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Reference TypeConference Paper (Conference Proceedings)
Last Update2017: administrator
Metadata Last Update2021: administrator
Citation KeyMirandaJrThomGira:2017:GeDaAn
TitleGeometric Data Analysis Based on Manifold Learning with Applications for Image Understanding
Access Date2021, Mar. 02
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Author1 Miranda Junior, Gastao Florencio
2 Thomaz, Carlos Eduardo
3 Giraldi, Gilson Antonio
Affiliation1 Department of Mathematics, Federal University of Sergipe, Aracaju, Brazil
2 Department of Electrical Engineering, FEI, Sao Bernardo do Campo, Brazil
3 Department of Mathematics and Computational Methods, National Laboratory for Scientific Computing, Petropolis, Brazil
EditorTorchelsen, Rafael Piccin
Nascimento, Erickson Rangel do
Panozzo, Daniele
Liu, Zicheng
Farias, Mylène
Viera, Thales
Sacht, Leonardo
Ferreira, Nivan
Comba, João Luiz Dihl
Hirata, Nina
Schiavon Porto, Marcelo
Vital, Creto
Pagot, Christian Azambuja
Petronetto, Fabiano
Clua, Esteban
Cardeal, Flávio
Conference NameConference on Graphics, Patterns and Images, 30 (SIBGRAPI)
Conference LocationNiterói, RJ
DateOct. 17-20, 2017
Book TitleProceedings
PublisherSociedade Brasileira de Computação
Publisher CityPorto Alegre
Tertiary TypeTutorial
History2017-09-09 11:31:11 :: -> administrator ::
2021-02-23 03:53:32 :: administrator -> :: 2017
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Is the master or a copy?is the master
Content Stagecompleted
Keywordsmanifold learning, statistical learning, Riemannian manifolds, image analysis, deep learning.
AbstractNowadays, pattern recognition, computer vision, signal processing and medical image analysis, require the managing of large amount of multidimensional image databases, possibly sampled from nonlinear manifolds. The complex tasks involved in the analysis of such massive data lead to a strong demand for nonlinear methods for dimensionality reduction to achieve efficient representation for information extraction. In this avenue, manifold learning has been applied to embed nonlinear image data in lower dimensional spaces for subsequent analysis. The result allows a geometric interpretation of image spaces with relevant consequences for data topology, computation of image similarity, discriminant analysis/classification tasks and, more recently, for deep learning issues. In this paper, we firstly review Riemannian manifolds that compose the mathematical background in this field. Such background offers the support to set up a data model that embeds usual linear subspace learning and discriminant analysis results in local structures built from samples drawn from some unknown distribution. Afterwards, we discuss topological issues in data preparation for manifold learning algorithms as well as the determination of manifold dimension. Then, we survey dimensionality reduction techniques with particular attention to Riemannian manifold learning. Besides, we discuss the application of concepts in discrete and polyhedral geometry for synthesis and data clustering over the recovered Riemannian manifold with emphasis in face images in the computational experiments. Next, we discuss promising perspectives of manifold learning and related topics for image analysis, classification and relationships with deep learning methods. Specifically, we discuss the application of foliation theory, discriminant analysis and kernel methods in curved spaces. Besides, we take differential geometry in manifolds as a paradigm to discuss deep generative models and metric learning algorithms.
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