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International audience; PurposeComputed tomography (CT) is a technique of choice to image bone structure at different scales. Methods to enhance the quality of degraded reconstructions obtained from low-dose CT data have shown impressive results recently, especially in the realm of supervised deep learning. As the choice of the loss function affects the reconstruction quality, it is necessary to focus on the way neural networks evaluate the correspondence between predicted and target images during the training stage. This is even more true in the case of bone microarchitecture imaging at high spatial resolution where both the quantitative analysis of bone mineral density (BMD) and bone microstructure is essential for assessing diseases such as osteoporosis. Our aim is thus to evaluate the quality of reconstruction on key metrics for diagnosis depending on the loss function that has been used for training the neural network.MethodsWe compare and analyze volumes that are reconstructed with neural networks trained with pixelwise, structural, and adversarial loss functions or with a combination of them. We perform realistic simulations of various low-dose acquisitions of bone microarchitecture. Our comparative study is performed with metrics that have an interest regarding the diagnosis of bone diseases. We therefore focus on bone-specific metrics such as bone volume and the total volume (BV and TV), resolution, connectivity assessed with the Euler number, and quantitative analysis of BMD to evaluate the quality of reconstruction obtained with networks trained with the different loss functions.ResultsWe find that using L1norm as the pixelwise loss is the best choice compared to L2 or no pixelwise loss since it improves resolution without deteriorating other metrics. Visual Geometry Group (VGG) perceptual loss, especially when combined with an adversarial loss, allows to better retrieve topological and morphological parameters of bone microarchitecture compared to Structural SIMilarity (SSIM) index. This however leads to a decreased resolution performance. The adversarial loss enhances the reconstruction performance in terms of BMD distribution accuracy.ConclusionsIn order to retrieve the quantitative and structural characteristics of bone microarchitecture that are essential for postreconstruction diagnosis, our results suggest to use L1norm as part of the loss function. Then, trade-offs should be made depending on the application: VGG perceptual loss improves accuracy in terms of connectivity at the cost of a deteriorated resolution, and adversarial losses help better retrieve BMD distribution while significantly increasing the training time.

International audience; In this paper, we present a method of choice of an adaptative regularization parameter for data corrupted by Poisson noise based on a bilevel approach. The forward operator considered is nonlinear. The existence and unicity of the smoothed lower level problem, the differentiability properties of the constraint, and the adjoint method used to calculate the gradient of the reduced functional are studied in detail. The variance of the KL functional for Poisson noise is also investigated. The method is applied to the spectral CT inverse problem. Better reconstruction results are obtained with the bilevel method of choice than with a scalar regularization parameter.

International audience; Super resolution problems are widely discussed in medical imaging. Spatial resolution of medical images are not sufficient due to the constraints such as image acquisition time, low irradiation dose or hardware limits. To address these problems, different super resolution methods have been proposed, such as optimization or learning-based approaches. Recently, deep learning methods become a thriving technology and are developing at an exponential speed. We think it is necessary to write a review to present the current situation of deep learning in medical imaging super resolution. In this paper, we first briefly introduce deep learning methods, then present a number of important deep learning approaches to solve super resolution problems, different architectures as well as up-sampling operations will be introduced. Afterwards, we focus on the applications of deep learning methods in medical imaging super resolution problems, the challenges to overcome will be presented as well

Regularized algorithms are the state-of-the-art in computed tomography, but they are also very demanding in computer resources. In this work we test two data-fidelity formulations and some associated algorithms for the resolution of the Total-Variation regularized tomographic problem. We compare their computational cost for a mixture of Poisson and Gaussian noises. We show that a recently proposed MAP-EM algorithm outperforms the TV-regularized SIRT and the Chambolle-Pock algorithms on synthetic data for the considered noise. We illustrate this result on experimental data from transmission electron microscopy.

International audience; In this work, we study the reconstruction of blood velocity with contrast enhanced computed tomography with tomographic projections perpendicular to the main flow field direction. The inverse problem is regularized with a convection-diffusion partial differential equation and with the Navier-Stokes equation. The velocity field is reconstructed together with the density field with a first-order adjoint method. The method is validated on a simple phantom. The addition of a physical constraint with a partial differential equation for the velocity field improves the reconstruction results.

International audience; Thanks to photon-counting detectors, spectral computerized tomography records energy-resolved data from which the chemical composition of a sample can be recovered. This problem, referred to as material decomposition, can be formulated as a nonlinear inverse problem. In previous work, we proposed to decompose the projection images using a regularized Gauss-Newton algorithm. To reduce further the ill-posedness of the problem, we propose here to consider equality and inequality constraints that are based on physical priors. In particular, we impose the positivity of the solutions as well the total mass in each projection image. In practice, we first decompose the projection images for each projection angle independently. Then, we reconstruct the sample slices from the decomposed projection images using a standard filtered-back projection algorithm. The constrained material decomposition problem is solved by the alternating direction method of multipliers (ADMM). We compare the proposed ADMM algorithm to the unconstrained Gauss-Newton algorithm in a numerical thorax phantom. Including constraints reduces the cross-talk between materials in both the decomposed projections and the reconstructed slices.

Over the last decade, dual-energy CT scanners have gone from prototypes to clinically available machines, and spectral photon counting CT scanners are following. They require a specific reconstruction process, consisting of two steps: material decomposition and tomographic reconstruction. Image-based methods perform reconstruction, then decomposition, while projection-based methods perform decomposition first, and then reconstruction. As an alternative, 'one-step inversion' methods have been proposed, which perform decomposition and reconstruction simultaneously. Unfortunately, one-step methods are typically slower than their two-step counterparts, and in most CT applications, reconstruction time is critical. This paper therefore proposes to compare the convergence speeds of five one-step algorithms. We adapted all these algorithms to solve the same problem: spectral photon-counting CT reconstruction from five energy bins, using a three materials decomposition basis and spatial regularization. The paper compares a Bayesian method which uses non-linear conjugate gradient for minimization (Cai et al 2013 Med. Phys. 40 111916-31), three methods based on quadratic surrogates (Long and Fessler 2014 IEEE Trans. Med. Imaging 33 1614-26, Weidinger et al 2016 Int. J. Biomed. Imaging 2016 1-15, Mechlem et al 2018 IEEE Trans. Med. Imaging 37 68-80), and a primal-dual method based on MOCCA, a modified Chambolle-Pock algorithm (Barber et al 2016 Phys. Med. Biol. 61 3784). Some of these methods have been accelerated by using μ-preconditioning, i.e. by performing all internal computations not with the actual materials the object is made of, but with carefully chosen linear combinations of those. In this paper, we also evaluated the impact of three different μ-preconditioners on convergence speed. Our experiments on simulated data revealed vast differences in the number of iterations required to reach a common image quality objective: Mechlem et al (2018 IEEE Trans. Med. Imaging 37 68-80) needed ten iterations, Cai et al (2013 Med. Phys. 40 111916-31), Long and Fessler (2014 IEEE Trans. Med. Imaging 33 1614-26) and Weidinger et al (2016 Int. J. Biomed. Imaging 2016 1-15) several hundreds, and Barber et al (2016 Phys. Med. Biol. 61 3784) several thousands. We also sum up other practical aspects, like memory footprint and the need to tune extra parameters.

International audience; Spectral computed tomography exploits energy-resolved detectors to recover the material composition of an object. Material decomposition is a challenging nonlinear and ill-posed inverse problem. While regularization improves the decomposition , the resulting material maps do not satisfy physical constraints (e.g., positivity). In this work, we propose a fast second-order algorithm for constrained material decomposition. The proposed constrained Gauss-Newton algorithm is compared to a standard (unconstrained) Gauss-Newton algorithm on two realistic numerical phantoms. An improved decomposition is obtained for both phantoms. We also found that the constraints must be enforced progressively during the iterations.

In this work, we study the reconstruction of blood velocity with contrast enhanced computed tomography with a tomographic projections perpendicular to the main flow field direction. The inverse problem is regularized with a convection-diffusion partial differential equation. The velocity field is reconstructed with first order and second order adjoint methods with a receding optimal control method and tested on simple phantoms.

In this work, we investigate an inverse problem approach to 3D super-resolution/segmentation for an application to the analysis of trabecular bone micro-architecture from in vivo 3D X-ray CT images. The problem is expressed as the minimization of a functional including a data term and a prior. We consider here a regularization term combining total variation (TV) and a double-well potential to enforce the quasi-binarity of the resulting image. Three different schemes to minimize this nonconvex functional are presented and compared. The methods are applied to experimental new high-resolution peripheral quantitative CT images (voxel size $$82\,\upmu \hbox {m}$$ ) and evaluated with respect to a micro-CT image at higher spatial resolution (voxel size $$41\,\upmu \hbox {m}$$ ) considered as a ground truth. Our results show that a combination of double-well functional and TV term improves the contrast and the quality of the restoration even if the connectivity may be degraded.

International audience; While standard computed tomography (CT) data do not depend on energy, spectral computed tomography (SPCT) acquire energy-resolved data, which allows material decomposition of the object of interest. Decompo-sitions in the projection domain allow creating projection mass density (PMD) per materials. From decomposed projections, a tomographic reconstruction creates 3D material density volume. The decomposition is made possible by minimizing a cost function. The variational approach is preferred since this is an ill-posed non-linear inverse problem. Moreover, noise plays a critical role when decomposing data. That is why in this paper, a new data fidelity term is used to take into account of the photonic noise. In this work two data fidelity terms were investigated: a weighted least squares (WLS) term, adapted to Gaussian noise, and the Kullback-Leibler distance (KL), adapted to Poisson noise. A regularized Gauss-Newton algorithm minimizes the cost function iteratively. Both methods decompose materials from a numerical phantom of a mouse. Soft tissues and bones are decomposed in the projection domain; then a tomographic reconstruction creates a 3D material density volume for each material. Comparing relative errors, KL is shown to outperform WLS for low photon counts, in 2D and 3D. This new method could be of particular interest when low-dose acquisitions are performed.

Le scanner X spectral est une nouvelle modalité d’imagerie qui permet de reconstruire les concentrations des matériaux constituant l’objet imagé. Les images 3D peuvent être obtenues en décomposant les projections en base de matériaux[1], avant l’étape de reconstruction tomographique. L’objectif de ce travail est de proposer une solution originale pour la décomposition en base de matériaux.

International audience; Purpose: Exploiting the X-ray measurements obtained in different energy bins, spectral computed tomography (CT) has the ability to recover the 3D description of a patient in a material basis. This may be achieved solving two subproblems, namely the material decomposition and the tomographic reconstruction problems. In this work, we address the material decomposition of spectral x-ray projection images, which is a non-linear ill-posed problem. Methods: Our main contribution is to introduce a material-dependent spatial regularization in the projection domain. The decomposition problem is solved iteratively using a Gauss-Newton algorithm that can benefit from fast linear solvers. A Matlab implementation is available online. The proposed regularized weighted least squares Gauss-Newton algorithm (RWLS-GN) is validated on numerical simulations of a thorax phantom made of up to five materials (soft tissue, bone, lung, adipose tissue and gadolinium), which is scanned with a 120 kV source and imaged by a 4-bin photon counting detector. To evaluate the method performance of our algorithm, different scenarios are created by varying the number of incident photons, the concentration of the marker and the configuration of the phantom. The RWLS-GN method is compared to the reference maximum likelihood Nelder-Mead algorithm (ML-NM). The convergence of the proposed method and its dependence on the regularization parameter are also studied.Results: We show that material decomposition is feasible with the proposed method and that it converges in few iterations. Material decomposition with ML-NM was very sensitive to noise, leading to decomposed images highly affected by noise and artifacts even for the best case scenario. The proposed method was less sensitive to noise and improved contrast-to-noise ratio of the gadolinium image. Results were superior to those provided by ML-NM in terms of image quality and decomposition was 70 times faster. For the assessed experiments, material decomposition was possible with the proposed method when the number of incident photons was equal or larger than 1e5 and when the marker concentration was equal or larger than 0.03 g/cm3.Conclusions: The proposed method efficiently solves the nonlinear decomposition problem for spectral CT, which opens up new possibilities such as material-specific regularization in the projection domain and a parallelization framework, in which projections are solved in parallel.

International audience; Context: Spectral Computed Tomography (SPCT) acquires energy-resolved projections that can be used to evaluate the concentrations of the different materials of an object. This can be achieved by decomposing the projection images in a material basis, in a first step, followed by a standard tomographic reconstruction step. However, material decomposition is a challenging non-linear ill-posed inverse problem, which is very sensitive to the photonic noise corrupting the data. Methods: In this paper, material decomposition is solved in a variational framework. We investigate two fidelity terms: a weighted least squares (WLS) term adapted to Gaussian noise and the Kullback-Leibler (KL) distance adapted to Poisson Noise. The decomposition problem is then solved iteratively by a regularized Gauss-Newton algorithm. Results: Simulations are performed in a numerical mouse phantom and the decompositions show that KL outperforms WLS for low numbers of photons, i.e., for high photonic noise. Conclusion: In this work, a new method for material decomposition in SPCT is presented. It is based on the KL distance to take into account the noise statistics. This new method could be of particular interest when low-dose acquisitions are performed.

International audience; The discrete tomographic reconstruction problem is generally considered for binary image. In this work, we consider the reconstruction of an image with more than two grey levels and compare two reconstruction methods. The first one is based on a classical TV regularization and the second one is a level-set regularization method. In this second method, the discrete tomographic problem is formulated as a shape optimization problem with several level-set functions and regularized with Total Variation-Sobolev terms. The two methods are applied to an image size of 128 × 128, with several additive Gaus-sian noises on the raw projection data and several number of projections.

International audience; The analysis of trabecular bone micro structure from in-vivo CT images is still limited due to insufficient spatial resolution. The goal of this work is to address both the problem of increasing the resolution of the image and of the segmenta-tion of the bone structure. To this aim, we investigate the joint super-resolution/segmentation problem by an approach based on the Mumford-Shah model. The validation of the method is performed on blurred, noisy and down-sampled images. A comparison of the reconstruction results with the Total Variation regularization is showed.

International audience; Spectral computed tomography (CT) exploits measurements from x-rays with different energies to obtain the 3D description of the patient in a material basis. It requires to solve two subproblems, namely the material decomposition and the tomographicreconstruction problems, either sequentially or jointly. In this work, we address the material decomposition problem, which an ill-posed non-linear problem. Our main contribution is to introduce a material-dependent spatial regularization scheme. The problem is solved iteratively using the Gauss-Newton’s method. The framework is validated on numerical experiments of a thorax phantom made of soft tissue, bone and gadolinium scanned with a 90 kV source and a 3-bin photon counting detector.

International audience; The investigation of bone fragility diseases, as osteoporosis, is based on the analysis of the trabecular bone microarchitecture. The aim of this paper is to improve the in-vivo trabecular bone segmentation and quantification by increasing the resolution of bone micro-architecture images. We propose a semi-blind joint super-resolution/segmentation approach based on a Total Variation regularization with a convex constraint. A comparison with the bicubic interpolation method and the non-blind version of the proposed method is shown. The validation is performed on blurred, noisy and down-sampled 3D synchrotron micro-CT bone images. Good estimates of the blur and of the high resolution image are obtained with the semi-blind approach. Preliminary results are obtained with the semi-blind approach on real HR-pQCT images.

International audience; In this work, we propose a stochastic level-set method to reconstruct binary tomography cross-sections from few projections. A first reconstruction image is obtained with a level-set regularization method. The reconstruction is then refined with a stochastic partial differential equation based on a Stratanovitch formulation. The reconstruction results are compared with the ones obtained with the classical simulated annealing method. The methods are tested on a complex bone μ- CT cross-section for different noise levels and number of projections. The best reconstruction results are obtained with the stochastic level set-method.

International audience; The phase retrieval process is a nonlinear ill-posed problem. The Fresnel diffraction patterns obtained with hard x-ray synchrotron beam can be used to retrieve the phase contrast. In this work, we present a convergence comparison of several nonlinear approaches for the phase retrieval problem involving regularizations with sparsity constraints. The phase solution is assumed to have a sparse representation with respect to an orthonormal wavelets basis. One approach uses alternatively a solution of the nonlinear problem based on the Fréchet derivative and a solution of the linear problem in wavelet coordinates with an iterative thresholding. A second method is the one proposed by Ramlau and Teschke which generalizes to a nonlinear problem the classical thresholding algorithm. The algorithms were tested on a 3D Shepp-Logan phantom corrupted by white Gaussian noise. The best simulation results are obtained by the first method for the various noise levels and initializations investigated. The reconstruction errors are significantly decreased with respect to the ones given by the classical linear phase retrieval approaches.

We present an iterative method for choosing the optimal regularization parameter for the linear inverse problem of Total Variation image deconvolution. This approach is based on the Morozov discrepancy principle and on an exponential model function for the data term. The Total Variation image deconvolution is performed with the Alternating Direction Method of Multipliers (ADMM). With a smoothed l2 norm, the differentiability of the value of the Lagrangian at the saddle point can be shown and an approximate model function obtained. The choice of the optimal parameter can be refined with a Newton method. The efficiency of the method is demonstrated on a blurred and noisy bone CT cross section.

The in-line X-ray phase contrast imaging technique relies on the measurement of the Fresnel diffraction intensity patterns associated to a phase shift induced by the object. The simultaneous recovery of the phase and of the absorption is an ill-posed nonlinear inverse problem. If the object is made up of several homogeneous materials, the absorption and the phase are proportional in each material. In this work, in order to include this a priori information, level set regularization methods are used to retrieve the two quantities. The algorithms are evaluated using simulated noisy data. Large decrease of the reconstruction errors is obtained.

International audience; Phase retrieval from Fresnel diffraction patterns is a nonlinear ill-posed inverse problem of paramount importance in various areas of applied physics. Recently a new non-linear iterative algorithm based on a Landweber method with an analytic calculation of the Frechet derivative adjoint has been proposed. In this work, we refine this scheme by introducing Fienup projectors in the algorithm. The new method was tested on noisy simulated data. Our method outperforms the approaches based on the linearization of the relation between the phase shift induced by the object and the diffracted intensity and the former non linear algorithm with no projectors.

International audience; Several methods of phase retrieval for in line phase tomography have already been investigated based on the linearization of the relation between the phase shift induced by the object and the diffracted intensity. They use the Transport Intensity Equation (TIE) or the Contrast Transfer Function (CTF), or mixed approaches. In this work, we present a non linear iterative approach using the Frechet derivative of the intensity recorded at a few number of propagation distances. The inverse problem is regularized with the smoothing L 2 norm of the phase gradient. The evaluation of the method was performed using a simple phase map, both with and without noise. Our approach outerperforms the linear methods on simulated noisy data up to high noise levels.

International audience; The first stages of the cavitation process in amorphous polymers submitted to an hydrostatic deformation in the glassy state are studied with coarse grain molecular dynamics simulations for various chain stiffness and intermolecular interactions strengths. For strong intermolecular interactions, the cavitation process is highly localized and the holes have a marked spherical symmetry. The cavitation regions are more diffuse for weaker intermolecular interactions or when the chain stiffness is increased. The mean Voronoï polyhedra volume and the disorder inside the polymer increase until the stress peak observed below the glass transition. High mobility regions are present before the stress peak that may act as nucleation sites for cavitation. The localization of these high mobility zones is enhanced for strong intermolecular interactions or low chain rigidity. Moreover, the velocity fluctuations are more marked in the vicinity of the holes. For strong intermolecular interactions, the holes are not randomly distributed throughout the system and the nucleation of cavities upon deformation occurs preferentially near the chain ends of the polymer.

International audience; The structural characterization of sintered polyoxymethylene was carried out to determine the mechanism of sintering. Thermal behavior and small-angle X-ray scattering (SAXS) of sintered samples reveal that an improvement in either crystal size or perfection, or both, takes place during the sintering process, in comparison with compression-molding from the melt. The loss modulus data also give evidence of crystal reorganization during sintering that improves the continuity of the crystalline phase and enhances its mechanical contribution. These findings eliminate the occurrence of major melting during sintering. However, the high mobility of the chains in the crystalline phase in the temperature range of the melting onset suggests that short range rearrangements of the chains in the crystalline lamellae are likely to occur during the time of the process. The proposed mechanism of sintering is that of crystallographic welding of crystals from neighboring particles, as a result of the combined effect of plastic flow and temperature during the hot compaction operation.

International audience; It has clearly been shown in the literature that the properties achieved by polymer clay nanocomposites are often related to their structures and to the states of dispersion of the silicate platelets in the polymer matrices. Unfortunately, up to date most techniques used in a standard procedure do not allow a correct interpretation of polymer-clay nanocomposite structure and dispersion. In a recent work, we proposed an image analysis procedure (I.A.P.) based on TEM/OM observations to characterize the clay dispersion in polymer clay nanocomposites. The I.A.P. allows a very fine description of the nanocomposites microstructure. Nevertheless this analysis method shows some limits like the representativity of the sample analyzed volume. The purpose of this work is to discuss about the accuracy of the parameters extracted from the I.A.P. We propose SAXS experimental developments for evaluating the thickness distribution of the clay tactoids. The good agreement between the results of the two techniques confirms the validity of the I.A.P. methodology. Moreover, other experiments were performed in order to understand the abnormally low platelet lengths and aspect ratios determined from TEM micrographs. Wet-STEM observations revealed that clay platelets were not broken during the extrusion process. And, low platelet lengths and aspect ratios were shown to originate from the preparation of the ultramicrotomed sections and from TEM projection effects induced by the clay platelet wavy shape.

International audience; Structural Approaches of Polyethylene Environmental Stress-Crack Resistance This work was aimed at understanding and predicting of the long-term mechanical behavior of polyethylene in relation to molecular architecture. An insight into the semi-crystalline microstructure and the chain topology consisting of intercrystalline tie chains and loose folding loops was the main objective of the study. A two-stage crystallization segregation was used for separating the chain species which can easily crystallize by regular folding from those which cannot. The natural draw ratio that develops beyond the yield point proved to be a highly sensitive parameter for probing the chain topology via the crystalline lamella fragmentation and chain unfolding under tensile drawing. Small-angle X-ray scattering provided semi-quantitative evaluation of the stacking disorder of the lamellar microstructure. The data from the three approaches provided correlations with chain architecture in very good agreement with the stress crack resistance dependency. These correlations were discussed in terms of crystallization kinetics and molecular topology. A practical issue of the work is to provide relevant tools for predicting stress cracking resistance.

International audience; A uniaxial deformation is performed on a styrene-butadiene elastomer and two nitril butadiene elastomers with two different crosslinking densities in their rubbery state. The slow a-relaxation process is investigated under constant deformation using low frequency dynamic mechanical analysis. No effect is observed in the stretched elastomers although microstructural changes are detected thanks to tensile tests and wide angle X-ray scattering experiments. A possible explanation of these results is that the inter-molecular interactions are not significantly modified in the stretched network.

Dielectric and mechanical spectroscopies have been used to study the dynamics in poly(butyl methacrylate) isomers (namely PnBMA, PiBMA and PtBMA). This analysis has revealed four relaxation processes: δ, γ, β and α and has allowed the observation of the evolution of molecular relaxation phenomena with the lateral chain steric hindrance. The α relaxation is associated with the glass transition of the isomers and depends strongly on the lateral chain structure. The secondary relaxation phenomena are very similar for the three isomers. Arrhenius diagrams have been plotted for the three isomers. The activation parameters have been calculated from Arrhenius equation for the sub-glass processes δ and γ whereas the dynamics are characterized by a Vogel–Fulcher–Tamman law for the α relaxations. With a deconvolution procedure for the α and β processes, a departure from the Arrhenius law of the β process in the vicinity of the merging between the α and β phenomena is observed for each isomer.

Molecular mobility in sorbitol and maltitol is studied with spectroscopic techniques and molecular dynamics simulations in order to evaluate the relative contributions of the intermolecular and intramolecular interactions involved in the relaxation processes. The results of the molecular dynamics simulations performed on the polyols in the bulk or in vacuum compares well with the results of the analysis of the relaxation diagrams in the framework of the Perez et al. model. They both imply that the difference in the relative contributions of the intermolecular and intramolecular interactions associated with the different chemical architectures of the two polyols must be taken into account. The intermolecular interactions cannot be neglected and they are stronger in sorbitol than in maltitol in relation with the linear structure of this polyol. The intramolecular barrier, higher in the maltitol molecule with a more complex structure, could be at the origin of the higher junction temperature between the α an...

Molecular mobility in sorbitol and maltitol is studied in order to understand their differences near the junction between the α and β relaxations. The molecular dynamics simulations performed on the polyols in their bulk state give support to the 13C NMR results and imply that the mobility of a carbon atom located at the extremity of the chain is higher than that of any other carbon. Moreover, the difference in carbon atoms mobility is greater within the sorbitol moiety of maltitol than in sorbitol and seems intimately related to the junction temperature of the α and β relaxation processes. The reorientation of the C–H vectors as probed by NMR is shown to be mainly the effect of conformation transitions in the case of a carbon atom located at the end of the chain.

Spectral tomodensitometry is a new emerging x-ray imaging modality. If the dual-energy principle was already known for quite some time, new developments on photon-counting detectors now allowing acquiring more energy bins than before. This modality allows reducing some artifacts presents in x-ray imaging, such as beam hardening, but mostly to decompose the data into the chemical composition of the imaged tissue. It also enables the use of new markers (i.e. gold) with an energic discontinuity. The use of these markers also allows to locate and quantify them in the patient, granting great potential for medical imaging. Decomposition in the projection domain followed by a tomographic reconstruction is a classical processing for those spectral data. However, decomposition methods in the projection domain are unstable for a high number of energy bins. Classical calibration technic is numerically unstable for more than two energy bins. This thesis aims to developed new material decomposition methods in the projections domains. After expressing the spectral forward model, the decomposition problem is expressed and dealt as a non-linear inverse problem. It will be solved by minimizing a cost function composed by a term characterizing the fidelity of the decomposition regarding the data and an \textit{a priori} of the decomposed material maps. We will firstly present an adaptation of the cost function that takes into account the Poissonian noise on the data. This formulation allows having better decomposed maps for a high level of noise than classical formulation. Then, two constrained algorithms will be presented. The first one, a projected Gauss-Newton algorithm, that enforces positivity on the decomposed maps, allows having better decomposed maps than an unconstrained algorithm. To improve the first algorithm, another one was developed that also used an egality constrain. The equality allows having images with fewer artifacts than before. These methods are tested on a numerical phantom of a mouse and thorax. To speed up the decomposition process, an automatic choice of parameters is presented, which allow faster decomposition while keeping good maps. Finally, the methods are tested on experimental data that are coming from a spectral scanner prototype.; La tomodensitométrie spectrale est une modalité d’imagerie par rayons X émergente. Si le principe de la double énergie est plus ancien, des développements récents sur des détecteurs à comptage de photons permettent d’acquérir des données résolues en énergie sur plusieurs plages. Cette modalité permet de réduire un certain nombre d’artéfacts classiques dont ceux liés au durcissement de spectre, mais surtout de remonter à la composition chimique des tissus. Les données spectrales permettent également d’utiliser de nouveaux agents de contraste (comme l’or par exemple) qui présentent des discontinuités énergétiques. La possibilité d’utiliser d’autres marqueurs et de quantifier leurs présences dans le patient donne à cette modalité un fort potentiel dans le domaine de l’imagerie médicale. Une approche classique pour le traitement des données spectrales est d’effectuer une décomposition en base de matériaux préalables à la reconstruction tomographique. Cependant, les méthodes de décomposition dans le domaine des projections avec un grand nombre de plages d’énergies n’en sont qu’à leurs débuts. Les techniques classiques par calibration, ne sont plus numériquement stables lorsqu’il y a plus de deux plages disponibles. Le but de cette thèse est de développer de nouvelles méthodes de décomposition des données spectrales dans le domaine des projections. Après avoir formalisé le problème direct de la tomodensitométrie spectrale, le problème de décomposition en base de matériaux sera exprimé et traité comme un problème inverse non linéaire. Il sera résolu en minimisant une fonction de coût incluant un terme caractérisant la fidélité de la décomposition par rapport aux données et un \textit{a priori} sur les cartes de matériaux projetées. Ces travaux présenteront tout d’abord une adaptation de la fonctionnelle prenant en compte la nature Poissonienne du bruit. Cette formulation permet d’obtenir de meilleures décompositions pour de forts niveaux de bruit par rapport à la formulation classique. Ensuite, deux algorithmes de minimisation incluant une contrainte de positivité additionnelle seront proposés. Le premier, un algorithme de Gauss-Newton projeté, permet d’obtenir des cartes rapidement et de meilleure qualité que des méthodes non contraintes. Pour améliorer les résultats du premier algorithme, une seconde méthode, de type ADMM, ajoute une contrainte d’égalité. Cette contrainte a permis de diminuer les artefacts présents dans l’image. Ces méthodes ont été évaluées sur des données numériques de souris et de thorax humain. Afin d’accélérer et de simplifier les méthodes, un choix automatique des hyperparamètres est proposé qui permet de diminuer fortement le temps de calcul tout en gardant de bonnes décompositions. Finalement, ces méthodes sont testées sur des données expérimentales provenant d’un prototype de scanner spectral.

Spectral tomodensitometry is a new emerging x-ray imaging modality. If the dual-energy principle was already known for quite some time, new developments on photon-counting detectors now allowing acquiring more energy bins than before. This modality allows reducing some artifacts presents in x-ray imaging, such as beam hardening, but mostly to decompose the data into the chemical composition of the imaged tissue. It also enables the use of new markers (i.e. gold) with an energic discontinuity. The use of these markers also allows to locate and quantify them in the patient, granting great potential for medical imaging. Decomposition in the projection domain followed by a tomographic reconstruction is a classical processing for those spectral data. However, decomposition methods in the projection domain are unstable for a high number of energy bins. Classical calibration technic is numerically unstable for more than two energy bins. This thesis aims to developed new material decomposition methods in the projections domains. After expressing the spectral forward model, the decomposition problem is expressed and dealt as a non-linear inverse problem. It will be solved by minimizing a cost function composed by a term characterizing the fidelity of the decomposition regarding the data and an a priori of the decomposed material maps. We will firstly present an adaptation of the cost function that takes into account the Poissonian noise on the data. This formulation allows having better decomposed maps for a high level of noise than classical formulation. Then, two constrained algorithms will be presented. The first one, a projected Gauss-Newton algorithm, that enforces positivity on the decomposed maps, allows having better decomposed maps than an unconstrained algorithm. To improve the first algorithm, another one was developed that also used an egality constrain. The equality allows having images with fewer artifacts than before. These methods are tested on a numerical phantom of a mouse and thorax. To speed up the decomposition process, an automatic choice of parameters is presented, which allow faster decomposition while keeping good maps. Finally, the methods are tested on experimental data that are coming from a spectral scanner prototype., La tomodensitométrie spectrale est une modalité d’imagerie par rayons X émergente. Si le principe de la double énergie est plus ancien, des développements récents sur des détecteurs à comptage de photons permettent d’acquérir des données résolues en énergie sur plusieurs plages. Cette modalité permet de réduire un certain nombre d’artéfacts classiques dont ceux liés au durcissement de spectre, mais surtout de remonter à la composition chimique des tissus. Les données spectrales permettent également d’utiliser de nouveaux agents de contraste (comme l’or par exemple) qui présentent des discontinuités énergétiques. La possibilité d’utiliser d’autres marqueurs et de quantifier leurs présences dans le patient donne à cette modalité un fort potentiel dans le domaine de l’imagerie médicale. Une approche classique pour le traitement des données spectrales est d’effectuer une décomposition en base de matériaux préalables à la reconstruction tomographique. Cependant, les méthodes de décomposition dans le domaine des projections avec un grand nombre de plages d’énergies n’en sont qu’à leurs débuts. Les techniques classiques par calibration, ne sont plus numériquement stables lorsqu’il y a plus de deux plages disponibles.Le but de cette thèse est de développer de nouvelles méthodes de décomposition des données spectrales dans le domaine des projections. Après avoir formalisé le problème direct de la tomodensitométrie spectrale, le problème de décomposition en base de matériaux sera exprimé et traité comme un problème inverse non linéaire. Il sera résolu en minimisant une fonction de coût incluant un terme caractérisant la fidélité de la décomposition par rapport aux données et un a priori sur les cartes de matériaux projetées. Ces travaux présenteront tout d’abord une adaptation de la fonctionnelle prenant en compte la nature Poissonienne du bruit. Cette formulation permet d’obtenir de meilleures décompositions pour de forts niveaux de bruit par rapport à la formulation classique. Ensuite, deux algorithmes de minimisation incluant une contrainte de positivité additionnelle seront proposés. Le premier, un algorithme de Gauss-Newton projeté, permet d’obtenir des cartes rapidement et de meilleure qualité que des méthodes non contraintes. Pour améliorer les résultats du premier algorithme, une seconde méthode, de type ADMM, ajoute une contrainte d’égalité. Cette contrainte a permis de diminuer les artefacts présents dans l’image. Ces méthodes ont été évaluées sur des données numériques de souris et de thorax humain. Afin d’accélérer et de simplifier les méthodes, un choix automatique des hyperparamètres est proposé qui permet de diminuer fortement le temps de calcul tout en gardant de bonnes décompositions. Finalement, ces méthodes sont testées sur des données expérimentales provenant d’un prototype de scanner spectral.

Spectral tomodensitometry is a new emerging x-ray imaging modality. If the dual-energy principle was already known for quite some time, new developments on photon-counting detectors now allowing acquiring more energy bins than before. This modality allows reducing some artifacts presents in x-ray imaging, such as beam hardening, but mostly to decompose the data into the chemical composition of the imaged tissue. It also enables the use of new markers (i.e. gold) with an energic discontinuity. The use of these markers also allows to locate and quantify them in the patient, granting great potential for medical imaging. Decomposition in the projection domain followed by a tomographic reconstruction is a classical processing for those spectral data. However, decomposition methods in the projection domain are unstable for a high number of energy bins. Classical calibration technic is numerically unstable for more than two energy bins. This thesis aims to developed new material decomposition methods in the projections domains. After expressing the spectral forward model, the decomposition problem is expressed and dealt as a non-linear inverse problem. It will be solved by minimizing a cost function composed by a term characterizing the fidelity of the decomposition regarding the data and an \textit{a priori} of the decomposed material maps. We will firstly present an adaptation of the cost function that takes into account the Poissonian noise on the data. This formulation allows having better decomposed maps for a high level of noise than classical formulation. Then, two constrained algorithms will be presented. The first one, a projected Gauss-Newton algorithm, that enforces positivity on the decomposed maps, allows having better decomposed maps than an unconstrained algorithm. To improve the first algorithm, another one was developed that also used an egality constrain. The equality allows having images with fewer artifacts than before. These methods are tested on a numerical phantom of a mouse and thorax. To speed up the decomposition process, an automatic choice of parameters is presented, which allow faster decomposition while keeping good maps. Finally, the methods are tested on experimental data that are coming from a spectral scanner prototype., La tomodensitométrie spectrale est une modalité d’imagerie par rayons X émergente. Si le principe de la double énergie est plus ancien, des développements récents sur des détecteurs à comptage de photons permettent d’acquérir des données résolues en énergie sur plusieurs plages. Cette modalité permet de réduire un certain nombre d’artéfacts classiques dont ceux liés au durcissement de spectre, mais surtout de remonter à la composition chimique des tissus. Les données spectrales permettent également d’utiliser de nouveaux agents de contraste (comme l’or par exemple) qui présentent des discontinuités énergétiques. La possibilité d’utiliser d’autres marqueurs et de quantifier leurs présences dans le patient donne à cette modalité un fort potentiel dans le domaine de l’imagerie médicale. Une approche classique pour le traitement des données spectrales est d’effectuer une décomposition en base de matériaux préalables à la reconstruction tomographique. Cependant, les méthodes de décomposition dans le domaine des projections avec un grand nombre de plages d’énergies n’en sont qu’à leurs débuts. Les techniques classiques par calibration, ne sont plus numériquement stables lorsqu’il y a plus de deux plages disponibles. Le but de cette thèse est de développer de nouvelles méthodes de décomposition des données spectrales dans le domaine des projections. Après avoir formalisé le problème direct de la tomodensitométrie spectrale, le problème de décomposition en base de matériaux sera exprimé et traité comme un problème inverse non linéaire. Il sera résolu en minimisant une fonction de coût incluant un terme caractérisant la fidélité de la décomposition par rapport aux données et un \textit{a priori} sur les cartes de matériaux projetées. Ces travaux présenteront tout d’abord une adaptation de la fonctionnelle prenant en compte la nature Poissonienne du bruit. Cette formulation permet d’obtenir de meilleures décompositions pour de forts niveaux de bruit par rapport à la formulation classique. Ensuite, deux algorithmes de minimisation incluant une contrainte de positivité additionnelle seront proposés. Le premier, un algorithme de Gauss-Newton projeté, permet d’obtenir des cartes rapidement et de meilleure qualité que des méthodes non contraintes. Pour améliorer les résultats du premier algorithme, une seconde méthode, de type ADMM, ajoute une contrainte d’égalité. Cette contrainte a permis de diminuer les artefacts présents dans l’image. Ces méthodes ont été évaluées sur des données numériques de souris et de thorax humain. Afin d’accélérer et de simplifier les méthodes, un choix automatique des hyperparamètres est proposé qui permet de diminuer fortement le temps de calcul tout en gardant de bonnes décompositions. Finalement, ces méthodes sont testées sur des données expérimentales provenant d’un prototype de scanner spectral.

Osteoporosis is a disease characterized by loss of bone mass and degradation of bone microarchitecture. Although osteoporosis is not a fatal disease, the fractures it causes can lead to serious complications (damage to vessels and nerves, infections, stiffness), sometimes accompanied with risk of death. The bone micro-architecture plays an important role for the diagnosis of osteoporosis. Two common CT devices to scan bone micro architecture is High resolution-peripheral Quantitative CT and Micro CT. The former device gives access to in vivo investigation, but its spatial resolution is inferior. Micro CT gives better spatial resolution, but it is constrained to ex vivo measurement. In this thesis, we attempt to improve the spatial resolution of high resolution peripheral CT images so that the quantitative analysis of the resolved images is close to the one given by Micro CT images. We started from the total variation regularization, to a combination of total variation and double-well potential to enhance the contrast of results. Then we consider to use dictionary learning method to recover more structure details. Afterward, a deep learning method has been proposed to solve a joint super resolution and segmentation problem. The results show that the deep learning method is very promising for future applications., L'ostéoporose est une maladie caractérisée par la perte de la masse osseuse et la dégradation de la micro-architecture osseuse. Bien que l'ostéoporose ne soit pas une maladie mortelle, les fractures qu'elle provoque peuvent entraîner de graves complications (lésions des vaisseaux et des nerfs, infections, raideur), parfois accompagnées de menaces de mort. La micro-architecture osseuse joue un rôle important dans le diagnostic de l'ostéoporose. Deux appareils de tomodensitométrie courants pour scanner la micro-architecture osseuse sont la tomodensitométrie quantitative périphérique à haute résolution et la tomodensitométrie microscopique. Le premier dispositif donne accès à l'investigation in vivo, mais sa résolution spatiale est inférieure. Le micro tomodensitomètre donne une meilleure résolution spatiale, mais il est contraint à une mesure ex vivo. Dans cette thèse, notre but est d'améliorer la résolution spatiale des images de tomodensitométrie périphérique à haute résolution afin que l'analyse quantitative des images résolues soit proche de celle donnée par les images de tomodensitométrie Micro. Nous sommes partis de la régularisation de la variation totale, à une combinaison de la variation totale et du potentiel de double puits pour améliorer le contraste des résultats. Ensuite, nous envisageons d'utiliser la méthode d'apprentissage par dictionnaire pour récupérer plus de détails sur la structure. Par la suite, une méthode d'apprentissage approfondi a été proposée pour résoudre un problème de super résolution et de segmentation joint. Les résultats montrent que la méthode d'apprentissage profond est très prometteuse pour les applications futures.

Discrete tomography reconstruction of bone microstructure is important in diagnosis of osteoporosis. One way to reduce the radiation dose and scanning time in CT imaging is to limit the number of projections. This method makes the reconstruction problem highly ill-posed. A common solution is to reconstruct only a finite number of intensity levels. In this work, we investigate only binary tomography reconstruction problem. First, we consider variational regularization methods. Two types of Total Variation (TV) regularization approaches minimized with the Alternate Direction of Minimization Method (ADMM) and two schemes based on Level-set (LS) regularization methods are applied to two experimental bone cross-section images acquired with synchrotron micro-CT. The numerical experiments have shown that good reconstruction results were obtained with TV regularization methods and that level-set regularization outperforms the TV regularization for large bone image with complex structures. Yet, for both methods, some reconstruction errors are still located on the boundaries and some regions are lost when the projection number is low. Local minima were obtained with these deterministic methods. Stochastic perturbations is a useful way to escape the local minima. As a first approach, a stochastic differential equation based on level-set regularization was studied. This method improves the reconstruction results but only modifies the boundaries between the 0 and 1 regions. Then partial stochastic differential equation obtained with the TV regularization semi-norm were studied to improve the stochastic level-set method. The random change of the boundary are performed in a new way with the gradient or wavelet decomposition of the reconstructed image. Random topological changes are included to find the lost regions in the reconstructed images. At the end of our work, we extended the TV regularization method to 3D images with real data on RTK (Reconstruction Toolkit). And we also extended the level-set to the multi-level cases., La reconstruction en tomographie discrète de la microstructure de l’os joue un role très important pour le diagnostic de l’ostéoporse, une maladie des os très fréquente. Le diagnostic clinique est basé sur l’absortiométrie duale de rayons X. Avec la tomographie de rayons X, une résolution spatiale élevée avec des images reconstruites in vivo requiert une dose d’irradiation élevée et un temps de balayage long, ce qui est dangereux pour le patient. Une des méthodes pour résoudre ce problème est de limiter le nombre de projections. Cependant, avec cette méthode le problème de reconstruction devient mal posé. Deux types de régularisation par Variation Totale minimisées avec la méthode Alternate Direction of Minimization Method (ADMM) et deux schémas basés sur les méthodes de régularisation Level-set sont appliquées à deux images d’os expérimentales acquises avec un synchrotron (pixel size: 15 μm). Des images de tailles variées et avec différents niveaux de bruit Gaussien additifs ajoutés aux projections sont utlisées pour étudier l’efficacité des méthodes de régularisation. Des minima locaux sont obtenus avec ces méthodes déterministes. Une approche globale d’optimisation est nécessaire pour améliorer les résultats. Des perturbations stochastiques peuvent être un moyen très utile pour échapper aux minima locaux. Dans une première approche, une équation différentielle stochastique basée sur la régularisation level-set est étudiée. Cette méthode améliore les résultats de reconstruction mais ne modifie que les frontières entre les régions 0 et 1. Ensuite une équation aux dérivées partielles stochastique est obtenue avec la régularisation TV pour améliorer la méthode stochastique level-set. A la fin de notre travail, nous avons étendu la méthode de régularisation à des images 3D avec des données réelles. Cette algorithme a été implémenté avec RTK. Nous avons aussi étendu l’approche level-set utilisée pour la tomographie binaire au cas multi-level.

The investigation of trabecular bone micro-architecture provides relevant information to determine the bone strength, an important parameter in osteoporosis investigation. While the spatial resolution of clinical CT is not sufficient to resolve the trabecular structure, the High Resolution peripheral Quantitative CT (HR-pQCT) has been developed to investigate bone micro-architecture in-vivo at peripheral sites (tibia and radius). Despite this considerable progress, the quantification of 3D trabecular bone micro-architecture in-vivo remains limited due to a lack of spatial resolution compared to the trabeculae size. The objective of this thesis is to propose new joint super-resolution/segmentation approaches for improving the quantitative analysis of in-vivo HR-pQCT images of the trabecular bone structure. To begin with, we have investigated 2D super-resolution methods based on Total Variation (TV) and Higher Degree Total Variation (HDTV) and Alternating Direction Method of Multipliers (ADMM) minimization. Afterwards, an iterative method combining the Morozov principle and the Newton method was proposed in order to estimate the TV regularization parameter. The proposed method provides a very good regularization parameter only in few iterations compared with the UPRE method that requires an extensive scanning of parameter values. Furthermore, we have developed a 3D joint super-resolution/segmentation method based on a TV a prior with a convex relaxation (TVbox). The validation of the proposed methods was made on experimental micro-CT bone images artificially deteriorated. The results showed an improvement of the bone parameters and 3D connectivity with the TVbox method. Moreover, we have investigated a semi-blind joint super-resolution/ segmentation approach aiming to estimate both the binary super-resolved image and the assumed Gaussian blurring kernel that is not known for the real HR-pQCT images. Results on micro-CT and HR-pQCT experimental bone images were presented. In conclusion, our work has shown that TV based regularization methods promise to improve the quantification of bone micro-architecture from HR-pQCT images.; L'analyse de la microstructure osseuse joue un rôle important pour étudier des maladies de l'os comme l'ostéoporose. Des nouveaux scanners périphériques haute résolution (HR-pQCT) permettent de faire des acquisitions de la micro-architecture osseuse in-vivo sur l'homme. Toutefois la résolution spatiale de ces appareils reste comparable à la taille des travées osseuses, ce qui limite leur analyse quantitative. L'objectif de cette thèse est de proposer de nouvelles approches jointes super-résolution/ segmentation pour une analyse quantitative plus fine des images HR-pQCT in-vivo de la structure osseuse trabéculaire. Dans une première étape nous nous sommes concentrés sur des méthodes 2D de super-résolution avec régularisation par variation totale (TV) puis par variation totale d'ordre plus élevé (Higher Degree TV), avec minimisation par un algorithme ADMM (Alternating Direction Method of Multipliers). Ensuite, nous avons proposé une méthode itérative combinant le principe de Morozov et la méthode de Newton pour estimer le paramètre de régularisation TV. Comparé à la méthode UPRE (Unbiased Predictive Risk Estimator), la méthode proposée est plus rapide et ne requiert pas un balayage exhaustif des valeurs des paramètres. Nous avons développé dans une deuxième étape une méthode de super-résolution/segmentation conjointe avec un a priori basé sur la Variation Totale et une relaxation convexe (Tvbox), qui permet d'améliorer les paramètres quantitatifs de l'os et de la connectivité 3D. La méthode a été validée sur des images expérimentales micro-CT déteriorées artificiellement. Finalement, en vue de l'application à des images réelles HR-pQCT, nous nous sommes intéressés à une approche conjointe semi-aveugle super-résolution/segmentation qui vise à estimer à la fois l'image binaire super-résolue et le noyau de convolution. Des résultats sur des images micro-CT et HR-pQCT sont présentés. En conclusion, notre travail montre que les méthodes d'optimisation basées sur la régularisation TV sont prometteurs pour améliorer la quantification de la micro-architecture osseuse sur des images HR-pQCT.

Discrete tomography reconstruction of bone microstructure is important in diagnosis of osteoporosis. One way to reduce the radiation dose and scanning time in CT imaging is to limit the number of projections. This method makes the reconstruction problem highly ill-posed. A common solution is to reconstruct only a finite number of intensity levels. In this work, we investigate only binary tomography reconstruction problem. First, we consider variational regularization methods. Two types of Total Variation (TV) regularization approaches minimized with the Alternate Direction of Minimization Method (ADMM) and two schemes based on Level-set (LS) regularization methods are applied to two experimental bone cross-section images acquired with synchrotron micro-CT. The numerical experiments have shown that good reconstruction results were obtained with TV regularization methods and that level-set regularization outperforms the TV regularization for large bone image with complex structures. Yet, for both methods, some reconstruction errors are still located on the boundaries and some regions are lost when the projection number is low. Local minima were obtained with these deterministic methods. Stochastic perturbations is a useful way to escape the local minima. As a first approach, a stochastic differential equation based on level-set regularization was studied. This method improves the reconstruction results but only modifies the boundaries between the 0 and 1 regions. Then partial stochastic differential equation obtained with the TV regularization semi-norm were studied to improve the stochastic level-set method. The random change of the boundary are performed in a new way with the gradient or wavelet decomposition of the reconstructed image. Random topological changes are included to find the lost regions in the reconstructed images. At the end of our work, we extended the TV regularization method to 3D images with real data on RTK (Reconstruction Toolkit). And we also extended the level-set to the multi-level cases., La reconstruction en tomographie discrète de la microstructure de l’os joue un role très important pour le diagnostic de l’ostéoporse, une maladie des os très fréquente. Le diagnostic clinique est basé sur l’absortiométrie duale de rayons X. Avec la tomographie de rayons X, une résolution spatiale élevée avec des images reconstruites in vivo requiert une dose d’irradiation élevée et un temps de balayage long, ce qui est dangereux pour le patient. Une des méthodes pour résoudre ce problème est de limiter le nombre de projections. Cependant, avec cette méthode le problème de reconstruction devient mal posé. Deux types de régularisation par Variation Totale minimisées avec la méthode Alternate Direction of Minimization Method (ADMM) et deux schémas basés sur les méthodes de régularisation Level-set sont appliquées à deux images d’os expérimentales acquises avec un synchrotron (pixel size: 15 μm). Des images de tailles variées et avec différents niveaux de bruit Gaussien additifs ajoutés aux projections sont utlisées pour étudier l’efficacité des méthodes de régularisation. Des minima locaux sont obtenus avec ces méthodes déterministes. Une approche globale d’optimisation est nécessaire pour améliorer les résultats. Des perturbations stochastiques peuvent être un moyen très utile pour échapper aux minima locaux. Dans une première approche, une équation différentielle stochastique basée sur la régularisation level-set est étudiée. Cette méthode améliore les résultats de reconstruction mais ne modifie que les frontières entre les régions 0 et 1. Ensuite une équation aux dérivées partielles stochastique est obtenue avec la régularisation TV pour améliorer la méthode stochastique level-set. A la fin de notre travail, nous avons étendu la méthode de régularisation à des images 3D avec des données réelles. Cette algorithme a été implémenté avec RTK. Nous avons aussi étendu l’approche level-set utilisée pour la tomographie binaire au cas multi-level.

The investigation of trabecular bone micro-architecture provides relevant information to determine the bone strength, an important parameter in osteoporosis investigation. While the spatial resolution of clinical CT is not sufficient to resolve the trabecular structure, the High Resolution peripheral Quantitative CT (HR-pQCT) has been developed to investigate bone micro-architecture in-vivo at peripheral sites (tibia and radius). Despite this considerable progress, the quantification of 3D trabecular bone micro-architecture in-vivo remains limited due to a lack of spatial resolution compared to the trabeculae size. The objective of this thesis is to propose new joint super-resolution/segmentation approaches for improving the quantitative analysis of in-vivo HR-pQCT images of the trabecular bone structure. To begin with, we have investigated 2D super-resolution methods based on Total Variation (TV) and Higher Degree Total Variation (HDTV) and Alternating Direction Method of Multipliers (ADMM) minimization. Afterwards, an iterative method combining the Morozov principle and the Newton method was proposed in order to estimate the TV regularization parameter. The proposed method provides a very good regularization parameter only in few iterations compared with the UPRE method that requires an extensive scanning of parameter values. Furthermore, we have developed a 3D joint super-resolution/segmentation method based on a TV a prior with a convex relaxation (TVbox). The validation of the proposed methods was made on experimental micro-CT bone images artificially deteriorated. The results showed an improvement of the bone parameters and 3D connectivity with the TVbox method. Moreover, we have investigated a semi-blind joint super-resolution/ segmentation approach aiming to estimate both the binary super-resolved image and the assumed Gaussian blurring kernel that is not known for the real HR-pQCT images. Results on micro-CT and HR-pQCT experimental bone images were presented. In conclusion, our work has shown that TV based regularization methods promise to improve the quantification of bone micro-architecture from HR-pQCT images., L'analyse de la microstructure osseuse joue un rôle important pour étudier des maladies de l'os comme l'ostéoporose. Des nouveaux scanners périphériques haute résolution (HR-pQCT) permettent de faire des acquisitions de la micro-architecture osseuse in-vivo sur l'homme. Toutefois la résolution spatiale de ces appareils reste comparable à la taille des travées osseuses, ce qui limite leur analyse quantitative. L'objectif de cette thèse est de proposer de nouvelles approches jointes super-résolution/ segmentation pour une analyse quantitative plus fine des images HR-pQCT in-vivo de la structure osseuse trabéculaire. Dans une première étape nous nous sommes concentrés sur des méthodes 2D de super-résolution avec régularisation par variation totale (TV) puis par variation totale d'ordre plus élevé (Higher Degree TV), avec minimisation par un algorithme ADMM (Alternating Direction Method of Multipliers). Ensuite, nous avons proposé une méthode itérative combinant le principe de Morozov et la méthode de Newton pour estimer le paramètre de régularisation TV. Comparé à la méthode UPRE (Unbiased Predictive Risk Estimator), la méthode proposée est plus rapide et ne requiert pas un balayage exhaustif des valeurs des paramètres. Nous avons développé dans une deuxième étape une méthode de super-résolution/segmentation conjointe avec un a priori basé sur la Variation Totale et une relaxation convexe (Tvbox), qui permet d'améliorer les paramètres quantitatifs de l'os et de la connectivité 3D. La méthode a été validée sur des images expérimentales micro-CT déteriorées artificiellement. Finalement, en vue de l'application à des images réelles HR-pQCT, nous nous sommes intéressés à une approche conjointe semi-aveugle super-résolution/segmentation qui vise à estimer à la fois l'image binaire super-résolue et le noyau de convolution. Des résultats sur des images micro-CT et HR-pQCT sont présentés. En conclusion, notre travail montre que les méthodes d'optimisation basées sur la régularisation TV sont prometteurs pour améliorer la quantification de la micro-architecture osseuse sur des images HR-pQCT.

The development of highly coherent X-ray sources offers new possibilities to image biological structures at different scales exploiting the refraction of X-rays. The coherence properties of the third-generation synchrotron radiation sources enables efficient implementations of phase contrast techniques. One of the first measurements of the intensity variations due to phase contrast has been reported in 1995 at the European Synchrotron Radiation Facility (ESRF). Phase imaging coupled to tomography acquisition allows three-dimensional imaging with an increased sensitivity compared to absorption CT. This technique is particularly attractive to image samples with low absorption constituents. Phase contrast has many applications, ranging from material science, palaeontology, bone research to medicine and biology. Several methods to achieve X-ray phase contrast have been proposed during the last years. In propagation based phase contrast, the measurements are made at different sample-to-detector distances. While the intensity data can be acquired and recorded, the phase information of the signal has to be "retrieved" from the modulus data only. Phase retrieval is thus an ill-posed nonlinear problem and regularization techniques including a priori knowledge are necessary to obtain stable solutions. Several phase recovery methods have been developed in recent years. These approaches generally formulate the phase retrieval problem as a linear one. Nonlinear treatments have not been much investigated. The main purpose of this work was to propose and evaluate new algorithms, in particularly taking into account the nonlinearity of the direct problem. In the first part of this work, we present a Landweber type nonlinear iterative scheme to solve the propagation based phase retrieval problem. This approach uses the analytic expression of the Fréchet derivative of the phase-intensity relationship and of its adjoint, which are presented in detail. We also study the effect of projection operators on the convergence properties of the method. In the second part of this thesis, we investigate the resolution of the linear inverse problem with an iterative thresholding algorithm in wavelet coordinates. In the following, the two former algorithms are combined and compared with another nonlinear approach based on sparsity regularization and a fixed point algorithm. The performance of theses algorithms are evaluated on simulated data for different noise levels. Finally the algorithms were adapted to process real data sets obtained in phase CT at the ESRF at Grenoble., Le développement de sources cohérentes de rayons X offre de nouvelles possibilités pour visualiser les structures biologiques à différentes échelles en exploitant la réfraction des rayons X. La cohérence des sources synchrotron de troisième génération permettent des implémentations efficaces des techniques de contraste de phase. Une des premières mesures des variations d'intensité dues au contraste de phase a été réalisée en 1995 à l'Installation Européenne de Rayonnement Synchrotron (ESRF). L'imagerie de phase couplée à l'acquisition tomographique permet une imagerie tridimensionnelle avec une sensibilité accrue par rapport à la tomographie standard basée sur absorption. Cette technique est particulièrement adaptée pour les échantillons faiblement absorbante ou bien présentent des faibles différences d'absorption. Le contraste de phase a ainsi une large gamme d'applications, allant de la science des matériaux, à la paléontologie, en passant par la médecine et par la biologie. Plusieurs techniques de contraste de phase aux rayons X ont été proposées au cours des dernières années. Dans la méthode de contraste de phase basée sur le phénomène de propagation l'intensité est mesurée pour différentes distances de propagation obtenues en déplaçant le détecteur. Bien que l'intensité diffractée puisse être acquise et enregistrée, les informations de phase du signal doivent être "récupérées" à partir seulement du module des données mesurées. L'estimation de la phase est donc un problème inverse non linéaire mal posé et une connaissance a priori est nécessaire pour obtenir des solutions stables. Si la plupart de méthodes d'estimation de phase reposent sur une linéarisation du problème inverse, les traitements non linéaires ont été très peu étudiés. Le but de ce travail était de proposer et d'évaluer des nouveaux algorithmes, prenant en particulier en compte la non linéarité du problème direct. Dans la première partie de ce travail, nous présentons un schéma de type Landweber non linéaire itératif pour résoudre le problème de la récupération de phase. Cette approche utilise l'expression analytique de la dérivée de Fréchet de la relation phase-intensité et de son adjoint. Nous étudions aussi l'effet des opérateurs de projection sur les propriétés de convergence de la méthode. Dans la deuxième partie de cette thèse, nous étudions la résolution du problème inverse linéaire avec un algorithme en coordonnées ondelettes basé sur un seuillage itératif. Par la suite, les deux algorithmes sont combinés et comparés avec une autre approche non linéaire basée sur une régularisation parcimonieuse et un algorithme de point fixe. Les performances des algorithmes sont évaluées sur des données simulées pour différents niveaux de bruit. Enfin, les algorithmes ont été adaptés pour traiter des données réelles acquises en tomographie de phase à l'ESRF à Grenoble.

The development of highly coherent X-ray sources offers new possibilities to image biological structures at different scales exploiting the refraction of X-rays. The coherence properties of the third-generation synchrotron radiation sources enables efficient implementations of phase contrast techniques. One of the first measurements of the intensity variations due to phase contrast has been reported in 1995 at the European Synchrotron Radiation Facility (ESRF). Phase imaging coupled to tomography acquisition allows threedimensional imaging with an increased sensitivity compared to absorption CT. This technique is particularly attractive to image samples with low absorption constituents. Phase contrast has many applications, ranging from material science, paleontology, bone research to medicine and biology. Several methods to achieve X-ray phase contrast have been proposed during the last years. In propagation based phase contrast, the measurements are made at different sample-to-detector distances. While the intensity data can be acquired and recorded, the phase information of the signal has to be "retrieved" from the modulus data only. Phase retrieval is thus an illposed nonlinear problem and regularization techniques including a priori knowledge are necessary to obtain stable solutions. Several phase recovery methods have been developed in recent years. These approaches generally formulate the phase retrieval problem as a linear one. Nonlinear treatments have not been much investigated. The main purpose of this work was to propose and evaluate new algorithms, in particularly taking into account the nonlinearity of the direct problem. In the first part of this work, we present a Landweber type nonlinear iterative scheme to solve the propagation based phase retrieval problem. This approach uses the analytic expression of the Fréchet derivative of the phase-intensity relationship and of its adjoint, which are presented in detail. We also study the effect of projection operators on the convergence properties of the method. In the second part of this thesis, we investigate the resolution of the linear inverse problem with an iterative thresholding algorithm in wavelet coordinates. In the following, the two former algorithms are combined and compared with another nonlinear approach based on sparsity regularization and a fixed point algorithm. The performance of theses algorithms are evaluated on simulated data for different noise levels. Finally the algorithms were adapted to process real data sets obtained in phase CT at the ESRF at Grenoble.; Le développement de sources cohérentes de rayons X offre de nouvelles possibilités pour visualiser les structures biologiques à différentes échelles en exploitant la réfraction des rayons X. La cohérence des sources synchrotron de troisième génération permettent des implémentations efficaces des techniques de contraste de phase. Une des premières mesures des variations d’intensité dues au contraste de phase a été réalisée en 1995 à l’Installation Européenne de Rayonnement Synchrotron (ESRF). L’imagerie de phase couplée à l’acquisition tomographique permet une imagerie tridimensionnelle avec une sensibilité accrue par rapport à la tomographie standard basée sur absorption. Cette technique est particulièrement adaptée pour les échantillons faiblement absorbante ou bien présentent des faibles différences d’absorption. Le contraste de phase a ainsi une large gamme d’applications, allant de la science des matériaux, à la paléontologie, en passant par la médecine et par la biologie. Plusieurs techniques de contraste de phase aux rayons X ont été proposées au cours des dernières années. Dans la méthode de contraste de phase basée sur le phénomène de propagation l’intensité est mesurée pour différentes distances de propagation obtenues en déplaçant le détecteur. Bien que l’intensité diffractée puisse être acquise et enregistrée, les informations de phase du signal doivent être "récupérées" à partir seulement du module des données mesurées. L’estimation de la phase est donc un problème inverse non linéaire mal posé et une connaissance a priori est nécessaire pour obtenir des solutions stables. Si la plupart de méthodes d’estimation de phase reposent sur une linéarisation du problème inverse, les traitements non linéaires ont été eux très peu étudiés. Le but de ce travail était de proposer et d’évaluer des nouveaux algorithmes, prenant en particulier en compte la non linéarité du problème direct. Dans la première partie de ce travail, nous présentons un schéma de type Landweber non linéaire itératif pour résoudre le problème de la récupération de phase. Cette approche utilise l’expression analytique de la dérivée de Fréchet de la relation phase-intensité et de son adjoint. Nous étudions aussi l’effet des opérateurs de projection sur les propriétés de convergence de la méthode. Dans la deuxième partie de cette thèse, nous étudions la résolution du problème inverse linéaire avec un algorithme en coordonnées ondelettes basé sur un seuillage itératif. Par la suite, les deux algorithmes sont combinés et comparés avec une autre approche non linéaire basée sur une régularisation parcimonieuse et un algorithme de point fixe. Les performances des algorithmes sont évaluées sur des données simulées pour différents niveaux de bruit. Enfin, les algorithmes ont été adaptés pour traiter des données réelles acquises en tomographie de phase à l’ESRF à Grenoble.

The development of highly coherent X-ray sources offers new possibilities to image biological structures at different scales exploiting the refraction of X-rays. The coherence properties of the third-generation synchrotron radiation sources enables efficient implementations of phase contrast techniques. One of the first measurements of the intensity variations due to phase contrast has been reported in 1995 at the European Synchrotron Radiation Facility (ESRF). Phase imaging coupled to tomography acquisition allows three-dimensional imaging with an increased sensitivity compared to absorption CT. This technique is particularly attractive to image samples with low absorption constituents. Phase contrast has many applications, ranging from material science, palaeontology, bone research to medicine and biology. Several methods to achieve X-ray phase contrast have been proposed during the last years. In propagation based phase contrast, the measurements are made at different sample-to-detector distances. While the intensity data can be acquired and recorded, the phase information of the signal has to be "retrieved" from the modulus data only. Phase retrieval is thus an ill-posed nonlinear problem and regularization techniques including a priori knowledge are necessary to obtain stable solutions. Several phase recovery methods have been developed in recent years. These approaches generally formulate the phase retrieval problem as a linear one. Nonlinear treatments have not been much investigated. The main purpose of this work was to propose and evaluate new algorithms, in particularly taking into account the nonlinearity of the direct problem. In the first part of this work, we present a Landweber type nonlinear iterative scheme to solve the propagation based phase retrieval problem. This approach uses the analytic expression of the Fréchet derivative of the phase-intensity relationship and of its adjoint, which are presented in detail. We also study the effect of projection operators on the convergence properties of the method. In the second part of this thesis, we investigate the resolution of the linear inverse problem with an iterative thresholding algorithm in wavelet coordinates. In the following, the two former algorithms are combined and compared with another nonlinear approach based on sparsity regularization and a fixed point algorithm. The performance of theses algorithms are evaluated on simulated data for different noise levels. Finally the algorithms were adapted to process real data sets obtained in phase CT at the ESRF at Grenoble.; Le développement de sources cohérentes de rayons X offre de nouvelles possibilités pour visualiser les structures biologiques à différentes échelles en exploitant la réfraction des rayons X. La cohérence des sources synchrotron de troisième génération permettent des implémentations efficaces des techniques de contraste de phase. Une des premières mesures des variations d'intensité dues au contraste de phase a été réalisée en 1995 à l'Installation Européenne de Rayonnement Synchrotron (ESRF). L'imagerie de phase couplée à l'acquisition tomographique permet une imagerie tridimensionnelle avec une sensibilité accrue par rapport à la tomographie standard basée sur absorption. Cette technique est particulièrement adaptée pour les échantillons faiblement absorbante ou bien présentent des faibles différences d'absorption. Le contraste de phase a ainsi une large gamme d'applications, allant de la science des matériaux, à la paléontologie, en passant par la médecine et par la biologie. Plusieurs techniques de contraste de phase aux rayons X ont été proposées au cours des dernières années. Dans la méthode de contraste de phase basée sur le phénomène de propagation l'intensité est mesurée pour différentes distances de propagation obtenues en déplaçant le détecteur. Bien que l'intensité diffractée puisse être acquise et enregistrée, les informations de phase du signal doivent être "récupérées" à partir seulement du module des données mesurées. L'estimation de la phase est donc un problème inverse non linéaire mal posé et une connaissance a priori est nécessaire pour obtenir des solutions stables. Si la plupart de méthodes d'estimation de phase reposent sur une linéarisation du problème inverse, les traitements non linéaires ont été très peu étudiés. Le but de ce travail était de proposer et d'évaluer des nouveaux algorithmes, prenant en particulier en compte la non linéarité du problème direct. Dans la première partie de ce travail, nous présentons un schéma de type Landweber non linéaire itératif pour résoudre le problème de la récupération de phase. Cette approche utilise l'expression analytique de la dérivée de Fréchet de la relation phase-intensité et de son adjoint. Nous étudions aussi l'effet des opérateurs de projection sur les propriétés de convergence de la méthode. Dans la deuxième partie de cette thèse, nous étudions la résolution du problème inverse linéaire avec un algorithme en coordonnées ondelettes basé sur un seuillage itératif. Par la suite, les deux algorithmes sont combinés et comparés avec une autre approche non linéaire basée sur une régularisation parcimonieuse et un algorithme de point fixe. Les performances des algorithmes sont évaluées sur des données simulées pour différents niveaux de bruit. Enfin, les algorithmes ont été adaptés pour traiter des données réelles acquises en tomographie de phase à l'ESRF à Grenoble.

The development of highly coherent X-ray sources offers new possibilities to image biological structures at different scales exploiting the refraction of X-rays. The coherence properties of the third-generation synchrotron radiation sources enables efficient implementations of phase contrast techniques. One of the first measurements of the intensity variations due to phase contrast has been reported in 1995 at the European Synchrotron Radiation Facility (ESRF). Phase imaging coupled to tomography acquisition allows threedimensional imaging with an increased sensitivity compared to absorption CT. This technique is particularly attractive to image samples with low absorption constituents. Phase contrast has many applications, ranging from material science, paleontology, bone research to medicine and biology. Several methods to achieve X-ray phase contrast have been proposed during the last years. In propagation based phase contrast, the measurements are made at different sample-to-detector distances. While the intensity data can be acquired and recorded, the phase information of the signal has to be "retrieved" from the modulus data only. Phase retrieval is thus an illposed nonlinear problem and regularization techniques including a priori knowledge are necessary to obtain stable solutions. Several phase recovery methods have been developed in recent years. These approaches generally formulate the phase retrieval problem as a linear one. Nonlinear treatments have not been much investigated. The main purpose of this work was to propose and evaluate new algorithms, in particularly taking into account the nonlinearity of the direct problem. In the first part of this work, we present a Landweber type nonlinear iterative scheme to solve the propagation based phase retrieval problem. This approach uses the analytic expression of the Fréchet derivative of the phase-intensity relationship and of its adjoint, which are presented in detail. We also study the effect of projection operators on the convergence properties of the method. In the second part of this thesis, we investigate the resolution of the linear inverse problem with an iterative thresholding algorithm in wavelet coordinates. In the following, the two former algorithms are combined and compared with another nonlinear approach based on sparsity regularization and a fixed point algorithm. The performance of theses algorithms are evaluated on simulated data for different noise levels. Finally the algorithms were adapted to process real data sets obtained in phase CT at the ESRF at Grenoble., Le développement de sources cohérentes de rayons X offre de nouvelles possibilités pour visualiser les structures biologiques à différentes échelles en exploitant la réfraction des rayons X. La cohérence des sources synchrotron de troisième génération permettent des implémentations efficaces des techniques de contraste de phase. Une des premières mesures des variations d’intensité dues au contraste de phase a été réalisée en 1995 à l’Installation Européenne de Rayonnement Synchrotron (ESRF). L’imagerie de phase couplée à l’acquisition tomographique permet une imagerie tridimensionnelle avec une sensibilité accrue par rapport à la tomographie standard basée sur absorption. Cette technique est particulièrement adaptée pour les échantillons faiblement absorbante ou bien présentent des faibles différences d’absorption. Le contraste de phase a ainsi une large gamme d’applications, allant de la science des matériaux, à la paléontologie, en passant par la médecine et par la biologie. Plusieurs techniques de contraste de phase aux rayons X ont été proposées au cours des dernières années. Dans la méthode de contraste de phase basée sur le phénomène de propagation l’intensité est mesurée pour différentes distances de propagation obtenues en déplaçant le détecteur. Bien que l’intensité diffractée puisse être acquise et enregistrée, les informations de phase du signal doivent être "récupérées" à partir seulement du module des données mesurées. L’estimation de la phase est donc un problème inverse non linéaire mal posé et une connaissance a priori est nécessaire pour obtenir des solutions stables. Si la plupart de méthodes d’estimation de phase reposent sur une linéarisation du problème inverse, les traitements non linéaires ont été eux très peu étudiés. Le but de ce travail était de proposer et d’évaluer des nouveaux algorithmes, prenant en particulier en compte la non linéarité du problème direct. Dans la première partie de ce travail, nous présentons un schéma de type Landweber non linéaire itératif pour résoudre le problème de la récupération de phase. Cette approche utilise l’expression analytique de la dérivée de Fréchet de la relation phase-intensité et de son adjoint. Nous étudions aussi l’effet des opérateurs de projection sur les propriétés de convergence de la méthode. Dans la deuxième partie de cette thèse, nous étudions la résolution du problème inverse linéaire avec un algorithme en coordonnées ondelettes basé sur un seuillage itératif. Par la suite, les deux algorithmes sont combinés et comparés avec une autre approche non linéaire basée sur une régularisation parcimonieuse et un algorithme de point fixe. Les performances des algorithmes sont évaluées sur des données simulées pour différents niveaux de bruit. Enfin, les algorithmes ont été adaptés pour traiter des données réelles acquises en tomographie de phase à l’ESRF à Grenoble.