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Though the wavelet threshold algorithm has been demonstrated to be a very effective tool to denoise the images with low levels of noise, it usually losses the power to well preserve meaningful details in images. To this end, this paper proposed an adaptive threshold denoising method for images that are usually interfered by noise on the Wireless Multimedia Sensor Network (WMSN). First, a scale parameter equation was defined according to different subband characteristics after the images were subject to wavelet decomposition, so as to determine the adaptive optimal threshold suitable for each scale level; Second, a new derivable threshold function was designed in this paper; Third, after comparison, proper wavelet basis function was selected for image denoising accordingly. Moreover, the test results on several test images proved the superiority of the proposed method over some classical methods in terms of PSNR and MSE.
wavelet transform, thresholding, WMSN, image denoising
With the rapid development of sensor hardware technology in recent years, the conventional sensor networks have been integrated with the sensing nodes of videos, images, and other multimedia information to form a novel distributed sensing monitoring network, the Wireless Multimedia Sensor Network (WMSN) [1]. WMSN is constructed based on multimedia sensor nodes, it can accurately and comprehensively monitor the environment within the applied range, therefore it has a broad application prospect in the field of largescale monitoring, especially the key area monitoring along traffic lines [2].
However, the diverse monitoring objects and the complex monitoring environments have determined that the WMSN images have rich details and the WMSN surveillance videos are easily affected by environmental noises such as ice, snow, rain and fog, electrical noises of the instrument and random noises during signal transmission, resulting in image quality decline [3]. Therefore, the effective denoising of acquired WMSN images has been a hot topic in related fields [46]. Wavelet transform can characterize both time and frequency domain features, and effectively determine the abnormal points in the images and their degree of abnormality, due to these advantages, it has been widely applied in the field of image denoising [711]. Among the many wavelet denoising methods, the wavelet threshold denoising method is the most commonlyused method [12]. However, the traditional threshold has a tendency to "overastringe" the wavelet coefficients, failing to separate the image and the noise at each scale to the greatest extent, thereby resulting in larger errors in reconstruction [1319]. In view of these issues, based on abovementioned research, this paper proposed an improved adaptive threshold method for WMSN image denoising.
The rest of this paper is organized as follows: in Section 2, the principle of wavelet transform is introduced; in Section 3, the wavelet threshold denoising method is explained; in Section 4, the test results are discussed; and in Section 5, the conclusion is drawn.
Assume signal $\psi (t)$ is a square integrable function, if its Fourier transform $\psi (\omega )$ satisfies the following condition:
$\int\limits_{R}{\frac{{{\left \psi (\omega ) \right}^{2}}}{\omega }}d\omega <\infty$ (1)
Then $\psi (t)$ is called a basic wavelet, and the wavelet basis function could be obtained by transforming according to the following formula:
${{\psi }_{a,b}}(t)={{\left a \right}^{\frac{1}{2}}}\psi (\frac{tb}{a})$ (2)
where, a is the scale parameter and b is the position parameter.
Assume $\psi $ is a basic wavelet, and $\{{{\psi }_{a,b}}\}$ is a continuous wavelet defined by formula (2), then the continuous wavelet transform of signal $f\in {{L}^{2}}$ is:
${{W}_{f}}(a,b)={{\left a \right}^{\frac{1}{2}}}\int\limits_{R}{f(t)\bar{\psi }(\frac{tb}{a})}dt$ (3)
Scale parameter a is used to adjust the frequency range of the wavelet, position parameter b is used to adjust the time domain position of the wavelet, and coefficient ${{\left a \right}^{\frac{1}{2}}}$ is used to achieve energy normalization.
The multiresolution analysis is to decompose the signal into a series of different levels of space. The largescale space corresponds to the general form of the signal, and the smallscale space corresponds to the details of the signal. When the scale changes from large to small, the overall conditions and information details of the signal could be observed [20].
If scale function $\varphi (t)\in {{L}^{2}}(R)$ satisfies the following condition:
${{\varphi }_{j.k}}(t)={{2}^{\frac{j}{2}}}\varphi ({{2}^{j}}tk)$ (4)
That is, after translation k (k is an integer) and scaling on scale j, this function can obtain a set with simultaneously changeable scales and positions, and the scale space V_{j} is defined as the space composed of all translation series ${{\varphi }_{j.k}}(t)$ on scale j.
${{V}_{j}}=span\{{{\varphi }_{j,k}}(t)\}(k\in Z)$ (5)
Suppose $f(t)\in {{V}_{j}}$ is any function, then there is:
$f(t)=\sum\limits_{k}{{{a}_{k}}{{\varphi }_{j,k}}(t)}={{2}^{\frac{j}{2}}}\sum\limits_{k}{{{a}_{k}}\varphi ({{2}^{j}}tk)}$ (6)
It can be seen from formula (6) that the domain and translation interval ${{2}^{j}}{{\tau }_{0}}$ of function ${{\varphi }_{j.k}}(t)$ both change with scale j, so the scale space contains both largescale slowlyvarying signals and smallscale signals. At this time, the signals are divided into the following two parts with scale j as the dividing line. All details above scale j are taken as the general form of the signals, and all details below scale j are taken as the detail part of the signals [21].
Wavelet transform can be used for image processing by extending from one dimension to two dimensions. By processing each row and column of the image using the Mallat algorithm, we can complete a twodimensional wavelet transform for one time [22].
It can be seen that wavelet analysis has the ability of local analysis and refinement, it can decompose signals or images into multiple scale components, and adopt time domain or space domain with corresponding thickness to sample the step size, so as to finely process the high frequency signals and coarsely process the low frequency signals. Compared with traditional signal analysis techniques, wavelet analysis can denoise images without significant loss [23].
In the transmission process, fluctuations in power supply voltage, electrostatic interference, poor grounding and other instrumental factors, and environmental interference would affect the quality of WMSN images. These noise signals generally exhibit flat broadband characteristics, so they can be considered as additional Gaussian white noise [24]. In this way, the image received by the terminal is the sum of the original signals and the additional noises [25].
In theory, a twodimensional signal containing noise can be represented by the following model:
$f(k)=r(k)+\sigma e(k)(k=0,1,2 \cdots)$ (7)
where, r(k) is the original image, e(k) is the noise signal, and f(k) is the terminal image superimposed with a lot of noises. Assuming that the amplitude of these noise signals follows the Gaussian distribution and the power spectral density follows the uniform distribution, then they can be replaced by the Gaussian noise with a noise level σ of 1 [26].
The basic method of denoising the signals using wavelet analysis is: first, the image is subject to twodimensional multilayer wavelet decomposition; second, the wavelet coefficient obtained by the decomposition is processed using thresholds and other forms; third, the signals are subject to wavelet reconstruction. In this way, signal denoising is completed [27].
The flow of the wavelet transform threshold denoising method is [28]:
(1) The signals are subject to wavelet transform to obtain wavelet coefficient x;
(2) The wavelet coefficient is subject to nonlinear threshold t to obtain the modified wavelet coefficient $\bar{x}$;
(3) The wavelet is subject to inverse transform to obtain the reconstructed signals.
In step 2, according to different thresholds, the threshold denoising methods could be divided into hard threshold method and soft threshold method, a brief introduction is as follows:
First, specify a threshold, compare the absolute value of the signal with the specified threshold, find points that are not greater than the threshold and set them to 0, as for the rest points that are greater than the threshold, they are set as the difference between the value of the point and the value of the threshold, this method is called the soft threshold method. While the hard threshold method is different, when comparing the absolute value of the signal with the specified threshold, it finds points that are greater than the threshold and retain them, as for the rest points that are less than or equal to the threshold, their values are set to 0, as shown in formulas (8) and (9) below.
${{s}_{1}}=\left\{ \begin{matrix} x & \left x \right>t \\ 0 & \left x \right\le t \\\end{matrix} \right.$ (8)
${{s}_{2}}=\left\{ \begin{matrix} sign(x)(\left x \rightt) & \left x \right>t \\ 0 & \left x \right\le t \\\end{matrix} \right.$ (9)
Although the wavelet denoising method can obtain approximate optimal estimates of the original images, and the denoised images are smoother, the threshold determined by the method is single, and the denoised images may exhibit visual distortion phenomena such as ringing artifact, pseudo Gibbs phenomenon, and blurred edges, etc. [29]. Therefore, the following adaptive threshold denoising method is adopted in this paper.
In this section, the proposed method is described. A novel derivable threshold function is designed to help reduce the constant deviation of wavelet coefficient caused by the discontinuous threshold function; in addition, suitable wavelet basis function is also determined through comparison.
4.1 A new threshold function
Wavelet threshold denoising methods include the hard threshold method and the soft threshold method. Although the difference between the wavelet coefficient x of the ideal signal (no noise) estimated by the hard threshold method and the wavelet coefficient $\hat{x}$ of the real signal is very small, due to the discontinuity of x at position t of the threshold, the estimated signal would generate additional oscillations at the singular point of the signal, which ultimately makes the denoised signal not as smooth as the ideal signal. As for the soft threshold method, although the obtained wavelet coefficient has good overall continuity, there is always a constant deviation between the estimated wavelet coefficient ${x}'$ of the ideal signal and the wavelet coefficient ${\hat{x}}'$ of the real signal, which leads to a lower degree of approximation between the denoised signal and the ideal signal, compromising the accuracy of signal reconstruction. A good threshold function should have two characteristics at the same time: one is that it makes the deviation between the estimated value and the actual value of the wavelet coefficient as small as possible; the other is that it has continuity in the wavelet domain.
The following formula is the new threshold function constructed in this study:
$\hat{x}=\left\{ \begin{matrix} \begin{matrix} sign(x)\cdot \frac{1}{{{e}^{\left x \right}}}\cdot \sqrt{\frac{{{x}^{2}}+{{(\left x \rightt)}^{2}}}{2}}, & \left x \right\ge t \\\end{matrix} \\ \begin{matrix} 0, & \left x \right\le t \\\end{matrix} \\\end{matrix} \right.$ (10)
For the $\hat{x}$ obtained solely by the soft threshold method, there is a constant deviation t between its absolute value and x, and this deviation should be reduced, but if the deviation is completely eliminated and set to 0, the situation exactly corresponds to the hard threshold method. It can be seen from formula (10) that $\sqrt{\frac{{{x}^{2}}+{{(\left x \rightt)}^{2}}}{2}}$ is the geometric mean value of x and $\left x \rightt$, $\frac{1}{{{e}^{\left x \right}}}$ is a moderator term. When $\left x \right\ge t$, with the increase of $\left x \right$,$\frac{1}{{{e}^{\left x \right}}}$ decreases continuously, which plays a role in dynamically moderating the threshold. The new threshold function can adaptively determine the attenuation of the wavelet coefficient, so that the loss of useful information in high frequency domain could be reduced, and the signaltonoise ratio of the reconstructed signal could be improved.
4.2 Determine the wavelet basis function
When processing WMSN images, different wavelet basis functions could be used as processing tools, and the obtained results are quite different. In order to get highprecision processing results, suitable wavelet basis functions must be chosen properly. Agarwal et al. [30] studied this problem and proposed five principles for wavelet selection in image processing, namely the orthogonality, regularity, support width, vanishing moment order and symmetry principles, which are also basic principles for wavelet selection in the engineering field.
At present, there are five basic wavelets that are commonly used in the field of WMSN image processing, and their properties are shown in Table 1.
Table 1. Comparison of five kinds of wavelet basis functions

Orthogonality 
Vanishing moments 
Scaling function 
Discrete transform 
Symmetry 
Highspeed algorithm 
haar 
Y 
N 
Y 
Y 
N 
Y 
mexh 
N 
N 
N 
N 
Y 
N 
morl 
N 
N 
N 
N 
Y 
N 
DBN 
Y 
Y 
Y 
Y 
N 
Y 
symN 
Y 
Y 
Y 
Y 
N 
Y 
Analysis of the transmission process of WMSN images shows that the image signal can be approximatively regarded as a kind of unstable signal modulated by the center frequency of the node and containing instantaneous and abrupt components, so the selected wavelet basis must conform to the abovementioned five principles; in order to facilitate implementation on a PC, discrete transform must be achieved. Based on the above factors, the sym4 wavelet had been selected as the wavelet basis function in this study.
To examine the performance of the proposed algorithm, four classical denoising algorithms and the proposed algorithm were subject to comparative experiments. In the meantime, in order to ensure the objectivity of the evaluation, peak signal to noise ratio (PSNR) and mean square error (MSE) had been taken as standards to evaluate the denoising effects of the algorithms. Larger PSNR and smaller MSE generally indicate better denoising effect, and higher degree of image feature retention. All tests were performed on an Inter (R) Core (TM) i5 PC with 16 GB RAM.
Figure 1. The original images: Test Image 1 (a), Test Image 2 (b), Test Image 3 (c), and Test Image 4 (d)
In the first test, the PSNR and MSE results obtained by the proposed method and the four classical methods (named Default, Global, VisuShrink, and NormalShrink, respectively) were compared. The tests were performed using four test images as shown in Figure 1. Each image had a size of 480×357 pixels, the horizontal and vertical resolution was 96 dpi, and the bit depth was 24. White noises with variations of 20, 30, 40 and 50 were added in the images. The results are given in Tables 25.
Table 2. Performance of the denoising methods of test image 1

σ=20 
σ=30 
σ=40 
σ=50 

PSNR 
MSE 
PSNR 
MSE 
PSNR 
MSE 
PSNR 
MSE 

Default 
26.7089 
138.7371 
25.6244 
178.0910 
24.5950 
225.7278 
23.5936 
284.2660 
Global 
27.4071 
118.1336 
25.9249 
166.1834 
24.7125 
219.6997 
23.6119 
283.0675 
VisuShrink 
26.9899 
130.0435 
25.1662 
197.9059 
23.5208 
289.0666 
21.9318 
416.7775 
NormalShrink 
24.3661 
237.9401 
22.3466 
378.8075 
20.8292 
537.2256 
19.6111 
711.1627 
Proposed 
29.4964 
73.0196 
27.2145 
123.4909 
25.3248 
190.8091 
23.7741 
272.6907 
Table 3. Performance of the denoising methods of test image 2

σ=20 
σ=30 
σ=40 
σ=50 

PSNR 
MSE 
PSNR 
MSE 
PSNR 
MSE 
PSNR 
MSE 

Default 
22.3611 
377.5430 
21.6585 
443.8392 
21.1543 
498.4874 
20.6470 
560.2461 
Global 
23.7380 
274.9663 
22.3634 
377.3468 
21.4703 
463.4987 
20.7769 
543.7342 
VisuShrink 
23.5279 
288.5945 
22.1107 
399.9587 
20.9619 
521.0651 
19.8194 
677.8555 
NormalShrink 
21.2498 
487.6367 
19.4918 
730.9712 
18.1446 
996.8226 
17.0462 
1283.7 
Proposed 
25.2100 
195.9225 
24.0410 
256.4368 
22.8648 
336.1987 
21.9228 
417.6383 
Table 4. Performance of the denoising methods of test image 3

σ=20 
σ=30 
σ=40 
σ=50 

PSNR 
MSE 
PSNR 
MSE 
PSNR 
MSE 
PSNR 
MSE 

Default 
26.3191 
151.7644 
25.3897 
187.9804 
24.4911 
231.1900 
23.6029 
283.6518 
Global 
26.8908 
133.0455 
25.5634 
180.6076 
24.5359 
228.8173 
23.6234 
282.3221 
VisuShrink 
26.7652 
136.9487 
25.4722 
184.4412 
24.2164 
246.2874 
22.8114 
340.3652 
NormalShrink 
22.7158 
347.9391 
20.1216 
632.2898 
18.0501 
1018.8 
16.3904 
1492.9 
Proposed 
28.2714 
96.8151 
26.5885 
142.6348 
25.0894 
201.4389 
23.8805 
266.0938 
Table 5. Performance of the denoising methods of test image 4

σ=20 
σ=30 
σ=40 
σ=50 

PSNR 
MSE 
PSNR 
MSE 
PSNR 
MSE 
PSNR 
MSE 

Default 
25.0348 
203.9861 
24.1379 
250.7809 
23.3943 
297.6149 
22.6556 
352.7934 
Global 
26.0216 
162.5247 
24.5782 
226.5980 
23.5312 
288.3765 
22.7067 
348.6705 
VisuShrink 
25.6977 
175.1105 
24.3006 
241.5551 
23.2572 
307.1541 
22.2822 
384.4661 
NormalShrink 
23.2568 
307.1829 
21.2546 
487.1041 
19.5947 
713.8501 
18.1831 
988.0361 
Proposed 
27.2681 
121.9736 
25.6948 
175.2277 
24.3248 
240.2150 
23.1729 
313.1809 
From the data in the Tables we can know that, for a same image, the results obtained by the proposed method were better than those of the other methods. For example, when white noise with a variance of 20 was added to test image 1, the PSNR result obtained by the proposed method was 29.4964 dB, which was increased by 2.7875 dB, 2.0893 dB, 2.5065 dB and 5.1303 dB respectively compared with the results obtained by Default, Global, VisuShrink and NormalShrink algorithms. In addition, the MSE result obtained by the proposed method was 73.0196, which was decreased by 65.7175, 45.1140, 57.0239 and 164.9205 respectively compared with the other four methods.
It was also found that the proposed method also outperformed the other four methods in different images. For instance, when white noise with a variance of 50 was added, compared with the other four methods, the PSNR result obtained by the proposed method increased by 0.5628 dB, 0.5079 dB, 1.4764 dB and 4.3391 dB, respectively.
In the second test, noises with different intensities (σ=15, 20, 25, 30) were added to the four test images, which were subject to denoising tests, and the test results are shown in Figure 2. The images in the first row are results obtained by the Default method; the images in the second row are results obtained by the Globa method; the images in the third row are results obtained by the VisuShrink method; the images in the fourth row are results obtained by the NormalShrink method; and the images in the fifth row are results obtained by the proposed method. The results showed that the images in the first four rows are relatively blurred, the edges of feature objects such as traffic signs and vehicles are not prominent, and there’s obvious ringing artifacts. The proposed method outperformed the other four methods in the visual effect of the denoised images, and it can also retain the edge features of the images.
In the final test, the time complexity of the five algorithms was compared and the test results are shown in Figure 3. From Figure 3(a) we can see that, when the noise level σ was 20 and 30, the time required by the proposed algorithm was slightly longer than that of the Global method and the NormalShrink method, but the processing time of a single picture was only slightly more than 1s. When the noise level σ was greater than 40, the time required by the proposed algorithm was shorter than the other four methods, this might be related to the test images. From Figures 3(b), 3(c) and 3(d) we can know that, when the noise level σ was 2050, the time required by the proposed algorithm was the shortest and it achieved good effects. Through comprehensive analysis we can see that, the proposed method is capable of realtime processing.
The reason for the good performance of the proposed algorithm is that, for wavelet coefficient of multilayer decomposition, the more the layers, the smaller the noise energy becomes. Due to the adaptability of the threshold, the noise and useful information in each layer could be separated to the greatest extent. In addition, this paper selected the sym4 wavelet to decompose the images. The Sym4 wavelet has larger support width, and higher vanishing moment and regularity. Larger order of vanishing moment means the wavelet energy is concentrated, and higher regularity means that the twodimensional images are smoother after reconstruction.
Figure 2. Visual effects of denoising results of four test images
Note: From left to right are test images 1, 2, 3, and 4 (σ=20). The first row: Default method; the second row: Global method; the third row: VisuShrink method; the fourth row: NormalShrink method; the final row: the proposed method.
(a)
(b)
(c)
(d)
Figure 3. Processing time required by five denoising methods for the four test images
Note: From left to right (a to d): results of test images 1, 2, 3 and 4, respectively.
This paper proposed an improved adaptive threshold denoising method suitable for WMSN image denoising. According to the different subband characteristics after the images were subject to wavelet decomposition, the scale parameter equation was defined to determine the adaptive optimal threshold suitable for each scale level; then combining with the proposed derivable threshold function and the optimal wavelet basis function selected by comparison, the images were subject to denoising processing. The proposed method was improved based the Donoho threshold denoising method, it could generate different thresholds suitable for each scale level and process different noise image signals. The test results showed that, the proposed method outperformed the classical methods in terms of visual effect and denoising performance, the method does not add time complexity and can be used for realtime processing.
This paper is supported by the Scientist Research Fund of Hunan Provincial Education Department under Grant No. 19B105, and the Scientist Research Project of Yiyang Technology Bureau under Grants No. YKZ(2016)51 & 2017YC01. The author would like to thank the anonymous reviewers for their insightful comments and suggestions, which have greatly improved this paper.
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