Classical Wavelet Research Articles

An Ultrasonic Signal Denoising Method for EMU Wheel Trackside Fault Diagnosis System Based on Improved Threshold Function

In the safety protection system of the railway electric multiple unit (EMU), the safety of the running part is extremely important. The daily detection of the internal hazard defects of the wheels in the running parts relies on a professional trackside fault online diagnosis system based on the ultrasonic sensor probe array data. However, the on-line ultrasonic diagnosis of EMU wheels is usually accompanied by various interference noises. The defect echo signals collected by the sensor probe array are weak and are easily submerged by noise, which makes it impossible to perform effective defect identification. This paper proposes an improved threshold function to overcome the discontinuous shortcomings of the classical wavelet soft threshold function and hard threshold function in view of the non-stationary characteristics of the ultrasonic detection signal of the EMU wheels. This paper proposes a sine-type threshold processing function. It is characterized by adopting gradual compression processing to denoise the echo signal of the ultrasonic sensor probe array. In order to verify the validity, the continuity at the threshold is observed through the linear space vector signal, and the algorithm is simulated and tested through the three-dimensional Gaussian echo mathematical model of the ultrasonic signal and the measured ultrasonic envelope signal. Experimental results show that the improved threshold function can suppress the noise in the ultrasonic echo data, improve the signal-to-noise ratio, and retain the waveform characteristics of the defect signal, which is conducive to defect recognition.

Hyperspectral Image Super-Resolution via Deep Progressive Zero-Centric Residual Learning.

This paper explores the problem of hyperspectral image (HSI) super-resolution that merges a low resolution HSI (LR-HSI) and a high resolution multispectral image (HR-MSI). The cross-modality distribution of the spatial and spectral information makes the problem challenging. Inspired by the classic wavelet decomposition-based image fusion, we propose a novel lightweight deep neural network-based framework, namely progressive zero-centric residual network (PZRes-Net), to address this problem efficiently and effectively. Specifically, PZRes-Net learns a high resolution and zero-centric residual image, which contains high-frequency spatial details of the scene across all spectral bands, from both inputs in a progressive fashion along the spectral dimension. And the resulting residual image is then superimposed onto the up-sampled LR-HSI in a mean-value invariant manner, leading to a coarse HR-HSI, which is further refined by exploring the coherence across all spectral bands simultaneously. To learn the residual image efficiently and effectively, we employ spectral-spatial separable convolution with dense connections. In addition, we propose zero-mean normalization implemented on the feature maps of each layer to realize the zero-mean characteristic of the residual image. Extensive experiments over both real and synthetic benchmark datasets demonstrate that our PZRes-Net outperforms state-of-the-art methods to a significant extent in terms of both 4 quantitative metrics and visual quality, e.g., our PZRes-Net improves the PSNR more than 3dB, while saving 2.3× parameters and consuming 15× less FLOPs. The code is publicly available at https://github.com/zbzhzhy/PZRes-Net.

Unsteady flow and heat transfer of tangent‐hyperbolic fluid: Legendre wavelet‐based analysis

AbstractThe objective of the current article is to explore the unsteady flow and heat transfer of magnetohydrodynamics tangent‐hyperbolic fluid flow over a stretching sheet. The governing flow model is transformed into a nonlinear set of ordinary differential equations by utilizing the appropriate similarity techniques. A new modification is introduced into the traditional Legendre wavelet method to obtain the results of the model mentioned above. The classic wavelet scheme is unable to find the solution for an infinite domain. Hence, we successfully extended it for an infinite domain and used it to attain the significant findings of the fluid problem. Additionally, the study of emerging parameters on temperature and velocity profiles is reported graphically. The velocity behavior is decreasing for the physical parameters, namely, power‐lax index, unsteadiness, Hartmann number, and Weissenberg number. The temperature profile is an increasing function for power‐law index and Eckert number while the behavior is the opposite for the Prandtl number. Moreover, a tabular form comparison of outcomes with existing literature, convergence, and error analysis is provided in our study, which confirms the credibility of the suggested method. The obtained results endorse the credibility and reliability of the proposed method; therefore, it could be extended for other nonlinear problems of complex nature.

Fractal geometry of wavelet decomposition in mechanical signature analysis

Multiple modes of vibration are usually incorporated in a single record of vibration measurement in condition monitoring of rotating machinery. Wavelet transform is an effective tool to detect and isolate transient fault features from other interfering modes. The conventional dyadic wavelet transform decomposes the signal into wavelet subspaces with distinct central frequencies and specific frequency bandwidths. In this paper, we propose a novel theory of centralized multiresolution analysis (CMR) and reveal the implicit fractal geometry properties in CMR. A concept of nested centralized wavelet packet space (NCWPS) is introduced to describe the self-similarity phenomenon in CMR. Within the theoretical framework, the classical dyadic wavelet packet is assimilated as a subordinated proper-subset of the augmented NCWPS. Moreover, the generalized CMR characterized by tunable and flexible frequency-scale topology configuration is established using harmonic wavelet transform. The CMR can be regarded as an improved transient signature dictionary. Therefore, the CMR is combined with an improved stationary signature dictionary to ensure enhanced performance in fault feature extraction in multiple modes coupled vibration measurements. The effectiveness of the proposed method is validated using numerical simulations, a rub-impact experiment, and a case study of vibration signal analysis in steel making industry.

Green\u2013Haar wavelets method for generalized fractional differential equations

The objective of this paper is to present two numerical techniques for solving generalized fractional differential equations. We develop Haar wavelets operational matrices to approximate the solution of generalized Caputo–Katugampola fractional differential equations. Moreover, we introduce Green–Haar approach for a family of generalized fractional boundary value problems and compare the method with the classical Haar wavelets technique. In the context of error analysis, an upper bound for error is established to show the convergence of the method. Results of numerical experiments have been documented in a tabular and graphical format to elaborate the accuracy and efficiency of addressed methods. Further, we conclude that accuracy-wise Green–Haar approach is better than the conventional Haar wavelets approach as it takes less computational time compared to the Haar wavelet method.

Wavelet Element Modelling for Inviscid Fluid-Solid Coupling Problem based on Partitioned Approach.

To provide a simple numerical formulation based on fixed grids, a wavelet element method for fluid–solid modelling is introduced in this work. Compared with the classical wavelet finite element method, the presented method can potentially handle more complex shapes. Considering the differences between the solid and fluid regions, a damping-like interface based on wavelet elements is designed, in order to ensure consistency between the two parts. The inner regions are constructed with the same wavelet function in space. In the time and spatial domains, a partitioned approach based on Jacobi iteration is combined with the pseudo-parallel calculation method. Numerical convergence analyses show that the method can serve as an alternative choice for fluid–solid coupling modelling.

An approximate wavelets solution to the class of variational problems with fractional order

In the present work, a generalized fractional integral operational matrix is derived by using classical Legendre wavelets. Then, a numerical scheme based on this operational matrix and Lagrange multipliers is proposed for solving variational problems with fractional order. This approach has been applied on some illustrative examples. The results obtained for these examples demonstrate that the suggested technique is efficient for solving variational problems with fractional order and gives a very perfect agreement with the exact solution. The results are depicted in graphical maps and data tables. The integral square error, maximum absolute error, and order of convergence have been evaluated to analyze the precision of the suggested method. The present scheme provides better and comparable results with some other existing approaches available in the literature.

Shifted Cubic Spline Wavelets with Two Vanishing Moments on the Interval and a Splitting Algorithm

This paper deals with the use of the first two vanishing moments for constructing cubic spline-wavelets orthogonal to polynomials of the first degree. A decrease in the supports of these wavelets is shown in comparison with the classical semiorthogonal wavelets. For splines with homogeneous Dirichlet boundary conditions of the second order, an algorithm of the shifted wavelet transform is obtained in the form of a solution of a tridiagonal system of linear equations with a strict diagonal dominance

Continuous Empirical Wavelets Systems

The recently proposed empirical wavelet transform was based on a particular type of filter. In this paper, we aim to propose a general framework for the construction of empirical wavelet systems in the continuous case. We define a well-suited formalism and then investigate some general properties of empirical wavelet systems. In particular, we provide some sufficient conditions to the existence of a reconstruction formula. In the second part of the paper, we propose the construction of empirical wavelet systems based on some classic mother wavelets.

A novel method to process surface electromyography signal for pedestrian lower limb motion pattern recognition

Surface electromyography (sEMG) signals mainly contain power line interference (PLI), white Gaussian noise (WGN), and baseline wandering (BW) noise. These noises lead to the problems of poor feature extraction performance and low recognition rate. In this paper, we propose a novel sEMG signal processing method named filtering and self-enhancement algorithm with classical wavelet (FSECW) algorithm, which denoises interference noises of raw signals and improve the pedestrian motion pattern recognition rate. The proposed FSECW algorithm contains two core parts: in the first part, the original signal is reconstructed after four-layer wavelet decomposition. This part decreases the BW noise and enhances the active segment; in the other part, band-pass filtering and lifting wavelet transformation are used to reduce noises inside and outside the sEMG signal band. Then two signals from the above parts are multiplied. Thus, the enhanced filtered signal of the active segment is obtained. After feature extraction, the algorithm uses classical machine learning algorithm for motion pattern recognition. Experimental results show that the proposed FSECW algorithm does not need to set different thresholds for different data sets with the same motion pattern. Moreover, it has better adaptability to conversions of different motion patterns.

Asymptotics of the Roots of Bernstein Polynomials Used in the Construction of Modified Daubechies Wavelets

This paper is devoted to the study of the asymptotics of roots of a sequence of Bernstein polynomials approximating a piecewise linear function. This sequence arises in the construction of modified compactly supported wavelets that, in contrast to classical Daubechies wavelets, preserve localization with the growth of smoothness. It is proved that the limiting curve for roots is the boundary of the domain of convergence of the Bernstein polynomials on the complex plane.

Learning traffic as a graph: A gated graph wavelet recurrent neural network for network-scale traffic prediction

Network-wide traffic forecasting is a critical component of modern intelligent transportation systems for urban traffic management and control. With the rise of artificial intelligence, many recent studies attempted to use deep neural networks to extract comprehensive features from traffic networks to enhance prediction performance, given the volume and variety of traffic data has been greatly increased. Considering that traffic status on a road segment is highly influenced by the upstream/downstream segments and nearby bottlenecks in the traffic network, extracting well-localized features from these neighboring segments is essential for a traffic prediction model. Although the convolution neural network or graph convolution neural network has been adopted to learn localized features from the complex geometric or topological structure of traffic networks, the lack of flexibility in the local-feature extraction process is still a big issue. Classical wavelet transform can detect sudden changes and peaks in temporal signals. Analogously, when extending to the graph/spectral domain, graph wavelet can concentrate more on key vertices in the graph and discriminatively extract localized features. In this study, to capture the complex spatial-temporal dependencies in network-wide traffic data, we learn the traffic network as a graph and propose a graph wavelet gated recurrent (GWGR) neural network. The graph wavelet is incorporated as a key component for extracting spatial features in the proposed model. A gated recurrent structure is employed to learn temporal dependencies in the sequence data. Comparing to baseline models, the proposed model can achieve state-of-the-art prediction performance and training efficiency on two real-world datasets. In addition, experiments show that the sparsity of graph wavelet weight matrices greatly increases the interpretability of GWGR.

Heart Sound Segmentation Using Bidirectional LSTMs With Attention.

This paper proposes a novel framework for the segmentation of phonocardiogram (PCG) signals into heart states, exploiting the temporal evolution of the PCG as well as considering the salient information that it provides for the detection of the heart state. We propose the use of recurrent neural networks and exploit recent advancements in attention based learning to segment the PCG signal. This allows the network to identify the most salient aspects of the signal and disregard uninformative information. The proposed method attains state-of-the-art performance on multiple benchmarks including both human and animal heart recordings. Furthermore, we empirically analyse different feature combinations including envelop features, wavelet and Mel Frequency Cepstral Coefficients (MFCC), and provide quantitative measurements that explore the importance of different features in the proposed approach. We demonstrate that a recurrent neural network coupled with attention mechanisms can effectively learn from irregular and noisy PCG recordings. Our analysis of different feature combinations shows that MFCC features and their derivatives offer the best performance compared to classical wavelet and envelop features. Heart sound segmentation is a crucial pre-processing step for many diagnostic applications. The proposed method provides a cost effective alternative to labour extensive manual segmentation, and provides a more accurate segmentation than existing methods. As such, it can improve the performance of further analysis including the detection of murmurs and ejection clicks. The proposed method is also applicable for detection and segmentation of other one dimensional biomedical signals.

Infrared and visible image fusion based on oversampled graph filter banks

The infrared image (RI) and visible image (VI) fusion method merges complementary information from the infrared and visible imaging sensors to provide an effective way for understanding the scene. The graph filter bank-based graph wavelet transform possesses the advantages of the classic wavelet filter bank and graph representation of a signal. Therefore, we propose an RI and VI fusion method based on oversampled graph filter banks. Specifically, we consider the source images as signals on the regular graph and decompose them into the multiscale representations with M-channel oversampled graph filter banks. Then, the fusion rule for the low-frequency subband is constructed using the modified local coefficient of variation and the bilateral filter. The fusion maps of detail subbands are formed using the standard deviation-based local properties. Finally, the fusion image is obtained by applying the inverse transform on the fusion subband coefficients. The experimental results on benchmark images show the potential of the proposed method in the image fusion applications.

Approximation of Functions by n-Separate Wavelets in the Spaces Lp(ℝ), 1 ≤ p ≤ ∞

We consider the orthonormal bases of n-separate MRAs and wavelets constructed by the author earlier. The classical wavelet basis of the space L2(ℝ) is formed by shifts and compressions of a single function ψ. In contrast to the classical case, we consider a basis of L2(ℝ) formed by shifts and compressions of n functions ψs, s = 1,...,n. The constructed n-separate wavelets form an orthonormal basis of L2(ℝ). In this case, the series $$\sum\nolimits_{s = 1}^n {\sum\nolimits_{j \in {\rm Z}} {\sum\nolimits_{k \in {\rm Z}} {f,\psi _{nj + s}^s >\psi _{nj + s}^s} } } $$ converges to the function f in the space L2(ℝ). We write additional constraints on the functions ϕs and ψs, s = 1,..., n, that provide the convergence of the series to the function f in the spaces Lp(ℝ), 1 ≤ p <- ∞, in the norm and almost everywhere.

Low complexity clutter removal in GPR images via lattice filters

A low complexity ground-penetrating radar (GPR) clutter removal method based on lattice filter (LF) decomposition is proposed. The GPR image is decomposed into several levels by LF where each level contains approximation and directional subbands. Due to its horizontal structure, the clutter is contained in a horizontal subband, while vertical and diagonal subbands represent the target information. Although the proposed approach seems similar to the classical undecimated wavelet decomposition, the whole process is accomplished in the spatial domain by simple mathematical operations and time delays, thus it has less complexity compared to the wavelet transform which needs the use of fast Fourier transform operations. Moreover, due to the recursive nature of the LFs, the proposed method is more robust to noise. It presents a better detection performance as indicated by the quantitative results. Our method is also compared with the conventional subspace–based methods for both simulated and real datasets, and visual and quantitative results validate the superiority of our proposed method.

Interpolatory filter banks and interpolatory wavelet packets

In classic wavelet sampling, research is generally focused on interpolatory bases in individual spaces. Consideration is seldom given to relations among different interpolatory bases, including those potentially related by filter banks. In this paper, we show that when an interpolatory basis exists in a space with a multiresolution analysis, this space can be decomposed orthogonally into two subspaces in which interpolatory bases also exist. Furthermore, similarly to standard orthogonal bases, these interpolatory bases can be related by pairs of filters, called interpolatory filter banks. For such interpolatory filter banks, it is shown that existence of interpolatory scaling functions generally leads to that of interpolatory wavelets and interpolatory wavelet packets. With this result, we further propose new algorithms for constructing such interpolatory wavelets/wavelet packets from corresponding scaling functions. In examples, our algorithms are applied to some typical wavelet spaces, demonstrating our theorems for interpolatory wavelets, interpolatory wavelet packets and interpolatory filter banks.

A Class of Warped Filter Bank Frames Tailored to Non-linear Frequency Scales

A method for constructing non-uniform filter banks is presented. Starting from a uniform system of translates, generated by a prototype filter, a non-uniform covering of the frequency axis is obtained by composition with a warping function. The warping function is a $${\mathcal {C}}^1$$-diffeomorphism that determines the frequency progression and can be chosen freely, apart from minor technical restrictions. The resulting functions are interpreted as filter frequency responses. Combined with appropriately chosen decimation factors, a non-uniform analysis filter bank is obtained. Classical Gabor and wavelet filter banks are special cases of the proposed construction. Beyond the state-of-the-art, we construct a filter bank adapted to a frequency scale derived from human auditory perception and families of filter banks that can be interpreted as an interpolation between linear (Gabor) and logarithmic (wavelet) frequency scales. We derive straightforward conditions on the prototype filter decay and the decimation factors, such that the resulting warped filter bank forms a frame. In particular, a simple and constructive method for obtaining tight frames with bandlimited filters is derived by invoking previous results on generalized shift-invariant systems.

A novel multiple impact factors based accuracy analysis approach for power quality disturbance detection

Nowadays, power quality problems are affecting people's daily life and production activities. With the aim to improve disturbance detection accuracy, a novel analysis approach based on multiple impact factors is proposed in this paper. First, a multiple impact factors analysis is implemented in which two perspectives, i.e., the wavelet analysis and disturbance features are considered simultaneously. Five key factors, including wavelet function, wavelet decomposition level, redundant algorithm, event type and disturbance intensity, start and end moment of disturbance, have been considered. Next, an impact factors based accuracy analysis algorithm is proposed, through which each factor's potential impact on disturbance location accuracy is investigated. Three transforms, i.e., the classic wavelet, lifting wavelet and redundant lifting wavelet are employed, and their superiority on disturbance location accuracy is investigated. Finally, simulations are conducted for verification. Through the proposed method, the wavelet based parameters can be validly selected in order to detect power quality disturbance accurately.

Novel Fractional Wavelet Packet Transform: Theory, Implementation, and Applications

The fractional wavelet transform (FRWT), which generalizes the classical wavelet transform and the well-known fractional Fourier transform, has recently been demonstrated as a powerful analytical tool for signal and image processing. However, this transform suffers from a relatively poor resolution in the high fractional frequency region, which results in difficulties in discriminating signals containing close high fractional frequency components. A simple but effective method to overcome this deficiency is the fractional wavelet packet transform (FRWPT). There exist several different definitions of the FRWPT in the literature. Unfortunately, these existing definitions do not generalize well the classical results for the conventional wavelet packet transform. The objective of this paper is to obtain a novel FRWPT that preserves the properties of its conventional counterpart. We first define the novel FRWPT and then discuss its related properties. Fractional wavelet packet subspaces are also constructed. Moreover, a recursive algorithm for implementing the proposed FRWPT is presented. Finally, we discuss potential applications of the proposed FRWPT.