@phdthesis{Tycowicz2014, author = {Tycowicz, Christoph von}, title = {Concepts and Algorithms for the Deformation, Analysis, and Compression of Digital Shapes}, doi = {10.17169/refubium-4697}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:188-fudissthesis000000096721-7}, year = {2014}, abstract = {This thesis concerns model reduction techniques for the efficient numerical treatment of physical systems governing the deformation behavior of geometrically complex shapes. We present new strategies for the construction of simplified, low-dimensional models that capture the main features of the original complex system and are suitable for use in interactive computer graphics applications. To demonstrate the effectiveness of the new techniques we propose frameworks for real-time simulation and interactive deformation-based modeling of elastic solids and shells and compare them to alternative approaches. In addition, we investigate differential operators that are derived from the physical models and hence can serve as alternatives to the Laplace-Beltrami operator for applications in modal shape analysis. Furthermore, this thesis addresses the compression of digital shapes. In particular, we present a lossless compression scheme that is adapted to the special characteristics of adaptively refined, hierarchical meshes.}, language = {en} } @phdthesis{Hanik2023, author = {Hanik, Martin}, title = {Geometric Data Analysis: Advancements of the Statistical Methodology and Applications}, publisher = {Refubium}, doi = {10.17169/refubium-39809}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:188-refubium-40087-8}, pages = {192}, year = {2023}, abstract = {Data analysis has become fundamental to our society and comes in multiple facets and approaches. Nevertheless, in research and applications, the focus was primarily on data from Euclidean vector spaces. Consequently, the majority of methods that are applied today are not suited for more general data types. Driven by needs from fields like image processing, (medical) shape analysis, and network analysis, more and more attention has recently been given to data from non-Euclidean spaces---particularly (curved) manifolds. It has led to the field of geometric data analysis whose methods explicitly take the structure (for example, the topology and geometry) of the underlying space into account. This thesis contributes to the methodology of geometric data analysis by generalizing several fundamental notions from multivariate statistics to manifolds. We thereby focus on two different viewpoints. First, we use Riemannian structures to derive a novel regression scheme for general manifolds that relies on splines of generalized B{\´e}zier curves. It can accurately model non-geodesic relationships, for example, time-dependent trends with saturation effects or cyclic trends. Since B{\´e}zier curves can be evaluated with the constructive de Casteljau algorithm, working with data from manifolds of high dimensions (for example, a hundred thousand or more) is feasible. Relying on the regression, we further develop a hierarchical statistical model for an adequate analysis of longitudinal data in manifolds, and a method to control for confounding variables. We secondly focus on data that is not only manifold- but even Lie group-valued, which is frequently the case in applications. We can only achieve this by endowing the group with an affine connection structure that is generally not Riemannian. Utilizing it, we derive generalizations of several well-known dissimilarity measures between data distributions that can be used for various tasks, including hypothesis testing. Invariance under data translations is proven, and a connection to continuous distributions is given for one measure. A further central contribution of this thesis is that it shows use cases for all notions in real-world applications, particularly in problems from shape analysis in medical imaging and archaeology. We can replicate or further quantify several known findings for shape changes of the femur and the right hippocampus under osteoarthritis and Alzheimer's, respectively. Furthermore, in an archaeological application, we obtain new insights into the construction principles of ancient sundials. Last but not least, we use the geometric structure underlying human brain connectomes to predict cognitive scores. Utilizing a sample selection procedure, we obtain state-of-the-art results.}, language = {en} }