Geometric Data Analysis Based on Manifold Learning with Applications for Image Understanding

Nowadays, 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.