## 4 Conclusions and Discussion  
Manual segmentation of the CC is often done slice by slice, and hence it is very time consuming and is also subject to intra and interobserver variability. In this paper, we presented an automatic and accurate method for segmentation of the CC that will save time and effort for radiologists by eliminating the need of manual segmentation. The method uses uncertainty information in the segmentation decisions by calculating the probabilities (membership values) of each voxel to belong to different clusters. The spatial contextual information is exploited.  
We demonstrated that sFCM outperformed K-means with the same metric for segmentation of the CC and sFCM with the root Euclidean metric as the preferred option. In this paper, the method was extended to a matrix-type of data (covariance matrix data) and was used for DTI classification (to classify images into white matter, grey matter, and cerebrospinal fluid).  
Further research about classification of different brain regions using this method will be carried out in future work. The CC in SCA2 subjects were compared with healthy subjects. We demonstrated that the determinant of diffusion tensors and DTI indices can be used to distinguish between SCA2 and healthy subjects and are sensitive to ageing effects. Hence, the determinant of diffusion tensors is useful for the detection of abnormalities in the CC and for surgical planning.  
The non-Euclidean metrics used in sFCM were the log-Euclidean and root Euclidean metrics as their calculations are straightforward. It would be interesting to use other non-Euclidean metrics, although numerical algorithms may need to be used. For example, the Procrustes size-and-shape metric may provide new segmentation results.  
A regularisation model was proposed for sFCM--currently, for the smooth part, the power parameter \(r_{1}=2\), and the regularisation parameter \(\lambda=0\). For various structures in the brain, the regularisation model would still need to be carefully tuned. It would be interesting to explore segmentation results when varying the parameters in the model. An efficient way to determine the reference tensor with the expected diffusion behaviour is needed for practical users.  
There are regions in the brain with more than one distinct fibre orientation captured in a single voxel. For example, two crossing fibres would need two diffusion tensors to model the diffusion behaviour at a single voxel. Tensor field segmentation becomes more challenging for the regions with crossing fibres. A basic and difficult question is how to measure the dissimilarity between a pair of tensors at one voxel and a pair of tensors at another voxel. To develop segmentation methods for DTI data containing crossing fibres is also of interest.  
Conceptualization, D.Z.; methodology, A.F., D.Z. and S.E.; software, S.E.; validation, A.F.; formal analysis, S.E.; investigation, S.E.; resources, A.F., D.Z. and S.E.; data curation, A.F., D.Z. and S.E.; writing--original draft preparation, S.E.; writing--review and editing, A.F., D.Z.; visualization, A.F., D.Z. and S.E.; supervision, A.F., D.Z.; project administration, D.Z.; funding acquisition, A.F. and D.Z. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Higher Education Innovation Funding-UK and the EPSRC IAA (UK).  
The SCA2 image data used in this paper is freely available from [23].  
**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.