TY - JOUR A1 - Nava-Yazdani, Esfandiar A1 - Hege, Hans-Christian A1 - von Tycowicz, Christoph T1 - A Hierarchical Geodesic Model for Longitudinal Analysis on Manifolds JF - Journal of Mathematical Imaging and Vision N2 - In many applications, geodesic hierarchical models are adequate for the study of temporal observations. We employ such a model derived for manifold-valued data to Kendall's shape space. In particular, instead of the Sasaki metric, we adapt a functional-based metric, which increases the computational efficiency and does not require the implementation of the curvature tensor. We propose the corresponding variational time discretization of geodesics and employ the approach for longitudinal analysis of 2D rat skulls shapes as well as 3D shapes derived from an imaging study on osteoarthritis. Particularly, we perform hypothesis test and estimate the mean trends. Y1 - 2022 U6 - https://doi.org/10.1007/s10851-022-01079-x VL - 64 IS - 4 SP - 395 EP - 407 ER - TY - JOUR A1 - Hanik, Martin A1 - Hege, Hans-Christian A1 - von Tycowicz, Christoph T1 - Bi-invariant Dissimilarity Measures for Sample Distributions in Lie Groups JF - SIAM Journal on Mathematics of Data Science N2 - Data sets sampled in Lie groups are widespread, and as with multivariate data, it is important for many applications to assess the differences between the sets in terms of their distributions. Indices for this task are usually derived by considering the Lie group as a Riemannian manifold. Then, however, compatibility with the group operation is guaranteed only if a bi-invariant metric exists, which is not the case for most non-compact and non-commutative groups. We show here that if one considers an affine connection structure instead, one obtains bi-invariant generalizations of well-known dissimilarity measures: a Hotelling $T^2$ statistic, Bhattacharyya distance and Hellinger distance. Each of the dissimilarity measures matches its multivariate counterpart for Euclidean data and is translation-invariant, so that biases, e.g., through an arbitrary choice of reference, are avoided. We further derive non-parametric two-sample tests that are bi-invariant and consistent. We demonstrate the potential of these dissimilarity measures by performing group tests on data of knee configurations and epidemiological shape data. Significant differences are revealed in both cases. Y1 - 2022 U6 - https://doi.org/10.1137/21M1410373 VL - 4 IS - 4 SP - 1223 EP - 1249 ER - TY - CHAP A1 - Paskin, Martha A1 - Dean, Mason A1 - Baum, Daniel A1 - von Tycowicz, Christoph T1 - A Kendall Shape Space Approach to 3D Shape Estimation from 2D Landmarks T2 - Computer Vision -- ECCV 2022 N2 - 3D shapes provide substantially more information than 2D images. However, the acquisition of 3D shapes is sometimes very difficult or even impossible in comparison with acquiring 2D images, making it necessary to derive the 3D shape from 2D images. Although this is, in general, a mathematically ill-posed problem, it might be solved by constraining the problem formulation using prior information. Here, we present a new approach based on Kendall’s shape space to reconstruct 3D shapes from single monocular 2D images. The work is motivated by an application to study the feeding behavior of the basking shark, an endangered species whose massive size and mobility render 3D shape data nearly impossible to obtain, hampering understanding of their feeding behaviors and ecology. 2D images of these animals in feeding position, however, are readily available. We compare our approach with state-of-the-art shape-based approaches both on human stick models and on shark head skeletons. Using a small set of training shapes, we show that the Kendall shape space approach is substantially more robust than previous methods and always results in plausible shapes. This is essential for the motivating application in which specimens are rare and therefore only few training shapes are available. Y1 - 2022 U6 - https://doi.org/10.1007/978-3-031-20086-1_21 SP - 363 EP - 379 PB - Springer Nature Switzerland ER - TY - GEN A1 - Paskin, Martha A1 - Baum, Daniel A1 - Dean, Mason N. A1 - von Tycowicz, Christoph T1 - A Kendall Shape Space Approach to 3D Shape Estimation from 2D Landmarks -- Source Code and Data N2 - Source code and novel dataset of basking shark head skeletons facilitating the reproduction of the results presented in 'A Kendall Shape Space Approach to 3D Shape Estimation from 2D Landmarks' - ECCV 2022. Y1 - 2022 U6 - https://doi.org/10.12752/8730 ER - TY - GEN A1 - Nava-Yazdani, Esfandiar A1 - Hanik, Martin A1 - Ambellan, Felix A1 - von Tycowicz, Christoph T1 - On Gradient Formulas in an Algorithm for the Logarithm of the Sasaki Metric N2 - The Sasaki metric is the canonical metric on the tangent bundle TM of a Riemannian manifold M. It is highly useful for data analysis in TM (e.g., when one is interested in the statistics of a set of geodesics in M). To this end, computing the Riemannian logarithm is often necessary, and an iterative algorithm was proposed by Muralidharan and Fletcher. In this note, we derive approximation formulas of the energy gradients in their algorithm that we use with success. T3 - ZIB-Report - 22-12 Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-87174 SN - 1438-0064 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 - TY - CHAP A1 - Sunkara, Vikram A1 - Rostami, Atefe A1 - von Tycowicz, Christoph A1 - Schütte, Christof T1 - Stop throwing away your Decoder; extract the learnt local coordinate system using Latent-XAI T2 - The 4th World Conference on Explainable Artificial Intelligence (XAI-2026) Y1 - 2026 ER - TY - CHAP A1 - Stokke, Jo Andersson A1 - Bergmann, Ronny A1 - Hanik, Martin A1 - von Tycowicz, Christoph T1 - p-Laplacians for Manifold-valued Hypergraphs T2 - Geometric Science of Information. GSI 2025 N2 - Hypergraphs extend traditional graphs by enabling the representation of N-ary relationships through higher-order edges. Akin to a common approach of deriving graph Laplacians, we define function spaces and corresponding symmetric products on the nodes and edges to derive hypergraph Laplacians. While this has been done before for Euclidean features, this work generalizes previous hypergraph Laplacian approaches to accommodate manifold-valued hypergraphs for many commonly encountered manifolds. Y1 - 2025 U6 - https://doi.org/10.1007/978-3-032-03924-8_17 VL - 16035 ER -