CSDK: A Chi-square distribution-Kernel method for image de-noising under the Internet of things big data environment

Abstract

Nowadays, Internet of things not only brings promising opportunities but also faces a lot of challenges. It attracts a lot of researchers’ attention and has important economic and social values. Internet of things plays a key role in the big data processing, especially in image field. Image de-noising still is a key problem in image pre-processing. Considering a given noisy image, the selection of thresholds should significantly affect the quality of the de-noising image. Although the state-of-the-art wavelet image de-noising methods perform better than other de-noising methods, they are not very effective for de-noising with different noises and with redundancy convergence time, sometimes. To mitigate the poor effect of traditional de-noising methods, this article proposes a new wavelet soft threshold based on the Chi-square distribution-Kernel method under the Internet of things big data environment. The new method alternates three minimization steps. First, the Chi-square distribution-Kernel model is constructed to find the customized threshold that corresponds to the de-noised image. Second, a freedom degree is considered, which is related to the customized wavelet coefficient of the Chi-square distribution-Kernel to be thresholded for image de-noising. Here, noisy image is first decomposed into many levels to obtain different frequency bands and the soft thresholding method based on Chi-square distribution-Kernel method is used to remove the noisy coefficients, by fixing the optimum threshold value using the proposed method. Third, the wavelet soft thresholding based on Chi-square distribution-Kernel method is adopted to handle the image de-noising, and a significant improvement is obtained by a specially developed Chi-square distribution-Kernel method. Finally, the experimental results illustrate that this computationally scalable algorithm achieves state-of-the-art de-noising performance in terms of peak signal-to-noise ratio, normalized mean square error, structural similarity, and subjective visual quality. It also shows a consistent accuracy, edge preservation, and detailed retention improvement compared to the classic de-noising algorithms.

Introduction

The Internet of things (IoT) generates a large amount of data, including geographic data, image data, medical data, and so on, which are stored and processed in cloud computing. In this article, we focus on the image data process. During the process of image formation, transmission, and processing, images are interfered by noise. Thus, the quality of the image can decrease. To remove or suppress the noise in the image and improve its quality, many de-noising methods are proposed, such as linear and nonlinear filtering, spectral analysis, and multi-resolution analysis. However, these traditional methods largely depend on explicit or implicit assumptions to properly separate the true signal from the random noise.

Over the past decade, wavelet analysis in the time domain and frequency domain, which has good localization properties and the multi-resolution analysis characteristics,^1–3 has received much attention from researchers in different areas, including pattern recognition, image de-noising, signal processing, and image compression. The wavelet analysis can effectively distinguish useful signal and noise, so it has become a notably effective image de-noising method.

At present, wavelet de-noising mainly includes three methods. ⁴ First, it adopts the wavelet’s singularity detection features ⁵ to separate the signal and the noise. Second, it uses a wavelet coefficient threshold function ⁶ to reduce the image noise. Third, the Bayesian criterion coefficient of the wavelet domain ⁷ is used for image noise reduction. The wavelet threshold shrinkage method is the most widely used in image de-noising because of its simplicity and effectiveness. The idea of wavelet threshold processing is derived from the Donoho and Johnstone ⁸ theory. Donoho first provided the general threshold de-noising formula based on an orthogonal wavelet transform, which made the complex de-noising problem easy to solve.

However, because of the lack of adaptability of the scale space, the threshold is difficult to determine. The result can lead to fuzzy image edge and poor de-noising performance. Thus, many scholars have introduced different wavelet coefficient scales and their corresponding threshold to reduce image noise, such as the hard threshold, ⁹ soft threshold, ¹⁰ VisuShrink threshold, ¹¹ improved sub-band adaptive SureShrink threshold, ¹² and NormalShrink threshold. ¹³ Although these de-noising algorithms can obtain good de-noising effect, much detail information is eliminated. The image quality seriously declines, and the pseudo Gibbs phenomenon may even be generated. To date, Wang et al. ¹⁴ proposed an optimized shape parameter method for image de-noising. Kadhim ¹⁵ presented a particle swarm optimization (PSO) algorithm to estimate the threshold value with no prior knowledge for these distributions. This process was achieved by implementing the PSO algorithm for kurtosis measuring of the residual noise signal to find an optimal threshold value, where the kurtosis function is maximal. Ji et al. ¹⁶ proposed a de-noising algorithm using the wavelet threshold method and exponential adaptive window width-fitting. Their method was divided into three parts. First, the wavelet threshold method was used to filter the white noise. Second, the data were segmented using a data window. Then, an exponential fitting algorithm was used to fit the attenuation curve of each window, and the data polluted by non-stationary electromagnetic noise were replaced with their fitting results. These methods have obtained good effects for image de-noising, but few works aim to improve the threshold function. Occasionally, their threshold functions are not better.

Our contributions in this article are as follows. We propose a new wavelet threshold function based on the Chi-square distribution-Kernel (CSDK) function for image de-noising. We also consider shape parameters on the wavelet coefficients to be thresholded. Hence, the soft transformation can achieve a high precision of the true signal until the noise is commendably separated by shrinkage. The wavelet soft thresholding based on CSDK is adopted to handle the image de-noising, and a significant improvement is obtained by a specially developed CSDK. To evaluate the performance of our new function, experiments were conducted on MATLAB to compare with other state-of-the-art methods. The results show that our new method performs better than other functions in terms of de-noising precision. Furthermore, the new function can enhance the image de-noising efficiency without the effect of layers or the number of image decompositions. New method can effectively remove noise and preserve the image details for de-noising image.

The remainder of this article is organized as follows. The Chi-square distribution related to this article is presented in section “Brief introduction of the chi-square distribution.” Section “Wavelet soft thresholding based on the CSDK method” illustrates the new threshold based on CSDK in detail, and section “Performance evaluation and analysis” presents the experimental results. The article is concluded in section “Conclusion.”

Brief introduction of the chi-square distribution

The presence of Gaussian white noise degrades images significantly and may hide important details and background on the images, leading to the loss of crucial information of original images.^17–20 Traditionally, the first step toward removing related noise in images is to understand its statistical properties. Despite the theoretical appeal and the analytical simplicity of the Gaussian model, images of some natural scenes such as fog deviate from the Gaussian distribution. To mitigate this situation, various distributions such as the Weibull distribution, ²¹ the log-normal distribution, ²² the k-distribution, ²³ and Cauchy distribution ²⁴ have been suggested.

However, in the above distributions, the log-normal distribution provides a convenient choice, but fails in modeling the lower half of the image histograms and overestimates the range of variation. Weibull distribution is an empirical model with limited theoretical justification. K-distribution is a successful model for SAR image despeckling, but not for this article’s testing data. Meanwhile, generalized Cauchy distribution (GCD) is a symmetric distribution with bell-shaped density function as the Gaussian distribution but with a greater probability mass in the tails. GCD is a peculiar distribution due to the difficulty of estimating its location parameter and its heavy tail. Because it has no mean, variance or higher moments defined, GCD has a long convergence time. To alleviate the above problems, this article utilizes CSDK method for image de-noising. The following is illustration for chi-square distribution.

Supposing that n independent random variables $({\xi }_1,{\xi }_2,\dots ,{\xi }_n)$ obey the Gaussian distribution, its sum of squares $Q=\sum _{i=1}^n{\xi }_i^2$ composes of a new random variable, which is named the chi-square distribution, where n is the freedom degree.^25,26

The probability density function (PDF) of the chi-square distribution is described as follows

$$f(\mathrm{PDF})=\{\begin{array}{cc}{f}_n(n)=\frac{{(1/2)}^{n/2}}{\Gamma (n/2)}{x}^{(n/2)-1}{e}^{(-x)/2}\hfill & \mathrm{if}x⩾0\hfill \\ \multicolumn{1}{c}{0}& \mathrm{otherwise}\hfill \end{array}$$

(1)

The cumulative distribution function (CDF) $F_n(x)$ of the chi-square distribution is defined as follows

$$F_n(x)=\frac{\gamma (n/2,x/2)}{\Gamma (k/2)}$$

(2)

where $\Gamma (·)$ and $\gamma (·)$ are gamma function and incomplete gamma function, respectively. k is a constant number. Figure 1 shows that the PDF and CDF can be changed with different freedom degrees. When $n\to \infty$ , the chi-square distribution is close to the Gaussian distribution as shown in Figure 1.

Figure 1.

Block diagram of PDF (left) and CDF (right).

Wavelet soft thresholding based on the CSDK method

The flow chart of our new image de-noising method is shown in Figure 2.

Figure 2.

Block diagram of new image de-noising method.

CSDK model construction

The main hypothesis of our CSDK model is that a combination of the structure Gaussian kernel of an image can significantly improve its reconstruction. The Gaussian kernel function has three important properties, which are conducive to image post-processing.

The Gaussian kernel function ²⁷ can be written as follows

$$y=\epsilon e^{-\frac{{(x-\mu )}^2}{2{\sigma }^2}}$$

(3)

where $\epsilon =1$ is the height of the function, $\mu$ is the center of curve in the x-axis, and $\sigma$ is the width. Figure 3 shows the curve of the Gaussian kernel function.

Figure 3.

Curve of the Gaussian kernel function.

Based on the principle of the Gaussian kernel function, we construct the CSDK model. The new function $\Re (x)$ is summarized as follows

$$\Re (x)=e^{-\left({(\frac{1}{(n/4){\int }_0^{\infty }{e}^{-x}\mathrm{dx}}(\frac{n}{x^2}-1)e^{-\frac{x}{2}}-\mu )}^2\right)/(2{\sigma }^2)}$$

(4)

In $\Re (x)$ , n > 0 is defined as the torsion resistance; $\mu$ and $\sigma$ are the shape parameters. If n varies, $\Re (x)$ will change as shown in Figure 4.

Figure 4.

Block diagram of CSDK function with different n.

Figure 4 shows that the CSDK retains the better properties of the chi-square distribution.

Wavelet soft thresholding based on CSDK

The new wavelet soft thresholding proposed in this article can be expressed as follows

$${\hat{w}}_{i,j}=\{\begin{array}{cc}\mathrm{sign}(w_{i,j})(|w_{i,j}|-\lambda \Re (w_{i,j}))\hfill & \mathrm{if}|w_{i,j}|⩾\lambda \hfill \\ \multicolumn{1}{c}{0}& \mathrm{otherwise}\hfill \end{array}$$

(5)

where $w_{i,j}$ is the wavelet coefficient and $\lambda$ is the threshold value. According to equation (5), a new function curve is drawn in Figure 5.

Figure 5.

Block diagram of wavelet soft thresholding based on CSDK.

The properties of the new function are as follows.

Theorem 1

Continuity

There is no breakpoint, so $f(\mathrm{ne}{w}_x)$ is a continuous function in its domain.

Proof

From its curve, we can know the domain, and the range of the function is $(-\infty ,+\infty )$ .

When $x>\lambda$

$$f(\mathrm{ne}{w}_x)=\mathrm{sign}(w_{i,j})(w_{i,j}-\lambda \Re (w_{i,j}))$$

(6)

Therefore, the right-hand limit of the function is

$$\begin{array}{cc}\multicolumn{1}{c}{lim\underset{x\to {\lambda }^{+}}{f(\mathrm{ne}{w}_x)}}& =lim\underset{x\to {\lambda }^{+}}{(x-\lambda \Re (x))}\hfill \\ \multicolumn{1}{c}{}& =x-\lambda e^0\hfill \\ \multicolumn{1}{c}{}& =0\hfill \end{array}$$

(7)

When $x<-\lambda$

$$f(\mathrm{ne}{w}_x)=\mathrm{sign}(w_{i,j})(-w_{i,j}-\lambda \Re (w_{i,j}))$$

(8)

Thus, the right-hand limit of the function is described as follows

$$\begin{array}{cc}\multicolumn{1}{c}{lim\underset{x\to {\lambda }^{-}}{f(\mathrm{ne}{w}_x)}}& =lim\underset{x\to {\lambda }^{-}}{(-x-\lambda \Re (x))}\hfill \\ \multicolumn{1}{c}{}& =-x-\lambda e^0\hfill \\ \multicolumn{1}{c}{}& =0\hfill \end{array}$$

(9)

When $-\lambda ⩽x⩽\lambda$

$$f(\mathrm{ne}{w}_x)\equiv 0$$

(10)

Considering equations (8)–(10), $lim\underset{x\to {\lambda }^{-}}{f(\mathrm{ne}{w}_x)}=lim\underset{x\to {\lambda }^{+}}{f(\mathrm{ne}{w}_x)}=limf(0)$ .

Thus, the new function is a continuous curve in its domain. Moreover, it compensates for the shortcomings of the hard threshold function.

Theorem 2

Monotonicity

$f(\mathrm{ne}{w}_x)$ is a monotonically increasing function in $(-\infty ,+\infty )$ , so $f(\mathrm{ne}{w}_x)$ is an increasing function in the domain of $(-\infty ,+\infty )$ .

Proof

When $x>\lambda$

$$f(\mathrm{ne}{w}_x)=\mathrm{sign}(w_{i,j})(x-\lambda e^{-\frac{{[\alpha ((x-\lambda )/\lambda )-\mu ]}^2}{2{\sigma }^2}})$$

(11)

The first derivative of $f(\mathrm{ne}{w}_x)$ is

$$f\prime (\mathrm{ne}{w}_x)=1+\frac{2x{\alpha }^2}{e^{{\alpha }^2{x}^2}}$$

(12)

Regardless of $\alpha$ , $f\prime (\mathrm{ne}{w}_x)>0$ .

We use the identical calculation method: if $x<-\lambda$ , similarly, $f\prime (\mathrm{ne}{w}_x)>0$ .

When $-\lambda <x<\lambda$

$$f(\mathrm{ne}{w}_x)\equiv 0$$

(13)

Therefore, $f(\mathrm{ne}{w}_x)$ , which is a monotonically increasing function, is proven.

Theorem 3

Differentiability

$f(\mathrm{ne}{w}_x)$ is differentiable.

Proof

The new function is continuous and monotonic, and its right and left limits are equal. Thus, it is differentiable.

Optimized threshold parameter $\lambda$

As we know, parameter $\lambda$ plays an important role in the wavelet threshold function. Donoho et al. ²⁸ proposed a common threshold formula

$$\lambda =\epsilon \sqrt{2\log (N)}$$

(14)

where $\epsilon$ is the noise variance, and N is the sampling length of the signal.

When multiple wavelet decompositions for an image are analyzed, the noise amplitude notably decreases with the increase in the number of image layers. However, the amplitude of image information increases. Therefore, this article proposes an optimized threshold parameter $\lambda$

$$\lambda =\frac{\epsilon \sqrt{2\log (N)}}{\log (1+e^j)}$$

(15)

where j denotes the layer of image decomposition. In this formula, if j increases, the optimized $\lambda$ gradually decreases. The improved $\lambda$ is superior to that in some state-of-the-art functions. For example, when N = 30, $\epsilon =0.6$ , j = 5, ${\lambda }_{[23]}=1.5649$ , and ${\lambda }_{\mathrm{our}}=0.3126$ , which is a good choice for the new wavelet threshold function.

In summary, our new wavelet threshold function includes the following ( $\alpha$ is an adaptive adjustment coefficient)

$$f_{{\hat{w}}_{i,j}}=\{\begin{array}{cc}\mathrm{sign}(w_{i,j})(|w_{i,j}|-\lambda e^{-\frac{{[\alpha ((x-\lambda )/\lambda )-\mu ]}^2}{2{\sigma }^2}})\hfill & \mathrm{if}|w_{i,j}|⩾\lambda \hfill \\ \multicolumn{1}{c}{0}& \mathrm{otherwise}\hfill \end{array}$$

(16)

$$\lambda =\frac{\epsilon \sqrt{2\log (N)}}{\log (1+e^j)}$$

(17)

Then, we study the effect of j on the new wavelet threshold function. Suppose that N = 30 and $\epsilon =0.6$ . When j = 2, 3, and 5, different curves are obtained as shown in Figures 6–8. When j is large, the effect of $\alpha$ is notably small, which can reduce the noise turbulence. Hence, our new function is effective. Algorithm 1 shows the process of our new method.

Figure 6.

Block diagram of CSDK, when j = 2.

Figure 7.

Block diagram of CSDK, when j = 3.

Figure 8.

Block diagram of CSDK, when j = 5.

Algorithm 1. The processes of CSDK

Input: Image A. Output: Image B after CSDK. 1. Adding noise into image A and obtaining the noisy image. 2. Selecting the wavelet function and layer of wavelet, executing wavelet decomposition on every scale and obtaining a set of wavelet coefficients $w_{i,j}$ . 3. Setting a threshold $\lambda$ . 4. If amplitudevalue >  $\lambda$ . 5. Adopting the CSDK to calculate ${\hat{w}}_{i,j}$ . 6. Else if amplitudevalue <  $\lambda$ . 7. ${\hat{w}}_{i,j}=0$ . 8. end. 9. Reconstructing the processed wavelet coefficients using the wavelet transform. 10. Output the recovered image.

Performance evaluation and analysis

In this section, experiments are conducted to demonstrate the effectiveness of the CSDK with MATLAB R2014b, Core i7 CPU, 8 GB memory, and Windows 10 platform environment. In section “Evaluation criterion,” the evaluation criterion and its function are introduced to evaluate our new method. First, we experimented with different parameter values in the new function and analyzed their effect on the wavelet threshold de-noising in section “Performance evaluation of different parameters for image de-noising.” Then, we made a comparison to state-of-the-art threshold functions to verify the effectiveness of our new method in section “Effect of parameter $\epsilon$ on image de-noising.” All experiments were conducted using the same software, hardware, and laptop.

Evaluation criterion

The image evaluation criterion contains two aspects: subjective evaluation and objective evaluation. In this section, we mainly evaluate the new function with objective evaluation.^29,30

In the new function, the shape parameter is adjusted to improve the effect on image de-noising. Two widely used indicators are employed to indicate the effect of image de-noising: the signal-to-noise ratio (SNR), normalized mean square error (NMSE), and structural similarity (SSIM). The SNR, NMSE, and SSIM are, respectively, calculated as shown in equations (18)–(20)

$$\mathrm{SNR}=10\ln \frac{\sum _{k=1}^{N}S{(k)}^2}{\sum _{k=1}^N{[X(k)-S(k)]}^2}$$

(18)

where N is the signal length; X(k) and S(k) are the noisy and original signals, respectively. Equation (18) shows that when the SNR is large, the processed image is close to the original image, which indicates that the image de-noising effect is good. Nevertheless, if the SNR is small, there is a large difference between the processed image and the original image, which reflects that the image de-noising effect is bad. Therefore, we use different shape parameters to verify our new method, and different SNRs can be obtained. According to the SNR, we can conclude a range of shape parameter (or its fixed value) with better effect on image de-noising

$$\mathrm{NMSE}=10\ln \frac{\sum _{k=1}^N{(S(k)-S\prime (k))}^2}{\sum _{k=1}^N|S(k){|}^2}$$

(19)

where $S\prime (k)$ is the filtered image

$$\mathrm{SSIM}=\frac{((2{\mu }_I{\mu }_{I\prime }+C_1)\times (2{\sigma }_{\mathrm{II}\prime }+C_2))}{(({\mu }_I^2+{\mu }_{I\prime }^2+C_1)\times ({\sigma }_I^2+{\sigma }_{I\prime }^2+C_2))}$$

(20)

where ${\mu }_I$ denotes the average intensity of image $I$ , ${\mu }_{I\prime }$ denotes the average intensity of image $I\prime$ , ${\sigma }_I$ and ${\sigma }_{I\prime }$ denote the variance of image $I$ and $I\prime$ , respectively. ${\sigma }_I$ gives the covariance of $I$ and $I\prime$ , and $C_1$ and $C_2$ are constants. SSIM is a recently proposed approach for measuring the similarity between two images, which is designed to improve on traditional metrics like peak signal-to-noise ratio (PSNR) and mean square error (MSE). Its value lies between [0, 1]. The value 1 signifies that the two images are same in all views.

Performance evaluation of different parameters for image de-noising

As we know, the shape parameter significantly affects the image de-noising. Thus, the shape parameter selection is notably important.

According to the principle of Gaussian kernel function, $\alpha =1$ . First, we study the effect of $\mu$ and $\sigma$ on the image de-noising. Assuming that N = 30, $\epsilon =0.6$ , and j = 0.2, we selected “Lena,”“Barbara,” and “Baboon” in international standard test images as the testing images. Figure 9 shows the original images and noisy images under Gaussian noise = 0.04, 0.3, 0.7. These results are shown in Figures 10(a)–(i) to 13(a)–(f) only when Gaussian noise = 0.04. Tables 1–3 are the detailed results including Gaussian noise = 0.04, 0.3, 0.7. I conducted the actual experiments extensively, and the results showed that the three noise numbers affected the results sharply. In these experiments, the Gaussian noisy and wavelet decomposition layers were held constant.

Figure 9.

Testing images. Original images: (a) Lena, (b) Baboon, and (c) Barbara. Noisy images with Gaussian noise = 0.04: (d) Lena, (e) Baboon, and (f) Barbara. Gaussian noise = 0.3: (g) Lena, (h) Baboon, and (i) Barbara. Gaussian noise = 0.7: (j) Lena, (k) Baboon, and (l) Barbara.

Figure 10.

Image Lena de-noising with different $\sigma$ and $\mu$ : (a) ${\sigma }^2=0.1$ , $\mu =0$ ; (b) ${\sigma }^2=0.3$ , $\mu =0$ ; (c) ${\sigma }^2=0.6$ , $\mu =0$ ; (d) ${\sigma }^2=0.1$ , $\mu =2$ ; (e) ${\sigma }^2=0.3$ , $\mu =2$ ; (f) ${\sigma }^2=0.6$ , $\mu =2$ ; (g) ${\sigma }^2=0.1$ , $\mu =5$ ; (h) ${\sigma }^2=0.3$ , $\mu =5$ ; (i) ${\sigma }^2=0.6$ , $\mu =5$ .

Figure 11.

Image Baboon de-noising with different $\sigma$ and $\mu$ : (a) ${\sigma }^2=0.1$ , $\mu =0$ ; (b) ${\sigma }^2=0.3$ , $\mu =0$ ; (c) ${\sigma }^2=0.6$ , $\mu =0$ ; (d) ${\sigma }^2=0.1$ , $\mu =2$ ; (e) ${\sigma }^2=0.3$ , $\mu =2$ ; (f) ${\sigma }^2=0.6$ , $\mu =2$ ; (g) ${\sigma }^2=0.1$ , $\mu =5$ ; (h) ${\sigma }^2=0.3$ , $\mu =5$ ; (i) ${\sigma }^2=0.6$ , $\mu =5$ .

Figure 12.

Image Barbara de-noising with different $\sigma$ and $\mu$ : (a) ${\sigma }^2=0.1$ , $\mu =0$ ; (b) ${\sigma }^2=0.3$ , $\mu =0$ ; (c) ${\sigma }^2=0.6$ , $\mu =0$ ; (d) ${\sigma }^2=0.1$ , $\mu =2$ ; (e) ${\sigma }^2=0.3$ , $\mu =2$ ; (f) ${\sigma }^2=0.6$ , $\mu =2$ ; (g) ${\sigma }^2=0.1$ , $\mu =5$ ; (h) ${\sigma }^2=0.3$ , $\mu =5$ ; (i) ${\sigma }^2=0.6$ , $\mu =5$ .

Figure 13.

Image Lena de-noising with different $\epsilon$ : (a) $\epsilon =0.1$ , (b) $\epsilon =0.2$ , (c) $\epsilon =0.4$ , (d) $\epsilon =0.6$ , (e) $\epsilon =0.8$ , and (f) $\epsilon =0.9$ .

Table 1.

SNR values of Lena with different $\sigma$ and $\mu$ .

Number	Parameter	SNR1	SNR2Gaussian noise = 0.04	SNR2Gaussian noise = 0.3	SNR2Gaussian noise = 0.7
1	${\sigma }^2=0.1$ , $\mu =0$	8.7816	19.1354	18.0942	17.3571
2	${\sigma }^2=0.3$ , $\mu =0$	8.8208	19.0018	17.8863	17.0914
3	${\sigma }^2=0.6$ , $\mu =0$	8.7995	18.9773	17.7144	16.8785
4	${\sigma }^2=0.1$ , $\mu =2$	8.8094	18.9644	17.5538	16.4927
5	${\sigma }^2=0.3$ , $\mu =2$	8.8078	18.9632	17.3361	16.2897
6	${\sigma }^2=0.6$ , $\mu =2$	8.8181	18.9609	17.1549	16.0884
7	${\sigma }^2=0.1$ , $\mu =5$	8.7876	18.9471	16.8955	15.9326
8	${\sigma }^2=0.3$ , $\mu =5$	8.8133	18.9415	16.6473	15.9021
9	${\sigma }^2=0.6$ , $\mu =5$	8.7987	18.9403	16.6128	15.8837

SNR: signal-to-noise ratio.

The best results are highlighted in bold and italic.

Table 2.

SNR values of Baboon with different $\sigma$ and $\mu$ .

Number	Parameter	SNR1	SNR2Gaussian noise = 0.04	SNR2Gaussian noise = 0.3	SNR2Gaussian noise = 0.7
1	${\sigma }^2=0.1$ , $\mu =0$	8.9211	14.2664	13.9876	13.1864
2	${\sigma }^2=0.3$ , $\mu =0$	8.9090	14.2487	13.9234	13.0816
3	${\sigma }^2=0.6$ , $\mu =0$	8.8923	14.2511	13.8461	12.9755
4	${\sigma }^2=0.1$ , $\mu =2$	8.8991	14.2587	13.8066	12.9259
5	${\sigma }^2=0.3$ , $\mu =2$	8.9094	14.2651	13.7109	12.8743
6	${\sigma }^2=0.6$ , $\mu =2$	8.8924	14.2479	13.6883	12.8022
7	${\sigma }^2=0.1$ , $\mu =5$	8.8923	14.1945	13.6112	12.7134
8	${\sigma }^2=0.3$ , $\mu =5$	8.9037	14.2653	13.5859	12.6918
9	${\sigma }^2=0.6$ , $\mu =5$	8.9083	14.2658	13.4931	12.6126

SNR: signal-to-noise ratio.

The best results are highlighted in bold and italic.

Table 3.

SNR values of Barbara with different $\sigma$ and $\mu$ .

Number	Parameter	SNR1	SNR2Gaussian noise = 0.04	SNR2Gaussian noise = 0.3	SNR2Gaussian noise = 0.7
1	${\sigma }^2=0.1$ , $\mu =0$	8.0997	15.8507	14.5927	13.9755
2	${\sigma }^2=0.3$ , $\mu =0$	8.0874	15.8216	14.4463	13.8129
3	${\sigma }^2=0.6$ , $\mu =0$	8.0864	15.8422	14.2209	13.5746
4	${\sigma }^2=0.1$ , $\mu =2$	8.0776	15.8391	14.1684	13.4557
5	${\sigma }^2=0.3$ , $\mu =2$	8.1023	15.8291	14.1529	13.4476
6	${\sigma }^2=0.6$ , $\mu =2$	8.1002	15.8367	14.1107	13.3851
7	${\sigma }^2=0.1$ , $\mu =5$	8.0792	15.8167	14.0834	13.2843
8	${\sigma }^2=0.3$ , $\mu =5$	8.0790	15.8252	13.9951	13.1859
9	${\sigma }^2=0.6$ , $\mu =5$	8.0852	15.8350	13.8649	13.1127

SNR: signal-to-noise ratio.

The best results are highlighted in bold and italic.

Table 1 shows that when $\mu$ and $\sigma$ vary, the SNR of the image with the new threshold function varies. When ${\sigma }^2=0.1$ and $\mu =0$ , SNR2 (Gaussian noise = 0.04) is 19.1354, which exceeds the value when ${\sigma }^2=0.3$ and $\mu =5$ . Similarly, when ${\sigma }^2=0.1$ and $\mu =2$ , SNR2 is the largest (18.9644) compared to when ${\sigma }^2=0.3$ and ${\sigma }^2=0.6$ (18.9632 and 18.9609, respectively). Furthermore, the average SNR is larger when $\mu =0$ and $\mu =2$ compared to when $\mu =5$ . When ${\sigma }^2=0.1$ and $\mu =0$ , the SNR is 19.1354, which is larger than that when ${\sigma }^2=0.1$ and $\mu =2$ (SNR = 18.9644). Meanwhile, Figures 11(a), 12(a), and 13(a) show the best de-noising effect. Therefore, when $\mu$ and $\sigma$ are minimal, the image de-noising effect is the best. Similarly, for Baboon and Barbara, the best SNRs with CSDK are 14.2664 and 15.8507 in Tables 2 and 3, respectively, which indicates that the results are better for smaller $\mu$ and $\sigma$ . Note that when Gaussian noise = 0.3 and Gaussian noise = 0.7, ${\sigma }^2=0.1$ , $\mu =0$ is the best choice.

Effect of parameter $\epsilon$ on image de-noising

The analyzed parameters were manually optimized for the best PSNR, which is the metric in our evaluation. Let $\mu =0$ , j = 5, ${\sigma }^2=0.1$ , and N = 30 in this section. We conducted six experiments to study the effect of shape parameter $\epsilon$ on the trend of the SNR, which are listed in Tables 4–6.

Table 4.

SNR values of Lena with different $\epsilon$ .

Number	Parameter	SNR1	SNR2Gaussian noise = 0.04	SNR2Gaussian noise = 0.3	SNR2Gaussian noise = 0.7
1	$\epsilon =0.1$	8.8158	18.9769	17.6859	17.3247
2	$\epsilon =0.2$	8.8052	18.9373	17.6321	17.2967
3	$\epsilon =0.4$	8.7992	18.9638	17.5847	17.1169
4	$\epsilon =0.6$	8.8120	18.9694	17.1365	17.0954
5	$\epsilon =0.8$	8.8097	18.9351	16.5846	16.8796
6	$\epsilon =0.9$	8.8241	18.9585	16.5787	16.8219

SNR: signal-to-noise ratio.

The best results are highlighted in bold and italic.

Table 5.

SNR values of Barbara with different $\epsilon$ .

Number	Parameter	SNR1	SNR2Gaussian noise = 0.04	SNR2Gaussian noise = 0.3	SNR2Gaussian noise = 0.7
1	$\epsilon =0.1$	8.1170	15.8529	14.6582	14.3213
2	$\epsilon =0.2$	8.1024	15.8523	14.6108	14.3106
3	$\epsilon =0.4$	8.0901	15.8218	14.5837	14.2885
4	$\epsilon =0.6$	8.0823	15.8295	14.5086	14.1907
5	$\epsilon =0.8$	8.0864	15.7821	14.4975	14.1537
6	$\epsilon =0.9$	8.0942	15.8028	14.4617	14.0662

SNR: signal-to-noise ratio.

The best results are highlighted in bold and italic.

Table 6.

SNR values of Baboon with different $\epsilon$ .

Number	Parameter	SNR1	SNR2Gaussian noise = 0.04	SNR2Gaussian noise = 0.3	SNR2Gaussian noise = 0.7
1	$\epsilon =0.1$	8.8960	14.2881	13.1968	12.9878
2	$\epsilon =0.2$	8.9035	14.2825	13.1724	12.9673
3	$\epsilon =0.4$	8.9075	14.2776	13.1309	12.8755
4	$\epsilon =0.6$	8.9026	14.2662	13.0859	12.8106
5	$\epsilon =0.8$	8.9001	14.2486	13.0554	12.7984
6	$\epsilon =0.9$	8.9154	14.2672	12.9975	12.7763

SNR: signal-to-noise ratio.

The best results are highlighted in bold and italic.

Obviously, with the increase in $\epsilon$ , the SNR of the new function is gradually reduced. The de-noising effects of images Lena, Baboon, Barbara with different $\epsilon$ are shown in Figures 13(a)–(f) to 15(a)–(f).

Figure 14.

Image Baboon de-noising with different $\epsilon$ : (a) $\epsilon =0.1$ , (b) $\epsilon =0.2$ , (c) $\epsilon =0.4$ , (d) $\epsilon =0.6$ , (e) $\epsilon =0.8$ , and (f) $\epsilon =0.9$ .

Figure 15.

Image Barbara de-noising with different $\epsilon$ : (a) $\epsilon =0.1$ , (b) $\epsilon =0.2$ , (c) $\epsilon =0.4$ , (d) $\epsilon =0.6$ , (e) $\epsilon =0.8$ , and (f) $\epsilon =0.9$ .

Comparison experiments

In this section, we compare our method with state-of-the-art threshold functions (the wavelet based including soft threshold function, NWT ³¹ and SDL ³² ) and other famous non-wavelet-based image de-noising methods (EAF, ³³ PNM, ³⁴ EGL, ³⁵ NLG ³⁶ ) to demonstrate the effectiveness of our new method. In the simulated study, three images were used for testing: “Lena,”“Baboon” and “Barbara.”

In all experiments, the parameters of the references were set according to the above analysis: $\mu =0$ , j = 5, ${\sigma }_1^2=0.1$ , N = 30, $\epsilon =0.1$ , and $\alpha =1$ . They were used for all comparison experiments. We have extensively tested these values as the criteria for image de-noising and find them succeed in virtually all cases.

In Lena test experiment, the de-noised image using the new method in Figure 16 is compared with the de-noised image of other de-noising methods. As observed, the new method successfully eliminates noise and obtains more accurate results than the other methods. Note that there are some spots in soft threshold function. SDL method indicates that the de-noising effect has better smoothness, but the image is fuzzy.

Figure 16.

Image Lena de-noising results and experiment contrast: (a) soft threshold, (b) NWT, (c) SDL, (d) EAF, (e) PNM, (f) EGL, (g) NLG, and (h) the proposed CSDK.

Similarly, in Baboon and Barbara experiments (Figures 17 and 18), the new method effectively removes the baseline noise and retrieves the images better compared to other de-noising methods.

Figure 17.

Image Baboon de-noising results and experiment contrast: (a) soft threshold, (b) NWT, (c) SDL, (d) EAF, (e) PNM, (f) EGL, (g) NLG, and (h) the proposed CSDK.

Figure 18.

Image Barbara de-noising results and experiment contrast: (a) soft threshold, (b) NWT, (c) SDL, (d) EAF, (e) PNM, (f) EGL, (g) NLG, and (h) the proposed CSDK.

Tables 7–9 show the SNR, NMSE, and SSIM of the original noisy and de-noised image with different de-noising methods. According to Table 7, SNR2, NMSE, and SSIM with our new method are 18.9692, –8.8180, and 0.9846, respectively, which are the largest values among the methods; the soft threshold function has the second largest SNR value (18.9387), followed by PNM (18.8827) and EAF (18.7109). Overall, the table shows that our new method obtains better effect than the other methods.

Table 7.

Comparison of Lena with different methods.

Method	SNR1	SNR2	NMSE	SSIM
Soft threshold	8.8078	18.9387	−10.8537	0.7264
NWT	14.4077	16.1786	−10.6529	0.7922
SDL	8.2921	17.6859	−10.5649	0.7916
EAF	8.6795	18.6653	−9.2416	0.8523
PNM	8.3549	18.6941	−8.9761	0.8467
EGL	8.5837	18.7109	−89178	0.8824
NLG	8.6714	18.8827	−8.8596	0.8927
CSDK	8.7965	18.9692	− 8.8180	0.9846

SNR: signal-to-noise ratio; NMSE: normalized mean square error; SSIM: structural similarity.

The best results are highlighted in bold and italic.

Table 8.

Comparison of Baboon with different methods.

Method	SNR1	SNR2	NMSE	SSIM
Soft threshold	8.9129	14.2760	−10.8795	0.8593
NWT	8.5807	14.3085	−10.6547	0.8654
SDL	8.2921	13.6859	−10.5466	0.8746
EAF	8.3164	14.2128	−9.9762	0.8922
PNM	8.3253	14.3107	−9.8524	0.9137
EGL	8.5508	14.3193	−9.7688	0.9248
NLG	8.5674	14.2984	−9.6417	0.9617
CSDK	8.9038	14.3692	− 8.1251	0.9835

SNR: signal-to-noise ratio; NMSE: normalized mean square error; SSIM: structural similarity.

The best results are highlighted in bold and italic.

Table 9.

Comparison of Barbara with different methods.

Method	SNR1	SNR2	NMSE	SSIM
Soft threshold	8.5746	15.2381	−10.81537	0.8601
NWT	8.6472	15.6812	−9.9054	0.8639
SDL	8.3451	15.8787	−9.7822	0.8854
EAF	8.3188	16.4827	−8.5897	0.9025
PNM	8.2643	16.5374	−8.3074	0.9437
EGL	8.7549	16.6548	−8.3576	0.9548
NLG	8.7654	16.7876	−8.5917	0.9627
CSDK	8.9277	16.8763	− 7.6548	0.9829

SNR: signal-to-noise ratio; NMSE: normalized mean square error; SSIM: structural similarity.

The best results are highlighted in bold and italic.

Similarly, Tables 8 and 9 imply that the new method outperforms the current de-noising methods and successfully recovers the desired images. The values of SNR, NMSE, and SSIM of the proposed adaptive de-noising algorithm are slightly higher than that of the other compared state-of-the-art methods. It shows that the proposed algorithm has a stronger ability to enhance the edges and the texture regions of the image, and preserve the smooth regions of the image while removing the noise.

To demonstrate the robustness of proposed method, we also select three medical images as shown in Figure 19. We add the same noise for them and obtain the de-nosing results shown in Tables 10–12. The results also show that our method is better than other de-noising methods in terms of medical images.

Figure 19.

Testing medical images: (a) image 1, (b) image 2, and (c) image 3.

Table 10.

Comparison of image 1 with different methods.

Method	SNR2	NMSE	SSIM
Soft threshold	16.0958	−9.5842	0.8791
NWT	16.2854	−9.5513	0.8936
SDL	16.7878	−9.4682	0.8937
EAF	16.9782	−8.9874	0.9205
PNM	17.3642	−8.8828	0.9473
EGL	17.6154	−8.5626	0.9745
NLG	17.6887	−8.1447	0.9762
CSDK	18.3452	− 7.2611	0.9867

SNR: signal-to-noise ratio; NMSE: normalized mean square error; SSIM: structural similarity.

The best results are highlighted in bold and italic.

Table 11.

Comparison of image 2 with different methods.

Method	SNR2	NMSE	SSIM
Soft threshold	16.1458	−9.7853	0.8127
NWT	17.5432	−8.9642	0.8548
SDL	17.8946	−8.6554	0.8911
EAF	17.9653	−8.3985	0.8928
PNM	18.1234	−8.2838	0.9157
EGL	18.5561	−7.8452	0.9539
NLG	18.5563	−7.4582	0.9654
CSDK	19.6586	− 6.5473	0.9759

SNR: signal-to-noise ratio; NMSE: normalized mean square error; SSIM: structural similarity.

The best results are highlighted in bold and italic.

Table 12.

Comparison of image 3 with different methods.

Method	SNR2	NMSE	SSIM
Soft threshold	17.5841	−8.6431	0.8358
NWT	18.4523	−7.8842	0.8648
SDL	18.9764	−7.4639	0.8723
EAF	19.1143	−7.2314	0.8982
PNM	19.2674	−6.5894	0.9177
EGL	20.5569	−6.3883	0.9394
NLG	20.6745	−6.1442	0.9681
CSDK	24.3674	− 5.7849	0.9876

SNR: signal-to-noise ratio; NMSE: normalized mean square error; SSIM: structural similarity.

The best results are highlighted in bold and italic.

Conclusion

This article proposed a new wavelet threshold function based on the CSDK function for image de-noising. The new idea was embedded in two aspects: introducing the CSDK function into the soft threshold function and improving the shape parameter $\lambda$ . In this article, we discussed our new function including the relation between j and $\alpha$ and the effect of $\epsilon$ . Then, proofs were provided to demonstrate its good properties. Finally, the proposed method was tested on three simulated images and medical images in comparison with several other popular de-noising methods. Both numerical and visual results demonstrate that the proposed method in this article can strongly remove noise and preserve the image details.

Direction of current and future work includes the following:

• Continuation on parameter $\lambda$ . $\lambda$ can be treated as a regularization parameter on the threshold function and universal adaptive local and non-local image de-noising methods can be studied.

• Speeding up the algorithm can be done using transfer learning method.

• Except wavelet soft thresholding, more state-of-the-art de-noising methods can be analyzed and improved.