N-ary Mathematical Morphology

International audienceMathematical morphology on binary images can be fully de-scribed by set theory. However, it is not sucient to formulate mathe-matical morphology for grey scale images. This type of images requires the introduction of the notion of partial order of grey levels, together with the denition of sup and inf operators. More generally, mathemati-cal morphology is now described within the context of the lattice theory. For a few decades, attempts are made to use mathematical morphology on multivariate images, such as color images, mainly based on the no-tion of vector order. However, none of these attempts has given fully satisfying results. Instead of aiming directly at the multivariate case we propose an extension of mathematical morphology to an intermediary situation: images composed of a nite number of independent unordered categories

Computing Histogram of Tensor Images using Orthogonal Series Density Estimation and Riemannian Metrics

This paper deals with the computation of the histogram of tensor images, that is, images where at each pixel is given a n by n positive definite symmetric matrix, SPD(n). An approach based on orthogonal series density estimation is introduced, which is particularly useful for the case of measures based on Riemannian metrics. By considering SPD(n) as the space of the covariance matrices of multivariate gaussian distributions, we obtain the corresponding density estimation for the measure of both the Fisher metric and the Wasserstein metric. Experimental results on the application of such histogram estimation to DTI image segmentation, texture segmentation and texture recognition are included

Kernel density estimation on spaces of Gaussian distributions and symmetric positive definite matrices

This paper analyses the kernel density estimation on spaces of Gaussian distributions endowed with different metrics. Explicit expressions of kernels are provided for the case of the 2-Wasserstein metric on multivariate Gaussian distributions and for the Fisher metric on multivariate centred distributions. Under the Fisher metric, the space of multivariate centred Gaussian distributions is isometric to the space of symmetric positive definite matrices under the affine-invariant metric and the space of univariate Gaussian distributions is isometric to the hyperbolic space. Thus kernel are also valid on these spaces. The density estimation is successfully applied to a classification problem of electro-encephalographic signals

Kernel density estimation on the Siegel space applied to radar processing

Main techniques of probability density estimation on Riemannian manifolds are reviewed in the case of the Siegel space. For computational reasons we chose to focus on the kernel density estimation. The main result of the paper is the expression of Pelletier's kernel density estimator. The method is applied to density estimation of reflection coefficients from radar observations

Morphologie, Géométrie et Statistiques en imagerie non-standard

Digital image processing has followed the evolution of electronic and computer science. It is now current to deal with images valued not in {0,1} or in gray-scale, but in manifolds or probability distributions. This is for instance the case for color images or in diffusion tensor imaging (DTI). Each kind of images has its own algebraic, topological and geometric properties. Thus, existing image processing techniques have to be adapted when applied to new imaging modalities. When dealing with new kind of value spaces, former operators can rarely be used as they are. Even if the underlying notion has still a meaning, a work must be carried out in order to express it in the new context.The thesis is composed of two independent parts. The first one, "Mathematical morphology on non-standard images", concerns the extension of mathematical morphology to specific cases where the value space of the image does not have a canonical order structure. Chapter 2 formalizes and demonstrates the irregularity issue of total orders in metric spaces. The main results states that for any total order in a multidimensional vector space, there are images for which the morphological dilations and erosions are irregular and inconsistent. Chapter 3 is an attempt to generalize morphology to images valued in a set of unordered labels.The second part "Probability density estimation on Riemannian spaces" concerns the adaptation of standard density estimation techniques to specific Riemannian manifolds. Chapter 5 is a work on color image histograms under perceptual metrics. The main idea of this chapter consists in computing histograms using local Euclidean approximations of the perceptual metric, and not a global Euclidean approximation as in standard perceptual color spaces. Chapter 6 addresses the problem of non parametric density estimation when data lay in spaces of Gaussian laws. Different techniques are studied, an expression of kernels is provided for the Wasserstein metric.Le traitement d'images numériques a suivi l'évolution de l'électronique et de l'informatique. Il est maintenant courant de manipuler des images à valeur non pas dans {0,1}, mais dans des variétés ou des distributions de probabilités. C'est le cas par exemple des images couleurs où de l'imagerie du tenseur de diffusion (DTI). Chaque type d'image possède ses propres structures algébriques, topologiques et géométriques. Ainsi, les techniques existantes de traitement d'image doivent être adaptés lorsqu'elles sont appliquées à de nouvelles modalités d'imagerie. Lorsque l'on manipule de nouveaux types d'espaces de valeurs, les précédents opérateurs peuvent rarement être utilisés tel quel. Même si les notions sous-jacentes ont encore un sens, un travail doit être mené afin de les exprimer dans le nouveau contexte. Cette thèse est composée de deux parties indépendantes. La première, « Morphologie mathématiques pour les images non standards », concerne l'extension de la morphologie mathématique à des cas particuliers où l'espace des valeurs de l'image ne possède pas de structure d'ordre canonique. Le chapitre 2 formalise et démontre le problème de l'irrégularité des ordres totaux dans les espaces métriques. Le résultat principal de ce chapitre montre qu'étant donné un ordre total dans un espace vectoriel multidimensionnel, il existe toujours des images à valeur dans cet espace tel que les dilatations et les érosions morphologiques soient irrégulières et incohérentes. Le chapitre 3 est une tentative d'extension de la morphologie mathématique aux images à valeur dans un ensemble de labels non ordonnés.La deuxième partie de la thèse, « Estimation de densités de probabilités dans les espaces de Riemann » concerne l'adaptation des techniques classiques d'estimation de densités non paramétriques à certaines variétés Riemanniennes. Le chapitre 5 est un travail sur les histogrammes d'images couleurs dans le cadre de métriques perceptuelles. L'idée principale de ce chapitre consiste à calculer les histogrammes suivant une approximation euclidienne local de la métrique perceptuelle, et non une approximation globale comme dans les espaces perceptuels standards. Le chapitre 6 est une étude sur l'estimation de densité lorsque les données sont des lois Gaussiennes. Différentes techniques y sont analysées. Le résultat principal est l'expression de noyaux pour la métrique de Wasserstein

Density estimation and modeling on symmetric spaces

In many applications, data and/or parameters are supported on non-Euclidean manifolds. It is important to take into account the geometric structure of manifolds in statistical analysis to avoid misleading results. Although there has been a considerable focus on simple and specific manifolds, there is a lack of general and easy-to-implement statistical methods for density estimation and modeling on manifolds. In this article, we consider a very broad class of manifolds: non-compact Riemannian symmetric spaces. For this class, we provide a very general mathematical result for easily calculating volume changes of the exponential and logarithm map between the tangent space and the manifold. This allows one to define statistical models on the tangent space, push these models forward onto the manifold, and easily calculate induced distributions by Jacobians. To illustrate the statistical utility of this theoretical result, we provide a general method to construct distributions on symmetric spaces. In particular, we define the log-Gaussian distribution as an analogue of the multivariate Gaussian distribution in Euclidean space. With these new kernels on symmetric spaces, we also consider the problem of density estimation. Our proposed approach can use any existing density estimation approach designed for Euclidean spaces and push it forward to the manifold with an easy-to-calculate adjustment. We provide theorems showing that the induced density estimators on the manifold inherit the statistical optimality properties of the parent Euclidean density estimator; this holds for both frequentist and Bayesian nonparametric methods. We illustrate the theory and practical utility of the proposed approach on the space of positive definite matrices

Wrapped statistical models on manifolds: motivations, the case SE(n), and generalization to symmetric spaces

International audienceWe address here the construction of wrapped probability densities on Lie groups and quotient of Lie groups using the exponential map. The paper starts by briefly reviewing the different approaches to build densities on a manifold and shows the interest of wrapped distributions. We then construct wrapped densities on SE(n) and discuss their statistical estimation. We conclude by an opening to the case of symmetric spaces

Fast and easy preparation of few-layered-graphene/magnesia powders for strong, hard and electrically conducting composites

Composite powders were prepared by the chemical vapor deposition (CH4/Ar atmosphere) of carbon in the form of 2e8 layers few-layered-graphene (FLG) covering the MgO powder grains, without any mixing step. The composites were consolidated to nearly full (99%) density by spark plasma sintering with no or little damage to the FLG. The FLG is located along the MgO grain boundaries, as opposed to be dispersed as discrete particles or flakes. This causes a dramatic hindrance of the MgO grain growth, the average grain size being considerably lower for the sample with 2.08 vol% carbon (200 nm) than for pure MgO(3.7 mm). The samples are investigated by Raman spectroscopy, scanning and transmission electron microscopy. The composites are electrically conducting with a percolation threshold below 0.56 vol%.Compared to pure MgO, the composites are simultaneously stronger (345 vs 200 MPa) and harder (9.8 vs 3.8 GPa). This could arise from reinforcement mechanisms such as crack-deflection and crack-bridging by FLG, but also from MgO grain refinement

Theoretical Limits on the Equation-of-State Parameter of Phantom Cosmology

We investigate the restrictions on the equation-of-state parameter of phantom cosmology, due to the minimum quantum gravitational requirements. We find that for all the examined $w_\Lambda(z)$-parametrizations and for arbitrary phantom potentials and spatial curvature, the phantom equation-of-state parameter is not restricted at all. This is in radical contrast with the quintessence paradigm, and makes phantom cosmology more robust and capable of constituting the underlying mechanism for dark energy.Comment: 7 pages, 7 figure

Future of IR: Emerging Techniques, Looking to the Future…and Learning from the Past

Innovation has been the cornerstone of interventional radiology since the early years of the founders, with a multitude of new therapeutic approaches developed over the last 50 years. What is the future holding for us? This article presents an overview of the in-coming developments that are catching on at this moment, particularly focusing on three items: the new applications of existing techniques, particularly embolotherapy and interventional oncology; the cutting-edge devices; the imaging technologies at the forefront of the image-guidance. Besides this, clinical vision and patient relation remain crucial for the future of the discipline