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La microscopie holographique en ligne permet l'observation d'échantillons microscopiques transparents, d’où son intérêt en microbiologie. L'information accessible de phase et d'absorption requiert des algorithmes de reconstruction robustes, à élaborer en lien étroit avec le design optique du montage. Ce chapitre propose de faire un état de l'art dans ce domaine, en s'appuyant principalement sur la méthodologie des problèmes inverses.

In-line digital holography is a simple yet powerful tool to image absorbing and/or phase objects. However, the holograms of interest are corrupted by the background signal due to unwanted scattering elements located in the optical path. Using only two holograms of the same object, shifted to different locations, an inverse problems approach is applied to jointly estimate the complex transmittance of the sample and the contribution of the interferent background signal at the sensor plane. Experimental results with stained bacteria are presented and show improved reconstructions of the sample while also accounting for the background contribution.

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In this paper, we introduce a deep learning model based on Wavenet to denoise periodic signals containing some strong discontinuities, where the dataset used for training contains only synthetic data. We introduce a new cost function using a total variation term. The synthetic data which contain strong discontinuities, are generated as the sum of a sine wave, a square signal and a white gaussian noise. This simple model is very time-efficient to compute and allows us to perform data generation for each training of the architecture instead of physically storing the dataset. We specifically apply this model to real friction signals obtained through a rotating tribological system. We also compared our method with an improved TV denoising algorithm.

In the context of inverse problems based 3D image reconstruction for tomographic diffractive microscopy, we propose a simulation study for evaluating the potential of the Generalized Stein Unbiased Risk Estimator for automatically tuning the regularization parameters.

We propose to reconstruct biological samples using calibration beads to perform numerical autofocusing in in-line holographic microscopy. The approach is based on a joint inversion using both a parametric and a regularized inversion method.

In this paper, we propose to present the general ingredients involved in an inverse problems methodology dedicated to the reconstruction of in-line holograms, and compare it with the classical Gercherg-Saxton or Fienup alternating projections strategies for phase retrieval [1,2,3]. An inverse approach [4,5] consists in retrieving an optimal solution to a reconstruction/estimation problem from a dataset, knowing an approximate model of its formation process. The problem is generally formulated as an optimization problem that aims at fitting the model to the data, while favoring a priori knowledge on the targeted information using regularizations and constraints. An appropriate resolution method has to be designed, based on a convex optimization framework. We develop the end-to-end inverse problems methodology on a case-study : the reconstruction of an in-line hologram of a collection of weakly dephasing objects. This simple problem allows us to explain current physical considerations (type of objects, diffraction physics) to derive the appropriate model, and to present classical constraints and regularizations that can be used in image reconstruction. Starting from these ingredients, we introduce a simple yet efficient method to solve this inverse problem, belonging to the class of proximal gradient algorithms [6,7]. A special focus is made on the connections between the numerous alternating projections strategies derived from Fienup’s phase retrieval technique and the inverse problems framework. In particular, an interpretation of Fienup’s algorithm as iterates of a proximal gradient descent for a particular cost function is given. We discuss the advantages provided by the inverse problems methodology. We illustrate both strategies on reconstructions from simulated and experimental holograms of micrometric beads. The results show that the transition from alternating projection techniques to the inverse problems formulation is straightforward and advantageous.

International audience; This paper includes a tutorial on how to reconstruct in-line holograms using an inverse problems approach, starting with modeling the observations, selecting regularizations and constraints, and ending with the design of a reconstruction algorithm. A special focus is made on the connections between the numerous alternating projection strategies derived from Fienup's phase retrieval technique and the inverse problems framework. In particular, an interpretation of Fienup's algorithm as iterates of a proximal gradient descent for a particular cost function is given. Reconstructions from simulated and experimental holograms of micrometric beads illustrate the theoretical developments. The results show that the transition from alternating projection techniques to the inverse problems formulation is straightforward and advantageous.

International audience; In-line digital holography is a simple and powerful tool to image absorbing and/or phase objects in numerous fields such as crystallography, biology or fluid mechanics. Nevertheless, this kind of interference imaging technique leads to a loss of the phase of the complex wave front on the sensor. This lack of phase information can be critical in the reconstruction process. Thus, the simplicity of the setup must be balanced by dedicated reconstruction algorithm to retrieve the object from its hologram, such as inverse approaches. In the case of simple objects for which an analytical model of propagation is known, parametric algorithms are very effective. But these approaches fail at reconstructing more complex objects, where non-parametric solutions must be involved. This may lead to a loss in precision or specificity. In this work we propose a new approach combining these two methods to take benefits from their own advantages. The object to reconstruction is split in two subparts. A part is described by a parametric model. The other part of the object is simulated via a non-parametric model. These two parts which interfere are jointly considered in the reconstruction algorithm by alternating parametric and non-parametric procedures. We apply this new technique to evaporating droplets where the high contrast fringes produced by the droplets tend to mask the fringes produced by the plume. With our method, both the droplet and the plume are jointly reconstructed.

Lensless color microscopy is a recent 3D quantitative imaging method allowing to retrieve physical parameters characterizing microscopic objects spread in a volume. The main advantages of this technique are related to its simplicity, compactness, low sensitivity of the setup to vibrations and the possibility to accurately characterize objects. The cost-effectiveness of the method can be further increased using low-end laser diodes as coherent sources and CMOS color sensor equipped with a Bayer filter array. However, the central wavelength delivered by this type of laser is generally known only with a limited precision and can evolve because of its dependence on temperature and power supply voltage. In addition, Bayer-type filters of conventional color sensors are not very selective, resulting in spectral mixing (crosstalk phenomenon) of signals from each color channel. Ignoring these phenomena leads to significant errors in holographic reconstructions. We have proposed a maximum likelihood estimation method to calibrate the setup (central wavelength of the laser sources and spectral mixing introduced by the Bayer filters) using spherical objects naturally present in the field of view or added (calibration objects). This calibration method provides accurate estimates of the wavelengths and of the crosstalk, with an uncertainty comparable to that of a high-resolution spectrometer. To perform the image reconstruction from color holograms following the self-calibration of the setup, we describe a regularized inversion method that includes a linear hologram formation model, sparsity constraints and an edge-preserving regularization. We show on holograms of calibrated objects that the self-calibration of the setup leads to an improvement of the reconstructions.

International audience; Phase retrieval reconstruction is a central problem in digital holography, with various applications in microscopy, biomedical imaging, fluid mechanics. In an in-line configuration, the particular difficulty is the non-linear relation between the object phase and the recorded intensity of the holograms, leading to high indeterminations in the reconstructed phase. Thus, only efficient constraints and a priori information, combined with a finer model taking into account the non-linear behaviour of image formation, will allow to get a relevant and quantitative phase reconstruction. Inverse problems approaches are well suited to address these issues, only requiring a direct model of image formation and allowing the injection of priors and constraints on the objects to reconstruct, and hence offer good warranties on the optimality of the expected solution. In this context, following our previous works in digital in-line holography, we propose a regularized reconstruction method that includes several physicallygrounded constraints such as bounds on transmittance values, maximum/minimum phase, spatial smoothness or the absence of any object in parts of the field of view. To solve the non-convex and non-smooth optimization problem induced by our modeling, a variable splitting strategy is applied and the closed-form solution of the sub-problem (the so-called proximal operator) is derived. The resulting algorithm is efficient and is shown to lead to quantitative phase estimation of micrometric objects on reconstructions of in-line holograms simulated with advanced models using Mie theory. Then we discuss the quality of reconstructions from experimental inline holograms obtained from two different applications of in-line digital holography: tracking of an evaporating droplet (size~100μm) and microscopy of bacterias (size~1μm). The reconstruction algorithm and the results presented in this proceeding have been initially published in [Jolivet et al., 2018].

International audience; Reconstruction of phase objects is a central problem in digital holography, whose various applications include microscopy, biomedical imaging, and fluid mechanics. Starting from a single in-line hologram, there is no direct way to recover the phase of the diffracted wave in the hologram plane. The reconstruction of absorbing and phase objects therefore requires the inversion of the non-linear hologram formation model. We propose a regularized reconstruction method that includes several physically-grounded constraints such as bounds on transmittance values, maximum/minimum phase, spatial smoothness or the absence of any object in parts of the field of view. To solve the non-convex and non-smooth optimization problem induced by our modeling, a variable splitting strategy is applied and the closed-form solution of the sub-problem (the so-called proximal operator) is derived. The resulting algorithm is efficient and is shown to lead to quantitative phase estimation on reconstructions of accurate simulations of in-line holograms based on the Mie theory. As our approach is adaptable to several in-line digital holography configurations, we present and discuss the promising results of reconstructions from experimental in-line holograms obtained in two different applications: the tracking of an evaporating droplet (size ∼ 100µm) and the microscopic imaging of bacteria (size ∼ 1µm).

International audience; We propose a new imaging platform based on lens-free time-lapse microscopy for 3D cell culture and its dedicated algorithm lying on a fully 3D regularized inverse problem approach. First 3D + t results are presented.

International audience; In conventional helical computed tomography (CT), the field-of-view is a cylinder centered on the axis of the helix. Here, we consider the situation where all measurement lines are blocked except those intersecting a small cylindrical region-ofinterest (ROI) not necessarily centered on the axis of the system. We address the question of image reconstruction inside the ROI. The patient boundary is assumed known, and we avoid the “interior problem” by assuming that the ROI includes part of the patient boundary. By applying analytic image reconstruction theory, we show that the entire cylindrical ROI can be reconstructed provided the pitch of the helix does not violate the well-known Tam-Danielsson detector condition. Using an iterative algorithm, we performed ROI reconstruction from simulated phantom data and from real patient data, and compared the results with fullfield reconstructions. Visually, the ROI reconstructed images perfectly matched the full-field reconstructions. However, there were small quantitative discrepancies near the interior boundaries of the ROIs, which we attribute to the known reduced stability at onesideoftheinversetruncatedHilberttransform.Inconclusion, we have demonstrated mathematically that accurate transverse ROI reconstruction is possible for helical CT, although care must be taken near the interior boundary to achieve quantitative accuracy.

International audience; We propose a three-dimensional (3D) imaging platform based on lens-free microscopy to perform multiangle acquisitions on 3D cell cultures embedded in extracellular matrices. Lens-free microscopy acquisitions present some inherent issues such as the lack of phase information on the sensor plane and a limited angular coverage. We developed and compared three different algorithms based on the Fourier diffraction theorem to obtain fully 3D reconstructions. These algorithms present an increasing complexity associated with a better reconstruction quality. Two of them are based on a regularized inverse problem approach. To compare the reconstruction methods in terms of artefact reduction, signal-to-noise ratio, and computation time, we tested them on two experimental datasets: an endothelial cell culture and a prostate cell culture grown in a 3D extracellular matrix with large reconstructed volumes up to ∼5 mm3∼5 mm3 with a resolution sufficient to resolve isolated single cells. The lens-free reconstructions compare well with standard microscopy.

In-line digital holography is a simple yet powerful tool to image absorbing and/or phase objects. Nevertheless, the loss of the phase of the complex wavefront on the sensor can be critical in the reconstruction process. The simplicity of the setup must thus be counterbalanced by dedicated reconstruction algorithms, such as inverse approaches, in order to retrieve the object from its hologram. In the case of simple objects for which the diffraction pattern produced in the hologram plane can be modeled using few parameters, a model fitting algorithm is very effective. However, such an approach fails to reconstruct objects with more complex shapes, and an image reconstruction technique is then needed. The improved flexibility of these methods comes at the cost of a possible loss of reconstruction accuracy. In this work, we combine the two approaches (model fitting and regularized reconstruction) to benefit from their respective advantages. The sample to be reconstructed is modeled as the sum of simple parameterized objects and a complex-valued pixelated transmittance plane. These two components jointly scatter the incident illumination, and the resulting interferences contribute to the intensity on the sensor. The proposed hologram reconstruction algorithm is based on alternating a model fitting step and a regularized inversion step. We apply this algorithm in the context of fluid mechanics, where holograms of evaporating droplets are analyzed. In these holograms, the high contrast fringes produced by each droplet tend to mask the diffraction pattern produced by the surrounding vapor wake. With our method, the droplet and the vapor wake can be jointly reconstructed.

International audience; In this paper we propose comparisons and correlations between analytic and iterative fan-beam reconstruction approaches when object's rigid motion and data truncation occur during a circular scan. Based on our recent work presenting an exact analytic reconstruction method, we are able to predict the field of theoretically reconstructible points and transform the problem from a dynamic to a static point a view where the source trajectory is virtually modified taking into account the known rigid motion. We implement the iterative reconstruction as the convex minimization of a data-fidelity term under non-negativity constraint and regularization to solve this virtually static inverse problem. We compare the reconstructed field of view by the two methods on 2D fan-beam Shepp-Logan phantom simulations. Our results show that both methods validate the predicted reconstructible zone and are in good correlation in terms of reconstruction quality. The iterative reconstruction also demonstrates that in certain cases it is possible to recover structures beyond the analytic strict frontier of reconstructibilty.

International audience; New microscopes are needed to help realize the full potential of 3D organoid culture studies bygathering large quantitative and systematic data over extended period of time while preserving the integrityof the living sample. On another hand, lensfree video microscopy is addressing these needs in the context of2D cell culture, providing label-free and non-phototoxic acquisition of large datasets. As scientistsroutinely adopt 3D culture techniques, the new challenging task is to extend lensfree microscopy techniquesto 3D structures.In order to image such large volume while preserving the ability to catch every single cell, we proposenew imaging platforms based on lensfree microscopy. We have built lensfree diffractive tomography setupsperforming multi-angle acquisitions of 3D organoid culture embedded in Matrigel® and developed adedicated 3D holographic reconstruction algorithm based on the Fourier diffraction theorem.With a first prototype we have been able to reconstruct a 3D volume as large as 21.5 mm^3 of a 3Dorganoid culture of prostatic RWPE1 cells in Matrigel® showing the ability of these cells to assemble in 3Dintricate cellular network at the mesoscopic scale. Comparisons with 2D images showed that it ispossible to detect single cells isolated from the main cellular structure.From results of the first prototype and numerical simulations, a second prototype wasdesigned to improve the reconstructions quality, especially on the vertical axis. First results on dust andbubbles lying at the bottom of a Petri® dish are promising.The adaptation of lensfree microscopy techniques to 3D organoid cultures imaging and thepresentation of the associated algorithms are the scopes of the present communication.

National audience; New microscopes are needed to help realize the full potential of 3D organoid culture studies by gathering large quantitative and systematic data over extended period of time while preserving the integrity of the living sample. In order to reconstruct large volume while keeping the ability to catch every single cell, we propose new imaging platforms based on lensfree microscopy, a technic which is addressing these needs in the context of 2D cell culture, providing label-free and non-phototoxic acquisition of large datasets. We have built lensfree diffractive tomography setups performing multi-angle acquisitions of 3D organoid culture embedded in Matrigel ® and developed dedicated 3D holographic reconstruction algorithms based on the Fourier diffraction theorem. Nonetheless, holographic setups do not record the phase of the incident wavefront and the biological samples in Petri dish strongly limit the angular coverage. These limitations introduces numerous artefacts in the sample reconstruction. We developed several methods to overcome them, such as multi wavelength imaging or iterative phase retrieval. The most promising technic currently developed is based on a regularized inverse problem approach directly performed on the 3D volume to reconstruct. 3D reconstructions were realized on several complex samples such as 3D networks or spheroids embedded in capsules with large reconstructed volumes up to ~25 mm^3 while still being able to identify single cells. To our knowledge, this is the first time that such an inverse problem approach is implemented in the context of lensfree diffractive tomography enabling to reconstruct large volume of unstained biological samples.

Lensfree imaging is an emerging microscopy technique based on in-line holography as invented by Gabor in 1948. Albeit the existence of the method since decades, the recent development of digital sensors, helped the realization of its full potential. Over the recent years, innovations and improvements in CMOS imaging technology design and fabrication have allowed to decrease the pixel pitch down to ∼1μm and the number of pixels has dramatically increased up to 250 million of pixels. As a result, the performance of lensfree microscopy, which features a bare CMOS sensor without any magnification optics, have tremendously increased while keeping the design simple, robust, and at a reasonable low cost. The detection ability improved from 10 μm (cell) in 2009, to 1 μm (bacteria) in 2010, down to 100 nm beads in 2012, paving the way to the detection of viruses in 2013.

In order to extend the analysis of the datasets produced by lensfree video microscopy we have implemented a cell tracking algorithm to combine and correlate cell motility to the previously devised metrics to quantify e.g. cell adhesion and spreading, cell division, and cell death. In this paper we present the assessment of these new methodology on experiments involving three different cell lines, namely 3T3 fibroblast cells, primary HUVEC cells and macrophage THP1 cells. We demonstrate that the good spatial resolution and the fast frame rate obtained with of our lensfree video microscope allows standard cell tracking algorithm to be computed. The results is the possibility to analyze thousands of cells successfully tracked over tens of hours. The results is the possibility to compare different cell cultures in terms of e.g. cell motility and cell confinement ration. Ultimately we managed to measure the doubling time at single cell level over a large number of N=235 cells tracked over two days.

In this paper, we discuss a new methodology based on lensfree imaging to perform wound healing assay with unprecedented statistics. Our video lensfree microscopy setup is a simple device featuring only a CMOS sensor and a semi coherent illumination system. Yet it is a powerful mean for the real-time monitoring of cultivated cells. It presents several key advantages, e.g. integration into standard incubator, compatibility with standard cell culture protocol, simplicity and ease of use. It can perform the follow-up in a large field of view (25 mm(2)) of several crucial parameters during the culture of cells i.e. their motility, their proliferation rate or their death. Consequently the setup can gather large statistics both in space and time. Here we uses this facility in the context of wound healing assay to perform label-free measurements of the velocities of the fronts of proliferation of the cell layer as a function of time by means of particle image velocimetry (PIV) processing. However, for such tissue growth experiments, the field of view of 25 mm(2) remains not sufficient and results can be biased depending on the position of the device with respect to the recipient of the cell culture. Hence, to conduct exhaustive wound healing assays, we propose to enlarge the field of view up to 10 cm(2) through a raster scan, by moving the source/sensor with respect to the Petri dish. We have performed acquisitions of wound healing assay (keratinocytes HaCaT) both in real-time (25 mm(2)) and in final point (10 cm(2)) to assess the combination of velocimetry measurements and final point wide field imaging. In the future, we aim at combining directly our extended field of view acquisitions (10 cm(2)) with real time ability inside the incubator.

In this paper, we discuss a new methodology based on lens-free imaging to perform wound healing assay with unprecedented statistics. Our video lens-free microscopy setup is a simple optical system featuring only a CMOS sensor and a semi coherent illumination system. Yet it is a powerful means for the real-time monitoring of cultivated cells. It presents several key advantages, e.g., integration into standard incubator, compatibility with standard cell culture protocol, simplicity and ease of use. It can perform the follow-up in a large field of view (25 mm 2 ) of several crucial parameters during the culture of cells i.e. their motility, their proliferation rate or their death. Consequently the setup can gather large statistics both in space and time. But in the case of tissue growth experiments, the field of view of 25 mm 2 remains not sufficient and results can be biased depending on the position of the device with respect to the recipient of the cell culture. Hence, to conduct exhaustive wound healing assay, here we propose to enlarge the field of view up to 10 cm 2 through two different approaches. The first method consists in performing a scan of the cell culture by moving the source/sensor couple and then stitch the stack of images. The second is to make an acquisition by scanning with a line scan camera. The two approaches are compared in term of resolution, complexity and acquisition time. Next we have performed acquisitions of wound healing assay (keratinocytes HaCaT) both in real-time (25 mm 2 ) and in final point (10 cm 2 ) to assess the combination of these two complementary modalities. In the future, we aim at combining directly super wide field of view acquisitions (>10 cm 2 ) with real time ability inside the incubator.

National audience; The study of in vitro cell populations is a major concern for biologists in the realms of drug screening and oncology for example. With standard microscopy techniques, crucial information at the mesoscopic scale is still difficult to reach due to reduced field of view and complexity to monitor cell behaviors in real time over large periods of time. Lensfree video microscopy has already overcome these difficulties in the context of 2D cell biology. Based on the in-line holography principle, it puts cells directly between a coherent light source and a sensor allowing large field of view, e.g. 30 mm 2. With the growth of 3D cell cultures, the new challenging task is to extend this imaging technique to acquisitions and 3D reconstructions of cellular culture. For this purpose, we have developed an experimental optical bench dedicated to lensfree diffractive tomography and studied several 3D reconstruction algorithms on acquisitions of 3D prostatic cell cultures.

International audience; L'étude de l'évolution et de l'organisation de populations de cellules cultivées in vitro intéresse les biologistes depuis plusieurs dizaines d'années. À ces fins, d'importants progrès ont été réalisés dans les méthodes d'imagerie à l'échelle microscopique. Cependant, certaines informations demeurent inaccessibles, notamment à l'échelle mésoscopique, en raison du champ de vue réduit, ainsi que la complexité et le coût pour réaliser des acquisitions hors incubateur en temps réel sur de longues périodes. En réponse à ces limitations, nous avons développé la vidéo-microscopie sans lentille, en plaçant directement les cellules vivantes sur un capteur numérique en regard d'une illumination cohérente selon le principe de l'holographie en ligne. Cette technique permet l'observation d'une culture cellulaire sur un large champ de vue (24 mm² soit plusieurs dizaines de milliers de cellules), et ce à l'intérieur même de l'incubateur, autorisant de surcroît des acquisitions dynamiques couvrant des périodes allant de quelques jours à plusieurs semaines. À partir des images holographiques brutes acquises, nous pouvons remonter aux images refocalisées par reconstruction numérique jusqu'à une résolution de 2µm. Le traitement de ces images donne accès à des niveaux d'information quantifiables allant de la cellule unique à l'organisation inter-individus de la population. Avec des premières études sur des cultures standard de cellules sur substrat 2D, nous sommes aujourd'hui en mesure, avec notre dispositif et la force de l'imagerie holographique, d'explorer et d'étudier la vie cellulaire en 3D, nous rapprochant un peu plus de la réalité physiologique des phénomènes biologiques.

International audience; Quantification of basic cell functions is a preliminary step to understand complex cellular mechanisms, for e.g., to test compatibility of biomaterials, to assess the effectiveness of drugs and siRNAs, and to control cell behavior. However, commonly used quantification methods are label-dependent, and end-point assays. As an alternative, using our lensfree video microscopy platform to perform high-throughput real-time monitoring of cell culture, we introduce specifically devised metrics that are capable of non-invasive quantification of cell functions such as cell-substrate adhesion, cell spreading, cell division, cell division orientation and cell death. Unlike existing methods, our platform and associated metrics embrace entire population of thousands of cells whilst monitoring the fate of every single cell within the population. This results in a high content description of cell functions that typically contains 25,000 - 900,000 measurements per experiment depending on cell density and period of observation. As proof of concept, we monitored cell-substrate adhesion and spreading kinetics of human Mesenchymal Stem Cells (hMSCs) and primary human fibroblasts, we determined the cell division orientation of hMSCs, and we observed the effect of transfection of siCellDeath (siRNA known to induce cell death) on hMSCs and human Osteo Sarcoma (U2OS) Cells.

We have developed “Lensfree video microscopy” to bring new perspectives to cell biology, with its ability to perform real-time cell culture monitoring, with a field of view of 24mm2 encompassing several thousand cells

International audience; Data modelization in tomography is a key point for iterative reconstruction. The design of the projector starts with the representation of the object of interest, decomposed on a discrete basis of functions. Standard models of projector such as ray driven, or more advanced models such as distance driven, use simple cubic voxels, which result in modelization errors due to their anisotropic behaviour. Moreover approximations made at the projection step increase these errors. Long, Fessler and Balter reduce approximation errors by projecting the cubic voxels more accurately. However anisotropy errors still hold. Spherically symmetric volume elements (blobs) eradicate them, but at the cost of increased complexity. We propose a compromise between these two approaches by using B-splines as basis functions. Their quasi-isotropic behaviour allows to avoid projection inconsistencies, while conserving local influence. Small approximations transform the exact footprint (projection of the basis function) into a separable function, which does not depend on the angle of projection, and is easier and faster to integrate on detector pixels. We obtain a more accurate projector, with no additional computation cost. Such an improvement is particularly of interest in the case of dynamic gated X-ray CT, which can be considered as a tomographic reconstruction problem with very few projection data, and for which we show some preliminary results, with an original method of iterative reconstruction, using spatio-temporal regularization of the "space + time" sequence, and making no use of motion estimation.

International audience; Data modelization in tomography is a key point for iterative reconstruction. The design of the projector, i.e. the numerical model of projection, is mostly influenced by the representation of the object of interest, decomposed on a discrete basis of functions. Standard projector models are voxel or ray driven; more advanced models such as distance driven, use simple staircase voxels, giving rise to modelization errors due to their anisotropic behaviour. Moreover approximations made at the projection step amplify these errors. Though a more accurate projection could reduce approximation errors, characteristic functions of staircase voxels constitute a too coarse basis for representing a continuous function. As a result, pure modelization errors still hold. Spherically symmetric volume elements (blobs) have already been studied to eradicate such errors, but at the cost of increased complexity, because they require some tuning parameters for adapting them to this use. We propose to use 3D B-splines, which are piecewise polynomials, as basis functions. When the degree of these polynomials is sufficiently high, they are very close from being with a spherical symmetry, i.e. blobs, avoiding projection inconsistencies, while keeping local influence and separability property. B-splines are considered, in sampling theory, as the almost optimal functions for the discretization of a continuous signal, not necessarily band-limited, potentially allowing to reduce the angular sampling of the data without any loss of quality. We show that the projection of B-splines can be approximated rather accurately by a separable function, independent from the angle of projection, easier to integrate on detector pixels. The higher the degree of the used B-splines, the better the quality of the approximation, but also the larger the number of required operations. Thanks to these approximations, a convenient tradeoff between the need of accuracy and a fast calculation can be obtained. This has resulted in the implementation of a more accurate numerical projector, which can deal with a reduced angular sampling without loss of performance. The additional computation cost is also efficiently reduced. We have studied the quality of enhancement involved by this projector on 2D iterative reconstructions of a Shepp-Logan phantom, from a small number of fan beam projections. Reconstructions have been performed by optimization methods, minimizing the squared data residuals with a regularization term, using an efficient Quasi-Newton optimization algorithm.

A challenge of adaptive optics (AO) on Extremely Large Telescopes (ELTs) is to overcome the difficulty of solving a huge number of equations in real time, especially when atmospheric tomography is involved. This is particularly the case for multi-conjugate or multi-objects AO systems. In addition, the quality of the wavefront estimation is crucial to optimize the performances of the future systems in a situation where measurements are missing and noises are correlated. The Fractal Iterative Method has been introduced as a fast iterative algorithm for minimum variance wavefront reconstruction and control on ELTs. This method has been successfully tested on Classical Single Conjugate AO systems on Octopus numerical simulator at ESO. But the minimum variance approach is expected to be mostly useful with atmospheric tomography. We present the first results obtained with FrIM in the context of atmospheric tomography. We recall the principle of the algorithm and we summarize the formalism used for modeling the measurements obtained from laser guide stars that entail spot elongation and tip/tilt indetermination, mixed with low order measurements from natural guide stars. We show the respective effects of tip/tilt indetermination, spot elongation, unseen modes on various configurations, as well as the usefulness of priors and correct noise models in the reconstruction. This analysis is essential for balancing the various errors that combine in a quite complex way and to optimize the configuration of the future AO systems for specific science cases and instrument requirements.

Atmospheric tomography is a key element for many adaptive optics architectures on which the com- ing Extremely Large Telescopes rely. When modeling atmospheric tomography, the samples of the wavefronts in the turbulent layers are not aligned with the subapertures after propagation down to the wavefront sensors (WFS). So the classical models of Shack-Hartmann WFS (e.g. Fried's model) include resampling before performing the standard slopes calculation. This operation introduces errors and only approximates the response of the WFS (error of 10 to 20% of the standard deviation of the slopes). We introduce a continuous model that represents the wavefronts on a basis of continuous functions in the turbulent layers. Propagation and gradient measurements are then computed analytically, without any resampling. Assuming the separability of the basis functions on the two dimensions, we obtain a sparse operator that can be factorized in two components. This factorization enhances the speed close to that of the standard approach using interpolation. We obtain a fast accurate continuous WFS model suitable to wide field adaptive optics systems on ELTs.

National audience; En tomographie, les méthodes de reconstruction itératives nécessitent une modélisation discrète du processus d'obtention des mesures. La représentation de l'objet d'intérêt est le point de départ à l'élaboration d'un modèle de projection tomographique sur le détecteur, précis et rapide. Les modèles conventionnels ray driven et distance driven, construits à partir d'indicatrices de voxels, ont l'inconvénient d'être fortement anisotropes. Nous proposons un modèle utilisant des fonctions de bases B-splines de degré adéquat, nous fournissant un projecteur quasiment isotrope. Une approximation de la projection par une B-spline séparable sur le détecteur conduit à un modèle efficace en géométrie parallèle et conique, de qualité supérieure aux modèles conventionnels. Nous montrons notamment que l'erreur de modélisation est améliorée d'un facteur 10 par rapport au modèle distance driven. Nous illustrons l'amélioration de la qualité de reconstruction apportée par notre modèle sur des simulations du fantôme de Shepp-Logan, en utilisant une méthode itérative de reconstruction régularisée.

National audience; La tomographie dynamique est la reconstruction, à partir de projections, d'objets induits d'un mouvement, le plus souvent périodique (e.g. le cycle respiratoire chez un patient). Le problème de reconstruction devient alors 4-D (3-D spatiale + temps), à données parcimonieuses puisqu'une projection ne correspondra qu'à un instant spécifique de la séquence 4-D d'un cycle (ou période). Nous traitons la reconstruction dynamique comme un problème inverse global avec un terme d'attache aux données utilisant la totalité des projections. Les paramètres estimés sont l'image 4-D d'un cycle dynamique de l'objet. Le modèle de reprojection est calé temporellement sur le cycle d'acquisition des projections grâce à un signal temporel 1-D décrivant l'évolution dynamique de l'objet, et sa périodicité. Une étape d'interpolation temporelle de la séquence 4-D sur les dates d'acquisition précède alors la projection standard à un instant donné. Nous injectons également une régularisation spatio-temporelle de l'objet sous forme d'une variation totale 4-D. La régularisation apporte alors la corrélation temporelle entre les différentes tranches reconstruites, et permet ainsi d'extraire au mieux l'information fournie par les données, sans aucune estimation ni compensation de mouvement. Nous faisons la démonstration de notre approche sur des reconstructions 2-D+t d'un fantôme mécanique acquises sur un scanner Cone-Beam. La régularisation spatio-temporelle apporte un gain sans équivoque sur la qualité des reconstructions dynamiques. Des premiers résultats 4-D (3-D+t) encourageants sont obtenus sur données cliniques d'un patient en respiration.

International audience; Holographie numérique en ligne, Problème Inverse, Super-résolution numérique, Reconstruction Couleur, Filtre de Bayer. RESUME Ces dernières années le secteur des capteurs « bas coût » profite d'un marché de plus en plus dynamique (notamment avec l'avènement du smartphone, de l'appareil photo numérique…). Ainsi des capteurs couleur peu onéreux, et ayant des tailles de pixels de l'ordre du micromètre permettent de repousser les performances de l'holographie numérique en ligne. De plus l'utilisation d'approches problèmes inverses a permis de lever certaines limites des méthodes de reconstruction holographique habituellement utilisées : présence d'images jumelles, artefacts dûs à la troncature (effet de bord…). Elles permettent également une amélioration de la précision de reconstruction [1,2,3]. Ces approches se basent sur un modèle de formation d'image linéaire, approximation satisfaisante dans le régime de la diffraction de Fresnel pour les milieux dilués. Afin d'améliorer la résolution des reconstructions holographiques, des travaux ont montré l'intérêt d'utiliser une pile d'hologrammes d'un objet translaté transversalement [3,4]. De leur côté les travaux [5,6,7] ont montré tout l'intérêt d'utiliser un montage opérant à plusieurs longueurs d'onde (sources Rouge, Vert, Bleu) avec un capteur couleur (suppression des aberrations chromatiques…). Nous proposons ici une méthode de reconstruction holographique RGB Super-Résolue basée sur une approche inverse non-paramétrique. Pour cela nous proposons de résoudre le problème sous contrainte de positivité. La méthode proposée alterne des étapes de reconstructions régularisées et d'estimation des translations entre hologrammes et la reconstruction courante. En termes de résultat, l'approche inverse super-résolue couleur proposée permet d'améliorer la résolution spatiale et le rapport signal à bruit des hologrammes reconstruits.