Contributions à l'étude de la classification spectrale et applications

La classification spectrale consiste à créer, à partir des éléments spectraux d'une matrice d'affinité gaussienne, un espace de dimension réduite dans lequel les données sont regroupées en classes. Cette méthode non supervisée est principalement basée sur la mesure d'affinité gaussienne, son paramètre et ses éléments spectraux. Cependant, les questions sur la séparabilité des classes dans l'espace de projection spectral et sur le choix du paramètre restent ouvertes. Dans un premier temps, le rôle du paramètre de l'affinité gaussienne sera étudié à travers des mesures de qualités et deux heuristiques pour le choix de ce paramètre seront proposées puis testées. Ensuite, le fonctionnement même de la méthode est étudié à travers les éléments spectraux de la matrice d'affinité gaussienne. En interprétant cette matrice comme la discrétisation du noyau de la chaleur définie sur l'espace entier et en utilisant les éléments finis, les vecteurs propres de la matrice affinité sont la représentation asymptotique de fonctions dont le support est inclus dans une seule composante connexe. Ces résultats permettent de définir des propriétés de classification et des conditions sur le paramètre gaussien. A partir de ces éléments théoriques, deux stratégies de parallélisation par décomposition en sous-domaines sont formulées et testées sur des exemples géométriques et de traitement d'images. Enfin dans le cadre non supervisé, le classification spectrale est appliquée, d'une part, dans le domaine de la génomique pour déterminer différents profils d'expression de gènes d'une légumineuse et, d'autre part dans le domaine de l'imagerie fonctionnelle TEP, pour segmenter des régions du cerveau présentant les mêmes courbes d'activités temporelles. ABSTRACT : The Spectral Clustering consists in creating, from the spectral elements of a Gaussian affinity matrix, a low-dimension space in which data are grouped into clusters. This unsupervised method is mainly based on Gaussian affinity measure, its parameter and its spectral elements. However, questions about the separability of clusters in the projection space and the spectral parameter choices remain open. First, the rule of the parameter of Gaussian affinity will be investigated through quality measures and two heuristics for choosing this setting will be proposed and tested. Then, the method is studied through the spectral element of the Gaussian affinity matrix. By interpreting this matrix as the discretization of the heat kernel defined on the whole space and using finite elements, the eigenvectors of the affinity matrix are asymptotic representation of functions whose support is included in one connected component. These results help define the properties of clustering and conditions on the Gaussian parameter. From these theoretical elements, two parallelization strategies by decomposition into sub-domains are formulated and tested on geometrical examples and images. Finally, as unsupervised applications, the spectral clustering is applied, first in the field of genomics to identify different gene expression profiles of a legume and the other in the imaging field functional PET, to segment the brain regions with similar time-activity curves

Similarity Detection for Free-Form Parametric Models

International audienceIn this article, we propose a framework for detecting local similarities in free-form parametric models, in particular on B-Splines or NURBS based B-reps: patches similar up to an approximated isometry are identified. Many recent articles have tackled similarity detection on 3D objects, in particular on 3D meshes. The parametric B-splines, or NURBS models are standard in the CAD (Computer Aided Design) industry, and similarity detection opens the door to interesting applications in this domain, such as model editing, objects comparison or efficient coding. Our contributions are twofold: we adapt the current technique called votes transformation space for parametric surfaces and we improve the identification of isometries. First, an orientation technique independent of the parameterization permits to identify direct versus indirect transformations. Second, the validation step is generalized to extend to the whole B-rep. Then, by classifying the isometries according to their fixed points, we simplify the clustering step. We also apply an unsupervised spectral clustering method which improves the results but also automatically estimates the number of clusters

Symmetry and Fourier descriptor : a hybrid feature for NURBS based B-Rep models retrieval

International audienceAs the number of models in 3D databases grows, an efficient 3D models indexing mechanism and a similarity measure to ease model retrieval are necessary. In this paper, we present a query-by-model framework for NURBS based B-Rep models retrieval that combines partial symmetry of the object and the Fourier shape descriptor of canonical 2D projections of the 3D models. In fact, most objects are composed by similar parts up to an isometry. By detecting the dominant partial symmetry of a given NURBS based B-Rep model, we define two canonical planes from which the Fourier descriptors are extracted to measure the similarity among 3D models

Alignement de modèles 3d paramétriques BRep basé sur la détection de symétries partielles. Application à l'indexation 3D

National audienceCet article présente une méthode originale d’alignement d’objets 3D modélisés par des B-Rep basés NURBS en identifiant les symétries partielles au sein de ces objets. L’alignement des objets 3D est une étape importante de pré-traitement pour la recherche et l’indexation : une méthode d’alignement fiable est nécessaire. Étant donné un modèle 3D, la pose normalisée de l’objet est définie par trois plans canoniques. Nous identifions le premier plan canonique par un algorithme efficace de détection de la symétrie partielle dominante en utilisant une approche de mise en correspondance des faces. Un autre algorithme basé sur l’aire de projection détermine les deux plans restants. Notre méthode est ensuite appliquée à la recherche des objets 3D dans une répertoire des modèles B-Rep basés NURBS

Spectral Clustering: interpretation and Gaussian parameter

Spectral clustering consists in creating, from the spectral elements of a Gaussian affinity matrix, a low-dimensional space in which data are grouped into clusters. However, questions about the separability of clusters in the projection space and the choice of the Gaussian parameter remain open. By drawing back to some continuous formulation, we propose an interpretation of spectral clustering with Partial Differential Equations tools which provides clustering properties and defines bounds for the affinity parameter

Numerical simulation of the thermal effects of localized fires

The thermal effects of a localized fire bellow a concrete slab with the length of 10 m and the thickness of 30 cm is simulated. The nonlinear equation of heat transfer was solved by finite differences using an implicit scheme. The appropriate mesh size in the direction corresponding to the heat flux resulting from the fire was defined. The temperature results of the two dimensional simulation does not depend on the dimension of the mesh size in the horizontal direction (perpendicular to the heat flux).info:eu-repo/semantics/publishedVersio

Segmentation of Dynamic PET Images with Kinetic Spectral Clustering

International audienceSegmentation is often required for the analysis of dynamic positron emission tomography (PET) images. However, noise and low spatial resolution make it a difficult task and several supervised and unsupervised methods have been proposed in the literature to perform the segmentation based on semi-automatic clustering of the time activity curves of voxels. In this paper we propose a new method based on spectral clustering that does not require any prior information on the shape of clusters in the space in which they are identified. In our approach, the p-dimensional data, where p is the number of time frames, is first mapped into a high dimensional space and then clustering is performed in a low-dimensional space of the Laplacian matrix. An estimation of the bounds for the scale parameter involved in the spectral clustering is derived. The method is assessed using dynamic brain PET images simulated with GATE and results on real images are presented. We demonstrate the usefulness of the method and its superior performance over three other clustering methods from the literature. The proposed approach appears as a promising pre-processing tool before parametric map calculation or ROI-based quantification tasks

3D+t segmentation of PET images using spectral clustering

International audienceSegmentation of dynamic PET images is often needed to extract the time activity curve (TAC) of regions. While clustering methods have been proposed to segment the PET sequence, they are generally either sensitive to initial conditions or favor convex shaped clusters. Recently, we have proposed a deterministic and automatic spectral clustering method (AD-KSC) of PET images. It has the advantage of handling clusters with arbitrary shape in the space in which they are identified. While improved results were obtained with AD-KSC compared to other methods, its use for clinical applications is constrained to 2D+t PET data due to its computational complexity. In this paper, we propose an extension of AD-KSC to make it applicable to 3D+t PET data. First, a preprocessing step based on a recursive principle component analysis and a Global K-means approach is used to generate many small seed clusters. AD-KSC is then applied on the generated clusters to obtain the final partition of the data. We validated the method with GATE Monte Carlo simulations of Zubal head phantom. The proposed approach improved the region of interest (ROI) definition and outperformed the K-means algorithm

Contributions to the study of spectral clustering and applications

La classification spectrale consiste à créer, à partir des éléments spectraux d'une matrice d'affinité gaussienne, un espace de dimension réduite dans lequel les données sont regroupées en classes. Cette méthode non supervisée est principalement basée sur la mesure d'affinité gaussienne, son paramètre et ses éléments spectraux. Cependant, les questions sur la séparabilité des classes dans l'espace de projection spectral et sur le choix du paramètre restent ouvertes. Dans un premier temps, le rôle du paramètre de l'affinité gaussienne sera étudié à travers des mesures de qualités et deux heuristiques pour le choix de ce paramètre seront proposées puis testées. Ensuite, le fonctionnement même de la méthode est étudié à travers les éléments spectraux de la matrice d'affinité gaussienne. En interprétant cette matrice comme la discrétisation du noyau de la chaleur définie sur l'espace entier et en utilisant les éléments finis, les vecteurs propres de la matrice affinité sont la représentation asymptotique de fonctions dont le support est inclus dans une seule composante connexe. Ces résultats permettent de définir des propriétés de classification et des conditions sur le paramètre gaussien. A partir de ces éléments théoriques, deux stratégies de parallélisation par décomposition en sous-domaines sont formulées et testées sur des exemples géométriques et de traitement d'images. Enfin dans le cadre non supervisé, le classification spectrale est appliquée, d'une part, dans le domaine de la génomique pour déterminer différents profils d'expression de gènes d'une légumineuse et, d'autre part dans le domaine de l'imagerie fonctionnelle TEP, pour segmenter des régions du cerveau présentant les mêmes courbes d'activités temporelles.The Spectral Clustering consists in creating, from the spectral elements of a Gaussian affinity matrix, a low-dimension space in which data are grouped into clusters. This unsupervised method is mainly based on Gaussian affinity measure, its parameter and its spectral elements. However, questions about the separability of clusters in the projection space and the spectral parameter choices remain open. First, the rule of the parameter of Gaussian affinity will be investigated through quality measures and two heuristics for choosing this setting will be proposed and tested. Then, the method is studied through the spectral element of the Gaussian affinity matrix. By interpreting this matrix as the discretization of the heat kernel defined on the whole space and using finite elements, the eigenvectors of the affinity matrix are asymptotic representation of functions whose support is included in one connected component. These results help define the properties of clustering and conditions on the Gaussian parameter. From these theoretical elements, two parallelization strategies by decomposition into sub-domains are formulated and tested on geometrical examples and images. Finally, as unsupervised applications, the spectral clustering is applied, first in the field of genomics to identify different gene expression profiles of a legume and the other in the imaging field functional PET, to segment the brain regions with similar time-activity curves