@article{HanikNavayazdanivonTycowicz2024, author = {Hanik, Martin and Navayazdani, Esfandiar and von Tycowicz, Christoph}, title = {De Casteljau's Algorithm in Geometric Data Analysis: Theory and Application}, volume = {110}, journal = {Computer Aided Geometric Design}, arxiv = {http://arxiv.org/abs/2402.07550}, doi = {10.1016/j.cagd.2024.102288}, pages = {102288}, year = {2024}, abstract = {For decades, de Casteljau's algorithm has been used as a fundamental building block in curve and surface design and has found a wide range of applications in fields such as scientific computing, and discrete geometry to name but a few. With increasing interest in nonlinear data science, its constructive approach has been shown to provide a principled way to generalize parametric smooth curves to manifolds. These curves have found remarkable new applications in the analysis of parameter-dependent, geometric data. This article provides a survey of the recent theoretical developments in this exciting area as well as its applications in fields such as geometric morphometrics and longitudinal data analysis in medicine, archaeology, and meteorology.}, language = {en} } @article{HanikSteidlvonTycowicz2024, author = {Hanik, Martin and Steidl, Gabriele and von Tycowicz, Christoph}, title = {Manifold GCN: Diffusion-based Convolutional Neural Network for Manifold-valued Graphs}, arxiv = {http://arxiv.org/abs/2401.14381}, year = {2024}, abstract = {We propose two graph neural network layers for graphs with features in a Riemannian manifold. First, based on a manifold-valued graph diffusion equation, we construct a diffusion layer that can be applied to an arbitrary number of nodes and graph connectivity patterns. Second, we model a tangent multilayer perceptron by transferring ideas from the vector neuron framework to our general setting. Both layers are equivariant with respect to node permutations and isometries of the feature manifold. These properties have been shown to lead to a beneficial inductive bias in many deep learning tasks. Numerical examples on synthetic data as well as on triangle meshes of the right hippocampus to classify Alzheimer's disease demonstrate the very good performance of our layers.}, language = {en} } @article{CaputoEmporioGiachettietal.2022, author = {Caputo, Ariel and Emporio, Marco and Giachetti, Andrea and Cristani, Marco and Borghi, Guido and D'Eusanio, Andrea and Le, Minh-Quan and Nguyen, Hai-Dang and Tran, Minh-Triet and Ambellan, Felix and Hanik, Martin and Navayazdani, Esfandiar and Tycowicz, Christoph von}, title = {SHREC 2022 Track on Online Detection of Heterogeneous Gestures}, volume = {107}, journal = {Computers and Graphics}, arxiv = {http://arxiv.org/abs/2207.06706}, doi = {10.1016/j.cag.2022.07.015}, pages = {241 -- 251}, year = {2022}, abstract = {This paper presents the outcomes of a contest organized to evaluate methods for the online recognition of heterogeneous gestures from sequences of 3D hand poses. The task is the detection of gestures belonging to a dictionary of 16 classes characterized by different pose and motion features. The dataset features continuous sequences of hand tracking data where the gestures are interleaved with non-significant motions. The data have been captured using the Hololens 2 finger tracking system in a realistic use-case of mixed reality interaction. The evaluation is based not only on the detection performances but also on the latency and the false positives, making it possible to understand the feasibility of practical interaction tools based on the algorithms proposed. The outcomes of the contest's evaluation demonstrate the necessity of further research to reduce recognition errors, while the computational cost of the algorithms proposed is sufficiently low.}, language = {en} } @article{HanikHegevonTycowicz2022, author = {Hanik, Martin and Hege, Hans-Christian and von Tycowicz, Christoph}, title = {Bi-invariant Dissimilarity Measures for Sample Distributions in Lie Groups}, volume = {4}, journal = {SIAM Journal on Mathematics of Data Science}, number = {4}, arxiv = {http://arxiv.org/abs/2402.12901}, doi = {10.1137/21M1410373}, pages = {1223 -- 1249}, year = {2022}, abstract = {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.}, language = {en} } @misc{NavaYazdaniHanikAmbellanetal.2022, author = {Nava-Yazdani, Esfandiar and Hanik, Martin and Ambellan, Felix and von Tycowicz, Christoph}, title = {On Gradient Formulas in an Algorithm for the Logarithm of the Sasaki Metric}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-87174}, year = {2022}, abstract = {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.}, language = {en} } @inproceedings{MyersUtpalaTalbaretal.2022, author = {Myers, Adele and Utpala, Saiteja and Talbar, Shubham and Sanborn, Sophia and Shewmake, Christian and Donnat, Claire and Mathe, Johan and Lupo, Umberto and Sonthalia, Rishi and Cui, Xinyue and Szwagier, Tom and Pignet, Arthur and Bergsson, Andri and Hauberg, S{\o}ren and Nielsen, Dmitriy and Sommer, Stefan and Klindt, David and Hermansen, Erik and Vaupel, Melvin and Dunn, Benjamin and Xiong, Jeffrey and Aharony, Noga and Pe'er, Itsik and Ambellan, Felix and Hanik, Martin and Navayazdani, Esfandiar and Tycowicz, Christoph von and Miolane, Nina}, title = {ICLR 2022 Challenge for Computational Geomerty \& Topology: Design and Results}, volume = {196}, booktitle = {Proceedings of Topology, Algebra, and Geometry in Learning}, publisher = {PMLR}, arxiv = {http://arxiv.org/abs/2206.09048}, pages = {269 -- 276}, year = {2022}, language = {en} } @inproceedings{SchadevonTycowiczHanik2025, author = {Schade, Johannes and von Tycowicz, Christoph and Hanik, Martin}, title = {Bi-invariant Geodesic Regression with Data from the Osteoarthritis Initiative}, booktitle = {Information Processing in Medical Imaging}, publisher = {Springer}, address = {Lecture Notes in Computer Science}, arxiv = {http://arxiv.org/abs/2502.11826}, doi = {10.1007/978-3-031-96628-6_4}, pages = {49 -- 63}, year = {2025}, abstract = {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.}, language = {en} } @inproceedings{StokkeBergmannHaniketal.2025, author = {Stokke, Jo Andersson and Bergmann, Ronny and Hanik, Martin and von Tycowicz, Christoph}, title = {p-Laplacians for Manifold-valued Hypergraphs}, volume = {16035}, booktitle = {Geometric Science of Information. GSI 2025}, arxiv = {http://arxiv.org/abs/2507.10335}, doi = {10.1007/978-3-032-03924-8_17}, year = {2025}, abstract = {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.}, language = {en} }