Image segmentation is an important step in image processing and analysis, pattern recognition, and machine vision. A few of algorithms based on level set have been proposed for image segmentation in the last twenty years. However, these methods are time consuming, and sometime fail to extract the correct regions especially for noisy images. Recently, neutrosophic set (NS) theory has been applied to image processing for noisy images with indeterminant information. In this paper, a novel image segmentation approach is proposed based on the filter in NS and level set theory

Guo, Yanhui

Şengür, Abdulkadir

English

Sengur, Abdulkadir

University of New Mexico

Neutrosophic Sets and Systems 
Volume 1 Article 8 
1-1-2013 
A Novel Image Segmentation Algorithm Based on Neutrosophic 
Filtering and Level Set 
Yanhui Guo 
Abdulkadir Sengur 
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Neutrosophic Sets and Systems, Vol. 1, 2013 
 
Yanhui Guo&Abdulkadir Şengür, A Novel Image Segmentation Algorithm Based on Neutrosophic Filtering and Level Set 
  
  
A Novel Image Segmentation Algorithm Based on 
Neutrosophic Filtering and Level Set 
Yanhui Guo1, and Abdulkadir Şengür2 
1 School of Science, Technology & Engineering Management, St. Thomas University, 16401 NW 37th Avenue 
Miami Gardens, Florida 33054, USA. E-mail: yguo@stu.edu 
2 Department of Electric and Electronics Engineering, Firat University, Elazig, Turkey. E-mail: ksengur@firat.edu.tr 
 
Abstract. Image segmentation is an important step in 
image processing and analysis, pattern recognition, and 
machine vision. A few of algorithms based on level set 
have been proposed for image segmentation in the last 
twenty years. However, these methods are time 
consuming, and sometime fail to extract the correct 
regions especially for noisy images. Recently, 
neutrosophic set (NS) theory has been applied to image 
processing for noisy images with indeterminant 
information. In this paper, a novel image segmentation 
approach is proposed based on the filter in NS and level 
set theory. At first, the image is transformed into NS 
domain, which is described by three membership sets (T, 
I and F). Then, a filter is newly defined and employed to 
reduce the indeterminacy of the image. Finally, a level 
set algorithm is used in the image after filtering operation 
for image segmentation. Experiments have been 
conducted using different images. The results 
demonstrate that the proposed method can segment the 
images effectively and accurately. It is especially able to 
remove the noise effect and extract the correct regions on 
both the noise-free images and the images with different 
levels of noise. 
 
Keywords: Image segmentation, Neutrosophic set, Directional alpha-mean filter, Level set.
1 Introduction 
Image segmentation is an essential process and is also one 
of the most difficult tasks in image processing field. It is 
defined as a process dividing an image into different 
regions such that each region is homogeneous, but the 
union of any two adjacent regions is not homogeneous.  
Image segmentation approaches are based on either 
discontinuity and/or homogeneity. The approaches based 
on discontinuity tend to partition an image by detecting 
isolated points, lines and edges according to the abrupt 
changes of the intensities. The approaches based on 
homogeneity include thresholding, clustering, region 
growing, and region splitting and merging [1]. 
Neutrosophy set (NS) provides a powerful tool to deal 
with the indeterminacy, and the indeterminacy is 
quantitatively described using a membership [2]. In 
neutrosophic set, a set A is described by three subsets: 
<A>, <Neut-A> and <Anti-A>, which is interpreted as 
truth, indeterminacy, and falsity set.  It provides a new tool 
to describe the image with uncertain information, which 
had been applied to image processing techniques [3], such 
as image segmentation, thresholding and denoise. 
In this paper, we proposed a novel image segmentation 
method based on NS theory. The image is mapped into NS 
domain and a new filter, directional alpha-mean filter is 
defined in NS domain, and used to remove the 
indeterminance on the image. Finally, the image on NS 
domain is segmented using the method based on level set 
active contour model. 
The remainder of this paper is organized as follows. 
The next section describes the neutrsosophic image, the 
directional alpha-mean filter, and the segmentation 
algorithm integrated with level set model. Section three 
reports the experiments and the relevant discussion. 
Concluding remarks are drawn in Section four. 
2 Proposed method 
2.1 Neutrosophic image 
An image might have a few indeterminate regions, 
such as noise, shadow, and boundary. It is hard for the 
classic sets to interpret the indeterminate regions on 
images clearly. In a neutrosophic set, a subset I, is named 
as indeterminate set and employed to interpret the 
indeterminacy in the image. 
A neutrosophic image is described by three 
membership sets T, I and F. The pixel P(i,j) in the image 
domain is transformed into the neutrosophic set domain, 
denoted as PNS(i,j) and PNS(i,j)={T(i,j), I(i,j), F(i,j) } (T, I 
and F are the membership values belonging to bright pixel 
set, indeterminate set and non-bright pixel set, 
respectively, which are defined as follows [4]: 
46
Neutrosophic Sets and Systems, Vol. 1, 2013 
Yanhui Guo&Abdulkadir Şengür, A Novel Image Segmentation Algorithm Based on Neutrosophic Filtering and Level Set 
min
max min
( , )
( , ) w
w w
wg i j gT i j
g g
−
=
−
   (1) 
min
max min
( , )
( , )
i j
I i j
δ δ
δ δ
−
=
−
    (2) 
( , ) 1 ( , )F i j T i j= −     (3) 
where ( , )wg i j  is the mean value of intensity in the local 
neighborhood, whose size is w w× . ( , )i jδ  is the 
absolute value of the difference between intensity ( , )g i j  
and its local mean value ( , )wg i j . The value of I measures 
the indeterminacy degree of PNS.  
 
2.2 Directional α-mean operation 
In [3], an α-mean operation was defined on a 
neutrosophic image, and it removed noise efficiently.  
However, it might blur the image and reduce the contrast, 
which could reduce the performance of the segmentation. 
To overcome this drawback, a directional α-mean 
operation (denoted as DAM) is newly proposed to remove 
the noise effect and preserve the edges at the same time. 
The function of the directional mean filter DMF  is 
defined as [5]: 
1
2
3
| ( , ) ( , ) |
( , ) ( , ) ( , )
( , ) ( , )
Th Tv
Th Tv
Tv Th
R G i j G i j
DMF i j R G i j G i j
R G i j G i j
σ
σ
σ
− ≤
= − >
− >
   (4) 
where ( , )ThG i j  and ( , )TvG i j  are the norm of the gradient 
at ( , )i j  of the subset T at the horizontal and vertical 
direction, respectively. σ  is a threshold value and selected 
as 0.01here.  
The directional α-mean filter DAMF  is defined using 
the subset T and I as: 
( , ) ( , )
( , )
( , ) ( , )
T i j I i j
DAMF i j
DMF i j I i j
α
α
<
=
≥
  
       (5) 
 
2.3 Level set 
Level set method was proposed in [6], and applied for 
image segmentation [7]. The level set method tracks the 
evolution of the boundaries between different objects, 
which are embedded as the zero level set. 
The level set active contour models can be divided into 
two classes: edge based and region based. The edge based 
model tries to find a curve with the maximum edge 
indicator value which can minimize the energy 
function ( )J C [8]: 
( ) ( ) ( )( )( ) 'J C C s g Im C s ds=        (6) 
where g() is an edge indicator function, C is the boundary, 
and it can be represented implicitly as the zero level set of a 
true positive function : Rφ Ω → , Ω is the domain of 
image. The evolution equation of boundary C can be 
derived as: 
( ) ( )
( ) ( )00, , ,   in 
div g Im v Im
t
x y x y
φ φφ φ
φ φ
   ∂ ∇
= ∇ +     ∂ ∇   
= Ω
        (7)                   
where v() is a term for increasing the evolution speed to 
reach the boundary. 
The region based model uses the inside/outside mean 
values to compose the energy function [9]: 
( ) ( ) ( )
1
2
1 2 1 2
2
1 1 1
2
2 2 2
, ,
                     
                     
S
S
F C c c L C S C
Im c dS
Im c dS
μ μ
λ
λ
= +
+ −
+ −


            
(8) 
where 1c and 2c are the mean intensities of the regions 
inside and outside the boundary C, respectively. L and S are 
the length of C and the area inside C. S1 and S2 are the 
region inside and outside of C, respectively. The associated 
level set flow can be represented as: 
( )
( ) ( )
( ) ( )
( )
2 2
1 1 2 2
00, , ,
0 
div v
t
Im c Im c
x y x y
n
φμφ δ φ φ
λ λ
φ φ
δ φ φ
φ
   ∇
− −∂    = ∇   ∂  
− + −  
= ∂ =
∇ ∂
       (9) 
where ()δ is the Dirac function, and n denotes the exterior 
normal to the boundary ∂Ω . div() is the divergence function 
on the image. 
Usually, edge based approaches often suffer from 
noise, especially, when the image has a low signal/noise 
ratio; while the region based approaches are more adaptive 
to noise or vanishing boundaries due to considering the 
entire information of the regions to build an energy 
function. 
2.4 Segmentation algorithm based on 
neutrosophic set and level set (NSLS) 
A segmentation algorithm is proposed based on the 
directional α-mean filter and level set on neutrosophic 
image. Firstly, the image is transferred into the NS domain. 
Then, the DAMF is processed in the NS image. Finally, the 
boundary of region is segmented using the level set active 
contour algorithm based on the region model. The energy 
function is defined using the T subset after DAMF 
processing.   
47
Neutrosophic Sets and Systems, Vol. 1, 2013 
 Yanhui Guo&Abdulkadir Şengür, A Novel Image Segmentation Algorithm Based on Neutrosophic Filtering and Level Set 
( ) ( ) ( )
1
2
1 2 1 2
2
1 1 1
2
2 2 2
, ,
                     
                     
S
S
F C c c L C S C
T c dS
T c dS
μ μ
λ
λ
= +
′+ −
′+ −


      (10) 
3 Experimental results and discussion 
To test the performance of the proposed method, a few 
of images and images with different noise levels are 
employed. The NFLS method is compared with that of the 
segmentation algorithm based on the traditional level set 
[9], which is noted as TLS. 
 
(a) Original image. (b) Segmentation result of NSLS at 
800 iterations. 
 
(c) Segmentation result of TLS at 
800 iterations. 
(d) Segmentation result of TLS at 
1500 iterations. 
Figure 1: Comparison result on the “liver tumour” image with different 
iterations. 
  
(a) Original image. (b) The segmentation 
result of NSLS at 50 
iterations. 
(c) The segmentation 
result of TLS at 50 
iterations. 
Figure 2: Comparison result on the “cells” image with same iterations. 
 
(a) Image with Gaussian noise 
(variance = 25). 
(c) Segmentation result of NSLS at 
1000 iterations. 
(d) Segmentation result of TLS at 
1500 iterations. 
(e) Segmentation result of TLS at 
2000 iterations. 
Figure 3: Comparison result on the noisy “liver tumour” image with 
different iterations.
 
 
(a) Image with Gaussian noise 
(variance = 25).
(b) Segmentation result of NSLS at 
50 iterations. 
 
(c) Segmentation result of TLS at 
50 iterations. 
(d) Segmentation result of LS at 
200 iterations. 
Figure 4: Comparison result on the noisy “three cells” image with 
different iterations.
 
An experiment is performed to compare the time 
consumption of NSLS and LS methods. The NSLS takes 
less than 33 seconds per image on average for an AMD 
Phenom(tm) 9500 Quad-core Processor, 2.2 GHz. Table 1 
compares the  computational time on different images for 
different algorithms. The NSLS takes less iteration and 
fewer CPU times than the TLS method. 
 
Image TLS NSLS 
Iteration CPU time (s) Iteration CPU time (s)
Liver tumour 1500 99.57 800  53.03
Cells 50 2.88 50  4.56 
Noisy liver tumour 2000 255.34 1000  66.01 
Noisy cells 200 13.19 50  4.56 
Table 1: Comparison of the CPU times on images with and without 
noise. 
 
From the comparisons, it can be seen clearly that the 
NFLS method has better performance on the image 
segmentation than the traditional level set method with 
high segmentation accuracy and low iteration time.  
 On noisy images, the NFLS segments the most objects 
with entire shape, while the performances of the traditional 
method are affected by the noise and some objects are 
divided into several regions. The results by the NSLS are 
smoother and more connected. Furthermore, the boundary’ 
48 
Neutrosophic Sets and Systems, Vol. 1, 2013 
Yanhui Guo&Abdulkadir Şengür, A Novel Image Segmentation Algorithm Based on Neutrosophic Filtering and Level Set 
position and orientation are more accurate. The outper-
formance benefits from the fact that the NSLS approach 
handles the indeterminacy of the images well and DAMF 
operation remove the effect of noise and other 
indeterminant information, and preserve the determinant 
information in NS domain. 
4 Conclusion 
In this paper, a novel image segmentation approach is 
proposed based on neutrosophic filtering and level set 
theory. The image is transformed into neutrosophic set 
domain, and described using three membership sets (T, I 
and F). The directional alpha-mean filter (DAMF) is 
employed to reduce the image’s indeterminacy, and the 
image is segmented on the T subset after DAMF 
processing using level set algorithm. The experimental 
results show that the proposed method can perform better 
on clear images and noisy images, due to the fact that the 
proposed approach can handle the indeterminacy of the 
images well. The proposed method can be used widely in 
many image processing applications. 
 
 
 
 
 
 
 
 
 
References 
  [1] H. D. Cheng, X. H. Jiang, Y. Sun and J. Wang. Color image 
segmentation: advances and prospects, Pattern Recognition 
34(12) (2001), 2259-2281. 
  [2] F. Smarandache. A Unifying Field in Logics: Neutrosophic 
Logic, Neutrosophy, Neutrosophic Set, Neutrosophic 
Probability. American Research Press, 2003. 
  [3] Y. Guo and H. D. Cheng, "New neutrosophic approach to  
image segmentation," Pattern Recognition, vol.42, pp. 587-
595, 2009. 
  [4] A. Sengur and Y. Guo. Color texture image segmentation 
based on neutrosophic set and wavelet transformation. 
Computer Vision and Image Understanding,vol. 115, no.8, 
pp.1134-1144, 2011. 
  [5] Y. Guo, H. D. Cheng, J. Tian and Y. Zhang. A novel 
approach to speckle reduction in ultrasound imaging. 
Ultrasound in Medicine & Biology, vol.35, 2009. 
  [6] S. Osher and J. A. Sethian, Front propagating with curvature 
dependent speed: Algorithms based on Hamilton Jacobi 
formulation. J. Comput. Phys., vol. 79 (1988), 12–49. 
  [7] R. Malladi, J. A. Sethian and B. C. Vemur. Shape modeling 
with front propagation: a level set approach. IEEE Trans. 
Patt. Anal. Mach. Intell. vol. 17(1995), 158-175. 
  [8] V. Caselles, R. Kimmel and G. Sapiro, “On geodesic active 
contours,” Int. J. Comput. Vis. vol. 22(1997), 61–79. 
  [9] T. F. Chan and L. A. Vese, Active contours without edges. 
IEEE Trans. Image Process. vol. 10(2001), 266– 277. 
Received: October 31, 2013.   Accepted: November 17, 2013. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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A Novel Image Segmentation Algorithm Based on Neutrosophic Filtering and Level Set

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A Novel Image Segmentation Algorithm Based on Neutrosophic Filtering and Level Set

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