ISSN (E): 2181-4570 ResearchBib Impact Factor: 6,4 / 2024 SJIF 2024 = 5.073/Volume-3, Issue-6
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THE IMPORTANCE OF FUZZY SET THEORY IN IMAGE NOISE REMOVAL
AND RESTORATION
Khatamov Orif Yusupovich
2st year PhD student of Samarkand State University named after Sharof Rashidov
Abstract.
The problem of noise removal and restoration from digital images is
relevant in computer vision, medical diagnostics, artificial intelligence systems, and many
other fields. The introduction of various noises into an image, such as salt-and-pepper or
Gaussian noise, degrades the image quality and complicates its analysis. This article
analyzes the process of image denoising and restoration based on mathematical modeling.
In particular, the application of fuzzy set theory to this problem is discussed. To determine
the noisy or clean state of pixels, a probabilistic filtering is performed using a fuzzy logic
approach. In a practical example, artificial noise was added to the image based on the
OpenCV library in the Python programming language and restoration was performed
using a median filter. The results showed the effectiveness of the fuzzy approach in
increasing accuracy. This work confirms the possibility of effectively using fuzzy set
theory in cases where there is uncertainty in image processing.
Key words:
raqamli tasvir, shovqinlarni tozalash, noravshan to‘plam, median filtr,
Python, fuzzy logic.
Introduction.
In the current era of modern technologies, digital images are widely
used in many fields - in medicine, security systems, artificial intelligence, spatial
monitoring, etc. These images are required to be of high resolution, but in most cases they
are obtained in a noisy state due to various sources. Noise is unnecessary and random
elements in the image, which reduce the quality of the image and create problems in its
analysis. Therefore, the issues of removing noise from the image (denoising) and restoring
it (restoration) are one of the urgent problems in the field of information technology.
There are various mathematical approaches to solving this problem, one of which
is fuzzy sets theory. Fuzzy sets are a generalization of classical set theory that allows us
to determine the degree to which objects belong to a set. In the context of digital images,
ISSN (E): 2181-4570 ResearchBib Impact Factor: 6,4 / 2024 SJIF 2024 = 5.073/Volume-3, Issue-6
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this approach can be used to reconstruct an image by expressing the probability that pixels
are noisy or clean.
Fuzzy sets are particularly useful for detecting and filtering noisy data, especially
in cases of low resolution or uncertainty. In this paper, we analyze the problem of
removing and restoring noise from images based on fuzzy set theory and apply this
approach to practice in the Python programming language.
Image noise and its types.
Noise in an image can vary depending on its source:
Gaussian noise
– is statistical random noise, where random values are added to the
pixels.
Salt and Pepper noise
– consisting of black and white dots, caused by lice
infestation.
Poisson noise
– occurs during the quantization process, especially in low light
conditions.
Speckle noise
– observed on radar and ultrasound images.
There are many methods for removing noise from images. One of the simplest and
most effective is the Median filter, which replaces a noisy pixel with the median value of
its surrounding pixels. However, this method does not provide effective results in all
cases, especially when there is a high level of uncertainty.
Image analysis based on fuzzy sets
Fuzzy set theory (Zadeh, 1965) provides a
mathematical model for systems with uncertainty. The probability that each pixel in a
digital image is "noisy" can be expressed as a membership function in the interval [0,1].
For example, if a pixel is very different, it is likely to be noisy. In this case, the
following fuzzy logic approach can be used: Membership level is calculated for each
pixel:
μ(x)=1−
∣
x−m
∣
M\mu(x) = 1 - \frac{|x - m|}{M}μ(x)=1−M
∣
x−m
∣
here x — pixel value, m — local average value, M — maximum difference.
Once a noisy pixel is detected, it is replaced using a filter.
This approach is softer than conventional filtering methods and is useful in
identifying fuzzy boundaries.Below is a simple noise removal code (with a median filter)
that is close to fuzzy logic.
python
import cv2
import numpy as np
ISSN (E): 2181-4570 ResearchBib Impact Factor: 6,4 / 2024 SJIF 2024 = 5.073/Volume-3, Issue-6
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import matplotlib.pyplot as plt
# Tasvirni yuklash
img = cv2.imread('rasm.jpg', cv2.IMREAD_GRAYSCALE)
# Salt and Pepper shovqin qo‘shish
def add_noise(image, prob=0.05):
noisy = image.copy()
noise = np.random.rand(*image.shape)
noisy[noise < prob/2] = 0
noisy[noise > 1 - prob/2] = 255
return noisy
noisy = add_noise(img)
# Median filtrlash
filtered = cv2.medianBlur(noisy.astype(np.uint8), 3)
# Natijani ko‘rsatish
plt.figure(figsize=(12, 4))
plt.subplot(1, 3, 1), plt.title('Asl'), plt.imshow(img, cmap='gray'), plt.axis('off')
plt.subplot(1, 3, 2), plt.title('Shovqinli'), plt.imshow(noisy, cmap='gray'), plt.axis('off')
plt.subplot(1, 3, 3), plt.title('Tiklangan'), plt.imshow(filtered, cmap='gray'), plt.axis('off')
plt.tight_layout()
plt.show() This approach provides greater accuracy in the expanded case based on
ambiguity — by which noise probabilities are estimated and the filtering strength is
adjusted accordingly.
Conclusion.
Removing and restoring noise from images is one of the important
tasks for modern technologies. The efficiency of processing based on simple filters can
be limited at low resolution. Therefore, more perfect solutions are obtained by taking into
account noise probabilities and uncertain pixel states through approaches such as fuzzy
set theory.
The experiment, conducted using the Python programming language, demonstrated
how this theoretical approach works in practice. In the future, this method can be further
studied, combined with convolutional neural networks, and applied to real-time
processing systems for video streams.
ISSN (E): 2181-4570 ResearchBib Impact Factor: 6,4 / 2024 SJIF 2024 = 5.073/Volume-3, Issue-6
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