Spline Wavelet Research Articles

Wavelet transform and deep learning-based obstructive sleep apnea detection from single-lead ECG signals.

Sleep apnea is a common sleep disorder. Traditional testing and diagnosis heavily rely on the expertise of physicians, as well as analysis and statistical interpretation of extensive sleep testing data, resulting in time-consuming and labor-intensive processes. To address the problems of complex feature extraction, data imbalance, and low model capacity, we proposed an automatic sleep apnea classification model (CA-EfficientNet) based on the wavelet transform, a lightweight neural network, and a coordinated attention mechanism. The signal is converted into a time-frequency image by wavelet transform and put into the proposed model for classification. The effects of input time window, wavelet transform type and data balancing on the classification performance are considered, and a cost-sensitive algorithm is introduced to more accurately distinguish between normal and abnormal breathing events. PhysioNet apnea ECG database was used for training and evaluation. The 3-min Frequency B-Spline wavelets transform of ECG signal was carried out, and Dice Loss was used to train the classification model of sleep breathing. The classification accuracy was 93.44%, sensitivity was 88.9%, specificity was 96.2% and most indexes were better than other related work.

High-Performance Position Control for Repetitive Tasks of Motor-Driven Servo Systems Based on Periodic Disturbance Observer

Repetitive position control of the motor-driven servo system (MDSS) plays an important role in numerous industrial processing and production. In this article, based on the Euler–Lagrange equation, a dynamics model was established by analyzing the relationship of the energy conversion to describe the motion of an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> -degree-of-freedom MDSS. For repetitive position control tasks of the considered system, a robust controller was designed to improve the performance of the closed-loop system, where the disturbance observer was employed to approximate the periodic dynamics of the system to improve both the transient and steady-state performances. The stability of the closed-loop system was analyzed based on the Lyapunov method. Moreover, to validate the control effect of the proposed method, the coordinated contouring task was considered for a dual-linear-motor-driven gantry system, where a B-spline wavelet series was employed as the periodic disturbance observer. The results of the comparative experiment show not only the effectiveness of the designed control scheme but also the superiority of the B-spline wavelet series on the approximation of periodic functions.

Quarklet characterizations for Triebel–Lizorkin spaces

In this paper we prove that under some conditions on the parameters the univariate Triebel–Lizorkin spaces Fr,qs(R) can be characterized in terms of quarklets. So for functions from Triebel–Lizorkin spaces we obtain a quarkonial decomposition as well as a new equivalent quasi-norm. For that purpose we use quarklets that are constructed by means of biorthogonal compactly supported Cohen–Daubechies–Feauveau spline wavelets, where the primal generator is a cardinal B-spline. Moreover we introduce some sequence spaces apposite to our quarklet system and study their properties. Finally we also obtain a quarklet characterization for the Triebel–Lizorkin–Morrey spaces Eu,r,qs(R).

Damage Identification in Reinforced Concrete Beams Using Wavelet Transform of Modal Excitation Responses

This study focuses on identifying damage in reinforced concrete (RC) beams using time-domain modal testing and wavelet analysis. A numerical model of an RC beam was used to generate various damage scenarios with different severities and locations. Acceleration time histories were recorded for both damaged and undamaged structures. Two damage indices, DI_MW and DI_SW, derived from the wavelet analysis, were employed to determine the location and severity of the damage. The results showed that different wavelet families and specific mother wavelets had varying effectiveness in detecting damage. The Daubechies wavelet family (db2, db6, and db9) detected damage at the center and sides of the RC beams due to good time and frequency localization. The Biorthogonal wavelet family (bior2.8 and bior3.1) provided improved time–frequency resolution. The Symlets wavelet family (sym2 and sym7) offered a balanced trade-off between time and frequency localization. The Shannon wavelet family (shan1-0.5 and shan1-0.1) exhibited good time localization, while the Frequency B-Spline wavelet family (fbsp2-1-0.1) excelled in frequency localization. Certain combinations of mother wavelets, such as shan1-0.5 with the DI_SW index, were highly effective in detecting damage. The DI_SW index outperformed DI_MW across different numerical models. Selecting appropriate wavelet analysis techniques, particularly utilizing shan1-0.5 in the DI_SW, proved effective for detecting damage in RC beams.

Peak Inelastic Displacement of Bilinear Systems in Support of Performance-Based Wind Design

A fundamental notion in building engineering is the equal displacement rule, which posits that the peak inelastic displacement of a system subjected to a ground motion excitation is approximately equal to the displacement of the same system responding elastically. The purpose of this study is to determine if the equal displacement rule can additionally be applied to wind excitations. To achieve this purpose, bilinear single-degree-of-freedom systems were subjected to B-spline wavelet excitations, Fejér–Korovkin wavelet excitations, and wind excitations derived from wind tunnel tests. The results showed the equal displacement rule generally held for excitations with neutral polarity. The frequency content of the excitation had a significant effect on the response because it shifted the location of the displacement-controlled region of the response spectrum. Duration had a mild effect for excitations with neutral polarity. The effect of stiffness and strength degradation due to gravity loads on the response was more pronounced for short-period structures. For regularly shaped buildings subjected to wind forces, the findings suggest that the equal displacement rule applies in the cross-wind direction however not in the along-wind direction.

Single-trial extraction of event-related potentials (ERPs) and classification of visual stimuli by ensemble use of discrete wavelet transform with Huffman coding and machine learning techniques

BackgroundPresentation of visual stimuli can induce changes in EEG signals that are typically detectable by averaging together data from multiple trials for individual participant analysis as well as for groups or conditions analysis of multiple participants. This study proposes a new method based on the discrete wavelet transform with Huffman coding and machine learning for single-trial analysis of evenal (ERPs) and classification of different visual events in the visual object detection task.MethodsEEG single trials are decomposed with discrete wavelet transform (DWT) up to the 4{th} level of decomposition using a biorthogonal B-spline wavelet. The coefficients of DWT in each trial are thresholded to discard sparse wavelet coefficients, while the quality of the signal is well maintained. The remaining optimum coefficients in each trial are encoded into bitstreams using Huffman coding, and the codewords are represented as a feature of the ERP signal. The performance of this method is tested with real visual ERPs of sixty-eight subjects.ResultsThe proposed method significantly discards the spontaneous EEG activity, extracts the single-trial visual ERPs, represents the ERP waveform into a compact bitstream as a feature, and achieves promising results in classifying the visual objects with classification performance metrics: accuracies 93.60pm {6.5}, sensitivities 93.55pm {4.5}, specificities 94.85pm {4.2}, precisions 92.50pm {5.5}, and area under the curve (AUC) 0.93pm {0.3} using SVM and k-NN machine learning classifiers.ConclusionThe proposed method suggests that the joint use of discrete wavelet transform (DWT) with Huffman coding has the potential to efficiently extract ERPs from background EEG for studying evoked responses in single-trial ERPs and classifying visual stimuli. The proposed approach has O(N) time complexity and could be implemented in real-time systems, such as the brain-computer interface (BCI), where fast detection of mental events is desired to smoothly operate a machine with minds.

Robust Control of Dual-Linear-Motor-Driven Gantry Stage for Coordinated Contouring Tasks Based on Feed Velocity

Contouring control is one of the most important tasks for the dual-linear-motor-driven gantry system (DLMDGS), whose control performance determines the in various precision machining equipment, mainly industrial machine tools. Nevertheless, there are some limitations in present results, including the insufficient consideration of system uncertainty and the approximation errors introduced by the computation of the path length. To overcome these issues, the effects of the rotational dynamic and uncertainties are studied and further refined in this article, and a robust control scheme with adaptive neural network identification is designed based on the desired contouring velocity. The method improves existing approaches and avoids the imperfection of the previous one, based on tracking control under global task coordinate frame. The effects of various environmental forces are considered and the B-spline wavelet neural network is employed to approximate unknown dynamic caused by the thrust ripples of the three motors. Some experiments on a real system verify the effectiveness and superiority of the proposed velocity-based contouring control method.

The Application of Splines of the Seventh Order Approximation to the Solution of Integral Fredholm Equations

There are various numerical methods for solving integral equations. Among the new numerical methods, methods based on splines and spline wavelets should be noted. Local interpolation splines of a low order of approximation have proved themselves well in solving differential and integral equations. In this paper, we consider the construction of a numerical solution to the Fredholm integral equation of the second kind using spline approximations of the seventh order of approximation. The support of the basis spline of the seventh order of approximation occupies seven grid intervals. We apply various modifications of the basis splines of the seventh order of approximation at the beginning, the middle, and at the end of the integration interval. It is assumed that the solution of the integral equation is sufficiently smooth. The advantages of using splines of the seventh order of approximation include the use of a small number of grid nodes to achieve the required error of approximation. Numerical examples of the application of spline approximations of the seventh order for solving integral equations are given.

Diagnosis of Covid-19 using Fractional B-spline Wavelet Transform in Lung Ultrasound images

Diagnosis of Covid-19 using Fractional B-spline Wavelet Transform in Lung Ultrasound images

WBEM-based analysis of band structures of solid-solid and fluid-fluid phononic crystals with frequency-independent fundamental solutions

A novel method is presented to determine the band structures of two-dimensional solid-solid and fluid-fluid phononic crystals (PCs) with square and triangular lattices using the wavelet-based boundary element method (WBEM). Both sweeping frequency technique and sweeping wave vector technique are used to determine the band structures of PCs. The fundamental solutions for integral equations established in primitive cell are independent of angular frequency, resulting in an avoidance of solving nonlinear eigenvalue problems and much reduction of computing time. The radial integration method and dual reciprocity method are respectively applied to handle the domain integral terms arising from the use of frequency-independent fundamental solutions, and to make the dimension reduction. The physical fields are approximated by B-spline wavelet on the interval and wavelet coefficients in wavelet space. It is proved that the wavelet coefficients still meet the relationship of their corresponding physical quantities, such as Bloch theorem. By means of the vanishing moment characteristic of wavelets, some small entries are produced in the system matrices which are further compressed into sparse ones, and the influence of sparse matrices on the results is discussed. Finally, the high computing speed and accuracy of the proposed method are shown in examples.

Wavelet-artificial neural network to predict the acetone sensing by indium oxide/iron oxide nanocomposites

This study applies a hybridized wavelet transform-artificial neural network (WT-ANN) model to simulate the acetone detecting ability of the Indium oxide/Iron oxide (In2O3/Fe2O3) nanocomposite sensors. The WT-ANN has been constructed to extract the sensor resistance ratio (SRR) in the air with respect to the acetone from the nanocomposite chemistry, operating temperature, and acetone concentration. The performed sensitivity analyses demonstrate that a single hidden layer WT-ANN with nine nodes is the highest accurate model for automating the acetone-detecting ability of the In2O3/Fe2O3 sensors. Furthermore, the genetic algorithm has fine-tuned the shape-related parameters of the B-spline wavelet transfer function. This model accurately predicts the SRR of the 119 nanocomposite sensors with a mean absolute error of 0.7, absolute average relative deviation of 10.12%, root mean squared error of 1.14, and correlation coefficient of 0.95813. The In2O3-based nanocomposite with a 15 mol percent of Fe2O3 is the best sensor for detecting acetone at wide temperatures and concentration ranges. This type of reliable estimator is a step toward fully automating the gas-detecting ability of In2O3/Fe2O3 nanocomposite sensors.

Truncated total variation in fractional B-spline wavelet transform for micro-CT image denoising.

In medical applications, computed tomography (CT) is widely used to evaluate various sample characteristics. However, image quality of CT reconstruction can be degraded due to artifacts. To propose and test a truncated total variation (truncation TV) model to solve the problem of large penalties for the total variation (TV) model. In this study, a truncated TV image denoising model in the fractional B-spline wavelet domain is developed to obtain the best solution. The method is validated by the analysis of CT reconstructed images of actual biological Pigeons samples. For this purpose, several indices including the peak signal-to-noise ratio (PSNR), structural similarity index (SSIM) and mean square error (MSE) are used to evaluate the quality of images. Comparing to the conventional truncated TV model that yields 22.55, 0.688 and 361.17 in PSNR, SSIM and MSE, respectively, using the proposed fractional B-spline-truncated TV model, the computed values of these evaluation indices change to 24.24, 0.898 and 244.98, respectively, indicating substantial reduction of image noise with higher PSNR and SSIM, and lower MSE. Study results demonstrate that compared with many classic image denoising methods, the new denoising algorithm proposed in this study can more effectively suppresses the reconstructed CT image artifacts while maintaining the detailed image structure.

Study on Propagation Characteristics of Guided Waves in Plate Structure via Semianalytical Wavelet Finite Element Method

The propagation characteristics of guided wave in waveguide structure plays an indispensable role in guided wave inspection. However, the analytical matrix method fails to solve the problems of wave propagation in waveguides with complex cross-section. The finite element method (FEM) has high computational cost for guided wave propagation problems. To overcome those shortcomings, a novel semi-analytical wavelet finite element (SAWFE) method is presented in this paper to study the guided wave propagation characteristics of plate structures. With the advantage of multi-scale and multi-resolution, the B-spline wavelet on the interval (BSWI) scale functions are adopted to construct one-dimensional BSWI elements and form the shape function. The element displacement field represented by the coefficients of wavelet expansions is transformed from wavelet space to physical space through the corresponding transformation matrix. Then, the constructed one-dimensional BSWI elements are employed to discretize the cross-section of waveguide structure, meanwhile, the wave displacement field in the wave propagation direction is described in an analytical way as a harmonic exponential function. The propagation characteristics of guided wave in aluminum plate and composite laminate are studied by using the proposed method, and the results show that the SAWFE method has high accuracy and efficiency in calculating dispersion curves and analyzing wave structure. The simulation analysis and experimental validation results further verify the effectiveness of the proposed method.

A three-dimensional B-spline wavelet finite element method for structural vibration analysis

In this study, a three-dimensional (3-D) B-spline wavelet finite element method (WFEM) is proposed by combining the wavelet theory and 3-D finite element method (FEM). Owing to the semi-orthogonality and multi-resolution analysis of spline wavelet over the solution domain, a cuboid with uniform distribution nodes is established for the 3-D B-spline wavelet on the interval (BSWI) element. The governing equations of vibration for the elastic structures are derived by introducing the 3-D B-spline wavelet theory into the displacement components of each node. A procedure for assembling the BSWI elements is presented by redefining the index element node functions. Numerical studies including the vibration analysis of solid and hollow beams and plates under different boundary conditions are performed to show the performance of the present method. A comprehensive comparison of the predicted vibration characteristics obtained by the present method, traditional FEM, and reference data is presented to verify the accuracy and efficiency of the 3-D B-spline WFEM. The core of the present method is involving the 3-D B-spline wavelet to replace the polynomial interpolation functions in the traditional FEM, which enhances computational accuracy during the vibration analysis of the elastic structures.

Elemental Determination in Stainless Steel via Laser-Induced Breakdown Spectroscopy and Back-Propagation Artificial Intelligence Network with Spectral Pre-Processing

Minor elements significantly influence the properties of stainless steel. In this study, a laser-induced breakdown spectroscopy (LIBS) technique combined with a back-propagation artificial intelligence network (BP-ANN) was used to detect nickel (Ni), chromium (Cr), and titanium (Ti) in stainless steel. For data pre-processing, cubic spline interpolation and wavelet threshold transform algorithms were used to perform baseline removal and denoising. The results show that this set of pre-processing methods can effectively improve the signal-to-noise ratio, remove the baseline of spectral baseline, reduce the average relative error, and reduce relative standard deviation of BP-ANN predictions. It indicates that BP-ANN combined with pre-processing methods has promising applications for the determination of Ni, Cr, and Ti in stainless steel with LIBS and improves prediction accuracy and stability.

Orthogonal systems of spline wavelets as unconditional bases in Sobolev spaces

AbstractWe exhibit the necessary range for which functions in the Sobolev spaces can be represented as an unconditional sum of orthonormal spline wavelet systems, such as the Battle–Lemarié wavelets. We also consider the natural extensions to Triebel–Lizorkin spaces. This builds upon, and is a generalization of, previous work of Seeger and Ullrich, where analogous results were established for the Haar wavelet system.

An adaptive stochastic Galerkin method based on multilevel expansions of random fields: Convergence and optimality

The subject of this work is a new stochastic Galerkin method for second-order elliptic partial differential equations with random diffusion coefficients. It combines operator compression in the stochastic variables with tree-based spline wavelet approximation in the spatial variables. Relying on a multilevel expansion of the given random diffusion coefficient, the method is shown to achieve optimal computational complexity up to a logarithmic factor. In contrast to existing results, this holds in particular when the achievable convergence rate is limited by the regularity of the random field, rather than by the spatial approximation order. The convergence and complexity estimates are illustrated by numerical experiments.

Robust Code Constructions Based on Bent Functions and Spline Wavelet Decomposition

The paper investigates new robust code constructions based on bent functions and spline–wavelet transformation. Implementation of bent functions in code construction increases the probability of error detection in the data channel and cryptographic devices. Meanwhile, the use of spline wavelet theory for constructing the codes gives the possibility to increase system security from the actions of an attacker. Presented constructions combine spline-wavelets functions and bent functions. Developed robust codes, compared to existing ones, have a higher parameter of the maximum error masking probability. Illustrated codes ensure the security of transmitted information. Some of the granted constructions were implemented on FPGA.

A Comparative Analysis of the Choice of Mother Wavelet Functions Affecting the Accuracy of Forecasts of Daily Balances in the Treasury Single Account

Improving the accuracy of cash flow forecasting in the TSA is key to fulfilling government payment obligations, minimizing the cost of maintaining the cash reserve, providing the absence of outstanding debt accumulation and ensuring investment in financial instruments to obtain additional income. This study aims to improve the accuracy of traditional methods of forecasting the time series compiled from the daily remaining balances in the TSAbased on prior decomposition using a discrete wavelet transform. The paper compares the influence of selecting a mother wavelet out of 570 mother wavelet functions belonging to 10 wavelet families (Haar;Dabeshies; Symlet; Coiflet; Biorthogonal Spline; Reverse Biorthogonal Spline; Meyer; Shannon; Battle-Lemarie; and Cohen–Daubechies–Feauveau) and the decomposition level (from 1 to 8) on the forecast accuracy of time series compiled from the daily remaining balances in the TSA in comparison with the traditional forecasting method without prior timeseries decomposition. The model with prior time series decomposition based on the Reverse Biorthogonal Spline Wavelet [5.5] mother wavelet function, upon the eighth iteration, features the highest accuracy, significantly higher than that of the traditional forecasting models. The choice of the mother wavelet and the decomposition level play an important role in increasing the accuracy of forecasting the daily remaining balances in the TSA.

Wavelet sparsity enhancement for extracting transient vibration signatures of bearing structural damages

Wavelet methods are widely used in mechanical transient vibration signature detection and fault diagnosis. Undesirable artifacts (e.g., spurious noise spikes and pseudo-Gibbs components) are serious issues for the mainstream methods, probably resulting in inaccurate analysis results. For this reason, a new wavelet sparsity enhancement methodology is proposed to achieve artifact-free extraction of bearing transient vibration signatures. The wavelet sparsity is enhanced through two aspects—that is, wavelet basis design and wavelet coefficient processing. Specifically, we use a new family of wavelets called fractional B-spline wavelets on bearing fault signal analysis and propose the adaptive undecimated fractional spline wavelet transform which addresses the shift-invariant problem and also permits to customize an optimal wavelet basis according to the signal itself adaptively. Meanwhile, we introduce a two-step wavelet processing method including a nonlinear operator and a generalized hard-thresholding (with symmetric or asymmetric thresholds determined automatically by wavelet coefficients at each level). Moreover, unlike the most existing wavelet methods, the approximation coefficients are also processed along with the detailed coefficients to remove the possible low-frequency noise. The final transient vibration signatures are reconstructed with the processed coefficients and would be sparser, more accurate, and almost free of artifacts. The validity of the methodology is verified with the analysis results of vibration signals measured from fault-injection experiment and industrial wind turbine transmission system. The comparisons highlight the advantages of the methodology over several common methods in suppressing artifacts and extracting the sparse transient vibration signatures of bearing structural damages.