Spline Wavelet Research Articles

On Five-Diagonal Splitting for Cubic Spline Wavelets with Six Vanishing Moments on a Segment

In this study, we use the vanishing property of the first six moments for constructing a splitting algorithm for cubic spline wavelets. First, we construct the corresponding wavelet space that satisfies the orthogonality conditions for all fifth-degree polynomials. Then, using the homogeneous Dirichlet boundary conditions, we adapt spaces to the closed interval. The originality of the study consists in obtaining implicit relations connecting the coefficients of the spline decomposition at the initial scale with the spline coefficients and wavelet coefficients at the nested scale by a tape system of linear algebraic equations with a non-degenerate matrix. After excluding the even rows of the system, in contrast to the case with two zero moments, the resulting transformation matrix has five (instead of three) diagonals. The results of numerical experiments on calculating the derivatives of a discrete function are presented.

Sparse representations for fault signatures via hybrid regularization in adaptive undecimated fractional spline wavelet transform domain

Recent studies on vibration-based machine diagnostics have highlighted the role played by the wavelet transform (WT). However, common WT-based denoising methods (e.g. wavelet thresholding and non-penalty regularization) are often challenging in attaining accurate sparse representations of fault signatures in practice due to artifacts. In this paper, a method via hybrid regularization in the adaptive undecimated fractional spline WT (AUFrSWT) domain is introduced to achieve the accurate sparse representations of fault signatures from strong noise environments. The method promotes wavelet sparsity by two aspects: wavelet basis design and wavelet coefficient processing. For the former, a new wavelet family, i.e. a fractional spline wavelet, is used, and the AUFrSWT is originally proposed to customize the optimal wavelet basis and meanwhile address the translation-invariant issue. For the latter, the wavelet coefficients are estimated by minimizing a single convex model function with hybrid regularization, where nonconvex arctangent penalty is employed for wavelet sparsity promoting and total variation is used to improve the reconstruction performance of the shrunken coefficients. The method is validated by the analysis of actual bearing and gear vibration data from fault-injection experiments. Analysis and comparison results show great potential for signal sparse representations and machinery fault diagnostics.

The regularization B-spline wavelet method for the inverse boundary problem of the Laplace equation from noisy data in an irregular domain

This study introduces a high accuracy and noise-robust numerical method, that is, the regularization B-spline wavelet method (RBSWM), for solving the inverse boundary problem of the Laplace equation with noisy data in an irregular domain. The problem that we consider is directly discretized by the B-spline wavelet scaling functions. To obtain a stable numerical solution of the problem for noisy data, the Tikhonov regularization technique augmented with the L-curve method is adopted. Numerical experiments demonstrate that the proposed method produces solutions with good stability and accuracy.

The Hilbert Transform of B-Spline Wavelets

Both the Hilbert transform and the B-spline wavelets are important tools in signal processing, which makes a study of relation between these two subjects be of practical significance. In particular, because the B-spline wavelets have good properties including vanishing moments, symmetry, compact support and so on, we focus on the Hilbert transform of B-spline wavelets in this letter. For this purpose, the B-spline wavelets of order $m$ is described in a piecewise polynomial form firstly. An important property of the Pascal triangle transform is then explored. Based on these results, an explicit form of the Hilbert transform of B-spline wavelet of order $m$ is established. Furthermore, the vanishing moments, symmetry and asymptote behavior of the Hilbert transform of B-spline wavelets are also discussed. To demonstrate the effectiveness of these results, two examples in the case of $m=3$ and $m=4$ are given and the graphs of these two B-spline wavelets as well as their Hilbert transforms are presented. These two cases of $m=3$ and $m=4$ provide two Hilbert transform pairs of wavelets, which can be used in digital image processing.

Uncertainty Quantification of Stochastic Epidemic SIR Models Using B-spline Polynomial Chaos

Real-life epidemic situations are modeled using systems of differential equations (DEs) by considering deterministic parameters. However, in reality, the transmission parameters involved in such models experience a lot of variations and it is not possible to compute them exactly. In this paper, we apply B-spline wavelet-based generalized polynomial chaos (gPC) to analyze possible stochastic epidemic processes. A sensitivity analysis (SA) has been performed to investigate the behavior of randomness in a simple epidemic model. It has been analyzed that a linear B-spline wavelet basis shows accurate results by involving fewer polynomial chaos expansions (PCE) in comparison to cubic B-spline wavelets. We have carried out our developed method on two real outbreaks of diseases, firstly, influenza which affected the British boarding school for boys in North England in 1978, and secondly, Ebola in Liberia in 2014. Real data from the British Medical Journal (influenza) and World Health Organization (Ebola) has been incorporated into the Susceptible-Infected-Recovered (SIR) model. It has been observed that the numerical results obtained by the proposed method are quite satisfactory.

Method for finding approximate solutions of elasticity equations using spline wavelets

Method for finding approximate solutions of elasticity equations using spline wavelets

Adaptive distance measurement method with blur of B-spline wavelet function

Adaptive distance measurement method with blur of B-spline wavelet function

Displacement Estimation of Self-Sensing Magnetic Bearings Based on Biorthogonal Spline Wavelet

By collecting the ripple current in the control coil, the self-sensing magnetic bearings can obtain rotor displacement information without a displacement sensor, which has the advantages of low cost and high integration. According to the problems of incomplete and inaccurate displacement information extraction caused by the traditional displacement estimation method using Fourier method to analyze the ripple current with non-stationary characteristics to estimate the rotor displacement with the Gibbs effect. Therefore, this paper proposes a rotor displacement estimation algorithm based on multiresolution filter bank biorthogonal spline wavelet for a magnetic bearing motor based on two-level switching power amplifier. This algorithm utilizes the biorthogonal spline wavelet with the characteristics of generalized linear phase and tight support, which can accurately demodulate the ripple current in the coil and extract the displacement information in the ripple current. Therefore, it is able to overcome the Gibbs effect in the traditional Fourier analysis estimation algorithm, to reduce the influence of ripple on rotor displacement estimation, and to improve the accuracy and stability of displacement estimation. In order to verify the correctness and effectiveness of the proposed algorithm, this paper establishes a simulation model of magnetic bearing in MATLAB Simulink and designs the proposed displacement estimator. Simulation results demonstrate that the proposed algorithm has higher accuracy and better stability than traditional displacement estimation algorithms. Experimental results show that the maximum deviation rate of the displacement estimation method proposed in this paper does not exceed 1% when the radial magnetic bearing gap is 0.5 mm, which can effectively estimate the rotor displacement.

On filter banks in spline wavelet transform on a non-uniform grid

On filter banks in spline wavelet transform on a non-uniform grid

Modeling of Solar Radiation Using the Wavelet Neural Network Model in Mataram City Lombok Island

Sunlight is a source of energy for living things in general. In reality, the intensity of solar radiation is an environmental parameter that has positive and negative impacts on human life in particular. Furthermore, the knowledge on the characteristics of solar radiation, including its distribution pattern, is considered by many circles, both policy-makers and researchers in the environmental field. This study aims to create a solar radiation model in response to meteorological factors such as wind speed, air pressure and temperature, humidity, and rainfall using the Wavelet Neural Network (WNN). The modeling of solar radiation in this study is carried out by simultaneously utilizing its advantages as a hybrid model that combines the neural network model and the wavelet method. These advantages through the learning process (supervised learning) are multiplied through the use of the wavelet transform as a pre-processing data method and two type wavelets function, B-spline and Morlet wavelets, as an activation function in the neural network learning process. The WNN model was analyzed in two cases of meteorological variables, which are with and without rainfall. The results based on the root of the mean square error (RMSE) indicator show that the WNN model in these two cases is quite accurate. Meanwhile, the other indicator shows that the interval of the data distribution from the model is within the actual range. This implies that the predicted intensity of the solar radiation will be in a safe position in its adverse effect when the model is used as a reference.

The property of linear fractional integro-differential equations on the basis of the collocation method and the operational matrices of cubic B-spline wavelets

In this paper, by using the collocation method the operational matrix of Caputo fractional derivative was constructed for the cubic B-spline scaling function and wavelets. We apply the operational matrix and the Collocation method to solve linear fractional Volterra integro-differential equations. By using the principal characteristic of this technique, we have the operational matrix, which is applied to solve any linear fractional Volterra integro-differential equation by reducing the time, computer memory occupation and convert to a system of linear equations. Finally, the validity and applicability of the new technique will be shown by some numerical examples and convergence analysis.

A two-step damage quantitative identification method for beam structures

The paper puts forward a novel method of the damage detection using singular value decomposition (SVD) and artificial bee colony (ABC) algorithm for beam structures. The proposed method includes two steps: damage localization and depth estimation. In the first step, singular value decomposition is used to decompose the trajectory matrix of the attractor reconstructed from the shape data to localize the damage. In the second step, B-spline wavelet on the interval (BSWI) finite element method is usually applied to construct a damage extent estimation database at the detected damage locations, which can reveal the relationship between natural frequencies and its damage extent. Then ABC algorithm is adopted to predict the damage depths by searching the frequency estimation database. Numerical simulations and experiment study are used to accurately verify the effectiveness of the proposed method for the case of cantilever beam.

Quasilinearized Semi-Orthogonal B-Spline Wavelet Method for Solving Multi-Term Non-Linear Fractional Order Equations

In the present article, we implement a new numerical scheme, the quasilinearized semi-orthogonal B-spline wavelet method, combining the semi-orthogonal B-spline wavelet collocation method with the quasilinearization method, for a class of multi-term non-linear fractional order equations that contain both the Riemann–Liouville fractional integral operator and the Caputo fractional differential operator. The quasilinearization method is utilized to convert the multi-term non-linear fractional order equation into a multi-term linear fractional order equation which, subsequently, is solved by means of semi-orthogonal B-spline wavelets. Herein, we investigate the operational matrix and the convergence of the proposed scheme. Several numerical results are delivered to confirm the accuracy and efficiency of our scheme.

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

A comparison study of wavelet transforms for the visualization of differentially methylated regions in DNA samples

DNA methylation analysis has become an important topic in the study of human health. DNA methylation analysis requires not only a specific treatment of DNA samples based on bisulfite, but also software tools for their analysis. Although many software tools have been developed and some tools for detecting differentially methylated regions (DMRs) in the DNA have been proposed, there is still a lack of tools for interactively displaying the DMRs found. In previous works, we proposed a new approach based on the Haar wavelet transform for the interactive visualization of DMRs at different scales. However, the nature of this wavelet produces signal overlapping which prevents an easy detection of visual differences between case and control signals in some cases. In this paper, we show a comparison study of different wavelet transforms which may solve this problem. The evaluation results show that some of the considered wavelet transforms are very prone to yield negative signal segments due to the signal inertia, which may lead to false DMR detection. However, the spline wavelet transform does not significantly suffer from this effect, clearly highlighting true DMRs. Also, the spline wavelet transform does not need to reconstruct the signal in the lower transformation levels, which improves interactivity.

Single-phase ground fault location method for distribution network based on traveling wave time-frequency characteristics

The traditional traveling wave fault location methods only consider the information about time domain or frequency domain characteristics, so it is difficult to ensure the location accuracy. Through the research of relationship between traveling wave characteristic parameters and frequency, this paper proposes a single-phase grounding fault location method based on traveling wave time-frequency characteristics. Firstly, determine the time during which the traveling wave reaches the busbar by using the quadratic B-spline wavelet analysis modulus maximum detection algorithm with denoising factor.. Then, according to the improved matrix pencil algorithm, extracte the frequency of traveling wave arriving at the bus measuring point, and the corresponding velocity of incident wave is obtained. Finally, the fault distance can be calculated using the location equation of modulus wave velocity difference. The simulation results show that this method proposed in this paper has higher accuracy than the traditional method, and in the terms of different grounding resistance, fault initial angle and fault distance, the error of fault location is smaller.

A new structure of spline wavelet transform based on adaptive directional lifting for efficient image coding

We present in this paper a modified structure of a polynomial spline wavelet transform based on adaptive directional lifting for image compression. The proposed method not only uses the polynomial splines as a tool for the construction of the appropriate filters seeing its efficiency as compared to other filters like the biorthogonal 9/7, but also adapts far better to the image-orientation features by carrying out a lifting-based prediction in local windows in the direction of high pixel correlation. The main purpose of this article is then to integrate the coefficients calculated by the best spline filter order into the adaptive directional lifting. The new method is designed to further reduce the magnitude of the high-frequency wavelet coefficients and preserve the detailed information of the original images more effectively. The numerical results demonstrate the efficiency of the proposed approach over the traditional lifting-based spline wavelet transform and the adaptive directional lifting with respect to both objective and subjective criteria for image compression applications.

Numerical solution of Blasius viscous flow problem using wavelet Galerkin method

In this article it is proposed to solve numerically the Blasius viscous flow problem of fluid mechanics using wavelet Galerkin method. The spline wavelet system is used for this purpose. It is assumed that the approximate solution lies in the space spanned by translates of the scaling function of the considered wavelet system. It is found that the numerical solution thus obtained is having high accuracy when compared to the benchmark solution given by Howarth.

Solving optimal control problems with integral equations or integral equations - differential with the help of cubic B-spline scaling functions and wavelets

Solving optimal control problems with integral equations or integral equations - differential with the help of cubic B-spline scaling functions and wavelets

A stochastic B-spline wavelet on the interval finite element method for beams

The current paper presents the formulation of stochastic B-spline wavelet on the interval (BSWI) based wavelet finite element method (WFEM) for beams wherein, the spatial variation of modulus of elasticity is modelled as a homogeneous random field. Stochastic beam element formulations based on both Euler-Bernoulli beam theory and Timoshenko beam theory are proposed. BSWI scaling functions are used for the discretization of the random field and the response statistics are obtained using the perturbation approach. Numerical examples are solved and the results from perturbation approach are compared with that obtained from Monte Carlo simulation (MCS). A parametric study is also done to understand the effect of different coefficient of variation (CV) values and correlation length parameters on the response statistics. The study concludes that the proposed BSWI WFEM based perturbation approach for beams produce accurate response statistics for values of CV less than 15%. A comparative study is carried out between the results obtained from the proposed stochastic WFEM with stochastic finite element method (SFEM) wherein the random field discretization is done using Lagrange shape functions. Furthermore, normalized computational times for the execution of perturbation approach and MCS based on WFEM are evaluated and compared with those obtained for SFEM.