Approach to image segmentation based on interval neutrosophic set

Ye Yuan; Yan Ren; Xiaodong Liu; Jing Wang

doi:10.3934/naco.2019028

Article Contents

2020, Volume 10, Issue 1: 1-11. Doi: 10.3934/naco.2019028

This issue Previous Article Next Article Numerical comparisons of smoothing functions for optimal correction of an infeasible system of absolute value equations

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

Received: April 30, 2018

Revised: March 31, 2019

Published: February 2020

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.

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Abstract

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.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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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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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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