TY - GEN A1 - Götschel, Sebastian A1 - von Tycowicz, Christoph A1 - Polthier, Konrad A1 - Weiser, Martin T1 - Reducing Memory Requirements in Scientific Computing and Optimal Control N2 - In high accuracy numerical simulations and optimal control of time-dependent processes, often both many time steps and fine spatial discretizations are needed. Adjoint gradient computation, or post-processing of simulation results, requires the storage of the solution trajectories over the whole time, if necessary together with the adaptively refined spatial grids. In this paper we discuss various techniques to reduce the memory requirements, focusing first on the storage of the solution data, which typically are double precision floating point values. We highlight advantages and disadvantages of the different approaches. Moreover, we present an algorithm for the efficient storage of adaptively refined, hierarchic grids, and the integration with the compressed storage of solution data. T3 - ZIB-Report - 13-64 KW - optimal control KW - trajectory storage KW - mesh compression KW - compression Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-42695 SN - 1438-0064 ER - TY - JOUR A1 - von Tycowicz, Christoph A1 - Ambellan, Felix A1 - Mukhopadhyay, Anirban A1 - Zachow, Stefan T1 - An Efficient Riemannian Statistical Shape Model using Differential Coordinates JF - Medical Image Analysis N2 - We propose a novel Riemannian framework for statistical analysis of shapes that is able to account for the nonlinearity in shape variation. By adopting a physical perspective, we introduce a differential representation that puts the local geometric variability into focus. We model these differential coordinates as elements of a Lie group thereby endowing our shape space with a non-Euclidean structure. A key advantage of our framework is that statistics in a manifold shape space becomes numerically tractable improving performance by several orders of magnitude over state-of-the-art. We show that our Riemannian model is well suited for the identification of intra-population variability as well as inter-population differences. In particular, we demonstrate the superiority of the proposed model in experiments on specificity and generalization ability. We further derive a statistical shape descriptor that outperforms the standard Euclidean approach in terms of shape-based classification of morphological disorders. Y1 - 2018 U6 - https://doi.org/10.1016/j.media.2017.09.004 VL - 43 IS - 1 SP - 1 EP - 9 ER - TY - GEN A1 - Nava-Yazdani, Esfandiar A1 - Hege, Hans-Christian A1 - von Tycowicz, Christoph T1 - A Geodesic Mixed Effects Model in Kendall's Shape Space T2 - Proc. 7th MICCAI workshop on Mathematical Foundations of Computational Anatomy (MFCA) 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 apply the approach for the estimation of group trends and statistical testing of 3D shapes derived from an open access longitudinal imaging study on osteoarthritis. T3 - ZIB-Report - 19-49 KW - Shape Space KW - Kendall Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-74621 SN - 1438-0064 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 - GEN A1 - Ambellan, Felix A1 - Zachow, Stefan A1 - von Tycowicz, Christoph T1 - A Surface-Theoretic Approach for Statistical Shape Modeling 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. T3 - ZIB-Report - 19-20 KW - Statistical shape analysis KW - Principal geodesic analysis KW - Lie groups KW - Classification KW - Manifold valued statistics Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-74497 SN - 1438-0064 ER - TY - GEN A1 - Ambellan, Felix A1 - Zachow, Stefan A1 - von Tycowicz, Christoph T1 - An as-invariant-as-possible GL+(3)-based Statistical Shape Model 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. T3 - ZIB-Report - 19-46 KW - Statistical shape analysis KW - Tangent principal component analysis KW - Lie groups KW - Classification KW - Manifold valued statistics Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-74566 SN - 1438-0064 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 - 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 - JOUR A1 - Nava-Yazdani, Esfandiar A1 - Hege, Hans-Christian A1 - Sullivan, T. J. A1 - von Tycowicz, Christoph T1 - Geodesic Analysis in Kendall's Shape Space with Epidemiological Applications JF - Journal of Mathematical Imaging and Vision N2 - We analytically determine Jacobi fields and parallel transports and compute geodesic regression in Kendall’s shape space. Using the derived expressions, we can fully leverage the geometry via Riemannian optimization and thereby reduce the computational expense by several orders of magnitude over common, nonlinear constrained approaches. The methodology is demonstrated by performing a longitudinal statistical analysis of epidemiological shape data. As an example application we have chosen 3D shapes of knee bones, reconstructed from image data of the Osteoarthritis Initiative (OAI). Comparing subject groups with incident and developing osteoarthritis versus normal controls, we find clear differences in the temporal development of femur shapes. This paves the way for early prediction of incident knee osteoarthritis, using geometry data alone. Y1 - 2020 U6 - https://doi.org/10.1007/s10851-020-00945-w VL - 62 IS - 4 SP - 549 EP - 559 ER - TY - CHAP A1 - Nava-Yazdani, Esfandiar A1 - Hege, Hans-Christian A1 - von Tycowicz, Christoph T1 - A Geodesic Mixed Effects Model in Kendall's Shape Space T2 - Proc. 7th MICCAI workshop on Mathematical Foundations of Computational Anatomy (MFCA) 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 apply the approach for the estimation of group trends and statistical testing of 3D shapes derived from an open access longitudinal imaging study on osteoarthritis. Y1 - 2019 U6 - https://doi.org/10.1007/978-3-030-33226-6_22 VL - 11846 SP - 209 EP - 218 ER - TY - JOUR A1 - Ambellan, Felix A1 - Zachow, Stefan A1 - von Tycowicz, Christoph T1 - Rigid Motion Invariant Statistical Shape Modeling based on Discrete Fundamental Forms JF - Medical Image Analysis 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. Additionally, as planar configurations form a submanifold in shape space, our representation allows for effective estimation of quasi-isometric surfaces flattenings. We evaluate the performance of our model w.r.t. shape-based classification of hippocampus and femur malformations due to Alzheimer's disease and osteoarthritis, respectively. In particular, we achieve state-of-the-art accuracies outperforming 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 biological shape variability, we carry out an analysis of specificity and generalization ability. Y1 - 2021 U6 - https://doi.org/10.1016/j.media.2021.102178 VL - 73 ER - TY - GEN A1 - Ambellan, Felix A1 - Zachow, Stefan A1 - von Tycowicz, Christoph T1 - Geodesic B-Score for Improved Assessment of Knee Osteoarthritis N2 - 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. T3 - ZIB-Report - 21-09 KW - Statistical shape analysis KW - Osteoarthritis KW - Geometric statistics KW - Riemannian manifolds Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-81930 SN - 1438-0064 ER - TY - CHAP A1 - Ambellan, Felix A1 - Zachow, Stefan A1 - von Tycowicz, Christoph T1 - Geodesic B-Score for Improved Assessment of Knee Osteoarthritis T2 - Proc. Information Processing in Medical Imaging (IPMI) N2 - 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. Y1 - 2021 U6 - https://doi.org/10.1007/978-3-030-78191-0_14 SP - 177 EP - 188 ER - TY - GEN A1 - Ambellan, Felix A1 - Lamecker, Hans A1 - von Tycowicz, Christoph A1 - Zachow, Stefan T1 - Statistical Shape Models - Understanding and Mastering Variation in Anatomy N2 - In our chapter we are describing how to reconstruct three-dimensional anatomy from medical image data and how to build Statistical 3D Shape Models out of many such reconstructions yielding a new kind of anatomy that not only allows quantitative analysis of anatomical variation but also a visual exploration and educational visualization. Future digital anatomy atlases will not only show a static (average) anatomy but also its normal or pathological variation in three or even four dimensions, hence, illustrating growth and/or disease progression. Statistical Shape Models (SSMs) are geometric models that describe a collection of semantically similar objects in a very compact way. SSMs represent an average shape of many three-dimensional objects as well as their variation in shape. The creation of SSMs requires a correspondence mapping, which can be achieved e.g. by parameterization with a respective sampling. If a corresponding parameterization over all shapes can be established, variation between individual shape characteristics can be mathematically investigated. We will explain what Statistical Shape Models are and how they are constructed. Extensions of Statistical Shape Models will be motivated for articulated coupled structures. In addition to shape also the appearance of objects will be integrated into the concept. Appearance is a visual feature independent of shape that depends on observers or imaging techniques. Typical appearances are for instance the color and intensity of a visual surface of an object under particular lighting conditions, or measurements of material properties with computed tomography (CT) or magnetic resonance imaging (MRI). A combination of (articulated) statistical shape models with statistical models of appearance lead to articulated Statistical Shape and Appearance Models (a-SSAMs).After giving various examples of SSMs for human organs, skeletal structures, faces, and bodies, we will shortly describe clinical applications where such models have been successfully employed. Statistical Shape Models are the foundation for the analysis of anatomical cohort data, where characteristic shapes are correlated to demographic or epidemiologic data. SSMs consisting of several thousands of objects offer, in combination with statistical methods ormachine learning techniques, the possibility to identify characteristic clusters, thus being the foundation for advanced diagnostic disease scoring. T3 - ZIB-Report - 19-13 Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-72699 SN - 1438-0064 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 - Ambellan, Felix A1 - Lamecker, Hans A1 - von Tycowicz, Christoph A1 - Zachow, Stefan ED - Rea, Paul M. T1 - Statistical Shape Models - Understanding and Mastering Variation in Anatomy T2 - Biomedical Visualisation N2 - In our chapter we are describing how to reconstruct three-dimensional anatomy from medical image data and how to build Statistical 3D Shape Models out of many such reconstructions yielding a new kind of anatomy that not only allows quantitative analysis of anatomical variation but also a visual exploration and educational visualization. Future digital anatomy atlases will not only show a static (average) anatomy but also its normal or pathological variation in three or even four dimensions, hence, illustrating growth and/or disease progression. Statistical Shape Models (SSMs) are geometric models that describe a collection of semantically similar objects in a very compact way. SSMs represent an average shape of many three-dimensional objects as well as their variation in shape. The creation of SSMs requires a correspondence mapping, which can be achieved e.g. by parameterization with a respective sampling. If a corresponding parameterization over all shapes can be established, variation between individual shape characteristics can be mathematically investigated. We will explain what Statistical Shape Models are and how they are constructed. Extensions of Statistical Shape Models will be motivated for articulated coupled structures. In addition to shape also the appearance of objects will be integrated into the concept. Appearance is a visual feature independent of shape that depends on observers or imaging techniques. Typical appearances are for instance the color and intensity of a visual surface of an object under particular lighting conditions, or measurements of material properties with computed tomography (CT) or magnetic resonance imaging (MRI). A combination of (articulated) statistical shape models with statistical models of appearance lead to articulated Statistical Shape and Appearance Models (a-SSAMs).After giving various examples of SSMs for human organs, skeletal structures, faces, and bodies, we will shortly describe clinical applications where such models have been successfully employed. Statistical Shape Models are the foundation for the analysis of anatomical cohort data, where characteristic shapes are correlated to demographic or epidemiologic data. SSMs consisting of several thousands of objects offer, in combination with statistical methods ormachine learning techniques, the possibility to identify characteristic clusters, thus being the foundation for advanced diagnostic disease scoring. Y1 - 2019 SN - 978-3-030-19384-3 SN - 978-3-030-19385-0 U6 - https://doi.org/10.1007/978-3-030-19385-0_5 VL - 3 IS - 1156 SP - 67 EP - 84 PB - Springer Nature Switzerland AG ET - 1 ER - TY - GEN A1 - Ambellan, Felix A1 - Hanik, Martin A1 - von Tycowicz, Christoph T1 - Morphomatics: Geometric morphometrics in non-Euclidean shape spaces N2 - 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. KW - shape analysis KW - geometric statistics KW - geometric morphometrics Y1 - 2021 U6 - https://doi.org/10.12752/8544 N1 - https://morphomatics.github.io/ ER - TY - GEN A1 - Nava-Yazdani, Esfandiar A1 - Hege, Hans-Christian A1 - von Tycowicz, Christoph A1 - Sullivan, T. J. T1 - A Shape Trajectories Approach to Longitudinal Statistical Analysis N2 - For Kendall’s shape space we determine analytically Jacobi fields and parallel transport, and compute geodesic regression. Using the derived expressions, we can fully leverage the geometry via Riemannian optimization and reduce the computational expense by several orders of magnitude. The methodology is demonstrated by performing a longitudinal statistical analysis of epidemiological shape data. As application example we have chosen 3D shapes of knee bones, reconstructed from image data of the Osteoarthritis Initiative. Comparing subject groups with incident and developing osteoarthritis versus normal controls, we find clear differences in the temporal development of femur shapes. This paves the way for early prediction of incident knee osteoarthritis, using geometry data only. T3 - ZIB-Report - 18-42 KW - shape space, shape trajectories, geodesic regression, longitudinal analysis, osteoarthritis Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-69759 SN - 1438-0064 ER - TY - JOUR A1 - Sipiran, Ivan A1 - Lazo, Patrick A1 - Lopez, Cristian A1 - Bagewadi, Nihar A1 - Bustos, Benjamin A1 - Dao, Hieu A1 - Gangisetty, Shankar A1 - Hanik, Martin A1 - Ho-Thi, Ngoc-Phuong A1 - Holenderski, Mike A1 - Jarnikov, Dmitri A1 - Labrada, Arniel A1 - Lengauer, Stefan A1 - Licandro, Roxane A1 - Nguyen, Dinh-Huan A1 - Nguyen-Ho, Thang-Long A1 - Pérez Rey, Luis A. A1 - Pham, Bang-Dang A1 - Pham, Minh-Khoi A1 - Preiner, Reinhold A1 - Schreck, Tobias A1 - Trinh, Quoc-Huy A1 - Tonnaer, Loek A1 - von Tycowicz, Christoph A1 - Vu-Le, The-Anh T1 - SHREC 2021: Retrieval of Cultural Heritage Objects JF - Computers and Graphics N2 - 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. Y1 - 2021 U6 - https://doi.org/10.1016/j.cag.2021.07.010 VL - 100 SP - 1 EP - 20 ER - 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 - JOUR A1 - Moewis, Philippe A1 - Kaiser, René A1 - Trepczynski, Adam A1 - von Tycowicz, Christoph A1 - Krahl, Leonie A1 - Ilg, Ansgar A1 - Holz, Johannes A1 - Duda, Georg T1 - 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 JF - Knee Surgery, Sports Traumatology, Arthroscopy N2 - 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. Y1 - 2021 U6 - https://doi.org/10.1007/s00167-021-06749-8 ER - TY - CHAP A1 - Mayer, Julius A1 - Baum, Daniel A1 - Ambellan, Felix A1 - von Tycowicz, Christoph T1 - A Soft-Correspondence Approach to Shape-based Disease Grading with Graph Convolutional Networks T2 - Proceedings of Machine Learning Research N2 - 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. Y1 - 2022 VL - 194 SP - 85 EP - 95 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 - GEN A1 - Nava-Yazdani, Esfandiar A1 - Hege, Hans-Christian A1 - von Tycowicz, Christoph T1 - A Hierarchical Geodesic Model for Longitudinal Analysis on Manifolds 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. T3 - ZIB-Report - 21-39 KW - Longitudinal KW - Hierarchical Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-85187 SN - 1438-0064 N1 - sumbitted to: Journal of Mathematical Imaging and Vision 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 - Türkseven, Doğa A1 - Rekik, Islem A1 - von Tycowicz, Christoph A1 - Hanik, Martin T1 - Predicting Shape Development: A Riemannian Method T2 - Shape in Medical Imaging N2 - 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. Y1 - 2023 U6 - https://doi.org/10.1007/978-3-031-46914-5_17 SP - 211 EP - 222 PB - Springer Nature ER - 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 - GEN A1 - Hanik, Martin A1 - von Tycowicz, Christoph T1 - Triangle meshes of shadow-recieving surfaces of ancient sundials N2 - 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. Y1 - 2022 U6 - https://doi.org/10.12752/8425 ER - TY - JOUR A1 - Hanik, Martin A1 - Ducke, Benjamin A1 - Hege, Hans-Christian A1 - Fless, Friederike A1 - von Tycowicz, Christoph T1 - Intrinsic shape analysis in archaeology: A case study on ancient sundials JF - Journal on Computing and Cultural Heritage N2 - 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. Y1 - 2023 U6 - https://doi.org/10.1145/3606698 VL - 16 IS - 4 SP - 1 EP - 26 ER - TY - JOUR A1 - Oehme, Stephan A1 - Moewis, Philippe A1 - Boeth, Heide A1 - Bartek, Benjamin A1 - Lippert, Annika A1 - von Tycowicz, Christoph A1 - Ehrig, Rainald A1 - Duda, Georg A1 - Jung, Tobias T1 - PCL insufficient patients with increased translational and rotational passive knee joint laxity have no increased range of anterior–posterior and rotational tibiofemoral motion during level walking JF - Scientific Reports Y1 - 2022 U6 - https://doi.org/10.1038/s41598-022-17328-3 VL - 12 IS - 1 SP - 1 EP - 11 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 - JOUR A1 - Navayazdani, Esfandiar A1 - Ambellan, Felix A1 - Hanik, Martin A1 - von Tycowicz, Christoph T1 - Sasaki Metric for Spline Models of Manifold-Valued Trajectories JF - Computer Aided Geometric Design N2 - 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 Bézier 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. Y1 - 2023 U6 - https://doi.org/10.1016/j.cagd.2023.102220 VL - 104 SP - 102220 ER - TY - JOUR A1 - Hanik, Martin A1 - Steidl, Gabriele A1 - von Tycowicz, Christoph T1 - Manifold GCN: Diffusion-based Convolutional Neural Network for Manifold-valued Graphs N2 - 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. Y1 - 2024 ER - TY - JOUR A1 - Kofler, Andreas A1 - Wald, Christian A1 - Kolbitsch, Christoph A1 - von Tycowicz, Christoph A1 - Ambellan, Felix T1 - Joint Reconstruction and Segmentation in Undersampled 3D Knee MRI combining Shape Knowledge and Deep Learning JF - Physics in Medicine and Biology N2 - 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. Y1 - 2024 U6 - https://doi.org/10.1088/1361-6560/ad3797 ER - TY - JOUR A1 - Oehme, Stephan A1 - Moewis, Philippe A1 - Boeth, Heide A1 - Bartek, Benjamin A1 - von Tycowicz, Christoph A1 - Ehrig, Rainald A1 - Duda, Georg A1 - Jung, Tobias T1 - Altered knee kinematics after posterior cruciate single-bundle reconstruction - a comprehensive prospective biomechanical in vivo analysis JF - Frontiers in Bioengineering and Biotechnology Y1 - 2024 U6 - https://doi.org/10.3389/fbioe.2024.1322136 VL - 12 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 - JOUR A1 - Hanik, Martin A1 - Navayazdani, Esfandiar A1 - von Tycowicz, Christoph T1 - De Casteljau's Algorithm in Geometric Data Analysis: Theory and Application JF - Computer Aided Geometric Design N2 - 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. Y1 - 2024 U6 - https://doi.org/10.1016/j.cagd.2024.102288 VL - 110 SP - 102288 ER -