EF2 – Digital Shapes



Spline Models for Shape Trajectory Analysis

Project Members

Project Duration

01.04.2019 – 31.12.2021

Located at



Cubic shape trajectory of a distal femur obtained by regression with Bézier curves

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 plan to develop parametric regression with Bézier splines on Riemannian manifolds to create widely applicable and efficient algorithms for the reconstruction and analysis of shape trajectories.
  • We aim at a hierarchical model that relies on spline trajectories to analyze mixed longitudinal and cross-sectional data that cannot be dealt with adequately/efficiently with existing approaches.

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.

Scientific details and results

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.

Regression analysis with Bézier splines leads to a minimization problem with respect to the control points. Since closed-form solutions do not exist in general, we apply optimization methods like Riemannian gradient descent. In order to compute the gradients, so-called concatenated adjoint Jacobi fields need to be derived. The de Casteljau algorithm, the construction of Bézier splines and an example of concatenated adjoint Jacobi fields on the 2-dimensional sphere are shown in the pictures below.

For a hierarchical model, not single shapes but curves in shape space are the objects of interest because subject-specific trends are viewed as perturbations of a population-average trajectory. For the development of our hierarchical model, we want to endow the space of spline trajectories in shape space with a functional-based metric. It promises increased computational efficiency as it does not require explicit evaluations of the curvature tensor (in contrast to other known metrics).

Receiving an award at MICCAI 2020 for the work, we introduce a novel approach for nonlinear regression on manifolds based on intrinsic Bézier splines in [1]. Therein, 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.

Realizing our goal, we developed a hierarichal model based on splines of Bézier curves in [4]. It is highly flexible and allows for groupwise analysis of higher-order (including cyclic) trends in Riemannian manifold. We successfully applied it longitudinal shape data of human knees.


To analyze data, like a set of shapes, statistical methods are used. Over the last decades many useful methods from the multivariate case have been generalized to data lying in Riemannian manifolds. Nevertheless, there are certain cases where they are not optimal. One way to model shapes of homogeneous objects (imagine, e.g., humans) is to view them as deformations of a reference template (a reference body in the human example). We then view its deformation (map) as the shape of a given object. In mathematical terms these are elements of Lie groups, which are manifolds that have a smooth (infinitely-often differentiable) group structure. To avoid biases in the analysis (e.g., due to the actual choice of reference or the way the data is computed) the underlying statistical methods should respect the group structure, that is, they should be as invariant/equivariant under the group operation as possible. Unfortunately, this is not true for statistical tools that utilize a Riemannian structure for several Lie groups (namely those that do not posses a bi-invariant Riemannian metric). Research for non-Riemannian statistical tools in Lie groups was thus initiated in recent years. In [2] we generalize two well-known test statistics in a way that is invariant under the group operations and show how they can be used for two sample hypothesis testing by locally comparing the mean shapes of right hippocampi of patients with beginning Alzheimer’s disease and a healthy control group. This paper won the best paper Award of the Shape in Medical Imaging workshop at the MICCAI 2020. The results of the test are shown in the picture below: colored areas show a significant (p<0.5) difference in shape (after applying FDR-correction of Benjamini and Hochberg).

We extendend these experiments, added a novel experiment, and proved several theoretical properties of the new measures in [3].


The regression code is available for Python as part of the Morphomatics data base.

Project Webpages

Selected Publications


Selected Pictures

Graphical Abstract of regression with a closed spline in a Shape Space with mitral valve data


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