TY - CHAP A1 - Hanik, Martin A1 - Hege, Hans-Christian A1 - von Tycowicz, Christoph T1 - Bi-invariant Two-Sample Tests in Lie Groups for Shape Analysis T2 - Shape in Medical Imaging N2 - We propose generalizations of the T²-statistics of Hotelling and the Bhattacharayya distance for data taking values in Lie groups. A key feature of the derived measures is that they are compatible with the group structure even for manifolds that do not admit any bi-invariant metric. This property, e.g., assures analysis that does not depend on the reference shape, thus, preventing bias due to arbitrary choices thereof. Furthermore, the generalizations agree with the common definitions for the special case of flat vector spaces guaranteeing consistency. Employing a permutation test setup, we further obtain nonparametric, two-sample testing procedures that themselves are bi-invariant and consistent. We validate our method in group tests revealing significant differences in hippocampal shape between individuals with mild cognitive impairment and normal controls. Y1 - 2020 U6 - https://doi.org/10.1007/978-3-030-61056-2_4 SP - 44 EP - 54 PB - Springer International Publishing CY - Cham ER - TY - CHAP A1 - Hanik, Martin A1 - Hege, Hans-Christian A1 - Hennemuth, Anja A1 - von Tycowicz, Christoph T1 - Nonlinear Regression on Manifolds for Shape Analysis using Intrinsic Bézier Splines T2 - Proc. Medical Image Computing and Computer Assisted Intervention (MICCAI) N2 - Intrinsic and parametric regression models are of high interest for the statistical analysis of manifold-valued data such as images and shapes. The standard linear ansatz has been generalized to geodesic regression on manifolds making it possible to analyze dependencies of random variables that spread along generalized straight lines. Nevertheless, in some scenarios, the evolution of the data cannot be modeled adequately by a geodesic. We present a framework for nonlinear regression on manifolds by considering Riemannian splines, whose segments are Bézier curves, as trajectories. Unlike variational formulations that require time-discretization, we take a constructive approach that provides efficient and exact evaluation by virtue of the generalized de Casteljau algorithm. We validate our method in experiments on the reconstruction of periodic motion of the mitral valve as well as the analysis of femoral shape changes during the course of osteoarthritis, endorsing Bézier spline regression as an effective and flexible tool for manifold-valued regression. Y1 - 2020 U6 - https://doi.org/10.1007/978-3-030-59719-1_60 SP - 617 EP - 626 PB - Springer International Publishing CY - Cham ER - TY - CHAP A1 - von Tycowicz, Christoph T1 - Towards Shape-based Knee Osteoarthritis Classification using Graph Convolutional Networks T2 - 2020 IEEE 17th International Symposium on Biomedical Imaging (ISBI 2020) N2 - We present a transductive learning approach for morphometric osteophyte grading based on geometric deep learning. We formulate the grading task as semi-supervised node classification problem on a graph embedded in shape space. To account for the high-dimensionality and non-Euclidean structure of shape space we employ a combination of an intrinsic dimension reduction together with a graph convolutional neural network. We demonstrate the performance of our derived classifier in comparisons to an alternative extrinsic approach. Y1 - 2020 U6 - https://doi.org/10.1109/ISBI45749.2020.9098687 ER -