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Total Variation Filter via Multiquadric Radial Basis Function Approximation Scheme for Additive Noise Removal

The Digital Total Variation (DTV) scheme is a digitized energy regularization scheme used for image denoising. This technique takes advantage of being applied to arbitrarily located data points and also has the edge detective property. This article aims to introduce a novel meshless scheme using DTV filtering and Radial Basis Functions (RBFs) to solve the associated equation with the DTV model numerically which results in the image denoising to remove additive noise from image information. This meshless algorithm based on local collocation and Multiquadric Radial Basis Function. These appearances allow this algorithm not only to remove the additive noise from images but also to resolve the discontinuities sharply. It is also noticed that the proposed meshless scheme is simple, fast, computationally effective, requires simply post-processing, and can be easily implemented mathematically. Experimental results confirm that the peak signal-to-noise ratio, the structural similarity, signal-to-noise ratio, the visual effect, and the computational performance of this new meshless scheme are improved compared with state-of-the-art denoising schemes. Furthermore, the proposed scheme can be applied to colour images as well.


I. INTRODUCTION
Image denoising is one of the most powerful aspects of image processing and computer vision. This paper concentrates on the additive noise removal. Image denoising is to remove noise from a noisy image, to restore the true image. Nevertheless, as noise, edge, and texture are high frequency elements, it is complicated to identify them in the process of denoising and the denoised images could necessarily lose some details. In general, recovering important information from noisy images in the process of noise removal to obtain high quality images is an essential problem nowadays [1].
The additive noise removal problem is modeled as The associate editor coordinating the review of this manuscript and approving it for publication was Gerardo Di Martino .
where z: ⊂ R −→ R 2 represents the given true image, z 0 is the noisy image with additive noise η. In literature, various nonlinear approaches have been utilized to tackle this problem, such as wavelet approaches [2]- [4], adaptive smoothing [5], [6], stochastic approaches [7], [8], anisotropic diffusion [9], [10]. Recently variational approaches have also been utilized to solve such problem, for instance [11]- [13]. In conventional variational approaches, the Euler-Lagrange PDE is obtained from the minimization functional of equation (1) is used for smooth solution of image denoising. But the Euler-Lagrange PDE is always a nonlinear and non-differential PDE equation. Therefore the classical numerical schemes unsuccessful to solve the PDE equation for the smooth solution, which produces staircase effects, textures, and degrading the fine details during the image denoising process. Some numerical scheme have been utilized by the researchers to minimize the above mentioned issues such as Augmented Lagrangian Method (ALM) [14], [15], VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ Dual Method (DM) [16], Variable Splitting Based Method (VSBM) [17], and Alternating Direction Method of Multipliers (ADMM) [18]. But Still there is a space for improvement. Various filter-based approaches have been introduced based on these IN detectors, for instance, adaptive center-weighted median filter [19], progressive switching median filter [20], adaptive weighted median filter (AWMF) [21], noise adaptive fuzzy switching median filter (NAFSMF) [22], modified decision-based unsymmetrical trimmed median filter (MDBUTMF) [23], and morphological mean filter (MMF) [24]. These filter-based approaches are smoothly useful in image denoising and reducing the staircase effect, textures and homogeneity in lightly contaminated noisy images. Nevertheless, these schemes are ineffective for reducing noise in highly corrupted images (i.e., noise frequency greater than or equal to 50 percent) because of the inaccuracy of order statistics.
Chan et al. [25] introduced a self-contained 'digital' (Digital total variation (DTV) )filtering theory for image denoising which is independent of the experience of PDEs and numerical solution. DTV filtering is acknowledged as a discrete variational approach that works on the general discrete domain of given data. This approach is more responsive to regular and irregular shaped domains, and the scattered data points can easily be managed. Another benefit of this scheme is to determine the discontinuities sharply without earlier information concerning edge locations due to its built-in edge detective property [25], [26], [28]. DTV filtering procedure is a general method in comparison with other pseudospectral post-processing schemes, so it does not require the location of data on the structured grid [29]- [31]. The DTV filtering approach has been used for the steady-state solutions of conservation of law computed by second-order Lax-Wendro methods [28], [30], [32]. The associated restoration equation to DTV functional of the model (1) is always nonlinear. Recently, some numerical schemes have been produced for the numerical solutions of DTV based nonlinear equation for smooth solutions i.e. to remove the noise and restore the accuracy of the given image information, for instance, see [25], [27]- [29], [31]. But when the noise variance becomes high, the fine details of the image can not be recovered properly, which is the principal disadvantage of the solution of DTV filtering by classical numerical techniques. To the best of our knowledge, few numerical schemes have been utilized to solve the DTV based nonlinear equation and to tickle this issue. In this study, we will introduce a meshless algorithm for the numerical solution of DTV based nonlinear equation to settle as the issues associated with image denoising.
Radial basis function (RBF) methods have become successful methods in recent years in approximation theory as well as in the numerical solution of PDEs. The most broadly applied RBF scheme for the latter class of problems is the RBF collocation method due to Kansa [33]- [35], known as the Kansa method. The popularity of the Kansa approach is due to its meshless applications which means that only a set of points is required in the discretization of the continuous problem. This contribution to the implementation of the method particularly easy, especially for problems in the complex shape domain, and in two and more dimensions. The solution of the nonlinear problem hence produces not only the coefficients in the RBF approximation but also a suitable value of the shape parameter [36]. Kansa method has also shown excellent efficiency compared to FDM [35], [37], pseudo-spectral method [38], and FEM [39]. Since, Kansa method is a domain type strategy, which has numerous features like the finite element approach for the approximated solution of the nonlinear equation. For further information of RBF collocation schemes, see [36], [40]- [44]. In this research study, we will use the BRF meshless collocation scheme (Kansa scheme) for the solution nonlinear equation arising in the DTV filtering based model [25]. Since, Kansa method is a domain type strategy, which has numerous features like the finite element approach for the approximation of the solution of the nonlinear equation. This proposed scheme will not only be helpful in image denoising and edge preservation but also be helpful in the minimization of staircase effect, preservation of textures, and fine details during the restoration process. The main purposes of the recommended meshless scheme in image restoration are; the RBF interpolation process used in the proposed scheme will preserve the edges and fine details in images while the smoothness property and the lack of dependence on a mesh or integration procedure will produce the best restoration performance regarding minimization of staircase effect and texture preservation.
The paper outlines are organized as follows. The detailed mathematical discussion of DTV filtering and RBFs approximation is given in section. The discussion of Chan et al. [25] model for image denoising having additive noise is also mentioned in section 2. The mesh-based scheme and proposed meshless scheme for the solution of the model [25] are discussed in section 22. Section 4 describes some numerical results to validate the performance of the proposed scheme. A comparison with existing state-of-art numerical schemes is described in section 5. Section 5 shows the detailed tabulated discussion of the sensitivity of parameters used in the proposed technique. Finally, the paper is concluded in the last section.

II. MATHEMATICAL BACKGROUND A. DIGITAL TOTAL VARIATION FILTERING
To define DTV filtering [25], let select a general graph [ ,D], containing a finite set of having different nodes (vertices) and a dictionary D of edges. In the graph, the size of D represents the total number of nodes (vertices). It is also assumed that the graph is associated and has no self-loops (no immediate edge from a vertex to itself). All the general vertices on the graph are named β 1 , β 2 , . . . . The notation β 1 ∼ β 2 . indicates that β 1 and β 2 are neighbors. All the neighbors of β 1 are represented by denotes the digital image, for any vertex β 1 , z β 1 signifies the z at vertex β 1 . At any vertex β 1 , the regularized local variation or strength function ∇ p z β 1 is defined as To avoid the regularized local variation from zero denominator, a small regularization parameter α is selected, thus the equation (2) can be re-written as As explained in [25], the edge derivative of z is given as under. ∂z where e denotes the edge of z along e and is represented by where e β 1 shows that β 1 is one node of e. For more information about DTV filtering, see [25], [29], [31].
The first DTV filtering based minimization model for image denoising having additive noise was presented by Chan et al. [25]. So, equation (1) by [25] can be re-written as for all β 1 ∈ , where z is restored image data, z 0 is noisy image data with additive noise data η. The minimization approach for equation (3) by [12], [25] is defined as; In the foregoing equation (4), the first term is the regularization term, while the second term is the data fidelity term, where γ and α are called the fitting and regularization parameters, respectively.

B. RADIAL BASIS FUNCTION APPROXIMATION
The RBF collocation scheme is defined as, a RBF function φ(x) with respect to the origin, φ(x) = φ(r) ∈ for the known data set {x j } and its distance from a given point (pixel) such that with φ(x − x j ) = φ(r j ) ∈ is satisfied. The function φ(x) = φ( x 2 ) is known as radial function. The different basis functions that are used in RBF collocation method are Multiquadric (MQ), Inverse Multiquadric (IMQ), Gaussian (GA), Polyharmonic spline, and Thin plate splines (TPS). For more information regarding these basis functions, see [45], [46].
The RBF collocation scheme is used on function f (x), x ∈ ∈ n including bounded domain to interpolate it smoothly. So, for the given selected interpolating values data centers points, the function is approximated as under.
where γ j are defined weights. To calculate γ j , the RBF scheme is defined as The above equation (6) gives a linear system of equations of order N × N which is written as under.
where γ = (γ 1 , γ 2 , . . . , γ N ) T is known as N × 1 undefined vector and to be defined, while b = (y 1 , y 2 , . . . , The polynomial term is added to the RBF approximation to ensure invertibility of the interpolated matrix C, in such case equation (5) can be defined as under.
with constraints with l i ∈ m−1 for i = 1, 2, . . . , M , in which m represents the polynomial space in which the polynomial is m in N variables are the total degree polynomials [45], which is given as The resultant solution of equations (7) and (8) through interpolation gives matrix system of (M +N )×(M +N ) equations which is given as under. The RBFs having shape parameter c and selection of shape parameter, also the RBFs having positive definiteness (PD) and conditionality positive definiteness (CPD) are explained in [45], [47]. The selection of shape parameter c in RBFs, positive definiteness in RBFs, and Conditionality positive definiteness CPD are described in [45], [47].

III. NUMERICAL SCHEMES
IIn this section, numerical schemes are reviewed for the numerical solution of the system of nonlinear equations connected with the DTV minimization functional (4).

A. MESH-BASED SCHEME M1
The primary DTV filter-based model was introduced by Chan et al. [25] for image restoration containing additive noise and produced some good recovery results on a 1-D graph and 2-D images (grayscale), and 3-D (color) images. The minimization functional for [25] from equation (4) is mathematically written as, where the first term is called regularization term, the second term is called data fidelity term and γ is the fitting parameter and α is regularized parameter. The fidelity parameter γ is used to set a balance between the denoised and smoothness of the denoised images which is often based on the level of noise added to the given image and the regularization parameter α is used to avoid a zero so its value is usually taken α = 10 −4 . The resultant nonlinear restoration equation from (9) by [25] is given as where and for a small fixed value of α i.e., α = 10 −4 . The above equation (10) can be fixed and re-defined as For further details, see in [25], [27], [28].

B. PROPOSED MESHLESS SCHEME M2
In this new subsection, we introduce the meshless scheme by applying the BRF collocation scheme for the numerical solution of DTV filter-based restoration equation (12) and to get resultant image z from the given noisy image z 0 selected in model (9). Consider {β j } Nc j=1 be Nc centers in the closed domain ⊆ 2 . For any RBF in 2D (2 Dimensional), the following equation will be satisfied, φ(r) = r 2 in 2 i.e r = (x, y).
For given pixel data center points {β j } Nc j=1 , the equation for RBF approximation with unaugmented polynomial term is given as below.
with undefined coefficients of θ j and are defined the given interpolation condition.
The RBF interpolation at Nc centers data is written as which results in Nc × Nc linear system of equations. This system is used to find the coefficients of Nc × 1 vector is called Nc × Nc matrix and is an invertible matrix [30], [47] due its positive definite application [43], [45]. Thus with θ represents a Nc × 1 order matrix. Again using the same above-mentioned operation on equation (14), the RBF interpolation at N evaluation points The resultant equation to obtain z at N data points using interpolation is given as, The combination of equations (15) and (16) results in the given equation.
which results in approximate solution of at any point inside the closed domain . Since equation (10) forms the minimization functional (4) of model [25] using DTV method and is given as where The steady state time marching equation from equation (18) is denoted by the equation below.
The Collocation method (Kansa scheme) is then used to solve the obtained restoration equation by the combination of the equations (17) and (19). Here Gauss-Jacobi iterative is used in Kansa scheme to solve resultant system of non linear equation which is given as follow.
where z and Here z β1 and z β2 represent the approximate values at vertices β1 and β2 respectively. The main application of the RBF scheme (Kansa method) is usually not necessary to satisfy the given restoration equation (21), so we are independent to choose any BRF. In literature, multiquadric (MQ) RBF gives good results in the RBF scheme (Kansa method) [34], [48] provided if a proper value for shape parameter is chosen [34]. The main utilization of the collocation scheme used on equation (21) is that the suggested meshless method displays the unique solution of (21) due to the MQ-RBF interpolation method employed in the collocation scheme which results in the preservation of the edges. Furthermore, the resultant smooth solution of equation (21) is due to the weighted mean determined in equation (15) using the interpolation process which is based on the Euclidian distance between a noisy pixel and other non-noisy pixels obtained from equation (14) in the selected frame. Consequently, the smooth solution of the equation (21) is responsible for image restoration, reducing the staircase effect, and preserving edges, textures, and image details. In our proposed meshless scheme M2 the value of shape parameter c plays an important role in smooth solution in terms of good image restoration result. In our meshless technique, c and λ are important parameters for good restoration results and their values are based on the image size and noise. The locality and adaptivity for smooth results by DTV filtering have already been discussed in [25]. Many research papers have been published by the researchers regarding the selection of the best value of shape parameter c for getting the smooth results used in RBF. Here, we will apply the extended Rippa's [49] algorithm proposed by Marjan [50] for the selection of the best value of MQ shape parameter c used in the proposed meshless scheme M2, which is shown in algorithm2. All entries in the error vector can be computed in a single statement in MATLAB. In order to determine a good value of the shape parameter as quickly as possible we have used the Matlab function fminbnd to find the minimum of the cost function for c The algorithm are chosen initially. In this article, the role of algorithm 2 in our meshless algorithm 1 is to find the optimal value of c to provide the best restoration performance in terms of PSNR value rather than error estimation.

IV. EXPERIMENTAL RESULTS
In this section, experimental analysis is demonstrated to validate the performance of the proposed meshless collocation scheme M2. To justify the performance of the meshless method M2, we perform several experiments on real and artificial grayscale and color images and compare the results with a classical method M1. The test images for experimental analysis are shown in Fig.1. For 2D case, N β can be defined in different ways. One way is to consider a point P neighbor of a node β consisting of all P points that are closest to β. In this paper, N β is defined in the best way. Dividing the region surrounding a point an into P regions of equal angle, N β is defined such that it consists of points in each region that are closest to β as discussed in [25]. In this research study, VOLUME 8, 2020 whereû is the selected image and u represents the denoised image, while M × N is the size of the selected image. The Structure Similarity (SSIM) index is a method for estimating the similarity between two images [51], [52]. It carries the necessary information for image quality evaluation utilizing the principle approach that the pixels have storing interdependency when they are spatially close. The SSIM metric is based on the intensity, contrast, and structure, and is determined as where µ u , µ v , σ u , σ v , σ uv indicate the mean, variance, and covariance on typical 8 × 8 square windows, which movies pixels by pixels in images u(i) and v(i), respectively. The two variables c 1 = k 1 L and c 2 = k 2 L are employed to stabilize the division with weak denominator. Here, L is the dynamic range of pixel value (e.g., 255 for 8-bit grayscale image), with k 1 = 0.01 and k 2 = 0.03 by default. As the SSIM metric is measured on various windows of an image, the mean SSIM (MSSIM) is used in this experiment to evaluate the overall image quality: where M is the number of local windows in the image while MSSIM ∈ [0, 1]. Higher MSSIM indicates better structural similarity between two images. Signal-to-noise ratio (SNR) is also applied to determine the visual quality of restoration of the restored image and is defined as: where u and n indicate the true image and noise, while u 0 and n 0 describe their mean values in the image domain . Repeatedly, the greater SNR value leads to better restoration results.
The stopping criteria for iterations in meshless scheme M2 is given by the following equation where shows the maximum acceptable error with = 10 −4 . In meshless scheme M2, Multiquadric radial basis function (MQ-RBF) is selected as a basis function. The formula for    MQ-RBF for each point (x i , y j ) is defined by the given equation.
All the four images are given in Figures 2, 3, 4, and 5, respectively. In all the three Figures mentioned for the three real images, (a) and (b) represent the true and degraded images while (c) and (d) described the resultant images by algorithms M1 and M2. In each case, it can be observed that the quality of the reconstructed images obtained using M2 is far better than the image quality of the restored images achieved utilizing method M1, without degrading the intrinsic jumps in the clean data, and also preserve the edges quite well due to the applications of DTV filter and the meshless applications due to the meshless characteristics of MQ-RBF approximation applied to the smooth of DTV based nonlinear VOLUME 8, 2020    Table 1. The bigger the PSNR and SSIM value results in good restoration performance. Thus, it can be seen from Table 1, that the PSNR and SSIM values utilized by using M2 are greater than that of M1 in each case which represents the good restoration performance of M2. The number of iterations required for convergence and the computational time obtained by the meshless scheme M2 is also smaller than of method M1 in each case which shows the accelerated denoising performance of introduced meshless scheme M2 over mesh-based scheme M1 because of the meshless application of the lack of dependency on a mesh or integration procedure connected with MQ-RBF for the solution of DTV based nonlinear restoration equation. The values of the two parameters i.e. (shape parameter c, the fitting parameter λ) selected for the M2 for the three test images 'Cameraman', 'Moon', and 'Peppers' in this experiment are (0.45, 0.315), (0.47, 0.274), (0.50, 0.259), and (0.52, 0.254), and respectively. The shape parameter c plays an essential role in image restoration in proposed meshless scheme M2 [32]. The fminbnd Matlab function has been used in this experiment to get the best value of c ∈ (c min , c max ) for the best restoration results shown in Table 1. In this test, c min = 0.37 and c max = 0.79 are chosen. Experiment 2: In this experimental test 2, the restoration results for two algorithms M1 and M2 are check on artificial and real images 'Ring', 'Lena', 'Bird', and 'Bridge' having salt and paper noise (slat and paper noise with mean value zero and standard deviation σ 2) that are shown in Figures 6, 7, 8, and 9, respectively. The noise levels in this test for 'Ring', 'Lena', 'Bird', and 'Bridge' are set to σ 2 = 0.08, σ 2 = 0.17, σ 2 = 0.21, and σ 2 = 0.24, respectively. Again for each case, the performance of the restored images and edge preservation property by Kansa method M2 (shown in Figures (6(d), (7(d), (8(d), and (9(d)) are much better than of traditional method M1 (shown in Figures (6(c), (7(c)), (8(c)) and (9(c), respectively). The PSNR and SSIM values, number of iterations, and CPU times for this experiment are also listed in Table 1, which shows better performance of the proposed algorithm M2 over algorithm     Table 1. In this test, c min = 0.37 and c max = 0.79 are selected.
Experiment 3: In the third experiment, the two schemes M1 and M2 are tested on 3D coluor images 'Peppers', 'Sanada', 'Man', and Roses' containing Gaussian noises that are displayed in Figures 10, 11, 12, and 13, respectively. It can be observed from Figure 10, 11, 12, and 13 that the proposed scheme M2 can successfully remove the noise without much smearing of the sharp edges from the two images 'Peppers', 'Sanada', 'Man', and 'Roses', (shown in Figures (10(d), (11(d), (12(d), and (13(d)), respectively, which only show the good image restoration performance (PSNR, SSIM values), reducing the staircase effect but also indicates the edge enhancement and texture preservation character of proposed method M2 over method M1 (shown in Figures (10(c), (11(c), (12(c), and (13(c)), respectively) even on color images due to the effectiveness of MQ-RBF Kansa method M2 for the smooth solution of complex images. Furthermore, it can be observed that M2 produces better purity of colors in    Figure 15 presents the SSIM and PSNR curves of the traditional mesh-based scheme M1 and the proposed meshless scheme M2 after different iterations. The convergence rate of the meshless scheme M2 is faster than the mesh-based scheme M1. Another judgment from Figure 15 is that the SSIM and PSNR values of the meshless scheme M2 are higher than the values of the mesh-based scheme M1. The comparison results confirm that the proposed method M2 reproduces the source image better and gives higher SSIM and PSNR in less iterations. The calculation time along with iterative numbers of these two methods with the same level SSIM and PSNR are recorded in Table 1.

V. COMPARSION WITH OTHER SCHEMES A. COMPARISON WITH ROF SCHEME (M3)
The primary TV-based model for image denoising possessing additive noise was introduced by Rudin-Osher-Fatemi (ROF) [11]. This scheme performed good results in terms of image restoration on 2D grayscale images. The TV-based minimization functional applied by ROF for the model (1) is presented as follows.
with |∇z| = z 2 x + z 2 y , where the two terms in equation (23) are called the regularization term and data fitting terms, while λ is the scale parameter which controls the similarities between the denoised and the original images which normally depends on upon the noise level. The resultant Euler Lagrange PDE received from model (23) is determined as below.

∂ ∂x
in , where ∂z ∂n = 0 on ∂ = . The time marching PDE from equation (24) is re-written as follow; for t > 0, and (x, y) ∈ R. The numerical solution of the PDE equation (25) has been discussed and explained in [11].

B. COMPARISON WITH DTV-FILTERING BASED SCHEME (M4)
Osher and Shen [53] introduced a different DTV filter-based model for image reconstruction having additive noise. The energy functional of model (1) by [53] is provided as below.
where ϕ(x) is a simple function and ϕ(x) = x 2−q for 0 ≤ q ≤ 2. The equation (26) results in given restoration equation.
The nonlinear recovery system of equations from equation (26) is given as follows.
λ + for all β 1 ∈ . The authors used the Gauss-Jacobi scheme to resolve the system of nonlinear equation (28) which is reproduced as below. where For further details, see [53]. Experiment 6: In this analysis, the recommended collocation scheme M2 is compared to schemes M3 and M4 for two real images 'Lena' and 'Peppers' holding additive Gaussian noises. Repeatedly, from the experimental outcomes, we can see that M2 is better in image recovery (PSNR), iterative numbers, and time of computation (CUP time) corresponded with schemes M3 and M4. Those reconstructed results are displayed in Figures 16, 17, and Table 2, respectively. Consequently, this analysis confirms the effectiveness of the meshless MQ-RBF applications used in M2.

C. A NEW METHOD FOR IMAGE RESTORATION IN THE PRESENCE OF IMPULSE NOISE (M5)
Yuan and Ghanem [54] has introduced a new scheme for a TV-based model for the removal of impulsive noise from the given image data and delivered good restoration results. The minimization functional for this model is defined as below.
arg min 0≤u,v≤ (34) VOLUME 8, 2020   where u describes the clean image, b is noisy or blurred image, K denotes the linear operator, a and b are the noise vectors, shows the elementwise product, and p is a parameter such that p = 1 for isotropic TV model and p = 2 for anisotropic TV model. Here, we choose K = 1. The contributors utilized the proximal ADMM strategy, which iteratively renewed the primal and dual variables of the augmented Lagrangian function of equation (34). For any auxiliary x ∈ 2n and y ∈ n , equation (34) can be rewritten as arg min 0≤u,v≤ Similarly, for £ β : n × n × 2n × n × 2n × n −→ be the augmented Lagrangian function of (35).
(·) determines the random-error map of the model, applying the Gaussian density (x i ) ∼ G(0, σ 2 ). The optimization procedure based on Sparsity, Density and Multimodality has been used by the authors to impulsive noise from noisy image. By employing the semisupervised cost function φ, the model (36) is redefined as The effect of the above cost function is to divide the model into three different stages for the solution.
In the first stage, φ is used to denoise the image corrupted by high-density noise. The functions φ 1 , φ 2 , and φ 3 are given as follows.
On the second stage, the number of images considered to be noise-free in the large-volume image set D is limited. Correspondingly, images identified to be clean in the training set are extremely sparse. In this situation, the fourth term of φ 4 is given as under.
In the third stage, the images in the large-volume image set D may be contaminated with various types of noises. Accordingly, the aforementioned property of average intensity shows that the average intensity of the reconstructed image set is similar to that of the clean pixels in the corrupted image set. This is shown in the fifth term of the stated cost function φ 5 is given as follows.
So the combination of all the equations results in total cost function and is given by the formula.

E. WEIGHTED COUPLE SPARSE REPRESENTATION WITH CLASSIFIED REGULARIZATION FOR IMPULSE NOISE REMOVAL (M7)
Chen et al. [56] proposed a model for image restoration having impulsive noise which is given as under.
(â,D,Ŷ ) = arg min where X shows the corrupt image, W is the weighted matrix generated by a noise detector, D ∈ R n×K (n < K ) is a redundant dictionary,R i,j matrix that is used to extract the (i, j) th √ n × √ n patch from the image, a is the representation coefficient expected to be sparse, and L is the sparse ratio. By alternating minimization method is utilized to solve the model (44) and is given by the given equation.
Similarly, the resultant restoration image can be calculated pixel by pixel by the given formula.
For more information, the researchers are refereed to [c] [56]. Experiment 7: In this experiment, we examine the reconstruction performance of the suggested meshless methods M2 with the latest state-of-art schemes M5, M6, and M7, respectively.
In the first example, the two schemes M2 and M5 are examined concerning image restoration performance on the real images 'peppers' and 'Lena' images containing impulsive noise for the same high noise levels and parameter values as chosen in [54]. All the reconstruction results are given in Figures 18 and 19, and displayed in Table 3, respectively. We can observe from the Figures 18 and 19, and Table 3, that the visual quality of image recovery (PSNR values) performance of the proposed meshless algorithm M2 is far better than scheme M5 concerning image restoration (PSNR values), minimization of staircase effect, textures and recovering edges. The importance of M2 for best reconstruction outcomes than M5 is due to the MQ-interpolation process which is responsible for edges preservation. Also due to the lack of dependence on a mesh or integration procedure and the Euclidian distance between a noisy pixel and other non-noisy pixel values applications used in the proposed scheme M2 are responsible for the unique smooth solution which results for qualified restoration results.
In the second example, the recommended scheme M2 is further examined with scheme M6 for the same high noise levels, image size, and parameters picked in [55] and are exhibited in Figures 20,21,22, and 23, and Table 4, respectively. In this illustration, repeatedly the suggested algorithm M2 produces more reliable restoration results than algorithm M6 due to the meshless utilization as discussed in the VOLUME 8, 2020     above-mentioned case first. More importantly, the homogeneous regions in the color images are also recovered sharply in this experiment.
In the third example, again the reconstruction performance of the meshless scheme M2 is excellent over scheme M7 [56] in image restoration completion presented in Figures 24, 25,    and 26, and displayed in Table 5, respectively. Repeatedly the meshless applicability of meshless scheme makes the meshless scheme M2 more active in image restoration in all perspectives than M7.
To summarize this experiment, we can observe that the achievements of image restoration concerning image restoration (PSNR values), minimization of staircase effect, textures and recovering edges of the recommended meshless scheme M2 is stable than M5, M6, and M7 due to the meshless application of MQ-RBF approach used in proposed method M2. Experiment 8: In this experiment, we have implemented the recommended algorithm M2 for varying additive Gaussian noise levels on 15 randomly selected images from the Berkeley images database BSD500 for image restoration in terms of the average PSNR SSIM values. All the information concerning the average PSNR and SSIM values is recorded in Table 6.

VI. SENSITIVITY ANALYSIS OF PARAMTERS
In this section, we will briefly discuss the selection of shape parameters shape parameter c and fitting parameter λ used in the proposed meshless scheme and their impact on image restoration. Since from experimental results, we can say the best-selected value chosen in meshless method M2 is quite complicated. Nevertheless, its best-selected values are adjusted and tuned according to the noise variance, image size, etc. It has been noticed that the range of values allowed is: c ∈ [0.37,0.79] and λ ∈ [0.245,0.502], for natural and artificial images. This shows that both the parameters c and λ VOLUME 8, 2020     used in M2 are valuable for enhancing the denoising performance. Likewise, the range of iterations to improve the PNSR results is [20,90]. Thus, the availability of information about the uncertainty of the denoising result on the user-chosen parameters is helpful to avoid incorrect decisions. For briefness, Tables 7 and 8

VII. CONCLUSION
In this article, the meshless collocation scheme was introduced in which DTV filter was applied in combination with Multiquadric RBF to solve nonlinear equation for the smooth solution and to remove the additive noise from the images. The proposed meshless scheme was investigated on various real and artificial images grayscale and color artificial images and the achieved results were compared with the existing traditional state-of-art schemes.
Our numerical experiments revealed that the proposed methodology produced not only good quality of image restoration but also sharply resolved the discontinuities and improved overall accuracy. It was observed that the performance of the meshless scheme was better in image restoration quality (PSNR, SSIM, SNR, minimization of staircase effect and preservation of edges, texture, and homogeneity), iteration numbers, and computation time required for convergence compared with existing methods. The parameter sensitivity analysis was also discussed.