## 2 Materials and Methods
### Soft Clustering for Covariance Matrices
Next, the membership \(w_{ij}\) in Equation (4) is replaced with the _spatial membership_\(z_{ij}\) as follows:  
\[\text{WCSS}=\sum_{j=1}^{n}\sum_{i=1}^{c}z_{ij}^{m}D(\mathbf{A}_{j},\bar{ \mathbf{A}}_{i})^{2},\text{ where} \tag{7}\]  
\[z_{ij}=\frac{w_{ij}^{p}h_{ij}^{q}}{\sum_{k=1}^{c}w_{kj}^{p}h_{kj}^{q}}. \tag{8}\]  
Here, the parameters \(p\) and \(q\) enable the fine control of the influence of the membership function \(w_{ij}\) and the function \(h_{ij}\), respectively. For example, in noisy regions \(q\) can be increased to increase the influence of neighbouring voxels. When the clustering region is homogenous, \(z_{ij}\) has similar value to \(w_{ij}\), whilst \(z_{ij}\) is smaller than \(w_{ij}\) if \(\mathbf{A}_{j}\) is considered 'noise' (i.e., it is not actually a member of the cluster \(i\)). FCM can be recovered from sFCM by choosing \(p=1\) and \(q=0\). The sFCM adapted for covariance matrices is described in Algorithm 2. The stopping condition \(\left\|z_{ij}^{(r)}-z_{ij}^{(r-1)}\right\|<\varepsilon\) in Step 2 of the sCFM algorithm captures that the change in spatial membership values has become suitably small. An alternative stopping criteria that can be used in Step 2 of the sFCM algorithm would be \(D(\bar{\mathbf{A}}_{i}^{(r)},\bar{\mathbf{A}}_{i}^{(r-1)})<\varepsilon\), capturing instead that the change in cluster centres has become suitably small, whilst taking account of the use of non-Euclidean metrics. The value of \(\varepsilon\) should be taken to be very small (e.g., it should be smaller than \(0.1\times 10^{-8}\), which is a possible value of diffusion in a given direction). Additionally, the paper presents a regularisation model for sFCM. The WCSS for the regularisation model is defined as  
\[\text{RWCSS}=\sum_{j=1}^{n}\sum_{i=1}^{c}z_{ij}^{m}D_{1}(\mathbf{A}_{j},\bar {\mathbf{A}}_{i})^{r_{1}}+\lambda\sum_{i=1}^{c}D_{2}(\mathbf{A}_{i}^{*},\bar {\mathbf{A}}_{i})^{r_{2}}, \tag{9}\]Here, \(\lambda>0\) is a regularisation parameter, \(r_{1},r_{2}\geq 0\), and \(A_{i}^{*}\) is a reference tensor representing the prescribed diffusion information for cluster \(i\). Users of the technique have the option to either define their own reference tensor for cluster \(i\) with the expressed diffusion behaviour or choose a representative tensor from the cluster as the reference tensor. The proposed regularisation model allows the user to integrate their prior knowledge into the segmentation scheme or to highlight and segment local structures. Note that \(D_{1}\) and \(D_{2}\) need not be the same and can be non-Euclidean. When \(r_{1}=2\), \(\lambda=0\), \(RWCSS\) simplifies, reducing to be equivalent to sFCM in [16]. In this work, after experimentation with the effects of varying the initial parameter values on segmentations of the CC, \(m=2\), \(p=2,q=1.5\) and \(\lambda=0\) (for simplicity) with a \(3\times 3\) window for \(2D\) segmentation and a \(3\times 3\times 3\) window for \(3D\) segmentation were chosen; these values yielded are used consistently for both the simulation studies and the real data experiment in the following sections. Finally, whilst Tables 1 and 2 show only the Euclidean, log, and root Euclidean distances and means, any other suitable distance and mean (i.e., Cholesky, Riemannian, or Procrustes) can be used with the algorithm.