Image Denoising Based on Implementing Threshold Techniques in Multi-Resolution Wavelet Domain and Spatial Domain Filters

The digital revolution has changed people's lives so that all aspects of our lives today have been affected by technology. One aspect of this revolution is the rapid technology development of digital images, which plays a critical role in some of the extensively utilized sectors, including the medical field, astronomy science, color/video processing, remote sensing, pattern recognition, and many other applications. However, the problem is obtaining images from sensors contaminated with what we refer to as "noise." The term "noise" in the digital image refers to damage that happens within the image or unwanted information that damages the quality contents of the original image and thus leads to the loss of its vital properties. It occurs as a result of acquisition and transmission processes or as a result of noise sources near image capturing devices, sensor misfocus, defective memory location, optical aberrations, atmospheric distortions, and motion of objects in the scene result in distorted image quality. A distorted image occurs due to exposure to various types of noise that can affect and damage the digital image, such as salt and pepper, Gaussian, Speckle, Poisson, and others noise [1, 2].

AWGN noise is one of several types of noise that can occur as a result of poor image acquisition quality, loud surroundings, or transmission mediums. In this research, AWGN noise has been added to all input images. It is essential to eliminate AWGN noise from the noisy image till it approximates the input image. Therefore, noise is one of the major sources of information in several applications and one of the significant constraints in image accuracy [3]. Denoising an image seeks to eliminate or reduce the amount of noise and attempt to retrieve valuable information from degraded and blurred images. As a result, the primary difficulty is to extract as much actual data as possible from the noisy image. Thus, denoising is a critical and necessary approach in digital image analysis and the appropriate first action to actually take before beginning any type of investigation, including image acquisition and understanding, pre-processing, classification, textures analysis, segmentation, features extraction, etc. [4, 5]. Researchers are continuously focusing on it to enhance visual assurance and the success of high-level vision tasks [6].

Additive random noise can be easily eliminated without difficulty when applying threshold techniques. The denoising of digital images affected by AWGN noise with WT techniques is a very effective procedure due to Its ability to capture the energy of a signal in a minimal number of energy transform values. Additionally, its ability to access time and frequency information concurrently makes it a powerful tool for signal processing, images denoising, and other applications in both continuous and discrete wavelet transform, which may result in small and less significant coefficients. The WT technique is advantageous for energy compression since it generates small and large coefficients as a result of noise and key image elements. Thresholding the low coefficients is doable without compromising the image's significant features [7].

Wavelet-based analysis techniques such as (DWT and SWT) are among the most frequently used denoising procedures; both focus on signal decomposition and band splitting into a few frequency bands. Thus, the procedure isolates the noisy image into various sub-band images, and then it associates the high-frequency wavelet coefficients as horizontal (H), vertical (V), and diagonal (D). DWT is commonly employed to eliminate noise from degraded images due to the ability to sparse representation for the primary image, which means that it has numerous coefficients close by zero value [8]. In the usual form, each coefficient (H, V, D) indicates the threshold by contrasting it with an estimate, referred to as a threshold value. Numerous threshold strategies have been investigated and used by researchers to ascertain the value of a threshold [9, 10].

In contrast, SWT decomposes the input signal into many levels using high-pass filters and low-pass with wavelet coefficients of equal length at each level. SWT implements a tree-structured algorithm similar to DWT, but it achieves and returns a better approximation result than DWT because the output signal is not decimated (without down-sampling), which DWT cannot achieve. As a result, a multi-layer SWT applies to overcome the limitation of the conventional wavelet transform [7].

This work concludes to introduce and apply a hybrid system designed to denoise the noise from the affected pixel-units that exist in the degraded digital image that has been affected by AWGN noise in terms of noise reduction and improved visual quality of the image (degraded image) in comparison to previous works. To achieve that, the concept use of both Median and Wiener filters as spatial domain filters are combined with a working mechanism based on selecting the threshold technique in both 2D-SWT and 2D-DWT into WT domain as a multi-resolution analysis technique in image processing at three levels of decomposition to achieve improved results of the noise reduction process. The advantage of adopting the hybrid method system is retaining the exact details and preserving the vital features of denoised images concerning AWGN noise elimination as much as possible and structure preservation. Furthermore, it compares hybrid noise removal mechanisms and applies several hard and soft threshold techniques (Hard, Sure shrink, Bayes shrink, and Penalized) for all case scenarios and various percentages of noise levels by calculating the PSNR value.

A denoising technique, also known as reduction of image noise, is a form of image processing that aims to eliminate the various forms of unwanted noise available in degraded digital images. The noise is defined as unwanted information and can contaminate original images. Noise occurs due to various sources, including failure to acquire or error in the data transfer process, storage and retrieval processes, and other causes. Therefore, there is a degradation in the visual quality of the digital image. Thus, the purpose of denoising from images is to eliminate noise value and recover original data from disturbing data while preserving the edges and other detailed features [5].

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Figure 1. The hierarchical description of various denoising techniques [7, 20]

In image processing, there are several denoising methods. The choice of the denoising technique used in digital images depends on the type of noise present in degraded images and also levels of noise ratios. According to the noise source, it can be roughly divided into two categories: external/internal noise [7]. The fundamental methods of image denoising can be divided into two major approaches: spatial domain filtering and wavelet domain methods, as presented in Figure 1. The methods included in the first approach are at the same level of image and depend on the direct treatment of image elements (pixels). In contrast, the second approach methods are based on modifying the coefficients of the sub-image to be processed [7, 20].

3.1 Noise

Noise is an important factor that interferes with human perception and understanding of information and hindering it. Since the noise itself is unpredictable, it can be considered as a random error that can be specified through probabilities and statistics. Therefore, image noise can be defined as a multidimensional random process that can be described by the probability distribution and the probability density function. According to the relation between the image and noise, image noise can be classified into two types: Additional noise, where the noise is not related to the original image and can be expressed in Eq. (1), and Multiplied noise, where the noise is related to the original image and can be expressed Eq. (2) [21, 22]:

$f(x, y)=g(x, y)+n(x, y)$     (1)

$f(x, y)=g(x, y) * n(x, y)$     (2)

where, f(x,y) represents the contaminated image (noise image), g(x,y) represents the original image, and n(x,y) represents the noise value.

3.2 Spatial domain filters

The phrase spatial domain filtering can be termed an effective technique to modify and enhance a digital image that has been degraded due to unwanted noise. Filtering or rather spatial domain operation capitalizes on the proposed value of a current pixel for its ability to handle directly on pixels of an image (x, y), which is usually affected by with itself pixels or with neighborhood pixels. In this case, filtering stands out as a neighborhood operation. The value of a subject pixel (x, y) in an output image is usually determined by applying algorithms to the values of the pixels in the neighborhood of corresponding input pixels [23, 24]. In a broader sense, a neighborhood pixel can be determined based on locations relative to that pixel. The filter response at this pixel is calculated via the filter mask at each pixel value (x, y) depending on a predetermined relationship (moving the filter mask from the pixel to another pixel value in an image).

The filtering process works by highlighting certain features and eliminating others that are unnecessary ones, with the goal of sharpening, smoothening, and enhancing the edges of an image. The spatial domain filtering is grouped into two types of filtering techniques: linear filter, in which the value of an output pixel is a linear combination of the values of the pixels in the input pixel's neighborhood. While nonlinear filter, its basic operation is to compute the median gray-level value in the neighborhood in which the filter is located. It also has greater performance than linear filtering in noise removal. The utilization of this concept has proven to be critical in image denoising. Thus, its use is recommended when one is interested in enhancing a quality image [25, 26].

3.3 Wavelet transform domain

In image denoising, WT is effective in enhancing the representation of signals that have a high degree of sparsity. This principle facilitates the non-linear wavelet signal estimation, which is called wavelet denoising. During transmission, images usually become corrupted. Cleaning or enhancing their quality requires the use of appropriate interventions. WT becomes handy because it enhances image purity and clarity. Precisely, WT has wavelet coefficients, which are dominated by noise. To achieve transformation, it is necessary to use an inverse wavelet transform. This approach leads to a reconstruction, which is characterized by minimal noise. The utilization of thresholding techniques becomes crucial at this phase. The reason for this claim is that thresholding is critical in eliminating noise from images [7].

This effective concept works by shrinking coefficients. The shrinking aspect is what aids in the elimination of noise from images. Noise ends up getting reduced regardless of the extent of damage on digital images. Both soft and hard thresholds are utilized in WT to aid in the attainment of positive outcomes. Small thresholds can easily bring forth outcomes that are close to the input value. As such, a significant threshold is required since it triggers results that have little or no noise at all [27].

Wavelet transformation is described by several authors as a mathematical technique that analyzes (or synthesizes) a signal in the time domain using various editions of an enlarged and translated base function named the wavelet prototype or the mother wavelet [28]. Besides, the wavelet transformation is a method that subsumes time and frequency fields and is exactly known as a non-stationary signal's time-frequency representation. In wavelet transform, the signal in the time domain is disintegrated (decomposed) to create two separate parts by carrying it via a high pass filter and low pass filter (low pass (L) and high pass (H) versions) [29]. Wavelet transform can be demonstrated by Eq. (3) as follows [30].

$X_{a, b}=\int_{-\infty}^{\infty} x(t) \psi_{a, b}(t) d t$     (3)

where, (x) indicates the real signal, (ψ) is an arbitrary mother wavelet, (a) denotes the scale, and (b) is the translation (X is the processed signal).

3.3.1 Hard thresholding

The main goal of image denoising is to achieve results that have no noise. The elimination of noise is necessary since it enhances the digital image’s quality. Hard thresholding becomes useful because it facilitates the elimination of noise at extreme levels. In this concept, the framelet coefficients, which are more significant than the preset threshold value gets retained. The remaining ones ate then made zero. The aspect of making them zero is what aids in reducing noise, hence making the images better [31]. The hard threshold is effective because it differs from the soft threshold, which is ineffective in reducing coefficient concentration. Image quality enhancement is thus easy when utilizing a hard threshold. The core aspect that explains why a hard threshold is critical in image denoising is turning coefficient to zero. The hard threshold is represented by the following Eq. (4) [32, 33].

$\widehat{W}_{j i}=\left\{\begin{array}{c}W_{j i} \text { When }\left|W_{j i}\right| \geq T \\ 0 \text { When }\left|W_{j i}\right|<T\end{array}\right.$      (4)

where, $\left(\widehat{W}_{j i}\right)$ means denoised wavelet coefficients, $\left(W_{j i}\right)$ denotes noisy wavelet coefficients, (i) is the location of the detail component, (j) is decomposition level, and (T) is the representative of the threshold value.

3.3.2 Sure shrink

Sure Shrink stands out as the other thresholding technique which is effective in image denoising. It can be termed as a combination of both universal and SURE thresholds. It has a core goal of minimizing the Mean Square Error which helps in the identification of a threshold in every sub-band. This outcome is arrived at through the concept called sub-band adaptive thresholding. Sure shrink is also adaptive to smoothness. As such, it is appropriate to state that it supports abrupt changes in an image. It is these changes that lead to the enhancement of image quality after the elimination of noise [34]. It is represented by the following Eq. (5) [29].

$M S E=\frac{1}{n^2} \sum_{x, y=1}^n(z(x, y)-s(x, y))^2$     (5)

where, z(x,y) is the estimate of the signal, s(x, y) is the original signal without noise, and (n) is the size of the signal.

Sure Shrink suppresses noise through thresholding the empirical wavelet coefficients [35]. The Sure Shrink threshold t* is denoted as:

$t^*=\min (t, \sigma \sqrt{2 \log n})$     (6)

where, (t) denotes the value that reduces SURE, (σ) is the noise variance computed from Eq. (6), and (n) is the size of the image.

Sure Shrink is following the rule of soft thresholding. The threshold employed is adaptive, i.e., a threshold level is set for each dyadic resolution level through reducing SURE for threshold estimates. It means that if the unknown function contains sudden changes or limits in the image, so does the reconstructed image [36].

3.3.3 Bayesian shrinkage

Bayesian shrinkage has been proposed by Chang et al. (2000) as an adaptive data-driven image denoising threshold by reducing the effects of sampling variation, which is based on the soft threshold estimate based on the hypothesis that the wavelet coefficients are the wavelet coefficients of the natural non-noise image are in the GGD [37, 38]. This method's objective is to predict a threshold estimate that reduces the possibility of Bayesian, which assists in digital image quality enhancement since all the unnecessary components get eliminated. It employs soft thresholding and is sub-band dependent, meaning thresholding is performed in the wavelet decomposition at each resolution band. The Bayes threshold is estimated as shown by Eq. (7) [39].

$\lambda_{\text {Bayes }}=\hat{\sigma}_n^2 / \hat{\sigma}_x$     (7)

where, $\left(\hat{\sigma}_n^2\right)$ is calculated by Eq. $(8),\left(\hat{\sigma}_x\right)$ is computed through WT coefficients applied in each sub-band. While $\left(\hat{\sigma}_x\right)$ value can be derived using Eq. (8) and (9) as shown below:

$\hat{\sigma}_x=\max \sqrt{\hat{\sigma}_y^2-\hat{\sigma}_n^2}, 0$     (8)

$\sigma_y=\frac{1}{n^2} \sum_{j, k=1}^n w^2 j, k$     (9)

The size of the subband under consideration is denoted by (n).

3.3.4 Penalized threshold

Birge and Massart introduced this threshold technique in 1997. It utilizes level-dependent thresholds obtained from the post-selection rules for WT coefficients. This procedure could arrange the detail coefficients in descending order, and then the succeeding equation threshold value (λ) could be derived in line with this arrangement, as indicated in the following Eq. (10) [40]:

$\lambda=\arg _t^{\min }\left[-\sum_{k=1}^t d_k^2+2 \sigma^2 t\left(\alpha+\ln \frac{n}{t}\right)\right]$; $t=1, \ldots, n$     (10)

where, (α>1) is the sparsity parameter.

Following the implementation of the proposed cases to five grayscale images of the same size will take into consideration as original images named (Cameraman, Lena, Butterfly, Peppers, and Fruits) in all process steps depending on the best PSNR values. Table 2 represents this evaluation focusing on the different five grayscale images with three levels of noise ratios (σ=10, σ=15, and σ=25). Furthermore, Table 2 shows that the best value of PSNR result and best-proposed method of denoising images are repeated frequently when applying 2D-SWT based on the Sure Shrink threshold technique with selected Haar, Biorthogonal, and Reverse biorthogonal Wavelet Filters Families (db1, bior1.1, and rbio1.1), which have a better PSNR value compared to one of the other cases (process steps) according to the process steps at each noise ratios (σ=10, σ=15 and σ=25), which mean that the PSNR value of denoising image, in these cases, are better than in other cases. While the best PSNR value at all over-tested cases for all images with all noise ratios equal is (32.5460). As a result, it can have concluded that it is better to use both of Sure Shrink threshold technique and the Reverse biorthogonal filter together to achieve more suitable results.

Based on Table 2, results were selected as the best result of denoising images in the proposed process based on two important properties to obtain this better outcome: the wavelet filters based on thresholding techniques with spatial domain filters which had an effective role in obtaining these results and conclusions.

The chart in Figure 8 shows the best results in all cases among PSNR values after evaluating the proposed methods of denoising images at noise ratio (σ=10, σ=15, and σ=25).

Furthermore, the comparison results of denoising technique between the proposed methods of the hybrid system in this work and the results of the related works respectively referred to as Related_1 Related_2, and Related_3, represented by the references [1, 15, 17] are shown in Tables 3, 4, 5, 6 and 7 with all noise ratios (σ=10, σ=15, and σ=25).

Table 2. The evaluation of the proposed method depending on the best PSNR values

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Figure 8. Best results in all cases among PSNR values after evaluating the proposed methods of denoising images at noise ratio (σ=10, σ=15, and σ=25)

Table 3. PSNR values of comparing the proposed case with related works for the Cameraman at noise ratios (σ=10, σ=15, and σ=25)

According to comparison results, the PSNR values for five grayscale images of the same size will take into consideration as an original image named (Cameraman, Lena, Butterfly, Peppers, and Fruits) in all process steps depending on the best PSNR values for three levels of noise ratios (σ=10, σ=15, and σ=25) between both the proposed cases and the results of the related works; the proposed cases have better PSNR values compared to related works indicated by the references [11, 15, 17].

The charts in Figures 9, 10, and 11 show how effective the hybrid system proposed as a solution is in removing the noise in images when compared to related work at each noise ratio (σ = 10, σ = 15, σ = 25).

Table 4. PSNR values of comparing the proposed case with related works for the Lena image at noise ratios (σ=10, σ=15, and σ=25)

Table 5. PSNR values of comparing the proposed case with related works for the Butterfly image at noise ratios (σ=10, σ=15, and σ=25)

Table 6. PSNR values of comparing the proposed case with related works for the Peppers image at noise ratios (σ=10, σ=15, and σ=25)

Table 7. PSNR values of comparing the proposed case with related works for the Fruits image at noise ratios (σ=10, σ=15, and σ=25)

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Figure 9. PSNR values of comparing the proposed case with related works for all images at noise ratios σ=10

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Figure ‎10. PSNR values of comparing the proposed case with related works for all images at noise ratios σ=15

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Figure ‎11. PSNR values of comparing the proposed case with related works for all images at noise ratios σ=25

Based on Tables 3, 4, 5, 6 and 7 comparing with Related_1 and Related_2 has a mechanism of applying hybrid denoising method by merging the 2D-DWT with median filter in the process of images denoising, it can be concluded that the Winer filter is more appropriate to be used with 2D-DWT in the hybrid denoising process than the median filter with 2D-DWT for the type AWGN noise. Compared to our proposed, by merging the 2D-SWT with spatial domain, it is observed that the proposed method is better in removing noise and increasing the quality of images compared to other related works and methods. While Related_3, which depends only on removing noise in the wavelet domain, it is observed that the proposed method is a better intern of increasing the quality of images by removing the noise in both domains (spatial and wavelet). The experimental assessment showed that the results of proposed cases of process steps gave an improvement in the denoising operation when compared to the results of related works.