# American Institute of Mathematical Sciences

March  2020, 10(1): 1-11. doi: 10.3934/naco.2019028

## 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

* 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.

Citation: Ye Yuan, Yan Ren, Xiaodong Liu, Jing Wang. Approach to image segmentation based on interval neutrosophic set. Numerical Algebra, Control & Optimization, 2020, 10 (1) : 1-11. doi: 10.3934/naco.2019028
##### References:

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##### References:
Lena1, Lena2, Pepper
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
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
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
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
], and the third part is the segmentation by interval neutrosophic set method">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
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
 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
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
 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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