@article{MoewisKaiserTrepczynskietal.2021, author = {Moewis, Philippe and Kaiser, Ren{\´e} and Trepczynski, Adam and von Tycowicz, Christoph and Krahl, Leonie and Ilg, Ansgar and Holz, Johannes and Duda, Georg}, title = {Patient specific resurfacing implant knee surgery in subjects with early osteoarthritis results in medial pivot and lateral femoral rollback during flexion: A retrospective pilot study}, journal = {Knee Surgery, Sports Traumatology, Arthroscopy}, doi = {10.1007/s00167-021-06749-8}, year = {2021}, abstract = {Purpose. Metallic resurfacing implants have been developed for the treatment of early, focal, small, condylar and trochlear osteoarthritis (OA) lesions. They represent an option for patients who are either too young to fulfill the criteria for total knee arthroplasty (TKA) or too old for biological treatment. Although relevant clinical evidence has been collected for different resurfacing types, the in vivo post-operative knee kinematics remains unknown. The present study aims to measure and analyse the knee joint kinematics in subjects with patient-specific Episealer implants Methods. Retrospective study design. Fluoroscopic analyses during high flexion activities (unloaded flexion-extension and loaded lunge) were conducted at >12 months post-surgery in ten Episealer knees. The post-operative knee joint kinematics was compared to equally assessed kinematic from ten healthy knees, twenty G-Curve TKA knees and 10 J-Curve knees. Pre- and postoperative clinical data of the Episealer knees were collected using a visual analog scale (VAS), the EQ 5d Health Questionnaire and the Knee Injury and Osteoarthritis Outcome Score (KOOS). Results. During unloaded flexion-extension and loaded lunge, the medial condyle in the Episealer knees remained relative stationary, indicating a medial pivot, while the lateral condyle translated consistently towards posterior. Similarly, reduced movement of the medial condyle and posterior translation of the lateral condyle was also observed in the healthy knees, although to a lesser extent. In contrast, the kinematics of both TKA cohorts during unloaded flexion-extension showed a tendency towards anterior displacement in the medial compartment, which led to significant differences in comparison with the Episealer knees. In the lateral compartment, a certain degree of femoral rollback was noted in the G-Curve TKA cohort. Improved scores were observed in the Episealer subjects between the preoperative and 1-year postoperative follow-up. Conclusion. At 12 months postsurgery, a physiological-like knee kinematics was observed in the group of patient-specific reconstructed chondral/osteochondral lesions by means of a resurfacing Episealer implant strategy. Considering that these patients are physically active and do not fulfill the criteria for TKA, the group is hard to be compared to TKA patients which usually are less active and more challenging. Nevertheless, the comparison to either healthy knee kinematics as well as to TKA reconstructed knees with different implant designs showed a more physiological-like kinematics in the resurfacing implants that seems more appropriate for such a patient group. Despite positive results, careful clinical follow-up of treated patients is recommended for the long-term OA progression. Further investigations need to be encouraged not only in larger patient groups but also in a prospective manner to assess the pre- to postoperative kinematic changes.}, 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} } @misc{AmbellanZachowvonTycowicz2021, author = {Ambellan, Felix and Zachow, Stefan and von Tycowicz, Christoph}, title = {Geodesic B-Score for Improved Assessment of Knee Osteoarthritis}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-81930}, year = {2021}, abstract = {Three-dimensional medical imaging enables detailed understanding of osteoarthritis structural status. However, there remains a vast need for automatic, thus, reader-independent measures that provide reliable assessment of subject-specific clinical outcomes. To this end, we derive a consistent generalization of the recently proposed B-score to Riemannian shape spaces. We further present an algorithmic treatment yielding simple, yet efficient computations allowing for analysis of large shape populations with several thousand samples. Our intrinsic formulation exhibits improved discrimination ability over its Euclidean counterpart, which we demonstrate for predictive validity on assessing risks of total knee replacement. This result highlights the potential of the geodesic B-score to enable improved personalized assessment and stratification for interventions.}, language = {en} } @misc{Paskin2022, type = {Master Thesis}, author = {Paskin, Martha}, title = {Estimating 3D Shape of the Head Skeleton of Basking Sharks Using Annotated Landmarks on a 2D Image}, year = {2022}, abstract = {Basking sharks are thought to be one of the most efficient filter-feeding fish in terms of the throughput of water filtered through their gills. Details about the underlying morphology of their branchial region have not been studied due to various challenges in acquiring real-world data. The present thesis aims to facilitate this, by developing a mathematical shape model which constructs the 3D structure of the head skeleton of a basking shark using annotated landmarks on a single 2D image. This is an ill-posed problem as estimating the depth of a 3D object from a single 2D view is, in general, not possible. To reduce this ambiguity, we create a set of pre-defined training shapes in 3D from CT scans of basking sharks. First, the damaged structures of the sharks in the scans are corrected via solving a set of optimization problems, before using them as accurate 3D representations of the object. Then, two approaches are employed for the 2D-to-3D shape fitting problem-an Active Shape Model approach and a Kendall's Shape Space approach. The former represents a shape as a point on a high-dimensional Euclidean space, whereas the latter represents a shape as an equivalence class of points in this Euclidean space. Kendall's shape space approach is a novel technique that has not yet been applied in this context, and a comprehensive comparison of the two approaches suggests this approach to be superior for the problem at hand. This can be credited to an improved interpolation of the training shapes.}, language = {en} } @phdthesis{Ambellan2022, author = {Ambellan, Felix}, title = {Efficient Riemannian Statistical Shape Analysis with Applications in Disease Assessment}, doi = {10.17169/refubium-36729}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:188-refubium-37016-3}, year = {2022}, abstract = {In this work, we address the challenge of developing statistical shape models that account for the non-Euclidean nature inherent to (anatomical) shape variation and at the same time offer fast, numerically robust processing and as much invariance as possible regarding translation and rotation, i.e. Euclidean motion. With the aim of doing that we formulate a continuous and physically motivated notion of shape space based on deformation gradients. We follow two different tracks endowing this differential representation with a Riemannian structure to establish a statistical shape model. (1) We derive a model based on differential coordinates as elements in GL(3)+. To this end, we adapt the notion of bi-invariant means employing an affine connection structure on GL(3)+. Furthermore, we perform second-order statistics based on a family of Riemannian metrics providing the most possible invariance, viz. GL(3)+-left-invariance and O(3)-right-invariance. (2) We endow the differential coordinates with a non-Euclidean structure, that stems from a product Lie group of stretches and rotations. This structure admits a bi-invariant metric and thus allows for a consistent analysis via manifold-valued Riemannian statistics. This work further presents a novel shape representation based on discrete fundamental forms that is naturally invariant under Euclidean motion, namely the fundamental coordinates. We endow this representation with a Lie group structure that admits bi-invariant metrics and therefore allows for consistent analysis using manifold-valued statistics based on the Riemannian framework. Furthermore, we derive a simple, efficient, robust, yet accurate (i.e. without resorting to model approximations) solver for the inverse problem that allows for interactive applications. Beyond statistical shape modeling the proposed framework is amenable for surface processing such as quasi-isometric flattening. Additionally, the last part of the thesis aims on shape-based, continuous disease stratification to provide means that objectify disease assessment over the current clinical practice of ordinal grading systems. Therefore, we derive the geodesic B-score, a generalization of the of the Euclidean B-score, in order to assess knee osteoarthritis. In this context we present a Newton-type fixed point iteration for projection onto geodesics in shape space. On the application side, we show that the derived geodesic B-score features, in comparison to its Euclidean counterpart, an improved predictive performance on assessing the risk of total knee replacement surgery.}, language = {en} } @inproceedings{MayerBaumAmbellanetal.2022, author = {Mayer, Julius and Baum, Daniel and Ambellan, Felix and von Tycowicz, Christoph}, title = {A Soft-Correspondence Approach to Shape-based Disease Grading with Graph Convolutional Networks}, volume = {194}, booktitle = {Proceedings of Machine Learning Research}, pages = {85 -- 95}, year = {2022}, abstract = {Shape analysis provides principled means for understanding anatomical structures from medical images. The underlying notions of shape spaces, however, come with strict assumptions prohibiting the analysis of incomplete and/or topologically varying shapes. This work aims to alleviate these limitations by adapting the concept of soft correspondences. In particular, we present a graph-based learning approach for morphometric classification of disease states that is based on a generalized notion of shape correspondences in terms of functional maps. We demonstrate the performance of the derived classifier on the open-access ADNI database for differentiating normal controls and subjects with Alzheimer's disease. Notably, our experiment shows that our approach can improve over state-of-the-art from geometric deep learning.}, language = {en} } @article{KoflerWaldKolbitschetal.2024, author = {Kofler, Andreas and Wald, Christian and Kolbitsch, Christoph and von Tycowicz, Christoph and Ambellan, Felix}, title = {Joint Reconstruction and Segmentation in Undersampled 3D Knee MRI combining Shape Knowledge and Deep Learning}, volume = {69}, journal = {Physics in Medicine and Biology}, number = {9}, doi = {10.1088/1361-6560/ad3797}, year = {2024}, abstract = {Task-adapted image reconstruction methods using end-to-end trainable neural networks (NNs) have been proposed to optimize reconstruction for subsequent processing tasks, such as segmentation. However, their training typically requires considerable hardware resources and thus, only relatively simple building blocks, e.g. U-Nets, are typically used, which, albeit powerful, do not integrate model-specific knowledge. In this work, we extend an end-to-end trainable task-adapted image reconstruction method for a clinically realistic reconstruction and segmentation problem of bone and cartilage in 3D knee MRI by incorporating statistical shape models (SSMs). The SSMs model the prior information and help to regularize the segmentation maps as a final post-processing step. We compare the proposed method to a state-of-the-art (SOTA) simultaneous multitask learning approach for image reconstruction and segmentation (MTL) and to a complex SSMs-informed segmentation pipeline (SIS). Our experiments show that the combination of joint end-to-end training and SSMs to further regularize the segmentation maps obtained by MTL highly improves the results, especially in terms of mean and maximal surface errors. In particular, we achieve the segmentation quality of SIS and, at the same time, a substantial model reduction that yields a five-fold decimation in model parameters and a computational speedup of an order of magnitude. Remarkably, even for undersampling factors of up to R=8, the obtained segmentation maps are of comparable quality to those obtained by SIS from ground-truth images.}, 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{NavaYazdaniHegevonTycowicz2022, author = {Nava-Yazdani, Esfandiar and Hege, Hans-Christian and von Tycowicz, Christoph}, title = {A Hierarchical Geodesic Model for Longitudinal Analysis on Manifolds}, volume = {64}, journal = {Journal of Mathematical Imaging and Vision}, number = {4}, doi = {10.1007/s10851-022-01079-x}, pages = {395 -- 407}, year = {2022}, abstract = {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.}, 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} } @inproceedings{PaskinDeanBaumetal.2022, author = {Paskin, Martha and Dean, Mason and Baum, Daniel and von Tycowicz, Christoph}, title = {A Kendall Shape Space Approach to 3D Shape Estimation from 2D Landmarks}, booktitle = {Computer Vision -- ECCV 2022}, publisher = {Springer Nature Switzerland}, arxiv = {http://arxiv.org/abs/2207.12687}, doi = {10.1007/978-3-031-20086-1_21}, pages = {363 -- 379}, year = {2022}, abstract = {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.}, language = {en} } @misc{PaskinBaumDeanetal.2022, author = {Paskin, Martha and Baum, Daniel and Dean, Mason N. and von Tycowicz, Christoph}, title = {A Kendall Shape Space Approach to 3D Shape Estimation from 2D Landmarks -- Source Code and Data}, doi = {10.12752/8730}, year = {2022}, abstract = {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.}, 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{SunkaraRostamivonTycowiczetal.2026, author = {Sunkara, Vikram and Rostami, Atefe and von Tycowicz, Christoph and Sch{\"u}tte, Christof}, title = {Stop throwing away your Decoder; extract the learnt local coordinate system using Latent-XAI}, booktitle = {The 4th World Conference on Explainable Artificial Intelligence (XAI-2026)}, year = {2026}, 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} }