Results

We will show simulation results obtained with the MAP estimator onturbulence-induced aberrations.

1. Choice of an Error Metric

Because we focus on the estimation of the wrapped phase φ[2π ], we havedeveloped a metric that quantifies the quality of the phase estimate within theinterval [−π, π ].

ε =√√√√ 1

A∑

(l,m)∈S

c(l, m)2 (57)

with c(l, m) =√∣∣ej (φtrue

l,m −φestimatedl,m ) − ⟨

ej (φtrue

l,m −φestimatedl,m )

⟩∣∣2, (58)

where A is the number of pixels in the pupil. c is the 2D error map thatindicates the difference between the true phase and its estimate at each pupilpixel. Note that this error metric is insensitive, as the data are, to any 2π -variation of the phase and to piston terms. A first-order expansion of Eq. (57)shows that, for small errors on the phase, ε corresponds to the standarddeviation of the residual phase.

2. Results

Data generation Twenty turbulent wave-fronts are obtained using a modalmethod (Roddier, 1990): the phase is expanded on the Zernike polynomialbasis and given Kolmogorov statistics. The strength of the turbulence is fixedby the ratio D/ro of the telescope diameter to the Fried parameter (seeSubsection I.B.3). For each of these turbulent phases, we compute a Shannon-sampled image. The image noise is a uniform Gaussian noise of 1%. Thedefocus amplitude between the two images is set to 2π radians, peak to valley.

60 MUGNIER ET AL.

FIGURE 23. Example of a turbulent phase estimate (D/ro = 30) from a point source object.The 2D error map representation has been multiplied by 10 compared to the others. The error term ε

is equal to λ/28.

Point source We first use a point source object. The strength of theturbulence is set to D/ro = 30 (this corresponds to very strong turbulenceconditions). The theoretical spatial standard deviation of the phase is thenequal to 2.7λ (Noll, 1976), which corresponds to a peak-to-valley amplitudeof 30π radians. Despite this high value, this case is a favorable situationbecause the object is known and the noise level is low. Accurate resultsare then obtained, even without regularization on the phase (i.e., the point-by-point phase is estimated using the maximum likelihood approach). Theaverage error ε on the 20 turbulent wave-fronts is equal to λ/30. An exampleof a phase estimate is shown Figure 23 (middle), to be compared to the truephase (left). The comparison of these two phases by visual inspection doesnot give a correct idea of the quality of the estimation. This is due to the factthat the phase estimate presents several 2π -jumps, whereas the true phase isunwrapped. The good quality of the estimate can be seen through the 2D errormap c (Figure 23, right) or in Figure 24, which shows the true phase and theestimated one both wrapped within [−π, π ].

Extended object The discrete extended object used here is a spiral galaxy(Figure 25). Three different strengths of turbulence are studied: low(D/ro = 4), medium (D/ro = 6), and strong turbulence levels (D/ro = 8).Phase estimates are obtained using the MAP approach of Eq. (55).

• Low turbulence level The strength of turbulence is set to D/ro = 4,corresponding to a peak-to-valley amplitude of 1.2π radians. The point-by-point phase estimates have been obtained using a focused image anda defocused one and the starting phase estimate of the iterative criterionminimization is zero. The average error ε on the 20 phase estimates isequal to λ/60. Figure 26 (middle) shows an example of a phase estimate for


FIGURE 24. Example of a turbulent phase estimate (D/ro = 30) from a point source object:left, the true phase within [−π, π ] and right, the phase estimate within [−π, π ].

FIGURE 25. Extended object used for simulations.

D/ro = 4, compared to the true phase (left). The right side of this figuredepicts the (10 times magnified) 2D error map c.

• Medium turbulence level The strength of turbulence is now set toD/ro = 6, corresponding to a peak-to-valley amplitude of 1.5π radians.The quality of the point-by-point phase estimates obtained with a focusedand a defocused image is poor (ε = λ/15). This is probably due tothe presence of local minima in the MAP criterion. Obviously, althoughwe have constructed a criterion such that all minima corresponding to a2π -variation of the phase are equivalent, there are other reasons for thepresence of local minima such as the nonlinear relationship between thephase and the image [see Eq. (1)]. To improve the estimation, we haveused an additional defocused image (with a defocus from the focused plane

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FIGURE 26. Example of a turbulent phase estimate (D/ro = 4) from the galaxy object. The2D error map representation has been multiplied by 10 compared to the others. The error term is equalto λ/60.

equal to 4π ). The reconstruction clearly shows an improvement when thethree images are used: the mean error ε becomes λ/40.

• Strong turbulence level Finally, we consider stronger conditions of tur-bulence with D/ro = 8, corresponding to a peak-to-valley amplitude of2π radians. The use of three images leads to a poor quality of estimation(i.e., the mean error ε is equal to λ/17). We propose, as suggested byJefferies et al. (2002), to improve the estimation by using a better startingphase estimate point for the criterion minimization. An initial estimate oflow-order Zernike polynomial coefficients (that exhibit greater power) isperformed: the phase is expanded on the first Zernike coefficients a4 to a6.After estimation of these few Zernike coefficients, the estimation of thepoint-by-point phase parameters is started using the corresponding phase asthe starting estimate for the minimization of the MAP criterion. The use ofthis starting point for the minimization makes the estimation more precise:the average error ε obtained using three images becomes λ/45. This startingpoint is closer to a global minimum than the null phase initially used.Figure 27 (middle) shows an example of a phase estimate for D/ro = 8,compared to the true phase (left). On the right side of this figure, the 2Derror map c is depicted.

To summarize, we have shown that the estimation of large aberrations from aknown object and for a low noise level does not require the use of an explicitregularization term on the phase. For extended objects, the use of the twoconventional images of the phase diversity perform well for low turbulencelevels. For medium turbulence levels (here D/ro = 6), the quality of thephase estimate is enhanced through the use of an additional defocused image.Finally, for higher turbulence cases, the accuracy is maintained if the point-by-point phase estimation is done with three images and is preceded by a first


FIGURE 27. Example of a turbulent phase estimate (D/ro = 8) from the galaxy object. The2D error map representation has been multiplied by 10 compared to the others. Three images havebeen used, and the initial phase estimate is obtained using a first estimation of the Zernike coefficientsa4 to a6. The error term is equal to λ/45.

modal estimation of the low spatial frequencies of the aberrated phase. Themeasurement of large aberrations by use of phase diversity is a problem thathas only been recently addressed. The results of the works referenced in thesepages together with the ones presented here are quite promising and openthe way to making phase diversity a practical wave-front sensor for adaptiveoptics.

IX. EMERGING APPLICATIONS: COPHASING OF

MULTIAPERTURE TELESCOPES

A. Background

The resolution of a telescope is ultimately limited by its aperture diameter.The latter is limited by current technology to about 10 m for ground-basedtelescopes and to a few meters for space-based telescopes because of volumeand mass considerations. Interferometry allows going beyond this limit; itconsists in making an array of subapertures interfere; the resulting instrumentis called an interferometer or a multiaperture telescope. So far, this techniquehas been used solely on ground-based instruments. The subapertures caneither be telescopes per se, as in astronomy (e.g., VLT interferometer, Navyprototype optical interferometer) or segments of a common primary mirror. Ifthese segments are adjacent, such as in the Keck telescope, the instrument isreferred to as a segmented telescope rather than an interferometer, even thoughit is conceptually one. Regarding high-resolution space-borne missions, in-terferometers are forecast in astronomy (with a segmented aperture for the

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James Webb Space Telescope and a diluted aperture for the Darwin9 mission,for instance) and can also be considered for Earth observation (Mugnier et al.,2004).

For a correct performance, the aperture of such an instrument must bephased to within a small fraction of the wavelength. A critical subsystem ofinterferometers is thus the cophasing sensor (CS), whose goal is to measurethe relative positioning (differential piston and tip-tilt) of the subapertures,which are the main sources of wave-front degradations, and possibly thehigher-order aberrations on each sub-aperture.

Differential piston and tip-tilt measurement has been studied extensivelyand demonstrated for distant ground-based telescopes. Most of the proposeddevices are based on a pupil-plane combination of the light coming froma given pair of subapertures. Because of their pupil-plane combination, thecontrast of interference fringes decreases quickly as the object extensionincreases, which makes these devices useless on very extended scenes suchas the Earth viewed from Space. Because of the pairwise light combination,these devices become impractical for instruments made of many subapertures.

The fact that phase diversity can be used as a CS on a segmented aperturetelescope was recognized very early (Paxman and Fienup, 1988). Addition-ally, in contrast with the above-mentioned devices, phase diversity enjoys twoappealing characteristics: first, it is appropriate for an instrument with a largenumber of subapertures, because the complexity of the hardware does notscale with the number of subapertures and remains essentially independent ofit. Second, it can be used on very extended objects.

This first property and the absence of noncommon-path aberrations (see,e.g., Subsection VII.B) are two strong motivations for the choice of phasediversity as a CS, even when looking at an unresolved source. Phase-diversityexperiments as a CS on a point source have been performed on the ground,notably to cophase the segments of the Keck telescope (Löfdahl et al., 1998b).Regarding space instruments, phase diversity has long been foreseen as acandidate of choice for the JWST segmented-aperture telescope (see, e.g.,Redding, 1998; Carrara et al., 2000; Lee et al., 2003), and has recently beenselected as the CS for the Darwin interferometer (Cassaing et al., 2003;Mocoeur et al., 2005).

In 1994 (Kendrick et al., 1994a, 1994b), phase diversity was experimentallyvalidated as a CS on an extended source for correcting static aberrations in realtime with a segmented mirror. Quite recently, the loop was also closed with aphase-diversity CS to correct static aberrations in the difficult framework of

9 Darwin is a forecast European Space Agency mission whose aim is to find and characterizeEarth-like planets. The instrument will be a so-called nulling interferometer, which cancels the lightcoming from the star in order to detect the planet—see, e.g., http://www.esa.int/science/darwin formore information.

http://www.esa.int/science/darwin


a six-telescope imaging interferometer using broadband light (Zarifis et al.,1999; Seldin and Paxman, 2000).

B. Experimental Results on an Extended Scene

As mentioned in the previous subsection, phase diversity can be used withextended scenes. Actually, for applications such as Earth observation fromSpace, the scenes extend beyond the field of view recorded by the image, andphase diversity is one of the very few possible CSs (Mugnier et al., 2004).We have designed, built, and validated a prototype phase-diversity CS for thisapplication. After a short presentation of this prototype and its testbed, wepresent its latest results.10 A more comprehensive presentation of the testbedalong with earlier results can be found in Sorrente et al. (2004) and Baron(2005).

A schematic view of the testbed, called BRISE (Banc Reconfigurabled’Imagerie sur Scènes Etendues) is shown on Figure 28. BRISE is mainly

FIGURE 28. Schematic view of the BRISE testbed and photograph of the deformable mirror(DEF), courtesy F. Cassaing. EXT: extended object; REF: reference point-source; ARC: arc lamp usedfor the illumination of EXT.

10 These experimental results are courtesy of I. Mocœur, after preliminary results by F. Baron.F. Cassaing is gratefully acknowledged for being the master architect of this prototype and testbedand B. Sorrente for overseeing their realization.

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composed of four modules (source, perturbation, detection, and control),described below.

The source module delivers two objects: an extended scene, which is anEarth scene on a high-resolution photographic plate illuminated by an arclamp, and a reference point source, which is the output of a monomode fiberfed with a He–Ne laser.

The perturbation module has three functions: it images the source on thedetector, defines the aperture configuration, and introduces calibrated aberra-tions; its main component is the DM, which performs the latter function. Inorder to introduce only piston and tip-tilt, we have chosen to manufacture aspecific segmented DM consisting of three planar mirrors mounted on piezo-actuated platforms by Physik Instrument, which have exactly these threedegrees of freedom.

The detection module is a water-cooled CCD camera that simultaneouslyrecords a focal-plane image and a defocused image of each of the two objectsto implement a phase-diversity CS. Figure 29 shows an experimental exampleof such an image. The control module drives the experiment.

Special care has been given to the control of errors that could limit CSperformance or the evaluation of the CS performance on extended objects.In particular, the two objects are observed simultaneously through very closepaths, to minimize the differential effects of field aberrations, vibrations, or

FIGURE 29. Focused (left) and defocused (right) experimental images of the extended scene(bottom) and reference point-source (top) objects. These images are recorded simultaneously ondifferent parts of the same detector and used for phase diversity.


air turbulence. A very accurate aberration calibration can thus be achievedthanks to the high SNR of the measurement obtained on the reference pointsource.

Figure 30 presents the piston measured at high photon level on a givensubaperture as a function of the piston effectively introduced by the DM,for the reference point source at λr = 633 nm and for the extended scene,illuminated with white light and a spectral filter of width 40 nm centeredaround λe = 650 nm. For each introduced piston, three measurementsare performed and reported on this figure. The point-source measurementsexhibit an excellent linearity between roughly −λr/2 and +λr/2, at whichpoints the expected modulo 2π -wrapping occurs. With the extended object,the curve is linear on a slightly smaller piston range. Some features aredifferent on this curve with respect to the one obtained with the referencepoint: the slope is not exactly unity, although this would not be a majorproblem in closed loop, and the sort of smooth wraparound that occurs around+λe/2 is somewhat surprising and currently interpreted as a consequence ofthe spectral bandwidth. Figure 31 shows the repeatability obtained on thepiston measurement with the extended object. The standard deviation of theestimated piston is, as expected, dominated by detector noise for low fluxesand then inversely proportional to the square root of the number of photons

FIGURE 30. Piston measured at high photon level on the first subaperture, as a function of thepiston effectively introduced by the DM.

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FIGURE 31. Repeatability obtained on the measurement of the piston on the first sub-aperturewith the extended object, as a function of the average photon level per pixel.

per pixel (photon-noise regime). It is, for instance, below 1 nm as soon as theaverage flux is above 1000 photo-electrons per pixel.

C. Conclusion

Both the literature cited in this section and the quantitative results presented init testify that phase diversity can be successfully used as a CS on segmented-aperture telescopes and on interferometers, for point sources as well asfor extended objects. Some challenges remain to be met before this use iswidespread. They are essentially the same as for single-aperture telescopewave-front sensing: the ability to sense large-amplitude aberrations and thereduction of the computing cost for use in real-time applications, be it thecompensation of atmospheric turbulence for ground-based instruments or thecompensation of environmental perturbations for space-based instruments.

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[Advances in Imaging and Electron Physics] Volume 141 || Phase Diversity: A Technique for Wave-Front Sensing and for Diffraction-Limited Imaging