Fuzzy Numbers and Analysis of Radiological Images

Abstract

Fuzzy sets are a mathematical notion that allows us to represent uncertainty by assigning a degree of membership to a value in a range. This degree of membership represents the degree to which a particular value belongs to the fuzzy set. Fuzzy numbers are fuzzy sets that satisfy specific conditions, and can be applied in the interpretation of the behavior of neural networks. In many real-world problems, the output of a neural network may not be a precise numerical value, but rather a range of values that are subject to uncertainty. In the context of neu-ral networks, fuzzy numbers can be used to represent uncertainty in the output of the network. Fuzzy numbers are used in applications where the representation of ambiguity and uncertainty in numerical data is explicitly desired. This paper will consider discretizations and defuzzifi-cations triangular fuzzy numbers with the use of MATLAB software. MATLAB functions were applied to the analysis of kidney images, where the fuzzy numbers represent the tendency value of the degree of redness, which serves as an indicator for classifying the image as normal or abnormal based on different characteristics.

Full text

Fuzzy Numbers and Analysis of Radiological
Images
Irma Ibriˇsimovi´c1[0000−0002−9361−0259], Bratislav Iriˇcanin2 [0000−0001−7457−7716],
Nataˇsa Milosavljevi´c3[0000−0003−4056−089X], Ljubo
Nedovi´c4[0000−0002−2650−3218], and Nebojˇsa Ralevi´c5 [0000−0002−3825−9822]
1Faculty of Science, Depertment of Mathematics, University of Tuzla, Bosnia and
Herzegovina http://mat.pmf.untz.ba/
Faculty of Technical Sciences, Department of Fundamental Sciences, University of
Novi Sad, Serbia
irmaibrisimovic94@gmail.com
http://ftn.uns.ac.rs/n1386094394/faculty-of-technical-sciences
2Faculty of Electrical Engineering, Department of Applied Mathematics, University
of Belgrade, Serbia
iricanin@etf.rs
https://www.etf.bg.ac.rs/en/departments/the-department-of-applied-mathematics
3Faculty of Agriculture, Department of Mathematics and Physics, University of
Belgrade, Serbia
natasam@agrif.bg.ac.rs https://www.agrif.bg.ac.rs/
4Faculty of Technical Sciences, Department of Fundamental Sciences, University of
Novi Sad, Serbia
nljubo@uns.ac.rs http://ftn.uns.ac.rs/n1386094394/faculty-of-technical-sciences
5Faculty of Technical Sciences, Department of Fundamental Sciences, University of
Novi Sad, Serbia
nralevic@uns.ac.rs http://ftn.uns.ac.rs/n1386094394/faculty-of-technical-sciences
Abstract. Fuzzy sets are a mathematical notion that allows us to rep-
resent uncertainty by assigning a degree of membership to a value in a
range. This degree of membership represents the degree to which a par-
ticular value belongs to the fuzzy set. Fuzzy numbers are fuzzy sets that
satisfy specific conditions, and can be applied in the interpretation of the
behavior of neural networks. In many real-world problems, the output
of a neural network may not be a precise numerical value, but rather a
range of values that are subject to uncertainty. In the context of neu-
ral networks, fuzzy numbers can be used to represent uncertainty in the
output of the network. Fuzzy numbers are used in applications where
the representation of ambiguity and uncertainty in numerical data is
explicitly desired. This paper will consider discretizations and defuzzifi-
cations triangular fuzzy numbers with the use of MATLAB R
O
software.
MATLAB R
O
functions were applied to the analysis of kidney images,
where the fuzzy numbers represent the tendency value of the degree of
redness, which serves as an indicator for classifying the image as normal
or abnormal based on different characteristics.
Keywords: fuzzy numbers, fuzzy number representation, defuzzifica-
tion, discretizations

2 Irma Ibriˇsimovi´c et al.
1 Introduction
This paper will describe the application of fuzzy number in the interpretation
of the work of neural networks with the use of MATLAB R
O
software. Medical
imaging is an essential tool for diagnosing and treating various medical condi-
tions. However, the interpretation of medical images can be a challenging task
due to the complexity and the variability of data. Fuzzy logic provides a powerful
framework for dealing with uncertainty and vagueness in data that is common
in medical imaging. Fuzzy numbers are a key component of fuzzy logic and can
be used to represent uncertain and vague data more precisely.
The first part of the paper refers to the definition of the fuzzy numbers and
methods[11] that are essential for implementation in the MATLAB R
O
software.
The second part of the paper concerns the application of triangular fuzzy num-
bers to medical diseases, more precisely in this case to kidney disease. Medical
image segmentation based on fuzzy numbers [25], fuzzy modeling for medical
image processing [21], behavioral neural networks on fuzzy systems [10] and tri-
angular fuzzy numbers for medical image segmentation [22] are key elements of
this paper, because the simulation of their application was done in MATLAB R
O
software. In the end, the analysis of the image was done and as an input it gives
graphic fuzzy numbers that show the tendency of the value of the degree of red-
ness, which represents an abnormality in the observed photographs (see Figure
2).
2 The definition of fuzzy number
A fuzzy number is a mathematical concept used in uncertainty theory. It is a
number that has an indeterminate value that is between two limits. In other
words, a fuzzy number is a number that has uncertainty in its value, which can
be described using a membership function. A membership function is a mathe-
matical tool used to describe uncertainty in values. It determines how much an
element is connected to a group. For the case of fuzzy numbers, the membership
function is used to describe how closely a value is related to a number. Fuzzy
numbers often are used in various fields, such as finance, engineering, data sci-
ence, etc. For example, in finance they are used for risk modelling and market
price estimation. In engineering, they are used to model uncertainty in design
and planning. In data science, they are used to analyze data that is not obtained
completely precise. There are various operations that can be applied to fuzzy
number, such as addition, subtraction, multiplication, and division, which are
performed using membership functions (see [1], [2],[3],[11]).
Definition 1. If X=∅and µA:X→[0,1], then the ordered pair (X, µA)is
called the fuzzy set A. Mapping µAis membership function and is often
identified with the fuzzy set.
Definition 2. Fuzzy set Ais normalized if µA(x) = 1,for someone x∈X.

Fuzzy Numbers and Analysis of Radiological Images 3
Definition 3. Set αA= [A]α={x∈X|µA(x)≥α}is called α-cut fuzzy set
A.
Definition 4. Fuzzy set Adefined above Ris convex if every α-cut is convex.
Definition 5. Fuzzy set Adefined above Ris fuzzy number if it is normalized
and convex.
Definition 6. The set supp A={x∈X|µA(x)>0}is called support of the
fuzzy set A.
Definition 7. The set ker A={x∈X|µA(x) = 1}is the kernel of the fuzzy
set A.
3 Methods
In this paper we focus on triangular fuzzy numbers which are commonly used
in fuzzy logic. We provide MATLAB R
O
implementations for the discretization
and defuzzification of these fuzzy numbers. Discretization involves converting a
fuzzy numbers into a set of discrete values, which can be useful for analysis and
interpretation. Defuzzification involves calculating a crisp value that represents
the central tendency of the fuzzy numbers, and can be useful for decision-making
and classification (see [5],[6],[8]).
Definition 8. The triangular fuzzy number Ais the fuzzy number the mem-
bership function of which is:
A(x) = 


x−a
b−a, a < x ≤b,
c−x
c−b, b < x < c,
0, x ≤aor c≤x
.
The triangular fuzzy number Ais often presented as an ordered triplet of num-
bers A= (a, b, c) (see [11]). Triangular fuzzy number whose core is {b}(a is not
necessary indicate support) is usually marked with ˆ
b.
Fig. 1: The membership function of triangular fuzzy numbers

4 Irma Ibriˇsimovi´c et al.
We apply these methods to medical images to demonstrate their usefulness in
analyzing and interpreting medical data. For example, we can use triangular
fuzzy numbers to represent the degree of redness (the observed area is marked
by red circles) in a certain area of a kidney image. We can then discretize this
fuzzy number and use the center of gravity method to calculate central tendency
of the fuzzy number, which can help in the classification of the kidney image as
normal or abnormal, based on the various features of the image.
Suppose we have a set of kidney images and we want to determine if a particular
image is normal or abnormal based on the degree of redness in a certain area
of the image. We can use fuzzy numbers to represent the degree of redness in
the area. Assume that the degree of redness can be represented by a triangular
fuzzy number with the membership function shown below, where a, b, and care
the lower, middle, and upper bounds of the fuzzy number. Suppose that we have
a kidney image with the following degree of redness in the area of interest. To
apply the center of gravity method, we first need to discretize the fuzzy number.
We can use the discrete set of values for the fuzzy number. Then we can calculate
the degree of membership of each discrete value using the triangular membership
function. The degree of membership of the value xin the support fuzzy number
can be calculated as follows
trimf (x;a, b, c, ) = max min x−a
b−a,c−x
c−b,0,
where trimf represents the label for triangular membership function. For exam-
ple, the degree of membership for the x= 6 and fuzzy number ˆ
7 = (5,7,9) is
calculated
µA(6) = max min 6−5
7−5,9−6
9−7,0= 0.5.
Next, we can calculate the center of
gravity (Cog) of the fuzzy number using the
following formula
Cog =sum(i∗µ(i))/sum(µ(i)),
where iis the index of the discrete value,
µ(i) is the degree of membership of the
value iin the fuzzy number, and the sum-
mation is over all discrete values.
Finally, we can use this central tendency
value to classify the kidney image as normal or abnormal based on some prede-
termined criteria. For example, if the central tendency value is less than a certain
threshold, we can classify the image as normal, otherwise, we can classify it as
abnormal (see [1],[7],[8],[9],[10]).
This is a simple example of how we can use fuzzy numbers and the center
of gravity method to analyze and interpret medical images. Fuzzy logic and its
related techniques can have a significant impact on medical images processing.

Fuzzy Numbers and Analysis of Radiological Images 5
Medical images often are complex, and there may be uncertainty and ambigu-
ity in the interpretation of image features. Fuzzy logic allows us to represent
this uncertainty and ambiguity through fuzzy numbers, which can then be dis-
cretized and defuzzified to provide more meaningful information. For example,
for the case of kidney images, the degree of redness in a certain area may be
an important indicator of kidney health. However, there may be variation in
the degree of redness, and it may be difficult to determine a precise threshold
for what constitutes abnormal redness. Fuzzy logic allows us to represent this
variation and uncertainty using fuzzy numbers, and we can then use techniques
such as the center of gravity method to extract meaningful information from the
fuzzy numbers (see [12],[15],[21]). Overall, the application of fuzzy logic to med-
ical images can improve the accuracy and the reliability of image analysis and
interpretation, which can ultimately lead to better diagnosis and consequently
better treatment of medical conditions.
To use the center of gravity method for a triangular fuzzy number, you can
use the following code in MATLAB R
O
:
- Define triangular fuzzy numbers and then define the range of values for dis-
cretization of triangular fuzzy number (a, b, c) with the feature a= (minrange), c =
(maxrange ) and bis randomly created between aand c.
- Calculate membership function values, where trimf(x, A) is a function to
calculate the membership function values for each value in the range µA=
trimf (x, A).
- Calculate the center of gravity cog =sum(x. ·µA)/sum(µA).
- We then define a range of values for discretization and use the trimf func-
tion, and finally, we use the center of gravity formula to calculate the central
tendency of the fuzzy number (see [4],[13],[14],[16],[17],[18],[19]).
An expansive mass in the left kidney can be seen in Figure 2, which is a
CT (Computer Tomography) scan of the abdomen. Pathohistologically, it was
confirmed that it was a renocellular carcinoma. For each description, it is rec-
ommended that the physician observe the following changes: location, shape,
size, limitation, intensity, and heterogeneity/homogeneity. Certainly, in order to
better distinguish changes during CT and MRI (Magnetic Resonance Imaging)
scans, it is recommended to use a contrast agent, and to scan the patient at
different time intervals determined by the speed of the passage of the contrast
agent through the vascular structures. The CT device is essentially based on the
various abilities of tissues to absorb or reflect X-rays, which are then analyzed on
detectors and presented to us in the form of images through computer process-
ing. The fuzzy numbers represents the degree of redness in the circled region of
the kidney image. The base of the fuzzy numbers represents the range of redness
values that can be observed, with the peak of the fuzzy number representing the
most likely or central value. The graph of the fuzzy numbers represents the degree
of uncertainty in the redness measurement (see [15],[20],[21],[22],[23],[24],[25]).
The discretization and defuzzification of the fuzzy numbers can then be per-
formed using methods such as the center of gravity method. This can provide

6 Irma Ibriˇsimovi´c et al.
(a) (b)
(c) (d)
Fig. 2: Original pictures
a central tendency value for the degree of redness, which can be used in the
classification of the kidney image as normal or abnormal based on the various
features of the image (see Figure 3).
(a) (b) (c) (d)
Fig. 3: Fuzzy number representing degree of redness on Figure 2
4 Conclusion
The application of fuzzy number in the interpretation of the work of neural
networks can be useful for assessing uncertainty in data and for understanding
the role of individual variables in decision-making process. Fuzzy logic makes it
possible to evaluate variables that are not completely defined.

Fuzzy Numbers and Analysis of Radiological Images 7
In this paper it was shown that MATLAB R
O
software can be a useful tool for im-
plementing fuzzy logic and neural networks. MATLAB R
O
offers many functions
and tools that allow users to easily model and analyze complex systems. This
paper presents examples of the application of these methods to areas of medical
imaging and improve the diagnosis and treatment of various medical conditions.
Our results demonstrate that the use of fuzzy number and their discretizations
and defuzzifications can provide a powerful tool for the analysis of medical im-
ages. By representing uncertain and vague data in a more precise manner, we
can improve the accuracy and the reliability of medical imaging analysis. The
MATLAB R
O
implementations are used in this paper to analyze and interpret
medical data with greater accuracy and reliability, demostrating that fuzzy logic
and neural networks can be successfully applied in different situations. The li-
brary of original medical images recorded/obtained in the radiology office in
Tuzla was analyzed. Based on the marked red part in Figure 2, determined by
the application of the mentioned procedure, the success of the answer to the
question whether the resulting changes in parts of the kidney are abnormal was
tested. The direction of further research will include the publication of adequate
results and comparison with appropriate research.
Finally, this paper provides useful guidelines and examples for the application of
fuzzy logic and neural networks in real-world situations. These tools are useful
in many fields, including engineering, science, economics, and others, where we
encounter uncertainties and complex systems.
Acknowledgements
The authors has been supported by the Ministry of Education, Science and
Technological Development of the Republic of Serbia through the project no.
451-03-47/2023-01/200156: ”Innovative scientific and artistic research from the
FTS (activity) domain”.
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