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Approach to image segmentation based on interval neutrosophic set
1.  Department of Mathematics, Dalian Maritime University, Dalian, 116026, P. R. China 
2.  College of Automation, Shenyang Aerospace University, Shenyang, 110136, P. R. China 
3.  Department of Mathematics, Dalian Maritime University, Dalian, 116026, P. R. China 
As a generalization of the fuzzy set and intuitionistic fuzzy set, the neutrosophic set (NS) have been developed to represent uncertain, imprecise, incomplete and inconsistent information existing in the real world. Now the interval neutrosophic set (INS) which is an expansion of the neutrosophic set have been proposed exactly to address issues with a set of numbers in the real unit interval, not just one specific number. After definition of concepts and operations, INS is applied to image segmentation. Images are converted to the INS domain, which is described using three membership interval sets: T, I and F. Then, in order to increase the contrast between membership and evaluate the indeterminacy, a fuzzy intensification for each element in the interval set is made and a score function in the INS is defined. Finally, the proposed method is employed to perform image segmentation using the traditional kmeans clustering. The experimental results on a variety of images demonstrate that the proposed approach can segment different sorts of images. Especially, it can segment "clean" images and images with various levels of noise.
References:
[1] 
M. R. Anderberg, Cluster analysis for applications, Probability & Mathematical Statistics New York Academic Press, 1 (1973), 347353. 
[2] 
K. T. Atanassov, Intuitionistic fuzzy sets, PhysicaVerlag HD, 20 (1986), 8796. doi: 10.1007/9783790818703. 
[3] 
K. T. Atanassov and G. Gargov, Interval valued intuitionistic fuzzy sets, PhysicaVerlag HD, (1999), 343–349. doi: 10.1016/01650114(89)902054. 
[4] 
C. Bai, D. Dhavale and J. Sarkis, Complex investment decisions using rough set and fuzzy Cmeans: An example of investment in green supply chains, European Journal of Operational Research, 248 (2016), 507521. doi: 10.1016/j.ejor.2015.07.059. 
[5] 
H. D. Cheng, Y. H. Chen and X. H. Jiang, Thresholding using twodimensionalh histogram and fuzzy entropy principle, IEEE Transactions on Image Processing, 9 (2000), 732735. doi: 10.1109/83.841949. 
[6] 
H. D. Cheng and Y. Guo, A new neutrosophic approach to image thresholding, New Mathematics & Natural Computation, 4 (2008), 291308. 
[7] 
C. Feng, D. Zhao and M. Huang, Image segmentation using CUDA accelerated nonlocal means denoising and bias correction embedded fuzzy Cmeans (BCEFCM), Signal Processing, 122 (2016), 164189. 
[8] 
Y. Guo, H. D. Cheng and W. Zhao et al., A novel image segmentation algorithm based on fuzzy Cmeans algorithm and neutrosophic set, JCIS2008 Proceedings, 2008. 
[9] 
Y. Guo and H. D. Cheng, New neutrosophic approach to image segmentation, Pattern Recognition, 42 (2009), 587595. 
[10] 
Y. Guo, A novel image edge detection algorithm based on neutrosophic set, Pergamon Press, Inc, 40 (2014), 325. 
[11] 
Y. Guo and A. Sengür, A novel image segmentation algorithm based on neutrosophic similarity clustering, Applied Soft Computing, 25 (2014), 391398. 
[12] 
Y. Guo, A. Sengür and J. Ye, A novel image thresholding algorithm based on neutrosophic similarity score, Measurement, 58 (2014), 175186. 
[13] 
Y. Guo and A. Sengür, NECM: Neutrosophic evidential Cmeans clustering algorithm, Neural Computing & Applications, 26 (2014), 111. 
[14] 
Y. Guo and A. Sengür, NCM: Neutrosophic Cmeans clustering algorithm, Pattern Recognition, 48 (2015), 27102724. 
[15] 
Y. Guo and Y. Akbulut et al., An efficient image segmentation algorithm using neutrosophic graph cut, Symmetry, 9 (2017), 185. 
[16] 
K. Hanbay and M. F. Talu, Segmentation of SAR images using improved artificial bee colony algorithm and neutrosophic set, Applied Soft Computing Journal, 21 (2014), 433443. 
[17] 
T. Kanungo, D. M. Mount and N. S. Netanyahu et al., An efficient kmeans clustering algorithm: Analysis and implementation, IEEE Transactions on Pattern Analysis & Machine Intelligence, 24 (2002), 881892. 
[18] 
X. Liu, The fuzzy theory based on AFS algebras and AFS structure, Journal of Mathematical Analysis & Applications, 217 (1998), 459478. doi: 10.1006/jmaa.1997.5718. 
[19] 
S. K. Pal and R. A. King, Image enhancement using smoothing with fuzzy sets, IEEE Transactions on Systems Man & Cybernetics, 11 (1981), 494501. 
[20] 
S. K. Pal and R. A. King, Image enhancement using fuzzy sets, Electronics Letters, 16 (1980), 376378. 
[21] 
A. Sengur and Y. Guo, Color texture image segmentation based on neutrosophic set and wavelet transformation, Computer Vision & Image Understanding, 115 (2011), 11341144. 
[22] 
J. Shan, H. D. Cheng and Y. Wang, A novel segmentation method for breast ultrasound images based on neutrosophic kmeans clustering, Medical Physics, 39 (2012), 5669. 
[23] 
F. Smarandache, A unifying field in logics: Neutrosophic logic, MultipleValued Logic, 8 (1999), 489503. 
[24] 
F. Smarandache, A unifying field in logics: Neutrsophic logic, neutrosophy, neutrosophic set, neutrosophic probability (Fourth edition), University of New Mexico, 332 (2002), 521. 
[25]  
[26] 
Turksen, Interval valued fuzzy sets based on normal forms, Fuzzy Sets & Systems, 20 (1986), 191210. doi: 10.1016/01650114(86)900771. 
[27] 
J. K. Udupa and S. Samarasekera, Fuzzy connectedness and object definition: Theory, algorithms, and applications in image segmentation, Academic Press Inc., 58 (1996), 246261. 
[28] 
H. Wang, P. Madiraju and Y. Zhang et al, Interval neutrosophic sets, Mathematics, 1 (2004), 274277. 
[29] 
L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338353. 
[30] 
H. Zhang, J. Wang and X. Chen, Interval neutrosophic sets and their application in multicriteria decision making problems, The Scientific World Journal, 2014 (2014), 645953. 
[31] 
X. Zhao and S. T. Wang, Neutrosophic image segmentation approach based on similarity, Application Research of Computers, 29 (2012), 23712374. 
show all references
References:
[1] 
M. R. Anderberg, Cluster analysis for applications, Probability & Mathematical Statistics New York Academic Press, 1 (1973), 347353. 
[2] 
K. T. Atanassov, Intuitionistic fuzzy sets, PhysicaVerlag HD, 20 (1986), 8796. doi: 10.1007/9783790818703. 
[3] 
K. T. Atanassov and G. Gargov, Interval valued intuitionistic fuzzy sets, PhysicaVerlag HD, (1999), 343–349. doi: 10.1016/01650114(89)902054. 
[4] 
C. Bai, D. Dhavale and J. Sarkis, Complex investment decisions using rough set and fuzzy Cmeans: An example of investment in green supply chains, European Journal of Operational Research, 248 (2016), 507521. doi: 10.1016/j.ejor.2015.07.059. 
[5] 
H. D. Cheng, Y. H. Chen and X. H. Jiang, Thresholding using twodimensionalh histogram and fuzzy entropy principle, IEEE Transactions on Image Processing, 9 (2000), 732735. doi: 10.1109/83.841949. 
[6] 
H. D. Cheng and Y. Guo, A new neutrosophic approach to image thresholding, New Mathematics & Natural Computation, 4 (2008), 291308. 
[7] 
C. Feng, D. Zhao and M. Huang, Image segmentation using CUDA accelerated nonlocal means denoising and bias correction embedded fuzzy Cmeans (BCEFCM), Signal Processing, 122 (2016), 164189. 
[8] 
Y. Guo, H. D. Cheng and W. Zhao et al., A novel image segmentation algorithm based on fuzzy Cmeans algorithm and neutrosophic set, JCIS2008 Proceedings, 2008. 
[9] 
Y. Guo and H. D. Cheng, New neutrosophic approach to image segmentation, Pattern Recognition, 42 (2009), 587595. 
[10] 
Y. Guo, A novel image edge detection algorithm based on neutrosophic set, Pergamon Press, Inc, 40 (2014), 325. 
[11] 
Y. Guo and A. Sengür, A novel image segmentation algorithm based on neutrosophic similarity clustering, Applied Soft Computing, 25 (2014), 391398. 
[12] 
Y. Guo, A. Sengür and J. Ye, A novel image thresholding algorithm based on neutrosophic similarity score, Measurement, 58 (2014), 175186. 
[13] 
Y. Guo and A. Sengür, NECM: Neutrosophic evidential Cmeans clustering algorithm, Neural Computing & Applications, 26 (2014), 111. 
[14] 
Y. Guo and A. Sengür, NCM: Neutrosophic Cmeans clustering algorithm, Pattern Recognition, 48 (2015), 27102724. 
[15] 
Y. Guo and Y. Akbulut et al., An efficient image segmentation algorithm using neutrosophic graph cut, Symmetry, 9 (2017), 185. 
[16] 
K. Hanbay and M. F. Talu, Segmentation of SAR images using improved artificial bee colony algorithm and neutrosophic set, Applied Soft Computing Journal, 21 (2014), 433443. 
[17] 
T. Kanungo, D. M. Mount and N. S. Netanyahu et al., An efficient kmeans clustering algorithm: Analysis and implementation, IEEE Transactions on Pattern Analysis & Machine Intelligence, 24 (2002), 881892. 
[18] 
X. Liu, The fuzzy theory based on AFS algebras and AFS structure, Journal of Mathematical Analysis & Applications, 217 (1998), 459478. doi: 10.1006/jmaa.1997.5718. 
[19] 
S. K. Pal and R. A. King, Image enhancement using smoothing with fuzzy sets, IEEE Transactions on Systems Man & Cybernetics, 11 (1981), 494501. 
[20] 
S. K. Pal and R. A. King, Image enhancement using fuzzy sets, Electronics Letters, 16 (1980), 376378. 
[21] 
A. Sengur and Y. Guo, Color texture image segmentation based on neutrosophic set and wavelet transformation, Computer Vision & Image Understanding, 115 (2011), 11341144. 
[22] 
J. Shan, H. D. Cheng and Y. Wang, A novel segmentation method for breast ultrasound images based on neutrosophic kmeans clustering, Medical Physics, 39 (2012), 5669. 
[23] 
F. Smarandache, A unifying field in logics: Neutrosophic logic, MultipleValued Logic, 8 (1999), 489503. 
[24] 
F. Smarandache, A unifying field in logics: Neutrsophic logic, neutrosophy, neutrosophic set, neutrosophic probability (Fourth edition), University of New Mexico, 332 (2002), 521. 
[25]  
[26] 
Turksen, Interval valued fuzzy sets based on normal forms, Fuzzy Sets & Systems, 20 (1986), 191210. doi: 10.1016/01650114(86)900771. 
[27] 
J. K. Udupa and S. Samarasekera, Fuzzy connectedness and object definition: Theory, algorithms, and applications in image segmentation, Academic Press Inc., 58 (1996), 246261. 
[28] 
H. Wang, P. Madiraju and Y. Zhang et al, Interval neutrosophic sets, Mathematics, 1 (2004), 274277. 
[29] 
L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338353. 
[30] 
H. Zhang, J. Wang and X. Chen, Interval neutrosophic sets and their application in multicriteria decision making problems, The Scientific World Journal, 2014 (2014), 645953. 
[31] 
X. Zhao and S. T. Wang, Neutrosophic image segmentation approach based on similarity, Application Research of Computers, 29 (2012), 23712374. 
range  
Lena1  21.9715  22.1662  21.5974  21.9076  21.1507  21.6422 
Lena2  22.8407  23.1224  22.5366  22.8503  22.3337  22.6582 
Pepper  20.9797  21.4961  20.4562  20.8171  20.1147  20.4185 
range  
Lena1  21.9715  22.1662  21.5974  21.9076  21.1507  21.6422 
Lena2  22.8407  23.1224  22.5366  22.8503  22.3337  22.6582 
Pepper  20.9797  21.4961  20.4562  20.8171  20.1147  20.4185 
noise  gaussian noise(1)  gaussian noise(2)  salt noise  speckle noise 
kmeans  19.1443  13.8363  22.6033  19.5381 
INI  21.2599  18.4293  21.1605  21.6448 
kmeans  19.1969  13.9415  23.3081  20.9689 
INI  22.1729  18.6085  22.4192  22.6730 
kmeans  18.6936  13.8596  21.4768  19.4912 
INI  20.6764  18.0361  20.7433  20.6611 
noise  gaussian noise(1)  gaussian noise(2)  salt noise  speckle noise 
kmeans  19.1443  13.8363  22.6033  19.5381 
INI  21.2599  18.4293  21.1605  21.6448 
kmeans  19.1969  13.9415  23.3081  20.9689 
INI  22.1729  18.6085  22.4192  22.6730 
kmeans  18.6936  13.8596  21.4768  19.4912 
INI  20.6764  18.0361  20.7433  20.6611 
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