# An Adaptive Boosting Algorithm for Image Denoising

Authors: Zhuang Fang; Xuming Yi; Liming Tang
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

Image denoising is an important problem in many fields of image processing. Boosting algorithm attracts extensive attention in recent years, which provides a general framework by strengthening the original noisy image. In such framework, many classical existing denoising algorithms can improve the denoising performance. However, the boosting step is fixed or nonadaptive; i.e., the noise level in iteration steps is set to be a constant. In this work, we propose a noise level estimation algorithm by combining the overestimation and underestimation results. Based on this, we further propose an adaptive boosting algorithm that excludes intricate parameter configuration. Moreover, we prove the convergence of the proposed algorithm. Experimental results that are obtained in this paper demonstrate the effectiveness of the proposed adaptive boosting algorithm. In addition, compared with the classical boosting algorithm, the proposed algorithm can get better performance in terms of visual quality and peak signal-to-noise ratio (PSNR).


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## Body

## 1. Introduction

Image denoising is a fundamental problem in image processing, computer vision, pattern recognition, and so on. Consider a noisy imagey modeled as(1)y=x+ϵ,where x is a clean image and ϵ is added white Gaussian noise with zero-mean and standard deviation σ. The goal of denoising is to resolve the clean image x in (1). To this end, many technique and methods, such as spatial adaptive filters, transform-domain methods, sparse representation, and processing of local patches have been explored to study this problem, which leads to state-of-the-art denoising algorithms (denoisers), including the NLM [1], K-SVD [2], EPLL [3], BM3D [4], BM3D-SAPCA [5], and LSSC [6].Although the algorithms mentioned above are effective in denoising application, the performance can be improved by employing some specific techniques, such as “Twicing”, Bregman iteration, and SAIF. In this paper, we focus on boosting skills originally used in machine learning, which improve denoising performance by multiple reusing [7–10] of weak denoisers to achieve strong ones. A general boosting algorithm [11] for image denoising can be expressed as(2)x^k+1=x^k+Dσy-x^k,where Dσ(·) is a denoiser and x^k is the denoised image of the kth iteration. Essentially, the boosting algorithm (2) is to repeatedly denoise the residuals, y-x^k, and add them back to the denoised image. However, Talebi et al. [10] pointed out that the number of iterations must be tuned carefully since the sequence obtained by (2) is not always convergent to the best restoration.To conquer this problem, Romano et al. [7] proposed another boosting algorithm, called SOS, which can be expressed as(3)x^k+1=Dσy+x^k-x^k.SOS strengthens the noisy image by adding portion of the denoised image x^k to the noisy image y before operating the next iteration denoiser and subtracting the same portion of the outcome. SOS has excellent performance in denoising application. In addition, it is guaranteed to converge to an optimal solution, which enables us to obtain stopping criteria easily.We note that the denoisersDσ(·) in (2) and (3) are invariant operators. In other words, the parameter σ in the denoisers is fixed regardless of the noise levels in the different iteration steps. The authors in [7] predefine a constant σ~ to estimate the initial noise level of y and then set σ=t(σ~)σ~ with t(σ~)>1 to estimate the noise level of y+x^k. In the following weaker denoising steps, the constant parameter σ is used in denoisers Dσ(·). Obviously, such scheme is not precise since x^ks are the progressively restoring images which lead to a decreasing noise level estimation of y+x^k with k increasing. To solve this problem, in this paper we propose an adaptive boosting algorithm for image denoising application. Adaptive denoisers Dσk(·) are used in boosting rather than an invariant operator Dσ(·), where the parameter σk is a noise level estimator of y+x^k.The estimation of the noise level plays a key role in our adaptive boosting algorithm. In the last decades, many methods on noise level estimation have been proposed [12–20]. Among these methods, the blind evaluation of noise level in textured images was widely studied [15, 16], which leads to many noise level estimation methods that were wildly used in processing the highly textured images. Different from the methods that were reported in [15, 16], in this paper, we focus on patch-based methods which have received a lot of attention due to the sound theoretical basis and promise performance, such as PCA [17], WT [18], and others [19, 20]. Chen et al. [19] pointed out that PCA and WT methods could lead to an underestimation of the noise level by using the Blom’s theorem [21]. Jiang et al. [20] proposed a noise level estimation method (called JZ in the following) based on the eigenvalues of covariance matrix of the flat patches. We in this paper prove that JZ method is an overestimation of the noise level. Combining WT underestimation and JZ overestimation, we propose a new estimator to obtain higher accurate estimation. The main contributions of this paper are as follows:(i)
We propose an adaptive boosting algorithm for image denoising application, in which an adaptive weaker denoiserDσk(·) is introduced rather than an invariant operator Dσ(·) often used in the traditional boosting.(ii)
We prove that JZ method is an overestimation of the noise level by using Blom’s theorem. In addition, combining WT underestimation and JZ overestimation, we propose a new estimator to obtain higher accurate estimation.(iii)
We prove the convergence of the proposed adaptive boosting algorithm. And some experiments are conducted to validate the proposed algorithm.This paper is organized as follows: in Section2, we present some related works concerning noise level estimation and image quality assessment. In Section 3, a noise level estimation algorithm and an adaptive boosting algorithm for image denoising application are proposed. In Section 4, the experiment procedure and the trial comparison between the proposed method and the initial denoising algorithm are described in detail. We summarize, conclude, and discuss the directions of future research in Section 5.


## 2. Previous Related Work

### 2.1. Eigenvalues and Noise Level

Liu et al. [18] gave a noise level estimation algorithm based on weak textured patches. According to their algorithm, the noise level of the image is estimated as follows:(4)σw2=min1⩽i⩽Newi,where ewi is the ith eigenvalues of ΣPw, in which Pw is a set of weak textured patches that are selected by Algorithm 1 that was reported in [18], and ΣPw is the covariance matrix of Pw. By selecting flat patches Pf in noisy image, Jiang et al. [20] gave another noise level estimation as follows:(5)σ^=min1⩽i⩽N⁡efiρ,where efi denotes the ith eigenvalues of ΣPf and ΣPf is the covariance matrix of Pf. In. (4) and (5), N is the element number of each patch. The basis of the above two methods is to extract a set of patches with Gaussian distribution. Chen et al. [19] proved the following theorems about the distribution of the eigenvalues.Lemma 1 (see [19]).
Given a set of random variables{yt[i]}t=1s, with each element following Gaussian distribution N(0,σ2) independently, the distribution of the noise estimation σ^i2=1/s∑t=1s‍yt[i]2 converges to the distribution N(σ2,2σ4/s) when s becomes sufficiently large.Lemma 2 (see [19]).
LetΦ(x)=1/2π∫-∞x‍e-t2/2dt denote the cumulative distribution function of a standard Gaussian distribution. Given that N independent random variables x1,x2,…,xN generated from the normal distribution N(σ2,ν2) with order x1≥x2≥⋯≥xN, then the expected value of xi can be approximated by E(xi)≈σ2+Φ-1(N-α+1-i/N-2α+1)ν with α=0.375.


### 2.2. Natural Image Quality Evaluator (NIQE)

The denoising algorithm is usually an iterative process in which the number of iterations needs to be selected such that the denoised image can achieve the best visual effect. For this problem, no-reference/blind image quality assessment models [22–26] are introduced. Recently, Mittal et al. [25] proposed a blind image quality assessment model called NIQE. In this method, the quality of a given image y is expressed as follows:(6)Dniqey=ν1-ν2TΣ1+Σ22-1ν1-ν2,where ν1, ν2 and Σ1, Σ2 are the mean vectors and covariance matrices of the natural MVG model and the distorted image’s MVG model. Compared with other methods, NIQE does not have to train on large databases of human opinions of distorted images and has low computation complexity. Thus, it is suited for determining the optimal numbers of iterations. More details about NIQE can be found in [25].


## 2.1. Eigenvalues and Noise Level

Liu et al. [18] gave a noise level estimation algorithm based on weak textured patches. According to their algorithm, the noise level of the image is estimated as follows:(4)σw2=min1⩽i⩽Newi,where ewi is the ith eigenvalues of ΣPw, in which Pw is a set of weak textured patches that are selected by Algorithm 1 that was reported in [18], and ΣPw is the covariance matrix of Pw. By selecting flat patches Pf in noisy image, Jiang et al. [20] gave another noise level estimation as follows:(5)σ^=min1⩽i⩽N⁡efiρ,where efi denotes the ith eigenvalues of ΣPf and ΣPf is the covariance matrix of Pf. In. (4) and (5), N is the element number of each patch. The basis of the above two methods is to extract a set of patches with Gaussian distribution. Chen et al. [19] proved the following theorems about the distribution of the eigenvalues.Lemma 1 (see [19]).
Given a set of random variables{yt[i]}t=1s, with each element following Gaussian distribution N(0,σ2) independently, the distribution of the noise estimation σ^i2=1/s∑t=1s‍yt[i]2 converges to the distribution N(σ2,2σ4/s) when s becomes sufficiently large.Lemma 2 (see [19]).
LetΦ(x)=1/2π∫-∞x‍e-t2/2dt denote the cumulative distribution function of a standard Gaussian distribution. Given that N independent random variables x1,x2,…,xN generated from the normal distribution N(σ2,ν2) with order x1≥x2≥⋯≥xN, then the expected value of xi can be approximated by E(xi)≈σ2+Φ-1(N-α+1-i/N-2α+1)ν with α=0.375.


## 2.2. Natural Image Quality Evaluator (NIQE)

The denoising algorithm is usually an iterative process in which the number of iterations needs to be selected such that the denoised image can achieve the best visual effect. For this problem, no-reference/blind image quality assessment models [22–26] are introduced. Recently, Mittal et al. [25] proposed a blind image quality assessment model called NIQE. In this method, the quality of a given image y is expressed as follows:(6)Dniqey=ν1-ν2TΣ1+Σ22-1ν1-ν2,where ν1, ν2 and Σ1, Σ2 are the mean vectors and covariance matrices of the natural MVG model and the distorted image’s MVG model. Compared with other methods, NIQE does not have to train on large databases of human opinions of distorted images and has low computation complexity. Thus, it is suited for determining the optimal numbers of iterations. More details about NIQE can be found in [25].


## 3. The Proposed Method

In this section, we discuss the theoretical foundation of the noise level estimation algorithm and give an estimation that linearly combines the overestimated and underestimated results to evaluate the noise levels. Then an adaptive boosting algorithm for image denoising is proposed. The details of these steps are presented in the following subsections.


### 3.1. Noise Level Estimation

Using Lemmas1 and 2 we can establish the following result.Corollary 3.
The estimation ofσw2 in (4) satisfies σw2<σ2. Particularly, if i0=N-α+1-Φ((ρ-1)s/2)(N-2α+1), then σf2>σ2, where(7)σf2=σ2+2σ4/sΦ-1N-α+1-i0/N-2α+1ρ,Proof.
Ifi>N+1/2, then N+1-2i<0. By adding N-2α+1 to both sides, we have 2N-2α+2-2i<N-2α+1; that is, N-α+1-i/N-2α+1<1/2. Then, we have Φ-1(N-α+1-i/N-2α+1)<0. Thus σ2+Φ-1(1-α/N-2α+1)ν<σ2, with ν=2σ4/s and i=N. In (4), the last eigenvalue is selected as the noise level; it follows that i=N. From Lemma 2, the following relationship holds:(8)σw2=σ2+Φ-11-αN-2α+1ν.Thus σw2<σ2.
Note thatN+1/2⩽⌊i0⌋⩽N-α+1-Φ((ρ-1)s/2)(N-2α+1). Then N-α+1-⌊i0⌋>Φ((ρ-1)s/2)(N-2α+1); that is, N-α+1-⌊i0⌋/N-2α+1>Φ((ρ-1)s/2). Because Φ is a monotonic function, we can obtain Φ-1(N-α+1-⌊i0⌋/N-2α+1)>(ρ-1)s/2. Then, Φ-1N-α+1-i0/N-2α+12/s>ρ-1. Finally, we multiply both sides of the above equation by σ2 and simplify it to obtain σf2=σ2+2/sσ2Φ-1(N-α+1-⌊i0⌋/N-2α+1)/ρ>σ2.Theorem 4.
Ifi0=N-α+1-Φ((ρ-1)s/2)(N-2α+1), then noise level can be estimated by(9)σ2=d1σf2+d2σw2d1+d2,where d1=-Φ-1(1-α/N-2α+1)2/s and d2=(1-ρ)+Φ-1(N-α+1-[i0]/N-2α+1)2/s/ρ.Proof.
From (7) and (8), we obtain the following:(10)σ2-σw2σf2-σ2=-Φ-11-α/N-2α+12/sσ21-ρ+Φ-1N-α+1-i0/N-2α+12/s/ρσ2.Let d1=-Φ-1(1-α/N-2α+1)2/s, d2=(1-ρ)+Φ-1(N-α+1-[i0]/N-2α+1)2/s/ρ. Then, (10) becomes(11)σ2-σw2σf2-σ2=d1d2.Therefore, σ2 can be solved by (11), that is,(12)σ2=d1σf2+d2σw2d1+d2.
The collection of weak textured patchesPw and the flat patches Pf may not be the same. We extract the flat patches from the weak textured patches and denote it by P. Then, both (7) and (8) hold on the collection P, and the proposed algorithm for noise level estimation can be described in Algorithm 1. In experiment, we set K=3 to ensure the efficiency of Algorithm 1 and obtain more accurate noise level. The results of the corresponding proof experiments can be seen in Section 4.1. In the following section, Algorithm 1 will be plugged in a new adaptive boosting algorithm.Algorithm 1:Estimating the noise variance.
Input: Noisy image y. Output Noise level σ^2. Initialize Estimate the initial noise level σ0, maximum number of iterations K. While  k<Kdo 1:k=k+1; 2: Using Algorthm1 in [18] to select weak textured patches Pw. 3: ForPw, use Algorithm 1 in [20] to select flat patches P. 4: For all the patches inP, calculate σf2 using Eq. (7). 5: For all the patches inP, calculate σw2 using Eq. (8). 6: Calculateσ^k+12 using Eq. (9). End while Return Noise level σ^2.


### 3.2. Adaptive Boosting Algorithm

Based on the analysis for denoiser in (3), the noise level is the main parameter in the iteration. We show that the denoiser can be improved by the following boosting procedure:( 1 ) Strengthen the signal by accumulating the previous denoised image to the noisy input image.( 2 ) Estimate the noise level σk of the strengthen image.( 3 ) Project the strengthen image to the range less than 1; i.e., divide the strengthen image by its infinite norm (maximum).( 4 ) Operate the denoiser Dcσk on the project image and back-project the range of the outcome to match the clean image x.The main equations that describe the above procedure can be written as(13)y^k=y^k-1+λx^k-1,x^k=Dcσky^k/y^k∞y^k∞1+k-1λ,where the infinite norm ·∞ represents the maximum value of all elements, σk is the noise level of y^k, the parameter c controls the strength of the denoiser, and λ controls the signal emphasis. Our full image denoising algorithm is given in Algorithm 2.Algorithm 2:Adaptive boosting algorithm for image denoising.
Input: Noisy image y, denoising operator D(⋅). Output: x^k An estimate for x Initialize: y0=y, k=0,c,λ, Dniqe(x^0)=Dniqe(y), x^0=0. Main Iteration: Increment k by 1 and perform the following steps 1.y^k=y^k-1+λx^k-1. 2.σk= the noise level of y^k and it is estimated by Algorithm 1. 3.u^k=Dcσk(y^k/∥y^k∥∞). 4.x^k=u^k∥y^k∥∞/1+(k-1)λ. Stopping Rule: If Dniqe(x^k)>Dniqe(x^k-1), stop. Otherwise, apply main iteration. Return: x^k is obtained after k iterations.


### 3.3. Convergence Analysis of Algorithm2

In this section, bounded denoiser and linear denoiser are introduced and some proposition are analysed, then the convergence of Algorithm2 is proved.Definition 5 ((bounded denoiser) [27]).
A bounded denoiser with a parameterσ is a function Dσ(x):Rn→Rn such that for any input x∈Rn,(14)Dσx-x22n≤Gσ2,for some universal constant G independent of n and σ.Definition 6 (linear denoiser).
For the given constantsσ, α, β, the denoiser Dσ(x) is a linear operator, if(15)Dσαx+βy=αDσx+βDσy.Milanfar [8] pointed out that many denoisers satisfy Definitions 5 and 6. Based on this, Romano et al. [7] proved the convergence of SOS algorithm by neglecting the nonlinear term. According to these definitions and the following properties, we prove the convergence of Algorithm 2.Proposition 7.
For the givenσ′ and σ′′, Dσ′(x)-Dσ′′(x)2⩽nG(σ′+σ′′).Proof.
SinceDσ(x) is a bounded denoiser, we have Dσ′x-Dσ′′x2=Dσ′x-x-Dσ′′x+x2⩽Dσ′x-x2+Dσ′′x-x2⩽nG(σ′+σ′′).In iterative equation (13), x^k is the kth iterative approximation of x. Denote the error between x^k and x by ϵk; that is, x^k=x+ϵk. Then it follows from the iterative equation y^k+1=y^k+λx^k that(16)y^k=y+λx^k-1+⋯+x^1.Substituting y=x+ϵ and x^k=x+ϵk into (16), we can get(17)y^k=1+λk-1x+ϵ+λϵk-1+⋯+ϵ1.Denote ϵ^k=ϵ+λ(ϵk-1+⋯+ϵ1). And let ϵ^=limk→∞ϵ^k. Using the central limit theorem, ϵ^ is a normal random variable with E(ϵ^)=0. Furthermore, we denote the standard deviation of ϵ^ by σ¯. This implies that y^k in (17) can be considered as clean portion [1+λ(k-1)]x and a zero-mean Gaussian white noise with standard deviation σ¯. Thus, the noise level of y^k estimated by Algorithm 1 is bounded. Then, we have the following proposition.Proposition 8.
There exists a constantM satisfying σk⩽M.Proposition 9.
For allk, x^k∞⩽y∞ and y^k∞⩽1+kλy∞.Proof.
Sincex^1=Dcσ1y^1/y^1∞y^1∞, we have x^1∞=Dcσ1y^1/y^1∞y^1∞∞, for Dcσk(y^k/y^k∞)∈[0,1]. Then x^1∞⩽y^1∞=y∞.
Assumex^k∞⩽y∞. Then x^k+1∞=Dcσk+1y^k+1/y^k+1∞y^k+1∞/1+kλ∞⩽y^k+1∞/1+kλ. Furthermore, noting that y^k+1=y^k+λx^k, we can get(18)x^k+1∞⩽y+λx^1+⋯+x^k1+kλ∞⩽y∞+λx^1∞+⋯+x^k∞1+kλ⩽y∞+kλy∞1+kλ=y∞.Therefore, the mathematical inductive method enables us to get x^k∞⩽y∞.
Sincey^k=y^k-1+λx^k-1, we have y^k∞=y^k-1+λx^k-1∞⩽y^k-1∞+λx^k-1∞⩽y^k-2∞+λx^k-2∞+λy∞⩽⋯⩽1+kλy∞.Proposition 10.
For any positive integerk∈Z+, the following inequalities hold:(19)x^k2⩽n1+kλ1+k-1λy∞,andy^k2⩽y2+nλ1+λk-1y∞.Proof.
For any giveny^k∈Rn, y^k=(y^k1,y^k2,…,y^kn), it holds that y^ki/y^k∞∈[0,1]. Therefore, for all noise level σk, we have(20)Dcσky^ky^k∞2⩽n.Noting that(21)x^k2=Dcσky^k/y^k∞y^k∞1+k-1λ2=11+k-1λDcσky^ky^k∞2y^k∞,and employing (20) and Proposition 9, we have(22)x^k2⩽n1+kλ1+k-1λy∞.Using the iteration y^k+1=y^k+λx^k, we have y^k=y^+λ(x^k-1+⋯+x^1). Then(23)y^k2⩽y2+λx^k-12+⋯+x^12.Substituting (22) into (23), we get(24)y^k2⩽y2+λn1+k-1λy∞1+k-2λ+n1+k-2λy∞1+k-3λ+⋯+n1+λy∞.Since 1+(k-1)λ/1+(k-2)λ<1-λ, then we have(25)y^k2⩽y2+nλ1+λk-1y∞.This completes the proof.Next, we further consider the convergence of Algorithm2.Theorem 11.
IfDσ(·) is bounded and linear denoiser, then the main iterations of Algorithm 2 are convergent; that is, x^k+1-x^k2→0 as k→∞.Proof.
It is easy to obtain(26)Dcσkx^k2=Dcσkx^k-x^k+x^k2⩽Dcσkx^k-x^k2+x^k2.Further applying Definition 5 and substituting (22) into (26), we have(27)Dcσkx^k2⩽nGcσk+n1+kλ1+k-1λy∞.Similarly,(28)Dcσky^k2=Dcσky^k-y^k+y^k2⩽Dcσky^k-y^k2+y^k2.Applying Definition 5 and substituting (25) into (28), we obtain(29)Dcσky^k2⩽nGcσk+y2+nλ1+λk-1y∞.On the other hand, the iteration y^k+1=y^k+λx^k leads to(30)11+kλDcσky^k+1-11+k-1λDcσky^k2=11+kλDcσky^k+λx^k-11+k-1λDcσky^k2=11+kλDcσky^k+λ1+kλDcσkx^k-11+k-1λDcσky^k2⩽11+kλDcσky^k-11+k-1λDcσky^k2+λ1+kλDcσkx^k2=λ1+kλ1+k-1λDcσky^k2+λ1+kλDcσkx^k2.Substituting (27) and (29) into (30), we have(31)11+kλDcσky^k+1-11+k-1λDcσky^k2⩽λnGcσk+y2+nλ1+λk-1y∞1+kλ1+k-1λ+λnGcσk+n1+kλ/1+k-1λy∞1+kλ.Finally, (32)x^k+1-x^k2=11+kλDcσk+1y^k+1-11+k-1λDcσky^k2=11+kλDcσk+1y^k+1-11+kλDcσky^k+1+11+kλDcσky^k+1-11+k-1λDcσky^k2⩽11+kλDcσk+1y^k+1-Dcσky^k+12+Dcσky^k+11+kλ-Dcσky^k1+k-1λ2.Applying Proposition 7 and substituting (31) into (32), we obtain(33)x^k+1-x^k2⩽nGcσk+1+σk1+kλ+λnGcσk+n1+kλ/1+k-1λy∞1+kλ+λnGcσk+y2+nλ1+λk-1y∞1+kλ1+k-1λ.From Proposition 8 both σk+1 and σk are all bounded. And since y2 and y∞ are finite, we can get x^k+1-x^k2→0 as k→∞, by taking the limit on both sides of (33).


### 3.4. Parameters Configuration

In Algorithm2, we select the state-of-the-art algorithms: BM3D and BM3D-SAPCA. The source code of these algorithms can be obtained from the original authors and we use the default parameters. When the denoiser is given, the parameters in Algorithm 2 are λ and c. In the remainder of this section, we mainly discuss the influence of λ and c on the final results.Consideringλ∈[0.3,0.9] and c∈[0.2,0.8] and discreting [0.3,0.9]×[0.2,0.8] by using a step-size 0.1×0.1, we get the parameters set {(λi,cj)}. Then, we introduce these parameters into the proposed Algorithm 2 and apply the proposed algorithm to noisy image. Figure 1(a) shows the relationship between (λ,c) and PSNR defined as 20log10(255/MSE), where MSE is the mean squared error between the original image and its denoised version. The relationship between (λ,c) and optimizing truncation iteration number is shown in Figure 1(b). It can be seen from Figure 1, when the parameters λ and c are small, we can get the smallest PSNR and the largest iteration number. In this case, the efficiency of the denoising is the lowest. When both λ and c achieve the highest value, the number of the iteration is the minimum, but the PSNR is not the maximum value. It is obvious that the PSNR has some volatility with λ. The iteration number of the λ-c area corresponding to the optimal PSNR value maintains consistency. In order to get the best denoising performance and reduce the computation complexity, suitable parameters λ and c must be determined. According to the results that are present in Figure 1, the PSNR is highest and the number of iterations is acceptable when λ=0.635 and c=0.472. Therefore, we select (0.635, 0.472) in the following experiment.Figure 1
Demonstration of the effect ofλ and c on the proposed algorithm outcome for the BM3D (σ=20).
(a)
Relationship amongλ, c, and PSNR (b)
Relationship amongλ, c, and iterations


## 3.1. Noise Level Estimation

Using Lemmas1 and 2 we can establish the following result.Corollary 3.
The estimation ofσw2 in (4) satisfies σw2<σ2. Particularly, if i0=N-α+1-Φ((ρ-1)s/2)(N-2α+1), then σf2>σ2, where(7)σf2=σ2+2σ4/sΦ-1N-α+1-i0/N-2α+1ρ,Proof.
Ifi>N+1/2, then N+1-2i<0. By adding N-2α+1 to both sides, we have 2N-2α+2-2i<N-2α+1; that is, N-α+1-i/N-2α+1<1/2. Then, we have Φ-1(N-α+1-i/N-2α+1)<0. Thus σ2+Φ-1(1-α/N-2α+1)ν<σ2, with ν=2σ4/s and i=N. In (4), the last eigenvalue is selected as the noise level; it follows that i=N. From Lemma 2, the following relationship holds:(8)σw2=σ2+Φ-11-αN-2α+1ν.Thus σw2<σ2.
Note thatN+1/2⩽⌊i0⌋⩽N-α+1-Φ((ρ-1)s/2)(N-2α+1). Then N-α+1-⌊i0⌋>Φ((ρ-1)s/2)(N-2α+1); that is, N-α+1-⌊i0⌋/N-2α+1>Φ((ρ-1)s/2). Because Φ is a monotonic function, we can obtain Φ-1(N-α+1-⌊i0⌋/N-2α+1)>(ρ-1)s/2. Then, Φ-1N-α+1-i0/N-2α+12/s>ρ-1. Finally, we multiply both sides of the above equation by σ2 and simplify it to obtain σf2=σ2+2/sσ2Φ-1(N-α+1-⌊i0⌋/N-2α+1)/ρ>σ2.Theorem 4.
Ifi0=N-α+1-Φ((ρ-1)s/2)(N-2α+1), then noise level can be estimated by(9)σ2=d1σf2+d2σw2d1+d2,where d1=-Φ-1(1-α/N-2α+1)2/s and d2=(1-ρ)+Φ-1(N-α+1-[i0]/N-2α+1)2/s/ρ.Proof.
From (7) and (8), we obtain the following:(10)σ2-σw2σf2-σ2=-Φ-11-α/N-2α+12/sσ21-ρ+Φ-1N-α+1-i0/N-2α+12/s/ρσ2.Let d1=-Φ-1(1-α/N-2α+1)2/s, d2=(1-ρ)+Φ-1(N-α+1-[i0]/N-2α+1)2/s/ρ. Then, (10) becomes(11)σ2-σw2σf2-σ2=d1d2.Therefore, σ2 can be solved by (11), that is,(12)σ2=d1σf2+d2σw2d1+d2.
The collection of weak textured patchesPw and the flat patches Pf may not be the same. We extract the flat patches from the weak textured patches and denote it by P. Then, both (7) and (8) hold on the collection P, and the proposed algorithm for noise level estimation can be described in Algorithm 1. In experiment, we set K=3 to ensure the efficiency of Algorithm 1 and obtain more accurate noise level. The results of the corresponding proof experiments can be seen in Section 4.1. In the following section, Algorithm 1 will be plugged in a new adaptive boosting algorithm.Algorithm 1:Estimating the noise variance.
Input: Noisy image y. Output Noise level σ^2. Initialize Estimate the initial noise level σ0, maximum number of iterations K. While  k<Kdo 1:k=k+1; 2: Using Algorthm1 in [18] to select weak textured patches Pw. 3: ForPw, use Algorithm 1 in [20] to select flat patches P. 4: For all the patches inP, calculate σf2 using Eq. (7). 5: For all the patches inP, calculate σw2 using Eq. (8). 6: Calculateσ^k+12 using Eq. (9). End while Return Noise level σ^2.


## 3.2. Adaptive Boosting Algorithm

Based on the analysis for denoiser in (3), the noise level is the main parameter in the iteration. We show that the denoiser can be improved by the following boosting procedure:( 1 ) Strengthen the signal by accumulating the previous denoised image to the noisy input image.( 2 ) Estimate the noise level σk of the strengthen image.( 3 ) Project the strengthen image to the range less than 1; i.e., divide the strengthen image by its infinite norm (maximum).( 4 ) Operate the denoiser Dcσk on the project image and back-project the range of the outcome to match the clean image x.The main equations that describe the above procedure can be written as(13)y^k=y^k-1+λx^k-1,x^k=Dcσky^k/y^k∞y^k∞1+k-1λ,where the infinite norm ·∞ represents the maximum value of all elements, σk is the noise level of y^k, the parameter c controls the strength of the denoiser, and λ controls the signal emphasis. Our full image denoising algorithm is given in Algorithm 2.Algorithm 2:Adaptive boosting algorithm for image denoising.
Input: Noisy image y, denoising operator D(⋅). Output: x^k An estimate for x Initialize: y0=y, k=0,c,λ, Dniqe(x^0)=Dniqe(y), x^0=0. Main Iteration: Increment k by 1 and perform the following steps 1.y^k=y^k-1+λx^k-1. 2.σk= the noise level of y^k and it is estimated by Algorithm 1. 3.u^k=Dcσk(y^k/∥y^k∥∞). 4.x^k=u^k∥y^k∥∞/1+(k-1)λ. Stopping Rule: If Dniqe(x^k)>Dniqe(x^k-1), stop. Otherwise, apply main iteration. Return: x^k is obtained after k iterations.


## 3.3. Convergence Analysis of Algorithm2

In this section, bounded denoiser and linear denoiser are introduced and some proposition are analysed, then the convergence of Algorithm2 is proved.Definition 5 ((bounded denoiser) [27]).
A bounded denoiser with a parameterσ is a function Dσ(x):Rn→Rn such that for any input x∈Rn,(14)Dσx-x22n≤Gσ2,for some universal constant G independent of n and σ.Definition 6 (linear denoiser).
For the given constantsσ, α, β, the denoiser Dσ(x) is a linear operator, if(15)Dσαx+βy=αDσx+βDσy.Milanfar [8] pointed out that many denoisers satisfy Definitions 5 and 6. Based on this, Romano et al. [7] proved the convergence of SOS algorithm by neglecting the nonlinear term. According to these definitions and the following properties, we prove the convergence of Algorithm 2.Proposition 7.
For the givenσ′ and σ′′, Dσ′(x)-Dσ′′(x)2⩽nG(σ′+σ′′).Proof.
SinceDσ(x) is a bounded denoiser, we have Dσ′x-Dσ′′x2=Dσ′x-x-Dσ′′x+x2⩽Dσ′x-x2+Dσ′′x-x2⩽nG(σ′+σ′′).In iterative equation (13), x^k is the kth iterative approximation of x. Denote the error between x^k and x by ϵk; that is, x^k=x+ϵk. Then it follows from the iterative equation y^k+1=y^k+λx^k that(16)y^k=y+λx^k-1+⋯+x^1.Substituting y=x+ϵ and x^k=x+ϵk into (16), we can get(17)y^k=1+λk-1x+ϵ+λϵk-1+⋯+ϵ1.Denote ϵ^k=ϵ+λ(ϵk-1+⋯+ϵ1). And let ϵ^=limk→∞ϵ^k. Using the central limit theorem, ϵ^ is a normal random variable with E(ϵ^)=0. Furthermore, we denote the standard deviation of ϵ^ by σ¯. This implies that y^k in (17) can be considered as clean portion [1+λ(k-1)]x and a zero-mean Gaussian white noise with standard deviation σ¯. Thus, the noise level of y^k estimated by Algorithm 1 is bounded. Then, we have the following proposition.Proposition 8.
There exists a constantM satisfying σk⩽M.Proposition 9.
For allk, x^k∞⩽y∞ and y^k∞⩽1+kλy∞.Proof.
Sincex^1=Dcσ1y^1/y^1∞y^1∞, we have x^1∞=Dcσ1y^1/y^1∞y^1∞∞, for Dcσk(y^k/y^k∞)∈[0,1]. Then x^1∞⩽y^1∞=y∞.
Assumex^k∞⩽y∞. Then x^k+1∞=Dcσk+1y^k+1/y^k+1∞y^k+1∞/1+kλ∞⩽y^k+1∞/1+kλ. Furthermore, noting that y^k+1=y^k+λx^k, we can get(18)x^k+1∞⩽y+λx^1+⋯+x^k1+kλ∞⩽y∞+λx^1∞+⋯+x^k∞1+kλ⩽y∞+kλy∞1+kλ=y∞.Therefore, the mathematical inductive method enables us to get x^k∞⩽y∞.
Sincey^k=y^k-1+λx^k-1, we have y^k∞=y^k-1+λx^k-1∞⩽y^k-1∞+λx^k-1∞⩽y^k-2∞+λx^k-2∞+λy∞⩽⋯⩽1+kλy∞.Proposition 10.
For any positive integerk∈Z+, the following inequalities hold:(19)x^k2⩽n1+kλ1+k-1λy∞,andy^k2⩽y2+nλ1+λk-1y∞.Proof.
For any giveny^k∈Rn, y^k=(y^k1,y^k2,…,y^kn), it holds that y^ki/y^k∞∈[0,1]. Therefore, for all noise level σk, we have(20)Dcσky^ky^k∞2⩽n.Noting that(21)x^k2=Dcσky^k/y^k∞y^k∞1+k-1λ2=11+k-1λDcσky^ky^k∞2y^k∞,and employing (20) and Proposition 9, we have(22)x^k2⩽n1+kλ1+k-1λy∞.Using the iteration y^k+1=y^k+λx^k, we have y^k=y^+λ(x^k-1+⋯+x^1). Then(23)y^k2⩽y2+λx^k-12+⋯+x^12.Substituting (22) into (23), we get(24)y^k2⩽y2+λn1+k-1λy∞1+k-2λ+n1+k-2λy∞1+k-3λ+⋯+n1+λy∞.Since 1+(k-1)λ/1+(k-2)λ<1-λ, then we have(25)y^k2⩽y2+nλ1+λk-1y∞.This completes the proof.Next, we further consider the convergence of Algorithm2.Theorem 11.
IfDσ(·) is bounded and linear denoiser, then the main iterations of Algorithm 2 are convergent; that is, x^k+1-x^k2→0 as k→∞.Proof.
It is easy to obtain(26)Dcσkx^k2=Dcσkx^k-x^k+x^k2⩽Dcσkx^k-x^k2+x^k2.Further applying Definition 5 and substituting (22) into (26), we have(27)Dcσkx^k2⩽nGcσk+n1+kλ1+k-1λy∞.Similarly,(28)Dcσky^k2=Dcσky^k-y^k+y^k2⩽Dcσky^k-y^k2+y^k2.Applying Definition 5 and substituting (25) into (28), we obtain(29)Dcσky^k2⩽nGcσk+y2+nλ1+λk-1y∞.On the other hand, the iteration y^k+1=y^k+λx^k leads to(30)11+kλDcσky^k+1-11+k-1λDcσky^k2=11+kλDcσky^k+λx^k-11+k-1λDcσky^k2=11+kλDcσky^k+λ1+kλDcσkx^k-11+k-1λDcσky^k2⩽11+kλDcσky^k-11+k-1λDcσky^k2+λ1+kλDcσkx^k2=λ1+kλ1+k-1λDcσky^k2+λ1+kλDcσkx^k2.Substituting (27) and (29) into (30), we have(31)11+kλDcσky^k+1-11+k-1λDcσky^k2⩽λnGcσk+y2+nλ1+λk-1y∞1+kλ1+k-1λ+λnGcσk+n1+kλ/1+k-1λy∞1+kλ.Finally, (32)x^k+1-x^k2=11+kλDcσk+1y^k+1-11+k-1λDcσky^k2=11+kλDcσk+1y^k+1-11+kλDcσky^k+1+11+kλDcσky^k+1-11+k-1λDcσky^k2⩽11+kλDcσk+1y^k+1-Dcσky^k+12+Dcσky^k+11+kλ-Dcσky^k1+k-1λ2.Applying Proposition 7 and substituting (31) into (32), we obtain(33)x^k+1-x^k2⩽nGcσk+1+σk1+kλ+λnGcσk+n1+kλ/1+k-1λy∞1+kλ+λnGcσk+y2+nλ1+λk-1y∞1+kλ1+k-1λ.From Proposition 8 both σk+1 and σk are all bounded. And since y2 and y∞ are finite, we can get x^k+1-x^k2→0 as k→∞, by taking the limit on both sides of (33).


## 3.4. Parameters Configuration

In Algorithm2, we select the state-of-the-art algorithms: BM3D and BM3D-SAPCA. The source code of these algorithms can be obtained from the original authors and we use the default parameters. When the denoiser is given, the parameters in Algorithm 2 are λ and c. In the remainder of this section, we mainly discuss the influence of λ and c on the final results.Consideringλ∈[0.3,0.9] and c∈[0.2,0.8] and discreting [0.3,0.9]×[0.2,0.8] by using a step-size 0.1×0.1, we get the parameters set {(λi,cj)}. Then, we introduce these parameters into the proposed Algorithm 2 and apply the proposed algorithm to noisy image. Figure 1(a) shows the relationship between (λ,c) and PSNR defined as 20log10(255/MSE), where MSE is the mean squared error between the original image and its denoised version. The relationship between (λ,c) and optimizing truncation iteration number is shown in Figure 1(b). It can be seen from Figure 1, when the parameters λ and c are small, we can get the smallest PSNR and the largest iteration number. In this case, the efficiency of the denoising is the lowest. When both λ and c achieve the highest value, the number of the iteration is the minimum, but the PSNR is not the maximum value. It is obvious that the PSNR has some volatility with λ. The iteration number of the λ-c area corresponding to the optimal PSNR value maintains consistency. In order to get the best denoising performance and reduce the computation complexity, suitable parameters λ and c must be determined. According to the results that are present in Figure 1, the PSNR is highest and the number of iterations is acceptable when λ=0.635 and c=0.472. Therefore, we select (0.635, 0.472) in the following experiment.Figure 1
Demonstration of the effect ofλ and c on the proposed algorithm outcome for the BM3D (σ=20).
(a)
Relationship amongλ, c, and PSNR (b)
Relationship amongλ, c, and iterations


## 4. Results

### 4.1. Noise Level Estimation Results

Usually, the value of Bias, Std, and RMSE are used to evaluate the noise estimation performance. We use well-known criteria: accuracy, reliability, and overall performance which have been considered in most literatures. In detail,(34)Biasσ^=Eσ-Eσ^,(35)Stdσ^=Eσ-Eσ^2,and(36)RMSEσ^=Bias2σ^+Std2σ^.We test our method on Tampere image datasets (TID2008) [28] which contains 25 images of size 512×384. All images in this datasets are disturbed by zero-mean Gaussian noise with different standard deviations σ = 5, 10, 15, 20, 25, 30, 40, 50, 60, 70, 80, 90, and 100. According to Figure 3(a), the estimated results are almost as same as the true noise level. As shown in Figure 3(b), the value of Bias, Std, and RMSE are all close to zero, which illustrates the accuracy of the Algorithm 1.


### 4.2. Denoising Results

#### 4.2.1. Objective Measurement Results

In this section, we provide detail results of the proposed methods. Based on the completely blind image quality assessment method NIQE, we always consider the following stopping rule:Dniqe(x^i)<Dniqe(x^i-1),i=1,2,…,k-1, and Dniqe(x^k)>Dniqe(x^k-1). The final outcome of Algorithm 2 is x^k. We evaluate the competing methods on standard image processing test dataset (SIPTD) including Forman, Lena, House, Fingerprint, and Peppers, whose scenes are shown in Figure 2. These images are extensively tested in image processing. Thus it is convenient to compare our method with other methods under the same condition. All the test images are corrupted by an additive zero-mean Gaussian noise with a variance σ2. The denoising performance is evaluated by using the PSNR, structural similarity (SSIM) [29], and dissimilar index (DSI) [30]. All results obtained by competing denoising method are shown in Tables 1 and 2. The results that appear in the σ0 column are obtained by applying the BM3D or BM3D-SAPCA on y using the accurate noise standard deviation. The results that appear in the SOS and “OURS” columns are obtained by applying SOS that was reported [7] and the proposed Algorithm 2, respectively. The best results for each image and noise level are highlighted. In contrast, Algorithm 2 achieves higher PSNR and SSIM than those of the other schemes. Compared with BM3D, PSNR obtained by Algorithm 2 achieves about 0.45dB improvement on the image Peppers with the noise level 50. The results presented in Table 3 indicate that the proposed approach achieves the highest average improvement of the PSNR and SSIM among all compared methods. Hence, for SIPTD dataset, the proposed algorithm is effective.Table 1
Comparison among the denoising results (PSNR, SSIM, and DSI) of BM3D and the proposed algorithm.
BM3D(λ=0.635, c=0.472)  σ Forman Lena House σ0 SOS OURS σ 0 SOS OURS σ 0 SOS OURS  10 37.24 37.24 37.24 35.87 35.87 35.87 36.65 36.65 36.65 0.9375 0.9368 0.9368 0.9149 0.9141 0.9154 0.9187 0.9163 0.9220 2.1859 2.3486 1.7016 3.1644 3.3521 2.8190 2.7045 2.8983 2.0538  20 34.43 34.43 34.50 32.98 32.99 33.01 33.67 33.69 33.67 0.9061 0.9085 0.9067 0.8764 0.8751 0.8771 0.8684 0.8657 0.8663 3.3650 3.7503 3.6073 4.7598 5.2280 5.0152 4.4826 4.9408 5.0619  25 33.44 33.46 33.60 32.02 32.04 32.07 32.79 32.80 32.85 0.8922 0.8978 0.8983 0.8599 0.8591 0.8619 0.8566 0.8555 0.8558 3.6520 4.2599 4.3283 5.4374 6.1221 6.2237 4.9677 5.5707 5.9420  50 30.04 30.13 30.18 28.78 28.81 29.04 29.38 29.42 29.79 0.8310 0.8453 0.8454 0.7860 0.7926 0.8014 0.7998 0.8057 0.8159 4.7899 5.3656 6.6166 10.191 10.757 11.867 7.7156 8.3724 9.3492  σ Fingerprint Pepper Average σ 0 SOS OURS σ 0 SOS OURS σ 0 SOS OURS  10 32.46 32.46 32.51 34.58 34.64 34.59 35.36 35.37 35.37 0.9690 0.9690 0.9682 0.9274 0.9267 0.9264 0.9336 0.9326 0.9338 0.7034 0.7205 0.4263 2.5305 2.6643 2.0317 2.2577 2.3968 1.8065  20 28.81 28.81 28.82 31.23 31.27 31.27 32.22 32.24 32.25 0.9305 0.9281 0.9308 0.8849 0.8855 0.8859 0.8933 0.8926 0.8934 6.2994 6.4851 5.3902 5.7494 6.1046 4.8925 4.9312 5.3018 4.7934  25 27.70 27.70 27.69 30.15 30.16 30.19 31.22 31.23 31.28 0.9121 0.9072 0.9122 0.8667 0.8678 0.8681 0.8775 0.8775 0.8793 10.310 10.657 10.260 7.6232 8.1311 7.0077 6.3980 6.9482 6.7523  50 24.34 24.36 24.53 26.32 26.38 26.83 27.77 27.82 28.07 0.8253 0.8143 0.8367 0.7700 0.7797 0.7952 0.8024 0.8075 0.8189 35.098 34.955 27.840 18.296 18.521 18.802 15.218 15.594 14.895Table 2
Comparison among the denoising results (PSNR, SSIM, and DSI) of BM3D-SAPCA and the proposed algorithm.
BM3D-SAPCA(λ=0.635; c=0.472)  σ Forman Lena House σ 0 SOS OURS σ 0 SOS OURS σ 0 SOS OURS  10 37.52 37.54 37.54 36.02 36.05 36.02 37.01 36.95 37.01 0.9396 0.9397 0.9398 0.9168 0.9168 0.9166 0.9274 0.9268 0.9279 1.8436 2.0052 1.5171 3.0206 3.2788 3.2277 2.2578 2.4728 2.2514  20 34.62 34.62 34.68 33.19 33.23 33.19 33.90 33.91 33.92 0.9054 0.9060 0.9055 0.8796 0.8804 0.8808 0.8727 0.8719 0.8735 3.4477 3.8614 3.8084 4.9624 5.5129 5.5065 4.6415 5.0781 5.0674  25 33.69 33.77 33.81 32.22 32.22 32.23 32.96 32.92 33.02 0.8917 0.8935 0.8935 0.8644 0.8656 0.8659 0.8588 0.8581 0.8585 3.8153 4.3223 4.5121 5.9266 6.5503 6.8346 5.4209 6.0023 6.1307  50 30.32 30.50 30.71 29.05 29.09 29.26 29.53 29.63 29.88 0.8402 0.8430 0.8497 0.8022 0.8053 0.8080 0.8045 0.8095 0.8162 5.2345 5.7885 7.3369 9.6687 10.7764 13.6531 7.5772 7.9921 9.72155  σ Fingerprint Peppers Average σ 0 SOS OURS σ 0 SOS OURS σ 0 SOS OURS  10 32.64 32.66 32.69 34.94 34.95 34.96 35.63 35.63 35.64 0.9703 0.9704 0.9706 0.9284 0.9284 0.9280 0.9365 0.9364 0.9366 0.7503 0.7795 0.8502 2.2488 2.4581 3.2900 2.0242 2.1989 2.2273  20 28.94 28.96 29.00 31.55 31.56 31.59 32.44 32.46 32.48 0.9328 0.9329 0.9329 0.8868 0.8875 0.8884 0.8955 0.8957 0.8962 6.8078 7.1170 7.4005 5.3362 5.9814 5.9419 5.0391 5.5102 5.5449  25 27.81 27.83 27.86 30.43 30.44 30.49 31.42 31.44 31.48 0.9145 0.9146 0.9135 0.8692 0.8699 0.8706 0.8797 0.8803 0.8804 11.629 12.163 13.8800 7.2262 7.8891 8.0591 6.8035 7.3855 7.8833  50 24.53 24.55 24.48 27.00 27.05 27.06 28.09 28.16 28.28 0.8354 0.8360 0.8286 0.7945 0.7982 0.8005 0.8154 0.8184 0.8206 36.143 39.1979 52.0200 16.5315 17.2333 20.4961 15.031 16.198 20.646Table 3
Comparison of average improvement between SOS and Algorithm2 on SIPTD dataset.
BM3D BM3D-SAPCA  PSNR PSNR  σ 10 20 25 50 10 20 25 50  SOS 0.01 0.01 0.01 0.05 0.01 0.02 0.02 0.07  Ours 0.01 0.03 0.06 0.30 0.27 0.22 0.01 0.04  SSIM SSIM  SOS 0.0005 0.0002 0.0009 0.0051 0.0001 0.0003 0.0006 0.0031  Ours 0.0007 0.0003 0.0018 0.0165 0.0202 0.0193 0.0001 0.0008  DSI DSI  SOS 0.1390 0.3705 0.5502 0.3759 0.1747 0.4710 0.5819 1.1667  Ours -0.4513 -0.1379 0.3543 -0.3234 0.2031 0.5058 1.0797 5.6146Figure 2
Visualization of the test images.
(a)
Foreman (b)
Lena (c)
House (d)
Fingerprint (e)
PeppersFigure 3
Noise level estimation on TID2008. (a) Noise level estimation results. (b) The evaluation performance of the noise level estimation, accuracy, reliability, and overall performance.
(a) (b)Moreover, we perform Algorithm2 on all the grass images from the MeasTex texture dataset [31] which contain rich texture features. The comparison results are presented in Table 4. Compared with SOS methods, the average improvements of PSNR and SSIM demonstrate significant comparative advantages for almost all noise levels. It is clear that the proposed algorithm offers better restoration of texture images.Table 4
Comparison of average improvement between SOS and Algorithm2 on MeasTex-grass dataset.
BM3D BM3D-SAPCA  PSNR PSNR  σ 10 20 25 50 10 20 25 50  SOS 0.01 0.01 0.01 0.03 0.01 0.02 0.02 0.07  Ours 0.01 0.02 0.05 0.24 0.01 0.04 0.06 0.19  SSIM SSIM  SOS 0.0005 0.0002 0.0009 0.0051 0.0001 0.0003 0.0006 0.0031  Ours 0.0007 0.0003 0.0018 0.0165 0.0001 0.0008 0.0007 0.0052  DSI DSI  SOS 0.1390 0.3705 0.5502 0.3759 0.0393 0.3872 0.6542 1.2431  Ours -0.4513 -0.1379 0.3543 -0.3234 -0.1778 -0.4219 -0.1181 4.9756Finally, we use DSI (the Matlab realization of the DSI metric are available at http://ponomarenko.info/flt.htm.) which obtains the largest spearman rank order correlation coefficient values to mean opinion scores to evaluate the visual quality of the denoised images. According to the results present in Tables3 and 4, the proposed Algorithm 2 obtained lower average DSI value for each denoiser on the considering datasets in most case.


#### 4.2.2. Detail Contrast

In this section, we compare the performance of considered denoising algorithms in preserving image details. Figure5 shows the fragments of noisy (σ=25) Baboon image and the corresponding images denoised by different denoisers. The first row is the original image and its noisy version. The second row shows the denoised result corresponding to different parameters: exact σ, SOS, and ours by using the denoiser BM3D. The third row is the denoised result obtained by the denoiser BM3D-SAPCA. The enlarge fragments in each subfigure are helpful for demonstrating the good quality of the denoised images of faithful detail preservation. Figures 5(f) and 5(i) show more details close to the original image.Figure6 shows the fragments of noisy (σ=20) fingerprint image and the corresponding images denoised by different denoisers. According to Figure 6, our approach outperforms BM3D, SOS, and BM3D-SAPCA, numerically and visually.Fragments of noisy (σ=50) and denoised Foreman are shown in Figure 7. For this relatively high level of noise, the proposed Algorithm 2 attains a good preservation of sharp details, such as the lines on the wall in Foreman image. Meanwhile the smooth regions, such as the hat in Foreman image, are also of a good preservation. All the denoised images obtained by using the proposed Algorithm 2 have the fewest disturbing artifacts.


#### 4.2.3. Computational Time

We discuss the efficiency of the proposed algorithm in this section. SOS and Algorithm2 in this paper need several iterations during the denoising application because these two algorithms may cost more time than the initial denoiser. Table 5 shows the results of computational time between SOS and Algorithm 2. Compared with SOS algorithm, Algorithm 2 has a certain advantage in processing time for higher noise level while BM3D is selected as the denoiser. According to the results presented in Table 5, the processing time of Algorithm 2 in this paper is lower than SOS for each noise level when the denoiser BM3D-SAPCA is used. Furthermore, the average processing time spent per image of Algorithm 2 is less than that of SOS.Table 5
Comparison of computational time (second) between SOS and Algorithm2.
SIPID dataset MeasTex-grass dataset  BM3D BM3D  σ 10 20 25 50 Ave. 10 20 25 50 Ave.  SOS 8.47 8.95 9.05 14.66 11.43 15.53 17.22 17.87 29.56 22.13  OURS 9.92 10.73 11.11 10.89 10.10 15.92 19.62 20.82 22.68 19.16  BM3D-SAPCA BM3D-SAPCA  SOS 181.3 175.3 169.9 253.5 202.4 234.8 310.3 285.4 456.3 363.2  OURS 118.0 112.7 110.8 133.0 114.3 268.5 242.9 228.8 290.6 248.6


## 4.1. Noise Level Estimation Results

Usually, the value of Bias, Std, and RMSE are used to evaluate the noise estimation performance. We use well-known criteria: accuracy, reliability, and overall performance which have been considered in most literatures. In detail,(34)Biasσ^=Eσ-Eσ^,(35)Stdσ^=Eσ-Eσ^2,and(36)RMSEσ^=Bias2σ^+Std2σ^.We test our method on Tampere image datasets (TID2008) [28] which contains 25 images of size 512×384. All images in this datasets are disturbed by zero-mean Gaussian noise with different standard deviations σ = 5, 10, 15, 20, 25, 30, 40, 50, 60, 70, 80, 90, and 100. According to Figure 3(a), the estimated results are almost as same as the true noise level. As shown in Figure 3(b), the value of Bias, Std, and RMSE are all close to zero, which illustrates the accuracy of the Algorithm 1.


## 4.2. Denoising Results

### 4.2.1. Objective Measurement Results

In this section, we provide detail results of the proposed methods. Based on the completely blind image quality assessment method NIQE, we always consider the following stopping rule:Dniqe(x^i)<Dniqe(x^i-1),i=1,2,…,k-1, and Dniqe(x^k)>Dniqe(x^k-1). The final outcome of Algorithm 2 is x^k. We evaluate the competing methods on standard image processing test dataset (SIPTD) including Forman, Lena, House, Fingerprint, and Peppers, whose scenes are shown in Figure 2. These images are extensively tested in image processing. Thus it is convenient to compare our method with other methods under the same condition. All the test images are corrupted by an additive zero-mean Gaussian noise with a variance σ2. The denoising performance is evaluated by using the PSNR, structural similarity (SSIM) [29], and dissimilar index (DSI) [30]. All results obtained by competing denoising method are shown in Tables 1 and 2. The results that appear in the σ0 column are obtained by applying the BM3D or BM3D-SAPCA on y using the accurate noise standard deviation. The results that appear in the SOS and “OURS” columns are obtained by applying SOS that was reported [7] and the proposed Algorithm 2, respectively. The best results for each image and noise level are highlighted. In contrast, Algorithm 2 achieves higher PSNR and SSIM than those of the other schemes. Compared with BM3D, PSNR obtained by Algorithm 2 achieves about 0.45dB improvement on the image Peppers with the noise level 50. The results presented in Table 3 indicate that the proposed approach achieves the highest average improvement of the PSNR and SSIM among all compared methods. Hence, for SIPTD dataset, the proposed algorithm is effective.Table 1
Comparison among the denoising results (PSNR, SSIM, and DSI) of BM3D and the proposed algorithm.
BM3D(λ=0.635, c=0.472)  σ Forman Lena House σ0 SOS OURS σ 0 SOS OURS σ 0 SOS OURS  10 37.24 37.24 37.24 35.87 35.87 35.87 36.65 36.65 36.65 0.9375 0.9368 0.9368 0.9149 0.9141 0.9154 0.9187 0.9163 0.9220 2.1859 2.3486 1.7016 3.1644 3.3521 2.8190 2.7045 2.8983 2.0538  20 34.43 34.43 34.50 32.98 32.99 33.01 33.67 33.69 33.67 0.9061 0.9085 0.9067 0.8764 0.8751 0.8771 0.8684 0.8657 0.8663 3.3650 3.7503 3.6073 4.7598 5.2280 5.0152 4.4826 4.9408 5.0619  25 33.44 33.46 33.60 32.02 32.04 32.07 32.79 32.80 32.85 0.8922 0.8978 0.8983 0.8599 0.8591 0.8619 0.8566 0.8555 0.8558 3.6520 4.2599 4.3283 5.4374 6.1221 6.2237 4.9677 5.5707 5.9420  50 30.04 30.13 30.18 28.78 28.81 29.04 29.38 29.42 29.79 0.8310 0.8453 0.8454 0.7860 0.7926 0.8014 0.7998 0.8057 0.8159 4.7899 5.3656 6.6166 10.191 10.757 11.867 7.7156 8.3724 9.3492  σ Fingerprint Pepper Average σ 0 SOS OURS σ 0 SOS OURS σ 0 SOS OURS  10 32.46 32.46 32.51 34.58 34.64 34.59 35.36 35.37 35.37 0.9690 0.9690 0.9682 0.9274 0.9267 0.9264 0.9336 0.9326 0.9338 0.7034 0.7205 0.4263 2.5305 2.6643 2.0317 2.2577 2.3968 1.8065  20 28.81 28.81 28.82 31.23 31.27 31.27 32.22 32.24 32.25 0.9305 0.9281 0.9308 0.8849 0.8855 0.8859 0.8933 0.8926 0.8934 6.2994 6.4851 5.3902 5.7494 6.1046 4.8925 4.9312 5.3018 4.7934  25 27.70 27.70 27.69 30.15 30.16 30.19 31.22 31.23 31.28 0.9121 0.9072 0.9122 0.8667 0.8678 0.8681 0.8775 0.8775 0.8793 10.310 10.657 10.260 7.6232 8.1311 7.0077 6.3980 6.9482 6.7523  50 24.34 24.36 24.53 26.32 26.38 26.83 27.77 27.82 28.07 0.8253 0.8143 0.8367 0.7700 0.7797 0.7952 0.8024 0.8075 0.8189 35.098 34.955 27.840 18.296 18.521 18.802 15.218 15.594 14.895Table 2
Comparison among the denoising results (PSNR, SSIM, and DSI) of BM3D-SAPCA and the proposed algorithm.
BM3D-SAPCA(λ=0.635; c=0.472)  σ Forman Lena House σ 0 SOS OURS σ 0 SOS OURS σ 0 SOS OURS  10 37.52 37.54 37.54 36.02 36.05 36.02 37.01 36.95 37.01 0.9396 0.9397 0.9398 0.9168 0.9168 0.9166 0.9274 0.9268 0.9279 1.8436 2.0052 1.5171 3.0206 3.2788 3.2277 2.2578 2.4728 2.2514  20 34.62 34.62 34.68 33.19 33.23 33.19 33.90 33.91 33.92 0.9054 0.9060 0.9055 0.8796 0.8804 0.8808 0.8727 0.8719 0.8735 3.4477 3.8614 3.8084 4.9624 5.5129 5.5065 4.6415 5.0781 5.0674  25 33.69 33.77 33.81 32.22 32.22 32.23 32.96 32.92 33.02 0.8917 0.8935 0.8935 0.8644 0.8656 0.8659 0.8588 0.8581 0.8585 3.8153 4.3223 4.5121 5.9266 6.5503 6.8346 5.4209 6.0023 6.1307  50 30.32 30.50 30.71 29.05 29.09 29.26 29.53 29.63 29.88 0.8402 0.8430 0.8497 0.8022 0.8053 0.8080 0.8045 0.8095 0.8162 5.2345 5.7885 7.3369 9.6687 10.7764 13.6531 7.5772 7.9921 9.72155  σ Fingerprint Peppers Average σ 0 SOS OURS σ 0 SOS OURS σ 0 SOS OURS  10 32.64 32.66 32.69 34.94 34.95 34.96 35.63 35.63 35.64 0.9703 0.9704 0.9706 0.9284 0.9284 0.9280 0.9365 0.9364 0.9366 0.7503 0.7795 0.8502 2.2488 2.4581 3.2900 2.0242 2.1989 2.2273  20 28.94 28.96 29.00 31.55 31.56 31.59 32.44 32.46 32.48 0.9328 0.9329 0.9329 0.8868 0.8875 0.8884 0.8955 0.8957 0.8962 6.8078 7.1170 7.4005 5.3362 5.9814 5.9419 5.0391 5.5102 5.5449  25 27.81 27.83 27.86 30.43 30.44 30.49 31.42 31.44 31.48 0.9145 0.9146 0.9135 0.8692 0.8699 0.8706 0.8797 0.8803 0.8804 11.629 12.163 13.8800 7.2262 7.8891 8.0591 6.8035 7.3855 7.8833  50 24.53 24.55 24.48 27.00 27.05 27.06 28.09 28.16 28.28 0.8354 0.8360 0.8286 0.7945 0.7982 0.8005 0.8154 0.8184 0.8206 36.143 39.1979 52.0200 16.5315 17.2333 20.4961 15.031 16.198 20.646Table 3
Comparison of average improvement between SOS and Algorithm2 on SIPTD dataset.
BM3D BM3D-SAPCA  PSNR PSNR  σ 10 20 25 50 10 20 25 50  SOS 0.01 0.01 0.01 0.05 0.01 0.02 0.02 0.07  Ours 0.01 0.03 0.06 0.30 0.27 0.22 0.01 0.04  SSIM SSIM  SOS 0.0005 0.0002 0.0009 0.0051 0.0001 0.0003 0.0006 0.0031  Ours 0.0007 0.0003 0.0018 0.0165 0.0202 0.0193 0.0001 0.0008  DSI DSI  SOS 0.1390 0.3705 0.5502 0.3759 0.1747 0.4710 0.5819 1.1667  Ours -0.4513 -0.1379 0.3543 -0.3234 0.2031 0.5058 1.0797 5.6146Figure 2
Visualization of the test images.
(a)
Foreman (b)
Lena (c)
House (d)
Fingerprint (e)
PeppersFigure 3
Noise level estimation on TID2008. (a) Noise level estimation results. (b) The evaluation performance of the noise level estimation, accuracy, reliability, and overall performance.
(a) (b)Moreover, we perform Algorithm2 on all the grass images from the MeasTex texture dataset [31] which contain rich texture features. The comparison results are presented in Table 4. Compared with SOS methods, the average improvements of PSNR and SSIM demonstrate significant comparative advantages for almost all noise levels. It is clear that the proposed algorithm offers better restoration of texture images.Table 4
Comparison of average improvement between SOS and Algorithm2 on MeasTex-grass dataset.
BM3D BM3D-SAPCA  PSNR PSNR  σ 10 20 25 50 10 20 25 50  SOS 0.01 0.01 0.01 0.03 0.01 0.02 0.02 0.07  Ours 0.01 0.02 0.05 0.24 0.01 0.04 0.06 0.19  SSIM SSIM  SOS 0.0005 0.0002 0.0009 0.0051 0.0001 0.0003 0.0006 0.0031  Ours 0.0007 0.0003 0.0018 0.0165 0.0001 0.0008 0.0007 0.0052  DSI DSI  SOS 0.1390 0.3705 0.5502 0.3759 0.0393 0.3872 0.6542 1.2431  Ours -0.4513 -0.1379 0.3543 -0.3234 -0.1778 -0.4219 -0.1181 4.9756Finally, we use DSI (the Matlab realization of the DSI metric are available at http://ponomarenko.info/flt.htm.) which obtains the largest spearman rank order correlation coefficient values to mean opinion scores to evaluate the visual quality of the denoised images. According to the results present in Tables3 and 4, the proposed Algorithm 2 obtained lower average DSI value for each denoiser on the considering datasets in most case.


### 4.2.2. Detail Contrast

In this section, we compare the performance of considered denoising algorithms in preserving image details. Figure5 shows the fragments of noisy (σ=25) Baboon image and the corresponding images denoised by different denoisers. The first row is the original image and its noisy version. The second row shows the denoised result corresponding to different parameters: exact σ, SOS, and ours by using the denoiser BM3D. The third row is the denoised result obtained by the denoiser BM3D-SAPCA. The enlarge fragments in each subfigure are helpful for demonstrating the good quality of the denoised images of faithful detail preservation. Figures 5(f) and 5(i) show more details close to the original image.Figure6 shows the fragments of noisy (σ=20) fingerprint image and the corresponding images denoised by different denoisers. According to Figure 6, our approach outperforms BM3D, SOS, and BM3D-SAPCA, numerically and visually.Fragments of noisy (σ=50) and denoised Foreman are shown in Figure 7. For this relatively high level of noise, the proposed Algorithm 2 attains a good preservation of sharp details, such as the lines on the wall in Foreman image. Meanwhile the smooth regions, such as the hat in Foreman image, are also of a good preservation. All the denoised images obtained by using the proposed Algorithm 2 have the fewest disturbing artifacts.


### 4.2.3. Computational Time

We discuss the efficiency of the proposed algorithm in this section. SOS and Algorithm2 in this paper need several iterations during the denoising application because these two algorithms may cost more time than the initial denoiser. Table 5 shows the results of computational time between SOS and Algorithm 2. Compared with SOS algorithm, Algorithm 2 has a certain advantage in processing time for higher noise level while BM3D is selected as the denoiser. According to the results presented in Table 5, the processing time of Algorithm 2 in this paper is lower than SOS for each noise level when the denoiser BM3D-SAPCA is used. Furthermore, the average processing time spent per image of Algorithm 2 is less than that of SOS.Table 5
Comparison of computational time (second) between SOS and Algorithm2.
SIPID dataset MeasTex-grass dataset  BM3D BM3D  σ 10 20 25 50 Ave. 10 20 25 50 Ave.  SOS 8.47 8.95 9.05 14.66 11.43 15.53 17.22 17.87 29.56 22.13  OURS 9.92 10.73 11.11 10.89 10.10 15.92 19.62 20.82 22.68 19.16  BM3D-SAPCA BM3D-SAPCA  SOS 181.3 175.3 169.9 253.5 202.4 234.8 310.3 285.4 456.3 363.2  OURS 118.0 112.7 110.8 133.0 114.3 268.5 242.9 228.8 290.6 248.6


## 4.2.1. Objective Measurement Results

In this section, we provide detail results of the proposed methods. Based on the completely blind image quality assessment method NIQE, we always consider the following stopping rule:Dniqe(x^i)<Dniqe(x^i-1),i=1,2,…,k-1, and Dniqe(x^k)>Dniqe(x^k-1). The final outcome of Algorithm 2 is x^k. We evaluate the competing methods on standard image processing test dataset (SIPTD) including Forman, Lena, House, Fingerprint, and Peppers, whose scenes are shown in Figure 2. These images are extensively tested in image processing. Thus it is convenient to compare our method with other methods under the same condition. All the test images are corrupted by an additive zero-mean Gaussian noise with a variance σ2. The denoising performance is evaluated by using the PSNR, structural similarity (SSIM) [29], and dissimilar index (DSI) [30]. All results obtained by competing denoising method are shown in Tables 1 and 2. The results that appear in the σ0 column are obtained by applying the BM3D or BM3D-SAPCA on y using the accurate noise standard deviation. The results that appear in the SOS and “OURS” columns are obtained by applying SOS that was reported [7] and the proposed Algorithm 2, respectively. The best results for each image and noise level are highlighted. In contrast, Algorithm 2 achieves higher PSNR and SSIM than those of the other schemes. Compared with BM3D, PSNR obtained by Algorithm 2 achieves about 0.45dB improvement on the image Peppers with the noise level 50. The results presented in Table 3 indicate that the proposed approach achieves the highest average improvement of the PSNR and SSIM among all compared methods. Hence, for SIPTD dataset, the proposed algorithm is effective.Table 1
Comparison among the denoising results (PSNR, SSIM, and DSI) of BM3D and the proposed algorithm.
BM3D(λ=0.635, c=0.472)  σ Forman Lena House σ0 SOS OURS σ 0 SOS OURS σ 0 SOS OURS  10 37.24 37.24 37.24 35.87 35.87 35.87 36.65 36.65 36.65 0.9375 0.9368 0.9368 0.9149 0.9141 0.9154 0.9187 0.9163 0.9220 2.1859 2.3486 1.7016 3.1644 3.3521 2.8190 2.7045 2.8983 2.0538  20 34.43 34.43 34.50 32.98 32.99 33.01 33.67 33.69 33.67 0.9061 0.9085 0.9067 0.8764 0.8751 0.8771 0.8684 0.8657 0.8663 3.3650 3.7503 3.6073 4.7598 5.2280 5.0152 4.4826 4.9408 5.0619  25 33.44 33.46 33.60 32.02 32.04 32.07 32.79 32.80 32.85 0.8922 0.8978 0.8983 0.8599 0.8591 0.8619 0.8566 0.8555 0.8558 3.6520 4.2599 4.3283 5.4374 6.1221 6.2237 4.9677 5.5707 5.9420  50 30.04 30.13 30.18 28.78 28.81 29.04 29.38 29.42 29.79 0.8310 0.8453 0.8454 0.7860 0.7926 0.8014 0.7998 0.8057 0.8159 4.7899 5.3656 6.6166 10.191 10.757 11.867 7.7156 8.3724 9.3492  σ Fingerprint Pepper Average σ 0 SOS OURS σ 0 SOS OURS σ 0 SOS OURS  10 32.46 32.46 32.51 34.58 34.64 34.59 35.36 35.37 35.37 0.9690 0.9690 0.9682 0.9274 0.9267 0.9264 0.9336 0.9326 0.9338 0.7034 0.7205 0.4263 2.5305 2.6643 2.0317 2.2577 2.3968 1.8065  20 28.81 28.81 28.82 31.23 31.27 31.27 32.22 32.24 32.25 0.9305 0.9281 0.9308 0.8849 0.8855 0.8859 0.8933 0.8926 0.8934 6.2994 6.4851 5.3902 5.7494 6.1046 4.8925 4.9312 5.3018 4.7934  25 27.70 27.70 27.69 30.15 30.16 30.19 31.22 31.23 31.28 0.9121 0.9072 0.9122 0.8667 0.8678 0.8681 0.8775 0.8775 0.8793 10.310 10.657 10.260 7.6232 8.1311 7.0077 6.3980 6.9482 6.7523  50 24.34 24.36 24.53 26.32 26.38 26.83 27.77 27.82 28.07 0.8253 0.8143 0.8367 0.7700 0.7797 0.7952 0.8024 0.8075 0.8189 35.098 34.955 27.840 18.296 18.521 18.802 15.218 15.594 14.895Table 2
Comparison among the denoising results (PSNR, SSIM, and DSI) of BM3D-SAPCA and the proposed algorithm.
BM3D-SAPCA(λ=0.635; c=0.472)  σ Forman Lena House σ 0 SOS OURS σ 0 SOS OURS σ 0 SOS OURS  10 37.52 37.54 37.54 36.02 36.05 36.02 37.01 36.95 37.01 0.9396 0.9397 0.9398 0.9168 0.9168 0.9166 0.9274 0.9268 0.9279 1.8436 2.0052 1.5171 3.0206 3.2788 3.2277 2.2578 2.4728 2.2514  20 34.62 34.62 34.68 33.19 33.23 33.19 33.90 33.91 33.92 0.9054 0.9060 0.9055 0.8796 0.8804 0.8808 0.8727 0.8719 0.8735 3.4477 3.8614 3.8084 4.9624 5.5129 5.5065 4.6415 5.0781 5.0674  25 33.69 33.77 33.81 32.22 32.22 32.23 32.96 32.92 33.02 0.8917 0.8935 0.8935 0.8644 0.8656 0.8659 0.8588 0.8581 0.8585 3.8153 4.3223 4.5121 5.9266 6.5503 6.8346 5.4209 6.0023 6.1307  50 30.32 30.50 30.71 29.05 29.09 29.26 29.53 29.63 29.88 0.8402 0.8430 0.8497 0.8022 0.8053 0.8080 0.8045 0.8095 0.8162 5.2345 5.7885 7.3369 9.6687 10.7764 13.6531 7.5772 7.9921 9.72155  σ Fingerprint Peppers Average σ 0 SOS OURS σ 0 SOS OURS σ 0 SOS OURS  10 32.64 32.66 32.69 34.94 34.95 34.96 35.63 35.63 35.64 0.9703 0.9704 0.9706 0.9284 0.9284 0.9280 0.9365 0.9364 0.9366 0.7503 0.7795 0.8502 2.2488 2.4581 3.2900 2.0242 2.1989 2.2273  20 28.94 28.96 29.00 31.55 31.56 31.59 32.44 32.46 32.48 0.9328 0.9329 0.9329 0.8868 0.8875 0.8884 0.8955 0.8957 0.8962 6.8078 7.1170 7.4005 5.3362 5.9814 5.9419 5.0391 5.5102 5.5449  25 27.81 27.83 27.86 30.43 30.44 30.49 31.42 31.44 31.48 0.9145 0.9146 0.9135 0.8692 0.8699 0.8706 0.8797 0.8803 0.8804 11.629 12.163 13.8800 7.2262 7.8891 8.0591 6.8035 7.3855 7.8833  50 24.53 24.55 24.48 27.00 27.05 27.06 28.09 28.16 28.28 0.8354 0.8360 0.8286 0.7945 0.7982 0.8005 0.8154 0.8184 0.8206 36.143 39.1979 52.0200 16.5315 17.2333 20.4961 15.031 16.198 20.646Table 3
Comparison of average improvement between SOS and Algorithm2 on SIPTD dataset.
BM3D BM3D-SAPCA  PSNR PSNR  σ 10 20 25 50 10 20 25 50  SOS 0.01 0.01 0.01 0.05 0.01 0.02 0.02 0.07  Ours 0.01 0.03 0.06 0.30 0.27 0.22 0.01 0.04  SSIM SSIM  SOS 0.0005 0.0002 0.0009 0.0051 0.0001 0.0003 0.0006 0.0031  Ours 0.0007 0.0003 0.0018 0.0165 0.0202 0.0193 0.0001 0.0008  DSI DSI  SOS 0.1390 0.3705 0.5502 0.3759 0.1747 0.4710 0.5819 1.1667  Ours -0.4513 -0.1379 0.3543 -0.3234 0.2031 0.5058 1.0797 5.6146Figure 2
Visualization of the test images.
(a)
Foreman (b)
Lena (c)
House (d)
Fingerprint (e)
PeppersFigure 3
Noise level estimation on TID2008. (a) Noise level estimation results. (b) The evaluation performance of the noise level estimation, accuracy, reliability, and overall performance.
(a) (b)Moreover, we perform Algorithm2 on all the grass images from the MeasTex texture dataset [31] which contain rich texture features. The comparison results are presented in Table 4. Compared with SOS methods, the average improvements of PSNR and SSIM demonstrate significant comparative advantages for almost all noise levels. It is clear that the proposed algorithm offers better restoration of texture images.Table 4
Comparison of average improvement between SOS and Algorithm2 on MeasTex-grass dataset.
BM3D BM3D-SAPCA  PSNR PSNR  σ 10 20 25 50 10 20 25 50  SOS 0.01 0.01 0.01 0.03 0.01 0.02 0.02 0.07  Ours 0.01 0.02 0.05 0.24 0.01 0.04 0.06 0.19  SSIM SSIM  SOS 0.0005 0.0002 0.0009 0.0051 0.0001 0.0003 0.0006 0.0031  Ours 0.0007 0.0003 0.0018 0.0165 0.0001 0.0008 0.0007 0.0052  DSI DSI  SOS 0.1390 0.3705 0.5502 0.3759 0.0393 0.3872 0.6542 1.2431  Ours -0.4513 -0.1379 0.3543 -0.3234 -0.1778 -0.4219 -0.1181 4.9756Finally, we use DSI (the Matlab realization of the DSI metric are available at http://ponomarenko.info/flt.htm.) which obtains the largest spearman rank order correlation coefficient values to mean opinion scores to evaluate the visual quality of the denoised images. According to the results present in Tables3 and 4, the proposed Algorithm 2 obtained lower average DSI value for each denoiser on the considering datasets in most case.


## 4.2.2. Detail Contrast

In this section, we compare the performance of considered denoising algorithms in preserving image details. Figure5 shows the fragments of noisy (σ=25) Baboon image and the corresponding images denoised by different denoisers. The first row is the original image and its noisy version. The second row shows the denoised result corresponding to different parameters: exact σ, SOS, and ours by using the denoiser BM3D. The third row is the denoised result obtained by the denoiser BM3D-SAPCA. The enlarge fragments in each subfigure are helpful for demonstrating the good quality of the denoised images of faithful detail preservation. Figures 5(f) and 5(i) show more details close to the original image.Figure6 shows the fragments of noisy (σ=20) fingerprint image and the corresponding images denoised by different denoisers. According to Figure 6, our approach outperforms BM3D, SOS, and BM3D-SAPCA, numerically and visually.Fragments of noisy (σ=50) and denoised Foreman are shown in Figure 7. For this relatively high level of noise, the proposed Algorithm 2 attains a good preservation of sharp details, such as the lines on the wall in Foreman image. Meanwhile the smooth regions, such as the hat in Foreman image, are also of a good preservation. All the denoised images obtained by using the proposed Algorithm 2 have the fewest disturbing artifacts.


## 4.2.3. Computational Time

We discuss the efficiency of the proposed algorithm in this section. SOS and Algorithm2 in this paper need several iterations during the denoising application because these two algorithms may cost more time than the initial denoiser. Table 5 shows the results of computational time between SOS and Algorithm 2. Compared with SOS algorithm, Algorithm 2 has a certain advantage in processing time for higher noise level while BM3D is selected as the denoiser. According to the results presented in Table 5, the processing time of Algorithm 2 in this paper is lower than SOS for each noise level when the denoiser BM3D-SAPCA is used. Furthermore, the average processing time spent per image of Algorithm 2 is less than that of SOS.Table 5
Comparison of computational time (second) between SOS and Algorithm2.
SIPID dataset MeasTex-grass dataset  BM3D BM3D  σ 10 20 25 50 Ave. 10 20 25 50 Ave.  SOS 8.47 8.95 9.05 14.66 11.43 15.53 17.22 17.87 29.56 22.13  OURS 9.92 10.73 11.11 10.89 10.10 15.92 19.62 20.82 22.68 19.16  BM3D-SAPCA BM3D-SAPCA  SOS 181.3 175.3 169.9 253.5 202.4 234.8 310.3 285.4 456.3 363.2  OURS 118.0 112.7 110.8 133.0 114.3 268.5 242.9 228.8 290.6 248.6


## 5. Conclusion and Discussion

In this paper, we proposed a new adaptive boosting denoising algorithm by plugging an accurate noise level estimation. Experience shows that the algorithm can improve performance of denoising algorithm which depended on noise level and can also keep the image’s edges and detail information well. Though it is a convenient tool for improving various denoising algorithm, there are still serval directions that we are interested in future works. Firstly, in order to get optimal output image, we use NIQE algorithm to assess the quality of output imagexk. When Dniqe(xk) are no longer decreasing, the final result of Algorithm 2 is given by (xk+1). Figure 4(a) shows the result; the highest PSNR is obtained when Dniqe(xk) first increase. By using Theorem 11, a straightforward stopping criterion (|x^k+1-x^k| is sufficiently small) can be obtained. Although this straightforward stopping criterion avoid the output of Algorithm 2 back to y, the PSNR does not always increase when |x^k+1-x^k| decrease. Figure 4(b) shows the relationship between PSNR and |x^k+1-x^k| with k; the smallest |x^k+1-x^k| does not give the optimal PSNR. Compared with Figure 4(a) and 4(b), we can find that the stopping rule in Algorithm 2 is helpful for determining the optimal numbers of iterations. Secondly, there are two main parameters λ and c in the boosting method. How to select these parameters for optimal denoising results needs further research. Finally, we hope that other image restoration problems, such as image deblurring/inpainting, can use similar embedding method.Figure 4
The comparing of stopping criterion. (a) The smallest NIQE value gives the optimal PSNR. (b) The smallest|xk+1-xk| does not give the optimal PSNR.
(a) (b)Figure 5
Visual and PSNR comparisons between standard denoising and proposed method of a120×120 cropped region from noisy image Baboon (σ=25). (a, b, c) are the original image and its noisy version, and (d–i) are the results when using the denoisers BM3D and BM3D-SAPCA, respectively.
(a)
Original image (b)
Clean patch (c)
Noise patch (d)
BM3D,σ0, 25.43 (e)
BM3D, SOS, 25.40 (f)
BM3D, ours, 25.50 (g)
BM3D-SAPCA,σ0, 25.68 (h)
BM3D-SAPCA, SOS, 25.66 (i)
BM3D-SAPCA, ours, 25.73Figure 6
Visual and PSNR comparisons between standard denoising and proposed method of a120×120 cropped region from noisy image fingerprint (σ=20). (a, b, c) are the original image and its noisy version and (d–i) are the results when using the denoisers BM3D and BM3D-SAPCA, respectively.
(a)
Original image (b)
Clean patch (c)
Noise patch (d)
BM3D,σ0, 28.81 (e)
BM3D, SOS, 28.81 (f)
BM3D, ours, 28.82 (g)
BM3D-SAPCA,σ0, 28.94 (h)
BM3D-SAPCA, SOS, 28.96 (i)
BM3D-SAPCA, ours, 29.00Figure 7
Visual and PSNR comparisons between standard denoising and proposed method of a120×120 cropped region from noisy image Foreman (σ=50). (a, b, c) are the original image and its noisy version and (d–i) are the results when using the denoisers BM3D and BM3D-SAPCA, respectively.
(a)
Original image (b)
Clean patch (c)
Noise patch (d)
BM3D,σ0, 30.04 (e)
BM3D, SOS, 30.13 (f)
BM3D, ours, 30.18 (g)
BM3D-SAPCA,σ0, 30.32 (h)
BM3D-SAPCA, sos, 30.50 (i)
BM3D-SAPCA, ours, 30.71



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*Source: 8365932-2019-02-18.xml*