@article{vonTycowiczAmbellanMukhopadhyayetal., author = {von Tycowicz, Christoph and Ambellan, Felix and Mukhopadhyay, Anirban and Zachow, Stefan}, title = {An Efficient Riemannian Statistical Shape Model using Differential Coordinates}, series = {Medical Image Analysis}, volume = {43}, journal = {Medical Image Analysis}, number = {1}, doi = {10.1016/j.media.2017.09.004}, pages = {1 -- 9}, abstract = {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.}, language = {en} } @inproceedings{AmbellanZachowvonTycowicz, author = {Ambellan, Felix and Zachow, Stefan and von Tycowicz, Christoph}, title = {An as-invariant-as-possible GL+(3)-based Statistical Shape Model}, 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)}, publisher = {Springer}, doi = {10.1007/978-3-030-33226-6_23}, pages = {219 -- 228}, abstract = {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.}, language = {en} } @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} } @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{AmbellanZachowvonTycowicz, author = {Ambellan, Felix and Zachow, Stefan and von Tycowicz, Christoph}, title = {Rigid Motion Invariant Statistical Shape Modeling based on Discrete Fundamental Forms}, series = {Medical Image Analysis}, volume = {73}, journal = {Medical Image Analysis}, doi = {10.1016/j.media.2021.102178}, abstract = {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.}, language = {en} } @inproceedings{AmbellanZachowvonTycowicz, author = {Ambellan, Felix and Zachow, Stefan and von Tycowicz, Christoph}, title = {Geodesic B-Score for Improved Assessment of Knee Osteoarthritis}, series = {Proc. Information Processing in Medical Imaging (IPMI)}, booktitle = {Proc. Information Processing in Medical Imaging (IPMI)}, doi = {10.1007/978-3-030-78191-0_14}, pages = {177 -- 188}, abstract = {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.}, language = {en} } @inproceedings{AmbellanZachowvonTycowicz, author = {Ambellan, Felix and Zachow, Stefan and von Tycowicz, Christoph}, title = {A Surface-Theoretic Approach for Statistical Shape Modeling}, series = {Proc. Medical Image Computing and Computer Assisted Intervention (MICCAI), Part IV}, volume = {11767}, booktitle = {Proc. Medical Image Computing and Computer Assisted Intervention (MICCAI), Part IV}, publisher = {Springer}, doi = {10.1007/978-3-030-32251-9_3}, pages = {21 -- 29}, abstract = {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.}, language = {en} } @incollection{AmbellanLameckervonTycowiczetal., author = {Ambellan, Felix and Lamecker, Hans and von Tycowicz, Christoph and Zachow, Stefan}, title = {Statistical Shape Models - Understanding and Mastering Variation in Anatomy}, series = {Biomedical Visualisation}, volume = {3}, booktitle = {Biomedical Visualisation}, number = {1156}, editor = {Rea, Paul M.}, edition = {1}, publisher = {Springer Nature Switzerland AG}, isbn = {978-3-030-19384-3}, doi = {10.1007/978-3-030-19385-0_5}, pages = {67 -- 84}, abstract = {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.}, language = {en} }