Kernel intuitionistic fuzzy entropy clustering for MRI image segmentation

Abstract

Fuzzy entropy clustering (FEC) is a variant of hard c-means clustering which utilizes the concept of entropy. However, the performance of the FEC method is sensitive to the noise and the fuzzy entropy parameter as it gives incorrect clustering and coincident cluster sometimes. In this work, a variant of the FEC method is proposed which incorporates advantage of intuitionistic fuzzy set and kernel distance measure termed as kernel intuitionistic fuzzy entropy c-means (KIFECM). While intuitionistic fuzzy set allows to handle uncertainty and vagueness associated with data, kernel distance measure helps to reveal the inherent nonlinear structures present in data without increasing the computational complexity. In this work, two popular intuitionistic fuzzy sets generators, Sugeno and Yager’s negation function, have been utilized for generating intuitionistic fuzzy sets corresponding to data. The performance of the proposed method has been evaluated over two synthetic datasets, Iris dataset, publicly available simulated human brain MRI dataset and IBSR real human brain MRI dataset. The experimental results show the superior performance of the proposed KIFECM over FEC, FCM, IFCM, UPCA, PTFECM and KFEC in terms of several performance measures such as partition coefficient, partition entropy, average segmentation accuracy, dice score, Jaccard score, false positive ratio and false negative ratio.

Derivation for partition matrix and cluster centroid

The Lagrangian for the objective function (23) of the proposed KIFECM method can be given as:

$$\begin{aligned} L= & {} \mathop \sum \nolimits _{i=1}^c \mathop \sum \nolimits _{j=1}^N\mu _{ij}(1 - K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_i)) \nonumber \\&+\, \frac{\sigma ^2}{c}\sum _{i=1}^c \mathop \sum \nolimits _{j=1}^N(\mu _{ij}\log \mu _{ij} - \mu _{ij}) \nonumber \\&+\, \lambda \mathop \sum \nolimits _{i=1}^c \mathop \sum \nolimits _{j=1}^N\mu _{ij}\log \mu _{ij} \nonumber \\&+\,\mathop \sum \nolimits _{i=1}^NY_j\left( 1-\mathop \sum \nolimits _{i=1}^c\mu _{ij}\right) \end{aligned}$$

(33)

where $Y_j$ is Lagrange multiplier $\forall j \in \{1,2, \dots , N\}$. The solution can be obtained after differentiating above with respect to $\mu _{ij}$, $\mu _{V}(v_{i})$, $\nu _{V}(v_{i})$ and $\pi _{V}(v_{i})$.

$$\begin{aligned}&\frac{\partial L}{\partial \mu _{ij}} =(1 - K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_i)) + \frac{\sigma ^2}{c}\log \mu _{ij} \nonumber \\&\qquad +\, \lambda (1 + \log \mu _{ij}) - Y_j = 0 \nonumber \\&\quad \Rightarrow \, c(1 - K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_i)) +(\sigma ^2 + c\lambda )\log \mu _{ij} \nonumber \\&\qquad +\,c\lambda - cY_j = 0 \nonumber \\&\quad \, \Rightarrow \, (\sigma ^2 + c\lambda )\log \mu _{ij} = cY_j - c(1 - K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_i))-c\lambda \nonumber \\&\quad \, \Rightarrow \, (\sigma ^2 + c\lambda )\log \mu _{ij} = cY_j - c[(1 - K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_i)) + \lambda ]\nonumber \\&\quad \,\Rightarrow \, \log \mu _{ij} = \frac{cY_j}{(\sigma ^2 + c\lambda )} -\frac{ c[(1 - K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_i)) + \lambda ]}{(\sigma ^2 + c\lambda )} \nonumber \\&\quad \Rightarrow \, \mu _{ij} = \exp \left( \frac{cY_j}{(\sigma ^2 + c\lambda )}\right) \nonumber \\&\qquad \exp \left( -\frac{ c[(1 - K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_i)) + \lambda ]}{(\sigma ^2 + c\lambda )}\right) \nonumber \\&\qquad {\mathrm{Elimination}}\ {\mathrm{of}}\ Y_j\ {\mathrm{with}}\ {\mathrm{the}}\ {\mathrm{constraint}}\ {\mathrm{will}}\ {\mathrm{leads}}\ {\mathrm{to}}\nonumber \\ \end{aligned}$$

(34)

$$\begin{aligned}&\quad \Rightarrow \, \mu _{ij}=\frac{\exp \left( -\frac{c[(1 - K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_i)) + \lambda ]}{(\sigma ^2 + c\lambda )}\right) }{\sum _{k=1}^c\exp \left( -\frac{c[(1 - K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_k)) + \lambda ]}{(\sigma ^2 + c\lambda )}\right) } \nonumber \\&\quad \,\Rightarrow \, \mu _{ij}=\frac{\exp \left( -\frac{[(1 - K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_i)) + \lambda ]}{\left( \frac{\sigma ^2}{c} + \lambda \right) }\right) }{\sum _{k=1}^c\exp \left( -\frac{[(1 - K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_k)) + \lambda ]}{\left( \frac{\sigma ^2}{c} + \lambda \right) }\right) } \nonumber \\&\quad \, \Rightarrow \, \mu _{ij}=\frac{\exp (-[(1 - K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_i)) + \lambda ])^{\frac{1}{\left( \frac{\sigma ^2}{c} + \lambda \right) }}}{\sum _{k=1}^c\exp (-[(1 - K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_k)) + \lambda ])^{\frac{1}{\left( \frac{\sigma ^2}{c} + \lambda \right) }}} \nonumber \\&\quad \, \Rightarrow \, \mu _{ij}=\frac{1}{\sum _{k=1}^c\frac{\exp ([(1 - K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_i)) + \lambda ])^{\frac{1}{\left( \frac{\sigma ^2}{c} + \lambda \right) }}}{\exp ([(1 - K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_k)) + \lambda ])^{\frac{1}{\left( \frac{\sigma ^2}{c} + \lambda \right) }}}} \nonumber \\&\quad \,\Rightarrow \mu _{ij} = \left\{ \mathop \sum \nolimits _{k=1}^c\left( \frac{\exp (1 - K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_i)+ \lambda )}{\exp (1 - K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_k)+\lambda )}\right) ^\frac{1}{\frac{\sigma ^2}{c}+\lambda }\right\} ^{-1}\nonumber \\ \end{aligned}$$

(35)

$$\begin{aligned}&\quad \Rightarrow \mu _{ij} = \left\{ \mathop \sum \nolimits _{k=1}^c\left( \frac{\exp (1 - K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_i))}{\exp (1 - K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_k))}\right) ^\frac{1}{\frac{\sigma ^2}{c}+\lambda }\right\} ^{-1}\nonumber \\ \end{aligned}$$

(36)

$$\begin{aligned}&\frac{\partial L}{\partial \mu _{V}(v_{i})} =\ \mathop \sum \nolimits _{j=1}^N\mu _{ij}K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_i)(\mu _{X}(x_j)\nonumber \\&\qquad -\mu _{V}(v_i))=0 \nonumber \\&\quad \Rightarrow \, \mathop \sum \nolimits _{j=1}^N\mu _{ij}K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_i)\mu _{X}(x_j)\nonumber \\&\qquad -\mathop \sum \nolimits _{j=1}^N\mu _{ij}K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_i)\mu _{V}(v_i)=0 \end{aligned}$$

(37)

$$\begin{aligned}&\quad \Rightarrow \, \mu _{V}(v_i)=\frac{\sum _{j=1}^N\mu _{ij}(K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_i)\mu _{X}(x_j) )}{\sum _{j=1}^N\mu _{ij}(K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_i) )} \end{aligned}$$

(38)

$$\begin{aligned}&\frac{\partial L}{\partial \nu _{V}(v_{i})} =\, \mathop \sum \nolimits _{j=1}^N\mu _{ij}K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_i)(\nu _{X}(x_j)-\nu _{V}(v_i))=0 \nonumber \\&\quad \Rightarrow \, \mathop \sum \nolimits _{j=1}^N\mu _{ij}K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_i)\nu _{X}(x_j)\nonumber \\&\qquad -\mathop \sum \nolimits _{j=1}^N\mu _{ij}K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_i)\nu _{V}(v_i)=0 \end{aligned}$$

(39)

$$\begin{aligned}&\quad \Rightarrow \, \nu _{V}(v_i)=\frac{\sum _{j=1}^N\mu _{ij}(K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_i)\nu _{X}(x_j) )}{\sum _{j=1}^N\mu _{ij}(K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_i))} \end{aligned}$$

(40)

$$\begin{aligned}&\frac{\partial L}{\partial \pi _{V}(v_{i})} {=}\,\mathop \sum \nolimits _{j=1}^N\mu _{ij}K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_i)(\pi _{X}(x_j){-}\pi _{V}(v_i)){=}0 \nonumber \\&\quad \Rightarrow \, \mathop \sum \nolimits _{j=1}^N\mu _{ij}K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_i)\pi _{X}(x_j)\nonumber \\&\qquad -\mathop \sum \nolimits _{j=1}^N\mu _{ij}K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_i)\pi _{V}(v_i)=0 \end{aligned}$$

(41)

$$\begin{aligned}&\quad \Rightarrow \, \pi _{V}(v_i)=\frac{\sum _{j=1}^N\mu _{ij}(K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_i)\pi _{X}(x_j) )}{\sum _{j=1}^N\mu _{ij}(K({\mathbf {x}}^{\mathrm{IFS}}_j,{\mathbf {v}}^{\mathrm{IFS}}_i) )} \end{aligned}$$

(42)