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Article

An Improved Image-Denoising Technique Using the Whale Optimization Algorithm

by
Pei Hu
1,
Yibo Han
1 and
Jeng-Shyang Pan
2,3,*
1
School of Computer and Software, Nanyang Institute of Technology, Nanyang 473004, China
2
College of Artificial Intelligence, Nanjing University of Information Science and Technology, Nanjing 210044, China
3
Department of Information Management, Chaoyang University of Technology, Taichung City 413310, Taiwan
*
Author to whom correspondence should be addressed.
Electronics 2025, 14(1), 145; https://doi.org/10.3390/electronics14010145
Submission received: 21 November 2024 / Revised: 17 December 2024 / Accepted: 27 December 2024 / Published: 1 January 2025
(This article belongs to the Special Issue Applications of Computational Intelligence, 3rd Edition)
Browse Figures
Figure 1
<p>The detailed steps for obtaining the optimized output of the denoised image.</p> "> Figure 2
<p>Flowchart of the proposed MWOA-based denoising algorithm.</p> "> Figure 3
<p>The test images.</p> "> Figure 4
<p>The test images for Gaussian noise.</p> "> Figure 5
<p>Results 1 of different denoising techniques for Gaussian noise.</p> "> Figure 6
<p>Results 2 of different denoising techniques for Gaussian noise.</p> "> Figure 7
<p>Results 3 of different denoising techniques for Gaussian noise.</p> "> Figure 8
<p>The test images for hybrid noise.</p> "> Figure 9
<p>Results 1 of different denoising techniques for hybrid noise.</p> "> Figure 10
<p>Results 2 of different denoising techniques for hybrid noise.</p> "> Figure 11
<p>Results 3 of different denoising techniques for hybrid noise.</p> ">
Versions Notes

Abstract

:
Images often suffer from various types of noise during their collection and transmission, such as salt-and-pepper, speckle, and Gaussian noise. The wavelet transform (WT) is widely utilized for denoising. However, the decomposition level and threshold significantly impact the quality of the resulting images, but they are difficult to set. This paper uses a modified whale optimization algorithm (MWOA) to optimize the parameters of the WT to achieve better image denoising. The MWOA is enhanced through position updates and mutation to improve the solution quality of WOA and enlarge the search space of the WT. In benchmark images, experimental comparisons with other optimization algorithms like WOA, adaptive cuckoo search (ACS), and social spider optimization (SSO) show that the proposed denoising method achieves superior results in terms of the peak signal-to-noise ratio (PSNR), mean square error (MSE), and structural similarity index (SSIM).

1. Introduction

Images have become important and irreplaceable information carriers in science and technology, industry, aerospace, medicine, education, and other fields [1,2]. However, they are often affected by noise during generation, acquisition, and transmission processes, such as impulse noise, speckle noise, and Gaussian noise [3,4]. These types of noise can degrade image quality, reduce visual appeal, and even seriously impact subsequent image processing. The study of image-denoising algorithms has a substantial impact on image detection and information recognition [5].
Image denoising, a technique that aims to remove noise from images, has become increasingly popular among researchers [6], improving the performance and enhancing the accuracy of results for technologies like image segmentation and recognition [7]. Traditional image-denoising methods are primarily categorized into spatial-domain methods and transform-domain methods according to the different denoising processing domains. Spatial-domain methods, including mean filtering, median filtering, and Gaussian filtering, directly analyze and process the pixel values of noisy images [8,9]. These methods may also cause a loss of edges and details because images tend to blur when all pixels are treated in the same way. Transform-domain denoising methods work by transforming noisy images from the spatial domain to the corresponding transform domain using the Fourier transform, wavelet transform (WT), sparse representation, and others [10,11].
The WT has been successfully applied to image denoising. WT-based denoising involves thresholding the wavelet coefficients before reconstructing the denoised image [12]. First, the optimal wavelet basis and decomposition levels are selected based on specific criteria, and then metaheuristic algorithms are used to optimize relevant parameters. Metaheuristic algorithms are utilized to discover approximate solutions to complex optimization problems for which traditional methods may not be efficient or feasible [13,14]. These algorithms draw inspiration from natural processes, and they are designed to effectively explore extensive search spaces [15,16].
Metaheuristic algorithms are one of the fastest-growing subfields in computational intelligence and have been widely used in various image-processing and feature-selection tasks due to their ability to efficiently search for optimal solutions in complex, high-dimensional problem spaces [17,18]. Images may be distorted or contaminated by noise at various stages of image acquisition, transmission, and storage. Therefore, the study of algorithms that can evaluate the perceptual quality of digital images through human-quality judgments is a hot topic in the literature. Varga developed an image quality assessment (IQA) technique to forecast the visual quality of images by optimizing the combination of multiple IQA metrics through the simulated annealing (SA) algorithm [19]. Mirjalili et al. highlighted the effectiveness of the genetic algorithm (GA) in optimizing reconstruction parameters and improving image fidelity [20]. Experiments have shown the significance of nature-inspired algorithms in advancing image processing, demonstrating their robustness, flexibility, and adaptability to various challenges in the field.
In wavelet-based noise reduction methods, the WT is applied to obtain coefficients. These coefficients must be adjusted with appropriate thresholds to preserve the key features and properties of an image and eliminate unnecessary components. The adjusted components are referred to as wavelet thresholded coefficients. Subsequently, an inverse WT is applied to these coefficients to produce a noise-free image. The whale optimization algorithm (WOA) is well known for its quick convergence, strong optimization ability, and stability. This paper employs WOA to optimize the wavelet parameters, and an improved image-denoising method based on WOA is proposed. The contributions of this study can be summarized as follows:
  • We construct a leader pool composed of the best solutions, providing a diverse set of top candidates to guide other individuals in the population. This structure enhances the algorithm’s ability to effectively explore promising regions of the solution space. A hierarchical learning strategy is implemented to systematically improve individual solutions, while random opposite learning is utilized to increase population diversity.
  • The modified WOA (MWOA) demonstrates significant improvements in image-denoising performance. Extensive comparisons with several algorithms on benchmark image datasets reveal that the MWOA achieves excellent results in terms of the peak signal-to-noise ratio (PSNR), mean square error (MSE), and structural similarity index (SSIM) when tested on images containing Gaussian and hybrid noise. The proposed MWOA not only achieves better denoising quality on noisy images but also exhibits strong adaptability and stability under different types and noise levels, making it highly suitable for image-denoising tasks.
The main structure of this paper is as follows. Section 2 introduces the latest research on image denoising using metaheuristic algorithms, while Section 3 presents the improved WOA-based image-denoising model. Section 4 covers the experiments and analysis of image denoising. Section 2 summarizes the work in this paper and discusses ideas for future research.

2. Related Works

Recently, it has become popular to apply optimization algorithms to image and signal processing. Metaheuristic algorithms can be applied to wavelet coefficients and obtain optimized wavelet thresholding components to enhance noise removal efficiency.
Traditional digital filters lack efficiency in achieving global robustness optimization. To overcome these limitations, Cao et al. adopted bio-inspired optimization algorithms as feasible filter design tools due to their simplicity and minimal number of parameters [21]. A denoising filter for images was developed through a novel guided-extraction band filter and a hybrid cuckoo particle swarm optimization (HCPSO) algorithm. Chauhan et al. designed a denoising filter to enhance subtle non-stationarities present in signals [22]. The filter coefficients were optimized using the mountain gazelle optimizer (MGO) algorithm based on maximizing the value of the fitness function to calculate the system’s impulse response. Gautam et al. devised a differential evolution (DE) algorithm that allows individuals to select better mutation strategies independently based on their accumulated historical experience [23]. Additionally, a parameter update mechanism with an archive was implemented to assign the appropriate control parameters to the strategies. The proposed algorithm demonstrated excellent performance in optimizing wavelet parameters for remote sensing image denoising. Jayapal and Subban proposed a denoising method that considers different types of noise and processes them using multiple adaptive filters with the lion optimization algorithm (LOA) [24]. The performance of the proposed denoising method was evaluated by varying the noise variance and comparing it to existing methods. Benhassine et al. aimed to effectively remove noise while retaining important image features [25]. They proposed a new image-denoising method based on the WT, which selects the optimal decomposition level and wavelet. A thresholding function is applied to the wavelet coefficients. The crow search algorithm (CSA) and social spider optimization (SSO) are utilized to optimize thresholding, and the inverse method of the WT is applied to reconstruct the denoised image. Satellite images transmitted to ground stations are frequently damaged by noise. Conventional denoising methods can remove noise components, but they often fail to preserve image quality, resulting in excessive blurring of image edges. To overcome these shortcomings, Asokan and Anitha developed a bilateral filter for image denoising and edge preservation [26]. They used PSO, cuckoo search (CS), and adaptive CS (ACS) to effectively denoise images without blurring edges. Golilarz et al. proposed the newly introduced harris hawks optimization (HHO) algorithm to denoise satellite images [27]. This algorithm consists of chaos, multi-population, and differential evolution strategies. In addition to improving quality and quantity, the algorithm is computationally efficient and reduces processing time. Alotaibi employed sparse representation (SR) to achieve image denoising and proposed a dynamic multi-group method and pre-learned dictionaries to reduce the time required for image denoising [28].
Noise degrades image quality and makes final diagnostic decisions challenging. Subhedar and Urooj introduced a WT method that utilizes particle swarm optimization (PSO) to denoise OCT images [29]. PSO automatically determines the hyperparameters based on image quality. The PSNR is used as the fitness function for the optimization problem. Aravindan and Seshasayanan proposed a method for denoising images using the WT and social spider optimization (SSO), which has low computational costs [30]. Initially, the WT is applied to the input medical images with various types of noise. Then, the SSO algorithm is utilized to optimize the wavelet coefficients. Finally, the inverse WT method is applied to the optimized coefficients. The denoised images were obtained and evaluated using the PSNR to determine the best performance. Vaiyapuri et al. proposed a multi-objective optimization approach to denoise medical images [31]. This technique uses the GA to optimize thresholds within a wavelet denoising framework. It can adapt to different types of noise images without requiring prior knowledge of imaging processes. It also balances the preservation of essential diagnostic details and noise reduction by utilizing the error factor and PSNR as objective functions. The electroencephalogram (EEG) is regarded as a valuable tool for managing neurological disorders, conducting medical diagnosis, and advancing cognitive research. However, EEG images are often contaminated by various artifacts. Narmada et al. used elephant herding optimization (EHO) to adaptively optimize the WT to facilitate epilepsy detection [32]. Medical imaging is a sensitive field, and its results significantly impact patients’ lives. However, images can become corrupted during medical imaging and processing. Vineeth et al. analyzed various optimization algorithms and identified their effectiveness in addressing the denoising of biomedical images [33]. These algorithms were implemented within a framework that involves finite impulse response (FIR) filtering. Subsequently, the results of the denoised images were compared visually and numerically to determine the most suitable algorithm for the task.
Deep learning methods have also been explored to reduce noise while preserving structural information and texture. Cell-image denoising is a challenging task that requires the high restoration of feature details. Current popular denoising methods based on convolutional neural networks (CNNs) suffer from blurred texture details after denoising, which is critical for cell-image denoising. Chen et al. first theoretically analyzed the cause of the blur problem and then proposed a denoising training framework based on Wasserstein generative adversarial networks (WGANs) [34]. This framework guides the denoising network to explore the distribution space of real clean images rather than that of blurred images and introduces feature information to solve this problem. Shan et al. proposed a new hyperspectral image-denoising algorithm, the dual deep CNN (DD-CNN) [35]. This DD-CNN consists of two CNNs: one extracts features from the target image, while the other extracts features from the reference image. A new activation function was introduced to activate both feature sets in the DD-CNN. This dual structure and new activation function were utilized to integrate the external features extracted from the reference image into the internal features of the target noisy image. Compared to internal denoising methods that only utilize features from the target noisy image, the DD-CNN extensively explores the similarities between the target noisy image and the clean reference image. Wang et al. proposed a method for image denoising that utilizes the BP neural network and an improved WOA [36]. The method introduces a nonlinear convergence factor and an adaptive weight coefficient to improve the optimization ability and convergence characteristics of the standard WOA. Subsequently, the improved WOA is utilized to optimize the initial weights and thresholds of the BP neural network to overcome the dependency in the construction process and shorten the training time of the neural network.
Based on related research, we understand that wavelet parameters affect image denoising. The proposed algorithm incorporates a leader pool and dynamic learning strategies to enhance the balance between exploration and exploitation. It searches for more diverse solutions and avoids premature convergence to suboptimal outcomes, a common limitation in many existing methods. Additionally, we integrate a random opposite learning method to increase population diversity, which has not been applied in previous WOA-based image-denoising methods.

3. Proposed MWOA-Based Image-Denoising Model

Color images are divided into three color channels, and the WT is applied to denoise the pixels in each channel. The final image is obtained by superimposing the denoised images from the three channels. The image-denoising model is illustrated in Figure 1.

3.1. Objective Function

The PSNR is a commonly used metric for judging image quality, especially in the fields of image compression and image denoising. The PSNR evaluates image quality by comparing the differences between the original and processed images. It is typically measured in decibels (dB), with higher values indicating that the processed image is closer to the original one. The PSNR is utilized as the objective function for image denoising in this study.
P S N R = 10 × log 10 L 2 MSE
M S E = 1 m × n ∑ i = 1 m ∑ j = 1 n [ I ( i , j ) − K ( i , j ) ] 2
where L represents the maximum pixel value of an image. I ( i , j ) and K ( i , j ) denote the pixel values at position ( i , j ) in the original and processed images, respectively, while m and n are the width and height of the image.

3.2. Modified Whale Optimization Algorithm

To simulate the shrinking encircling mechanism and the spiral update of whales, WOA uses the following method to complete the position update:
X i ( t + 1 ) = X i * ( t ) − A × D i f ( r a n d ( ) < 0.5 ) D * × e ( a 2 − 1 ) × r a n d ( ) + 1 × c o s ( 2 π ( ( a 2 − 1 ) × r a n d ( ) + 1 ) ) + X i * ( t ) e l s e
D * = | X i * ( t ) − X i ( t ) |
A = 2 × a × r a n d ( ) − 1
D = | C   ×   X i * ( t ) − X i ( t ) |
C = 2 × r a n d ( )
where X i ( t ) represents the position of individual i at iteration t. a and a 2 are two coefficients, which are usually set to 2. X * is the global optimum. If | A | > = 1 , X * is also replaced with the position of a random individual during the first condition in Equation (3).
To improve the performance of WOA in image denoising, we mainly modify it by forming the leader pool, updating the position, and introducing population mutation.

3.2.1. Construction of the Leader Pool

WOA uses the global optimal solution to guide population updates. This approach accelerates convergence and facilitates the rapid concentration of the population in a certain area of the solution space. It also increases the risk of becoming trapped in a local optimum. In the gray wolf optimizer (GWO) [37], three leader wolves guide the population, and the algorithm explores more spaces. Inspired by GWO, we construct a leader pool, consisting of three optimal solutions, to improve population diversity and the exploration capability. It balances the depth and breadth of the search.

3.2.2. Position Update

Individuals in the population are randomly paired and compared, and the winners learn from the optimal solutions in the leader pool. The losers learn from the winners, as shown in Algorithm 1.
The fifth line of the algorithm determines whether i 1 and i 2 are solutions within the leader pool. If not, they are updated according to the original WOA algorithm. If they are, they are updated as described below.
The optimal solutions direct individuals to move toward them during the algorithm’s search process. According to Equation (3), when the population conducts a local search around the optimal individuals, especially during position updates through the spiral mechanism, the optimal individuals remain unchanged. Therefore, an opposite learning strategy is introduced to find the optimal solutions that participate in the competition.
X i = U B − L B − X i
where U B and L B are the upper and lower bounds of the search space. Equation (8) represents the process of opposite position learning. By generating the opposite solutions of the current population and letting them compete, the algorithm can move closer to the global optimal solution. If a solution undergoes reverse learning twice, it will be the same as the original solution. We adopt a random opposite learning strategy to avoid this situation and enhance diversity, as shown in Equation (9):
X i = U B − L B − r a n d ( ) × X i
The randomness in Equation (9) ensures that the optimal solutions explore a larger space during multiple learning updates.
Algorithm 1: Position update
Electronics 14 00145 i001

3.2.3. Population Mutation

After updating the positions of an individual, we assess whether the current objective function value is superior to the previous one. If it does not improve, the original position will remain. If half of the population remains unchanged, the unchanged individuals are required to utilize Equation (9) to move from their current position and enhance the algorithm’s diversity.
The size of the population is 18, which includes the wavelet transform parameters required for each color channel: the number of decomposition layers (1–4), the thresholds for each layer (0–2), and the type of wavelet used (nine types). Their dimensions are 4, 1, and 1, respectively, and the process of the MWOA is shown in Figure 2.

3.2.4. Computational Complexity

In each iteration, the MWOA evaluates the objective function for each individual in the population. The time complexity of the MWOA is O ( T · N · f ( ) ) , where T is the number of iterations, N is the population size, and f ( ) represents the complexity of evaluating the objective function.

4. Experimental Results and Analysis

To evaluate the denoising capability of the MWOA, we experimentally compared the proposed algorithm with the latest image-denoising algorithms, ACS [26] and SSO [30], as well as the original WOA [38]. ACS is a denoising method created by Asokan and Anitha in 2020 that employs adaptive cuckoo search-based optimal bilateral filtering to denoise images. SSO is an algorithm proposed by Aravindan and Seshasayanan in 2022 that utilizes social spider optimation (SSO) to optimize the WT parameters for image denoising. We used the CHASEDB1 [39] and BSD500 [40] datasets to test the performance of these algorithms. These datasets are commonly used to evaluate image-denoising algorithms, especially in the fields of medical and natural images.
The experiments were conducted on a Windows 10 platform, with an Intel i7-6700 CPU and 12 GB of memory. The population size for the algorithms was set to 20, with a maximum of 100 iterations. Table 1 shows the key parameter settings for the optimization algorithms.
Gaussian noise is the most common type of noise and follows a Gaussian (or normal) distribution. Since Gaussian noise is continuous, it affects all pixels in an image uniformly, typically resulting in a blurred image where fine details are obscured.
Salt-and-pepper noise is a type of discrete noise characterized by certain pixels in an image being randomly replaced with either maximum values (white “salt” dots) or minimum values (black “pepper” dots). This noise appears as sudden bright or dark pixels in images, with significant contrast compared to the surrounding pixels. Since salt-and-pepper noise is sparse, most pixels retain their original values.
Speckle noise is a type of multiplicative noise caused by the interference of multiple reflected waves. Unlike simple additive noise, it affects images in a multiplicative manner, amplifying or diminishing pixel values in certain areas.

4.1. Experimental Analysis of Gaussian Noise

Figure 3 shows the original images, for which Img1–Img10 were taken from the BSD500 dataset, while Img11–Img15 came from the CHASEDB1 dataset. Figure 4 displays the noisy images.
In this experiment, the images were corrupted by Gaussian noise with a variance of 0.2, and then they were denoised. For ease of reading, the optimal solutions obtained by the algorithms are prominently displayed in bold font. Table 2 shows the PSNR values for different denoising methods. The metaheuristic algorithms demonstrated excellent denoising capabilities, and they all achieved high PSNR values (the objective function). The MWOA exhibited superior performance over ACS, SSO, and WOA in Img2, Img4, Img5, Img7, Img8, Img11, Img12, and Img13, while SSO was the most efficient in the other seven images. The average ranks of ACS, SSO, WOA, and the MWOA were 2.67, 2, 3.73, and 1.6, respectively. From the average ranks, it can be concluded that the MWOA was better than the comparison algorithms. Although the MWOA and SSO achieved the best objective function values in the test images, the performance of SSO fluctuated significantly, particularly in Img4 and Img12, where it was inferior to ACS, WOA, and the MWOA. The MWOA exhibited excellent solution performance and stability, and its objective function value was second only to SSO in Img1, Img3, Img6, Img9, Img10, Img14, and Img15. The PSNR results show that the MWOA effectively preserved the overall image quality and details during denoising.
Table 3 shows the average image-denoising times for the algorithms. The operational efficiency of ACS was significantly superior to that of SSO, WOA, and the MWOA. WOA had the shortest runtime in Img10. Although the MWOA required more time, it was only 3.6% to 9.2% longer than the shortest time, which was still within an acceptable range. The denoising time of the MWOA was shorter than that of SSO in ten images. In Img1–Img8, the algorithms were quicker than in Img9–Img15. Img9–Img15 had the largest image sizes, which is one of the main factors that affect denoising efficiency.
To evaluate the effectiveness of the denoising methods, the MSE and SSIM values were computed, in addition to the PSNR (the objective function), as shown in Table 4 and Table 5. The MWOA and SSO outperformed the other algorithms in terms of the MSE in eight and seven images, respectively. Although the MWOA performed worse than SSO in Img1, Img3, Img6, Img9, Img10, Img14, and Img15, it still ranked second. The MSE of SSO was inferior to that of the comparison algorithms in Img4, Img7, and Img15. Regarding the SSIM, the MWOA performed the best, followed by SSO, ACS, and WOA. The MWOA effectively preserved important structural details and textures in images.
For visual analysis, the results of different denoising techniques are shown in Figure 5, Figure 6 and Figure 7. After adding Gaussian noise to the images, their recognition was severely affected. After denoising by the algorithms, the images were significantly improved in terms of clarity and recognizability.
WOA often produced blurry results, as shown in Img1 and Img6. This blurriness suggests that WOA has difficulty striking a balance between noise suppression and the preservation of fine details due to the parameter tuning of its wavelet shrinkage threshold. The residual noise in ACS is prominent in Img2 and Img8, while the mosaic effect in Img1 and the blurriness in Img2 and Img8 indicate that SSO excessively suppressed high-frequency components, which are essential for preserving edges and details. The MWOA performed significantly better in maintaining clarity and reducing noise, especially in challenging cases like Img7, Img13, and Img15. The results show that the MWOA demonstrates adaptive parameter optimization and robust processing of wavelet coefficients. SSO and WOA tend to lose fine details, as illustrated in Img2 and Img8. While the outlines of the animals are preserved, the internal textures appear washed out. There is excessive emphasis on denoising without adequately enhancing local details. The MWOA presents superior retention of vascular details in retinal images (e.g., Img13–Img15), and it has an exceptional ability to manage small-scale wavelet coefficients.
A leader pool was established to enable the population to benefit from a wider variety of high-quality solutions rather than being guided by a single leader. This pool improves the search space exploration, so it is particularly important when optimizing complex wavelet coefficients that require precise adjustments. The population is randomly paired and compared, where the winner learns from the best solution in the leader pool, and the loser learns from the winner. The modified learning process balances exploration and exploitation, and each individual can adapt based on their interactions with others in the population. This approach ensures efficient local search and minimizes the chance of the algorithm being stuck in suboptimal solutions, which results in better performance in optimizing WT coefficients. These strategies enhance diversity by encouraging individuals to move away from search spaces that may be over-exploited. In the context of the WT, the optimization process can explore a broader range of wavelet coefficients and improve overall denoising performance.

4.2. Experimental Analysis of Hybrid Noise

Figure 8 displays the noisy images. The images are degraded by hybrid noise with a variance of 0.2, including Gaussian noise, speckle noise, and salt-and-pepper noise.
Table 6 presents the PSNR values for the denoising techniques. SSO performed well in eight images, while the MWOA outperformed the comparison algorithms in seven images. Although SSO produced more images with the best PSNR value than the MWOA, the MWOA displayed greater stability. The average ranks of ACS, SSO, WOA, and the MWOA were 2.87, 1.93, 3.6, and 1.6, respectively. Based on the average ranks, the MWOA exhibited the best performance. The PSNR values for hybrid noise were not as good as those for Gaussian noise, which indicates that complex noise affects the algorithms’ denoising ability.
Table 7 shows the denoising times of the images under hybrid noise. ACS once again demonstrated excellent efficiency. The MWOA performed second best, with the shortest running times in Img1 and Img7. While the metaheuristic algorithms displayed different efficiencies in image denoising, these differences were not significant. They all achieved satisfactory denoising results within a limited number of iterations. The MWOA was compared with other algorithms with regard to computation time for different image sizes and noise levels. The results showed that while the proposed method achieved superior image quality, it also incurred a higher computational cost. This raises an important question about the balance between improving image quality and managing computational demands. When dealing with smaller images or images with lower noise levels, the algorithms required relatively short computation times. However, as image size and noise levels increased, the MWOA showed a significant rise in computation time compared to WOA, ACS, and SSO. The advanced optimization process required more complex operations to achieve higher-quality results. In addition to computation time, it is important to consider memory requirements. The MWOA typically demanded more memory to store intermediate results and population data. WOA and ACS were relatively memory-efficient due to their simpler algorithmic structures, while the MWOA and SSO required additional resources, especially when handling larger images or more complex noise conditions.
Table 8 presents the MSE values of the algorithms for hybrid noise. Although the MSE values were not very good, the MWOA was superior to the comparison algorithms in Img1, Img3, Img4, Img10, Img11, Img12, and Img13. Although SSO performed well, it was less effective in Img1, Img4, Img10, and Img12. On average, the MWOA ranked the highest, followed by SSO, ACS, and WOA.
The algorithms were highly efficient in terms of the SSIM (as shown in Table 9), and the performance of ACS, SSO, WOA, and the MWOA was excellent in one, six, two, and six images, respectively. For Img8, the images obtained were quite different from the original ones, while for images Img1, Img2, and Img4, the images obtained were close to the original ones.
The hybrid noise in the images not only caused blurriness but also changed the colors. The denoised images displayed significant improvements in details and appearances compared to the noisy ones, as illustrated in Figure 9, Figure 10 and Figure 11. Except for Img5 and Img8, the algorithms significantly enhanced the visual quality of the images. The MWOA performed the best. In Img3, the zebra stripes were less distinct in the denoised images produced by SSO and WOA. These algorithms failed to preserve edge details during the denoising process. As shown in Img12, the image produced by ACS exhibited noticeable blurring. Suboptimal threshold settings during wavelet decomposition were the cause of this, as smoothing suppresses noise but also eliminates fine details. The image produced by SSO displayed blurring in Img14, particularly in areas with high-frequency details (like textures). This was caused by insufficient preservation of smaller wavelet coefficients during optimization. The MWOA effectively balanced noise suppression and detail preservation through optimized wavelet thresholds, especially in Img13 and Img15. The mosaic blocks produced by WOA and ACS in the central region of Img15 were likely caused by sudden changes in wavelet coefficients. These artifacts occur when algorithms disproportionately emphasize low-frequency components and fail to handle edge regions accurately. The MWOA excelled at preserving complex details, such as the vascular structures in Img13, and exhibited a remarkable ability to control fine-scale wavelet coefficients. The MWOA effectively reduced ringing artifacts by managing the transition between low- and high-frequency components, specifically in Img15. The MWOA was also more effective in dealing with image color changes caused by hybrid noise (as seen in Img5 and Img8). The algorithm minimizes color shifts by optimizing the choice of wavelet basis functions and thresholds to ensure uniform processing of color channels.
The MWOA’s improvement strategies made it possible to explore the solution space in a more diverse manner and enhanced its ability to maintain diversity and adapt to complex image noise. The algorithm could better explore and exploit the search space to obtain optimal denoising parameters. In contrast, SSO and ACS lacked this structured approach, making them more prone to converging at local minima, which resulted in inferior denoising performance. Consequently, SSO and ACS had lower PSNR values, higher MSE values, and lower SSIM values.

4.3. Parameter Sensitivity Analysis

This section evaluates the impact of the population size, number of iterations, and mutation rate on the denoising performance of the MWOA. Table 10 presents the evaluation metrics for different combinations of population size and iteration count.
Although the first combination outperformed the other two, the differences were not statistically significant. The final optimization results were influenced by the balance between exploration and exploitation, regardless of whether the population size was small (e.g., 10) or large (e.g., 40). A small population size restricts the exploration of the search space, while a large population wastes computational resources, especially when the number of iterations is insufficient. The performance of the configurations was similar. Each configuration achieved a reasonably good local optimum; however, different settings may lead to different convergence paths and times, with minimal impact on the final outcome. To achieve optimal image-denoising performance, it is necessary to fine-tune the parameters through experiments to find the optimal combination.
Table 11 presents the performance evaluation of the MWOA under different mutation rates. The mutation rates of 0.25, 0.5, and 0.75 represent 25%, 50%, and 75% of individuals, respectively, using the random position opposite learning strategy. The MWOA performed better at a mutation rate of 0.5 compared to mutation rates of 0.25 and 0.75. A mutation rate of 0.5 achieved a favorable balance between exploration and exploitation. However, a mutation rate of 0.25 was more likely to become trapped in local optima, especially in complex search spaces with multiple local extrema. A mutation rate of 0.75 led to a large number of individuals being randomized. Although this enhanced population exploration, convergence was compromised by the excessively high proportion of mutations.

5. Conclusions

This study proposes a wavelet-based image-denoising method where the wavelet control parameters are quickly obtained using the MWOA. By adding different types of noise to images, the performance of the proposed method is compared with that of existing algorithms like WOA, SSO, and ACS. The experimental results demonstrate that the wavelet parameter optimization based on the modified WOA outperforms comparison methods. Although the runtime of the MWOA may be slightly longer than that of WOA, the increase in computation time is relatively minor and does not hinder its applicability in practical image-denoising tasks. In applications such as medical imaging, satellite imagery, and other areas, where high-quality image denoising is essential, the slightly higher computational cost of the MWOA can be viewed as a worthwhile trade-off for its improved performance in maintaining image quality. The MWOA outperforms SSO and ACS in terms of denoising efficiency for smaller images and remains competitive with WOA for larger images. It is a practical choice for numerous real-world scenarios, particularly when focusing on high-quality results over computational speed. This MWOA can be further enhanced by using different objective functions and exploring multi-scale detail enhancement to improve the entropy value in images.

Author Contributions

Conceptualization, P.H.; Data curation, Y.H.; Investigation, J.-S.P.; Methodology, P.H.; Project administration, J.-S.P.; Software, P.H.; Supervision, Y.H.; Validation, Y.H.; Writing—original draft, P.H.; Writing—review and editing, J.-S.P. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported by the Research on the Key Technology of Damage Identification Methods of Dam Concrete Structures based on Transformer Image Processing (242102521031), the Research on Situational Awareness and Behavior Anomaly Prediction of Social Media Based on Multimodal Time-Series Graphs (232102520004), and the Research on Key Methods for Multimodal Sentiment Data Analysis (2024KJGG0069).

Data Availability Statement

Data are contained within the article. Further inquiries can be directed to the corresponding author. The CHASEDB1 dataset can be downloaded from https://www.kaggle.com/datasets/khoongweihao/chasedb1 (accessed on 21 December 2024), and the BSD500 dataset can be downloaded from https://www2.eecs.berkeley.edu/Research/Projects/CS/vision/grouping/resources.html (accessed on 21 December 2024).

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. The detailed steps for obtaining the optimized output of the denoised image.
Figure 1. The detailed steps for obtaining the optimized output of the denoised image.
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Figure 2. Flowchart of the proposed MWOA-based denoising algorithm.
Figure 2. Flowchart of the proposed MWOA-based denoising algorithm.
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Figure 3. The test images.
Figure 3. The test images.
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Figure 4. The test images for Gaussian noise.
Figure 4. The test images for Gaussian noise.
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Figure 5. Results 1 of different denoising techniques for Gaussian noise.
Figure 5. Results 1 of different denoising techniques for Gaussian noise.
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Figure 6. Results 2 of different denoising techniques for Gaussian noise.
Figure 6. Results 2 of different denoising techniques for Gaussian noise.
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Figure 7. Results 3 of different denoising techniques for Gaussian noise.
Figure 7. Results 3 of different denoising techniques for Gaussian noise.
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Figure 8. The test images for hybrid noise.
Figure 8. The test images for hybrid noise.
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Figure 9. Results 1 of different denoising techniques for hybrid noise.
Figure 9. Results 1 of different denoising techniques for hybrid noise.
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Figure 10. Results 2 of different denoising techniques for hybrid noise.
Figure 10. Results 2 of different denoising techniques for hybrid noise.
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Figure 11. Results 3 of different denoising techniques for hybrid noise.
Figure 11. Results 3 of different denoising techniques for hybrid noise.
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Table 1. The key parameters of the comparison algorithms.
Table 1. The key parameters of the comparison algorithms.
AlgorithmParameters
WOA and MWOAa = 2; a2 = −1; b = 1;
ACSbeta = 3/2;
SSOfpl = 0.65; fpu = 0.9;
Table 2. The fitness values of the algorithms for Gaussian noise.
Table 2. The fitness values of the algorithms for Gaussian noise.
ImageACSSSOWOAMWOA
Img10.63440.63480.63430.6344
Img20.67550.67640.67240.6764
Img30.62680.62760.62650.6270
Img40.62230.62230.62240.6225
Img50.66190.66230.66130.6629
Img60.67300.67550.67230.6752
Img70.64130.64110.64120.6413
Img80.59410.59470.59370.5948
Img90.63820.63860.6380.6383
Img100.61150.61160.61120.6115
Img110.60550.60560.60470.6056
Img120.60290.60250.60260.6030
Img130.60470.60430.60410.6048
Img140.60660.60670.60640.6067
Img150.59970.59970.59950.5997
Table 3. The average running times of the algorithms for Gaussian noise (seconds).
Table 3. The average running times of the algorithms for Gaussian noise (seconds).
ImageACSSSOWOAMWOA
Img1199.7458209.7178205.4640208.3324
Img2186.3825191.6464192.5559194.9633
Img3190.3968217.1379197.9497199.2488
Img4193.1905205.6128198.1487202.6699
Img5185.0479194.2200192.4226195.4791
Img6178.1643199.4078196.6583194.5241
Img7178.0879181.2288189.5961187.7872
Img8183.8261194.6155195.1808194.1937
Img9354.2321382.5654373.0415374.5746
Img10360.5241360.1280356.6657369.3679
Img11351.1338361.1323384.2498374.1855
Img12350.0619361.3373360.4179361.1006
Img13353.3241367.6729365.8955366.8991
Img14351.8643369.1498361.3729361.7909
Img15377.5266386.4970389.4868386.2755
Table 4. The MSE values of the algorithms for Gaussian noise.
Table 4. The MSE values of the algorithms for Gaussian noise.
ImageACSSSOWOAMWOA
Img10.09910.09860.09930.0990
Img20.06180.06110.06420.0611
Img30.10810.10710.10860.1079
Img40.11390.11390.11370.1136
Img50.07220.07180.07280.0714
Img60.06360.06170.06420.0619
Img70.09160.09170.09160.0915
Img80.15760.15650.15830.1564
Img90.09490.09440.09500.0948
Img100.12900.12880.12940.1290
Img110.13820.13800.13940.1380
Img120.14240.14300.14290.1422
Img130.13940.14020.14050.1393
Img140.13650.13620.13680.1363
Img150.14780.14770.14810.1477
Table 5. The SSIM values of the algorithms for Gaussian noise.
Table 5. The SSIM values of the algorithms for Gaussian noise.
ImageACSSSOWOAMWOA
Img10.80100.80200.79730.8011
Img20.72580.73240.68610.7345
Img30.60840.60700.58900.6193
Img40.73780.71930.72970.7451
Img50.84700.84590.84200.8563
Img60.84260.86220.83310.8608
Img70.60190.60150.59910.6092
Img80.39350.40230.38290.4051
Img90.83050.83310.82830.8316
Img100.46980.47160.47130.4689
Img110.58290.58350.57930.5836
Img120.59720.59590.59610.5977
Img130.56360.56150.56040.5642
Img140.53900.54070.53630.5406
Img150.54830.54860.54700.5484
Table 6. The fitness values of the algorithms for hybrid noise.
Table 6. The fitness values of the algorithms for hybrid noise.
ImageACSSSOWOAMWOA
Img10.50790.50790.50790.5079
Img20.49280.49280.49280.4928
Img30.50280.50290.50280.5029
Img40.51390.51390.51390.5139
Img50.51390.51390.51390.5139
Img60.50020.50020.50020.5002
Img70.50110.50110.50110.5011
Img80.49580.49580.49580.4958
Img90.50850.50850.50850.5085
Img100.50570.50570.50570.5057
Img110.52130.52140.52130.5214
Img120.52690.52690.52680.5269
Img130.52190.52200.52190.5220
Img140.51650.51650.51650.5166
Img150.52940.52950.52940.5295
Table 7. The average running times of the algorithms for hybrid noise (seconds).
Table 7. The average running times of the algorithms for hybrid noise (seconds).
ImageACSSSOWOAMWOA
Img1198.4014200.6925203.4161195.4663
Img2186.0418197.8694202.4309198.8629
Img3187.7699191.7490197.0326195.0406
Img4192.5261201.3504203.0051194.5544
Img5185.8746206.9397195.2177192.2871
Img6179.3382183.7928187.7695184.4568
Img7186.6075191.6183186.5130182.5112
Img8184.9921192.9573194.5810192.0973
Img9342.4972355.0359355.1487350.1446
Img10344.9363346.5270359.6982351.4045
Img11349.0379357.9138361.4281360.8077
Img12343.6023362.5241368.0353368.7195
Img13363.5080361.1031365.1787365.4785
Img14358.5476377.9083381.8342367.9906
Img15367.8351390.6076371.6104381.6882
Table 8. The MSE values of the algorithms for hybrid noise.
Table 8. The MSE values of the algorithms for hybrid noise.
ImageACSSSOWOAMWOA
Img10.42510.42510.42500.4250
Img20.50550.50550.50550.5055
Img30.45070.45050.45080.4505
Img40.39690.39690.39690.3969
Img50.39680.39680.39680.3968
Img60.46460.46460.46460.4646
Img70.45960.45950.45970.4595
Img80.48850.48850.48860.4885
Img90.42200.42200.42200.4220
Img100.43610.43610.43610.4361
Img110.36410.36410.36440.3641
Img120.34180.34170.34210.3416
Img130.36170.36160.36190.3616
Img140.38480.38500.38490.3846
Img150.33170.33160.33200.3316
Table 9. The SSIM values of the algorithms for hybrid noise.
Table 9. The SSIM values of the algorithms for hybrid noise.
ImageACSSSOWOAMWOA
Img10.52100.52080.52160.5213
Img20.54750.55080.54760.5476
Img30.35040.35910.34810.3542
Img40.50630.50610.50570.5068
Img50.41200.41210.41310.4130
Img60.54610.54560.54590.5449
Img70.49650.49790.49700.4940
Img80.25870.25940.25710.2589
Img90.38760.38790.38760.3878
Img100.32880.32860.32830.3289
Img110.42810.42810.42730.4282
Img120.47360.47370.47310.4738
Img130.40440.40440.40420.4047
Img140.34230.34150.34210.3430
Img150.41890.41920.41830.4191
Table 10. Comparative analysis of different combinations of the MWOA.
Table 10. Comparative analysis of different combinations of the MWOA.
CombinationPopulation SizeIterationPSNRMSESSIM
C120100121211
C2102008811
C3405010108
Table 11. Comparative analysis of different mutate rates of the MWOA.
Table 11. Comparative analysis of different mutate rates of the MWOA.
MutationPSNRMSESSIM
0.25002
0.5222217
0.758811
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Hu, P.; Han, Y.; Pan, J.-S. An Improved Image-Denoising Technique Using the Whale Optimization Algorithm. Electronics 2025, 14, 145. https://doi.org/10.3390/electronics14010145

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Hu P, Han Y, Pan J-S. An Improved Image-Denoising Technique Using the Whale Optimization Algorithm. Electronics. 2025; 14(1):145. https://doi.org/10.3390/electronics14010145

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Hu, Pei, Yibo Han, and Jeng-Shyang Pan. 2025. "An Improved Image-Denoising Technique Using the Whale Optimization Algorithm" Electronics 14, no. 1: 145. https://doi.org/10.3390/electronics14010145

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Hu, P., Han, Y., & Pan, J. -S. (2025). An Improved Image-Denoising Technique Using the Whale Optimization Algorithm. Electronics, 14(1), 145. https://doi.org/10.3390/electronics14010145

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