# Unsupervised Image Classification Based on Fully Fuzzy Voronoi Tessellation
## 3 Proposed FVT-FCM Algorithm
### Parameter Estimation
Thus, we can obtain the solution of parameter \(\mu_{ij}\), that is,  
\[\mu_{ij}=\left(\sum\limits_{j=1}^{m}\frac{\left[d_{ij}^{2}+\eta\sum\limits_{ l=1}^{k}w_{jl}^{\lambda_{2}}\left(\mathbf{I}_{i}-\mathbf{V}_{l}\right)^{2}\right]^{ \frac{1}{\lambda_{1}-1}}}{\left[d_{ij}^{2}+\eta\sum\limits_{l=1}^{k}w_{jl}^{ \lambda_{2}}\left(\mathbf{I}_{i}-\mathbf{V}_{l}\right)^{2}\right]^{\frac{1}{\lambda_{1 }-1}}}\right)^{-1} \tag{30}\]  
(2) For parameter \(w_{jl}\),  
\[\frac{\partial L(\mathbf{\mu},\mathbf{W},\mathbf{C},\mathbf{V})}{\partial w_{jl}}=\eta\lambda_ {2}w_{jl}^{\lambda_{2}-1}{\sum\limits_{i=1}^{n}w_{ij}^{\lambda_{1}}\left(\mathbf{ I}_{i}-\mathbf{V}_{l}\right)^{2}}+\rho_{j}=0 \tag{31}\]  
Thus, we can obtain,  
\[w_{jl}=\frac{\left(-\rho_{j}\right)^{\frac{1}{\lambda_{2}-1}}}{ \left(\eta\lambda_{2}\sum\limits_{i=1}^{n}\mu_{ij}^{\lambda_{1}}\left(\mathbf{I}_ {i}-\mathbf{V}_{l}\right)^{2}\right)^{\frac{1}{\lambda_{2}-1}}} \tag{32}\]  
According to \(\sum\limits_{l=1}^{k}w_{jl}=1,j=1,\ldots,m\), then,\[\sum\limits_{l^{\prime}=1}^{k}\frac{\left(-\rho_{j}\right)\frac{1}{\lambda_{2}-1}}{ \left(\eta\lambda_{2}\sum\limits_{i=1}^{n}\mu_{ij}^{\lambda_{1}}(\mathbf{I}_{i}-\bm {V}_{l^{\prime}})^{2}\right)\overline{\lambda_{2}-1}}=1,\rightarrow\left(-\rho_ {j}\right)\frac{1}{\lambda_{2}-1}=\left[\sum\limits_{l^{\prime}=1}^{k}\left( \eta\lambda_{2}\sum\limits_{i=1}^{n}\mu_{ij}^{\lambda_{1}}(\mathbf{I}_{i}-\mathbf{V}_ {l^{\prime}})^{2}\right)^{-\frac{1}{\lambda_{2}-1}}\right]^{-1} \tag{33}\]  
Substitute Equation (33) into Equation (32). Thus, we can obtain the solution of parameter \(w_{jl}\), that is,  
\[w_{jl}=\left(\sum\limits_{l^{\prime}=1}^{k}\frac{D_{jl}\frac{1}{\lambda_{2}-1 }}{D_{jl^{\prime}}\frac{1}{\lambda_{2}-1}}\right)^{-1}=\left(\sum\limits_{l^ {\prime}=1}^{k}\frac{\left(\sum\limits_{i=1}^{n}\mu_{ij}^{\lambda_{1}}(\mathbf{I} _{i}-\mathbf{V}_{l})^{2}\right)\overline{\lambda_{2}-1}}{\left(\sum\limits_{i=1}^ {n}\mu_{ij}^{\lambda_{1}}(\mathbf{I}_{i}-\mathbf{V}_{l^{\prime}})^{2}\right)\overline{ \lambda_{2}-1}}\right)^{-1} \tag{34}\]  
(3) For parameter \(\mathbf{C}_{j}\), \(\mathbf{C}_{j}=(a_{j},b_{j})\)  
\[\frac{\partial L(\mathbf{\mu},\mathbf{W},\mathbf{C},\mathbf{V})}{\partial a_{j}}=\sum \limits_{i=1}^{n}\mu_{ij}^{\lambda_{1}}2(x_{i}-a_{j})=0 \tag{35}\]  
\[\frac{\partial L(\mathbf{\mu},\mathbf{W},\mathbf{C},\mathbf{V})}{\partial u_{ij}}=\lambda_{1} \mu_{ij}^{\lambda_{1}-1}d_{ij}^{2}+\eta\lambda_{1}\mu_{ij}^{\lambda_{1}-1}{ \sum\limits_{l=1}^{k}w_{jl}^{\lambda_{2}}\left(\mathbf{I}_{i}-\mathbf{V}_{l} \right)^{2}}+\gamma_{i}=0 \tag{27}\]0  
Thus, the solution of parameter \(\mathbf{C}_{j}\) is,  
\[\frac{\partial L(\mathbf{\mu},\mathbf{W},\mathbf{C},\mathbf{V})}{\partial u_{ij}}=\lambda_{1} \mu_{ij}^{\lambda_{1}-1}d_{ij}^{2}+\eta\lambda_{1}\mu_{ij}^{\lambda_{1}-1}{ \sum\limits_{l=1}^{k}w_{jl}^{\lambda_{2}}\left(\mathbf{I}_{i}-\mathbf{V}_{l} \right)^{2}}+\gamma_{i}=0 \tag{27}\]1  
(4) For parameter \(\mathbf{V}_{l}\),  
\[\frac{\partial L(\mathbf{\mu},\mathbf{W},\mathbf{C},\mathbf{V})}{\partial u_{ij}}=\lambda_{1} \mu_{ij}^{\lambda_{1}-1}d_{ij}^{2}+\eta\lambda_{1}\mu_{ij}^{\lambda_{1}-1}{ \sum\limits_{l=1}^{k}w_{jl}^{\lambda_{2}}\left(\mathbf{I}_{i}-\mathbf{V}_{l} \right)^{2}}+\gamma_{i}=0 \tag{27}\]2  
Thus, the solution of parameter \(\mathbf{V}_{l}\) is,  
\[\frac{\partial L(\mathbf{\mu},\mathbf{W},\mathbf{C},\mathbf{V})}{\partial u_{ij}}=\lambda_{1} \mu_{ij}^{\lambda_{1}-1}d_{ij}^{2}+\eta\lambda_{1}\mu_{ij}^{\lambda_{1}-1}{ \sum\limits_{l=1}^{k}w_{jl}^{\lambda_{2}}\left(\mathbf{I}_{i}-\mathbf{V}_{l} \right)^{2}}+\gamma_{i}=0 \tag{27}\]3  
Finally, considering the transitivity of fuzzy, the final label needs to be determined by both pixel level fuzzy membership \(\mu_{ij}\) and polygon level fuzzy membership \(w_{jl}\). According to the theory of fuzzy comprehensive decision analysis in the field of fuzzy mathematics, the matmul product is used in this paper.