@inproceedings{HanikHegevonTycowicz, author = {Hanik, Martin and Hege, Hans-Christian and von Tycowicz, Christoph}, title = {Bi-invariant Two-Sample Tests in Lie Groups for Shape Analysis}, series = {Shape in Medical Imaging}, booktitle = {Shape in Medical Imaging}, publisher = {Springer International Publishing}, address = {Cham}, doi = {10.1007/978-3-030-61056-2_4}, pages = {44 -- 54}, abstract = {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.}, language = {en} } @inproceedings{HanikHegeHennemuthetal., author = {Hanik, Martin and Hege, Hans-Christian and Hennemuth, Anja and von Tycowicz, Christoph}, title = {Nonlinear Regression on Manifolds for Shape Analysis using Intrinsic B{\´e}zier Splines}, series = {Proc. Medical Image Computing and Computer Assisted Intervention (MICCAI)}, booktitle = {Proc. Medical Image Computing and Computer Assisted Intervention (MICCAI)}, publisher = {Springer International Publishing}, address = {Cham}, doi = {10.1007/978-3-030-59719-1_60}, pages = {617 -- 626}, abstract = {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 B{\´e}zier 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 B{\´e}zier spline regression as an effective and flexible tool for manifold-valued regression.}, language = {en} } @misc{AmbellanHanikvonTycowicz, author = {Ambellan, Felix and Hanik, Martin and von Tycowicz, Christoph}, title = {Morphomatics: Geometric morphometrics in non-Euclidean shape spaces}, doi = {10.12752/8544}, abstract = {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.}, language = {en} } @article{SipiranLazoLopezetal., author = {Sipiran, Ivan and Lazo, Patrick and Lopez, Cristian and Bagewadi, Nihar and Bustos, Benjamin and Dao, Hieu and Gangisetty, Shankar and Hanik, Martin and Ho-Thi, Ngoc-Phuong and Holenderski, Mike and Jarnikov, Dmitri and Labrada, Arniel and Lengauer, Stefan and Licandro, Roxane and Nguyen, Dinh-Huan and Nguyen-Ho, Thang-Long and P{\´e}rez Rey, Luis A. and Pham, Bang-Dang and Pham, Minh-Khoi and Preiner, Reinhold and Schreck, Tobias and Trinh, Quoc-Huy and Tonnaer, Loek and von Tycowicz, Christoph and Vu-Le, The-Anh}, title = {SHREC 2021: Retrieval of Cultural Heritage Objects}, series = {Computers and Graphics}, volume = {100}, journal = {Computers and Graphics}, doi = {10.1016/j.cag.2021.07.010}, pages = {1 -- 20}, abstract = {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.}, language = {en} } @inproceedings{HanikHegevonTycowicz, author = {Hanik, Martin and Hege, Hans-Christian and von Tycowicz, Christoph}, title = {A Nonlinear Hierarchical Model for Longitudinal Data on Manifolds}, series = {2022 IEEE 19th International Symposium on Biomedical Imaging (ISBI)}, booktitle = {2022 IEEE 19th International Symposium on Biomedical Imaging (ISBI)}, doi = {10.1109/ISBI52829.2022.9761465}, pages = {1 -- 5}, abstract = {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 B{\´e}zier 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.}, language = {en} }