Rayleigh-Monte Carlo Method for Image Noise Detection and Elimination

doi:10.21203/rs.3.rs-2962357/v1

The issue of identifying areas of distortion in images is a common problem encountered in digital media processing. Traditional filtering methods often rely on setting thresholds that are dependent on the specific image being processed, leading to imprecise results. Adaptive filters have been proposed as a solution, but they are not without their technical difficulties. This paper presents a study on the effectiveness of the Rayleigh distribution method in identifying distortion areas in images. Monte Carlo estimation was used to estimate the distribution of image content, and the results showed that the Rayleigh distribution is highly efficient at identifying areas of distortion. The filtering process was then applied only to the regions with deformation, while ensuring that all other features of the image were preserved. The novelty of this study lies in the use of the Rayleigh distribution method to identify areas of distortion in images. The purpose of this study was to improve the quality of images by using a more precise filtering method that only targets distorted areas. The theoretical framework of this study is based on the application of the Rayleigh distribution method in digital media processing. The approach used in this study involved the use of Monte Carlo estimation to determine the effectiveness of the Rayleigh distribution method. The findings of this study show that the Rayleigh distribution method is highly effective in identifying areas of distortion in images, leading to higher quality images than traditional filtering methods. The research findings have practical implications for digital media processing industries, where the quality of images is paramount. By using the Rayleigh distribution method, images can be processed more efficiently, leading to improved quality and reduced processing time. Socially, improved image quality can have a positive impact on areas such as medical imaging, where accurate and high-quality images are crucial for diagnosis and treatment. Overall, this study offers an original and effective solution to the problem of identifying areas of distortion in images. By using the Rayleigh distribution method, images can be processed more precisely, leading to higher quality results. The use of Monte Carlo estimation provides a solid foundation for future research in this area.

Rayleigh Distribution

Monte Carlo Simulation

Image Processing

Noise is a common phenomenon that affects images captured by digital cameras, video cameras, and medical imaging devices. Image noise can be caused by multiple factors such as sensor limitations, low light conditions, and transmission errors. Image noise can impact the quality of an image and can make it hard to interpret. Therefore, image noise detection and elimination is necessary for image processing and analysis. Mathematics is the foundation of image processing due to its diverse possibilities of application in all fields of scientific and practical study. Applied mathematics plays an important role in engineering and applied studies [1].

Impulse noise is a type of image noise that occurs randomly in an image. Impulse noise can be caused by a malfunctioning sensor or transmission errors. Impulse noise is characterized by a sudden change in pixel intensity, resulting in the presence of black or white dots in the image. Classical median filter support is a popular technique used to remove impulse noise from images. However, the classical median filter support method is computationally expensive and may not be efficient for processing large images [2].

Statistics is concerned with the process of collecting, presenting, summarizing and extracting data results. Data consistency is one of the most important factors that help draw conclusions and solve problems through the charts that are created [3]. The process of collecting data and arranging it in graphical forms is a key factor in the applications of physics, social and human sciences, politics and economics [4]. Information is classified to produce evidence, indicators, and clues through collecting and summarizing data and arranging them statistically in the form of charts [5].

Performing statistical analysis requires an understanding of the data and its context and a solid grasp of basic inference theory [6]. Schools of statistical inference focus on several different principles, the principle of repeated sampling, which depends on the arithmetic mean, and the principle of data collection, which depends on maintaining sufficient data [7, 8]. The classic iterative sampling methods are considered one of the most important methods used, but the Bayesian methods have become more common now as a result of the development of computers and the emergence of smart arithmetic methods [9].

The Rayleigh distribution is a probability distribution used to describe the magnitude of a random variable with phase and amplitude. The Rayleigh distribution is commonly used in image processing to model the magnitude of image noise. In the Rayleigh-Monte Carlo method, the Rayleigh distribution is used to simulate image noise. Monte Carlo estimation is a numerical method used to approximate the probability of an event by generating random samples from a given probability distribution. In the Rayleigh-Monte Carlo method, Monte Carlo estimation is used to estimate the probability of detecting image noise.

These classifications are based mainly on the predictions of statistical models, which may not correspond to reality [10]. This was treated using Monte Carlo simulation, which is a multiple probability simulation [11]. The Monte Carlo method is a mathematical technique used to estimate the probable outcomes of uncertain events that relies on finding a probability distribution that fits the frequency gradations. The Monte Carlo method simulates changes with the same distribution and the same real values [12].

Monte Carlo simulation is used because of its high accuracy in prediction. Monte Carlo simulation results in a set of possible outcomes and the probability of occurrence of each outcome depending on the increase in the number of inputs, resulting in a more accurate prediction [13–15].

The topic of digital image enhancement is one of the areas of digital image processing due to the explosive growth of information technology technologies and their applications in all medical, industrial and military fields [16–18].

The Rayleigh-Monte Carlo method for image noise detection and elimination is essential in digital image processing. The method's ability to simulate image noise accurately and estimate the probability of detecting image noise makes it a valuable tool for image analysis. The Rayleigh-Monte Carlo method can be used to evaluate the performance of image processing algorithms in the presence of image noise. Moreover, the method can be used to optimize image processing algorithms to achieve better results. In a recent study, the Rayleigh-Monte Carlo method was used to evaluate the performance of a de-noising algorithm in the presence of image noise. The study showed that the de-noising algorithm achieved better results when tested with the Rayleigh-Monte Carlo method compared to other noise simulation methods. The Rayleigh-Monte Carlo method was also used to optimize the parameters of a de-noising algorithm, resulting in improved de-noising performance.

This paper presents a new design for removing impulse noise from color images using statistical methods. Rayleigh distribution is used to detect noise points in the corrupted images with the help of Monte Carlo simulation estimation. The Classic filter only applies to distorted areas and does not filter out undistorted areas. Experimental results show higher quality images compared to conventional filtering methods.

In conclusion, the Rayleigh-Monte Carlo method for image noise detection and elimination is an essential tool in digital image processing. The method's ability to simulate image noise accurately and estimate the probability of detecting image noise makes it a valuable tool for image analysis. The Rayleigh-Monte Carlo method can be used to evaluate the performance of image processing algorithms in the presence of image noise and optimize these algorithms to achieve better results. With the increasing importance of digital image processing in various fields, the Rayleigh-Monte Carlo method is an area of ongoing research, and future studies will undoubtedly uncover new applications and improvements of the method.

The industrial and engineering administration relies in its statistical analysis on the normal distribution, which is similar in shape to the shape of a bell [19]. The normal distribution curve is one of the most common curves in use [20, 21]. In some statistical processes the data distributions follow a behaviour that constitutes an abnormal distribution [22] (Fig. 1).

In real life, the data may follow abnormal distributions such as the Weibull distribution [for several reasons [23]:

If the data is skewed.
When the data is bimodal or multivariate.
When the data is not enough.
Incorrect graph of the data.

In these cases, it is necessary to search for an appropriate distribution commensurate with the nature and form of the information [24]. In this research, a special case of the Weibull distribution was studied, which is the Rayleigh distribution method. The Rayleigh distribution is one of the widely used continuous probability distributions in various fields, including:

Theories of calculating the path of signals in communications.
theories of the study of natural sciences.
Feasibility studies.
Analysis of medical images.

The Rayleigh distribution is a special case of the Weibull distribution. The Whipple distribution depends on two degrees of freedom, which corresponds to the Kay distribution, while the Rayleigh distribution can be used using one degree of freedom, which makes the Rayleigh distribution simpler [25].

The probability density function of the Rayleigh distribution is given in the following form:

$$f\left(x\right)=\frac{x{e}^{-\frac{{x}^{2}}{\left(2{s}^{2}\right)}}}{{s}^{2}}, \text{f}\text{o}\text{r} \text{x}\in \left[\left.(0,{\infty }\right)\right.] \text{a}\text{n}\text{d} \text{p}\text{a}\text{r}\text{a}\text{m}\text{e}\text{t}\text{e}\text{r} \text{s}$$

With mean

$$\mu =s\sqrt{\frac{\pi }{2}}$$

And Variance

$${\sigma }^{2}=\frac{4-\pi }{2}{s}^{2}$$

As the shape parameter increases, the distribution widens. After calculating µ and σ² from the spatial data of the experimental sample we need to find the scale parameter from the relation:$s=\frac{\sqrt{2 }\mu }{\sqrt{\pi }}$

or

$$s=\sqrt{\frac{2{\sigma }^{2}}{4-\pi }}$$

The Rayleigh distribution relates the overall magnitude of a vector to its directional component such that all components are vectors in two-dimensional orthogonality [26, 27]. When each part of it has its own correlation with equal variance and Gaussian with zero mean the overall shape can be described by Rayleigh distribution. The shape of the curve becomes more Gaussian with an increase in the number of data N. The peak of the curve when taking out the probability distribution shifts to the right. When N → ∞, the Rayleigh channel becomes Gaussian (Fig. 2).

Impulse noise appears on the grayscale images in the form of black and white dots as a result of the image sensors not distinguishing the basic information during the period of information transmission or decoding processing operations. In color images, this noise appears in the form of colored dots that differ in the topography of the image, which leads to the loss of the natural features of the image according to the degree or percentage of the resulting noise. We resort to the filtering process to re-smooth the image after removing the noise. One of the most important basic requirements when resorting to the nomination process is to take into account the following

Not losing important information, especially borders and edges.
Clear appearance or good visual effect.

Median filtering is one of the most widely used algorithms for impulse noise removal. Medium filtering belongs to the field of nonlinear filtering techniques. The idea of the median filter depends on replacing the pixel value with one of the values surrounding the pixel after arranging it in ascending or descending order and taking the middle value based on the principle of statistical sorting. Signal-to-noise ratio (PSNR) is used to measure the degree of image noise, as well as when measuring the filter performance when removing noise from an image.

The detection process of salt and pepper splash noise is particularly visible to the naked eye as the values around pixels differ greatly from their normal values. A disadvantage of using a filter is that it alters some features that are not affected by the noise processed by the filter. Switching filters has been demonstrated when dealing with noise [28], where the filters are only directed at the areas affected by noise. One of the problems with switching filters is choosing the right algorithm to detect distorted areas in an image.

Using the Rayleigh distribution method, select the areas distorted by the noise, apply an averaging filter to them, and leave the rest of the image unfiltered. This method is based on two stages:

Phase noise detection using Rayleigh distribution.
Predict skewed point values using a mean filter and correct for missing values.

A Monte Carlo simulation is applied to a Rayleigh distribution to fit the topology of the data (see Fig. 2). We compared the results of the conventional method using the MF median filter and the proposed method using the Rayleigh filter for RDF detection.

The results were significantly improved compared to the baseline results as shown in the table below

Tables 1 and 2 show the filtering comparison between MF and RDF in PSNR in the presence of 5% and 10% impulse noise potential applied to standard test patterns. The detection rate is also shown in the table to measure the detection performance of the Rayleigh distribution for locating the pulse noise. The detection result is very close to the defective pixel. The small difference in the amount of detection is because the noisy pixel values are very close to the values of the surrounding samples, which are considered as noiseless pixels. In fact, undetected pixels do not block the eye and are not visible. The

A vision check is shown in Fig. 3. Figure 3 shows that the MF filter does a good job of removing noise by blurring areas free of noise, while the RDF removes noise while leaving fine details and textures intact. The result of the baboon image was chosen because of its complexity and demonstrates the robustness of the filter.

Table 1

filter performance with the presence of 5% noise.
Input Image	PSNR for 5% noise probability applied			Detection %
Input Image	Noisy Image	MF output	RDF output	Detection %
Baboon	13.61	15.57	26.83	93.27
Boats	13.36	21.52	32.01	90.00
Lena	13.52	23.42	33.75	93.35
Peppers	13.22	25.18	34.33	93.87
Tiffany	12.74	28.89	38.44	90.00

Table 2

filter performance with the presence of 10% noise.
Input Image	PSNR for 10% noise probability applied			Detection %
Input Image	Noisy Image	MF output	RDF output	Detection %
Baboon	10.58	15.41	23.94	88.39
Boats	10.38	20.90	28.27	85.27
Lena	10.46	22.85	30.39	88.47
Peppers	10.24	24.33	31.00	86.08
Tiffany	9.82	28.26	35.13	77.63

An important problem with switching type filters is the actual identification of distortion areas. Most of the previous work had to set a threshold depending on the nature of the image being processed. The process of using adaptive filters may guarantee some solutions, but there are still some technical problems. The process of studying the nature of the distribution of the image content that was carried out in this research proved the effectiveness of the Rayleigh distribution in determining distortion areas with high efficiency. Therefore, the filtering process was used on the regions deformed only, while ensuring that all remaining image features were preserved. As a final result, higher-quality images were obtained compared to the use of conventional filters.

Acknowledgments

We would like to thank in this occasion "Al-Nusour University College" for their generous and positive financial and moral support for this work.

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No competing interests reported.

Rayleigh-Monte Carlo Method for Image Noise Detection and Elimination

Status:

Version 1

Abstract

Figures

1. Introduction

2. Rayleigh Distribution

3. Experimental Results

4. Conclusion

Declarations

References

Additional Declarations

Status:

Version 1