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Ce document présente les lois elliptiques d'un point de vue géométrique. Après avoir rappelé les notions élémentaires de géométrie nécessaires à la compréhension de ce document, nous introduisons la notion de lois elliptiques et une caractérisation de la loi normale (certainement la meilleure caractérisation). Nous finissons ce document par l'inférence des lois elliptiques.

International audience

International audience

Ce document présente les démonstrations du théorème de Bolzano-Weierstrass (version suite et version filtre) ainsi que du théorème de Tychonov dans le cas de produit généraux de compacts (dans des espaces non métriques).

Ce document présente ce qu'il faut savoir sur la logique du premier ordre ainsi que la très intéressante théorie de Zermelo-Fraenkel.

International audience

International audience; Purpose: Accurate tumor delineation in positron emission tomography (PET) images is crucial in oncology. Although recent methods achieved good results, there is still room for improvement regarding tumors with complex shapes, low signal-to-noise ratio, and high levels of uptake heterogeneity.Methods: The authors developed and evaluated an original clustering-based method called spatial positron emission quantification of tumor—Automatic Lp-norm estimation (SPEQTACLE), based on the fuzzy C-means (FCM) algorithm with a generalization exploiting a Hilbertian norm to more accurately account for the fuzzy and non-Gaussian distributions of PET images. An automatic and reproducible estimation scheme of the norm on an image-by-image basis was developed. Robustness was assessed by studying the consistency of results obtained on multiple acquisitions of the NEMA phantom on three different scanners with varying acquisition parameters. Accuracy was evaluated using classification errors (CEs) on simulated and clinical images. SPEQTACLE was compared to another FCM implementation, fuzzy local information C-means (FLICM) and fuzzy locally adaptive Bayesian (FLAB).Results: SPEQTACLE demonstrated a level of robustness similar to FLAB (variability of 14% ± 9% vs 14% ± 7%, p = 0.15) and higher than FLICM (45% ± 18%, p < 0.0001), and improved accuracy with lower CE (14% ± 11%) over both FLICM (29% ± 29%) and FLAB (22% ± 20%) on simulated images. Improvement was significant for the more challenging cases with CE of 17% ± 11% for SPEQTACLE vs 28% ± 22% for FLAB (p = 0.009) and 40% ± 35% for FLICM (p < 0.0001). For the clinical cases, SPEQTACLE outperformed FLAB and FLICM (15% ± 6% vs 37% ± 14% and 30% ± 17%, p < 0.004).Conclusions: SPEQTACLE benefitted from the fully automatic estimation of the norm on a case-by-case basis. This promising approach will be extended to multimodal images and multiclass estimation in future developments.

International audience; Accurate and robust tumor delineation using PET images is crucial for the extraction of features in therapy response monitoring and radiotherapy target definition. Manual delineation suffers from high inter- and intra-operator variability. Amongst the methods previously proposed, fuzzy c-means (FCM) has been popular due to its flexibility and low computational complexity. The objective of this work was to develop a modified FCM approach to handle PET images specific spatial resolution and noise properties.SPEQTACLE (Spatial Positron Emission image Quantification of Tumor-AutomatiC Lp-norm Estimation) is based on FCM in which the Euclidian norm is replaced by a more appropriate one on a case-by-case basis. More specifically an automated scheme was developed to estimate the optimal norm, as well as the number of clusters for each image, with the user input reduced to locating the tumor. SPEQTACLE was compared to the standard FCM (FCM1) and the kernel-regularized FCM (FCM2), as well as the Fuzzy Locally Adaptive Bayesian (FLAB) as a state-of-the-art method. The evaluation was carried out on a database of simulated and real clinical PET images with a wide range of tumor sizes and shapes, contrast, noise level, voxel sizes and intra-tumor tracer heterogeneity.In terms of accuracy, SPEQTACLE consistently outperformed other FCM implementations (errors 9.8±5.3% vs. 28±14% and 20±15% for FCM1 and FCM2 respectively) as well as FLAB in the majority of cases (15.6%±13.1%). Improved robustness and reproducibility of the algorithm was also achieved thanks to its ability to automatically estimate the optimal parameters on an image-by-image basis without compromising computational time and algorithm convergence.SPEQTACLE outperformed state-of-the-art methods, especially in challenging cases of low contrast, high noise, complex shapes and high intra-tumor heterogeneity. It is fully automatic and computationally efficient. Finally SPEQTACLE may be easily extended to multimodal PET/CT/MRI analysis which will be investigated in the future.

The objective of this thesis is to propose different - more general - models than the classical hidden Markov chain, which is often used in Bayesian inference of signals or images. The different extensions of this model aim to take into account different properties of the signal, such as its non gaussian behaviour or semi-markovianity of the hidden process. In a Bayesian inference context, we have two random processes X and Y , where the realisation y of Y is observed and the realisation x of X has to be estimated. The relationship between the two processes X and Y is modeled by the probability distribution p(x, y). In the classical model of hidden Markov chains with independent noise, the distribution p(x) is that of a Markov chain and the distribution p(y|x) is that of a vector whose marginal distributions are independent conditionally to x. Although this model can be used in several applications, it fails to model all situations of dependence. The first model we propose belongs to the Markov triplet models, in which we consider a third process U such that the triplet (X, U, Y ) is a Markov chain. In the proposed model, this auxiliary process models the semi-markovianity of X, and so we are able to take into account the possible non-markovianity of the hidden process. In a second model, we consider long dependence within the observations and we propose an original algorithm to estimate the parameters of this model. We also study different models taking simultaneously into account semi-markovianity of the hidden data, long dependence and non-stationarity. Finally, the non-Gaussian properties of noise are taken into account through the introduction of copulas. The viability of different models is also confirmed through experimentation. In the last part of this thesis, we are studying how the segmentation obtained by a Bayesian method can be used in detecting targets in the radar signal. The original detector that we implement uses statistical difference between a signal and the signals received from its surroundings. To achieve this, we define a distance between distributions that is used in detection. Through a series of tests, we show how the new detector thus implemented produces better results than the classical detector in a very noisy environment., L'objectif de cette thèse est de proposer différents modèles généralisant le modèle classique des chaînes de Markov cachées à bruit indépendant couramment utilisé en inférence bayésienne de signaux. Les diverses extensions de ce modèle visent à l'enrichir et à prendre en compte différentes propriétés du signal, comme le caractère non gaussien du bruit, ou la nature semi-markovienne du signal caché. Dans un problème d'inférence bayésienne, nous disposons de deux processus aléatoires X et Y , on observe la réalisation y de Y et nous estimons la réalisation cachée x de X. Le lien existant entre les deux processus est modélisé par la distribution de probabilité p(x, y). Dans le modèle classique des chaînes de Markov cachées à bruit indépendant, la distribution p(x) est celle d'une chaîne de Markov et la distribution p(y|x) est celle de marginales indépendantes conditionnellement à x. Bien que ce modèle puisse être utilisé dans de nombreuses applications, il ne parvient pas à modéliser toutes les situations de dépendance. Le premier modèle que nous proposons est de type “chaînes de Markov triplet”, on considère ainsi un troisième processus U tel que le triplet (X, U, Y ) soit une chaîne de Markov. Dans le modèle proposé, ce processus auxiliaire modélise la semi-markovianité de X ; on parvient ainsi à prendre en compte la non markovianité éventuelle du processus caché. Dans un deuxième modèle, nous considérons des observations à dépendance longue et nous proposons un algorithme d'estimation original des paramètres de ce modèle. Nous étudions par ailleurs différents modèles prenant en compte simultanément la semi-markovianité des données cachées, la dépendance longue dans les observations ou la non stationnarité des données cachées. Enfin, la nature non nécessairement gaussienne du bruit est prise en compte via l'introduction des copules. L'intérêt des différents modèles proposés est également validé au travers d'expérimentations. Dans la dernière partie de cette thèse, nous étudions également comment la segmentation obtenue par une méthode bayésienne peut être utilisée dans la détection de cibles dans le signal radar. Le détecteur original que nous implémentons utilise la différence de statistiques entre un signal recu et les signaux recus de son voisinage. Le détecteur ainsi implémenté s'avère donner de meilleurs résultats en présence de fort bruit que le détecteur habituellement utilisé en traitement radar.

International audience; Unsupervised segmentation of non-stationary hidden Markov chains with copulas