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Manifold learning and applications to shape and image processing

Abstract : The amount of data is continuously increasing through online databases such as Flicker1. Not only is the amount of stored data increasing constantly but also the data itself is highly complex. The need for smart algorithms is obvious. Recently, manifold learning has made a strong entry into the computer vision community. This method provides a powerful tool for the analysis of high-dimensional complex data. Manifold learning is based on the assumption that the degrees of freedom of the data are much smaller than the dimension of the data space itself. More specifically, these methods try to recover a submanifold embedded in a high-dimensional space which can even be dimensionally infinite as in the case of shapes. The output of such an algorithm is a mapping into a new space (commonly referred to as feature space) where the analysis of data becomes easier. Often this mapping can be thought of as a parameterization of the dataset. In the first part of this thesis, we introduce the concepts and theory of metric spaces providing the theoretical background to manifold learning. Once the necessary tools are introduced, we will present a survey on linear and non-linear manifold learning algorithms and compare their weak and strong points. In the second part, we will develop two applications using manifold learning techniques. In both applications manifold learning is applied to recover or approximate the metric on the original space data space. In this way distance between points in the original space can be computed using the metric in the feature space. This allows the formulation of distance based optimization problems. In this spirit, we tackle a first problem known under the name of Pre-image problem. We will look at this problem in the context of Kernel PCA and diffusion maps. It turns out, that previous Pre-image methods based on Kernel PCA used a wrong normalization in centred feature space. We propose a more subtle normalization improving previously proposed algorithm for the computation of the Pre-image. We then look at the same problem in the context of di
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Submitted on : Monday, March 29, 2010 - 8:00:00 AM
Last modification on : Monday, March 29, 2010 - 8:00:00 AM
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  • HAL Id : pastel-00005860, version 1



Nicolas Thorstensen. Manifold learning and applications to shape and image processing. Mathematics [math]. Ecole des Ponts ParisTech, 2009. English. ⟨pastel-00005860⟩



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