# WP2 - Random shapes

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 - Subprojects

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

## Research questions

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.

### Selected references

- Hauberg, S., Feragen, A., Enficiaud, R. & Black, M.J. (2016): Scalable robust principal component analysis using Grassmann averages.
*IEEE T. Pattern Anal.***38**, 2298-2311. - 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. - Liptrot, M.G. & Lauze, F.B. (2016): Rotationally invariant clustering of diffusion MRI data using spherical harmonics.
*Proceedings of SPIE Medical Imaging***9784.** - 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.