01.01.2019 – 31.12.2021
In many applications manifold-valued data arises in increasingly large numbers and sophisticated tools for its analysis are needed. As an important example, detailed shape representations of 2D/3D objects that are elements of a shape space (a high-dimensional quotient manifold) are obtained with the help of modern imaging technology. Using suitable Riemannian metrics, distances between shapes can be defined, which, in turn, allow for statistical analysis. A frequently encountered situation is that instead of a static shape, one is interested in how it varies with respect to some parameter (e.g. time). The corresponding mathematical object is a trajectory in shape space. They are of great importance for longitudinal studies whose ever-growing number has fueled research for regression analysis in Riemannian manifolds. So far the focus has been on geodesic models, that is, it was assumed that the assembled data spreads along generalized straight lines. In this project, we want to extend these methods to analyze data that is distributed more broadly. Furthermore, in order to cope with correlations within the data, hierarchical models are needed. These allow, for example, for the correct separation of time and population effects in medical studies where repeated measurements from several individuals are made.
Therefore, we envisage the following research questions in our project
Spline models for shape trajectory analysis:
We will mainly focus on 2 applications in order to validate our models: archaeology and cardiology.
In archaeology dating of finds is one of the biggest problems, partly because written sources are very rare. For example, for buildings and ceramics, the characteristics of their ornamentation are used to approximate their age. We want to represent the development of ornaments by shape trajectories and investigate whether this can help to answer dating questions.
In cardiology, we plan to focus on the investigation of valve diseases. These have reached endemic features (more than 12% of the people older than 70 develop one). In patients with mitral valve insufficiency (MI), the valve’s leaflets do not close fully or prolapse into the left atrium during systole. Blood then flows back lowering the heart’s efficiency. A cascade of further diseases can be triggered as a result. Presently, only a few of the parameters of the 4D data that can be obtained with echocardiography or computed tomography scans are used for diagnosis and therapy planning. We want to analyze the change of shape of the mitral valve during the cardiac cycle to identify new MI indicators.
We base our methods on manifold-valued Bézier curves. Being generalized polynomial curves, they are constructed via the generalized de Casteljau algorithm from control points; these are elements of the manifold whose number determines the degree of the curve. Several such segments can be easily joined to a differentiable spline.
In (Hanik et al. 2020) we introduce a novel approach for nonlinear regression on manifolds based on intrinsic Bézier splines. Moreover, we apply our method to distal femora (upper knee bone) and mitral valve data. Discrete samples of the reconstructed shape trajectories are displayed in the first picture on this website and the one below.
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