Bayesian Multi-Energy Computed Tomography reconstruction approaches based on decomposition models

La tomographie par rayons X multi-énergétiques (MECT) permet d'obtenir plus d'information concernant la structure interne de l'objet par rapport au scanner CT classique. Un de ses intérêts dans l’imagerie médicale est d'obtenir les images de fractions d’eau et d’os. Dans l'état de l'art, les intérêts de MECT n'est pas encore découvert largement. Les approches de reconstruction existantes sont très limitées dans leurs performances. L'objectif principal de ce travail est de proposer des approches de reconstruction de haute qualité qui pourront être utilisés dans la MECT afin d’améliorer la qualité d’imagerie.Ce travail propose deux approches de reconstruction bayésiennes. La première est adaptée au système avec un détecteur discriminant en énergie. Dans cette approche, nous considérons que les polychromaticités de faisceaux sont négligeables. En utilisant le modèle linéaire de la variance et la méthode d'estimation maximum à postériori (MAP), l'approche que nous avons proposé permets de prendre en compte les différents niveaux de bruit présentés sur les mesures multi-énergétiques. Les résultats des simulations montrent que, dans l'imagerie médicale, les mesures biénergies sont suffisantes pour obtenir les fractions de l'eau et de l'os en utilisant l'approche proposée. Des mesures à la troisième énergie est nécessaire uniquement lorsque l'objet contient des matériaux lourdes. Par exemple, l’acier et l'iode. La deuxième approche est proposée pour les systèmes où les mesures multi-énergétiques sont obtenues avec des faisceaux polychromatiques. C'est effectivement la plupart des cas dans l'état actuel du practice. Cette approche est basée sur un modèle direct non-linéaire et un modèle bruit gaussien où la variance est inconnue. En utilisant l’inférence bayésienne, les fractions de matériaux de base et de la variance d'observation pourraient être estimées à l'aide de l'estimateur conjoint de MAP. Sous réserve à un modèle a priori Dirac attribué à la variance, le problème d'estimation conjointe est transformé en un problème d'optimisation avec une fonction du coût non-quadratique. Pour le résoudre, l'utilisation d'un algorithme de gradient conjugué non-linéaire avec le pas de descente quasi-optimale est proposée.La performance de l'approche proposée est analysée avec des données simulées et expérimentales. Les résultats montrent que l'approche proposée est robuste au bruit et aux matériaux. Par rapport aux approches existantes, l'approche proposée présente des avantages sur la qualité de reconstruction.Multi-Energy Computed Tomography (MECT) makes it possible to get multiple fractions of basis materials without segmentation. In medical application, one is the soft-tissue equivalent water fraction and the other is the hard-matter equivalent bone fraction. Practical MECT measurements are usually obtained with polychromatic X-ray beams. Existing reconstruction approaches based on linear forward models without counting the beam polychromaticity fail to estimate the correct decomposition fractions and result in Beam-Hardening Artifacts (BHA). The existing BHA correction approaches either need to refer to calibration measurements or suffer from the noise amplification caused by the negative-log pre-processing and the water and bone separation problem. To overcome these problems, statistical DECT reconstruction approaches based on non-linear forward models counting the beam polychromaticity show great potential for giving accurate fraction images.This work proposes a full-spectral Bayesian reconstruction approach which allows the reconstruction of high quality fraction images from ordinary polychromatic measurements. This approach is based on a Gaussian noise model with unknown variance assigned directly to the projections without taking negative-log. Referring to Bayesian inferences, the decomposition fractions and observation variance are estimated by using the joint Maximum A Posteriori (MAP) estimation method. Subject to an adaptive prior model assigned to the variance, the joint estimation problem is then simplified into a single estimation problem. It transforms the joint MAP estimation problem into a minimization problem with a non-quadratic cost function. To solve it, the use of a monotone Conjugate Gradient (CG) algorithm with suboptimal descent steps is proposed.The performances of the proposed approach are analyzed with both simulated and experimental data. The results show that the proposed Bayesian approach is robust to noise and materials. It is also necessary to have the accurate spectrum information about the source-detector system. When dealing with experimental data, the spectrum can be predicted by a Monte Carlo simulator. For a variety of materials, less than 5% estimation errors are observed on their average decomposition fractions.The proposed approach is a statistical reconstruction approach based on a non-linear forward model counting the full beam polychromaticity and applied directly to the projections without taking negative-log. Compared to the approaches based on linear forward models and the BHA correction approaches, it has advantages in noise robustness and reconstruction accuracy

A parameter inversion approach based on a MCMC sampling method in Eddy-Current Testing

International audienceIn Eddy Current Testing (ECT), we are often interested in getting details about the flaw presented in the inspected parts, such as size, location, shape, etc. For this objective, inversion approaches are employed to estimate the flaw parameters. Among the variety of inversion approaches, Bayesian methods have great potential in giving accurate estimations. However, their computational costs are usually very high. This makes them unsuitable for practical applications. Since most of the Bayesian estimation approaches need the help of a Markov Chain Monte Carlo (MCMC) method or an iterative optimization algorithm to get the final estimations of flaw parameters, the forward model is required to be employed many times in a Bayesian approach. This is the main reason why they are so expensive in terms of the computational cost. In this work, we first propose to use a metamodeling method to approximate the tradi-tional forward model. In such a way, the Posterior Mean (PM) estimation method can best reduce the computational cost. To solve the PM estimation problem, a random walk MCMC method is proposed. The metamodel can highly reduces the computational cost while the MCMC sampling method characterizes the estimation uncertainty at the same time. This makes the combined approach become very attractive in practical applications. Simulations and laboratory controlled experiments are conducted to test the performance and robustness of the method. The results show that one can estimate accurately the flaw parameters by using this Bayesian inversion method, no matter it is a dimension charac-terization problem or a flaw location problem. The results also show that the estimation uncertainty depends upon the size of the flaw. In general, small flaws have poor estimation certainty. As for the computational time, it depends exponentially on the dimension of the unknown flaw parameters

Choice of flaw models in eddy-current testing by using nested sampling

Mini-symposium A13 "Bayesian Inverse Problems and Data Assimilation", V. H. Hoang and A. Stuart as organizersInternational audienceIn Eddy-Current Testing (ECT), we are interested in obtaining information about the flaws possibly present in the inspected parts. Choice of model is a general problem that one can run into when analyzing the shapes, the number of flaws and furthermore the influence of nuisance parameters, like the lift-off. In the framework of Bayesian inference, we need to calculate the evidences of the concerned models in order to choose between (or among) models. Nested Sampling (NS) shows the possibility of approximating the evidence with reasonable computational cost. This contribution proposes a model choice method based on an improved NS algorithm the aim of which is to get independent samples with hard constraint on the likelihood value in a more efficient way. It works for models who have Gaussian-like likelihood distributions. By analyzing the model evidences approximated by the NS algorithm and the final active samples, this method makes it possible to assign the correct model for the flaw of concern and meanwhile to estimate the corresponding flaw parameters. Simulations have been conducted to validate this method. The results confirm its computational efficiency and model choice reliability insisting the fact that metamodels contribute to the efficiency in the most complex cases

Joint Reduce of Metal and Beam Hardenning Artifacts using Multi-Energy MAP Approach in X-ray Computed Tomography

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Influence of partially known parameter on flaw characterization in Eddy Current Testing by using a random walk MCMC method based on metamodeling

International audienceFirst, we present the implementation of a random walk Metropolis-within-Gibbs (MWG) sampling method in flaw characterization based on a metamodeling method. The role of metamodeling is to reduce the computational time cost in Eddy Current Testing (ECT) forward model calculation. In such a way, the use of Markov Chain Monte Carlo (MCMC) methods becomes possible. Secondly, we analyze the influence of partially known parameters in Bayesian estimation. The objective is to evaluate the importance of providing more specific prior information. Simulation results show that even partially known information has great interest in providing more accurate flaw parameter estimations. The improvement ratio depends on the parameter dependence and the interest shows only when the provided information is specific enough

Choice of flaw models in eddy-current testing by using nested sampling

Mini-symposium A13 "Bayesian Inverse Problems and Data Assimilation", V. H. Hoang and A. Stuart as organizersInternational audienceIn Eddy-Current Testing (ECT), we are interested in obtaining information about the flaws possibly present in the inspected parts. Choice of model is a general problem that one can run into when analyzing the shapes, the number of flaws and furthermore the influence of nuisance parameters, like the lift-off. In the framework of Bayesian inference, we need to calculate the evidences of the concerned models in order to choose between (or among) models. Nested Sampling (NS) shows the possibility of approximating the evidence with reasonable computational cost. This contribution proposes a model choice method based on an improved NS algorithm the aim of which is to get independent samples with hard constraint on the likelihood value in a more efficient way. It works for models who have Gaussian-like likelihood distributions. By analyzing the model evidences approximated by the NS algorithm and the final active samples, this method makes it possible to assign the correct model for the flaw of concern and meanwhile to estimate the corresponding flaw parameters. Simulations have been conducted to validate this method. The results confirm its computational efficiency and model choice reliability insisting the fact that metamodels contribute to the efficiency in the most complex cases