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 - 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 - 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 - Ambellan, Felix A1 - Zachow, Stefan A1 - von Tycowicz, Christoph T1 - An as-invariant-as-possible GL+(3)-based Statistical Shape Model T2 - Proc. 7th MICCAI workshop on Mathematical Foundations of Computational Anatomy (MFCA) N2 - We describe a novel nonlinear statistical shape model basedon differential coordinates viewed as elements of GL+(3). We adopt an as-invariant-as possible framework comprising a bi-invariant Lie group mean and a tangent principal component analysis based on a unique GL+(3)-left-invariant, O(3)-right-invariant metric. Contrary to earlier work that equips the coordinates with a specifically constructed group structure, our method employs the inherent geometric structure of the group-valued data and therefore features an improved statistical power in identifying shape differences. We demonstrate this in experiments on two anatomical datasets including comparison to the standard Euclidean as well as recent state-of-the-art nonlinear approaches to statistical shape modeling. Y1 - 2019 U6 - https://doi.org/10.1007/978-3-030-33226-6_23 VL - 11846 SP - 219 EP - 228 PB - Springer ER - TY - CHAP A1 - Ambellan, Felix A1 - Zachow, Stefan A1 - von Tycowicz, Christoph T1 - A Surface-Theoretic Approach for Statistical Shape Modeling T2 - Proc. Medical Image Computing and Computer Assisted Intervention (MICCAI), Part IV N2 - We present a novel approach for nonlinear statistical shape modeling that is invariant under Euclidean motion and thus alignment-free. By analyzing metric distortion and curvature of shapes as elements of Lie groups in a consistent Riemannian setting, we construct a framework that reliably handles large deformations. Due to the explicit character of Lie group operations, our non-Euclidean method is very efficient allowing for fast and numerically robust processing. This facilitates Riemannian analysis of large shape populations accessible through longitudinal and multi-site imaging studies providing increased statistical power. We evaluate the performance of our model w.r.t. shape-based classification of pathological malformations of the human knee and show that it outperforms the standard Euclidean as well as a recent nonlinear approach especially in presence of sparse training data. To provide insight into the model’s ability of capturing natural biological shape variability, we carry out an analysis of specificity and generalization ability. Y1 - 2019 U6 - https://doi.org/10.1007/978-3-030-32251-9_3 VL - 11767 SP - 21 EP - 29 PB - Springer ER - TY - CHAP A1 - Hanik, Martin A1 - Hege, Hans-Christian A1 - von Tycowicz, Christoph T1 - A Nonlinear Hierarchical Model for Longitudinal Data on Manifolds T2 - 2022 IEEE 19th International Symposium on Biomedical Imaging (ISBI) N2 - Large longitudinal studies provide lots of valuable information, especially in medical applications. A problem which must be taken care of in order to utilize their full potential is that of correlation between intra-subject measurements taken at different times. For data in Euclidean space this can be done with hierarchical models, that is, models that consider intra-subject and between-subject variability in two different stages. Nevertheless, data from medical studies often takes values in nonlinear manifolds. Here, as a first step, geodesic hierarchical models have been developed that generalize the linear ansatz by assuming that time-induced intra-subject variations occur along a generalized straight line in the manifold. However, this is often not the case (e.g., periodic motion or processes with saturation). We propose a hierarchical model for manifold-valued data that extends this to include trends along higher-order curves, namely Bézier splines in the manifold. To this end, we present a principled way of comparing shape trends in terms of a functional-based Riemannian metric. Remarkably, this metric allows efficient, yet simple computations by virtue of a variational time discretization requiring only the solution of regression problems. We validate our model on longitudinal data from the osteoarthritis initiative, including classification of disease progression. Y1 - 2022 U6 - https://doi.org/10.1109/ISBI52829.2022.9761465 SP - 1 EP - 5 ER - TY - CHAP A1 - Myers, Adele A1 - Utpala, Saiteja A1 - Talbar, Shubham A1 - Sanborn, Sophia A1 - Shewmake, Christian A1 - Donnat, Claire A1 - Mathe, Johan A1 - Lupo, Umberto A1 - Sonthalia, Rishi A1 - Cui, Xinyue A1 - Szwagier, Tom A1 - Pignet, Arthur A1 - Bergsson, Andri A1 - Hauberg, Søren A1 - Nielsen, Dmitriy A1 - Sommer, Stefan A1 - Klindt, David A1 - Hermansen, Erik A1 - Vaupel, Melvin A1 - Dunn, Benjamin A1 - Xiong, Jeffrey A1 - Aharony, Noga A1 - Pe’er, Itsik A1 - Ambellan, Felix A1 - Hanik, Martin A1 - Navayazdani, Esfandiar A1 - Tycowicz, Christoph von A1 - Miolane, Nina T1 - ICLR 2022 Challenge for Computational Geomerty & Topology: Design and Results T2 - Proceedings of Topology, Algebra, and Geometry in Learning Y1 - 2022 VL - 196 SP - 269 EP - 276 PB - PMLR ER - TY - CHAP A1 - Schade, Johannes A1 - von Tycowicz, Christoph A1 - Hanik, Martin T1 - Bi-invariant Geodesic Regression with Data from the Osteoarthritis Initiative T2 - Information Processing in Medical Imaging N2 - Many phenomena are naturally characterized by measuring continuous transformations such as shape changes in medicine or articulated systems in robotics. Modeling the variability in such datasets requires performing statistics on Lie groups, that is, manifolds carrying an additional group structure. As the Lie group captures the symmetries in the data, it is essential from a theoretical and practical perspective to ask for statistical methods that respect these symmetries; this way they are insensitive to confounding effects, e.g., due to the choice of reference coordinate systems. In this work, we investigate geodesic regression---a generalization of linear regression originally derived for Riemannian manifolds. While Lie groups can be endowed with Riemannian metrics, these are generally incompatible with the group structure. We develop a non-metric estimator using an affine connection setting. It captures geodesic relationships respecting the symmetries given by left and right translations. For its computation, we propose an efficient fixed point algorithm requiring simple differential expressions that can be calculated through automatic differentiation. We perform experiments on a synthetic example and evaluate our method on an open-access, clinical dataset studying knee joint configurations under the progression of osteoarthritis. Y1 - 2025 U6 - https://doi.org/10.1007/978-3-031-96628-6_4 SP - 49 EP - 63 PB - Springer CY - Lecture Notes in Computer Science ER -