Linear Diophantine Fuzzy Sets:
Image Edge Detection Techniques based on Similarity Measures
BA
S
¸
AK
ALDEM
I
˙
R
1
, EL
I
˙
F
G
U
¨
NER2, HAL
I
˙
S
A
YG
U
¨
N
2
1Department of Mathematics,
Afyon Kocatepe University,
Ahmet Necdet Sezer Campus,
TURKEY
2Department of Mathematics,
Kocaeli University,
Umuttepe Campus, 41380,
TURKEY
Abstract: - In the digital imaging process, fuzzy logic provides many advantages, including uncertainty
management, adaptability to variations, noise tolerance, and adaptive classification. One of the techniques of digital
image processing is the edge detection. The edge detection process is an essential tool to segment the foreground
objects from the image background. So, it facilitates subsequent analysis and comprehension of the image’s
underlying structural properties. This process can be moved on with the notion of fuzzy sets and their
generalizations. The concept of Linear Diophantine fuzzy sets is a generalization of fuzzy sets where reference
parameters correspond to membership and non-membership grades. This study aims to apply linear Diophantine
fuzzy sets (LDFSs) to edge detection of images. The novelty of this paper is twofold. The first one is that we
conduct a comprehensive evaluation to ascertain the similarity values using the linear Diophantine fuzzy similarity
measure by leveraging the gray normalized membership values associated with fundamental edge detection
techniques. The other is to modify the image pixels into the LDFSs and then filter the images by using the
presented similarity measure operators given in the LDFS environment.
Key-Words: - Image processing, Edge detection, Filtering, Distance, Similarity, Linear Diophantine fuzzy sets.
Received: July 27, 2023. Revised: October 26, 2023. Accepted: December 12, 2023. Published: December 31, 2023.
1 Introduction
In 1965, fuzzy set (FS) theory was given in [1], to
overcome ambiguous and uncertain information.
Then, FS theory has been applied in different areas
from economics, information sciences, computer
sciences, and medical sciences to the social sciences.
While the authors applied this theory effectively,
some of them have stated that there are some
situations where this theory is inadequate. For this
reason, different generalizations of FS theory have
been presented such as intuitionistic fuzzy set (IFS),
Pythagorean fuzzy set (PyFS), picture fuzzy set
(PFS), spherical fuzzy set (SFS), and LDFSs. These
generalized fuzzy sets have been the subject of
numerous studies in the literature, and their
applications have been extensively explored in
various domains. The authors in [2], [3], [4], [5], [6],
[7], [8], have successfully applied these theories to
the decision-making problems. Also, distance,
similarity, and entropy measures have been defined
in the mentioned environment and applied to real-life
problems, [9], [10], [11], [12].
The notion of IFSs was introduced in [13],
according to this theory, every element is defined by
a degree of membership and a degree of non-
membership, with the condition that the total of these
levels does not surpass one. There are a lot of
applications of IFSs such as decision-making, image
processing, expert systems, pattern recognition, and
multi-criteria decision analysis, [14], [15], [16], [17].
Then a more generalized version of IFSs, named
LDFSs, was given in [18], by incorporating reference
parameters to offer a more flexible and efficient
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approach. The theory of LDFSs expands the range of
membership and non-membership degrees through
the use of reference parameters. There are many
application areas where LDFSs find practical use that
have been described in the current literature [19],
[20], [21], [22]. These studies cover the extent and
various domains and include a wide variety of
problem-solving. Moreover, some works on image
processing can be found in [23], [24]. In this work,
we aim to give an application of LDFSs to image
processing that efficiently provides numerous
benefits such as effectively handling uncertainty,
adapting to variations, tolerating noise and enable
adaptive classification. These benefits significantly
increase the precision and adaptability of image
processing, enabling more accurate and flexible
handling of visual data. Image processing consists of
multiple stages and edge detection is a crucial first
step. Accurate edge detection is crucial for the
correct execution of the subsequent stages. Therefore,
we first establish some new similarity measures and
show that an image can be represented as LDFSs.
Then, using the similarity measures, we show that the
edge detection process can be effectively managed
and executed using LDFS principles. In conclusion,
with this work we can show that image processing is
a suitable application area for theoretical
mathematics topics.
2 Preliminaries
2.1
Linear Diophantine Fuzzy Sets
In this subsection, we give the concepts of FSs, IFSs,
and LDFSs. Then, we recall some distance and
similarity measures for LDFSs. Furthermore, we give
the definition of the Minkowski distance measure
which is a generalization of the mentioned distance
measures.
Definition 1
,
[1], [13],
Let U be a non-empty set.
(i) A fuzzy set (FS)
F
on U is given by
where
is the function that
repre
sent the membership of
to the
.
(ii) An intuitionistic fuzzy set (IFS) A on U
is
given by
where
are the functions that
represent the membership and non-membership
function of
to the
, respectively, satisfying
for all
.
Remark 2 Each FS can be taken as an IFS by
considering the non-membership function
. So, the collection of FSs is the
subset of the collection of IFSs.
As a more general form, the notion of LDFSs is
described in [18], in the following manner:
Definition 3, [18], Let be a non-empty set. A
LDFS on is given by
where are the functions that
represent the membership and non-membership
function of to the , respectively, and
denotes the reference parameters
value satisfying , for all
, with . This reference
parameters help us to define or classify a particular
system.
The hesitation value is calculated by
for all .
We use the pair to denote the linear
Diophantine fuzzy number (LDFN) if the conditions
and are satisfied.
The collection of all LDFSs on will be represented
by L.
Remark 4 Each IFS can be taken as an LDFS. That
is, if is an IFS and
the parameter values satisfies
, then we have that
Hence the set
satisfying
and , for all , is
an LDFS.
We obtain the following results from Remark 2
and Remark 4:
Corollary 5 If is a
FS, then we have that the set
satisfying is an LDFS.
Definition 6, [18], A LDFS on L of the form
is called absolute
LDFS and is called
null (empty) LDFS.
Definition 7, [18], Let L. Then the
complement of , represented by , is given by
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.
Algebraic operations between LDFSs were defined in
[18], as follows:
Definition 8, [18], Let L and .
Then,
•
•
•
,
•
.
Proposition 9, [18], For LDFSs and
, , , and are also LDFSs.
The concept of distance measurement is a very
important tool because it shows how different or far
away two objects are from each other. The definition
of distance measure in LDFS environment was given
as follows:
Definition 10, [20], Let be a non-empty set. Then a
mapping is said to be a
distance measure on LDFSs if the following
conditions hold for all L:
• ,
• ,
• if ,
• If then
and .
Theorem 11, [20], Let ,
and the mappings
defined as follows:
Then the mappings and are distance
measures on LDFSs. These distance measures are
called the normalized Hamming distance and the
normalized Euclidean distance, respectively.
In the following, we define the Minkowski
distance between LDFSs:
Definition 12 Let ,
and the mapping defined
as follows:
.
Then the mapping is a distance measure on
LDFSs. This distance measure is called the
normalized Minkowski distance.
Note that, in the definition of the normalized
Minkowski distance, if and , then we
have the definitions of the normalized Hamming
distance and the normalized Euclidean distance,
respectively.
Definition 13, [20], Let be a non-empty set. Then a
mapping is said to be a
similarity measure on LDFSs if the following
conditions hold for all :
• ,
• ,
• if ,
• If then
and .
Next, we show that a similarity measure can be
induced via distance measure:
Theorem 14 Let be a decreasing
function and be a distance measure on LDFSs.
Suppose that the mapping
is defined by
Then the
mapping
is a similarity measure on LDFSs. This
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similarity measure is said to be an f-similarity
measure based on the distance measure on LDFSs.
Now, we give some specific examples to demonstrate
the Theorem 14.
Example 15 (1) Let be defined by
and be a distance measure on
LDFSs. Then, we obtain
(2) Let be defined by
and be a distance measure on LDFSs. Then, we
obtain
(3) Let be defined by
and be a distance measure on LDFSs. Then, we
obtain
Using the distance measures introduced and
described above, we can obtain similarity measures
to use in the process of edge detection as follows;
Corollary 16 Let and
. Then the similarity measures
and
are
obtained as follows:
=(
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/(
2.2 Basic Edge Detection Techniques
Edge detection plays a crucial role in various image-
related tasks such as image processing, image
analysis, image pattern recognition, and computer
vision techniques. The outcome of the edge detection
process on an image provides a collection of
connected curves that represent object boundaries,
surface markings, and variations in surface
orientation. By applying an edge detection algorithm
to an image, it becomes possible to significantly
reduce the data volume for processing, effectively
filtering out less relevant information while
preserving the essential structural characteristics of
the image. Successful execution of the edge detection
step allows for a simplified interpretation of the
information contained in the original image.
Brief information about the commonly used edge
detection techniques Sobel, Prewitt, LoG (Laplacian
of Gaussian), Canny and Roberts are given below
and these techniques are applied for a grey level
image on MATLAB and given in Figure 1, Figure 2,
Figure 3, Figure 4 and Figure 5.
In the Sobel technique, edges are identified by
employing a image filter in the local
neighborhood. This technique gives some smoothing
effect against the random noise of the image, making
the edges appear thicker and brighter.
Fig. 1: The results of Sobel technique
The Prewitt technique which is very similar to
the Sobel technique estimates edge detection by
utilizing a simplified image filter in the local
neighborhood.
Fig. 2: The results of Prewitt technique
The Laplacian technique detects edges by
searching for points where the second derivative
crosses zero. The second derivative is more effective
in capturing fine details compared to the first
derivative. However, it has the disadvantage of being
highly sensitive to noise. Therefore, to mitigate this,
a Gaussian filter is applied beforehand to remove
noise.
Fig. 3: The results of LoG technique
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The Canny technique is designed to reduce noise,
emphasize actual edges, and increase the sensitivity
of edge detection. It involves several steps to identify
edges. Firstly, it uses a Gaussian filter to reduce the
details and noise in the image, finds the edge
candidates by determining the direction and
magnitude of the gradient for each pixel, and then
selects the most suitable edges by removing the weak
pixels with the thresholding method.
Fig. 4: The results of Canny technique
The Roberts method employs two separate 2x2
convolutional masks to compute the gradient
magnitude within the image. This involves applying
these masks to specific pixel neighborhoods through
a process known as convolution, enabling the
derivation of gradients across the image.
Fig. 5: The results of Roberts technique
To facilitate a comparative analysis among the
aforementioned edge detection techniques, each
method is individually employed on the grayscale
version of a different image. Subsequently, a
collective presentation of the outcomes is provided in
the combined Figure 6 for easy comparison.
Fig. 6: Comparison of edge detection techniques
Fuzzification of an image is particularly useful
when working with algorithms or techniques that
require a specific range of inputs, such as image
processing algorithms based on normalized values.
Generally, fuzzification of an image involves
rescaling pixel values to the range [0, 1], this can be
made in various ways using different membership
functions. This process allows for easier analysis and
processing of image data while ensuring consistency
and compatibility with image processing techniques
and algorithms.
The above-mentioned techniques are employed
on the image after fuzzification, and the resulting
outputs are displayed in Figure 7.
Fig. 7: Edge detection for fuzzified image
3 Modification of an Image to LDFS
The process of modification of an image into LDFS
entails the incorporation of the specific characteristic
of being LDFN into the data, wherein said data
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possesses pixel values falling within the predefined
range of [0, 1]. For example, if image A satısfies the
condition where
is its th pixel and are randomly
chosen, then we can define each pixel of A as
. Therefore,
it can be argued that every image has the potential to
be expressed and characterized within the LFDS
framework.
Implementing such a modification facilitates the
seamless utilization of operators specifically defined
for LDFS, enabling straightforward integration into
various operations and processes applied to the
image.
The modified version of the Lena image and
traditional edge detection filters applied can be seen
in Figure 8 and Figure 9 for different and .
Fig. 8: Edge detection for modified image with
and
Fig. 9: Edge detection for modified image with
and
The modification has some disadvantages as well
as advantages. As seen in Figure 8 and Figure 9, this
process does not work in harmony with the LoG filter
for the selected alpha and beta values and does not
give a clear output. However, compatible alpha and
beta values can be found. The results obtained from
the LoG filter for some values of and are given
in Figure 10 below.
Fig. 10: Results of LoG filter: (a) and
, (b) and , (c)
and
On the other hand, it is imperative to
acknowledge its performance, mainly when operating
with filters rooted in matrix-based methodologies.
This noteworthy observation underscores the
remarkable compatibility between this approach and
the application of matrix-oriented filtering
techniques.
4 An Application of the LDFSs
Similarity for Edge Detection
The similarities between each edge detection filter
were calculated based on the similarity measures
obtained from the above-mentioned distance
measures and by using random parameters.
Different outcomes are obtained when employing
various parameters, which are randomly
chosen. To illustrate, specified values for
each filter given in Table 1 are used and the
corresponding outcomes are presented in Table 2,
Table 3 and Table 4.
Table 1. Parameters for filters
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Table 2. Similarity results under
,
and
Table 3. Similarity results under
,
and
Table 4. Similarity results under
,
and
The similarities between the fuzzified images
obtained from each edge detection technique were
calculated with the defined similarity measures.
According to the results obtained, it is seen that
the Sobel and Prewitt filters are largely similar to
each other. This similarity gives close results even
if the alpha and beta parameters change. Both
filters are popular image processing methods used
for edge detection. Both perform a gradient
calculation to detect the horizontal and vertical
edges in the image and use the differences between
pixel values to determine the boundaries. However,
there are also some differences. For example, the
Sobel filter does more computation than the Prewitt
filter and may therefore have a higher
computational cost. Also, the Sobel filter may have
a better edge redirection ability than the Prewitt
filter. In general, however, Sobel and Prewitt filters
give similar results and are interchangeable.
We can apply the above calculations on the
modified image and filters, similarly. The results
obtained by using the alpha and beta values in
Table 1 are shown in Table 5, Table 6 and Table 7.
Table 5. Similarity results of modified image
filters under
,
and
Table 6. Similarity results of modified image
filters under
,
and
Table 7. Similarity results of modified image
filters under
,
and
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5 Conclusion
In this paper, we combine the structure of LDFSs
with edge detection techniques using the presented
similarity measures defined on LDFSs to generalize
the fuzzy edge detection domain. We give
numerical results of this technique on some images,
analyze the validity of the techniques, and show
these results in tables. Different generalizations of
FS theory can be applied to the edge detection
process for future work. Some different operators
can be created to be used in this process. Moreover,
LDFSs can be applied to problems such as image
denoising and recognition.
Acknowledgement:
The authors are thankful to the anonymous referees
for their valuable suggestions.
References:
[1] Yang, H., Song, H., Li, W., Qin, K., Shi, H.,
Jiao, Q.: Social image annotation based on
image captioning. WSEAS Transactions on
Signal Processing, 18, 109-115 (2022),
https://doi.org/10.37394/232014.2022.18.15.
[2] Aydoğdu, E., Güner, E., Aldemir, B., Aygün,
H.: Complex spherical fuzzy TOPSIS based
on entropy. Expert Systems with
Applications, 215, 119331 (2023), doi:
10.1016/j.eswa.2022.119331.
[3] Aydoğdu, E., Aldemir, B., Güner, E., Aygün,
H.: A novel entropy measure with its
application to the COPRAS method in
complex spherical fuzzy environment.
Informatica, 1-33 (2023), doi: 10.15388/23-
INFOR539.
[4] Dursun, M., Goker, N.: Fuzzy cognitive map
methodology for evaluating material
selection criteria. Engineering World, 2, 15-
19 (2020).
[5] Jurio, A., Paternain, D., Bustince, H., Guerra,
C., Beliakov, G.: A construction method of
Atanassov’s intuitionistic fuzzy sets for
image processing. In 2010 5th IEEE
International Conference Intelligent Systems
pp. 337-342, (2010), doi:
10.1109/IS.2010.5548390.
[6] Kilic, M., Dursun, M., Goker, N.: Evaluation
of R&D projects using fuzzy MCDM
method. Engineering World, 2, 150-154
(2020).
[7] Mahmood, T., Rehman, U.U.: A novel
approach towards bipolar complex fuzzy sets
and their applications in generalized
similarity measures. International Journal of
Intelligent Systems, 37.1: 535-567 (2022).
doi: 10.1002/int.22639.
[8] Srebrenkoska, S., Apostolova, A., Dzidrov,
M., Krstev, D.: Application of Analytical
Hierarchy Process (AHP) in the selection of a
flexible production system. Engineering
World, 5, 138-143 (2023),
https://doi.org/10.37394/232025.2023.5.15.
[9] Cuong, B. C., Kreinovich, V.: Picture fuzzy
sets-A new concept for computational
intelligence problems. In 2013 third world
congress on information and communication
technologies (WICT 2013), pp. 1-6, (2013),
doi: 10.1109/WICT.2013.7113099.
[10] Ejegwa P.A.: Distance and similarity
measures for Pythagorean fuzzy sets.
Granular Computing, 5(2): 225-238 (2020),
doi: 10.1007/s41066-018-00149-z.
[11] Xuecheng, L.: Entropy, distance measure and
similarity measure of fuzzy sets and their
relations. Fuzzy Sets and Systems, 52(3), 305-
318 (1992), doi: 10.1016/0165-
0114(92)90239 Z.
[12] Zadeh, L.A.: Fuzzy sets. Information and
Control, 8(3), 338-353 (1965).
doi:10.1016/S0019-9958(65)90241-X
[13] Atanassov, K.: Intuitionistic fuzzy sets. Fuzzy
Sets and Systems, 20(1), 87-96 (1986), doi:
10.1016/s0165-0114(86)80034-3.
[14] Bustince, H., Barrenechea, E., Pagola, M.,
Orduna, R.: Image thresholding computation
using Atanassov’s intuitionistic fuzzy sets.
Journal of Advanced Computational
Intelligence and Intelligent Informatics,
11(2), 187-194 (2007), doi:
10.20965/jaciii.2007.p0187.
[15] De, S.K., Biswas, R., Roy, A.R.: An
application of intuitionistic fuzzy sets in
medical diagnosis. Fuzzy Sets and Systems,
117(2), 209-213 (2001), doi: 10.1016/S0165-
0114(98)00235-8.
[16] Liu, S., Zhang, J., Niu, B., Liu, L., He, X.: A
novel hybrid multi-criteria group decision-
making approach with intuitionistic fuzzy
sets to design reverse supply chains for
COVID-19 medical waste recycling
channels. Computers and Industrial
Engineering, 169, 108228 (2022), doi:
10.1016/j.cie.2022.108228.
[17] Xu, Z.S., Chen, J.: An overview of distance
and similarity measures of intuitionistic
fuzzy sets. International Journal of
Uncertainty, Fuzziness and Knowledge-
Based Systems, 16(04), 529-555 (2008), doi:
WSEAS TRANSACTIONS on SIGNAL PROCESSING
DOI: 10.37394/232014.2023.19.22
Basak Aldemir, Elif Guner, Halis Aygun
E-ISSN: 2224-3488
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Volume 19, 2023
10.1142/S0218488508005406.
[18] Riaz M., Hashmi M.R.: Linear Diophantine
fuzzy set and its applications towards multi-
attribute decision-making problems, Journal
of Intelligent and Fuzzy Systems, 37(4):
5417-5439 (2019), doi: 10.3233/JIFS-
190550.
[19] Aldring, J., Santhoshkumar, S., Ajay, D.: A
decision-making approach using linear
diophantine fuzzy sets with Dombi
operations. In International Conference on
Intelligent and Fuzzy Systems. Cham:
Springer International Publishing, 684-692
(2022). doi:10.1007/978-3-031-09176-6-76.
[20] Gül S., Aydoğdu A.: Novel distance and
entropy definitions for linear Diophantine
fuzzy sets and an extension of TOPSIS
(LDF-TOPSIS). Expert Systems, 40(1):
e13104 (2023), doi:10.1111/exsy.13104.
[21] Kamacı, H.: Linear Diophantine fuzzy
algebraic structures. Journal of Ambient
Intelligence and Humanized Computing,
12(11), 10353-10373, (2021).
[22] Mohammad, M.M.S., Abdullah, S., Al-
Shomrani, M.M.: Some linear Diophantine
fuzzy similarity measures and their
application in decision-making problem.
IEEE Access, 10, 29859-29877, (2022), doi:
10.1109/ACCESS.2022.3151684.
[23] Greer, K.: Recognising image shapes from
image parts, not neural parts. WSEAS
Transactions on Signal Processing, 19, 77-82
(2023),
https://doi.org/10.37394/232014.2023.19.9.
[24] Shanthi S.A., Valarmathi R.: Edge detection
on fuzzy near sets, Materials Today:
Proceedings, 51: 2504-2511 (2022), doi:
10.1016/j.matpr.2021.12.120.
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