@inproceedings{HanikHegevonTycowicz2020, author = {Hanik, Martin and Hege, Hans-Christian and von Tycowicz, Christoph}, title = {Bi-invariant Two-Sample Tests in Lie Groups for Shape Analysis}, booktitle = {Shape in Medical Imaging}, publisher = {Springer International Publishing}, address = {Cham}, arxiv = {http://arxiv.org/abs/2008.12195}, doi = {10.1007/978-3-030-61056-2_4}, pages = {44 -- 54}, year = {2020}, abstract = {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.}, language = {en} } @article{HanikDemirtaşGharsallaouietal.2022, author = {Hanik, Martin and Demirta{\c{s}}, Mehmet Arif and Gharsallaoui, Mohammed Amine and Rekik, Islem}, title = {Predicting cognitive scores with graph neural networks through sample selection learning}, volume = {16}, journal = {Brain Imaging and Behavior}, arxiv = {http://arxiv.org/abs/2106.09408}, doi = {10.1007/s11682-021-00585-7}, pages = {1123 -- 1138}, year = {2022}, abstract = {Analyzing the relation between intelligence and neural activity is of the utmost importance in understanding the working principles of the human brain in health and disease. In existing literature, functional brain connectomes have been used successfully to predict cognitive measures such as intelligence quotient (IQ) scores in both healthy and disordered cohorts using machine learning models. However, existing methods resort to flattening the brain connectome (i.e., graph) through vectorization which overlooks its topological properties. To address this limitation and inspired from the emerging graph neural networks (GNNs), we design a novel regression GNN model (namely RegGNN) for predicting IQ scores from brain connectivity. On top of that, we introduce a novel, fully modular sample selection method to select the best samples to learn from for our target prediction task. However, since such deep learning architectures are computationally expensive to train, we further propose a \emph{learning-based sample selection} method that learns how to choose the training samples with the highest expected predictive power on unseen samples. For this, we capitalize on the fact that connectomes (i.e., their adjacency matrices) lie in the symmetric positive definite (SPD) matrix cone. Our results on full-scale and verbal IQ prediction outperforms comparison methods in autism spectrum disorder cohorts and achieves a competitive performance for neurotypical subjects using 3-fold cross-validation. Furthermore, we show that our sample selection approach generalizes to other learning-based methods, which shows its usefulness beyond our GNN architecture.}, language = {en} } @inproceedings{HanikHegevonTycowicz2022, author = {Hanik, Martin and Hege, Hans-Christian and von Tycowicz, Christoph}, title = {A Nonlinear Hierarchical Model for Longitudinal Data on Manifolds}, booktitle = {2022 IEEE 19th International Symposium on Biomedical Imaging (ISBI)}, arxiv = {http://arxiv.org/abs/2202.01180}, doi = {10.1109/ISBI52829.2022.9761465}, pages = {1 -- 5}, year = {2022}, abstract = {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 B{\´e}zier 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.}, language = {en} } @misc{AmbellanHanikvonTycowicz2021, author = {Ambellan, Felix and Hanik, Martin and von Tycowicz, Christoph}, title = {Morphomatics: Geometric morphometrics in non-Euclidean shape spaces}, doi = {10.12752/8544}, year = {2021}, abstract = {Morphomatics is an open-source Python library for (statistical) shape analysis developed within the geometric data analysis and processing research group at Zuse Institute Berlin. It contains prototype implementations of intrinsic manifold-based methods that are highly consistent and avoid the influence of unwanted effects such as bias due to arbitrary choices of coordinates.}, language = {en} } @article{VeldhuijzenVeltkampIkneetal.2024, author = {Veldhuijzen, Ben and Veltkamp, Remco C. and Ikne, Omar and Allaert, Benjamin and Wannous, Hazem and Emporio, Marco and Giachetti, Andrea and LaViola Jr, Joseph J. and He, Ruiwen and Benhabiles, Halim and Cabani, Adnane and Fleury, Anthony and Hammoudi, Karim and Gavalas, Konstantinos and Vlachos, Christoforos and Papanikolaou, Athanasios and Romanelis, Ioannis and Fotis, Vlassis and Arvanitis, Gerasimos and Moustakas, Konstantinos and Hanik, Martin and Nava-Yazdani, Esfandiar and von Tycowicz, Christoph}, title = {SHREC 2024: Recognition Of Dynamic Hand Motions Molding Clay}, volume = {123}, journal = {Computers \& Graphics}, doi = {10.1016/j.cag.2024.104012}, pages = {104012}, year = {2024}, abstract = {Gesture recognition is a tool to enable novel interactions with different techniques and applications, like Mixed Reality and Virtual Reality environments. With all the recent advancements in gesture recognition from skeletal data, it is still unclear how well state-of- the-art techniques perform in a scenario using precise motions with two hands. This paper presents the results of the SHREC 2024 contest organized to evaluate methods for their recognition of highly similar hand motions using the skeletal spatial coordinate data of both hands. The task is the recognition of 7 motion classes given their spatial coordinates in a frame-by-frame motion. The skeletal data has been captured using a Vicon system and pre-processed into a coordinate system using Blender and Vicon Shogun Post. We created a small, novel dataset with a high variety of durations in frames. This paper shows the results of the contest, showing the techniques created by the 5 research groups on this challenging task and comparing them to our baseline method.}, 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} } @inproceedings{TuerksevenRekikvonTycowiczetal.2023, author = {T{\"u}rkseven, Doğa and Rekik, Islem and von Tycowicz, Christoph and Hanik, Martin}, title = {Predicting Shape Development: A Riemannian Method}, booktitle = {Shape in Medical Imaging}, publisher = {Springer Nature}, arxiv = {http://arxiv.org/abs/2212.04740}, doi = {10.1007/978-3-031-46914-5_17}, pages = {211 -- 222}, year = {2023}, abstract = {Predicting the future development of an anatomical shape from a single baseline observation is a challenging task. But it can be essential for clinical decision-making. Research has shown that it should be tackled in curved shape spaces, as (e.g., disease-related) shape changes frequently expose nonlinear characteristics. We thus propose a novel prediction method that encodes the whole shape in a Riemannian shape space. It then learns a simple prediction technique founded on hierarchical statistical modeling of longitudinal training data. When applied to predict the future development of the shape of the right hippocampus under Alzheimer's disease and to human body motion, it outperforms deep learning-supported variants as well as state-of-the-art.}, language = {en} } @article{HanikDuckeHegeetal.2023, author = {Hanik, Martin and Ducke, Benjamin and Hege, Hans-Christian and Fless, Friederike and von Tycowicz, Christoph}, title = {Intrinsic shape analysis in archaeology: A case study on ancient sundials}, volume = {16}, journal = {Journal on Computing and Cultural Heritage}, number = {4}, arxiv = {http://arxiv.org/abs/2305.18960}, doi = {10.1145/3606698}, pages = {1 -- 26}, year = {2023}, abstract = {The fact that the physical shapes of man-made objects are subject to overlapping influences—such as technological, economic, geographic, and stylistic progressions—holds great information potential. On the other hand, it is also a major analytical challenge to uncover these overlapping trends and to disentagle them in an unbiased way. This paper explores a novel mathematical approach to extract archaeological insights from ensembles of similar artifact shapes. We show that by considering all shape information in a find collection, it is possible to identify shape patterns that would be difficult to discern by considering the artifacts individually or by classifying shapes into predefined archaeological types and analyzing the associated distinguishing characteristics. Recently, series of high-resolution digital representations of artifacts have become available. Such data sets enable the application of extremely sensitive and flexible methods of shape analysis. We explore this potential on a set of 3D models of ancient Greek and Roman sundials, with the aim of providing alternatives to the traditional archaeological method of "trend extraction by ordination" (typology). In the proposed approach, each 3D shape is represented as a point in a shape space—a high-dimensional, curved, non-Euclidean space. Proper consideration of its mathematical properties reduces bias in data analysis and thus improves analytical power. By performing regression in shape space, we find that for Roman sundials, the bend of the shadow-receiving surface of the sundials changes with the latitude of the location. This suggests that, apart from the inscribed hour lines, also a sundial's shape was adjusted to the place of installation. As an example of more advanced inference, we use the identified trend to infer the latitude at which a sundial, whose location of installation is unknown, was placed. We also derive a novel method for differentiated morphological trend assertion, building upon and extending the theory of geometric statistics and shape analysis. Specifically, we present a regression-based method for statistical normalization of shapes that serves as a means of disentangling parameter-dependent effects (trends) and unexplained variability. In addition, we show that this approach is robust to noise in the digital reconstructions of the artifact shapes.}, language = {en} } @article{SipiranLazoLopezetal.2021, author = {Sipiran, Ivan and Lazo, Patrick and Lopez, Cristian and Bagewadi, Nihar and Bustos, Benjamin and Dao, Hieu and Gangisetty, Shankar and Hanik, Martin and Ho-Thi, Ngoc-Phuong and Holenderski, Mike and Jarnikov, Dmitri and Labrada, Arniel and Lengauer, Stefan and Licandro, Roxane and Nguyen, Dinh-Huan and Nguyen-Ho, Thang-Long and P{\´e}rez Rey, Luis A. and Pham, Bang-Dang and Pham, Minh-Khoi and Preiner, Reinhold and Schreck, Tobias and Trinh, Quoc-Huy and Tonnaer, Loek and von Tycowicz, Christoph and Vu-Le, The-Anh}, title = {SHREC 2021: Retrieval of Cultural Heritage Objects}, volume = {100}, journal = {Computers and Graphics}, doi = {10.1016/j.cag.2021.07.010}, pages = {1 -- 20}, year = {2021}, abstract = {This paper presents the methods and results of the SHREC'21 contest on a dataset of cultural heritage (CH) objects. We present a dataset of 938 scanned models that have varied geometry and artistic styles. For the competition, we propose two challenges: the retrieval-by-shape challenge and the retrieval-by-culture challenge. The former aims at evaluating the ability of retrieval methods to discriminate cultural heritage objects by overall shape. The latter focuses on assessing the effectiveness of retrieving objects from the same culture. Both challenges constitute a suitable scenario to evaluate modern shape retrieval methods in a CH domain. Ten groups participated in the contest: thirty runs were submitted for the retrieval-by-shape task, and twenty-six runs were submitted for the retrieval-by-culture challenge. The results show a predominance of learning methods on image-based multi-view representations to characterize 3D objects. Nevertheless, the problem presented in our challenges is far from being solved. We also identify the potential paths for further improvements and give insights into the future directions of research.}, language = {en} } @misc{HanikvonTycowicz2022, author = {Hanik, Martin and von Tycowicz, Christoph}, title = {Triangle meshes of shadow-recieving surfaces of ancient sundials}, doi = {10.12752/8425}, year = {2022}, abstract = {This repository contains triangle meshes of the shadow-recieving surfaces of 13 ancient sundials; three of them are from Greece and 10 from Italy. The meshes are in correspondence.}, language = {en} } @article{NavayazdaniAmbellanHaniketal.2023, author = {Navayazdani, Esfandiar and Ambellan, Felix and Hanik, Martin and von Tycowicz, Christoph}, title = {Sasaki Metric for Spline Models of Manifold-Valued Trajectories}, volume = {104}, journal = {Computer Aided Geometric Design}, arxiv = {http://arxiv.org/abs/arXiv:2303.17299}, doi = {10.1016/j.cagd.2023.102220}, pages = {102220}, year = {2023}, abstract = {We propose a generic spatiotemporal framework to analyze manifold-valued measurements, which allows for employing an intrinsic and computationally efficient Riemannian hierarchical model. Particularly, utilizing regression, we represent discrete trajectories in a Riemannian manifold by composite B{\´e}zier splines, propose a natural metric induced by the Sasaki metric to compare the trajectories, and estimate average trajectories as group-wise trends. We evaluate our framework in comparison to state-of-the-art methods within qualitative and quantitative experiments on hurricane tracks. Notably, our results demonstrate the superiority of spline-based approaches for an intensity classification of the tracks.}, language = {en} } @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} }