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.

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References

  1. Alipour S, Shanbehzadeh J (2014) Fast automatic medical image segmentation based on spatial kernel fuzzy c-means on level set method. Mach Vis Appl 25(6):1469–1488

    Google Scholar 

  2. Anbeek P, Vincken KL, van Osch MJ, Bisschops RH, van der Grond J (2004) Automatic segmentation of different-sized white matter lesions by voxel probability estimation. Med Image Anal 8(3):205–215

    Google Scholar 

  3. Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96

    MATH  Google Scholar 

  4. Atanassov KT (2003) Intuitionistic fuzzy sets: past, present and future. In: EUSFLAT conference, pp 12–19

  5. Balafar M (2014) Fuzzy c-mean based brain mri segmentation algorithms. Artif Intell Rev 41(3):441–449

    Google Scholar 

  6. Balafar MA, Ramli AR, Saripan MI, Mashohor S (2010) Review of brain mri image segmentation methods. Artif Intell Rev 33(3):261–274

    Google Scholar 

  7. Benaichouche A, Oulhadj H, Siarry P (2013) Improved spatial fuzzy c-means clustering for image segmentation using pso initialization, mahalanobis distance and post-segmentation correction. Dig Signal Process 23(5):1390–1400

    MathSciNet  Google Scholar 

  8. Berry MW, Castellanos M (2004) Survey of text mining. Comput Rev 45(9):548

    Google Scholar 

  9. Bezdek JC (1981) Objective function clustering. In: Pattern recognition with fuzzy objective function algorithms. Springer, pp 43–93

  10. Bezdek JC, Keller JM, Krishnapuram R, Kuncheva LI, Pal NR (1999) Will the real iris data please stand up? IEEE Trans Fuzzy Syst 7(3):368–369

    Google Scholar 

  11. Bustince H, Kacprzyk J, Mohedano V (2000) Intuitionistic fuzzy generators application to intuitionistic fuzzy complementation. Fuzzy Sets Syst 114(3):485–504

    MathSciNet  MATH  Google Scholar 

  12. Chaira T (2011) A novel intuitionistic fuzzy c means clustering algorithm and its application to medical images. Appl Soft Comput 11(2):1711–1717

    Google Scholar 

  13. Chen S, Zhang D (2004) Robust image segmentation using fcm with spatial constraints based on new kernel-induced distance measure. IEEE Trans Syst Man Cybern Part B Cybern 34(4):1907–1916

    Google Scholar 

  14. Chen X, Nguyen BP, Chui CK, Ong SH (2016) Automated brain tumor segmentation using kernel dictionary learning and superpixel-level features. In: 2016 IEEE international conference on systems, man, and cybernetics (SMC). IEEE, pp 002547–002552

  15. Chuang KS, Tzeng HL, Chen S, Wu J, Chen TJ (2006) Fuzzy c-means clustering with spatial information for image segmentation. Comput Med Imaging Graph 30(1):9–15

    Google Scholar 

  16. Cocosco CA, Kollokian V, Kwan RKS, Pike GB, Evans AC (1997) Brainweb: online interface to a 3d mri simulated brain database. In: NeuroImage. Citeseer

  17. Cover TM (1965) Geometrical and statistical properties of systems of linear inequalities with applications in pattern recognition. IEEE Trans Electron Comput 3:326–334

    MATH  Google Scholar 

  18. Derrac J, García S, Molina D, Herrera F (2011) A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms. Swarm Evol Comput 1(1):3–18

    Google Scholar 

  19. Fisher RA (1936) The use of multiple measurements in taxonomic problems. Ann Hum Genet 7(2):179–188

    Google Scholar 

  20. Friedman M (1937) The use of ranks to avoid the assumption of normality implicit in the analysis of variance. J Am Stat Assoc 32(200):675–701

    MATH  Google Scholar 

  21. Graves D, Pedrycz W (2010) Kernel-based fuzzy clustering and fuzzy clustering: a comparative experimental study. Fuzzy Sets Syst 161(4):522–543

    MathSciNet  Google Scholar 

  22. Hai-Jun F, Xiao-Hong W, Han-Ping M, Bin W (2011) Fuzzy entropy clustering using possibilistic approach. Procedia Eng 15:1993–1997

    Google Scholar 

  23. Held K, Kops ER, Krause BJ, Wells WM, Kikinis R, Muller-Gartner HW (1997) Markov random field segmentation of brain mr images. IEEE Trans Med Imaging 16(6):878–886

    Google Scholar 

  24. Hoffman EJ, Huang SC, Phelps ME (1979) Quantitation in positron emission computed tomography: 1.Effect of object size. J Comput Assist Tomogr 3(3):299–308

    Google Scholar 

  25. Huang CW, Lin KP, Wu MC, Hung KC, Liu GS, Jen CH (2015) Intuitionistic fuzzy c-means clustering algorithm with neighborhood attraction in segmenting medical image. Soft Comput 19(2):459–470

    Google Scholar 

  26. Iakovidis D, Pelekis N, Kotsifakos E, Kopanakis I (2008) Intuitionistic fuzzy clustering with applications in computer vision. In: Advanced concepts for intelligent vision systems. Springer, pp 764–774

  27. Iman RL, Davenport JM (1980) Approximations of the critical region of the fbietkan statistic. Commun Stat Theory Methods 9(6):571–595

    MATH  Google Scholar 

  28. Jain AK, Murty MN, Flynn PJ (1999) Data clustering: a review. ACM Comput Surv CSUR 31(3):264–323

    Google Scholar 

  29. Ji ZX, Sun QS, Xia DS (2014) A framework with modified fast fcm for brain mr images segmentation (retraction of vol 44, pg 999, 2011). Pattern Recognit 47(12):3979–3979

    Google Scholar 

  30. Kahali S, Sing JK, Saha PK (2018) A new fuzzy clustering algorithm for brain mr image segmentation using gaussian probabilistic and entropy-based likelihood measures. In: 2018 international conference on communication, computing and internet of things (IC3IoT). pp 54–59. https://doi.org/10.1109/IC3IoT.2018.8668139

  31. Kannan S, Devi R, Ramathilagam S, Takezawa K (2013) Effective fcm noise clustering algorithms in medical images. Comput Biol Med 43(2):73–83

    Google Scholar 

  32. Krinidis S, Chatzis V (2010) A robust fuzzy local information c-means clustering algorithm. IEEE Trans Image Process 19(5):1328–1337

    MathSciNet  MATH  Google Scholar 

  33. Krishnapuram R, Keller JM (1993) A possibilistic approach to clustering. IEEE Trans Fuzzy Syst 1(2):98–110

    Google Scholar 

  34. Kumar D, Verma H, Mehra A, Agrawal RK (2018) A modified intuitionistic fuzzy c-means clustering approach to segment human brain mri image. Multimed Tools Appl. https://doi.org/10.1007/s11042-018-5954-0

    Article  Google Scholar 

  35. Li RP, Mukaidono M (1999) Gaussian clustering method based on maximum-fuzzy-entropy interpretation. Fuzzy Sets Syst 102(2):253–258

    MathSciNet  MATH  Google Scholar 

  36. Li C, Huang R, Ding Z, Gatenby JC, Metaxas DN, Gore JC (2011) A level set method for image segmentation in the presence of intensity inhomogeneities with application to mri. IEEE Trans Image Process 20(7):2007–2016

    MathSciNet  MATH  Google Scholar 

  37. Li C, Zhao H, Xu Z (2018) Kernel c-means clustering algorithms for hesitant fuzzy information in decision making. Int J Fuzzy Syst 20(1):141–154

    MathSciNet  Google Scholar 

  38. Lin KP (2013) A novel evolutionary kernel intuitionistic fuzzy \( c \)-means clustering algorithm. IEEE Trans Fuzzy Syst 22(5):1074–1087

    Google Scholar 

  39. Lloyd S (1982) Least squares quantization in pcm. IEEE Trans Inf Theory 28(2):129–137

    MathSciNet  MATH  Google Scholar 

  40. Mercer J (1909) Xvi. functions of positive and negative type, and their connection the theory of integral equations. Philos Trans R Soc Lond A 209(441–458):415–446

    MATH  Google Scholar 

  41. Muller KR, Mika S, Ratsch G, Tsuda K, Scholkopf B (2001) An introduction to kernel-based learning algorithms. IEEE Trans Neural Netw 12(2):181–201

    Google Scholar 

  42. Murofushi T, Sugeno M (2000) Fuzzy measures and fuzzy integrals. In: Grabisch M, Murofushi T, Sugeno M (eds) Fuzzy measures and integrals. Theory and applications. Heidelberg, Physica-Verlag, pp 3–41

    Google Scholar 

  43. Nayak J, Naik B, Behera H (2015) Fuzzy c-means (fcm) clustering algorithm: a decade review from 2000 to 2014. In: Computational intelligence in data mining-volume 2. Springer, pp 133–149

  44. Olabarriaga SD, Smeulders AW (2001) Interaction in the segmentation of medical images: a survey. Med Image Anal 5(2):127–142

    Google Scholar 

  45. Pal NR, Pal SK (1992) Higher order fuzzy entropy and hybrid entropy of a set. Inf Sci 61(3):211–231

    MathSciNet  MATH  Google Scholar 

  46. Pal NR, Pal K, Keller JM, Bezdek JC (2005) A possibilistic fuzzy c-means clustering algorithm. IEEE Trans Fuzzy Syst 13(4):517–530

    Google Scholar 

  47. Pelekis N, Iakovidis DK, Kotsifakos EE, Kopanakis I (2008) Fuzzy clustering of intuitionistic fuzzy data. Int J Bus Intell Data Min 3(1):45–65

    Google Scholar 

  48. Pham DL, Xu C, Prince JL (2000) Current methods in medical image segmentation. Ann Rev Biomed Eng 2(1):315–337

    Google Scholar 

  49. Pham TX, Siarry P, Oulhadj H (2018) Integrating fuzzy entropy clustering with an improved pso for mri brain image segmentation. Appl Soft Comput 65:230–242

    Google Scholar 

  50. Qiu C, Xiao J, Yu L, Han L, Iqbal MN (2013) A modified interval type-2 fuzzy c-means algorithm with application in mr image segmentation. Pattern Recognit Lett 34(12):1329–1338

    Google Scholar 

  51. Reddick WE, Glass JO, Cook EN, Elkin TD, Deaton RJ (1997) Automated segmentation and classification of multispectral magnetic resonance images of brain using artificial neural networks. IEEE Trans Med Imaging 16(6):911–918

    Google Scholar 

  52. Rohlfing T, Brandt R, Menzel R, Maurer CR (2004) Evaluation of atlas selection strategies for atlas-based image segmentation with application to confocal microscopy images of bee brains. NeuroImage 21(4):1428–1442

    Google Scholar 

  53. Rui Y, Huang TS, Chang SF (1999) Image retrieval: current techniques, promising directions, and open issues. J Vis Commun Image Represent 10(1):39–62

    Google Scholar 

  54. Sato M, Lakare S, Wan M, Kaufman A, Nakajima M (2000) A gradient magnitude based region growing algorithm for accurate segmentation. In: 2000 International Conference on image processing, 2000. Proceedings, vol 3. IEEE, pp 448–451

  55. Schaaf T, Kemp T (1997) Confidence measures for spontaneous speech recognition. In: International conference on acoustics, speech, and signal processing, 1997. ICASSP-97, IEEE, vol 2. IEEE, pp 875–878

  56. Szmidt E, Kacprzyk J (2000) Distances between intuitionistic fuzzy sets. Fuzzy Sets Syst 114(3):505–518

    MathSciNet  MATH  Google Scholar 

  57. Tran D, Wagner M (2000) Fuzzy entropy clustering. In: The ninth IEEE international conference on fuzzy systems, 2000. FUZZ IEEE 2000, vol 1. IEEE, pp 152–157

  58. Verma H, Agrawal R (2015) Possibilistic intuitionistic fuzzy c-means clustering algorithm for mri brain image segmentation. Int J Artif Intell Tools 24(05):1550,016

    Google Scholar 

  59. Verma H, Agrawal RK, Kumar N (2014) Improved fuzzy entropy clustering algorithm for mri brain image segmentation. Int J Imaging Syst Technol 24(4):277–283

    Google Scholar 

  60. Verma H, Agrawal R, Sharan A (2016) An improved intuitionistic fuzzy c-means clustering algorithm incorporating local information for brain image segmentation. Appl Soft Comput 46:543–557

    Google Scholar 

  61. Vlachos IK, Sergiadis GD (2007) The role of entropy in intuitionistic fuzzy contrast enhancement. In: International fuzzy systems association world congress. Springer, pp 104–113

  62. Vovk U, Pernus F, Likar B (2007) A review of methods for correction of intensity inhomogeneity in mri. IEEE Trans Med Imaging 26(3):405–421

    Google Scholar 

  63. Wang L, Chen Y, Pan X, Hong X, Xia D (2010) Level set segmentation of brain magnetic resonance images based on local gaussian distribution fitting energy. J Neurosci Methods 188(2):316–325

    Google Scholar 

  64. Wang Z, Song Q, Soh YC, Sim K (2013) An adaptive spatial information-theoretic fuzzy clustering algorithm for image segmentation. Computer Vis Image Underst 117(10):1412–1420

    Google Scholar 

  65. Xu Z, Wu J (2010) Intuitionistic fuzzy c-means clustering algorithms. J Syst Eng Electron 21(4):580–590

    Google Scholar 

  66. Yager RR (1979) On the measure of fuzziness and negation part i: membership in the unit interval. Int J General Syst 5(4):221–229

    MATH  Google Scholar 

  67. Yager RR (1980) On the measure of fuzziness and negation. ii. Lattices. Inf Control 44(3):236–260

    MathSciNet  MATH  Google Scholar 

  68. Yager RR (2000) On the entropy of fuzzy measures. IEEE Trans Fuzzy Syst 8(4):453–461

    MathSciNet  Google Scholar 

  69. Yang MS, Wu KL (2006) Unsupervised possibilistic clustering. Pattern Recognit 39(1):5–21

    Google Scholar 

  70. Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353

    MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Department of Science and Technology (PURSE), New Delhi, and Council of Scientific & Industrial Research, New Delhi, 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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Derivation for partition matrix and cluster centroid

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)

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Kumar, D., Agrawal, R.K. & Verma, H. Kernel intuitionistic fuzzy entropy clustering for MRI image segmentation. Soft Comput 24, 4003–4026 (2020). https://doi.org/10.1007/s00500-019-04169-y

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Keywords

  • Intuitionistic fuzzy sets
  • Fuzzy entropy clustering
  • Kernel distance measure
  • Image segmentation
  • Magnetic resonance imaging