10 research outputs found

    Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements

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    A preliminary appeared as INRIA RR-5093, January 2004.International audienceIn medical image analysis and high level computer vision, there is an intensive use of geometric features like orientations, lines, and geometric transformations ranging from simple ones (orientations, lines, rigid body or affine transformations, etc.) to very complex ones like curves, surfaces, or general diffeomorphic transformations. The measurement of such geometric primitives is generally noisy in real applications and we need to use statistics either to reduce the uncertainty (estimation), to compare observations, or to test hypotheses. Unfortunately, even simple geometric primitives often belong to manifolds that are not vector spaces. In previous works [1, 2], we investigated invariance requirements to build some statistical tools on transformation groups and homogeneous manifolds that avoids paradoxes. In this paper, we consider finite dimensional manifolds with a Riemannian metric as the basic structure. Based on this metric, we develop the notions of mean value and covariance matrix of a random element, normal law, Mahalanobis distance and XÂČ law. We provide a new proof of the characterization of Riemannian centers of mass and an original gradient descent algorithm to efficiently compute them. The notion of Normal law we propose is based on the maximization of the entropy knowing the mean and covariance of the distribution. The resulting family of pdfs spans the whole range from uniform (on compact manifolds) to the point mass distribution. Moreover, we were able to provide tractable approximations (with their limits) for small variances which show that we can effectively implement and work with these definitions

    Heterogeneous System in Organic Synthesis: A Review

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    Periphery Decorated and Core Initiated Neutral and Polyanionic Borane Large Molecules: Forthcoming and Promising Properties for Medicinal Applications

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    Dendrimers and Fullerenes

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    Architecture of Polymers: Topological Structure–Properties Relationship

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