# A Fuzzy Consensus Clustering Algorithm for MRI Brain Tissue Segmentation
## 2 Methodology
### Segmentation
#### 2.2.2 Generalized Spatial Kernel FCM (GSKFCM)  
The generalized Spatial Kernel FCM (GSKFCM) [29] is another variant of the conventional FCM. Even though RSKFCM overcomes the limitations of the FCM, the performance is not good because it injects neighborhood information only into the objective function. However, the distance function plays a vital role in computing the membership value. Thus, the addition of neighborhood information can increase the performance. The RSKFCM also assumes all features have equal importance. However, in a real-world problem, all the features may not be equally important. GSKFCM overcomes these limitations by injecting the weighted neighbourhood information into the distance function and employing the Gaussian kernel as the distance metric.  
The aim of the GSKFCM is to minimize the objective function shown in Equation (10).  
\[J=2\sum\limits_{i=1}^{c}\sum\limits_{j=1}^{n}z_{ij}^{m}d_{new}^{2}\big{(}x_{j},v_ {i}\big{)} \tag{10}\]  
where \(z_{ij}\) is the GSKFCM membership function, and it is computed as:  
\[z_{ij}=\frac{1}{\sum\limits_{k=1}^{c}\left(\frac{d_{new}^{2}\big{(}x_{j},v_{i} \big{)}}{d_{new}^{2}\big{(}x_{j},v_{k}\big{)}}\right)^{\frac{1}{(n-1)}}} \tag{11}\]  
where \(d_{new}\) is the GSKFCM distance function which incorporates the neighbourhood function into the distance function, and it is computed as:  
\[d_{new}^{2}\big{(}x_{j},v_{i}\big{)}=d^{2}\big{(}x_{j},v_{i}\big{)}\ f\big{(}p _{ij}\big{)} \tag{12}\]  
where, \(d^{2}\big{(}x_{j},v_{i}\big{)}\) is the Gaussian Kernel distance function shown in Equation (4), and \(f(p_{ij})=\frac{1}{p_{ij}}\) is the neighbourhood function.  
The GSKFCM considers the neighbourhood information and computes the membership value associated with each voxel as the weighted sum of the traditional FCM membership value and the membership value of the \(N_{k}\) neighbour points. The neighbourhood function \((p_{ij})\) is defined as:  
\[p_{ij}=\sum\limits_{k=0}^{N_{k}}h\big{(}x_{j},x_{k}\big{)}g(u_{ik}) \tag{13}\]  
where \(N_{k}\) is the number of neighbourhood voxels, \(g(u_{ik})=u_{ik}\) is the membership function (Equation (7)), \(h(x_{j},x_{k})\) is the distance function which is computed as:  
\[h\big{(}x_{j},x_{k}\big{)}=\left(\sum\limits_{l=0}^{N_{k}}\frac{d^{2}\big{(}x _{j},x_{k}\big{)}}{d^{2}\big{(}x_{j},x_{l}\big{)}}\right)^{-1} \tag{14}\]  
Substituting Equation (14) in Equation (13), the neighbourhood function becomes:  
\[p_{ij}=\sum\limits_{k=0}^{N_{k}}g(u_{ik})\Bigg{(}\sum\limits_{l=0}^{N_{k}} \frac{d^{2}\big{(}x_{j},x_{k}\big{)}}{d^{2}\big{(}x_{j},x_{l}\big{)}}\Bigg{)} ^{-1} \tag{15}\]  
Substituting Equation (12) in Equation (11), the membership function \(z_{ij}\) becomes,  
\[z_{ij}=\left(\sum\limits_{k=1}^{c}\Bigg{(}\frac{d^{2}\big{(}x_{j},v_{i}\big{)} \ f(p_{ij})}{d^{2}\big{(}x_{j},v_{k}\big{)}\ f\big{(}p_{jk}\big{)}}\Bigg{)}^{ \frac{1}{(n-1)}}\right)^{-1} \tag{16}\]  
\[=\frac{\left(\sum\limits_{k=1}^{c}\left(\frac{d^{2}\big{(}x_{j},v_{i}\big{)}}{ d^{2}\big{(}x_{j},v_{k}\big{)}}\right)^{\frac{1}{(n-1)}}\right)^{-1}f^{\frac{1}{1-n}} \big{(}p_{ij}\big{)}}{\sum\limits_{k=1}^{c}\Bigg{(}\frac{c}{\sum\limits_{l=1} ^{c}\left(\frac{d^{2}\big{(}x_{j},v_{i}\big{)}}{d^{2}\big{(}x_{j},v_{l}\big{)} }\right)^{\frac{1}{n-1}}\Bigg{)}^{-1}f^{\frac{1}{1-n}}\Big{(}p_{jk}\big{)}}} \tag{17}\]where \(\left(\sum\limits_{k=1}^{c}\left(\frac{d^{2}(x_{j},v_{j})}{d^{2}(x_{j},v_{k})} \right)^{\frac{1}{m-1}}\right)^{-1}=u_{ij}\). Then the membership function \(z_{ij}\) becomes  
\[z_{ij}=\frac{u_{ij}\,f^{\frac{1}{1-m}}\left(p_{ij}\right)}{\sum\limits_{k=1}^{c }u_{jk}\,f^{\frac{1}{1-m}}\left(p_{jk}\right)} \tag{18}\]  
Similar to FCM and RSK FCM, GSKFCM operates as an iterative process by updating membership and cluster center value. The cluster centers are updated using Equation (19)  
\[v_{i}=\frac{\sum\limits_{j=1}^{n}z_{ij}^{m}K\left(x_{j},v_{i}\right)x_{j}}{ \sum\limits_{j=1}^{n}z_{ij}^{m}K\left(x_{j},v_{i}\right)} \tag{19}\]  
GSKFCM decides the label based on the maximum membership value.