Analysis and application of modified methods of wavelet threshold functions

In order to improve the effects of denoising, this paper introduces the basic principles of wavelet threshold denoising and traditional structures threshold functions. Meanwhile, it proposes wavelet threshold function and fixed threshold formula which are both improved here. First, this paper studies the problems existing in the traditional wavelet threshold functions and introduces the adjustment factors to construct the new threshold function basis on soft threshold function. Then, it studies the fixed threshold and introduces the logarithmic function of layer number of wavelet decomposition to design the new fixed threshold formula. Finally, this paper uses hard threshold, soft threshold, Garrote threshold, and improved threshold function to denoise different signals. And the paper also calculates signal-to-noise (SNR) and mean square errors (MSE) of the hard threshold functions, soft thresholding functions, Garrote threshold functions, and the improved threshold function after denoising. Theoretical analysis and experimental results showed that the proposed approach could improve soft threshold functions with constant deviation and hard threshold with discontinuous function problems. The proposed approach could improve the different decomposition scales that adopt the same threshold value to deal with the noise problems, also effectively filter the noise in the signals, and improve the SNR and reduce the MSE of output signals.

The relevant properties of the selected classical threshold function and selected threshold were compared and analyzed for the methods of noisy-image wavelet threshold de-noising. On the basis of this, a new wavelet threshold de-noising function was given in this paper. The new defined threshold function overcomes the shortcoming which the hard threshold function and the soft threshold function on higher derivatives are discontinuous by adding a variable. Theoretical analysis and experimental results show that, the constructed wavelet threshold de-noising function possess the better adaptability and de-noising effect. In the case of strong Gauss noise for the image, relatively soft threshold method, the PSNR of a new threshold de-noising method can be improved from 4dB to 6dB.

Using wavelet analysis for non-stationary signal de-noising of electro-mechanical system is considered to be the best approach, and wavelet threshold de-noising method is the most simple method that needs the minimum amount of calculation. But this method in selecting threshold functions needs to be improved. Based on different domestic and foreign methods of improving threshold function, propose an improved bivariate threshold function. According to the simulation of non-stationary signal de-noising, the results show that the optimal de-noising results of different signals exist by taking different parameters. Compared with all the de-noising effects, application of the bivariate threshold function considering signal-to-noise ratio and mean square error is superior to the traditional soft and hard threshold functions. At the same time, it can significantly improve the filtering precision, and reserve the main signal details while effectively removing the noise well.

In the wavelet packet threshold denoising algorithm, retaining as many of the original signal wavelet packet coefficients as possible is necessary while eliminating noise. Due to the lack of an adaptive parameter, the traditional soft and hard threshold functions cannot be adjusted adaptively, and the useful signal is always mixed with noise, which leads to insecurity of the Internet of Things (IoT) application. Therefore, this paper discusses the significance and feasibility of sample entropy as an adaptive parameter in the wavelet packet threshold function. Furthermore, a new threshold function is proposed to improve the traditional wavelet packet denoising algorithm. The algorithm can process the wavelet packet coefficients differently for different noise backgrounds to achieve better denoising effect, thereby improving the security of the IoT. The simulation results show that compared with the classical wavelet packet threshold function, the proposed improved algorithm achieves better denoising effect and is beneficial to enhance the security of IoT applications.

Acoustic emission (AE) detection, as a non-electrical detection method, is very suitable for effective fault detection of power equipment with a strong electromagnetic field. However, the AE signal collected at industrial sites often contains a lot of interference noise, affecting the analysis and prediction of faults. In this study, a wavelet denoising method based on a new threshold function is proposed, to achieve a noise reduction in the low signal-to-noise ratio (SNR) signals. Simulation experiment results show that the proposed threshold function not only overcomes the shortcomings of the discontinuous hard threshold function, but also solves the constant deviation of the soft threshold function. What’s more, the proposed function achieves a good adaptability. When SNR = 10 dB: The SNR of the new threshold is 20.6622, and the RMSE is 0.0026. The SNR of the hard threshold is 20.2246 and the RMSE is 0.0027, compared with the traditional hard threshold method, the SNR of the new threshold is increased by 2.16% and the root mean square error (RMSE) is reduced by 3.7%; the SNR of the soft threshold is 15.5656, and the RMSE is 0.0047, compared with the traditional soft threshold method, the new threshold has a 32.74% increase in SNR and a 40.43% reduction in RMSE. When SNR = -10 dB: The SNR of the new threshold is 4.2602, and the RMSE is 0.0172. The SNR of the hard threshold is 3.8558 and the RMSE is 0.0182, compared with the traditional hard threshold method, the SNR of the new threshold is increased by 10.49% and the RMSE is reduced by 5.49%; the SNR of the soft threshold is 2.1625, and the RMSE is 0.0212, compared with the soft threshold method, SNR is improved by 97% and RMSE is reduced by 18.87%. Performance analyses have proved that the improved wavelet denoising method can obtain a good noise reduction effect. It is very helpful for AE signal analysis with the generally low SNR, which can improve the accuracy of failure identification in subsequent acts.

In order to overcome the discontinuance of the hard thresholding function and the defect of seriously slashing singularity in the soft thresholding function, improve the denoising effect and detect the transformer partial discharge signal more accurately, in this paper an improved wavelet threshold denoising method is put forward through analyzing the interference noise of transformer partial discharge signals and studying various wavelet threshold denoising method, especially the wavelet threshold denoising method that overcomes the shortcomings of the hard and soft threshold. Simulation results show that the denoising effect of this method has been greatly improved than the traditional hard and soft threshold method. This method can be widely used in practical transformer partial discharge signal denoising.

Unlike inflexible structure of soft and hard threshold function, a unified linear matrix form with flexible structure for threshold function is proposed. Based on the unified linear flexible structure threshold function, both supervised and unsupervised subband adaptive denoising frameworks are established. To determine flexible coefficients, a direct mean-square error (MSE) minimization is conducted in supervised denoising while Stein’s unbiased risk estimate as a MSE estimate is minimized in unsupervised denoising. The SURE rule requires no hypotheses or a priori knowledge about clean signals. Furthermore, we discuss conditions to obtain optimal coefficients for both supervised and unsupervised subband adaptive denoising frameworks. Applying an Odd-Term Reserving Polynomial (OTRP) function as concrete threshold function, simulations for polynomial order, denoising performance, and noise effect are conducted. Proper polynomial order and noise effect are analyzed. Both proposed methods are compared with soft and hard based denoising technologies—VisuShrink, SureShrink, MiniMaxShrink, and BayesShrink—in denoising performance simulation. Results show that the proposed approaches perform better in both MSE and signal-to-noise ratio (SNR) sense.

In this paper, we propose an image denoising method based on nonsubsampled contourlet transform (NSCT) with successive quadratic programming (SQP) optimization. This method can obtain the optimal threshold for each subband without the priori information of the noise variance using SQP optimization and generalized cross validation (GCV) criterion. After the threshold is determined, a nonlinear threshold function is applied to overcome the inadequate of soft threshold and hard threshold function. The experimental results show that the proposed method has a better performance than other contourlet-based image denoising methods and outperforms on both visual quality and peak signal-to-noise ratio (PSNR).

Based on multi-resolution analysis of wavelet, this article is aimed at building a new soft threshold function for wavelet de-noising, to overcome the discontinuous disadvantage of the hard threshold function. In the area adjacent to the threshold, continuously adjustable nonlinear functions are introduced in piecewise to process the wavelet coefficients more carefully in this area. Consequently, large deviation caused by super-compression of wavelet coefficients when using traditional soft threshold could be avoided, and the nonlinearity of the system is able to be effectively kept. Using both signal to noise ratio (SNR) and mean square error (MSE) as the evaluation indicators, simulation results show that the improved method is more effective than the method based on traditional hard and soft threshold.

Broiler sound signal filtering method based on improved wavelet denoising and effective pulse extraction

ABSTRACTModulation recognition in underwater acoustic (UWA) signals is challenging due to the intricate marine environment and substantial underwater noise. Wavelet‐based denoising lacks adaptivity and can be affected by the wavelet function, the number of decomposition layers, and the threshold function. Although the denoising method based on deep learning has achieved a good denoising effect, it fails to integrate with the physical model and lacks certain theoretical support. To address these problems, this paper proposes a deep fusion network for signal denoising, named WDNet, based on wavelet denoising theory and deep learning techniques. We initialize the tap coefficients of the wavelet decomposition and reconstruction filters as learnable parameter matrices and use the soft threshold function as the activation function so as to realize the decomposition, thresholding, and reconstruction of the signal. The filter and threshold are adjusted adaptively by backpropagation to achieve optimal signal denoising. Simulation results demonstrate that our model achieves a higher signal‐to‐noise ratio (SNR) gain and lower root mean square error (RMSE) compared to other methods. After denoising, the recognition rate of UWA modulation signals significantly improves.

In view of the wheel state monitoring, the wheel vibration signal is often used to reflect the wheel state information, but the original vibration signal contains a lot of noise. In order to extract the effective wheel information, it is necessary to reduce the noise of the original vibration signal. The problems existing in the traditional wavelet threshold denoising method include unreasonable threshold setting, lack of continuity of hard threshold function and fixed deviation of wavelet coefficients processed by soft threshold function. In this paper, the traditional wavelet threshold denoising method is improved from threshold selection and threshold function, and an improved wavelet threshold denoising method is proposed. This method is used to reduce the noise of the measured wheelset vibration signal and compared with the traditional wavelet denoising method from evaluation indexes and graphical results. The results show that the improved wavelet threshold denoising method has better noise reduction effect.

The measurement of downhole engineering parameters is greatly disturbed by the working environment. Effective de-noising methods are required for processing logging-while-drilling (LWD) acquisition signals, in order to obtain downhole engineering parameters accurately and effectively. In this paper, a new de-noising method for measuring downhole engineering parameters was presented, based on a feedback method and a wavelet transform threshold function. Firstly, in view of the mutability and density of downhole engineering data, an improved wavelet threshold function was proposed to de-noise the signal, so as to overcome the shortcomings of data oscillation and deviation caused by the traditional threshold function. Secondly, due to the unknown true value, traditional single denoising effect evaluation cannot meet the requirements of quality evaluation very well. So the root mean square error (RMSE), signal-to-noise ratio (SNR), smoothness (R) and fusion indexs (F) are used as the evaluation parameters of the de-noising effect, which can determine the optimal wavelet decomposition scale and the best wavelet basis. Finally, the proposed method was verified based on the measured downhole data. The experimental results showed that the improved wavelet de-noising method could reduce all kinds of interferences in the LWD signal, providing reliable measurement for analyzing the working status of the drilling bit.

In this article, we developed and analyzed a thresholding method in which soft thresholding estimators are independently expanded by empirical scaling values. The scaling values have a common hyper-parameter that is an order of expansion of an ideal scaling value that achieves hard thresholding. We simply call this estimator a scaled soft thresholding estimator. The scaled soft thresholding is a general method that includes the soft thresholding and non-negative garrote as special cases and gives an another derivation of adaptive LASSO. We then derived the degree of freedom of the scaled soft thresholding by means of the Stein's unbiased risk estimate and found that it is decomposed into the degree of freedom of soft thresholding and the reminder connecting to hard thresholding. In this meaning, the scaled soft thresholding gives a natural bridge between soft and hard thresholding methods. Since the degree of freedom represents the degree of over-fitting, this result implies that there are two sources of over-fitting in the scaled soft thresholding. The first source originated from soft thresholding is determined by the number of un-removed coefficients and is a natural measure of the degree of over-fitting. We analyzed the second source in a particular case of the scaled soft thresholding by referring a known result for hard thresholding. We then found that, in a sparse, large sample and non-parametric setting, the second source is largely determined by coefficient estimates whose true values are zeros and has an influence on over-fitting when threshold levels are around noise levels in those coefficient estimates. In a simple numerical example, these theoretical implications has well explained the behavior of the degree of freedom. Moreover, based on the results here and some known facts, we explained the behaviors of risks of soft, hard and scaled soft thresholding methods.

Wavelet de-noising method is often used in the processing of magnetotelluric (MT) signal, and the wavelet hard and soft threshold de-noising method are the most popular although there is much room for improvement in the threshold function selection and threshold determination. A new adaptive wavelet threshold de-noising method was proposed by selecting a new threshold function and presenting an adaptive method for obtaining optimal threshold based on the multi-resolution Stein unbiased risk estimation. New threshold function and an adaptive method to determine threshold were discussed, and the principle and implementation of the algorithm were given. The simulated signal and the measured MT data contaminated by impulse interference were analyzed, and the obtained results were compared with those of the conventional wavelet hard and soft threshold de-noising methods. The results show that the proposed method overcomes the defects of the traditional wavelet soft and hard threshold due to a new threshold function, and a new method to determine the threshold of each layer is applied and provides an adaptive method for filtering MT data in the wavelet domain that requires a minimum of human intervention. The presented de-noising method is very suitable for suppressing the impulse interference for MT data and can get higher signal-to-noise ratio than the traditional wavelet threshold de-noising methods. After de-noising, the accurate data is loaded for further impedance estimation and geological interpretation.