## 1 Introduction  
Brain image classification and region segmentation methods are crucial components in medical applications. The Corpus Callosum (CC) is a great fibre bundle in the white matter of the brain that connects the two hemispheres of the brain. A change in the size and shape of the CC can be indicator of a brain abnormality [1, 2, 3], and hence an accurate segmentation of the CC is important in the diagnosis of disease and for surgical planning. Automating the production of accurate segmentations is a crucial step in saving a significant amount of clinician time in practice by removing the need for manual segmentation. Diffusion Tensor Imaging (DTI) is an advanced magnetic resonance imaging technique that measures the Brownian displacements of water molecules in each voxel in the brain [4], providing unique information about biological tissues in the brain. A diffusion tensor (DT) is a 3 \(\times\) 3 real positive semi-definite covariance matrix [5] that describes the mobility of molecules in each direction. Therefore, the dimensions of a DTI dataset are five for 2D images and six for 3D images (i.e., including the spatial location dimensions). Analysing the DTI data itself, as 4D or 5D, retains all the information from the data with no loss of diffusivity information. The space of positive semi-definite covariance matrices arises in many applications, such as in image and longitudinal data analysis, and thus methods developed for covariance matrices are more widely applicable than medical imaging. To measure distance in the space of positive semi-definite covariance matrices, various non-Euclidean metrics [6; 7] have been proposed as alternatives to the Euclidean metric to avoid violations of the positive semi-definiteness criterion in the course of extrapolation, for example, and since Euclidean averaging is prone to swelling (i.e., inflation of the determinant [7]). Computing a mean using certain non-Euclidean metrics (e.g., Procrustes and Riemannian) require numerical solutions to be solved, whilst other non-Euclidean metrics (e.g., log Euclidean, root Euclidean, and Cholesky) do not; not requiring numerical methods tends to be computational faster, which is an important criterion for practical medical applications due to the large data sets. Diffusion Tensor indices, such as Fractional Anisotropy (FA) and Mean Diffusivity (MD), are sensitive to brain abnormalities; the FA values of the CC have been reported to be decreased and MD values to be increased in abnormal brains as compared to healthy brains [8; 9; 10]. Spinocerebellar ataxia 2 (SCA2) is a particular hereditary neurodegenerative disorder [11] caused by progressive dysfunction of the cerebellum and the brain stem, and it is characterized by progressive problems with movement. DTI has been found to be useful for assessing microstructural changes in the brain white matter, which are associated with SCA2 [12; 13; 14]. The authors in [15] used FreeSurfer software for the segmentation of the cerebrum and the brainstem-cerebellum, and performed a histogram analysis of the DTI indices of FA, MD, radial diffusivity (RD), axial diffusivity (AX), and mode of anisotropy (MO). The FA and MO values were decreased whilst the MD, RD, and AX values were increased in SCA2 subjects compared to healthy brains. Thus, all of these indices can be used for disease diagnosis from suitably segmented images. Traditional image processing methods use scalar and vector-valued data for data analysis. In particular, spatial fuzzy c-means (sFCM) improves the segmentation of images (in comparison with K-means) by using the neighbouring information to minimize the effect of noise [16]. The standard sFCM algorithm uses Euclidean distance and the Euclidean mean to cluster sets of vectors and needs adaptation to be applicable for use with matrix-based data.