Examinando por Materia "Differential geometry"

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    Hessian eigenfunctions for triangular mesh parameterization
    (SciTePress, 2016-02-27) Mejia, D.; Ruiz OE; Cadavid, C.; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAE
    Hessian Locally Linear Embedding (HLLE) is an algorithm that computes the nullspace of a Hessian functional H for Dimensionality Reduction (DR) of a sampled manifold M. This article presents a variation of classic HLLE for parameterization of 3D triangular meshes. Contrary to classic HLLE which estimates local Hessian nullspaces, the proposed approach follows intuitive ideas from Differential Geometry where the local Hessian is estimated by quadratic interpolation and a partition of unity is used to join all neighborhoods. In addition, local average triangle normals are used to estimate the tangent plane TxM at x ? M instead of PCA, resulting in local parameterizations which reflect better the geometry of the surface and perform better when the mesh presents sharp features. A high frequency dataset (Brain) is used to test our algorithm resulting in a higher rate of success (96.63%) compared to classic HLLE (76.4%). © Copyright 2016 by SCITEPRESS - Science and Technology Publications, Lda. All rights reserved.
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    Ítem
    Hessian eigenfunctions for triangular mesh parameterization
    (SciTePress, 2016-02-27) Mejia, D.; Ruiz OE; Cadavid, C.; Mejia, D.; Ruiz OE; Cadavid, C.; Universidad EAFIT. Departamento de Ciencias; Matemáticas y Aplicaciones
    Hessian Locally Linear Embedding (HLLE) is an algorithm that computes the nullspace of a Hessian functional H for Dimensionality Reduction (DR) of a sampled manifold M. This article presents a variation of classic HLLE for parameterization of 3D triangular meshes. Contrary to classic HLLE which estimates local Hessian nullspaces, the proposed approach follows intuitive ideas from Differential Geometry where the local Hessian is estimated by quadratic interpolation and a partition of unity is used to join all neighborhoods. In addition, local average triangle normals are used to estimate the tangent plane TxM at x ? M instead of PCA, resulting in local parameterizations which reflect better the geometry of the surface and perform better when the mesh presents sharp features. A high frequency dataset (Brain) is used to test our algorithm resulting in a higher rate of success (96.63%) compared to classic HLLE (76.4%). © Copyright 2016 by SCITEPRESS - Science and Technology Publications, Lda. All rights reserved.
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    A q-exponential statistical Banach manifold
    (ACADEMIC PRESS INC ELSEVIER SCIENCE, 2013-02-15) Loaiza, G.; Quiceno, H. R.; Loaiza, G.; Quiceno, H. R.; Universidad EAFIT. Departamento de Ciencias; Matemáticas y Aplicaciones
    Let µ be a given probability measure and Mµ the set of µ-equivalent strictly positive probability densities. In this paper we construct a Banach manifold on Mµ, modeled on the space L 8(p{dot operator}µ) where p is a reference density, for the non-parametric q-exponential statistical models (Tsallis's deformed exponential), where 0<q<1 is any real number. This family is characterized by the fact that when q?1, then the non-parametric exponential models are obtained and the manifold constructed by Pistone and Sempi is recovered, up to continuous embeddings on the modeling space. The coordinate mappings of the manifold are given in terms of Csiszár's F-divergences; the tangent vectors are identified with the one-dimensional q-exponential models and q-deformations of the score function. © 2012 Elsevier Ltd.