# 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

Feragen, A., Lo, P., de Bruijne, M., Nielsen, M. and Lauze, F. (2013): Toward a theory of statistical tree-shape analysis. *IEEE T. Pattern Anal.* **35**, 2008-2021.

Hauberg, S., Feragen, A., Enficiaud, R. and Black, M.J. (2016): Scalable robust principal component analysis using Grassmann averages. *IEEE T. Pattern Anal.* **38**, 2298-2311.

Liptrot, M. and Lauze, F. (2014): A model-free unsupervised method to cluster brain tissue directly from DWI volumes. *Proceedings of the Annual Meeting of the International Society for Magnetic Resonance in Medicine, ISMRM.*

Liptrot, M. and Lauze, F. (2016): Rotationally invariant clustering of diffusion MRI data using spherical harmonics. To appear in *Proceedings of SPIE Medical Imaging 2016.*

Sommer, S.H. (2015): Anisotropic distributions on manifolds: template estimation and most probable paths. In *Proceedings from the conference Information Processing in Medical Imaging (IPMI)*, Sabhal Mor Ostaig, Isle of Skye, UK*.* Springer LNCS **9123**, pp. 193-204.

Sommer, S.H., Nielsen, M., Darkner, S. and Pennec, X. (2013): Higher-order momentum distributions and locally affine LDDMM registration. *SIAM Journal on Imaging Sciences* **6**, 341-367.