Sparse optical interferometric arrays of many apertures can produce direct images in the densified-pupil mode, also called “hypertelescope” mode. Pending the introduction of adaptive optics for cophasing, indirect images can also be reconstructed with speckle imaging techniques. But adaptive phasing is preferable, when a sufficiently bright guide star is available. Several wave sensing techniques, by-products of those used on monolithic telescopes for some of them, are potentially usable. For cophased direct images of very faint sources in the absence of a natural guide star, a modified form of the Laser Guide Star techniques demonstrated on conventional and segmented telescopes is described. Preliminary testing in laboratory suggests further investigation. Recorded images, assumed co-phased, are also improvable post-detection with optical aperture-synthesis techniques such as Earth rotation synthesis, where data from successive exposures are combined incoherently. Nevertheless, the gain becomes modest if hundreds of sub-apertures are used. Image deconvolution techniques are also applicable, if suitably modified as demonstrated by Aime et al. (2012), and Mary (2012). Their modified deconvolution algorithms can extend the Direct Imaging Field (also called Clean Field) of hypertelescopes. More sub-apertures at given collecting area, implying that their size is reduced, improve the direct-imaging performance. The predictable trend thus favors systems combining hundreds of sub-apertures of modest size, if workable designs can be evolved. One such design, the “Ubaye Hypertelescope” entering the initial testing phase in the southern Alps, has a fixed spherical meta-mirror with a 57 m effective aperture, expandable to 200 m. Preliminary results suggest that larger versions, whether spherical or active paraboloidal, can reach a kilometric aperture size at terrestrial sites having a suitable concave topography. In space, hypertelescope meta-apertures spanning up to 100 000 km are in principle feasible in the form of a flotilla of mirrors, driven by micro-thrusters or by the radiation pressure of the Sun or lasers.

In this paper I review some of the fundamental aspects of optical long baseline interferometry. I present the principles of image formation, the main difficulties and the ways that have been opened for high angular resolution imaging. I also review some of the recent aspects of the science program developed on the VEGA/CHARA interferometer.

In a first part of the paper we give a simple introduction to the free space propagation of light at the level of a Master degree in Physics. The presentation promotes linear filtering aspects at the expense of fundamental physics. Following the Huygens-Fresnel approach, the propagation of the wave writes as a convolution relationship, the impulse response being a quadratic phase factor. We give the corresponding filter in the Fourier plane. As an illustration, we describe the propagation of wave with a spatial sinusoidal amplitude, introduce lenses as quadratic phase transmissions, discuss their Fourier transform properties and give some properties of Soret screens. Classical diffractions of rectangular diaphragms are also given here. In a second part of the paper, the presentation turns into the use of external occulters in coronagraphy for the detection of exoplanets and the study of the solar corona. Making use of Lommel series expansions, we obtain the analytical expression for the diffraction of a circular opaque screen, giving thereby the complete formalism for the Arago-Poisson spot. We include there shaped occulters. The paper ends up with a brief application to incoherent imaging in astronomy.

This course/paper deals with adaptive optics in the framework of astronomical imaging. It does not pretend to be an exhaustive course of astronomical adaptive optics. It is rather intended to give an introductory overview of it, from my very partial point-of-view.

We propose in this paper an introduction to the wavefront coding technique for incoherent imaging. Wavefront coding introduces image processing in the conception of an imaging system. It consists in introducing controlled aberrations in the optics able to reduce, after processing, some defaults of the optical system such as defocus, chromaticity. We present the basis of wavefront coding and illustrate them on two images with different characteristics: a spoke pattern and a galaxy image.

This paper concentrates on the control aspects of Adaptive Optics (AO) systems and includes a prior exposure to linear control systems from the “classical” point of view. The AO control problem is presented and the well-established optimized modal gain integral control approach is discussed. The design of a controller from a modern control point of view is addressed by means of a linear quadratic Gaussian control methodology. The proposed approach emphasizes the ability of the adaptive optics loop to reject the atmospheric aberration. We derive a diagonal state space system which clearly separates the dynamics of the plant (deformable mirror & wavefront sensor) from the disturbance dynamics (atmospheric model). This representation facilitates the numerical resolution of the problem. A frequency analysis is carried out to check performance and robustness specifications of the multiple-input multiple-output feedback system. The effectiveness of the approach is demonstrated through numerical experiments.

The VLTI (Very Large Telescope Interferometer) makes available milli-arcsecond-scale observations in the infrared. It offers new possibilities for constraining stellar structures such as polar jets, equatorial disks and rotationally-flattened photospheres of Be stars. Such constraints allows us to better estimate the stellar fundamental parameters and refine the mechanisms such as mass loss, pulsation and magnetism that govern the variability and evolution of these stars.

In this paper we present a chromatic semi-analytical model of fast rotators, which allows us to study the dynamics and the interaction between the photosphere and the wind of fast rotating stars of O, B, A and F spectral types. Our simple analytical model addresses the oblateness, inclination and position angle of the rotation axis of the star. It produces iso-velocity maps and intensity maps. It includes line profiles, limb-darkening and the von Zeipel effect and the non-radial pulsations.

SCIROCCO+: Simulation Code of Interferometric-observations for ROtators and CirCumstellar Objects including Non-Radial Pulsations, includes all the parameters cited above in order to be fast, powerful and light simulation tool in high angular resolution of rotating objects.

In the course of our VLTI young stellar object PIONIER imaging program, we have identified a strong visibility chromatic dependency that appeared in certain sources. This effect, rising value of visibilities with decreasing wavelengths over one base, is also present in previous published and archival AMBER data. For Herbig AeBe stars, the H band is generally located at the transition between the star and the disk predominance in flux for Herbig AeBe stars. We believe that this phenomenon is responsible for the visibility rise effect. We present a method to correct the visibilities from this effect in order to allow “gray” image reconstruction software, like Mira, to be used. In parallel we probe the interest of carrying an image reconstruction in each spectral channel and then combine them to obtain the final broadband one. As an illustration we apply these imaging methods to MWC158, a (possibly Herbig) B[e] star intensively observed with PIONIER. Finally, we compare our result with a parametric model fitted onto the data.

Image reconstruction from interferometric data is an inverse problem. Owing to the sparse spatial frequency coverage of the data and to missing Fourier phase information, one has to take into account not only the data but also prior constraints. Image reconstruction then amounts to minimizing a joint criterion which is the sum of a likelihood term to enforce fidelity to the data and a regularization term to impose the priors. To implement strict constraints such as normalization and non-negativity, the minimization is performed on a feasible set. When the complex visibilities are available, image reconstruction is relatively easy as the joint criterion is convex and finding the solution is similar to a deconvolution problem. In optical interferometry, only the powerspectrum and the bispectrum can be measured and the joint criterion is highly multi-modal. The success of an image reconstruction algorithm then depends on the choice of the priors and on the ability of the optimization strategy to find a good solution among all the local minima.

Compared to optical astronomy, millimetre radio astronomy experiences not only a different and complementary aspect of the universe but also different perturbations and limitations from Earth’s atmosphere that are mostly imposed by atmospheric water vapour and its dynamics. After discussing the physics behind the refractive index variations and possible correction schemes, a small introduction into the basics of radio interferometry and image reconstruction with the CLEAN algorithm is given.

Present soil moisture and ocean salinity maps retrieved by remote sensing are characterized by a coarse spatial resolution. Hydrological, meteorological and climatological applications would benefit greatly from a better spatial resolution. Owing to the dimensions of the satellite structure and to the degradation of the instrument’s radiometric sensitivity, such improvement cannot be achieved with classical interferometry. Then, in order to achieve this goal an original concept for passive interferometric measurements is described. This concept should allow to achieve a much finer spatial resolution, which can be further improved with the application of disaggregation methods. The results will then allow the integration of global soil moisture maps into hydrological models, a better management of water resources at small scales and an improvement in spatial precision for various applications.

This article first provides a historical and detailed introduction to the image formation models for diluted pupils array and their densified versions called hypertelescopes. We propose in particular an original derivation showing that densification using a periscopic setting like in Michelson’s 20 −  foot interferometer, or using inverted Galilean telescopes are fully equivalent. After a review based on previous reference studies (Tallon & Tallon-Bosc 1992; Labeyrie 1996; Aime 2008 and Aime et al. 2012), the introductory part ends with a tutorial section for simulating optical interferometric images produced by cophased arrays. We illustrate in details how the optical image formation model can be used to simulate hypertelescopes images, including sampling issues and their effects on the observed images.

In a second part of the article, we address the issue of restoring hypertelescope images and present numerical illustrations obtained for classical (constrained Maximum Likelihood) methods. We also provide a detailed survey of more recent deconvolution methods based on sparse representations and of their spread in interferometric image reconstruction.

The last part of the article is dedicated to two original and numerical studies. The first study shows by Monte Carlo simulations that the restoration quality achieved by constrained ML methods applied to photon limited images obtained from a diluted array on a square grid, or from a densified array (without spectral aliasing) on a grid, are essentially equivalent. The second study shows that it is possible to recover in hypertelescopes images quasi point sources that are not only far outside the clean field, but also superimposed on the replicas of other objects. This is true at least for the considered pupil array and in the limit of vanishing noise.

The image restoration is today an important part of the astrophysical data analysis. The denoising and the deblurring can be efficiently performed using multiscale transforms. The multiresolution analysis constitutes the fundamental pillar for these transforms. The discrete wavelet transform is introduced from the theory of the approximation by translated functions. The continuous wavelet transform carries out a generalization of multiscale representations from translated and dilated wavelets. The à trous algorithm furnishes its discrete redundant transform. The image denoising is first considered without any hypothesis on the signal distribution, on the basis of the a contrario detection. Different softening functions are introduced. The introduction of a regularization constraint may improve the results. The application of Bayesian methods leads to an automated adaptation of the softening function to the signal distribution. The MAP principle leads to the basis pursuit, a sparse decomposition on redundant dictionaries. Nevertheless the posterior expectation minimizes, scale per scale, the quadratic error. The proposed deconvolution algorithm is based on a coupling of the wavelet denoising with an iterative inversion algorithm. The different methods are illustrated by numerical experiments on a simulated image similar to images of the deep sky. A white Gaussian stationary noise was added with three levels. In the conclusion different important connected problems are tackled.

In this paper, we consider the inverse problem of restoring an unknown signal or image, knowing the transformation suffered by the unknowns. More specifically we deal with transformations described by a linear model linking the unknown signal to an unnoisy version of the data. The measured data are generally corrupted by noise. This aspect of the problem is presented in the introduction for general models. In Section 2, we introduce the linear models, and some examples of linear inverse problems are presented. The specificities of the inverse problems are briefly mentionned and shown on a simple example. In Section 3, we give some information on classical distances or divergences. Indeed, an inverse problem is generally solved by minimizing a discrepancy function (divergence or distance) between the measured data and the model (here linear) of such data. Section 4 deals with the likelihood maximization and with their links with divergences minimization. The physical constraints on the solution are indicated and the Split Gradient Method (SGM) is detailed in Section 5. A constraint on the inferior bound of the solution is introduced at first; the positivity constraint is a particular case of such a constraint. We show how to obtain strictly, the multiplicative form of the algorithms. In a second step, the so-called flux constraint is introduced, and a complete algorithmic form is given. In Section 6 we give some brief information on acceleration method of such algorithms. A conclusion is given in Section 7.

We describe recently proposed algorithms, denoted scaled gradient projection (SGP) methods, which provide efficient and accurate reconstructions of astronomical images. We restrict the presentation to the case of data affected by Poisson noise and of nonnegative solutions; both maximum likelihood and Bayesian approaches are considered. Numerical results are presented for discussing the practical behaviour of the SGP methods.

This article deals with the problem of minimization of a general cost function under non-negativity and flux conservation constraints. The proposed algorithm is founded on the Split Gradient Method (SGM) adapted here to solve the Non Negative Matrix Factorization (NMF). We show that SGM can be easily regularized, allowing to introduce some physical constraints. Finally, to validate the algorithm, we propose an example of application to hyperspectral data unmixing.

In this paper, we describe two fully Bayesian algorithms that have been previously proposed to unmix hyperspectral images. These algorithms relies on the widely admitted linear mixing model, i.e. each pixel of the hyperspectral image is decomposed as a linear combination of pure endmember spectra. First, the unmixing problem is addressed in a supervised framework, i.e., when the endmembers are perfectly known, or previously identified by an endmember extraction algorithm. In such scenario, the unmixing problem consists of estimating the mixing coefficients under positivity and additivity constraints. Then the previous algorithm is extended to handle the unsupervised unmixing problem, i.e., to estimate the endmembers and the mixing coefficients jointly. This blind source separation problem is solved in a lower-dimensional space, which effectively reduces the number of degrees of freedom of the unknown parameters. For both scenarios, appropriate distributions are assigned to the unknown parameters, that are estimated from their posterior distribution. Markov chain Monte Carlo (MCMC) algorithms are then developed to approximate the Bayesian estimators.

In this paper we present a method for hyper-spectral image restoration for integral field spectrographs (IFS) data. We specifically address two topics: (i) the design of a fast approximation of spectrally varying operators and (ii) the comparison between two kind of regularization functions: quadratic and spatial sparsity functions. We illustrate this method with simulations coming from the Multi Unit Spectroscopic Explorer (MUSE) instrument. It shows the clear increase of the spatial resolution provided by our method as well as its denoising capability.

Spectral unmixing is an important issue to analyze remotely sensed hyperspectral data. This involves the decomposition of each mixed pixel into its pure endmember spectra, and the estimation of the abundance value for each endmember. Although linear mixture models are often considered because of their simplicity, there are many situations in which they can be advantageously replaced by nonlinear mixture models. In this chapter, we derive a supervised kernel-based unmixing method that relies on a pre-image problem-solving technique. The kernel selection problem is also briefly considered. We show that partially-linear kernels can serve as an appropriate solution, and the nonlinear part of the kernel can be advantageously designed with manifold-learning-based techniques. Finally, we incorporate spatial information into our method in order to improve unmixing performance.