Nonconvex Regularization with Multi-Weighted Strategy for Real Color Image Denoising

. Most existing image denoising methods commonly assume that the image is contaminated by additive white Gaussian noise (AWGN). However, real-world color image noise exhibits more complicated distribution properties, making it challenging to develop an accurate model. Consequently, denoising methods designed for AWGN often fail to achieve satisfactory performance on real-world images. In this paper, we present a novel multi-channel optimization model for real-world color images denoising within the multi-weighted Schatten p -norm minimization. Our proposed model utilizes the weighted Schatten p -norm as the regularization term, while the data fdelity term employs two weight matrices to balance the noise level across channels and regions. Besides, it helps to preserve as much detail as possible in the recovered image while removing noise. Although our proposed model is nonconvex and has no analytical solution, an accurate and efcient optimization algorithm is established based on the alternating direction method of multipliers (ADMMs) framework. Finally, we demonstrate the superior performance of our proposed method over existing state-of-the-art models on three real image datasets.


Introduction
During the process of image acquisition and dissemination, the quality of the acquired images is infuenced by a plethora of factors, including environmental conditions and equipment characteristics [1].Tese can give rise to noise artifacts that hinder the efective extraction of information from the imagery, ultimately limiting its utility in subsequent processing tasks, such as image classifcation [2,3], segmentation [4,5], and recognition [6,7].In general, image denoising is to recover a clean image x from a noisy image y � x + n, where n is usually assumed to be the additive white Gaussian noise (AWGN), endowed with a constant distribution intensity [8][9][10][11][12][13].However, AWGN is an idealized model that assumes a specifc probability distribution and intensity for noise, which may not accurately refect realworld situations.In practice, realistic noise has a complex distribution that is signal independent and infuenced by various parameters of the capture device [14].Terefore, real image denoising is a challenging task that requires more attention.
In the past two decades, various methods have been proposed for denoising images, which can be broadly classifed into the following categories: fltering-based [11,15], sparse coding-based [7,[16][17][18], low-rank-based [12,[19][20][21][22], deep learning-based [13,[23][24][25][26][27], and hybrid methods [28,29].Te majority of existing methods are intended to remove AWGN or reduce noise in grayscale images.Unfortunately, Xu et al. [20] demonstrated that applying gray image denoising directly to color images does not yield satisfactory results.Tis is because the noise levels between image channels tend to difer.In the last 10 years, deep learning approaches have shown promising results in denoising color images afected by AWGN.However, these methods are highly dependent on the training dataset and often fail to generalize well to real-world images.As a result, there is a need to develop a more robust denoising method based on a more accurate model that can efectively tackle the challenges posed by real-world color image denoising.
Recently, the low-rank minimization model [30][31][32][33][34] has received considerable attention due to its superior performance in image denoising.Since image is considered to have low-rank property, numerous methods have been proposed based on this property.However, in the realm of low-rank minimization models, the widely employed convex nuclear norm minimization leads to overshrinking of large singular values containing more dominant information, thus yielding biased solutions.To counter this limitation, researchers have delved into nonconvex optimization techniques, aiming to more precisely capture the inherent low-rank characteristics of the image.For instance, Gu et al. proposed the weighted nuclear norm minimization (WNNM) model [12] and applied it to image denoising.It ensures information preservation by weighting the singular values so that larger one will be less penalized.However, WNNM still tends to overshrink the rank component, resulting in bad performance when noise level is high.To overcome this limitation, WSNM employs weighted Schatten p-norm instead of weighted nuclear norm, which provides a better approximation of the rank function.By setting the power parameter p appropriately, WSNM [35] ofers greater fexibility and exhibits superior image denoising performance compared to WNNM.Unfortunately, WNNM and WSNM are initially developed for grayscale image denoising.Considering that noise level in the R, G, and B channels of color images, Xu et al. [20] proposed the MCWNNM, which extends WNNM to perform color image denoising.To address the same issues encountered with WNNM, Huang et al. [21] proposed the MCWSNM and achieved excellent results on real-world color image denoising.Shan et al. [36] introduced a novel multi-channel optimization framework for color image denoising, utilizing nuclear norm minus Frobenius norm minimization.Furthermore, Chen et al. [37] innovatively represented color images as pure quaternion matrices and then applied low-rank constraints to these constructed quaternion matrices, efectively harnessing the inherent cross-channel correlation within color channels.
Real-world images captured by CCD or CMOS cameras exhibit complex characteristics.Tese characteristics may be signal dependent, which can vary with diferent camera settings (such as ISO, shutter speed, and aperture).While MCWNNM and MCWSNM have shown promising results in real-world image denoising, both of them do not consider the variation in noise level across patches in the real-world image.To address the above issues, we introduce a new weight matrix to balance the noise diferences across various patches and propose a novel image denoising model based on multi-weighted Schatten p-norm minimization.Our proposed model has three main merits.First, we use the weighted Schatten p-norm as the rank surrogate to accurately approximate the rank function, which helps to preserve as much detail as possible in the recovered image while removing noise.Second, we incorporate two weight matrices in our proposed model to balance the noise between different channels and patches.Tis makes our model more realistic and the recovered images closer to clean images.Tird, we utilize the alternating direction method of multipliers (ADMMs) framework to decompose the nonconvex problem into several subproblems that can be easily calculated.Tis ensures the efectiveness and efciency of our proposed method.Besides, we make a detailed analysis of the proposed algorithm, including complexity analysis, convergence analysis, and its theoretical proof.Finally, extensive experiments are conducted to demonstrate the superior performance of our proposed method in real-world color image denoising compared to existing state-of-the-art models.
Tis paper is structured as follows.In Section 2, we present the related work, including the application of lowrank matrix approximation methods to image denoising and real-world image denoising.Section 3 presents the proposed multi-weighted Schatten p-norm minimization model, the optimization method, and the theoretical analysis of the proposed algorithm.In Section 4, the denoising performance of our proposed method is compared with some state-of-the-art methods on three real image datasets.Finally, we summarize our work in Section 5.

Related Work
In this section, we frst introduce the methods of low-rank matrix approximation in image denoising and then discuss the approaches for real-world image denoising.

Existing Rank Minimization Methods for Color Image
Denoising.Te color image denoising model based on lowrank minimization relies on the assumption that natural color images exhibit nonlocal similarity (NSS) [38].In [19], Dong et al. demonstrated that low-rank minimization can be efectively utilized in image denoising by establishing NSS.NSS suggests that a given pixel or patch of an image can be well represented by similar pixels or patches located elsewhere in the image, rather than just those adjacent neighbors.Terefore, by exploiting NSS, a latent clean matrix can be obtained by performing a low-rank minimization procedure on each patch-generated matrix and then it can be merged to create a noise-free image.
Among the low-rank minimization models, nuclear norm minimization (NNM) [39,40] is the most commonly used method.However, it treats all singular values equally, which can lead to overshrinking of large singular values that contain more dominant information.To address this issue, Gu et al. proposed the weighted nuclear norm minimization (WNNM) model [12], which can be formulated as where X and Y denote the matrix composed of similar patches in the latent clean image and noisy image, respectively.‖X‖ w, * �  r i�1 w i σ i (X) is the weighted nuclear norm of matrix X, w � [w 1 , w 2 , . . ., w n ] ⊤ is the nondescending weight vector with 0 ≤ w 1 ≤ w 2 ≤ . . .≤ w n , and Frobenius norm of X. σ is the standard deviation of added AWGN noise.
Although WNNM has been shown to achieve good denoising results, several improved models have been proposed [20,35,36,41,42].In [20], Xu et al. proposed the MCWNNM, extending WNNM to perform color image denoising, which can be formulated as with W � (σ −1 r I, σ −1 g I, σ −1 b I).W is the weight matrix characterizing the noise levels in diferent channels.I is the unit matrix.σ r , σ g , σ b are the standard deviations of noise in R, G, and B channels, respectively.In [21], Huang et al. proposed the MCWSNM with the same weighted strategy, which can be formulated as where weighted Schatten p-norm ‖X‖ p w,Sp � ( r i�1 w i σ p i ) 1/p and power p ∈ (0, 1].
Currently, MCWSNM is considered a highly competitive method for color image denoising.Building upon the MCWSNM, we introduce a new weight matrix to balance the noise diferences across various patches and propose a novel model for real-world color image denoising.

Real-World Image Denoising.
Over the last few decades, researchers have extensively studied the denoising of AWGN.However, it is essential to acknowledge that AWGN is an artifcial addition that is used to simulate noise in realworld scenarios.It is assumed to follow a specifc probability distribution, which may not necessarily refect the actual characteristics of noise in the real world as it often remains unknown.As a result, real-world image denoising methods can be categorized into two groups based on whether or not they rely on the noise levels as input.
We frst discuss methods that require the noise level as input.Since real-world noise is typically unknown, numerous methods have been proposed to estimate the noise level [43][44][45].Denoising color images has been a challenging task, and several methods have been proposed to tackle this problem.For instance, color block-matching 3D (CBM3D) [46] frst converts the sRGB image into the luminancechromaticity space and applies BM3D separately to each channel.However, the color space conversion complicates the structure of the noise and does not fully leverage the correlation between the channels.LSSC [16] was the frst method to suggest combining sparse coding with NSS for image denoising.To suppress sparse coding noise, Zhang et al. [18] introduced the concept of sparse coding noise and shifted the focus of image restoration.Nevertheless, the efcacy of sparse coding methods remains contingent upon the selection of a suitable dictionary, while the task of identifying the optimal dictionary to accurately represent a signal or image patch remains a formidable challenge.By utilizing the inherent low-rank characteristic of the NSS generating matrix, the low-rank model attains remarkable performance in image denoising.Considering that the noise level varies in each channel of a color image, MCWNNM [20], based on WNNM [12], introduces a weight matrix to balance the noise in diferent channels of color images.In addition, MCWSNM [21] introduces a weighted Schatten pnorm to tackle the issue of overshrinking singular values of the weighted nuclear norm and adds fexibility to the denoising process.However, these methods do not consider the patchwise variation of noise in the spatial domain.
We then discuss methods that do not require an input noise level, commonly referred to as blind denoising.For instance, Lebrun et al. [47] proposed a blind image algorithm that can operate on arbitrary digital images as input.Tis method estimates a signal and frequency-dependent (SFD) noise and subsequently denoises the image by employing a multi-scale adaptation of a nonlocal Bayesian denoising method.Xu et al. [48] employed a nonlocal selfsimilarity prior to denoise patch group-(PG-) based images, using nonparametric Bayesian dictionary learning to extract potentially clean structures from PG variations.In addition, a number of deep learning-based blind denoising methods have been proposed.Wang et al. [49] developed an endto-end structured multi-column convolutional neural network (MC-Net) to train a denoiser using known noise levels, which was then utilized to estimate the noise levels.Guo et al. [26] combined synthetic noise data with real data to train the convolutional blind denoising network (CBDNet), resulting in improved generalization ability of model and enhanced denoising efectiveness.While deep learning has undoubtedly secured a dominant role in computer vision, it also comes with its own set of challenges compared with traditional methods.Tese include heavy dependence on datasets, leading to potential overftting, and a lack of rigorous theoretical foundations.
Our approach introduces two weight matrices to model the distinct noise distribution across channels and spatial patches in real-world images, aiming to further improve the denoising performance of real-world color images.Moreover, we provide the theoretical underpinnings of our proposed method and conduct a convergence analysis for the proposed optimization algorithm.

Multi-Weighted Schatten p-Norm Minimization for Real-World Image Denoising
In this section, we frst present the weighted Schatten p-norm minimization model for color image denoising and then establish an efcient algorithm to solve the proposed model by using the framework of ADMM.

Problem Formulation.
Te problem of real color image denoising aims to recover the clean image x cm for its noisy observation y cm : International Journal of Intelligent Systems where c ∈ r, g, b   is the index of R, G, and B channels in color image, m ∈ 1, 2, . . ., M { } is the index of the patch number, and n cm refers to noise level in m-th patch of channel c.
In the past decade, low-rank minimization methods have demonstrated remarkable denoising performance by exploiting the low-rank property of redundant nonlocal similarity over an image.To denoise a noisy image y cm , we identify a set of key patches sized p × p with a fxed interval across the image.For each key patch, we select M most similar patches, including itself, in a search window.Te similarity between two patches is measured using the Euclidean distance.For each similar patch, we reshape it as a column vector v cm ∈ R p 2 , where v � y, x, n   denotes the noisy image, the clean image, and the noise, respectively.By arranging the M vectors in each channel, we can obtain a matrix b ] denotes the matrix combined by all channels.Ten, we use Y � X + N to represent a matrix stacking by M similar patches in the noisy image, where Y denotes the noisy image, X is the latent clean image, and N is the noise values.Te entire clean image can then be generated by all denoised patches.
Te distribution of noise in real-world images varies not only in each color channel but also in each patch.To address this issue efectively, we propose a multi-weighted Schatten p-norm minimization model for denoising real-world color images.Te proposed model is defned as follows: where W 1 represents the weight matrix employed to characterize the diverse noise levels in R, G, and B channels and W 2 refers to the weight matrix for describing diferent noise levels between individual patches.Te defnition of W 1 and W 2 is discussed in the next subsection.According to the defnition in [21], we set the weight of the weighted Schatten p-norm to be w i � c �� M √ /(σ i (X) + ϵ) based on empirical evidence.Here, c > 0 is a constant, and ϵ � 10 − 16 is a parameter that is used to prevent division by zero.M is the number of similar patches, and σ i (X) represents the i-th singular values of X.

Defnition of Weighted Matrices.
Te values of W 1 and W 2 are determined using the maximum a posteriori (MAP) estimation framework [50].To obtain the MAP estimate of matrix X, given the observation matrix Y, we maximize the log posterior probability as follows: Here, the likelihood term P(X | Y) is used to model the statistics of the noise.As posited by Leung et al. [50], it can be assumed that the noise is independently and identically (i.i.d.) distributed in every patch and channel, following a Gaussian distribution.Tus, we have n c ∼N(0, σ c ) and n m ∼N(0, σ m ), where n c and n m denote the noise levels in channel c and the m-th patch, respectively.According to the log-linear model, the relationship between σ c and σ m can be modeled as m where l 1 + l 2 � 1. Empirically, we consider σ c and σ m are of equal signifcance, so we set σ c � σ m � 1/2 and defne the noise of m-th patch in channel c as n cm � y cm − x cm ∼N(0, σ cm ).Terefore, P(X | Y) can be formulated as Since the minimum Schatten p-norm property is imposed on underlying latent clean image X, we let P(X) be Putting ( 7) and ( 8) into (6), we obtain with

Optimization Algorithm.
Tere is no analytical solution to the proposed model ( 5) due to its nonconvex property.Fortunately, we can utilize the alternating direction method of multipliers (ADMMs) framework to decompose the problem into a set of subproblems that can be solved iteratively.To achieve this, we introduce an auxiliary variable Z, which allows us to rewrite model (5) as follows: min Te augmented Lagrangian function of (10) can be formulated as 4 International Journal of Intelligent Systems where Λ is the augmented Lagrange multiplier and ρ > 0 is the parameter for penalty.Ten, the subproblem can be achieved by minimizing each variable in turn while fxing the other variables.Consequently, problem (10) can be solved via following steps.
(1) Updating X: to obtain X, we derive it from the right part of ( 12), which yields the following equation: Equation ( 16) is a standard Sylvester's equation which aims to fnd possible matrix X from given matrices A, B, and C that satisfes the following equation: Let in (17).By using Kronecker product ⊗ and the vectorization operator vec, we can rewrite Sylvester's equation in the form Ten, we can get the solution of (12) as (2) Updating Z: (13) is a weighted Schatten p-norm minimization problem.Previous research [21] has shown that if the weights w i   adhere to a nondescending permutation, then the optimization problem can be transformed into independent nonconvex subproblems of the l p -norm.Tese subproblems can be solved efciently using the generalized soft-thresholding (GST) algorithm.Tis procedure is summarized as Algorithm 1.
(3) Updating Λ: In the ADMM framework, the role of Λ is pivotal in managing constraints.It orchestrates updates, gradually aligning them with the optimization problem's constraints, thereby facilitating the algorithm in approximating the ultimate solution.Λ is updated as follows: Here, ρ k is a positive constant, often termed as the step size or penalty parameter, while X k+1 and Z k+1 correspond to the variables computed at iteration step k + 1. Te essence of this formulation lies in updating the Lagrange multiplier Λ progressively to better align with the constraints of the optimization problem.By adapting ρ k and accounting for variations in the current variable X k+1 , this iterative update gradually refnes the Lagrange multiplier, thereby inching closer to the optimal solution.(4) Updating ρ: Te update parameter ρ in the ADMM algorithm is to regulate the convergence rate of the Lagrange multipliers.Mathematically, the update of ρ follows this formula: International Journal of Intelligent Systems Here, μ ≥ 0 is a step-size parameter, ρ k corresponds to the parameter at the current iteration step, and ρ max represents a predetermined upper threshold.Tis formula dynamically adjusts the parameter ρ used to penalize the constraints.By judiciously adapting this parameter, the algorithm strives for a more efcient approximation of the problem's solution.
Based on the above analysis, we summarize the optimization procedure for solving the proposed multi-weighted Schatten p-norm minimization model in Algorithm 2, and the fowchart of our proposed algorithm is shown in Figure 1.

Algorithm Analysis.
We frst discuss the computational complexity of our proposed algorithm.Te computational cost of a single iteration in Algorithm 2 comprises two primary parts.Te frst part is updating X. Te primary computation here involves computing specifc functions in Sylvester's equation, which incurs a cost of O(max M 3 , (3p 2 ) 3  ).Te second component involves updating Z. Te primary cost here arises from the SVD operation in Algorithm 1, which costs O(max p 4 M, p 2 M 2   + p 2 r 4 M).Given the parameter settings, we consider M to be signifcantly larger than p and r.Tus, the overall computational complexity of our proposed algorithm is O(K 1 M 3 ).
We then present a theorem that provides theoretical guarantee for the convergence of Algorithm 2.
Theorem 1. Suppose the weight vector w i is sorted in a nondescending order and the parameter ρ k is unbounded.Under these conditions, the sequences X k , Z k , and Λ k generated by Algorithm 2 satisfy the following:

Proof
(1) We frst prove that the sequence of the augmented Lagrangian multiplier Λ k   is upper bounded.We let k .Ten we have: where J is the number of iterations in the GST algorithm.Tus, Λ k   is bounded.
(3) We next prove that the sequences of X k and Z k are upper bounded.

International Journal of Intelligent Systems
Start Input: Noisy image y, estimate noise levels, iteration numbers K 2 , ADMM iteration numbers K 1 .
Output: x k2 and Z k are also bounded.Since X k+1 � Z k+1 + (Λ k+1 − Λ k )/ρ k , we can deduce that the sequence X k   is upper bounded as well.As a result, X k , Z k   have at least one accumulation point.Ten, we obtain Consequently, the convergence criterion of equation (a) is proved.(4) In the fourth step, we prove the convergence criterion of equation (b).As defned in Section 3.3 and (1) while isStop � � False do (2) update X by ( 12); (3) update Z by ( 13); (4) update Λ by ( 14); (5) update ρ by ( 15 International Journal of Intelligent Systems (5) At last, we prove the convergence criterion of equation (c).
Tis completes the proof of Teorem 1. Teorem 1 explains that the sequences of variables generated by Algorithm 2 are bounded and converge to their respective stationary points.Consequently, the convergence of Algorithm 2 can be established.

Experimental Results
In our study, we conduct an extensive evaluation of the proposed model on three real noisy color image datasets.To assess the efectiveness of our proposed model, we compared it with seven state-of-the-art models, including CBM3D [46], MCWNNM [20], MCWSNM [21], Denoising Convolutional Neural Network (DnCNN) [13], Fast and Flexible Denoising Network (FFDNet) [24], Guided Image Denoising (GID) [51], and Neat Image (NI) [52].Notably, all experiments (except NI) are conducted on a desktop computer running MATLAB R2020a, with a Windows 11 operating system, 2.90 GHz Intel i7-10700 CPU, and 16 GB RAM.Meanwhile, NI uses the latest release version available on its ofcial website, and the experiments are implemented in standalone software on the same device.
Te next content is organized as follows.In Section 4.1, we present the details of the three used real color image datasets, and the evaluation criteria of the denoised image are discussed in Section 4.2.Ten, we present the parameter settings of the proposed algorithm in Section 4.3.Finally, in Section 4.4, we show the denoised results of all compared experiments.

Description of Datasets.
Obtaining ground-truth data for a dataset of real color images can be challenging due to the unavoidable noise generated during the image acquisition process.Currently, one of the most widely used methods for generating ground-truth is to capture multiple photos of the same scene under diferent camera settings and lighting conditions and then average them to produce a high-quality image that serves as the ground truth.
In order to evaluate the performance of image denoising, we utilize three widely recognized and commonly used color image datasets.Each of these datasets is distinct in its characteristics and presents unique challenges for denoising algorithms.Te detailed description of these datasets is as follows.
(1) Dataset 1: Te Noise Clinic (https://www.ipol.im/pub/art/2015/125/) dataset [47] comprises numerous realistic noise images captured in an uncontrolled outdoor environment, featuring both gray and color images.In our experiments, we selected 12 real color images, as illustrated in Figure 2, and cropped them into squares of equal size.However, it should be noted that this dataset does not have a ground-truth reference.
( PolyU-Real-World-Noisy-Images-Dataset) dataset [54] is a sizable repository of real-world noisy images, each of which has been paired with a reasonably obtained corresponding "ground-truth" image.Similar to dataset 2, the fundamental concept of this dataset is to capture a constant scene multiple times and generate a mean image to be the approximate representation of the ground truth.Tis dataset is available in JPG format with a size of 512 × 512.
By utilizing these datasets, we can comprehensively evaluate the performance of our image denoising method and compare it with other state-of-the-art denoising approaches.

Assessment Criteria for Denoising Results
. We evaluate the efcacy of denoising methods using two widely used standards: peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM), which are defned in equations (30) and (31).Higher PSNR and SSIM values indicate a better denoising performance in comparison to the approximated ground truth.However, it should be noted that for dataset 1, the evaluation can only be based on visual inspection due to the unavailability of the ground truth.PSNR(x, y) � 10 × log 10 MAX 2 where is the mean square error of x and y.MAX I is the maximum possible pixel value for the image.For the color image in our experiments, the value is set to 255.x, y denotes the pixel value matrix of clean image and denoised image with size of m × n. μ x , μ y and σ 2 x , σ 2 y are mean value and variance of x and y, respectively.σ xy is the covariance of x and y.

Experimental Settings.
Te crucial parameters in our method are confgured as follows.Specifcally, for dataset 1, we set the number of iterations K 2 � 1, while K 2 � 2 for the second and third datasets.Additionally, the parameter p for the weighted Schatten p-norm is set to p � 0.9 and p � 0.99 for the frst and second/third datasets, respectively.Te remaining parameters are maintained consistently across all datasets, where the patch size is 7 × 7, the number of iterations K 1 � 10, the size of the search window is 40 × 40, the number of similar patches is M � 70, the constant for the weight vector c � 2 � 2 √ , and the updating parameter μ is set to 1.001.
Apart from NI, which is a standalone software program, all other denoising methods require the input noise level to be specifed.In the experiments, we utilize Chen's method [45] to predict the noise level of the real image, which is then used as input to the denoising methods.For methods such as CBM3D that only require a single noise level, we set this value as σ � ������������� � mean(σ 2 r , σ 2 g , σ 2 b )

􏽱
. Regarding the noise standard deviation of patches in our algorithm, we initialize it as σ m � σ and then update it using the following equation: where ‖•‖ 2 is the Euclidean norm.International Journal of Intelligent Systems the eye area, and MCWNNM overly smooths the image.In contrast, our proposed method obtains the best visual results with abundant details. Figure 5 displays an image captured by a Canon 5D camera with ISO � 3200.It is evident from the results that CBM3D, DnCNN, and FFDNet failed to efectively remove the noise.NI just removes a small part of the noise.GID removes most of the noise, while a small portion of noise still remained.Tough MCWNNM and MCWSNM manage to attain respectable denoising outcomes, they compromise texture details noticeably.In contrast, our proposed algorithm secures the highest numerical results, successfully eliminating noise while simultaneously conserving a higher degree of intricate details. Figure 6 depicts the image captured by Nikon D800.Te results show that CBM3D, NI, DnCNN, and FFDNet did not achieve satisfactory denoising results.In addition, GID yields a blurring efect.In comparison, while the denoising outcomes of MCWNNM and MCWSNM are commendable, they encounter challenges in recovering intricate details.Conversely, our algorithm excels in both detail recovery and denoising efectiveness, yielding the highest PSNR value.

Experiments on the
Te comparison of denoising methods on datasets 2 and 3 is facilitated by the availability of ground truth, enabling numerical evaluation of their performance.Tables 1 and 2 present the PSNR and SSIM values of the competing models, respectively, with the highest values highlighted in bold.It can be seen that our proposed model outperforms other models in terms of PSNR and SSIM for most images in both datasets.However, it should be noted that FFDNet, which is designed as a deep neural network for color image denoising, does not perform well on both datasets, indicating that neural network methods can be overly reliant on the training dataset.Additionally, NI, a commercial software program, exhibits varying levels of denoising performance across diferent images.CBM3D and DnCNN, on the other hand, are methods specially designed to remove AWGN noise and are therefore not well suited for denoising real color images.GID is a denoising method tailored for real color images, but its performance only outperforms other models on a few images.Te most similar models to our proposed method are MCWNNM and MCWSNM, and the experimental results demonstrate that by considering the noise diferences between diferent patches, superior denoising performance can be achieved.

Conclusion
Te noise distribution in real-world images varies not only between color channels but also spatially within each patch.Building upon this observation, in this paper, we proposed a new real-world color image denoising model.Te model employed a multi-weighted strategy to denoise images by utilizing nonconvex weighted Schatten p-norm minimization.Specifcally, a new weight matrix was introduced into MCWSNM to balance noise diferences between patches.Since the proposed model is nonconvex, and without an analytical solution, we developed an accurate and efcient optimization algorithm based on ADMM framework to solve the proposed model.In addition, the theoretical convergence analysis of the optimization algorithm was also provided.Finally, the experiments on three real-world noisy image datasets demonstrated superior performance of the proposed model compared to state-of-the-art methods.In our future work, we will focus on modeling more complex statistical properties of noise and further improving real-world color image denoising.

) Dataset 2 :
To improve the training of their model, the authors in[53] created a real image dataset (http:// snam.ml/research/ccnoise),comprising 11 indoor scenes, with 500 images each.To obtain noise-free images, the authors computed the mean image of 10 International Journal of Intelligent Systems each scene as the ground truth.Due to the large size of the original images (7360 × 4912 pixels), the dataset was cropped to 15 smaller images with a size of 512 × 512 for the experiment.Tis dataset is available in PNG format and includes generated ground-truth references for evaluation purposes.(3) Dataset 3: Te PolyU (https://github.com/csjunxu/

Figure 3 :
Figure 3: Denoised images of the real noisy image "frog" in dataset 1 by diferent methods.Te ground truth of the noisy input is not available.

Figure 4 :
Figure 4: Denoised images of the real noisy image "dog" in dataset 1 by diferent methods.Te ground truth of the noisy input is not available.

Table 2 :
14NR (dB)and SSIM results of all competing models on dataset 3.Te frst column of the table represents the camera settings.14InternationalJournal of Intelligent Systems