@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} } @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} } @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} }