# Improved Multiscale Learning Dictionary Image Denoising Algorithm in Curved Wave Domain

Authors: Jing Mao; Shunyuan Yu; Jie Chen; Shuo Chen
Journal: Scientific Programming
 (2023)

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## Abstract

It was proposed to develop a better multiscale learning dictionary picture de-noising technique. The approach improves the adaptive threshold curvilinear transform, which can divide an image into different scale information and be used to build a curvilinear multiscale learning dictionary. The method finished the dictionary and sparse coefficient updates in the picture through circular iterations and then superimposed the matching curvilinear wave domain image blocks and performed the curvilinear inverse transform to generate the denoised image. The test was carried out by adding additive Gaussian noise to a standard grayscale image, and the results revealed that the peak signal-to-noise ratio of the grayscale image de-noising result of this paper’s method was improved by 56.6% on average, and the structural similarity was improved by 0.44 on average, compared to the conventional de-noising algorithm. It was determined that the enhanced approach preserved the picture’s edge and texture information well, that image quality was greatly improved, and that the algorithm’s execution efficiency was superior to that of the conventional de-noising algorithm.


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## Body

## 1. Introduction

To better read and recognize the legitimate information in a picture, the noise included in it must be eliminated, but the detailed information such as edges and textures of the image structure must be preserved as much as possible. Hence, image de-noising was carried out using several approaches today. For example, in the spatial domain, there were median filtering methods [1, 2], Gaussian filtering methods [3], nonlocal mean filtering methods [4], bilateral filtering methods [5], etc. In the transform domain, there were wavelet thresholding methods [6], singular value decomposition methods [7], curvilinear wave transform methods [8], BM3D algorithms [9], etc. However, the spatial domain de-noising method was limited by the principle, and it was more difficult to remove the noise in complex images. The transform domain de-noising method made empirical assumptions on the transform domain coefficients, and it was difficult for a single fixed basis function to adapt to complex image features, which can easily damage the detailed information of image features and make the image blurred. Therefore, digital images require an efficient noise removal method that can both better eliminate image noise and retain maximum image edge and texture detail information [10].In recent years, there have been more de-noising methods in the field of digital images, and Pei et al. proposed a multiinformation combined dictionary method in 2018 for improving the problem that conventional dictionary learning methods easily lead to blurred de-noising results [11], and the de-noising results need to be improved due to the poor anti-interference ability of the algorithm. Khmag et al. in 2018 used noisy images and clustered batches of Hidden Markov Models (HMMs) to remove noise from images, but the results need to be improved because the NLM algorithm is computationally intensive and time-consuming when estimating pixels [12]. Tao et al. fused dictionary learning and a variational model in 2019, which can improve the structural similarity of the resultant images [13]. However, the de-noising effect was poor when the noise was strong. Lu et al. proposed a gradient-domain nonparametric Bayesian dictionary learning method in 2020, which was better for image detail protection [14]. However, the optimal model parameters of the algorithm were difficult to determine, making the de-noising results less than ideal. The curvilinear wave transform has directional and multiscale characteristics, which can better represent the local features of the signal; however, the determination of the appropriate threshold was more difficult, and the de-noising results need to be improved. Khmag et al. in 2022 used a generative adversarial network model and a semisoft thresholding method to achieve image noise removal, but it was susceptible to interference from external factors when carrying out noise removal due to the inability to produce diverse results when considering independent samples alone [15].To address the abovementioned situation, this paper combined the advantages of the K-SVD algorithm and curvilinear wave transform, improved the dictionary training and construction method of the K-SVD algorithm, constructed multiscale learning dictionaries in the curvilinear wave domain, created multiscale adaptive dictionaries, suppressed the noise in different scale cases in multiple iterative updates, and had good de-noising ability while protecting effective information such as effective edges and textures of images. The proposed method can overcome the limitation of sparse image representation at a single scale and made the new dictionary the advantage of multiscale analysis, which can protect the effective information of the image, especially the small-scale details and edges, and other features will not be oversmoothed so that the noise and effective information at different scales can be identified and distinguished. The dictionary and the sparse coefficients were updated by iterative cycles to remove the noise at different scales so that the effective information of the image can be expressed adaptively and completely.A series of simulations showed that the revised de-noising algorithm can successfully improve the de-noising effect, better protect picture features such as edges and textures, and significantly improve image quality.


## 2. Curve-Wave Transform and Dictionary Learning Principle

### 2.1. Curve-Wave Transformation

The curvilinear transform was first proposed by Candes et al. [16] and has undergone two developments. At present, the second-generation curve-wave transform is widely used, and its expression is(1)cj,l.k=f,φj,l,k=∫R2fxφj,l,kx¯dx.In formula (1), j,l.k represent scale, angle, and position parameters, respectively; f represents the objective function; φj,l,k represent the curvilinear function; and the expression in the frequency domain is(2)cj,l,k=12π2∫f^ωUjRθLωi<xkj,l,ωdω.In formula (2), f^ω represents the objective function in the frequency domain, Uj represents the frequency window, and RθL represents the angle matrix. For the image size of M × N, the scale layers of the curvilinear transform are(3)s=log2M×N−3.The curvilinear coefficients may be split into three layers after the curvilinear wave transform: a large-scale coefficient layer, a medium-scale coefficient layer, and a fine small-scale layer. The image-effective signal was represented by a few large-scale curvilinear coefficients, whereas the noisy signal was mixed in the medium- and small-scale curvilinear coefficients [17]. As an example, following the curvilinear wave transforms, an image with a size of 256256 had its curvilinear wave coefficients separated into five scales, and their scales, orientations, and other information are given in Table 1.Table 1
Scale coefficients and structural information of 256 × 256 images.
LayerScaleDirectionCoefficient of structureLarge scaleC(1)132 × 32Meso scaleC(2)32 (4 × 8)16 × 1212 × 1616 × 1212 × 16C(3)32 (4 × 8)32 × 2222 × 3232 × 2222 × 32C(4)64 (4 × 16)64 × 2222 × 6464 × 2222 × 64Small scaleC(5)1256 × 256The curvilinear coefficients in formula (1) were thresholded, the small-value coefficients corresponding to the noisy signal were discarded, the large-value coefficients corresponding to the valid signal were retained, and the de-noising results were obtained by reconstructing the valid signal using the curvilinear inverse transform.


### 2.2. K-SVD Dictionary Learning Algorithm

The K-SVD algorithm mainly included two steps: sparse coding and dictionary updating. For a given original noise-free imageI and noisy image X, set learning dictionary D, the de-noising of sparse representation of images can be expressed as an optimization problem, which can be expressed as(4)a^ij,D^,I^=argminaij,D,IλI−X22+∑ijuij+∑ijDaij−RijX22.Here,a^ij represents the optimal sparse representation coefficient, I^ represents the image after de-noising, λ and uij represent the balance parameter, aij represents the sparse representation coefficient corresponding to the image containing noise, and Rij represents the image block matrix.When solving formula (4), it needed to set up an initial learning dictionary D0=dk0;k=1,2,⋯,K. The initial dictionary was usually composed of random matrix or discrete cosine transform coefficients, where k was the number of iterations. Then, the sparse coding coefficient is calculated, and the updated sparse representation coefficient formula is(5)a^ij=argminaij∑i,juijaij0+∑i,jD0aij−RijX22.The Least Angle Regression (LARS) algorithm was used to solve the abovementioned formula and obtain the sparse representation coefficient of the image. Then, the updated dictionary formula is(6)D^=argminD∑i,juijaij0+∑i,jD0aij−RijX22.Formula (6) was adopted to complete dictionary updating, and then the sparse coefficient matrices a^ij and D^ were fixed. Then, the objective function of formula (4) can be further transformed into the following:(7)I^=argminXλI−X22+∑i,jDaij−RijX22.Rewrite formula (7) into matrix form, and then the denoised image expression is(8)I^=λE+∑i,jRijTRij−1λX+∑ijRijTDaij.In formula (8), E represents the identity matrix, and I^ represents the image result after de-noising. Since the image’s effective signal had good sparsity while the noise did not, according to this feature, K-SVD achieved the separation of the effective signal and the noise signal by reconstructing the sparse representation signal of the image.


## 2.1. Curve-Wave Transformation

The curvilinear transform was first proposed by Candes et al. [16] and has undergone two developments. At present, the second-generation curve-wave transform is widely used, and its expression is(1)cj,l.k=f,φj,l,k=∫R2fxφj,l,kx¯dx.In formula (1), j,l.k represent scale, angle, and position parameters, respectively; f represents the objective function; φj,l,k represent the curvilinear function; and the expression in the frequency domain is(2)cj,l,k=12π2∫f^ωUjRθLωi<xkj,l,ωdω.In formula (2), f^ω represents the objective function in the frequency domain, Uj represents the frequency window, and RθL represents the angle matrix. For the image size of M × N, the scale layers of the curvilinear transform are(3)s=log2M×N−3.The curvilinear coefficients may be split into three layers after the curvilinear wave transform: a large-scale coefficient layer, a medium-scale coefficient layer, and a fine small-scale layer. The image-effective signal was represented by a few large-scale curvilinear coefficients, whereas the noisy signal was mixed in the medium- and small-scale curvilinear coefficients [17]. As an example, following the curvilinear wave transforms, an image with a size of 256256 had its curvilinear wave coefficients separated into five scales, and their scales, orientations, and other information are given in Table 1.Table 1
Scale coefficients and structural information of 256 × 256 images.
LayerScaleDirectionCoefficient of structureLarge scaleC(1)132 × 32Meso scaleC(2)32 (4 × 8)16 × 1212 × 1616 × 1212 × 16C(3)32 (4 × 8)32 × 2222 × 3232 × 2222 × 32C(4)64 (4 × 16)64 × 2222 × 6464 × 2222 × 64Small scaleC(5)1256 × 256The curvilinear coefficients in formula (1) were thresholded, the small-value coefficients corresponding to the noisy signal were discarded, the large-value coefficients corresponding to the valid signal were retained, and the de-noising results were obtained by reconstructing the valid signal using the curvilinear inverse transform.


## 2.2. K-SVD Dictionary Learning Algorithm

The K-SVD algorithm mainly included two steps: sparse coding and dictionary updating. For a given original noise-free imageI and noisy image X, set learning dictionary D, the de-noising of sparse representation of images can be expressed as an optimization problem, which can be expressed as(4)a^ij,D^,I^=argminaij,D,IλI−X22+∑ijuij+∑ijDaij−RijX22.Here,a^ij represents the optimal sparse representation coefficient, I^ represents the image after de-noising, λ and uij represent the balance parameter, aij represents the sparse representation coefficient corresponding to the image containing noise, and Rij represents the image block matrix.When solving formula (4), it needed to set up an initial learning dictionary D0=dk0;k=1,2,⋯,K. The initial dictionary was usually composed of random matrix or discrete cosine transform coefficients, where k was the number of iterations. Then, the sparse coding coefficient is calculated, and the updated sparse representation coefficient formula is(5)a^ij=argminaij∑i,juijaij0+∑i,jD0aij−RijX22.The Least Angle Regression (LARS) algorithm was used to solve the abovementioned formula and obtain the sparse representation coefficient of the image. Then, the updated dictionary formula is(6)D^=argminD∑i,juijaij0+∑i,jD0aij−RijX22.Formula (6) was adopted to complete dictionary updating, and then the sparse coefficient matrices a^ij and D^ were fixed. Then, the objective function of formula (4) can be further transformed into the following:(7)I^=argminXλI−X22+∑i,jDaij−RijX22.Rewrite formula (7) into matrix form, and then the denoised image expression is(8)I^=λE+∑i,jRijTRij−1λX+∑ijRijTDaij.In formula (8), E represents the identity matrix, and I^ represents the image result after de-noising. Since the image’s effective signal had good sparsity while the noise did not, according to this feature, K-SVD achieved the separation of the effective signal and the noise signal by reconstructing the sparse representation signal of the image.


## 3. De-Noising Algorithm for Multiscale Learning Dictionary

In this paper, based on the curvilinear wave transform and K-SVD algorithm, a learning dictionary de-noising algorithm with multiscale features were proposed, which combined the advantages of the curvilinear wave transform and K-SVD algorithm to effectively improve the de-noising results and protects details such as edges and textures. The algorithm’s principle is as follows.


### 3.1. Curvilinear Transform and Threshold Preprocessing

The input image was subjected to a discrete curvilinear transform, and the curvilinear coefficients at different scales were thresholded and updated. The updated coefficients are(9)xj,l,k=Hτj,l,k∗cj,l,k.In formula (9), x represents the updated curvilinear coefficient, c represents the curvilinear coefficient before the update, Hτ represents the threshold function, and j,l,k represent scale, angle, and position parameters, respectively. In this paper, the improved Bayes soft threshold algorithm was used to preprocess the noise coefficient of the curving coefficient, and the threshold calculation steps under various scales are as follows:(1)
Estimated the noise variance at each scaleσ2(10)σj,l,k=mediancj,l,k0.6745.(2)
Estimated the variance of the curvature coefficient of the imageσx2(11)σxj,l,k=max∑c2j,l,kn2−σ2j,l,k,0.Then, the final threshold expression is(12)xj,l,k=σ2j,l,kσxj,l.k,σx>0,maxxj,l,k,σx≤0.However, the real noise variance was difficult to estimate, and it usually took several debugs to obtain reliable threshold parameters. Therefore, a sparse inversion model is established in this paper to solve the optimal threshold.First, a noise-containing model was established, and its expression is(13)x,φj,l,k+n,φj,l,k=cj,l,k.Here,x represents effective image information, n represents noise information, φj,l,k represent the curvilinear function, and c represents the curvilinear coefficient. When the noise information σ is known, the effective information x of the image can be obtained by solving the following formula to achieve noise removal. The expression is as follows:(14)minx1s.t.x,φj,l,k−cj,l,k2≤σ.In formula (14), σ represents the noise variance. Since it was difficult to estimate the real noise of the actual image, a sparse inversion model was established in this paper for solving the problem, and its expression is as follows:(15)min12x,φj,l,k−cj,l,k22+λx1,λ>0.Here,λ is the regularization parameter, and the effective information after de-noising is obtained by solving the formula (15).The key to achieving the abovementioned goal was determining the appropriate and optimal regularization parameters. In general, when the image contained weak noise, a smaller value can be selected; when there was strong noise, a larger value should be selected; and the best de-noising effect can be achieved only when the optimal value is selected. Therefore, for curvilinear coefficients at different scales, different ones should be selected to effectively remove the noise information at this scale.For formula (15), this paper used an improved iterative approximation with a soft thresholding method to find the regularization parameter λ. First, apply a larger λ parameter to solve formula (15), obtain the initial solution for the next iteration, and then gradually reduce the regularization parameter λ and solve until the optimal regularization parameter is obtained. To achieve an adaptive threshold finding, this paper defined an intermediate variable parameter with the following formula:(16)Pk=logxk1∙logx,φj,l,k−cj,l,k2.According to formula (16), as the regularization parameter gradually decreased, xk1 increased and x,φj,l,k−cj,l,k2 decreased, the intermediate variable parameter Pk increased first and then decreased. The experiment showed that the corresponding threshold was the optimal threshold only when Pk was the maximum value.


### 3.2. Initialization of the Online Dictionary

According to the sparse representation theory, the curvilinear coefficient matrix of each scale in formula (12) was partitioned into n pieces according to a given specification, and each small piece of data became a sample xi.Before dictionary learning, an initial learning dictionary was set for each layer of curvilinear wave coefficients, then the initial dictionary at the current scale, angle, and position could be set asDj,l,k0=dt0;t=1,2,⋯,K, under which t was the number of iterations. This initial dictionary was typically set up as a random matrix.To avoid the occurrence of small values in the sparse representation coefficient matrix, the binary norm of the initial dictionary column vector is usually constrained, and the constraint formula is(17)∅≜Dj,l,k0∈Rm×kdtTdt≤1.Here,∅ represents the convex set of the lexicographical matrix, Rm×k represents the m×k real matrix, dt represents the column vectors of the lexicographical matrix, and dtT represents the transposition of the column vectors of the lexicographical matrix.


### 3.3. Online Dictionary Learning and Updating

#### 3.3.1. Cyclic Sparse Coding

According to formula (5), LARS was used to calculate the sparse coding coefficient:(18)αtj,l,k≜argmin12xtj,l,k−Dt−1j,l,kα22+λα1,t=1,2,⋯,m.Here,αt represents the sparse coefficient calculated under the current number of cycles, xt represents the curving coefficient matrix under the current number of cycles, α represents the sparse coefficient of the curving coefficient xj,l,k of the original noisy image, and m represents the set number of cycles.


#### 3.3.2. Intermediate Update

(19)Atj,l,k=12∫i=1tαiαiT.Atis the intermediate variable, and A=a1,a2,⋯,ak∈Rk×k. The purpose was to facilitate dictionary updates, and the initial quantity was usually set to zero matrices. During each iteration, the data block xt matrix of the current cycle number contained several valid signals xt1,xt2,⋯,xtη, then, after the iterative update, is(20)Atj,l,k=βAt−1j,l,k+∑i=1ηαtiαtiT,β=tη+1−ηtη+1,t<η,η2+t−2η+1η2+t−η+1,t≥η.Defined intermediate variablesBt, and Bt=b1,b2,⋯,bk∈Rm×k. The initial quantity was usually set to zero matrices. In each iteration process, the data block xt matrix of the current cycle number contains η effective signals, and then the iteration update quantity is(21)Btj,l,k=βBt−1j,l,k+∑i=1ηxαtiT.


#### 3.3.3. Dictionary Update

The formulas (20) and (21) were used to complete dictionary updating, and the dictionary of the current iteration number t is(22)Dtj,l,k≜argmin1t∑i=1t12xj,l,k−Dj,l,kαi22+λαi1=argmin1t12trDj,l,kTDj,l,kAt−trDj,l,kTBt.Solved formula (22) and differentiate both sides concerning D, then, formula (22) became(23)12Dj,l,kAtj,l,kE+Atj,l,kET−Btj,l,k=0.Here,E was the identity matrix and the At was the symmetric matrix, then formula (23) can be further deduced as(24)Dj,l,kAtj,l,k−Btj,l,k=0.Taking the column vectors of the dictionary as the target, took the column derivatives one by one, then, formula (24) became(25)∂DAt−Bt∂dj=Ajj00Ajj⋯0⋯0⋮⋮00⋱⋮⋯Ajj=Ajjl.Combining formula (25) with the Newton iteration method, the updated value of the m column of the dictionary can be obtained as follows:(26)ym,t=1Ammbm−Dam+dm,t−1.Normalizing the above-updated values, the dictionary value of the finalm column after updating is(27)dm,t=1maxym,t2,1ym,t.The loop ended with the final dictionary of the current scale, angle, and positionDTj,l,k.


### 3.4. De-Noising

Finally, the sparse coefficientαt was multiplied by the final dictionary DT, and the meander coefficient after de-noising is(28)ATj,l,k=xj,l,k−DTj,l,kαtj,l,k.


### 3.5. Inverse Transformation of Meander

If you superimpose the curve-wave coefficients and perform the inverse curve-wave transformation, then the image result after final processing can be obtained, and the inverse transformation formula is(29)I^=∑j,j,kATj,l,kφj,l,k.The algorithm flow is shown in the following Figure1.Figure 1
Diagram of the de-noising algorithm flow of the multiscale learning dictionary.The corresponding pseudocode of the algorithm is shown in Figure2.Figure 2
Pseudocode of the algorithm.


## 3.1. Curvilinear Transform and Threshold Preprocessing

The input image was subjected to a discrete curvilinear transform, and the curvilinear coefficients at different scales were thresholded and updated. The updated coefficients are(9)xj,l,k=Hτj,l,k∗cj,l,k.In formula (9), x represents the updated curvilinear coefficient, c represents the curvilinear coefficient before the update, Hτ represents the threshold function, and j,l,k represent scale, angle, and position parameters, respectively. In this paper, the improved Bayes soft threshold algorithm was used to preprocess the noise coefficient of the curving coefficient, and the threshold calculation steps under various scales are as follows:(1)
Estimated the noise variance at each scaleσ2(10)σj,l,k=mediancj,l,k0.6745.(2)
Estimated the variance of the curvature coefficient of the imageσx2(11)σxj,l,k=max∑c2j,l,kn2−σ2j,l,k,0.Then, the final threshold expression is(12)xj,l,k=σ2j,l,kσxj,l.k,σx>0,maxxj,l,k,σx≤0.However, the real noise variance was difficult to estimate, and it usually took several debugs to obtain reliable threshold parameters. Therefore, a sparse inversion model is established in this paper to solve the optimal threshold.First, a noise-containing model was established, and its expression is(13)x,φj,l,k+n,φj,l,k=cj,l,k.Here,x represents effective image information, n represents noise information, φj,l,k represent the curvilinear function, and c represents the curvilinear coefficient. When the noise information σ is known, the effective information x of the image can be obtained by solving the following formula to achieve noise removal. The expression is as follows:(14)minx1s.t.x,φj,l,k−cj,l,k2≤σ.In formula (14), σ represents the noise variance. Since it was difficult to estimate the real noise of the actual image, a sparse inversion model was established in this paper for solving the problem, and its expression is as follows:(15)min12x,φj,l,k−cj,l,k22+λx1,λ>0.Here,λ is the regularization parameter, and the effective information after de-noising is obtained by solving the formula (15).The key to achieving the abovementioned goal was determining the appropriate and optimal regularization parameters. In general, when the image contained weak noise, a smaller value can be selected; when there was strong noise, a larger value should be selected; and the best de-noising effect can be achieved only when the optimal value is selected. Therefore, for curvilinear coefficients at different scales, different ones should be selected to effectively remove the noise information at this scale.For formula (15), this paper used an improved iterative approximation with a soft thresholding method to find the regularization parameter λ. First, apply a larger λ parameter to solve formula (15), obtain the initial solution for the next iteration, and then gradually reduce the regularization parameter λ and solve until the optimal regularization parameter is obtained. To achieve an adaptive threshold finding, this paper defined an intermediate variable parameter with the following formula:(16)Pk=logxk1∙logx,φj,l,k−cj,l,k2.According to formula (16), as the regularization parameter gradually decreased, xk1 increased and x,φj,l,k−cj,l,k2 decreased, the intermediate variable parameter Pk increased first and then decreased. The experiment showed that the corresponding threshold was the optimal threshold only when Pk was the maximum value.


## 3.2. Initialization of the Online Dictionary

According to the sparse representation theory, the curvilinear coefficient matrix of each scale in formula (12) was partitioned into n pieces according to a given specification, and each small piece of data became a sample xi.Before dictionary learning, an initial learning dictionary was set for each layer of curvilinear wave coefficients, then the initial dictionary at the current scale, angle, and position could be set asDj,l,k0=dt0;t=1,2,⋯,K, under which t was the number of iterations. This initial dictionary was typically set up as a random matrix.To avoid the occurrence of small values in the sparse representation coefficient matrix, the binary norm of the initial dictionary column vector is usually constrained, and the constraint formula is(17)∅≜Dj,l,k0∈Rm×kdtTdt≤1.Here,∅ represents the convex set of the lexicographical matrix, Rm×k represents the m×k real matrix, dt represents the column vectors of the lexicographical matrix, and dtT represents the transposition of the column vectors of the lexicographical matrix.


## 3.3. Online Dictionary Learning and Updating

### 3.3.1. Cyclic Sparse Coding

According to formula (5), LARS was used to calculate the sparse coding coefficient:(18)αtj,l,k≜argmin12xtj,l,k−Dt−1j,l,kα22+λα1,t=1,2,⋯,m.Here,αt represents the sparse coefficient calculated under the current number of cycles, xt represents the curving coefficient matrix under the current number of cycles, α represents the sparse coefficient of the curving coefficient xj,l,k of the original noisy image, and m represents the set number of cycles.


### 3.3.2. Intermediate Update

(19)Atj,l,k=12∫i=1tαiαiT.Atis the intermediate variable, and A=a1,a2,⋯,ak∈Rk×k. The purpose was to facilitate dictionary updates, and the initial quantity was usually set to zero matrices. During each iteration, the data block xt matrix of the current cycle number contained several valid signals xt1,xt2,⋯,xtη, then, after the iterative update, is(20)Atj,l,k=βAt−1j,l,k+∑i=1ηαtiαtiT,β=tη+1−ηtη+1,t<η,η2+t−2η+1η2+t−η+1,t≥η.Defined intermediate variablesBt, and Bt=b1,b2,⋯,bk∈Rm×k. The initial quantity was usually set to zero matrices. In each iteration process, the data block xt matrix of the current cycle number contains η effective signals, and then the iteration update quantity is(21)Btj,l,k=βBt−1j,l,k+∑i=1ηxαtiT.


### 3.3.3. Dictionary Update

The formulas (20) and (21) were used to complete dictionary updating, and the dictionary of the current iteration number t is(22)Dtj,l,k≜argmin1t∑i=1t12xj,l,k−Dj,l,kαi22+λαi1=argmin1t12trDj,l,kTDj,l,kAt−trDj,l,kTBt.Solved formula (22) and differentiate both sides concerning D, then, formula (22) became(23)12Dj,l,kAtj,l,kE+Atj,l,kET−Btj,l,k=0.Here,E was the identity matrix and the At was the symmetric matrix, then formula (23) can be further deduced as(24)Dj,l,kAtj,l,k−Btj,l,k=0.Taking the column vectors of the dictionary as the target, took the column derivatives one by one, then, formula (24) became(25)∂DAt−Bt∂dj=Ajj00Ajj⋯0⋯0⋮⋮00⋱⋮⋯Ajj=Ajjl.Combining formula (25) with the Newton iteration method, the updated value of the m column of the dictionary can be obtained as follows:(26)ym,t=1Ammbm−Dam+dm,t−1.Normalizing the above-updated values, the dictionary value of the finalm column after updating is(27)dm,t=1maxym,t2,1ym,t.The loop ended with the final dictionary of the current scale, angle, and positionDTj,l,k.


## 3.3.1. Cyclic Sparse Coding

According to formula (5), LARS was used to calculate the sparse coding coefficient:(18)αtj,l,k≜argmin12xtj,l,k−Dt−1j,l,kα22+λα1,t=1,2,⋯,m.Here,αt represents the sparse coefficient calculated under the current number of cycles, xt represents the curving coefficient matrix under the current number of cycles, α represents the sparse coefficient of the curving coefficient xj,l,k of the original noisy image, and m represents the set number of cycles.


## 3.3.2. Intermediate Update

(19)Atj,l,k=12∫i=1tαiαiT.Atis the intermediate variable, and A=a1,a2,⋯,ak∈Rk×k. The purpose was to facilitate dictionary updates, and the initial quantity was usually set to zero matrices. During each iteration, the data block xt matrix of the current cycle number contained several valid signals xt1,xt2,⋯,xtη, then, after the iterative update, is(20)Atj,l,k=βAt−1j,l,k+∑i=1ηαtiαtiT,β=tη+1−ηtη+1,t<η,η2+t−2η+1η2+t−η+1,t≥η.Defined intermediate variablesBt, and Bt=b1,b2,⋯,bk∈Rm×k. The initial quantity was usually set to zero matrices. In each iteration process, the data block xt matrix of the current cycle number contains η effective signals, and then the iteration update quantity is(21)Btj,l,k=βBt−1j,l,k+∑i=1ηxαtiT.


## 3.3.3. Dictionary Update

The formulas (20) and (21) were used to complete dictionary updating, and the dictionary of the current iteration number t is(22)Dtj,l,k≜argmin1t∑i=1t12xj,l,k−Dj,l,kαi22+λαi1=argmin1t12trDj,l,kTDj,l,kAt−trDj,l,kTBt.Solved formula (22) and differentiate both sides concerning D, then, formula (22) became(23)12Dj,l,kAtj,l,kE+Atj,l,kET−Btj,l,k=0.Here,E was the identity matrix and the At was the symmetric matrix, then formula (23) can be further deduced as(24)Dj,l,kAtj,l,k−Btj,l,k=0.Taking the column vectors of the dictionary as the target, took the column derivatives one by one, then, formula (24) became(25)∂DAt−Bt∂dj=Ajj00Ajj⋯0⋯0⋮⋮00⋱⋮⋯Ajj=Ajjl.Combining formula (25) with the Newton iteration method, the updated value of the m column of the dictionary can be obtained as follows:(26)ym,t=1Ammbm−Dam+dm,t−1.Normalizing the above-updated values, the dictionary value of the finalm column after updating is(27)dm,t=1maxym,t2,1ym,t.The loop ended with the final dictionary of the current scale, angle, and positionDTj,l,k.


## 3.4. De-Noising

Finally, the sparse coefficientαt was multiplied by the final dictionary DT, and the meander coefficient after de-noising is(28)ATj,l,k=xj,l,k−DTj,l,kαtj,l,k.


## 3.5. Inverse Transformation of Meander

If you superimpose the curve-wave coefficients and perform the inverse curve-wave transformation, then the image result after final processing can be obtained, and the inverse transformation formula is(29)I^=∑j,j,kATj,l,kφj,l,k.The algorithm flow is shown in the following Figure1.Figure 1
Diagram of the de-noising algorithm flow of the multiscale learning dictionary.The corresponding pseudocode of the algorithm is shown in Figure2.Figure 2
Pseudocode of the algorithm.


## 4. Algorithm Testing and Application

To verify the algorithm presented in this paper, a standard grayscale image of 512 × 512 containing additive noise was used for testing. The test machine was a Windows system. The code was compiled and developed using the open-source library Python. To prevent the threshold value from changing too quickly and missing the extreme value of the intermediate variables, the exponential method was used to reduce the threshold value during the continuation of the iteration, and then the iteration was continued. Through the test, it can be seen that the number of cycles was usually around the number of layers of the scale of the curvilinear transform to search for the optimal threshold, and usually, the maximum number of cycles can be set to two times the number of layers of the scale of the curvilinear transform. Therefore, the operation efficiency was mainly influenced by the multiscale layers.The original noise-free and noise-containing images are shown in Figure3, and then tested using the K-SVD algorithm, the bilateral filtering method, the nonlocal mean filtering method, the curvilinear wave transform algorithm, the BM3D algorithm, the literature [13] method, the literature [14] method, the literature [15] method, and the method in this paper. Using the noisy image shown in Figure 2 as the input image, the peak signal-to-noise ratio (PSNR) and structural similarity (SSIM) values of the de-noising results of different methods were counted, and the results are shown in Table 2.Figure 3
Original image without noise and image with noise. (a) Original noiseless image. (b) Images containingσ=20 Gaussian white noise. (c) Original noiseless image. (d) Images containing σ=20 Gaussian white noise.
(a)(b)(c)(d)Table 2
Peak signal-to-noise ratio–structural similarity of simulation results.
σPSNR (dB)-SSIMK-SVDBilateral filteringNonlocal mean filteringCurvilinear transform algorithmBM3DLiterature [13]Literature [14]Literature [15]Methods in this paperBird1032.35–0.8233.15–0.8333.45–0.8533.63–0.8434.79–0.8934.21–0.8534.35–0.8734.56–0.8834.75–0.892028.42–0.7529.33–0.7729.54–0.7829.89–0.8031.16–0.8330.13–0.7930.52–0.8130.66–0.8131.15–0.823025.32–0.7026.54–0.7227.35–0.7427.88–0.7529.13–0.7928.12–0.7628.45–0.7628.89–0.7729.11–0.784024.31–0.6726.11–0.6926.57–0.7026.62–0.7127.68–0.7426.87–0.7127.11–0.7127.34–0.7227.69–0.745023.76–0.6224.36–0.6425.21–0.6525.54–0.6626.88–0.7125.98–0.6726.23–0.6826.55–0.6926.87–0.70Boat1032.22–0.84532.32–0.85432.78–0.89133.55–0.88633.67–0.89234.18–0.91134.21–0.90834.16–0.95834.42–0.9632029.12–0.69729.32–0.73229.64–0.79830.11–0.81530.19–0.81930.38–0.84230.33–0.83830.53–0.92831.18–0.9353028.05–0.62228.21–0.65228.32–0.68728.21–0.70128.34–0.71128.64–0.73228.65–0.72928.57–0.89028.88–0.8924026.11–0.51226.34–0.54726.32–0.61226.65–0.0.60726.89–0.65527.18–0.67227.11–0.66827.38–0.67327.52–0.6805024.12–0.48924.56–0.53125.12–0.55925.32–0.56825.53–0.61126.12–0.63126.06–0.62926.33–0.65526.41–0.661It can be seen from Table2that in 90 groups of test results, the average improvement of PSNR and SSIM of K-SVD was 39.7% and 0.36, respectively. The average PSNR and SSIM of bilateral filtering were improved by 45.5% and 0.38, respectively. The average PSNR and SSIM of nonlocal mean filtering are improved by 49% and 0.394, respectively. The average PSNR and SSIM of the curve transformation results were increased by 50.1% and 0.402, respectively. BM3D results showed an average increase of 56.62% in PSNR and 0.442 in SSIM. The average improvement of PSNR and SSIM in the literature [13] was 52% and 0.406, respectively. The average PSNR and SSIM of the literature [14] improved by 53.7% and 0.418, respectively. The average improvement of PSNR and SSIM in the literature [15] was 54.8% and 0.424, respectively. The PSNR and SSIM of this paper method were improved by 56.6% and 0.436 on average. Among the above results, the results with the highest PSNR and SSIM were generated by the BM3D method, and the results in this paper were the closest to those of the BM3D method.To better evaluate the quality of the images, a completely nonreference algorithm (Natural image quality evaluator, NIQE) can be used to evaluate the de-noising results, and the expert evaluation values (percentage system) were counted together, then the evaluation values of the nine methods average are shown in Table3. It can be seen from Table 3 that the quality of the de-noised image obtained using the de-noising algorithm proposed in this paper was the same as that of the image obtained using the BM3D de-noising algorithm, which can retain the details and texture features of the image relatively well.Table 3
Evaluation value of the de-noise method.
De-noise methodNIQE valueExpert rated valuesOriginal noisy image5.3365K-SVD algorithm6.7271Bilateral filtering algorithm6.8575Nonlocal mean algorithm7.1182Curvilinear transform algorithm7.3583BM3D algorithm7.8785Literature [13] algorithm7.4879Literature [14] algorithm7.6183Literature [15] algorithm7.7181Algorithm of this paper7.8586Then, the noisy images shown in Figure3(d) are denoised using different methods, and the results of the nine treatments are shown in Figure 4. For a detailed analysis of the detail retention, the images of the boxed area in Figure 4 are displayed enlarged, and the enlargement results were located at the top left position of the respective image.Figure 4
Results of different de-noising methodsσ=20. (a) Results of K-SVD filtering and de-noising. (b) Bilateral filtering de-noising results. (c) Nonlocal mean filtering results. (d) The result of a curvilinear wave transforms de-noising. (e) De-noising results of the BM3D method. (f) De-noising results of the method in the literature [13]. (g) De-noising results of the method in the literature [14]. (h) De-noising results of the method in the literature [15]. (i) The de-noising results of the method in this paper.
(a)(b)(c)(d)(e)(f)(g)(h)(i)To further verify the effectiveness of the algorithm proposed in this paper, a standard data set was used for testing, and some of the results are shown in Figures5 and 6. Figure 5(a) shows the original noise-free image, and Figure 5(b) shows the noisy image. Figure 6 shows the effect of different methods of de-noising.Figure 5
Original image without noise and image with noiseσ=40. (a) Original noiseless image. (b) Images containing σ=40 Gaussian white noise.
(a)(b)Figure 6
Results of different de-noising methods. (a) Results of K-SVD filtering and de-noising. (b) Bilateral filtering de-noising results. (c) Nonlocal mean filtering results. (d) The result of curvilinear wave transforms. (e) De-noising results of the BM3D method. (f) De-noising results of the method in the literature [13]. (g) De-noising results of the method in the literature [14]. (h) De-noising results of the method in the literature [15]. (i) The de-noising results of the method in this paper.
(a)(b)(c)(d)(e)(f)(g)(h)(i)From Figures4 and 6, it can be seen that the image edge information was blurred and texture details were damaged in the de-noising result of the K-SVD method. In the bilateral filtering result, the phenomenon of “ringing” appeared. In the nonlocal mean de-noising result, the degree of de-noising was good, but the edges were blurred. The noise removal effect of the curvilinear transform was slightly better than that of the K-SVD results, bilateral filtering results, and nonlocal mean results, but the local details of the noise removal results were still blurred and some “line” noise existed. The details were shown in Figures 4(d) and 6(d). The results of the literature [13], literature [14], and literature [15] showed that the image quality was significantly better compared to the results of the abovementioned four methods, but the quality of the local details of the images was still blurry or “ringing” compared to the results of the BM3D method and this paper. The processing results of the method in this paper and the BM3D results were the best. The edge/texture details in the results of both methods were better maintained, and the overall image was closest to the original noise-free image, as shown in Figures 4(e)/6(e) and Figures 4(i)/6(i).In the following, the time spent on different images using different processing methods was counted, and the average time was obtained using 30 experiments, respectively, as shown in Figure7. It can be seen from Figure 7 that the running time of the proposed method in this paper was shorter and the efficiency of the algorithm was higher compared to other de-noising methods. From Figure 7, it can be seen that the execution efficiency of the de-noising algorithm proposed in this paper was better than several other de-noising algorithms.Figure 7
Running time statistics of different de-noising methods.


## 5. Summary

This paper proposes an improved multiscale learning dictionary de-noising algorithm by combining the adaptive nature of the K-SVD method with the advantages of multiscale and directionality of the curvilinear wave transform, and standard gray-scale image testing confirmed that the method in this paper had good de-noising ability and detail preservation effects such as edge and texture. The results were better than the conventional algorithm, the de-noising results were similar to the BM3D results, and the de-noising ability was about the same as the BM3D technique. However, in terms of computing efficiency, it was much superior than the BM3D technique, which had a certain extension impact on picture processing. In this research, the original picture was separated into three scales for dictionary generation, and the dictionary and sparse coefficients were updated using thresholding. In fact, the scale was still restricted for more complex pictures, and the following work will need to work on various scales with adaptive thresholds under varied scale coefficients.



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*Source: 1111301-2023-01-05.xml*