Issue 
A&A
Volume 551, March 2013



Article Number  A138  
Number of page(s)  14  
Section  Astronomical instrumentation  
DOI  https://doi.org/10.1051/00046361/201220318  
Published online  08 March 2013 
Simultaneous exoplanet detection and instrument aberration retrieval in multispectral coronagraphic imaging
^{1} ONERA – The French Aerospace Lab, 92322 Châtillon, France
email: marie.ygouf@obs.ujfgrenoble.fr
^{2} UJFGrenoble 1/CNRSINSU, Institut de Planétologie et d’Astrophysique de Grenoble (IPAG) UMR 5274, 38041 Grenoble, France
Received: 31 August 2012
Accepted: 16 November 2012
Context. Highcontrast imaging for the detection and characterization of exoplanets relies on the instrument’s capability to block out the light of the host star. Some current postprocessing methods for calibrating out the residual speckles use information redundancy offered by multispectral imaging but do not use any prior information on the origin of these speckles.
Aims. We investigate whether additional information on the system and image formation process can be used to more finely exploit the multispectral information.
Methods. We developed an inversion method in a Bayesian framework that is based on an analytical imaging model to estimate both the speckles and the object map. The model links the instrumental aberrations to the speckle pattern in the image focal plane, distinguishing between aberrations upstream and downstream of the coronagraph.
Results. We propose and validate several numerical techniques to handle the difficult minimization problems of phase retrieval and achieve a contrast of 10^{6} at 0.2 arcsec from simulated images, in the presence of photon noise.
Conclusions. This opens up the the possibility of tests on real data where the ultimate performance may override the current techniques if the instrument has good and stable coronagraphic imaging quality. This paves the way for new astrophysical exploitations or even new designs for future instruments.
Key words: techniques: high angular resolution / techniques: image processing / planets and satellites: detection
© ESO, 2013
1. Introduction
Groundbased instruments have now demonstrated the capability of detecting planetary mass companions (Chauvin et al. 2004; Lagrange et al. 2010; Marois et al. 2008) around bright host stars. By combining adaptive optics (AO) system and coronagraphs, some first direct detections from the ground have been possible in favorable cases, at large separations and in young systems when lowmass companions are still warm ( ≥ 1000 K) and therefore not too faint. There is a very strong astrophysical case to improve the highcontrast detection capability (10^{5} for a young giant planet to 10^{10} for an earthlike planet in the near infrared) very close to stars (<0.1′′ to 1′′).
Several instruments will be capable of performing multispectral imaging and will allow characterizing the planets by measuring their spectra. This is the case of GPI (Gemini; Graham et al. 2007), Palm 3000 (Palomar; Hinkley et al. 2011), SCExAO (Subaru; Martinache & Guyon 2009), SPHERE (VLT; Beuzit et al. 2008), and several others that will follow, such as EPICS (EELT; Kasper et al. 2008). By combining extreme adaptive optics (ExAO) and more accurate coronagraphs than before, the level of star light cancellation is highly improved, leading to a better signaltonoise ratio. Even so, the residual host star light is affected by the instrument aberrations and forms a pattern of intensity variations or “speckle noise” on the final image. Part of the speckles cannot be calibrated because they evolve on various time scales (neither fast enough to smooth down to a halo nor stable enough to remove) and for this reason, these “quasistatic speckles” are one of the main limitations for highcontrast imaging.
Several authors have discussed the challenge posed by the elimination of speckle noise in highcontrast multispectral images. It can be done by postprocessing, after the best possible observations. Because images are highly spectrally correlated, one can use the wavelength dependence of the speckles to subtract them. In the particular case of coronagraphic multispectral imaging, only some empirical methods have been developed to subtract the speckle field from the image in the focal plane.
Fig. 1
Evolution of the speckle field with the wavelength. Simulated images at 950, 1306, and 1647 nm for a 10^{3} stellar flux over planet flux contrast. The dynamic range is adapted to the visualization. The speckle field moves with the wavelength but not with the planet position. 

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We propose an alternative approach based on a parametrized imaging model for the postprocessing of multispectral coronagraphic imaging corrected by an extreme AO system in the nearinfrared domain. The aberrations and bright companions at small separations are estimated jointly in a Bayesian framework. In particular, it is possible to take advantage of prior information such as a knowledge on the aberration levels. This type of approach will be all the more efficient as the instruments improve with lower or more stable aberrations and more efficient coronagraphs.
In Sect. 2, we explain how previous methods used the information redundancy to suppress the speckles in highcontrast imaging. Then, we describe the advantages of a joint Bayesian estimation of the aberrations in the pupil plane and of the planet map, based on a parametrized model of coronagraphic imaging. Section 3 presents the longexposure coronagraphic imaging model that is used to simulate the images and to restore them. We also study the case of an approximate model. Section 4 describes the proposed Bayesian joint estimation method as well as theoretical and numerical problems of an alternating restoration algorithm. In particular, we address the strong minimization difficulties associated to the aberration estimation and we propose some solutions. In Sect. 5, our method is validated by restoring images simulated with a perfect coronagraph.
2. Postprocessing speckle subtraction and multispectral imaging
Several empirical postprocessing methods have already been proposed to overcome the problem of detection limitation caused by the quasistatic speckles. Some of these methods used the wavelength dependence of the speckle pattern (Fig. 1) to estimate it and subtract it from the image, while preserving both the flux and spectrum of the planet.
Racine et al. (1999) suggested to subtract two images at different wavelengths to eliminate both the pointspread function (PSF) and the speckle field in noncoronagraphic images. The main limitation of this simultaneous differential imaging (SDI) method comes from the residuals caused by the evolution of the general PSF profile and of the speckle pattern with wavelength. These residuals can be reduced by increasing the number of images used for the speckle field subtraction. Marois et al. (2000) showed with their double difference method that adding another image to the SDI theoretically improves the signaltonoise ratio in the final image of the restored companion. The case of multispectral images has been tackled by Sparks & Ford (2002), who described the socalled spectral deconvolution method in the framework of spacebased observations for an instrument combining a coronagraph and an integralfield spectrometer (IFS). The method, subsequently improved and tested on groundbased noncoronagraphic data by Thatte et al. (2007), is entirely based on a speckle intensity fit by loworder polynomials as a function of wavelength in the focal plane. More recently, Crepp et al. (2011) combined this method with the LOCI algorithm, which is based on a linear combination of images (Lafrenière et al. 2007). They tested this approach to restored onsky images from the Project 1640 IFS on the Palomar telescope. These methods are applicable to any optical system and in particular to those with coronagraphs. However, it is challenging to prevent the planet signals from being eliminated with the speckles because the planet presence is not explicitly modeled.
In addition, some information on the measurement system can be very useful to distinguish a planet from the speckle field. Burke & Devaney (2010) combined classical empirical techniques of differential imaging with a multiwavelength phase retrieval method to estimate the aberration pattern in the pupil plane with a simple imaging model without a coronagraph. This multiwavelength phase retrieval is nicknamed wavelength diversity (Gonsalves 1982). Information diversity is obtained here by different wavelengths whereas it is obtained by introducing a known phase, e.g. defocus, in phase diversity. But in contrast to the phase diversity, the wavelength diversity does not remove the phase sign ambiguity. In Burke & Devaney (2010), the inversion algorithm is based on a maximumlikelihood estimator, which measures the discrepancy between the data and an imaging model. The minimization of this estimator is all the more difficult as the number of unknowns to estimate is high. This problem is overcome by the sparse parametrization of the unknown phases φ_{λ} through the opticalpatherrors (or aberrations) δ, assuming that the former are achromatic: . This allows one to exploit jointly the images at all wavelengths to estimate the aberrations efficiently: the map of the unknown opticalpatherrors δ is common to all wavelengths. The number of unknowns is thus limited and the problem constrained. In the present case, Burke’s wavelength diversity method does not apply readily, because it assumes noncoronagraphic imaging, whereas we consider the highly nonlinear case of a coronagraphic imaging model.
That is why we propose to take advantage of a combined use of wavelength diversity applied in a case of a coronagraphic imaging model, and a Bayesian inversion to jointly estimate the aberrations in the pupil plane and the planet map. The joint estimation aims at taking up the challenge of preserving the planets signal. An advantage of the Bayesian inversion is that it can potentially include an important regularization diversity to constrain the problem, using for example prior information on the noise, the planet map (position, spectrum, etc.) or the aberrations. In the Bayesian framework, the criterion to be minimized is the sum of two terms: the data fidelity term, which measures the distance between the data and the imaging model, and one or more penalty terms. An important difficulty is to define a realistic coronagraphic imaging model, that depends on parameters (e.g. aberrations) that can be either calibrated beforehand or estimated from the data.
3. Parametric model for multispectral coronagraphic imaging
To carry out the Bayesian inversion, we need a parametric direct model of coronagraphic imaging. This direct model will also be useful to simulate our test images.
Fig. 2
Optical scheme of a coronagraphic imager. The upstream and downstream static aberrations, and the adopted notations are denoted δ_{u} and δ_{d}. denote focal plane complex amplitudes, whereas Ψ_{i}(ρ) denotes pupil plane amplitudes. 

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We used a nonlinear analytical expression of the coronagraphic image as proposed by Sauvage et al. (2010), with an explicit role of the optical aberrations before and after the coronagraph, and turbulence residuals. This model assumes that the coronagraph is “perfect” in the sense that the coherent energy is perfectly canceled out. The presence of upstream aberration however, will result in remaining intensity from the star in the image. The aberrations, or opticalpatherrors, δ, are assumed to be achromatic as an approximation. The most recent spectroimagers take increasing care to avoid any source of chromatism, such as outofpupil aberrations, down to a level compatible with contrasts higher than 10^{6}. The variable represents the angular position in the focal plane in radians and the variable is the angular position in the pupil plane in radians^{1}. Finally, corresponds to a spatial position in the pupil plane in meters.
We recall and discuss this model below. In particular, we estimate its simplified expression in the asymptotic case of very low phase, with its secondorder Taylor expansion. This simplified expression helps to understand the explicit way in which each type of aberration impacts the image. It also helps to identify some important ambiguities with different sets of phases that can produce similar images, which will guide the subsequently selected approach for phase retrieval. We also estimate the departure from this lowphase approximation when the phase grows and discuss the validity of this approximation in a SPHERElike case.
3.1. Imaging model
We assume that for an AOcorrected coronagraphic image at the wavelength λ, the direct model is the following sum of three terms, separating the residual coronagraphic stellar halo, the circumstellar source (for which the impact of coronagraph is neglected), and noise n_{λ}: (1)where is the data, i.e., the image to which we have access, is the star flux at wavelength lambda and , the noncoronagraphic PSF, which can be estimated separately. Solving the inverse problem is finding the unknowns, namely the object and the speckle field , which we also call the “coronagraphic PSF”.
3.2. Longexposure coronagraphic PSF model
A model description of directly depends on the turbulence residuals and optical wavefront errors. After previous works to model noncoronagraphic PSFs (Perrin et al. 2003) and coronagraphic PSFs (Cavarroc et al. 2006; Soummer et al. 2007), Sauvage et al. (2010) proposed an analytical expression for the coronagraphic image with a distinction between upstream and downstream aberrations (cf. Eq. (16) in Appendix B). The considered optical system is composed of a telescope, a perfect coronagraph, and a detector plane (cf. Fig. 2). Some residual turbulent aberrations δ_{r}(ρ,t) are introduced in the telescope pupil plane. δ_{r}(ρ,t) is assumed to be temporally zeromean, stationary and ergodic. Because we only consider exposure times that are long with respect to turbulence timescales, these turbulent aberrations contribute only through their statistical spatial properties: power spectral density S_{δr}(α) or structure function D_{φr}. The static aberrations are separated into two contributions: the aberrations upstream of the coronagraph δ_{u}(ρ), in the telescope pupil plane and the aberrations downstream of the coronagraph δ_{d}(ρ) in the Lyot Stop pupil plane . The perfect coronagraph is defined as an optical device that subtracts a centered Airy pattern of maximal energy from the electromagnetic field. Finally, the coronagraphic PSF depends on three parameters that define our system: the aberration maps δ_{u}, δ_{d} and, the residual phase structure function D_{φr}.
3.2.1. Approximate longexposure coronagraphic model in the lowphase regime
Because the analytical expression for is a highly nonlinear function of the aberrations (Sauvage et al. 2010), we derived and studied the relevance of an approximate expression for this model (Ygouf et al. 2010). Approximate coronagraphic imaging models have been derived in several works. Cavarroc et al. (2006) have developed a shortexposure expression and showed by simulations that the main limitation comes from the static aberrations and particularly the aberrations upstream of the coronagraph. Here, we consider a longexposure imaging model and confirm analytically the dominance of the upstream aberrations. Soummer et al. (2007) have developed a twoterm expression with one static and one turbulent term. Nevertheless, these terms are not explicitly linked to the aberrations, which is what we are interested in.
Fig. 3
Speckle noncentrosymmetry. Evolution of the level of residuals with respect to the rms value of the upstream quasistatic aberrations in an image simulated with the model of Sauvage et al. These residuals correspond to the antisymmetric part of the image. 

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Assuming that all phases are small and that the spatial means of φ_{u}(ρ) and φ_{d}(ρ) are equal to zero on the aperture, we derive a secondorder Taylor expansion of expression 24 of Sauvage et al. (2010): (2)where and are the Fourier transforms of the downstream pupil and upstream aberrations and P[δ_{r}(λρ,t)] denotes the piston of the aberration map δ_{r}(λρ,t). is a corrective term that compensates for the fact that δ_{r}(λρ,t) is stationary and thus nonpistonfree on the aperture at every instant. is the Airy pattern formed by the pupil .
This approximate expression brings physical insight into the longexposure coronagraphic PSF model of Sauvage et al.:

The speckle pattern scales radially in λ within the approximate model and evolves as 1/λ^{2} in intensity in the data cube. It is consistent with the analysis of Sparks & Ford (2002), who performed fits of loworder polynomials as a function of the wavelength after rescaling radially.

The approximate expression can be separated into one static and one turbulent term. This is consistent with the analysis of Soummer et al. (2007) with the advantage that these terms depend on the parameters of interest. The turbulent term is simply the turbulent aberration power spectral density, as seen at the resolution of the instrument, i.e., convolved by the output pupil Airy pattern. The static term is explicitly a function of the upstream aberrations.

The downstream aberrations do not appear in the static term. This confirms that the role of the aberrations upstream and downstream of the coronagraph is very different and that upstream aberrations are dominant in the final image.

Four equivalent upstream aberration sets, δ_{u}(ρ), δ_{u}(−ρ), − δ_{u}(ρ) and − δ_{u}( − ρ), which we call quasiequivalent aberration maps in the following, lead to the same image (cf. Appendix A). This item is discussed in more detail in Sect. 4.4.2.

By using this approximate expression for in the imaging model (1), we can see that there is a degeneracy between the value of the star flux and the rms value of the aberration map, if there is no turbulence. Indeed, without turbulent aberrations, the approximate model multiplied by the star flux can be written as (3)This is discussed in Sect. 4.4.1 in greater depth.
3.2.2. Discussion
Because of the complexity of the longexposure coronagraphic PSF model of Sauvage et al., we first considered using the approximate model in our inversion algorithm to decrease the number of unknowns to estimate and simplify the criterion to minimize. But a study of this approximate model showed that the resulting image is too different from the one simulated with the Sauvage et al. expression: computing the root mean square of the difference between the two images leads to a substantial error of typically 30% in SPHERElike conditions (Ygouf et al. 2010).
Fig. 4
Speckle noncentrosymmetry. In the same dynamic range: (left) simulated image, (center) 180 degree rotated version of the image and (right) residuals after combination of the image with the 180 degree rotated version of itself. The gain on the rms value after the subtraction is about 6. 

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Fig. 5
Speckle dilation. In the same dynamic range: (left) image at 950 nm rescaled at 1650 nm and multiplied by the γ coefficient, image at 1650 nm (center) before and (right) after speckle subtraction. The gain on the rms value after the subtraction is 2.5. 

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Speckle noncentrosymmetry.
A substantial part of this error arises because in the approximate model, the quasistatic speckles are purely centrosymmetric. Thus, if we eliminate the centrosymmetric part of the image by combining it with a 180 degree rotated version of itself as follows: (4)there are no residuals with this model, i.e. i_{antisym} = 0. But this is not the case with a more physically realistic image such as the one simulated with the model of Sauvage et al. The level of residuals after such a subtraction is determined by the quantity of upstream aberrations, as we can see in Fig. 3. For example, with 30 nm rms of upstream aberrations, the level of residuals is about six times lower than the rms value of the simulated image (cf. Fig. 4). Thus, for a significant quantity of upstream aberrations, using the model of Sauvage et al. rather than the approximate model for an inversion, should lead to fewer residuals on the final image.
Speckle dilation.
Another difference between the two models tips the balance toward the model of Sauvage et al.. Indeed, in the approximate model the speckle dilation is powered by the 1/λ^{2} factor. If we subtract an image at 950 nm from its 1650 nmdilation, there are no residuals with this model. The same operation with the model of Sauvage et al. (cf. Fig. 5) shows some residuals, at a level 2.5 lower than the rms value of the simulated image, which attests that it is not a pure speckle dilation. To do this, we took the images at the minimum and maximum wavelengths and rescaled the image at 950 nm with respect to the image at 1650 nm. Finally, we performed the following spectral differences between the two images: i_{diff1650} = i_{1650 nm} − γi_{950 nm}, where γ is a coefficient that minimizes the squared difference i_{max} − γi_{min}^{2}, and is given by Cornia et al. (2010): (5)A fine model of the speckle field must be able to account for deviations form centrosymmetry and for deviations from a radial scaling of the speckles proportional to the wavelength. That is why we adopted the model of Sauvage et al. rather than the approximate one in the inversion.
3.2.3. Assumptions on the longexposure coronagraphic PSF model
The information we obtained from the approximate model study helped us define some key assumptions for the success of the speckle field estimation with the Sauvage et al. longexposure coronagraphic PSF model.
Because they have quite a different impact on the final image, it is important to distinguish the aberrations upstream and downstream of the coronagraph. The effect of the downstream aberrations is lower than that of the upstream aberrations, and furthermore, in predicted systems such as SPHERE, they are expected to be much more stable and easier to calibrate than upstream aberrations. Additionally, because we consider longexposure images, the residual turbulent aberrations will be averaged to form a smooth halo that is easily distinguishable from a planet. Furthermore, the statistical quantity D_{φr}, which characterizes this halo, will be measured through the AO system wavefront sensor (Véran et al. 1997). Therefore, we here assumed that both the static downstream aberrations and the residual turbulent aberrations are calibrated and known. This decreases the number of unknowns because the only aberration map to estimate in order to access the coronagraphic PSF is the quasistatic upstream aberrations. We therefore denote the longexposure coronagraphic PSF by instead of to underline the fact that δ_{d} and D_{φr} are assumed to be known.
An advantage of our approach is that these assumptions can evolve. The formalism will allow us to refine our method if we finally decide to estimate either the downstream aberrations or the residual turbulent aberrations. Thus, we can slowly increase the complexity of the problem in anticipation of using real data from SPHERE or from another instrument.
4. Joint estimation of wavefront and object algorithm and minimization strategy
This section introduces the criterion to be minimized (Sect. 4.1) as well as the regularization elements that were used to constrain the problem for the present validations (Sect. 4.2). The minimization algorithm is then described, stressing the two stages that constitute its core (Sect. 4.3). One of these stages presents some convergence difficulties. A minimization strategy is described in Sect. 4.4. The chosen optimizer in described in Sect. 4.5.
4.1. Definition of the criterion to be minimized
Following the Bayesian inverse problem approach, solving the inverse problem consists of finding the unknowns, firstly the object characteristics , secondly the parameters of the speckle field and , which are the most likely given the data and our prior information about the unknowns. This can be reduced to minimizing the following criterion: (6)This criterion is the sum of two terms: the data fidelity term, which measures the distance between the data and the imaging model, and a nonexhaustive list of regularization terms on our unknowns R_{o}, R_{f ∗ }, R_{δ}. We consider that the noise is the sum of a photonic contribution and a detector contribution, that it is white and approximately Gaussian, which is a valid approximation for the flux as considered in this application. The noise variance is assumed to be known here and if it were not, it could be estimated as (Mugnier et al. 2004), where is the photon noise variance and is the detector noise variance previously calibrated. It is assumed that the noise is not correlated from pixel to pixel or between images.
The star flux at each wavelength can be analytically estimated from the criterion provided the regularization on flux is quadratic or absent. In the latter case, the maximum likelihood solution being given by , we obtain: (7)Thanks to this analytical expression, the criterion to be minimized is that of Eq. (6) with replaced by , which will be denoted by and depends explicitly on o_{λ} and δ_{u} only.
Estimating both the object and the aberrations from a single image is a highly underdetermined problem. Any diffractionsized feature in the halo can be interpreted either as a circumstellar pointsource or as a part of the stellar speckle halo. This ambiguity can be decreased by using multispectral images but it is not sufficient. It is thus necessary to regularize the problem by adding more constraints. This is the role of the regularization terms. In this paper, we study the case of constraints on the object that is sufficient for the simulated images we used, but we should keep in mind that it is also possible to use constraints on the aberration map.
4.2. Regularization terms and constraints
We describe below the regularization terms that we used for the validation tests of this paper. Other regularizations could be chosen depending on the kind of images to be processed.
4.2.1. Regularization on the object R_{o}
A regularization on the object is fundamental to help compensating for the degeneracy that exists in the inversion between the aberrations and the object. By penalizing the energy in the object map, it favors the energy in the aberration map and prevents the speckles from being mistaken for a planet.
The regularization term R_{o} includes the prior spatial and spectral information we have on the object. We chose here an L1L2 white spatial regularization, which assumes independence between the pixels (Meimon et al. 2009) because we are mainly looking for point sources. We used an L1L2 regularization rather than a true L1 regularization to keep a differentiable criterion, which simplifies the minimization problem. The spectral prior is based on smoothness of the object spectrum. We currently assume that the object is white (constant spectrum) but as the final aim is to extract some spectra, for future validations we will use a L2 correlated spectral regularization (Thiébaut & Mugnier 2006), which will involve the differences between the spectrum at neighboring wavelength at each pixel and will enforce smoothness on the object spectrum.
The regularization on the object is necessary to obtain a sparse object. Without the regularization many residuals remain on the estimated object.
4.2.2. Positivity and support constraints on the object
The object intensity map is a set of positive values, which is an important prior information. One should therefore enforce a positivity constraint on the object. This constraint can be implemented in various ways, such as criterion minimization under the positivity constraint, reparameterization of the object, or explicit modification of the a priori probability distribution (e.g., addition of an entropic term). Mugnier et al. (2004) have found that the best way to ensure positivity, with respect both to speed and to not introducing local minima, is to directly minimize the criterion under this constraint. We proceeded similarly.
Because the star light is concentrated around the optical axis, the flux is essentially estimated on this very bright region (cf. Eq. (7)). But if there are too many residuals on the object, the flux estimation can be biased. Thus, imposing, as is physically meaningful, that the object is null very close to the star, in a region of typically 3λ/D radius, helps us estimate the star flux accurately.
4.2.3. Regularization on the star flux R_{f∗}
To make the minimization more robust, it can be useful to constrain the flux estimation to physical values. Indeed, if there are no turbulent aberrations, there is an indetermination between the star flux value and the phase rms value in the approximate model of Eq. (13). Consequently, for aberrations with very low rms values, the coronagraphic PSF is close to zero and thus the analytical flux estimate given by Eq. (6) can diverge. The presence of known turbulent aberrations naturally constrains the flux value, but to make the method more robust, we chose to prevent the flux from diverging. In practice, we regularized the flux estimation by the following quadratic metric: . This metric can be interpreted as a Gaussian prior low on the flux, but its role is not as essential for the criterion minimization as that of the L1L2 regularization and the positivity constraint. This leads to the following expression for the analytic star flux: (8)In practice, we chose a very high standard deviation σ_{f,λ} = 100 × ∑ _{α}i_{λ}, to avoid biasing the flux. With this standard deviation, we can choose any mean flux, for example, f_{0} = 0. This is sufficient to avoid the division by zero in the flux computation and thus the flux divergence.
4.3. Iterative algorithm
The structure of the joint criterion of Eq. (6) prompted us to adopt an estimation of wavefront and object that alternates between estimation of the aberrations, assuming that the object is known (multispectral phase retrieval) and estimation of the object assuming that the aberrations are known (multispectral deconvolution).
Multispectral deconvolution.
For given aberrations, we define the following intermediate data where the (assumed known) stellar halo is subtracted, keeping then only the circumstellar object as seen in classical imaging: . By inserting these intermediate data into the the criterion of Eq. (6), we obtain: (9)which shows that the problem at hand is a nonmyopic multispectral deconvolution of images . The chosen regularization leads to a convex criterion (Mugnier et al. 2004) and thus to a unique solution for a given set of aberrations and a given object regularization.
Phase retrieval.
If we replace the intermediate data into the the criterion of Eq. (6), we obtain: (10)which shows that the problem at hand is essentially a phase retrieval problem. In this phase retrieval stage, the combination of a high number of parameters to estimate (typically 10^{3}, see Sect. 5) and of a highly nonconvex criterion complicates the problem. To avoid local minima, several numerical solutions resulting from a fine understanding of the imaging process are necessary and are described below.
4.4. Phase retrieval: dealing with local minima
4.4.1. Choice of an appropriate starting point: very small random phase
To keep the computation time reasonable, we used the local descent algorithm described in Sect. 4.5 to minimize the criterion. Because the latter is highly nonconvex, the chosen starting point so that we are fully in the conditions where the Taylor expansion developed in 3.2.1 is valid and where the criterion is less nonconvex. It allows the algorithm to avoid many wrong directions, and thus many local minima. As the algorithm converges, the upstream aberration rms value increases toward its true value and a gradual nonlinearity of the model is gradually introduced.
We tested the phase retrieval capability of our algorithm with respect to the chosen starting point, assuming that there is no object to estimate. We give the rms value of the estimated upstream aberration map and the rms value of the difference between the simulated and the estimated maps estimated as follows: (11)The inversion is performed with one spectral channel and without turbulence. Figure 6 compares some estimated upstream aberration maps (a, b, c) to the simulated one (“true”):

(a)
Using a random aberration map with the same rms value as thetrue aberrations (30 nm at950 nm) as a starting point does not help infinding the global minimum. Indeed, the algorithm convergesvery quickly toward a local minimum and the estimatedaberration map (rms^{a} = 307 nm at 950 nm) is completely different from the simulated one (rms^{diff,a} = 10^{3}%).

(b)
Using a zeroaberration map as a starting point does not work either. This is probably because the approximate model is an even function. For this particular starting point, the gradient is null, which leads to some convergence difficulties. The rms of the difference between the two maps is about 1.4 × 10^{4}%. The estimated aberration pattern (rms^{b} = 4069 nm at 950 nm) seems to show that the algorithm does not explore the high frequencies.

(c)
The solution we propose is to use as a starting point for the minimization a nonnull random aberration map with a low rms value compared to those of the “true” simulated aberration map. In practice, we chose an rms value about 10^{8} times lower than the “true” value. This leads to a correct estimation of the aberration map (rms^{c} = 30.2 nm at 950 nm) with an rms of the difference between the two maps of about 0.6%.
If we plot the same results for images simulated with turbulence, the conclusions are not the same. The convergence to an aberration map that resembles the true one and has similar rms value is easier:

(a’)
By using a random aberration map with the same rms value asthe true aberrations (30 nm at950 nm) as a starting point, the rms value of theestimated aberration map (30.7 nm) is close tothe true value. After a careful inspection, the estimated aberrationmap turned out to be similar to the opposite of the simulated one.This is discussed in more detail in 4.4.2.

(b’)
Using a zeroaberration map as a starting point leads to a good pattern and a good rms value of aberration map (30.1 nm).

(c’)
Using a nonnull random aberration map with a low rms value as a starting point also leads to a good pattern and a good rms value of the aberration map (30.1 nm).
For the (b’) and (c’) cases, the difference between the two maps is bigger than before (17%) but the estimated aberration map is a sufficiently good starting point for the alternating minimization.
The choice of an appropriate starting point seems not to be as essential with turbulent aberrations as it was without turbulent aberrations. Indeed, the presence of turbulent aberrations raises the ambiguity that exists between the value of the star flux value and the rms value of the upstream aberration map. Nevertheless, even if we assume that there are turbulent aberrations in the following, we chose to use a random aberration map with a low rms value as a starting point of the phase retrieval because it allows us to avoid some local minima by linearizing the highly nonlinear model used in the inversion.
Fig. 6
Choice of an appropriate starting point. Estimated upstream aberration maps with one spectral channel for three different starting points. [i] Without turbulent aberrations in the simulated images. [ii] With turbulent aberrations in the simulated images. From left to right, with a dynamic range adapted to the visualization: “true” simulated aberration map, (a) and (a’) estimated aberrations with a random aberration map (rms value of the simulated aberrations) as starting point, (b) and (b’) estimated aberrations with a zero aberration map as starting point, and (c) and (c’) estimated aberrations with a random aberration map (rms value 10^{8} times lower than the “true” one) as starting point. The estimation is performed with a regularization on the star flux. 

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Fig. 7
Estimated upstream aberrations for the four quasiequivalent aberration maps as starting points. From left to right, with the same dynamic range: , , , . The image simulation is performed with one spectral channel in the presence of turbulent aberrations. 

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4.4.2. Avoiding some local minima by testing quasiequivalent starting points
In the approximate model, four different aberration maps can give the same image (cf. Eq. (15) in Appendix A). This means that, from a given starting point, the minimization algorithm can take four different but equivalent directions from the approximate model point of view. But from the point of view of the model used in the inversion, this is not the case because it depends on downstream aberrations, which break the symmetry. That is why we call them “quasiequivalent” aberrations maps (cf. 3.2.1).
The idea is then to explore the several regions offered by the four different quasiequivalent aberrations maps to determine which of these solutions gives the smallest criterion. To do this, we performed an initialization step where the very small random phase is taken as a starting point. A first phase retrieval stage was performed with this starting point, leading to a first estimated aberration map denoted by . Then, the three other quasiequivalent aberration maps , and were taken as starting points for three other phase retrieval stages. This led to three more estimated aberration maps denoted by , and .
Figure 7 shows the four estimated aberration maps at the end of the initialization step. These estimated aberration maps are compared to the simulated one (Fig. 6d). The final aberration map chosen as a starting point for the alternating algorithm is the one that gives the minimum value for criterion J of Eq. (6): (12)In practice, quite often and in particular in this simulation, it turns out that the chosen set of aberrations is also the one whose rms value (rms^{b’} = 30.1 nm at 950 nm) is closest to the ‘true” phase (Fig. 6i. “true” and Fig. 6ii. “true”).
Fig. 8
Block diagram of the algorithm used for the joint estimation of the object map and the upstream aberrations. 

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4.4.3. Avoiding some local minima in the multispectral inversions by taking the previously estimated aberration map as starting point
We used a local descent algorithm to minimize the criterion. For this, the gradients are computed for each explored direction and the computation is all the longer as there are spectral channels. That is why it is useful to perform several inversions by gradually increasing the number of spectral channels. This also helps us avoid some local minima. We begin by an inversion with one spectral channel. Then, we add some more spectral channels for new inversions and each time, we take the previous estimated aberration map as a starting point.
Because an inversion with only one spectral channel sometimes leads to a local minimum, it is useful to also test the four quasiequivalent starting points with two spectral channels (cf. Sect. 4.4.2).
4.5. VMLM optimizer
To minimize the criterion, we chose the variable metric with limited memory and bounds (VMLMB) method (Thiébaut 2002). Updated from the BFGS variable metric method (Press et al. 2007), it is usable for a problem of large dimensionality. Moreover, it offers the possibility to constrain these parameters. This makes this method a good tool for many inversion problems in high angular resolution (Meimon et al. 2009; Gratadour et al. 2005). It is available from http://wwwobs.univlyon1.fr/labo/perso/eric.thiebaut/optimpack.html.
4.6. Summary of the developed algorithm
Figure 8 summarizes the different steps of the developed algorithm. The choice of a very small random phase as a starting point is essential because it avoids falling into some local minima (Sect. 4.4.1). An initialization phase is performed. It consists in running the algorithm for the four quasiequivalent solutions (Sect. 4.4.2). The solution that leads to the lowest criterion value is selected. Then, the minimization core is performed, alternating between the aberration estimation, assuming that the object is known (multispectral phase retrieval), and the object estimation, assuming that the aberrations are known (nonmyopic multispectral deconvolution). Regularization terms and constraints prevent the algorithm from falling into other local minima (Sect. 4.2.3). Iterations are performed (Sect. 4.3) until the stopping rule of the algorithm is verified. The chosen optimizer is the VMLMB (Thiébaut 2002, Sect. 4.5).
5. Validation of the inversion method by simulations
In this section, we validate the exoplanet detection capabilities of our inversion method. After giving the numerical simulation conditions, we investigate the estimation quality of the aberrations and the object as a function of the number of images at different wavelengths used. We also study the algorithm robustness with respect to the simulated images and with respect to the starting point we use. Finally, we study the effect of the bandwidth on the quality of the object estimation.
5.1. Simulation hypothesis
From a data cube of six images simulated with the image formation model of Eq. (1) and the Sauvage et al. (2010) analytical expression for coronagraphic imaging (cf. Eq. (16) in Appendix B), we jointly estimated the speckle field and the object map. We chose pixel indicator functions as the basis for the phase rather than, e.g., a truncated basis of Zernike polynomials, to model and reconstruct phases with a high spatial frequency content. The hypotheses are typical of a SPHERElike instrument: upstream δ_{u} and downstream δ_{d} aberrations simulated with standard deviation of 30 nm (cf. their power spectral densities in Fig. 10), starplanet angular separations of 0.2 and 0.4 arcsec, contrasts, i.e. ratio of star flux over planet flux of 10^{5}, 10^{6} and 10^{7}, a [950 nm; 1647 nm] spectral bandwidth and an integrated flux of 4 × 10^{11} on the data cube in presence of photon noise and a transmission (throughput and quantum efficiency) of 10%, corresponding to the observation of a 6magnitude star for 25 min with the VLT. We used 128 × 128 pixels to simulate our images, Shannonsampled at 950 nm. This results in a number of unknowns to estimate for the aberration map of about 3 × 10^{3}. If we add the unknowns to estimate for the object map, which is 16 × 10^{3}, the total number of unknowns is about 2 × 10^{4}. Figure 9 shows the simulated objet map (9(a)) and the associated image in the focal plane (9(b)). For an easier visualization, we represent the images in the focal plane and not the object map in the following. Figure 9 shows the simulated aberration map (9(c)) and the associated image of the speckle field in the focal plane (9(d)).
Fig. 9
Simulated images at λ = 950nm. a) Simulated object map and b) associated image in the focal plane. The following planets are simulated: one with a staroverplanet contrast of 10^{5} at a separation of 0.2 arcsec (planet 1), two with staroverplanet contrast of 10^{6} at separations of 0.2 (planet 2) and 0.4 (planet 3), respectively, and one with a staroverplanet contrast of 10^{7} at a separation of 0.4 arcsec (planet 4). The image in the focal plane is obtained by convolving the object map o_{λ} by the noncoronagraphic psf . c) Simulated aberrations and d) associated image of the speckle field in the image focal plane. The image is given by the coronagraphic PSF . 

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Fig. 10
Power spectral densities of the simulated aberrations. The upstream and downstream aberrations are randomly generated according to a f^{2} spectrum and with an rms value of 30 nm. The downstream aberrations are not corrected by the adaptive optics. The upstream aberrations are corrected by the adaptive optics up to 20 cycles/pupil. The residuals are due to the noncommon path aberrations. Below 4 cycles/pupil, these noncommon path aberrations are corrected but there are some residuals from rotative optics. 

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5.2. Algorithm robustness and performance studies
We jointly estimated the upstream quasistatic aberration map and the object map with multispectral data. To study the robustness of the method we have developed, we ran several simulations in a Monte Carlolike manner. Both different simulated images (5.2.1) and different starting point (5.2.2) were used for the inversion. The results of the inversions with two and six spectral channels were compared to study the effect of the redundancy of information offered by the multispectral imaging.
5.2.1. Different simulated phases
We applied our method to ten images simulated with different random upstream aberration maps to assess the algorithm robustness. Figure 11 shows the estimated images of the object from these images for a twospectral channel inversion (a) and a sixspectral channel inversion (b). These results bring several conclusions, both on the robustness of the method and the multispectral redundancy.
Fig. 11
Robustness study on the simulated phases. With the same dynamic range, at 950 nm: estimated planet images from different images, simulated with ten different randomly generated upstream aberration maps. 

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Robustness of the method.
In one out of ten cases, some residuals from the speckle field remain on the object. The level of this residual prevents one from detecting any planet. In the other nine cases, the level of the residuals is negligible, which allows one to detect at least one planet. These observations, independent of the number of spectral channels used for the inversion, shows that the method is relatively robust, given the minimization difficulties we met.
Multispectral redundancy.
We can see some significant differences between the twospectral channel inversion and the sixspectral channel inversion:

The planet with a contrast of 10^{5} is detected in eight out of ten cases with two spectral channels and in nine out of ten cases with six spectral channels.

Only one planet with a contrast of 10^{6} is detected with two spectral channels in three out of ten cases. The same planet is detected in nine out of ten cases with six spectral channels.

The other planet with a contrast of 10^{6} is detected in three out of ten cases with six spectral channels.

Furthermore, the estimation of the flux of the planets is more accurate with six spectral channels. The inversion shown in Fig. 12 is representative of a large body of simulated tests and is performed with two and six spectral channels. The simulated and estimated image of the object maps for these two different inversions are represented as the errors on the estimated planet flux when the planet is effectively detected.
These results show that multispectral redundancy helps us detect the planets. However, we note that even if the convergence problems were solved with our minimization strategy in most cases, there are still challenges to be overcome to arrive at a perfectly robust algorithm.
Fig. 12
Multispectral redundancy. With the same dynamic range, at 950 nm, from left to right: simulated planet image and estimated ones with a two and sixspectral channel inversion. The errors on the planet flux estimations are computed when the planet is detected. 

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5.2.2. Different starting points
For each of the ten previous simulated phases, we took ten different random aberration maps with the same level of rms value as starting points to demonstrate than any random aberration map, provided it is small, allows us to find a good solution. Figure 13 shows the estimated images of the object for the ten different random aberration maps, used as starting points, for one phase.
The starting point does not have a strong impact on the planet detection. In all cases, the three brightest planets are detected. In only one out of ten cases, a false planet is detected. Despite all the attention given to the initialization of the algorithm, the choice of starting point can have, to a limited extent, an effect on detection.
Fig. 13
Robustness study on the aberrations maps used as starting points. With the same dynamic range, at 950 nm: estimated planet images from one image with different random aberration maps used as starting points. 

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5.3. Bandwidth effect
We study here the bandwidth effect on the planet image estimation quality. We selected the following spectral bandwidths: [950 nm; 1050 nm], [950 nm; 1150 nm], [950 nm; 1350 nm], and [950 nm; 1650 nm]. The two last bandwidths correspond to operational modes of the SPHEREIFS instrument, whereas the other bandwidths are produced for the purpose of our study’s completeness. Figure 14 compares the estimated planet image maps for these different inversions. If the planet with a contrast of 10^{5} is detected each time, only the two broader bandwidths (700 and 400 nm) allow one to detect the two planets with a contrast of 10^{6}. With a 200 and a 100 nm bandwidth, the planet with a contrast of 10^{6} which is at a separation of 0.4 arcsec is detected, whereas the one that is at a separation of 0.2 arcsec is not.
The detection performance increases with the bandwidth because the incorporated spectral information helps the algorithm to distinguish between speckles and planets. Indeed, from one wavelength to another, the amplitude of a speckle movement is proportional to the wavelength difference and to the radial position of the speckle. For a given spectral bandwidth, it is therefore easier to detect a planet that is far away from the star than a planet than is close to the star.
Fig. 14
Bandwidth effect. With the same dynamic range, at 950 nm, from left to right: estimated planet images with a 700, 400, 200, and 100 nmbandwidth inversion. 

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6. Conclusion
We have proposed an original method that jointly estimates the object (multispectral deconvolution) and the aberrations (multispectral phase retrieval) for the new generation of planet finders. For the first time, a fine parametric model of coronagraphic imaging, describing the instrument response, is used for the inversion of simulated multispectral images in a solid statistical framework. Even though the model remains a simplification of reality, in particular when assuming achromatic wavefront errors, it goes much further than only assuming that the spatial speckle pattern essentially scales with wavelength. We have shown that the secondorder approximation of the imaging model has the same behavior as the oftenused model for the problem at hand: the speckle pattern is centrosymmetric, it scales with the wavelength, and it is not dependent on the downstream aberrations. The departure from this case quickly becomes very significant as the phase grows. It is clear that the ability of a finite phase to produce nonsymmetric speckles induces a strong ambiguity between the estimates of the aberrations and the object, even if the latter is a pointlike companion.
To set up our method, we developed an iterative algorithm. With only one spectral channel, this joint estimation is an underdetermined problem, as we emphasized before. This underdetermination results mathematically in a degeneracy of the global minimum. A multispectral inversion raises this underdetermination but it is still possible to fall into local minima. Because of the high nonlinearity of the coronagraphic imaging analytical model and the number of unknowns to estimate (about 10^{3} in our case), the phase retrieval, even if it is multispectral, remains a difficult problem. We setup a minimization approach that remains quite fast (without systematically exploring the whole parameter space in a search for global minimum) but that is still relatively robust: extensive tests showed the success in converging to a good solution in 90% of the cases in a systematic manner, and in the other cases, the failure appears obviously in the results with no risk of confusion and can motivate tests with alternative criterion minimization approaches. We obtained these convergence capabilities of the algorithm by bringing original solutions to the minimization difficulties of the phase retrieval, inspired by studying the imaging model. One element of the solution is to use a small random aberration map as starting point. Another element is to explore the several directions offered by the quasiequivalent aberration maps.
A wide variety of prior information, either about the system (aberrations, flux, noise) or about the object of interest, can be used to constrain the problem. The choice of a Bayesian approach allows this flexibility. In particular, a regularization that minimizes the energy on the object map helped us in separating the aberrations from the object and in decreasing the speckle noise in the reconstructed object map.
The restoration of images simulated with a perfect coronagraph is very encouraging for the extraction of planetary signals at levels that begin to be astrophysically interesting. We demonstrated the efficiency of the method even with only two spectral channels, by achieving a contrast of 10^{5} at 0.2 arcsec. Multispectral redundancy improves the detection, which allowed us to achieve a contrast of 10^{6} at 0.2 arcsec with six spectral channels.
We therefore believe that this approach will be quite powerful when we are faced with experimental data. This deserves to be studied, as well as how the performance will evolve in the cases of images simulated with a nonperfect coronagraph, real images from the SPHERE instrument in the lab, or real images from an instrument onsky. Eventually, this method could be used to improve the performance of the existing multispectral imaging instruments, providing better astrophysical exploitations. Now that we demonstrated that we can manage the difficulties linked to the criterion minimization, we can now focus on its applicability, and on adding more prior information, starting with the full set of information that can be obtained from the instrument calibrations.
The lessons learned by applying the method could also facilitate the approach for the design of future instruments such as EPICS for the European Extremely Large Telescope (Kasper et al. 2008), and the definition of their calibration procedures.
Appendix A: Indetermination on the estimated aberrations from an image simulated with our approximate model
We show here that four sets of upstream aberrations give the same image for our approximate model. We rewrite the expression below as a function of φ_{u} = (2π/λ) × δ_{u} and without the variables for better readability: (13)We consider here the static term and rewrite it in the form of a correlation. For this, we consider two functions and g = δ_{u} of the two variables ρ_{x} and ρ_{y} and denote .
By using the definition of the correlation Γ_{fg}(ρ) and the convolution of the two functions f(ρ) and g(ρ), and the properties we obtain This yields (14)The properties of the autocorrelation and lead to and This yields (15)which means that the upstream aberration sets δ_{u}(ρ), δ_{u}( − ρ), − δ_{u}(ρ) and − δ_{u}( − ρ) are equivalent with respect to the approximate model, because they give the same image. This is true even in the presence of the turbulent term .
Appendix B: Indetermination on the estimated aberrations from an image simulated with the Sauvage et al. model
In classical imaging, i.e. “noncoronagraphic imaging”, the sign of the even part of the phase cannot be deduced from only one image in the focal plane (Blanc 2002). In other words, if we denote φ_{e} and φ_{o}, the even and the odd parts of the phase, the two phases φ = φ_{e} + φ_{o} and φ′ = −φ_{e} + φ_{o} give the same image. In this appendix, we show that this is also the case for the Sauvage et al. expression (Sauvage et al. 2010) in coronagraphic imaging, if we assume that the sign of the even part changes for all phase errors.
The expression of Sauvage et al. is (16)with , , and φ_{tot}(ρ) = φ_{r}(ρ) + φ_{u}(ρ) + φ_{d}(ρ). TF[.] denotes the Fourier transform. represents the mean Strehl ratio during observation, such as with φ(ρ,t) = φ_{r}(ρ,t) + φ_{u}(ρ).
The first term of Sauvage et al.’s expression is the classical case of noncoronagraphic PSF, which is wellknown (Blanc 2002). The term stays identical whatever the sign of the even part of the phase.
The second term of Sauvage et al.’s expression is the product of two factors: and . The latter stays identical whatever the sign of the even part of the phase. We take the following phase φ′ = −φ_{e} + φ_{o} and calculate the corresponding , assuming that ρ′′ = −ρ: is then independent of the sign of the even part of the phase. Thus, the product is also independent of the sign of the even part of the phase.
We study now the third term . Assuming that φ_{d} = (φ_{d})_{e} + (φ_{d})_{o} and :
In the same way as the previous demonstration, we can show that . Under the effect of the transformation φ → φ′, the different terms become
In other words, When we take the conjugate of a complex number, only the sign of the imaginary part changes. Because we take the real part of this expression, changing the sign of the even part of the phase does not change the term.
To conclude, like in classical imaging, changing the sign of the even part of the phase does not change the image in the focal plane. This means that two sets of aberrations give the same image. But if we assume like in this communication that the downstream aberrations are fixed and known, this removes the degeneracy.
Acknowledgments
This research is supported by the “Groupement d’Intérêt Scientifique” PHASE.
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All Figures
Fig. 1
Evolution of the speckle field with the wavelength. Simulated images at 950, 1306, and 1647 nm for a 10^{3} stellar flux over planet flux contrast. The dynamic range is adapted to the visualization. The speckle field moves with the wavelength but not with the planet position. 

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In the text 
Fig. 2
Optical scheme of a coronagraphic imager. The upstream and downstream static aberrations, and the adopted notations are denoted δ_{u} and δ_{d}. denote focal plane complex amplitudes, whereas Ψ_{i}(ρ) denotes pupil plane amplitudes. 

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In the text 
Fig. 3
Speckle noncentrosymmetry. Evolution of the level of residuals with respect to the rms value of the upstream quasistatic aberrations in an image simulated with the model of Sauvage et al. These residuals correspond to the antisymmetric part of the image. 

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In the text 
Fig. 4
Speckle noncentrosymmetry. In the same dynamic range: (left) simulated image, (center) 180 degree rotated version of the image and (right) residuals after combination of the image with the 180 degree rotated version of itself. The gain on the rms value after the subtraction is about 6. 

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In the text 
Fig. 5
Speckle dilation. In the same dynamic range: (left) image at 950 nm rescaled at 1650 nm and multiplied by the γ coefficient, image at 1650 nm (center) before and (right) after speckle subtraction. The gain on the rms value after the subtraction is 2.5. 

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In the text 
Fig. 6
Choice of an appropriate starting point. Estimated upstream aberration maps with one spectral channel for three different starting points. [i] Without turbulent aberrations in the simulated images. [ii] With turbulent aberrations in the simulated images. From left to right, with a dynamic range adapted to the visualization: “true” simulated aberration map, (a) and (a’) estimated aberrations with a random aberration map (rms value of the simulated aberrations) as starting point, (b) and (b’) estimated aberrations with a zero aberration map as starting point, and (c) and (c’) estimated aberrations with a random aberration map (rms value 10^{8} times lower than the “true” one) as starting point. The estimation is performed with a regularization on the star flux. 

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In the text 
Fig. 7
Estimated upstream aberrations for the four quasiequivalent aberration maps as starting points. From left to right, with the same dynamic range: , , , . The image simulation is performed with one spectral channel in the presence of turbulent aberrations. 

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In the text 
Fig. 8
Block diagram of the algorithm used for the joint estimation of the object map and the upstream aberrations. 

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In the text 
Fig. 9
Simulated images at λ = 950nm. a) Simulated object map and b) associated image in the focal plane. The following planets are simulated: one with a staroverplanet contrast of 10^{5} at a separation of 0.2 arcsec (planet 1), two with staroverplanet contrast of 10^{6} at separations of 0.2 (planet 2) and 0.4 (planet 3), respectively, and one with a staroverplanet contrast of 10^{7} at a separation of 0.4 arcsec (planet 4). The image in the focal plane is obtained by convolving the object map o_{λ} by the noncoronagraphic psf . c) Simulated aberrations and d) associated image of the speckle field in the image focal plane. The image is given by the coronagraphic PSF . 

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In the text 
Fig. 10
Power spectral densities of the simulated aberrations. The upstream and downstream aberrations are randomly generated according to a f^{2} spectrum and with an rms value of 30 nm. The downstream aberrations are not corrected by the adaptive optics. The upstream aberrations are corrected by the adaptive optics up to 20 cycles/pupil. The residuals are due to the noncommon path aberrations. Below 4 cycles/pupil, these noncommon path aberrations are corrected but there are some residuals from rotative optics. 

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In the text 
Fig. 11
Robustness study on the simulated phases. With the same dynamic range, at 950 nm: estimated planet images from different images, simulated with ten different randomly generated upstream aberration maps. 

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In the text 
Fig. 12
Multispectral redundancy. With the same dynamic range, at 950 nm, from left to right: simulated planet image and estimated ones with a two and sixspectral channel inversion. The errors on the planet flux estimations are computed when the planet is detected. 

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In the text 
Fig. 13
Robustness study on the aberrations maps used as starting points. With the same dynamic range, at 950 nm: estimated planet images from one image with different random aberration maps used as starting points. 

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In the text 
Fig. 14
Bandwidth effect. With the same dynamic range, at 950 nm, from left to right: estimated planet images with a 700, 400, 200, and 100 nmbandwidth inversion. 

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In the text 
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