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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 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; 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; 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.

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.