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Superpixel-based brain tumor segmentation in MR images using an extended local fuzzy active contour model

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Abstract

In this paper, to deal with poor boundaries in the presence of noise and heterogeneity of magnetic resonance (MR) images, a new region-based fuzzy active contour model based on techniques of curve evolution is introduced for the brain tumor segmentation. On the other hand, since brain MR images intrinsically contain significant amounts of dark areas such as cerebrospinal fluid, therefore for properly declining the heterogeneity of classes and better segmentation results, the proposed fuzzy energy-based function has been extended to consider three distinct regions; target, dark tissues with a dark background and the rest of the foreground. Moreover, due to the inevitable dependency of pixel-based models on the initial contour, artifact, and inhomogeneity of MR images, we have used superpixels as basic atomic units not only to reduce the sensitivity to the mentioned factors but also to reduce the computational cost of the algorithm. Results show that the proposed method outperforms the accuracy of the state-of-the-art models in both real and synthetic brain MR images.

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References

  1. Abbasi S, Tajeripour F (2017) Detection of brain tumor in 3D MRI images using local binary patterns and histogram orientation gradient. Neurocomputing 219:526–535

    Google Scholar 

  2. Abood LK et al (2015) Automatic brain tumor segmentation from MRI images using region growing algorithm. Int J Sci Res 6(5):1592–1595

    Google Scholar 

  3. Achanta R, Shaji A, Smith K, Lucchi A, Fua P, Süsstrunk S (2012) Slic superpixels compared to state-of-the-art superpixel methods. IEEE Trans Pattern Anal Mach Intell (TPAMI) 34(11):2274–2282

    Google Scholar 

  4. Amirmoezzi Y, Salehi S, Parsaei H, Kazemi K, Torabi Jahromi A (2019) A knowledge-based system for brain tumor segmentation using only 3D FLAIR images. Australas Phys Eng Sci Med 42(2):529–540

    Google Scholar 

  5. Anitha V, Murugavalli S (2016) Brain tumour classificationusing two-tier classifier with adaptive segmentation technique. IET Comput Vis 10(1):9–17

    Google Scholar 

  6. Bakas S, Akbari H, Sotiras A, Bilello M, Rozycki M, Kirby JS, Freymann JB, Farahani K, Davatzikos C (2017) Advancing the cancer genome atlas glioma mri collections with expert segmentation labels and radiomic features. Sci Data 4:170117

    Google Scholar 

  7. Bakas S, Reyes M, Jakab A, Bauer S, Rempfler M, Crimi A, et al (2018) Identifying the Best Machine Learning Algorithms for Brain Tumor Segmentation, Progression Assessment, and Overall Survival Prediction in the BRATS Challenge. arXiv preprint arXiv:1811.02629

  8. Bauer S et al (2011) Fully automatic segmentation of brain tumor images using support vector machine classification in combination with hierarchical conditional random field regularization. Med. Image Comput. Comput.-Assisted Intervent -MICCAI, Springer, pp, 354–361

  9. Caselles V et al (1977) Geodesic active contours. Int J Comput Vis 22(1):61–79

    MATH  Google Scholar 

  10. Chan T, Vese L (2001) Active contours without edges. IEEE Trans Image Process 10(2):266–277

    MATH  Google Scholar 

  11. Chithra PL, Dheepa G (2020) Di-phase midway convolution and deconvolution network for brain tumor segmentation in MRI images. Int J Imaging Syst Technol:1–13

  12. Cordier N et al (2013) Patch-based segmentation of brain tissues. In: Proc. MICCAI-BRATS

  13. Demirhan A, Toru M, Guler I (2015) Segmentation of tumor and edema along with healthy tissues of brain using wavelets and neural networks. IEEE J Biomed Health Inf 19(4):1451–1458

    Google Scholar 

  14. Dice LR (1945) Measures of the amount of ecologic association between species. Ecology 26:297–302

    Google Scholar 

  15. Fang J, Liu H, Liu H, Zhang L, Liu J (2016) Localized patch-based fuzzy active contours for image segmentation. Comput Math Methods Med 2016:1–14

    MATH  Google Scholar 

  16. Festa J et al (2013) Automatic brain tumor segmentation of multisequence mr images using random decision forests. In: Proc. MICCAI-BRATS, pp 23–26

  17. Ibrahim RW, Hasan AM, Jalab HA (2018) A new deformable model based on fractional Wright energy function for tumor segmentation of volumetric brain MRI scans. Computer Methods Programs Biomed 163:21–28

    Google Scholar 

  18. Ilhan U, Ilhan A (2017) Brain tumor segmentation based on a new threshold approach. Procedia Comput Sci 120:580–587

    Google Scholar 

  19. Ilunga-Mbuyamba E, Avina–Cervantes JG, Garcia–Perez A, Romero–Troncoso RJ, Aguirre–Ramos H, Cruz–Aceves I, Chalopin C (2017) Localized active contour model with background compensation applied on automatic MR brain tumor segmentation. Neurocomputing 220:84–97

    Google Scholar 

  20. Iscan Z, Dokur Z, Ölmez T (2010) Tumor detection by using Zernike moments on segmented magnetic resonance brain images. Expert Syst Appl 37(3):2540–2549

    Google Scholar 

  21. Jaccard P (1908) Nouvelles recherches sur la distribution florale. Bulletin de la Societe Vaudoise des Sciences Naturelles 44:223–270

    Google Scholar 

  22. Jiang J, Wu Y, Huang M, Yang W, Chen W, Feng Q (2013) 3D brain tumor segmentation in multimodal MR images based on learning population-and patient-specific feature sets. Computer Med Imag Graph 37(7–8):512–521

    Google Scholar 

  23. Kass M, Witkin A, Terzopoulos D (1988) Snakes: active contour model. Int J Comput Vis 1(4):321–331

    Google Scholar 

  24. Kermi A, Andjouh K, Zidane F (2018) Fully automated brain tumour segmentation system in 3D-MRI using symmetry analysis of brain and level sets. IET Image Process 12(11):1964–1971

    Google Scholar 

  25. Khotanlou H et al (2007) Automatic brain tumor segmentation using symmetry analysis and deformable models. Int. conf. on adv. in pattern recognit. ICAPR, Kolkata, India, pp 198–202

  26. Kistler M et al (2013) The virtual skeleton database: an open access repository for biomedical research and collaboration. J Med Internet Res 15(11)

  27. Krinidis S, Chatzis V (2009) Fuzzy energy-based active contour. IEEE Trans Image Process 18(12):2747–2755

    MathSciNet  MATH  Google Scholar 

  28. Lankton S, Tannenbaum A (2008) Localizing region-based active contours. IEEE Trans Image Process 17(11):2029–2039

    MathSciNet  MATH  Google Scholar 

  29. Lok KH et al (2017) Fast and robust brain tumor segmentation using level set method with multiple image information. J X-ray Sci Technol 25(2):1–12

    MathSciNet  Google Scholar 

  30. Lv H, Wang Z, Fu S, Zhang C, Zhai L, Liu X (2017) A robust active contour segmentation based on fractional-order differentiation and fuzzy energy. IEEE Access 5:7753–7761

    Google Scholar 

  31. Meier R et al (2013) A hybrid model for multimodal brain tumor segmentation. In: Proc. MICCAI BRATS, pp 31–37

  32. Menze BH, Jakab A, Bauer S, Kalpathy-Cramer J, Farahani K, Kirby J, Burren Y, Porz N, Slotboom J, Wiest R, Lanczi L, Gerstner E, Weber MA, Arbel T, Avants BB, Ayache N, Buendia P, Collins DL, Cordier N, Corso JJ, Criminisi A, Das T, Delingette H, Demiralp C, Durst CR, Dojat M, Doyle S, Festa J, Forbes F, Geremia E, Glocker B, Golland P, Guo X, Hamamci A, Iftekharuddin KM, Jena R, John NM, Konukoglu E, Lashkari D, Mariz JA, Meier R, Pereira S, Precup D, Price SJ, Raviv TR, Reza SMS, Ryan M, Sarikaya D, Schwartz L, Shin HC, Shotton J, Silva CA, Sousa N, Subbanna NK, Szekely G, Taylor TJ, Thomas OM, Tustison NJ, Unal G, Vasseur F, Wintermark M, Ye DH, Zhao L, Zhao B, Zikic D, Prastawa M, Reyes M, van Leemput K (2015) The multimodal brain tumor image segmentation benchmark. IEEE Trans Med Imag 34(10):1993–2024

    Google Scholar 

  33. Mumford D, Shah J (1989) Optimal approximation by piecewise smooth function and associated variational problems. Commun Pure Appl Math 42(5):577–685

    MathSciNet  MATH  Google Scholar 

  34. Otsu N (1979) A threshold selection method from gray-level histograms. IEEE Trans Syst Man Cybern 9(1):62–66

    Google Scholar 

  35. Perona P, Malik J (1990) Scale-space and edge detection using anisotropic diffusion. IEEE Trans Pattern Anal Mach Intell 12(7):629–639

    Google Scholar 

  36. Sachdeva J, Kumar V, Gupta I, Khandelwal N, Ahuja CK (2012) A novel content-based active contour model for brain tumor segmentation. Magnetic Reson Imag 30(5):694–715

    Google Scholar 

  37. Saha BN, Ray N, Greiner R, Murtha A, Zhang H (2012) Quick detection of brain tumors and edemas: a bounding box method using symmetry. Computer Med Imag Graph 36(2):95–107

    Google Scholar 

  38. Sethian JA (1995) Theory, algorithms and applications of level set methods for propagating interfaces. Acta Numerica 5:309–395

    MathSciNet  MATH  Google Scholar 

  39. Sheela CJJ, Suganthi G (2020) Morphological edge detection and brain tumor segmentation in magnetic resonance (MR) images based on region growing and performance evaluation of modified fuzzy C-means (FCM) algorithm. Multimedia Tools Appl 79(25):17483–17496

    Google Scholar 

  40. Shivhare SN, Kumar N, Singh N (2019) A hybrid of active contour model and convex hull for automated brain tumor segmentation in multimodal MRI. Multimedia Tools Appl 78(24):34207–34229

    Google Scholar 

  41. Shyu K et al (2012) Global and local fuzzy energy-based active contours for image segmentation. Nonlinear Dynam 67(2):1559–1578

    MathSciNet  MATH  Google Scholar 

  42. Soltaninejad M, Yang G, Lambrou T, Allinson N, Jones TL, Barrick TR, Howe FA, Ye X (2017) Automated brain tumour detection and segmentation using superpixel-based extremely randomized trees in FLAIR MRI. Int J Comput Assist Radiol Surg 12(2):183–203

    Google Scholar 

  43. Song B, Chan T (2002) A Fast Algorithm for Level Set Based Optimization. Univ. California, Los Angeles, Tech. Rep. CAM 02–68

  44. Sun L et al (2019) Brain tumor segmentation and survival prediction using multimodal MRI scans with deep learning. Front Neurosci 13(810)

  45. Tarkhaneh O, Shen H (2019) An Adaptive differential evolution algorithm to optimal multi-level thresholding for MRI brain image segmentation. Expert Syst Appl 138

  46. Thaha MM et al (2019) Brain tumor segmentation using convolutional neural networks in MRI images. J Med Syst 43(294)

  47. Usman K, Rajpoot K (2017) Brain tumor classification from multimodality MRI using wavelets and machine learning. Pattern Anal Appl 20(3):871–881

    MathSciNet  Google Scholar 

  48. Wadhwa A, Bhardwaj A, Singh Verma V (2019) A review on brain tumor segmentation of MRI images. Magnetic Reson Imag 61:247–259

    Google Scholar 

  49. Wang T et al (2009) Fluid vector flow and applications in brain tumor segmentation. IEEE Trans Biomed Eng 56(3):781–789

    Google Scholar 

  50. Wu Y, Ma W, Gong M, Li H, Jiao L (2015) Novel fuzzy active contour model with kernel metric for image segmentation. Appl Soft Comput 34:301–311

    Google Scholar 

  51. Xu C, Prince J (1998) Snakes, shapes and gradient vector flow. IEEE Trans Image Process 7(3):359–369

    MathSciNet  MATH  Google Scholar 

  52. Yezzi JA et al (2002) A fully global approach to image segmentation via coupled curve evolution equations. J Vis Comm Image Rep 13(1):195–216

    Google Scholar 

  53. Yu C et al (2012) Statistical asymmetry-based brain tumor segmentation from 3D MR images. In Proceedings of the Int Conf on Bio-inspired Sys and Signal Process, pp 527–533

  54. Zeineldin RA et al (2020) DeepSeg: deep neural network framework for automatic brain tumor segmentation using magnetic resonance FLAIR images. Int J Comput Assist Radiol Surg 15(9)

  55. Zhao L et al (2013) Automatic brain tumor segmentation with mef on supervoxels. In: Proc. MICCAI-BRATS, pp 51–54.

  56. Zhao X et al (2017) A deep learning model integrating FCNNs and CRFs for brain tumor segmentation. Med Image Anal 43:98–111

    Google Scholar 

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Authors and Affiliations

  1. Department of Electrical Engineering, University of Guilan, Rasht, Iran

    Niloufar Alipour & Reza P. R. Hasanzadeh

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  1. Niloufar Alipour
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  2. Reza P. R. Hasanzadeh
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Correspondence to Reza P. R. Hasanzadeh.

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Appendix

Appendix

This appendix contains the proof of energy differences (16) used by the proposed algorithm. Let us Consider a pixel, P, with intensity value I(P) and membership degrees \( {u}_{o_1} \) and \( {u}_{o_2} \). If we calculate the new membership degrees \( {u}_{n_1} \) and \( {u}_{n_2} \) using (15) for the point P and change its old membership degrees to the new values, the values of vi, i = 1,2,3 will be changed to new ones: \( \tilde{v}_{i},i \) = 1,2,3. The new values of vi, i = 1,2,3 can be calculated as:

$$ \tilde{v}_{i}(x)=\frac{\int_{\varOmega_y}W\left(x,y\right).{u}_i{(y)}^m\ I(y)\ dy}{\int_{\varOmega_y}W\left(x,y\right).{u}_i{(y)}^m\ dy}=\kern0.5em \frac{\sum_{\varOmega_y}W\left(x,y\right).{\left[{u}_i(y)\right]}^mI(y)+\left({u_{n_i}}^m\ I(P)-{u_{o_i}}^m\ I(P)\right)}{\sum_{\varOmega_y}W\left(x,y\right).{\left[{u}_i(y)\right]}^m+\left({u_{n_i}}^m-{u_{o_i}}^{\mathrm{m}}\right)} = \frac{s_i(x)\ {v}_i(x)+I(P)\left({u_{n_i}}^m-{u_{o_i}}^m\right)}{s_i(x)+\left({u_{n_i}}^m-{u_{o_i}}^m\right)}=\kern0.5em \frac{v_i(x)\ \left({s}_i(x)+{u_{n_i}}^m-{u_{o_i}}^m\ \right)-{v}_i(x)\ \left({u_{n_i}}^m-\kern0.5em {u_{o_i}}^m\right)+I(P)\left({u_{n_i}}^m-\kern0.5em {u_{o_i}}^m\right)}{s_i(x)+\left({u_{n_i}}^m-\kern0.5em {u_{o_i}}^m\right)}=\kern0.5em {v}_i(x)+\frac{{u_{n_i}}^m-{u_{o_i}}^m}{s_i(x)+\left({u_{n_i}}^m-\kern0.5em {u_{o_i}}^m\right)}\ \left(I(P)-{v}_i(x)\right) $$
(21)

where \( {s}_i(x)={\sum}_{\varOmega_y}W\left(x,y\right).{\left[{u}_i(y)\right]}^m,\kern0.5em i=1,2 \). In a similar way, it is proven that the new \( {\overset{\sim }{v}}_3 \) is given by:

$$ {\overset{\sim }{v}}_3(x)={v}_3(x)+\frac{{\left(1-{u}_{n_1}-{u}_{n_2}\right)}^m-{\left(1-{u}_{o_1}-{u}_{o_2}\right)}^m}{s_3(x)+{\left(1-{u}_{n_1}-{u}_{n_2}\right)}^m-{\left(1-{u}_{o_1}-{u}_{o_2}\right)}^m}\ \left(I(P)-{v}_3(x)\right) $$
(22)

where \( {s}_3(x)={\sum}_{\varOmega_y}W\left(x,y\right).{\left[1-{u}_1(y)-{u}_2(y)\right]}^m \).

Thus, the changed values \( \Delta {v}_i=\tilde{v}_{i}-{v}_i \) for the point P can be easily computed using formulation (21) for i = 1, 2 and (22) for i = 3. Furthermore, changing the membership degrees of point P to the new values will lead to a change in the model energy. Assuming that the new energy is denoted by \( \overset{\sim }{F} \):

(23)

We will separately examine \( {\overset{\sim }{A}}_1 \) and \( {\overset{\sim }{B}}_1 \) to formulate our result. Therefore,

$$ {\overset{\sim }{A}}_1={\sum}_{\varOmega_y}W\left(x,y\right).{\left[\tilde{u}_{i}(y)\right]}^m\ {\left(I(y)-\tilde{v}_{i}(x)\right)}^2={\sum}_{\varOmega_y}W\left(x,y\right)\ \left[{\left[{u}_i(y)\right]}^m\ {\left(I(y)-\tilde{v}_{i}(x)\right)}^2+\left({u_{n_i}}^m-{u_{0_i}}^m\right)\ {\left(I(P)-\tilde{v}_{i}(x)\right)}^2\right] $$
(24)

By substituting (21) into (24), we obtain following equation:

$$ {\overset{\sim }{A}}_1=\sum \limits_{\varOmega_y}W\left(x,y\right).\left[{\left[{u}_i(y)\right]}^m\ {\left(I(y)-{v}_i(x)-\frac{{u_{n_i}}^m-{u_{0_i}}^m}{s_i(x)+\left({u_{n_i}}^m-{u_{0_i}}^m\right)}\ \left(I(P)-{v}_i(x)\right)\right)}^2+\kern0.5em \left({u_{n_i}}^m-{u_{0_i}}^m\right){\left(I(P)-{v}_i(x)-\frac{{u_{n_i}}^m-{u_{0_i}}^m}{s_i(x)+\left({u_{n_i}}^m-{u_{0_i}}^m\right)}\right)}^2\right]=\sum \limits_{\varOmega_y}W\left(x,y\right).\left[{\left[{u}_i(y)\right]}^m\ \left[{\left(I(y)-{v}_i(x)\right)}^2+{\left(\frac{{u_{n_i}}^m-{u_{0_i}}^m}{s_i(x)+\left({u_{n_i}}^m-{u_{0_i}}^m\right)}\right)}^2\ {\left(I(P)-{v}_i(x)\right)}^2-2\ \left(I(y)-{v}_i(x)\right)\ \left(\frac{{u_{n_i}}^m-{u_{0_i}}^m}{s_i(x)+\left({u_{n_i}}^m-{u_{0_i}}^m\right)}\right)\ \left(I(P)-{v}_i(x)\right)\right]\right]+\sum \limits_{\varOmega_y}W\left(x,y\right).\left[\left({u_{n_i}}^m-{u_{0_i}}^m\right)\ {\left(I(P)-{v}_i(x)\right)}^2\ \frac{{\left[{s}_i(x)\right]}^2}{{\left({s}_i(x)+\left({u_{n_i}}^m-{u_{0_i}}^m\right)\right)}^2}\right]={A}_1+{s}_i(x){\left(\frac{{u_{n_i}}^m-{u_{0_i}}^m}{s_i(x)+\left({u_{n_i}}^m-{u_{0_i}}^m\right)}\ \left(I(P)-{v}_i(x)\right)\right)}^2-2\left({s}_i(x){v}_i(x)-{s}_i(x){v}_i(x)\right)\left(\frac{{u_{n_i}}^m-{u_{0_i}}^m}{s_i(x)+\left({u_{n_i}}^m-{u_{0_i}}^m\right)}\right)\left(I(P)-{v}_i(x)\right)+\left({u_{n_i}}^m-{u_{0_i}}^m\right)\ {\left(\frac{s_i(x)\ \left(I(P)-{v}_i(x)\right)}{s_i(x)+\left({u_{n_i}}^m-{u_{0_i}}^m\right)}\right)}^2={A}_1+{s}_i(x)\ \left({u_{n_i}}^m-{u_{0_i}}^m\right)\ {\left(\frac{I(P)-{v}_i(x)}{s_i(x)+\left({u_{n_i}}^m-{u_{0_i}}^m\right)}\right)}^2\ \left[\left({u_{n_i}}^m-{u_{0_i}}^m\right)+{s}_i(x)\right]={A}_1+{s}_i(x)\ \left(\frac{{u_{n_i}}^m-{u_{0_i}}^m}{s_i(x)+{u_{n_i}}^m-{u_{0_i}}^m}\right)\ {\left(I(P)-{v}_i(x)\right)}^2 $$
(25)

For the image domain Ωx:

$$ \overset{\sim }{A}=\sum \limits_{\varOmega_x}{\overset{\sim }{A}}_1=\sum \limits_{\varOmega_x}\left({A}_1+{s}_i(x)\ \left(\frac{{u_{n_i}}^m-{u_{0_i}}^m}{s_i+{u_{n_i}}^m-{u_{0_i}}^m}\right)\ {\left(I(P)-{v}_i(x)\right)}^2\right)=A+\sum \limits_{\varOmega_x}\left({s}_i(x)\ \left(\frac{{u_{n_i}}^m-{u_{0_i}}^m}{s_i(x)+{u_{n_i}}^m-{u_{0_i}}^m}\right)\ {\left(I(P)-{v}_i(x)\right)}^2\right) $$
(26)

In a similar way, it can be shown that:

$$ \overset{\sim }{B}=B+\sum \limits_{\varOmega_x}\left({s}_3(x)\ \left(\frac{{\left(1-{u}_{n_1}-{u}_{n_2}\right)}^m-{\left(1-{u}_{0_1}-{u}_{0_2}\right)}^m}{s_i(x)+{\left(1-{u}_{n_1}-{u}_{n_2}\right)}^m-{\left(1-{u}_{0_1}-{u}_{0_2}\right)}^m}\right)\ {\left(I(P)-{v}_i(x)\right)}^2\right) $$
(27)

Combining (23), (26) and (27), the new total energy functional is given by

$$ \overset{\sim }{F}=\sum \limits_{i=1}^2\left(A+\sum \limits_{\varOmega_x}\left({s}_i(x)\ \left(\frac{{u_{n_i}}^m-{u_{0_i}}^m}{s_i(x)+{u_{n_i}}^m-{u_{0_i}}^m}\right)\ {\left(I(P)-{v}_i(x)\right)}^2\right)\right)+B+\sum \limits_{\varOmega_x}\left({s}_3(x)\ \left(\frac{{\left(1-{u}_{n_1}-{u}_{n_2}\right)}^m-{\left(1-{u}_{0_1}-{u}_{0_2}\right)}^m}{s_3(x)+{\left(1-{u}_{n_1}-{u}_{n_2}\right)}^m-{\left(1-{u}_{0_1}-{u}_{0_2}\right)}^m}\right)\ {\left(I(P)-{v}_i(x)\right)}^2\right)\overset{\sim }{F}=F+\sum \limits_{i=1}^2\sum \limits_{\varOmega_x}\left({s}_i(x)\ \left(\frac{{u_{n_i}}^m-{u_{0_i}}^m}{s_i(x)+{u_{n_i}}^m-{u_{0_i}}^m}\right)\ {\left(I(P)-{v}_i(x)\right)}^2\right)+\sum \limits_{\varOmega_x}\left({s}_3(x)\ \left(\frac{{\left(1-{u}_{n_1}-{u}_{n_2}\right)}^m-{\left(1-{u}_{0_1}-{u}_{0_2}\right)}^m}{s_i(x)+{\left(1-{u}_{n_1}-{u}_{n_2}\right)}^m-{\left(1-{u}_{0_1}-{u}_{0_2}\right)}^m}\right)\ {\left(I(P)-{v}_i\right)}^2\right) $$
(28)

Therefore,

$$ \Delta F=\overset{\sim }{F}-F=\sum \limits_{i=1}^2\sum \limits_{\varOmega_x}\left({s}_i(x)\ \left(\frac{{u_{n_i}}^m-{u_{0_i}}^m}{s_i(x)+{u_{n_i}}^m-{u_{0_i}}^m}\right)\ {\left(I(P)-{v}_i(x)\right)}^2\right)+\sum \limits_{\varOmega_x}\left({s}_3(x)\ \left(\frac{{\left(1-{u}_{n_1}-{u}_{n_2}\right)}^m-{\left(1-{u}_{0_1}-{u}_{0_2}\right)}^m}{s_3(x)+{\left(1-{u}_{n_1}-{u}_{n_2}\right)}^m-{\left(1-{u}_{0_1}-{u}_{0_2}\right)}^m}\right)\ {\left(I(P)-{v}_i(x)\right)}^2\right) $$
(29)

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Alipour, N., Hasanzadeh, R.P.R. Superpixel-based brain tumor segmentation in MR images using an extended local fuzzy active contour model. Multimed Tools Appl 80, 8835–8859 (2021). https://doi.org/10.1007/s11042-020-10122-1

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