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- Institut fÃ¼r Mathematik und Informatik (3) (remove)

The goal of this doctoral thesis is to create and to implement methods for fully automatic segmentation applications in magnetic resonance images and datasets. The work introduces into technical and physical backgrounds of magnetic resonance imaging (MRI) and summarizes essential segmentation challenges in MRI data including technical malfunctions and ill-posedness of inverse segmentation problems. Theoretical background knowledge of all the used methods that are adapted and extended to combine them for problem-specific segmentation applications are explained in more detail. The first application for the implemented solutions in this work deals with two-dimensional tissue segmentation of atherosclerotic plaques in cardiological MRI data. The main part of segmentation solutions is designed for fully automatic liver and kidney parenchyma segmentation in three-dimensional MRI datasets to ensure computer-assisted organ volumetry in epidemiological studies. The results for every application are listed, described and discussed before important conclusions are drawn. Among several applied methods, the level set method is the main focus of this work and is used as central segmentation concept in the most applications. Thus, its possibilities and limitations for MRI data segmentation are analyzed. The level set method is extended by several new ideas to overcome possible limitations and it is combined as important part of modularized frameworks. Additionally, a new approach for probability map generation is presented in this thesis, which reduces data dimensionality of multiple MR-weightings and incorporates organ position probabilities in a probabilistic framework. It is shown, that essential organ features (i.e. MR-intensity distributions, locations) can be well represented in the calculated probability maps. Since MRI data are produced by using multiple MR- weightings, the used dimensionality reduction technique is very helpful to generate a single probability map, which can be used for further segmentation steps in a modularized framework.

The geometric arena here is a smooth manifold of dimension n equipped with a Riemannian or pseudo-Riemannian metric and an affine connection. Field theories following from a variational principle are considered on this basis. In this context, all invariants which are quadratic in the curvature are determined. The work derives several manifestly covariant formulas for the Euler-Lagrange derivatives or the field equations. Some of these field theories can be interpreted as gravitational theories alternatively to EinsteinÂ´s general relativity theory. The work also touches the difficult problem to define and to calculate energy and momentum of a gravitational field.

Independence is a basic concept of probability theory and statistics. In a lot of fields of sciences, dependency of different variables is gained lots of attention from scientists. A measure, named information dependency, is proposed to express the dependency of a group of random variables. This measure is defined as the Kullback-Leibler divergence of a joint distribution with respect to a product-marginal distribution of these random variables. In the bivariate case, this measure is known as mutual information of two random variables. Thus, the measure information dependency has a strong relationship with the Information Theory. The thesis aims to give a thorough study of the information dependency from both mathematical and practical viewpoints. Concretely, we would like to research three following problems: 1. Proving that the information dependency is a useful tool to express the dependency of a group of random variables by comparing it with other measures of dependency. 2. Studying the methods to estimate the information dependency based on the samples of a group of random variables. 3. Investigating how the Independent Component Analysis problem, an interesting problem in statistics, can be solved using information dependency.