Composição de coordenadas normais de Rieman locais e geometria poliedral em aprendizado de variedades com aplicações de teoria de folheações

• Autor(a) principal: Miranda Junior, Gastão Florêncio
• Data de Publicação: 2015
• Tipo de documento: Tese
• Idioma: por
• Título da fonte: Biblioteca Digital de Teses e Dissertações do LNCC
• Texto Completo: https://tede.lncc.br/handle/tede/226
• topic: Reconhecimento de padrões; Geometria poliedral; Pattern recognition; Polyhedral geometry; CNPQ::CIENCIAS EXATAS E DA TERRA::CIENCIA DA COMPUTACAO::TEORIA DA COMPUTACAO::LINGUAGEM FORMAIS E AUTOMATOS

Manifold learning techniques have been used for dimensionality reduction in applications involving pattern recognition, data mining and computer vision. This thesis describes recent works that we have done in this area as well as perspectives for future works. First, we propose a methodology called Local Riemannian Manifold Learning (LRML), which recovers the topology and geometry of the manifold using local systems of normal coordinates computed through the exponential application. The LRML strategy has the advantage of reducing the accumulation of errors during the process of manifold learning. However, the obtained parameterization can not be used as an unambiguous representation space. Furthermore, the synthesis process needs domain triangulation in the parameter space to be efficiently performed. We address this drawback of LRML using a composition procedure to structure the neighborhoods of normal coordinates building a global representation space that locally preserves radial geodesic distances. Besides, we add a geometric structure based on triangulations obtaining an efficient methodology to the synthesis process. We also explored discrete geometry concepts for generation of piecewise linear manifolds for data analysis. In the computational experiments we verify the efficiency of the LRML combined with the composition process and discrete geometry framework for the synthesis and data mining. We explored the application of foliation theory for images of human faces with multiple facial expressions. We conclude that this approach is a promising one for the study of the geometry and topology of the space of human face images.