## 1 Introduction
To cluster sets of covariance matrices, such as diffusion tensors, a K-means algorithm was adapted for use by the authors in [17]; using a single manually segmented image as ground truth, the log Euclidean and Riemannian metrics were found to provide the most accurate segmentations of the CC (using accuracy and specificity as performance measures). As a significant step forward, a spatial fuzzy c-means (sFCM) clustering method for covariance data (so that we do not loose information from the diffusion tensors), which effectively segment DT images by accounting for the noise, partial voluming, magnetic field inhomogeneity, and other imaging artefacts, was developed in the paper. To retain the non-Euclidean nature of DTs, the efficiently computable log and root Euclidean metrics are used to estimate the mean DT for each cluster. The results are compared with a baseline of K-means to the suitably adapted sFCM algorithm for clustering sets of covariance matrices. Simulation studies are designed to test the performance of the algorithms and the metrics in the face of increasing noise levels, which is an important aspect of automated segmentations in this domain; since simulations ensure that the CC as ground truth is determined, performance measures (the accuracy, specificity, sensitivity, precision, Gmean, and F-measure) can be calculated, and sFCM with the efficiently computable metrics compared to the baseline K-means algorithm with the same metrics. The findings indicate that sFCM is better than K-means with the same metric for almost all noise levels and performance measures; almost always sFCM with the root Euclidean metric is the preferred option. To further explore use in practice, the segmentation method is also applied to the CC for real brain data from healthy and Spinocerebellar ataxia 2 (SCA2) subjects. This is unannotated real data (i.e., the CC has not already been annotated/indicated by a clinician), and thus one can make use of the fact that the CC is known to be a well-connected single white tract in order to determine the efficacy of the segmentation method in practice; since there should not be extra small regions or outliers appearing in the same cluster, disconnected voxels in the segmentation results can be considered as noise. The findings indicate that the number of voxels that are incorrectly labelled as being within the CC is significantly smaller using sFCM versus K-means in this real data application. The paper also presents additional results that point to future utility and developments, such as: the determinant can be used to distinguish healthy and SCA2 subjects, the determinant can be used as a DTI index, which distinguishes between healthy and SCA2 subjects and is sensitive to ageing effects, and that the method can be used for more general classification tasks. In terms of presentation, the base algorithms FCM and sFCM, adapted for covariance matrices, are described in Section 2. Basic matrix operators of a positive definite covariance matrix are described in Appendix A. The main content of the set-up and results obtained using the simulation studies and real brain data experiments are presented in Section 3. A block diagram indicating an overview of the processing steps and software used is presented in Appendix B. Our conclusions are presented in Section 4.