Les images médicales sont utilisées afin de représenter l'anatomie. Le caractère non- linéaire d'imagerie médicale rendent leur analyse difficile. Dans cette thèse, nous nous intéressons à l'analyse d'images médicales du point de vue de la théorie statistique de l'apprentissage. Tout d'abord, nous examinons méthodes de régularisation. Dans cette direction, nous introduisons une nouvelle méthode de régularisation, la k-support regularized SVM. Cet algorithme étend la SVM régularisée `1 à une norme mixte de toutes les deux normes `1 et `2. Ensuite, nous nous intéressons un problème de comparaison des graphes. Les graphes sont une technique utilisée pour la représentation des données ayant une structure héritée. L'exploitation de ces données nécessite la capacité de comparer des graphes. Malgré le progrès dans le domaine des noyaux sur graphes, les noyaux sur graphes existants se concentrent à des graphes non-labellisés ou labellisés de façon discrète, tandis que la comparaison de graphes labellisés par des vecteurs continus, demeure un problème de recherche ouvert. Nous introduisons une nouvelle méthode, l'algorithme de Weisfeiler-Lehman pyramidal et quantifié afin d'aborder le problème de la comparaison des graphes labellisés par des vecteurs continus. Notre algorithme considère les statistiques de motifs sous arbre, basé sur l'algorithme Weisfeiler-Lehman ; il utilise une stratégie de quantification pyramidale pour déterminer un nombre logarithmique de labels discrets. Globalement, les graphes étant des objets mathématiques fondamentaux et les méthodes de régularisation étant utilisés pour contrôler des problèmes mal-posés, notre algorithmes pourraient appliqués sur un grand éventail d'applications. / Medical images have been used to depict the anatomy or function. Their high-dimensionality and their non-linearity nature makes their analysis a challenging problem. In this thesis, we address the medical image analysis from the viewpoint of statistical learning theory. First, we examine regularization methods for analyzing MRI data. In this direction, we introduce a novel regularization method, the k-support regularized Support Vector Machine. This algorithm extends the 1 regularized SVM to a mixed norm of both `1 and `2 norms. We evaluate our algorithm in a neuromuscular disease classification task. Second, we approach the problem of graph representation and comparison for analyzing medical images. Graphs are a technique to represent data with inherited structure. Despite the significant progress in graph kernels, existing graph kernels focus on either unlabeled or discretely labeled graphs, while efficient and expressive representation and comparison of graphs with continuous high-dimensional vector labels, remains an open research problem. We introduce a novel method, the pyramid quantized Weisfeiler-Lehman graph representation to tackle the graph comparison problem for continuous vector labeled graphs. Our algorithm considers statistics of subtree patterns based on the Weisfeiler-Lehman algorithm and uses a pyramid quantization strategy to determine a logarithmic number of discrete labelings. We evaluate our algorithm on two different tasks with real datasets. Overall, as graphs are fundamental mathematical objects and regularization methods are used to control ill-pose problems, both proposed algorithms are potentially applicable to a wide range of domains.

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Medical images have been used to depict the anatomy or function. Their high-dimensionality and their non-linearity nature makes their analysis a challenging problem. In this thesis, we address the medical image analysis from the viewpoint of statistical learning theory. First, we examine regularization methods for analyzing MRI data. In this direction, we introduce a novel regularization method, the k-support regularized Support Vector Machine. This algorithm extends the 1 regularized SVM to a mixed norm of both '1 and '2 norms. We evaluate our algorithm in a neuromuscular disease classification task. Second, we approach the problem of graph representation and comparison for analyzing medical images. Graphs are a technique to represent data with inherited structure. Despite the significant progress in graph kernels, existing graph kernels focus on either unlabeled or discretely labeled graphs, while efficient and expressive representation and comparison of graphs with continuous high-dimensional vector labels, remains an open research problem. We introduce a novel method, the pyramid quantized Weisfeiler-Lehman graph representation to tackle the graph comparison problem for continuous vector labeled graphs. Our algorithm considers statistics of subtree patterns based on the Weisfeiler-Lehman algorithm and uses a pyramid quantization strategy to determine a logarithmic number of discrete labelings. We evaluate our algorithm on two different tasks with real datasets. Overall, as graphs are fundamental mathematical objects and regularization methods are used to control ill-pose problems, both proposed algorithms are potentially applicable to a wide range of domains.