In this work package we study random shapes that reside in non-linear spaces. An important example is the **tree-spaces**, arising naturally in modelling of anatomical networks. The field of non-linear statistics has close connections to functional data analysis, and random topology and graphs.

While there is a deep understanding of the mathematical and computational aspects of many data types living in non-linear spaces, a detailed understanding of **random variation in non-linear spaces** and how to handle randomness statistically is largely missing.

**WP2.1:** Deformation modelling and statistics of deformations**WP2.2:** Modelling and inference in non-linear spaces**WP2.3:** Applications in diffusion weighted imaging

In the second funding period of CSGB, we focus on projects in (a) deformation modelling and statistics of deformations and (b) modelling and inference in non-linear spaces.

- Based on the large deformation diffeomorphic metric mapping (LDDMM) approach to deformation modelling, we aim at developing a stochastic framework for
**deformation groups**. In a Bayesian analysis of the deformation models, we want to use prior distributions that arise from transition densities of stochastic differential equations. - The general aim is to develop statistical tools for data in non-linear spaces such as
**manifolds**,**stratified spaces**and general**metric spaces**. The study of tree-spaces, initiated in the first funding period of CSGB, is continued. As a new challenge, inspired by voxel-based analysis of diffusion-weighted imaging (DWI) data, we want to develop statistical inference for mixture models for multi-tensors.

In the second funding period, we aim at using the developed mixture models for representing local diffusion orientation distributions that take **crossing fiber diffusions** into account. Mixture models have appeared previously, but crucial issues such as matching and aligning of modes have not been addressed.

- Holm, A.N., Feragen, A., Dela Haije, T. & Darkner, S. (2018): Deterministic group tractography with local uncertainty quantification.
*Computational Diffusion MRI (CDMRI) Workshop at MICCAI 2018*. - Kasenburg, N., Darkner, S., Hahn, U., Liptrot, M.G. & Feragen, A. (2016): Structural parcellation of the Thalamus using shortest-path tractography.
*Proceedings of the International Symposium on Biomedical Imaging (ISBI)*, 559-563. - Kühnel, L. & Sommer, S.H. (2017): Stochastic development regression on non-linear manifolds. In Proceedings of Information Processing in Medical Imaging: 25
^{th}International Conference (IPMI) 2017. Lecture Notes in Computer Science**10265**, pp. 53-64. Springer. - Kühnel, L., Sommer, S., Pai, A. & Raket, L.L. (2017): Most likely separation of intensity and warping effects in image registration.
*Siam J. Imaging Sci.***10**, 578-601. - Sommer, S.H. (2019): An infinitesimal model for principal component analysis of manifold valued data.
*Sankhya A***81**, 37–62. - Sommer, S.H. & Svane, A.M. (2017): Modelling anisotropic covariance using stochastic development and sub-Riemannian frame bundle geometry.
*J. Geom. Mech.***9**, 391-410.