Morphomatics: Geometric morphometrics in non-Euclidean shape spaces
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.
The source code is freely available on GitHub.
Walk through the tutorials live on 
Citation
If you use Morphomatics in your academic projects, we politely ask you to aknowledge Morphomatics in your manuscript and to cite each of the publications (listed below) that are relevant for your use. You may cite the library in general using this BibTeX entry:
@misc{Morphomatics,
title = {Morphomatics: Geometric morphometrics in non-Euclidean shape spaces},
author = {Felix Ambellan and Martin Hanik and Christoph von Tycowicz},
note = {https://morphomatics.github.io/},
year = {2021},
doi = {10.12752/8544}
}
Of course, if you want, we will also list a reference to your work/project/etc. here.
Riemannian shape spaces
We provide a novel Riemannian framework for shape analysis based on differential coordinates that naturally belong to Lie groups and effectively describe local changes in shape. Performing intrinsic calculus on this representation allows for fast computations while, at the same time, accounts for the nonlinearity in shape variation. The rich structure of the derived shape space yields highly discriminative shape descriptors providing a compact representation that is amenable to learning algorithms.
References:
Christoph von Tycowicz, Felix Ambellan, Anirban Mukhopadhyay, and Stefan Zachow:
An Efficient Riemannian Statistical Shape Model using Differential Coordinates.
Medical Image Analysis, Volume 43, January 2018.
Felix Ambellan, Stefan Zachow, and Christoph von Tycowicz:
Rigid Motion Invariant Statistical Shape Modeling Based on Discrete Fundamental Forms.
Medical Image Analysis, Volume 73, January 2021.
Regression in Riemannian manifolds
Regression methods are central to multivariate statistics. We provide a consistent generalization of spline regression (including polynomial regression) to Riemannian manifolds that is based on Bézier curves. It allows to model not only geodesic (i.e., generalized linear) relationships but also (amongst others) saturated or even cyclic dependencies. It can therefore be applied to a wide class of real-world applications with manifold-valued features, including but not limited to applications of shape analysis.
References:
Martin Hanik, Hans-Christian Hege, Anja Hennemuth, Christoph von Tycowicz:
Nonlinear Regression on Manifolds for Shape Analysis using Intrinsic Bézier Splines.
Proc. Medical Image Computing and Computer Assisted Intervention (MICCAI), 2020.
Hierarchical Models and the Geometry of Bézier Splines
Longitudinal studies are common, e.g., in medical and pharmaceutical research. They yield data with high correlation between some samples and (almost) no correlation between others. Hierarchical models are an adequate choice of modelling the relationships underlying such data. With the ‘‘Bézierfold’’ (the manifold of Bézier spline through a given base manifold), Morphomatics provides the means for a hierarchical modeling of manifold-valued data (e.g., from longitudinal studies that monitor shape developments). For an introductory example see the corresponding tutorial.
References:
Martin Hanik, Hans-Christian Hege, Christoph von Tycowicz:
A Nonlinear Hierarchical Model for Longitudinal Data on Manifolds.
Proc. International Symposium on Biomedical Imaging (ISBI), 2022.
Dissimilarity measures for sample distributions in Lie groups
Multivariate statistical indices are not influenced by common shifts of the data; this is highly desirable as the latter are fundamental symmetries of Euclidean space that do not change relationships between the samples. Lie groups also possess symmetries, but Riemannian statistical tools respect them only in special cases. (For example, they do not respect them for data in the group of rigid-body transformations SE(3) in 3-space.) Morphomatics offers methods (a mean and dissimilarity measures for sets of samples) that extend symmetry-awareness to all Lie groups. They can be used, e.g., for two sample tests of SE(3)-valued data that do not depend on the initial place and orientation in 3-space of the underlying objects.
References:
Martin Hanik, Hans-Christian Hege, Christoph von Tycowicz:
Bi-Invariant Dissimilarity Measures for Sample Distributions in Lie Groups.
SIAM Journal on Mathematics of Data Science, Volume 4, Issue 4, 2022.
Applications
Several applications have been successively developed based on algorithms from Morphomatics:
- Manifold-Valued Graph Neural Networks [Hanik et al. 24]
- Predicting Future Shape [Türkseven et al. 23]
- Analysis of Ancient Sundials [Hanik et al. 23]
- Hurrican Track Analysis [Nava-Yazdani et al. 23]
- 3D-from-2D Shape Estimation [Paskin et al. 22]
- Predicting Cognitive Scores [Hanik et al. 22]
- Geodesic B-Score [Ambellan et al. 21]
- Thin-Volume Visualization [Herter et al. 21]
- Jerash Silver Scroll Unfolding [Baum et al. 21]