A modified intuitionistic fuzzy c-means clustering approach to segment human brain MRI image


Fuzzy c-means (FCM) is one of the prominent method utilized for medical image segmentation. In literature intuitionistic fuzzy c-means (IFCM) is suggested which is based on intuitionistic fuzzy sets (IFSs) theory to handle uncertainty and vagueness associated with real data. The objective function of which is defined using the hesitation degree along with membership degree. However, instead of solving the objective function analytically, the approximate solution is obtained using FCM. In this paper, we have proposed a modified intuitionistic fuzzy c-means algorithm (MIFCM) and solved analytically the objective function of the MIFCM method using Lagrange method of undetermined multiplier. To incorporate hesitation degree, two parametric intuitionistic fuzzy complements namely Sugeno’s negation function and Yager’s negation function are investigated. The performance of the MIFCM method is compared with three intuitionistic fuzzy clustering methods and the FCM on two publicly available MRI dataset and a synthetic dataset. The performance measures (average segmentation accuracy, dice score, jaccard score, false negative ratio and false positive ratio) are used to compare the performance of the MIFCM method with three variants of intuitionistic fuzzy clustering methods and the FCM. Experimental results demonstrate the superior performance of the MIFCM method over others.

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The authors would like to thank CSIR (Grant no. 09/263(1016)/2014-EMR-I) and DST PURSE for the financial support. The authors are also thankful to the anonymous reviewers for their constructive suggestions.

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Correspondence to Dhirendra Kumar.

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Appendix A: Derivation for the membership value and cluster center

Appendix A: Derivation for the membership value and cluster center

The Lagrangian for the objective function (18) can be given as

$$\begin{array}{@{}rcl@{}} L &=& {\sum}_{i = 1}^{c} \sum\limits_{j = 1}^{N} \left( \frac{(\lambda + 1)\mu_{ij}}{1+\lambda{\mu_{ij}}}\right)^{m}\| {\mathbf{x}}_{j}^{IFS} - {\mathbf{v}}_{i}^{IFS}\|^{2} + {\sum}_{j = 1}^{N} Y_{j} (1 - {\sum}_{i = 1}^{c} \mu_{ij}) \end{array} $$
$$\begin{array}{@{}rcl@{}} \frac{\partial L}{\partial \mu_{ij}} &=& m \left( \frac{(\lambda + 1)\mu_{ij}}{1+\lambda{\mu_{ij}}}\right)^{m-1} \left[\frac{(\lambda + 1)}{1+\lambda{\mu_{ij}}} - \frac{\lambda (\lambda + 1)\mu_{ij}}{(1+\lambda{\mu_{ij}})^{2}}\right] \left( \|{\mathbf{x}}_{j}^{IFS} - {\mathbf{v}}_{i}^{IFS}\|^{2} \right) - Y_{j} = 0\\ &\Rightarrow& m \left( \frac{(\lambda + 1)}{1+\lambda{\mu_{ij}}}\right)^{m} {\mu}_{ij}^{m-1} \left[ \frac{1}{(1+\lambda{\mu_{ij}})}\right] \left( \|{\mathbf{x}}_{j}^{IFS} - {\mathbf{v}}_{i}^{IFS}\|^{2} \right) - Y_{j} = 0\\ &\Rightarrow& {\mu}_{ij}^{m-1} = \left( \frac{Y_{j}}{m(\lambda + 1)^{m}}\right) \left[\frac{(1+\lambda{\mu_{ij}})^{m + 1}}{\left( \|{\mathbf{x}}_{j}^{IFS} - {\mathbf{v}}_{i}^{IFS}\|^{2} \right)}\right]\\ &\Rightarrow& \mu_{ij} = \left( \frac{Y_{j}}{m(\lambda + 1)^{m}}\right)^{\frac{1}{m-1}} \left[\frac{(1+\lambda{\mu_{ij}})^{\frac{m + 1}{m-1}}}{\left( \|{\mathbf{x}}_{j}^{IFS} - {\mathbf{v}}_{i}^{IFS}\|^{2} \right)^{\frac{1}{m-1}}}\right]\\ &\Rightarrow& {\sum}_{i = 1}^{c} \mu_{ij} = {\sum}_{i = 1}^{c} \left( \frac{Y_{j}}{m(\lambda + 1)^{m}}\right)^{\frac{1}{m-1}} \left[\frac{(1+\lambda{\mu_{ij}})^{\frac{m + 1}{m-1}}}{\left( \|{\mathbf{x}}_{j}^{IFS} - {\mathbf{v}}_{i}^{IFS}\|^{2} \right)^{\frac{1}{m-1}}}\right]= 1\end{array} $$
$$\begin{array}{@{}rcl@{}} &\Rightarrow& \left( \frac{Y_{j}}{m(\lambda + 1)^{m}}\right)^{\frac{1}{m-1}} = \frac{1}{\sum\limits_{i = 1}^{c} \frac{(1+\lambda{\mu_{ij}})^{\frac{m + 1}{m-1}}}{\left( \|{\mathbf{x}}_{j}^{IFS} - {\mathbf{v}}_{i}^{IFS}\|^{2} \right)^{\frac{1}{m-1}}}}\\ &\Rightarrow& \mu_{ij} = \frac{ \frac{(1+\lambda{\mu_{ij}})^{\frac{m + 1}{m-1}}}{\left( \|{\mathbf{x}}_{j}^{IFS} - {\mathbf{v}}_{i}^{IFS}\|^{2} \right)^{\frac{1}{m-1}}}}{\sum\limits_{l = 1}^{c} \frac{(1+\lambda{u_{lj}})^{\frac{m + 1}{m-1}}}{\left( \|{\mathbf{x}}_{j}^{IFS} - \mathbf{v}^{IFS}_{l}\|^{2}\right)^{\frac{1}{m-1}}}}\\ &\Rightarrow& \mu_{ij} = \frac{1}{\sum\limits_{l = 1}^{c} \left( \frac{\|{\mathbf{x}}_{j}^{IFS} - {{\mathbf{v}}_{i}^{IFS}}\|}{\|\mathbf{x}_{j}^{IFS} - {\mathbf{v}^{IFS}_{l}}\|}\right)^{\frac{2}{m-1}} \left( \frac{1 + \lambda u_{lj}}{1 + \lambda \mu_{ij}}\right)^{\frac{m + 1}{m-1}}} \end{array} $$
$$\begin{array}{@{}rcl@{}} \frac{\partial L}{\partial \mu_{V}(v_{i})} &=& {\sum}_{j = 1}^{N} \left( \frac{(\lambda + 1)\mu_{ij}}{1+\lambda{\mu_{ij}}}\right)^{m} (\mu_{X}(x_{j})-\mu_{V}(v_{i}))= 0 \end{array} $$
$$\begin{array}{@{}rcl@{}} &\Rightarrow& {\sum}_{j = 1}^{N} \left( \frac{(\lambda + 1)\mu_{ij}}{1+\lambda{\mu_{ij}}}\right)^{m} \mu_{X}(x_{j}) - {\sum}_{j = 1}^{N} \left( \frac{(\lambda + 1) \mu_{ij}}{1+\lambda{\mu_{ij}}}\right)^{m}\mu_{V}(v_{i})= 0\\ &\Rightarrow& \mu_{V}(v_{i}) = \frac{{\sum}_{j = 1}^{N} \left( \frac{(\lambda + 1) \mu_{ij}}{1+\lambda{\mu_{ij}}}\right)^{m} \mu_{X}(x_{j})}{{\sum}_{j = 1}^{N} \left( \frac{(\lambda + 1)\mu_{ij}}{1+\lambda{\mu_{ij}}}\right)^{m}} \end{array} $$
$$\begin{array}{@{}rcl@{}} \frac{\partial L}{\partial \nu_{V}(v_{i})} &=& {\sum}_{j = 1}^{N} \left( \frac{(\lambda + 1)\mu_{ij}}{1+\lambda{\mu_{ij}}}\right)^{m} (\nu_{X}(x_{j})-\nu_{V}(v_{i}))= 0 \end{array} $$
$$\begin{array}{@{}rcl@{}} &\Rightarrow& {\sum}_{j = 1}^{N} \left( \frac{(\lambda + 1)\mu_{ij}}{1+\lambda{\mu_{ij}}}\right)^{m} \nu_{X}(x_{j}) - {\sum}_{j = 1}^{N} \left( \frac{(\lambda + 1) \mu_{ij}}{1+\lambda{\mu_{ij}}}\right)^{m}\nu_{V}(v_{i})= 0\\ &\Rightarrow& \nu_{V}(v_{i}) = \frac{{\sum}_{j = 1}^{N} \left( \frac{(\lambda + 1)\mu_{ij}}{1+\lambda{\mu_{ij}}}\right)^{m} \nu_{X}(x_{j})}{{\sum}_{j = 1}^{N} \left( \frac{(\lambda + 1) \mu_{ij}}{1+\lambda{\mu_{ij}}}\right)^{m}} \end{array} $$
$$\begin{array}{@{}rcl@{}} \frac{\partial L}{\partial \pi_{V}(v_{i})} &=& {\sum}_{j = 1}^{N} \left( \frac{(\lambda + 1)\mu_{ij}}{1+\lambda{\mu_{ij}}}\right)^{m} (\pi_{X}(x_{j})-\pi_{V}(v_{i}))= 0 \end{array} $$
$$\begin{array}{@{}rcl@{}} &\Rightarrow& {\sum}_{j = 1}^{N} \left( \frac{(\lambda + 1)\mu_{ij}}{1+\lambda{\mu_{ij}}}\right)^{m} \pi_{X}(x_{j}) - {\sum}_{j = 1}^{N} \left( \frac{(\lambda + 1) \mu_{ij}}{1+\lambda{\mu_{ij}}}\right)^{m}\pi_{V}(v_{i})= 0\\ &\Rightarrow& \pi_{V}(v_{i}) = \frac{{\sum}_{j = 1}^{N} \left( \frac{(\lambda + 1)\mu_{ij}}{1+\lambda{\mu_{ij}}}\right)^{m} \pi_{X}(x_{j})}{{\sum}_{j = 1}^{N} \left( \frac{(\lambda + 1)\mu_{ij}}{1+\lambda{\mu_{ij}}}\right)^{m}} \end{array} $$

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Kumar, D., Verma, H., Mehra, A. et al. A modified intuitionistic fuzzy c-means clustering approach to segment human brain MRI image. Multimed Tools Appl 78, 12663–12687 (2019). https://doi.org/10.1007/s11042-018-5954-0

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  • Intuitionistic fuzzy sets
  • Fuzzy c-means
  • Intuitionistic fuzzy c-means
  • Hesitation degree
  • Image segmentation
  • Magnetic resonance imaging