Statistical Analysis and Optimal Transport for Euclidean and Manifold-Valued Data

Beschreibung

This PhD thesis provides contributions to different mathematical areas, including robust statistical estimation methods, manifold-valued image and data processing, and optimal transport problems. The first part of the thesis deals with the maximum likelihood estimation of the parameters of (multivariate) Cauchy and Student-t distributions. We present existence and uniqueness results as well as efficient algorithms for their computation. We apply these algorithms to define generalized myriad filters that we use in a nonlocal, patch-based method to denoise images corrupted by different kind of noise, including additive Cauchy and Gaussian noise. The second part of the thesis addresses the denoising of manifold-valued data. In order to formulate an analogue of the classical Gaussian white noise model on manifolds, we first propose different approaches for the definition of a normal distribution on manifolds. With the help of one of the approaches we formulate in a second step a stochastic noise model for manifold-valued data and propose an estimator imitating the construction of the Euclidean Minimum Mean Squared Error estimator. We use this estimator to derive a nonlocal, patch-based algorithm for denoising manifold-valued data. Finally, in the third part of the thesis we consider different optimal transport problems for color images. In a first approach, we interpret an RGB color image as discretized, three-dimensional density function of a probability measure, where the third dimension contains the color information. In a second model, which is not limited to color images, we exploit the framework of so-called lifted optimal transport to define measures based on color images living in different color spaces. For all presented topics and questions we develop mathematical models and provide a detailed analysis of their properties. Various numerical examples and experiments illustrating different aspects of the proposed models complete the PhD thesis.

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Friederike Johanna Laus

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statistical estimation methods manifold-valued data processing optimal transport

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