Why wavelet denoising

The main source of random noise is the random interference during the sampling process of data collector, which superimposes a wideband random noise on the original VPVS [ 22 ].

The original VPVS has two main features: the dominant frequency and the main frequency range. According to the two main features, two original VPVS models are established; one is a sine signal with dominant frequency, and the other is sweep signal with the main frequency range.

The denoising method can be proved, if the two original VPVS models are retained relatively complete after the denoising method. The trend is associated with the inherent test conditions, which can be collected in the condition that the engine and generator are inoperative. The random noise has similar characteristics with Gaussian white noise and can be expressed by Gaussian white noise. In order to clarify the signal details clearly, the number of sampling points is selected as The original sine VPVS model is defined as follows: where is the dominant vibration frequency and is the amplitude of the original sine VPVS model, which is selected as 1.

The SNR is 4. The main vibration frequency range is as follows: where is the initial frequency, is the target frequency. The analysis of power spectral and energy distribution is essential to design the wavelet denoising algorithm flow.

The power spectral distribution of Gaussian white noise is uniform distribution [ 23 ]. The energy of each part is calculated as follows [ 24 ]: where is the energy of the wavelet coefficients in the decomposition level and includes the approximate coefficients and the detail coefficients , , and.

The energy distribution of each parts is shown in Figure 9. It shows that the energy of the original VPVS models mainly distributes in the approximate coefficients and the detail coefficients.

However, the energy of the VPVS models distributes in all the wavelet coefficients, which is influenced by the trend and Gaussian white noise. The energy of trend mainly distributes in the approximate coefficients. The energy of Gaussian white noise distributes in the all the wavelet coefficients and decreases with the increasing decomposition level.

In conclusion, the wavelet denoising algorithm flow can be divided into two parts: eliminating the influence of trend on the approximate coefficients and eliminating the influence of Gaussian white noise on the detail coefficients. Each part is decomposed and reconstructed, respectively, the detail coefficients of both parts are processed by unsupervised learning of TNN, the approximate coefficients of are reserved, and the approximate coefficients of are set to zero; the denoised signal is obtained through the superposition of denoised and.

Six denoising approaches are used for VPVS denoising. According to the theoretical background mentioned in Section 2. According to the multiresolution threshold method [ 25 ], the initial threshold of unsupervised learning is set as follows: where the noise variance estimation is calculated as follows:. In this paper, the thresholds of soft thresholding method and hard thresholding method are set as functions 19 and The initial threshold of TNN method is set as functions 19 and 20 , and the threshold is selected by functions 12 and In addition, the VPVS denoising flow is applied in proposed thresholding method.

According to Figure 11 , for time-domain analysis, the profile of the sine VPVS model denoised by the proposed thresholding method is closer to the profile of original sine VPVS model. The proposed denoising method is efficient for denoising sine VPVS polluted by the trend and Gaussian white noise.

According to Figure 12 , for time-domain analysis, the profile of the sweep VPVS model denoised by the proposed thresholding method is closer to the profile of original sweep VPVS model.

The proposed denoising method is efficient for denoising sweep VPVS polluted by the trend and Gaussian white noise. In conclusion, the proposed denoising method is efficient for denoising both sine and sweep VPVS model polluted by the trend and Gaussian white noise and performs better in the smoothness and integrity of both sine and sweep VPVS model.

The apparatus of vehicle platform vibration test is shown in Figure It consists of three main parts: signal measurement section, signal collection section, and vehicle equipment. The signal measurement section is the piezoelectric accelerometer of model A26PCB. The signal collection section includes a data collector and a laptop. The vehicle equipment includes a precise optical measuring instrument and a vehicle platform.

In the vehicle platform vibration test, the accelerometer is placed in the sensitive vibration position, and the sensitive axis of the accelerometer is perpendicular to the measuring plane. The power spectral analysis of the measured signal is shown in Figure All denoising methods mentioned above are applied to the measured signal; the result is shown in Figure However, the frequency component can be the evaluation criterion instead according to the characteristics of the VPVS.

Compared with the measured signal in Figure 15 , the signal denoised by the proposed thresholding method has some superior characteristics. In conclusion, compared with other thresholding methods, the proposed thresholding method performs better in both low-frequency part and high-frequency part and reserves the VPVS more completely.

The results show that the proposed thresholding method is efficient for denoising VPVS polluted by the trend and random noise.

In this paper, a VPVS denoising method is presented. In the future, the rapidness of the proposed VPVS denoising method should be considered, so that the method can be portable to hardware platform to achieve real-time denoising and larger applications. This is an open access article distributed under the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article of the Year Award: Outstanding research contributions of , as selected by our Chief Editors. Read the winning articles. Journal overview. Special Issues. The measured image y is, thus.

In the first step, discrete wavelet transform is applied to the noisy image y , to get the approximation and wavelet coefficients. In the second step, approximation and wavelet coefficients are denoised by using an adaptive dictionary learned on the set of extracted patches from wavelet representation of the noisy image, by the K-SVD as a dictionary learning algorithm.

According to 19 and the property of wavelet transform, we have:. By maximum a posteriori estimation, denoising of Wy is equivalent to the energy minimization problem:. The second and the third terms are the sparse prior of wavelet coefficients. When D is known, we can solve 23 by two steps. This step is called sparse coding. Returning to 23 , we need to solve:. This expression says the averaging of the denoised patches and is called patches averaging step.

Given the updated, we can repeat the sparse coding stage, working on the already denoised patches. Once this is done, a new averaging should be calculated, and so on, and so forth.

In practice, the dictionary D is unknown and we can get it by learning which is same to [ 23 ]. After fusion dictionary learning in 23 , we get the denoising model as follows:.

The following sequence defines our proposed algorithm:. In this section, we aim to demonstrate the advantages and the performance that our proposed wavelet denoising approach based on unsupervised learning model has. Our proposed algorithm is evaluated and compared with four representative and state-of-the-art denoising algorithms: the wavelet thresholding approach for image denoising, the sparse representation-based K-SVD denoising method in the image domain, the TV denoising method, and the BM3D denoising method.

The SSIM index is defined as:. The comparison of quantitative results is shown in Tables 1 and 2. A visual comparison is shown in Figs. The results show that the proposed method outperforms the three denoising methods: wavelet thresholding, TV, and K-SVD, in all cases. The statistical results of the five methods are also recorded in the SSIM measure as shown in Table 2. Based on these results, we can deduce that our proposed method is always better than the other denoising methods, which assures the efficiency of our algorithm.

Let us then focus on the visual quality evaluation of these denoising algorithms. The Figs. Under high noise levels, the results show that the edges are well preserved, the textures and more details are better restored, and least artifacts exist in the result of our proposed method. The performance of the wavelet thresholding approach, the K-SVD, and the TV denoising method is the worst in various noise levels; it is easy to see that the restored images contain too many artifacts, the image edges and flat areas are blurred, and a number of details and textures are lost.

It also found in the experiments that the proposed method has much better visual quality than the BM3D method that produces too many artificial ringing effects which are caused by stacking the image blocks and erroneous grouping.

The average time required by the other algorithms to complete their executions is given as follows: the wavelet thresholding approach for image denoising: 2 to 5 s, the sparse representation-based K-SVD denoising method in the image domain: 15 s to 2 min, the TV denoising method: 12 to 30 s, and the BM3D denoising method: 5 to 15 s. The proposed algorithm is slower than the other denoising methods, due to the use of the K-SVD as a dictionary learning algorithm that needs many iteration steps.

Our proposed method performs better than the state-of-the-art denoising methods, due to the merits of the wavelet transform and to the use of an adaptive dictionary devoted to noise reduction instead of using the thresholding operator. We propose in this work a new method for image denoising. The approach taken aims at exploiting the merits of the wavelet transform: sparsity, multi-resolution structure, similarity with the human visual system, and good localization properties both in space and frequency, to adapt an unsupervised dictionary learning algorithm for creating a dictionary devoted to eliminate the useless information while keeping most significant ones.

Experiments illustrate that the proposed method achieves a better performance than some other well-developed denoising methods, especially in PSNR, SSIM index, and visual effects. In future work, we will be focused more on reducing the computational cost of the proposed model. The benchmark used to demonstrate the effectiveness of the proposed approach is composed of five standard images used for image processing. These images are frequently found in literature. Google Scholar. Boyat, B. Joshi, A review paper: noise models in digital image processing.

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Mandadelis, A. Tzortzis, et al. I Piraeus Appl. IX 1 , 97— Grace Chang, B. Yu, M. Vattereli, Adaptive wavelet thresholding for image denoising and compression. IEEE Trans. Image Process. Donoho, I. Johnstone, Adapting to unknown smoothness via wavelet shrinkage. Donoho, De-noising by soft-thresholding. Gupta, M. Ahmad, Brain MR image denoising based on wavelet transform.

Fedak, A. Nakonechnyy, in Technical Transaction on Electrical Engineering. Kozintsev, K. Ramchandran, P Moulin, Low-complexity image denoising based on statistical modeling of wavelet coefficients. Chang, B. Vetterli, Spatially adaptive wavelet thresholding with context modeling for image denoising. Pizurica, W. Philips, I. Lamachieu, et al. Donoho, Wedgelets: nearly minimax estimation of edges. Candes, D. Donoho, Recovering edges in ill-posed inverse problems: optimality of curvelet frames.

Donoho, New tight frames of curvelets and the problem of approximating piecewise C2 image with piecewise C2 edges. Pure Appl. Pennec, S. Gaussian noise tends to be represented by small values in the wavelet domain and can be removed by setting coefficients below a given threshold to zero hard thresholding or shrinking all coefficients toward zero by a given amount soft thresholding.

In this example, we illustrate two different methods for wavelet coefficient threshold selection: BayesShrink and VisuShrink. The VisuShrink approach employs a single, universal threshold to all wavelet detail coefficients. This threshold is designed to remove additive Gaussian noise with high probability, which tends to result in overly smooth image appearance.

You can also denoise the signal using the undecimated wavelet transform. Denoise the signal again down to level 4 using the undecimated wavelet transform. You see that in both cases, wavelet denoising has removed a considerable amount of the noise while preserving the sharp features in the signal. This is a challenge for Fourier-based denoising. In Fourier-based denoising, or filtering, you apply a lowpass filter to remove the noise. However, when the data has high-frequency features such as spikes in a signal or edges in an image, the lowpass filter smooths these out.

You can also use wavelets to denoise signals in which the noise is nonuniform. Import and examine a portion of a signal showing electricity consumption over time. The signal appears to have more noise after approximately sample Accordingly, you want to use different thresholding in the initial part of the signal.

You can use cmddenoise to determine the optimal number of intervals to denoise and denoise the signal. In this example, use the 'db3' wavelet and decompose the data down to level 3. Two intervals were identified.