@misc{NavaYazdaniHegevonTycowicz, author = {Nava-Yazdani, Esfandiar and Hege, Hans-Christian and von Tycowicz, Christoph}, title = {A Geodesic Mixed Effects Model in Kendall's Shape Space}, series = {Proc. 7th MICCAI workshop on Mathematical Foundations of Computational Anatomy (MFCA)}, journal = {Proc. 7th MICCAI workshop on Mathematical Foundations of Computational Anatomy (MFCA)}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-74621}, 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 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.}, language = {en} } @article{NavaYazdaniHegeSullivanetal., author = {Nava-Yazdani, Esfandiar and Hege, Hans-Christian and Sullivan, T. J. and von Tycowicz, Christoph}, title = {Geodesic Analysis in Kendall's Shape Space with Epidemiological Applications}, series = {Journal of Mathematical Imaging and Vision}, volume = {62}, journal = {Journal of Mathematical Imaging and Vision}, number = {4}, doi = {10.1007/s10851-020-00945-w}, pages = {549 -- 559}, abstract = {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.}, language = {en} } @inproceedings{NavaYazdaniHegevonTycowicz, author = {Nava-Yazdani, Esfandiar and Hege, Hans-Christian and von Tycowicz, Christoph}, title = {A Geodesic Mixed Effects Model in Kendall's Shape Space}, series = {Proc. 7th MICCAI workshop on Mathematical Foundations of Computational Anatomy (MFCA)}, volume = {11846}, booktitle = {Proc. 7th MICCAI workshop on Mathematical Foundations of Computational Anatomy (MFCA)}, doi = {10.1007/978-3-030-33226-6_22}, pages = {209 -- 218}, 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 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.}, language = {en} } @article{NavaYazdaniPolthier, author = {Nava-Yazdani, Esfandiar and Polthier, Konrad}, title = {De Casteljau's Algotithm on Manifolds}, series = {Computer Aided Geometric Design}, volume = {30}, journal = {Computer Aided Geometric Design}, number = {7}, publisher = {CAGD}, doi = {10.1016/j.cagd.2013.06.002}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-69096}, pages = {722 -- 732}, abstract = {This paper proposes a generalization of the ordinary de Casteljau algorithm to manifold-valued data including an important special case which uses the exponential map of a symmetric space or Riemannian manifold. We investigate some basic properties of the corresponding B{\´e}zier curves and present applications to curve design on polyhedra and implicit surfaces as well as motion of rigid body and positive definite matrices. Moreover, we apply our approach to construct canal and developable surfaces.}, language = {en} } @article{ApprichDieterichHoelligetal., author = {Apprich, Christian and Dieterich, Annegret and H{\"o}llig, Klaus and Nava-Yazdani, Esfandiar}, title = {Cubic spline approximation of a circle with maximal smoothness and accuracy}, series = {Computer Aided Geometric Design}, volume = {56}, journal = {Computer Aided Geometric Design}, publisher = {Elsevier B.V.}, doi = {10.1016/j.cagd.2017.05.001}, pages = {1 -- 3}, abstract = {We construct cubic spline approximations of a circle which are four times continuously differentiable and converge with order six.}, language = {en} } @inproceedings{MyersUtpalaTalbaretal., 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}, series = {Proceedings of Topology, Algebra, and Geometry in Learning}, volume = {196}, booktitle = {Proceedings of Topology, Algebra, and Geometry in Learning}, publisher = {PMLR}, pages = {269 -- 276}, language = {en} } @misc{NavaYazdaniHegevonTycowiczetal., author = {Nava-Yazdani, Esfandiar and Hege, Hans-Christian and von Tycowicz, Christoph and Sullivan, T. J.}, title = {A Shape Trajectories Approach to Longitudinal Statistical Analysis}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-69759}, abstract = {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.}, language = {en} } @article{CaputoEmporioGiachettietal., 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}, series = {Computers and Graphics}, volume = {107}, journal = {Computers and Graphics}, doi = {10.1016/j.cag.2022.07.015}, pages = {241 -- 251}, 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} } @misc{NavaYazdaniHanikAmbellanetal., 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}, 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} } @misc{NavaYazdaniHegevonTycowicz, author = {Nava-Yazdani, Esfandiar and Hege, Hans-Christian and von Tycowicz, Christoph}, title = {A Hierarchical Geodesic Model for Longitudinal Analysis on Manifolds}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-85187}, 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{NavaYazdaniHegevonTycowicz, author = {Nava-Yazdani, Esfandiar and Hege, Hans-Christian and von Tycowicz, Christoph}, title = {A Hierarchical Geodesic Model for Longitudinal Analysis on Manifolds}, series = {Journal of Mathematical Imaging and Vision}, volume = {64}, journal = {Journal of Mathematical Imaging and Vision}, number = {4}, doi = {10.1007/s10851-022-01079-x}, pages = {395 -- 407}, 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{NavayazdaniAmbellanHaniketal., author = {Navayazdani, Esfandiar and Ambellan, Felix and Hanik, Martin and von Tycowicz, Christoph}, title = {Sasaki Metric for Spline Models of Manifold-Valued Trajectories}, series = {Computer Aided Geometric Design}, volume = {104}, journal = {Computer Aided Geometric Design}, doi = {10.1016/j.cagd.2023.102220}, pages = {102220}, 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{HanikNavayazdanivonTycowicz, author = {Hanik, Martin and Navayazdani, Esfandiar and von Tycowicz, Christoph}, title = {De Casteljau's Algorithm in Geometric Data Analysis: Theory and Application}, series = {Computer Aided Geometric Design}, volume = {110}, journal = {Computer Aided Geometric Design}, doi = {10.1016/j.cagd.2024.102288}, pages = {102288}, 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{Navayazdani, author = {Navayazdani, Esfandiar}, title = {On Geodesics in the Spaces of Constrained Curves}, series = {Journal of Differential Geometry and its Applications}, journal = {Journal of Differential Geometry and its Applications}, abstract = {In this work, we study the geodesics of the space of certain geometrically and physically motivated subspaces of the space of immersed curves endowed with a first order Sobolev metric. This includes elastic curves and also an extension of some results on planar concentric circles to surfaces. The work focuses on intrinsic and constructive approaches.}, language = {en} } @inproceedings{Navayazdani, author = {Navayazdani, Esfandiar}, title = {Elastic Analysis of Augmented Curves and Constrained Surfaces}, series = {Proc. of IAPR Third International Conference on Discrete Geometry and Mathematical Morphology}, booktitle = {Proc. of IAPR Third International Conference on Discrete Geometry and Mathematical Morphology}, publisher = {Springer}, doi = {https://link.springer.com/chapter/10.1007/978-3-031-57793-2_27}, pages = {353 -- 363}, language = {en} }