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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.

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; 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; 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; 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.