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 - 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 - 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 - 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 - 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 - 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 - 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 -