American Institute of Mathematical Sciences

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doi: 10.3934/naco.2019028

Approach to image segmentation based on interval neutrosophic set

Ye Yuan 1, , Yan Ren 2,, , Xiaodong Liu 3, and  Jing Wang 3,

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

* Corresponding author: Yan Ren, renyan1108@hotmail.com

Received  May 2018 Revised  April 2019 Published  May 2019

Fund Project: The first author is supported by NSF grant No.61602321, Aviation Science Foundation with No. 2017ZC54007, Science Fund Project of Liaoning Province Education Department with No. L201614, Natural Science Fund Project of Liaoning Province with No.20170540694, the Fundamental Research Funds for the Central Universities (No.3132018228) and the Doctor Startup Foundation of Shenyang Aerospace University13YB11

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 k-means 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.

Keywords: Neutrosophic sets, Interval neutrosophic sets, k-means algorithm, Image segmentation.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Ye Yuan, Yan Ren, Xiaodong Liu, Jing Wang. Approach to image segmentation based on interval neutrosophic set. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2019028
References:
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M. R. Anderberg, Cluster analysis for applications, Probability & Mathematical Statistics New York Academic Press, 1 (1973), 347-353.   Google Scholar

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K. T. Atanassov, Intuitionistic fuzzy sets, Physica-Verlag HD, 20 (1986), 87-96.  doi: 10.1007/978-3-7908-1870-3.  Google Scholar

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H. D. Cheng and Y. Guo, A new neutrosophic approach to image thresholding, New Mathematics & Natural Computation, 4 (2008), 291-308.   Google Scholar

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C. Feng, D. Zhao and M. Huang, Image segmentation using CUDA accelerated non-local means denoising and bias correction embedded fuzzy C-means (BCEFCM), Signal Processing, 122 (2016), 164-189.   Google Scholar

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Y. Guo, H. D. Cheng and W. Zhao et al., A novel image segmentation algorithm based on fuzzy C-means algorithm and neutrosophic set, JCIS-2008 Proceedings, 2008. Google Scholar

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Y. Guo and H. D. Cheng, New neutrosophic approach to image segmentation, Pattern Recognition, 42 (2009), 587-595.   Google Scholar

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Y. Guo, A novel image edge detection algorithm based on neutrosophic set, Pergamon Press, Inc, 40 (2014), 3-25.   Google Scholar

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Y. Guo and A. Sengür, A novel image segmentation algorithm based on neutrosophic similarity clustering, Applied Soft Computing, 25 (2014), 391-398. Google Scholar

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Y. Guo, A. Sengür and J. Ye, A novel image thresholding algorithm based on neutrosophic similarity score, Measurement, 58 (2014), 175-186. Google Scholar

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Y. Guo and A. Sengür, NECM: Neutrosophic evidential C-means clustering algorithm, Neural Computing & Applications, 26 (2014), 1-11. Google Scholar

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Y. Guo and A. Sengür, NCM: Neutrosophic C-means clustering algorithm, Pattern Recognition, 48 (2015), 2710-2724. Google Scholar

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Y. Guo and Y. Akbulut et al., An efficient image segmentation algorithm using neutrosophic graph cut, Symmetry, 9 (2017), 185. Google Scholar

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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), 433-443.   Google Scholar

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T. Kanungo, D. M. Mount and N. S. Netanyahu et al., An efficient k-means clustering algorithm: Analysis and implementation, IEEE Transactions on Pattern Analysis & Machine Intelligence, 24 (2002), 881-892. Google Scholar

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X. Liu, The fuzzy theory based on AFS algebras and AFS structure, Journal of Mathematical Analysis & Applications, 217 (1998), 459-478.  doi: 10.1006/jmaa.1997.5718.  Google Scholar

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S. K. Pal and R. A. King, Image enhancement using smoothing with fuzzy sets, IEEE Transactions on Systems Man & Cybernetics, 11 (1981), 494-501.   Google Scholar

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S. K. Pal and R. A. King, Image enhancement using fuzzy sets, Electronics Letters, 16 (1980), 376-378.   Google Scholar

[21]

A. Sengur and Y. Guo, Color texture image segmentation based on neutrosophic set and wavelet transformation, Computer Vision & Image Understanding, 115 (2011), 1134-1144.   Google Scholar

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J. Shan, H. D. Cheng and Y. Wang, A novel segmentation method for breast ultrasound images based on neutrosophic k-means clustering, Medical Physics, 39 (2012), 5669. Google Scholar

[23]

F. Smarandache, A unifying field in logics: Neutrosophic logic, Multiple-Valued Logic, 8 (1999), 489-503.   Google Scholar

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F. Smarandache, A unifying field in logics: Neutrsophic logic, neutrosophy, neutrosophic set, neutrosophic probability (Fourth edition), University of New Mexico, 332 (2002), 5-21.   Google Scholar

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F. Smarandache, Neutrosophic Topologies, 1999. Google Scholar

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Turksen, Interval valued fuzzy sets based on normal forms, Fuzzy Sets & Systems, 20 (1986), 191-210.  doi: 10.1016/0165-0114(86)90077-1.  Google Scholar

[27]

J. K. Udupa and S. Samarasekera, Fuzzy connectedness and object definition: Theory, algorithms, and applications in image segmentation, Academic Press Inc., 58 (1996), 246-261.   Google Scholar

[28]

H. Wang, P. Madiraju and Y. Zhang et al, Interval neutrosophic sets, Mathematics, 1 (2004), 274-277.   Google Scholar

[29]

L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.   Google Scholar

[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. Google Scholar

[31]

X. Zhao and S. T. Wang, Neutrosophic image segmentation approach based on similarity, Application Research of Computers, 29 (2012), 2371-2374.   Google Scholar

show all references

References:
[1]

M. R. Anderberg, Cluster analysis for applications, Probability & Mathematical Statistics New York Academic Press, 1 (1973), 347-353.   Google Scholar

[2]

K. T. Atanassov, Intuitionistic fuzzy sets, Physica-Verlag HD, 20 (1986), 87-96.  doi: 10.1007/978-3-7908-1870-3.  Google Scholar

[3]

K. T. Atanassov and G. Gargov, Interval valued intuitionistic fuzzy sets, Physica-Verlag HD, (1999), 343–349. doi: 10.1016/0165-0114(89)90205-4.  Google Scholar

[4]

C. Bai, D. Dhavale and J. Sarkis, Complex investment decisions using rough set and fuzzy C-means: An example of investment in green supply chains, European Journal of Operational Research, 248 (2016), 507-521.  doi: 10.1016/j.ejor.2015.07.059.  Google Scholar

[5]

H. D. Cheng, Y. H. Chen and X. H. Jiang, Thresholding using two-dimensionalh histogram and fuzzy entropy principle, IEEE Transactions on Image Processing, 9 (2000), 732-735.  doi: 10.1109/83.841949.  Google Scholar

[6]

H. D. Cheng and Y. Guo, A new neutrosophic approach to image thresholding, New Mathematics & Natural Computation, 4 (2008), 291-308.   Google Scholar

[7]

C. Feng, D. Zhao and M. Huang, Image segmentation using CUDA accelerated non-local means denoising and bias correction embedded fuzzy C-means (BCEFCM), Signal Processing, 122 (2016), 164-189.   Google Scholar

[8]

Y. Guo, H. D. Cheng and W. Zhao et al., A novel image segmentation algorithm based on fuzzy C-means algorithm and neutrosophic set, JCIS-2008 Proceedings, 2008. Google Scholar

[9]

Y. Guo and H. D. Cheng, New neutrosophic approach to image segmentation, Pattern Recognition, 42 (2009), 587-595.   Google Scholar

[10]

Y. Guo, A novel image edge detection algorithm based on neutrosophic set, Pergamon Press, Inc, 40 (2014), 3-25.   Google Scholar

[11]

Y. Guo and A. Sengür, A novel image segmentation algorithm based on neutrosophic similarity clustering, Applied Soft Computing, 25 (2014), 391-398. Google Scholar

[12]

Y. Guo, A. Sengür and J. Ye, A novel image thresholding algorithm based on neutrosophic similarity score, Measurement, 58 (2014), 175-186. Google Scholar

[13]

Y. Guo and A. Sengür, NECM: Neutrosophic evidential C-means clustering algorithm, Neural Computing & Applications, 26 (2014), 1-11. Google Scholar

[14]

Y. Guo and A. Sengür, NCM: Neutrosophic C-means clustering algorithm, Pattern Recognition, 48 (2015), 2710-2724. Google Scholar

[15]

Y. Guo and Y. Akbulut et al., An efficient image segmentation algorithm using neutrosophic graph cut, Symmetry, 9 (2017), 185. Google Scholar

[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), 433-443.   Google Scholar

[17]

T. Kanungo, D. M. Mount and N. S. Netanyahu et al., An efficient k-means clustering algorithm: Analysis and implementation, IEEE Transactions on Pattern Analysis & Machine Intelligence, 24 (2002), 881-892. Google Scholar

[18]

X. Liu, The fuzzy theory based on AFS algebras and AFS structure, Journal of Mathematical Analysis & Applications, 217 (1998), 459-478.  doi: 10.1006/jmaa.1997.5718.  Google Scholar

[19]

S. K. Pal and R. A. King, Image enhancement using smoothing with fuzzy sets, IEEE Transactions on Systems Man & Cybernetics, 11 (1981), 494-501.   Google Scholar

[20]

S. K. Pal and R. A. King, Image enhancement using fuzzy sets, Electronics Letters, 16 (1980), 376-378.   Google Scholar

[21]

A. Sengur and Y. Guo, Color texture image segmentation based on neutrosophic set and wavelet transformation, Computer Vision & Image Understanding, 115 (2011), 1134-1144.   Google Scholar

[22]

J. Shan, H. D. Cheng and Y. Wang, A novel segmentation method for breast ultrasound images based on neutrosophic k-means clustering, Medical Physics, 39 (2012), 5669. Google Scholar

[23]

F. Smarandache, A unifying field in logics: Neutrosophic logic, Multiple-Valued Logic, 8 (1999), 489-503.   Google Scholar

[24]

F. Smarandache, A unifying field in logics: Neutrsophic logic, neutrosophy, neutrosophic set, neutrosophic probability (Fourth edition), University of New Mexico, 332 (2002), 5-21.   Google Scholar

[25]

F. Smarandache, Neutrosophic Topologies, 1999. Google Scholar

[26]

Turksen, Interval valued fuzzy sets based on normal forms, Fuzzy Sets & Systems, 20 (1986), 191-210.  doi: 10.1016/0165-0114(86)90077-1.  Google Scholar

[27]

J. K. Udupa and S. Samarasekera, Fuzzy connectedness and object definition: Theory, algorithms, and applications in image segmentation, Academic Press Inc., 58 (1996), 246-261.   Google Scholar

[28]

H. Wang, P. Madiraju and Y. Zhang et al, Interval neutrosophic sets, Mathematics, 1 (2004), 274-277.   Google Scholar

[29]

L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.   Google Scholar

[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. Google Scholar

[31]

X. Zhao and S. T. Wang, Neutrosophic image segmentation approach based on similarity, Application Research of Computers, 29 (2012), 2371-2374.   Google Scholar

Figure 1.  Lena1, Lena2, Pepper
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Figure 2.  The first part is the original image, the middle part is the segmentation by the traditional k-means algorithms, and the third part is the segmentation by interval neutrosophic set method
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Figure 3.  For Lena1 image, the first part is the original image with gaussian noise (mean is $ 0 $ and variance is $ 0.01 $), the middle part is the segmentation by the traditional k-means algorithms, and the third part is the segmentation by interval neutrosophic set method
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Figure 4.  For Lena2 image, the first part is the original image with gaussian noise (mean is 0 and variance is 0.01), the middle part is the segmentation by the traditional k-means algorithms, and the third part is the segmentation by interval neutrosophic set method
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Figure 5.  For Pepper image, the first part is the original image with gaussian noise (mean is $ 0 $ and variance is $ 0.01 $), the middle part is the segmentation by the traditional k-means algorithms, and the third part is the segmentation by interval neutrosophic set method
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Figure 6.  For Lena image, the first part is the original image, the middle part is the segmentation by the approach in [9], and the third part is the segmentation by interval neutrosophic set method
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Table 1.  The value of PSNR in different ranges for the three images ($ W = w\ast w $ is a collection of image pixels)
range $ w_{1}=3 $ $ w_{1}=3 $ $ w_{1}=3 $ $ w_{1}=3 $ $ w_{1}=3 $ $ w_{1}=3 $
$ w_{2}=3 $ $ w_{2}=4 $ $ w_{2}=5 $ $ w_{2}=6 $ $ w_{2}=7 $ $ w_{2}=8 $
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
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range $ w_{1}=3 $ $ w_{1}=3 $ $ w_{1}=3 $ $ w_{1}=3 $ $ w_{1}=3 $ $ w_{1}=3 $
$ w_{2}=3 $ $ w_{2}=4 $ $ w_{2}=5 $ $ w_{2}=6 $ $ w_{2}=7 $ $ w_{2}=8 $
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
Table 2.  The value of PSNR for the three images with different noises
noise gaussian noise(1) gaussian noise(2) salt noise speckle noise
k-means 19.1443 13.8363 22.6033 19.5381
INI 21.2599 18.4293 21.1605 21.6448
k-means 19.1969 13.9415 23.3081 20.9689
INI 22.1729 18.6085 22.4192 22.6730
k-means 18.6936 13.8596 21.4768 19.4912
INI 20.6764 18.0361 20.7433 20.6611
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noise gaussian noise(1) gaussian noise(2) salt noise speckle noise
k-means 19.1443 13.8363 22.6033 19.5381
INI 21.2599 18.4293 21.1605 21.6448
k-means 19.1969 13.9415 23.3081 20.9689
INI 22.1729 18.6085 22.4192 22.6730
k-means 18.6936 13.8596 21.4768 19.4912
INI 20.6764 18.0361 20.7433 20.6611
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