Vanishing Wavelet Moments Research Articles

Non-Destructive Diagnostics of Concrete Beams Strengthened with Steel Plates Using Modal Analysis and Wavelet Transform.

Externally bonded reinforcements are commonly and widely used in civil engineering objects made of concrete to increase the structure load capacity or to minimize the negative effects of long-term operation and possible defects. The quality of adhesive bonding between a strengthened structure and steel or composite elements is essential for effective reinforcement; therefore, there is a need for non-destructive diagnostics of adhesive joints. The aim of this paper is the detection of debonding defects in adhesive joints between concrete beams and steel plates using the modal analysis approach. The inspection was based on modal shapes and their further processing with the use of continuous wavelet transform (CWT) for precise debonding localization and imaging. The influence of the number of wavelet vanishing moments and the mode shape interpolation on damage imaging maps was studied. The results showed that the integrated modal analysis and wavelet transform could be successfully applied to determine the exact shape and position of the debonding in the adhesive joints of composite beams.

Construction of Dual Multiple Knot B-Spline Wavelets on Interval

The paper is concerned with the construction of biorthogonal multiple knot B-spline(MKBS) scaling functions and multiple knot B-spline wavelet (MKBSW) basis functions. The resulting bases are on the unit interval with the desired number of vanishing wavelet moments. Moreover, in order to be the same the general structure of MKBS and MKBSW basis functions, a new basis of MKBS is introduced. This arises a significant decrease in the computational costs.

Wavelet analysis of uniformly time-modulated processes

ABSTRACTUniformly time-modulated processes (UTMP) offer a simple, though convenient model for many non stationary time series. However, in many applications, dependency is considered as a disturbing factor. For the UTMP using Weierstrass representation of modulating function, we demonstrated that the correlation between coefficients in wavelet domain is a decreasing function of number of wavelet vanishing moments and tends to zero in progressive rate as we move from finer scales to coarser scales.

Study on Optimal Selection of Wavelet Vanishing Moments for ECG Denoising

The frequency characteristics of wavelets and the vanishing moments of wavelet filters are both important parameters of wavelets. Clarifying the relationship between the wavelet frequency characteristics and the vanishing moments of the wavelet filter can provide a theoretical basis for selecting the best wavelet. In this paper, the frequency characteristics of wavelets were analyzed by mathematical modeling, the mathematical relationship between wavelet frequency characteristics and vanishing moments was clarified, the optimal wavelet base function was selected hierarchically according to the amplitude frequency characteristics of ECG signal, and an accurate notch filter was realized according to the frequency characteristics of the noise. The experimental results showed that the optimal orthogonal wavelet analysis for the ECG signals with different frequency characteristics could make the high frequency energy distribution sparser, and the method proposed in this paper could effectively preserve the singularity of the signal and reduce the signal distortion.

Identification of complex sound sources produced by gear units

Noise source visualisation represents an important tool for technical acoustics. Many techniques of noise source visualisation have been developed, based on a specific noise source in a specific type of acoustic environment. A new visualisation method of complex noise sources is presented, using an acoustic camera and a new algorithm. Different transient acoustical phenomena can be noted. Additionally, a new family of biorthogonal wavelets is applied to determine fault in gears. The new wavelets are a generalisation of biorthogonal wavelet systems. Smoothness is controlled independently in the analysis. For the optimisation of the synthesis bank, discrete finite variation is used. Differentiability is measured, for which a large number of vanishing wavelet moments is necessary, in favour of a smoothness measure based on the fact that a finite depth of the filter bank tree is in most case related to practical applications.

Construction of optimally conditioned cubic spline wavelets on the interval

The paper is concerned with a construction of new spline-wavelet bases on the interval. The resulting bases generate multiresolution analyses on the unit interval with the desired number of vanishing wavelet moments for primal and dual wavelets. Both primal and dual wavelets have compact support. Inner wavelets are translated and dilated versions of well-known wavelets designed by Cohen, Daubechies, and Feauveau. Our objective is to construct interval spline-wavelet bases with the condition number which is close to the condition number of the spline wavelet bases on the real line, especially in the case of the cubic spline wavelets. We show that the constructed set of functions is indeed a Riesz basis for the space L 2 ([0, 1]) and for the Sobolev space H s ([0, 1]) for a certain range of s. Then we adapt the primal bases to the homogeneous Dirichlet boundary conditions of the first order and the dual bases to the complementary boundary conditions. Quantitative properties of the constructed bases are presented. Finally, we compare the efficiency of an adaptive wavelet scheme for several spline-wavelet bases and we show a superiority of our construction. Numerical examples are presented for the one-dimensional and two-dimensional Poisson equations where the solution has steep gradients.

Lifting construction based on Bernstein bases and application in image compression

A novel framework for the construction of biorthogonal wavelets based on Bernstein bases with an arbitrary order of vanishing moments using the lifting scheme is proposed. We explore the field of application of it in still image compression. The major contributions of this work can be summarised highlighting the following three aspects. First and foremost, we propose an algorithm that is used to increase the vanishing moments of wavelets from biorthogonal symmetrical wavelets based on Bernstein bases by the lifting scheme. An iterative algorithm for designing the lifting scheme is proposed, which is based on the relationship between the vanishing moments of the wavelet and multiples of zeros of z = 1. The authors provide formulas of the lifting scheme for the construction of wavelets with an arbitrary order of vanishing moments. In addition, the lifting scheme is the shortest among the lifting schemes with the same order of vanishing moments increasing and, more importantly, it is the only one possible. Second, to guarantee the symmetry of the lifting (dual lifting) biorthogonal filters, explicit formulas of the lifting scheme with an arbitrary order of vanishing moments are introduced, which simultaneously have the above two characteristics. With our method, a new family of the parameterisation with symmetry of filters and the related library of biorthogonal symmetric waveforms are presented. Finally, we present a new transform rule aiming at image compression and its corresponding algorithm. Applying the parameterisation of filters constructed in this paper, by adjusting their coefficients, we can realise the transform rule and obtain a new transform. We explore the possibility of applying the presented transforms in image compression at different compression rates, and the results of the experiments prove to be comparable with the CDF9/7 and several state-of-the-art wavelet transforms.

Determining cracks in gears using adaptive wavelet transform approach

A new family of biorthogonal wavelets used to identify fault in gears is dealt with. The fault detection analyses and methods presented in this paper are based on signals created by a faultless gear and by a gear with a crack in the tooth root, caused through operating conditions. The new wavelets are a generalization of biorthogonal wavelet systems. In the analysis, smoothness is controlled independently and discrete finite variation is used to optimise the synthesis bank. The procedure measure dispenses with a measure of differentiability, requiring a large number of vanishing wavelet moments in favour of a smoothness measure that is based on the fact that, in most practical applications, only a finite depth of the filter bank tree is involved.

Adaptive multiresolution finite element method based on second generation wavelets

A distinguishing feature of second generation wavelets is that it can be custom designed depending on applications. Based on second generation wavelets, a multiresolution finite element method is discussed, and its adaptive algorithm is constructed. The hierarchical approximation spaces for finite element analysis are produced. The finite element equation is scale-decoupled via eliminating all coupling in the stiffness matrix of element across scales, then resolved in different spaces independently. The coarse solution can be obtained in the coarse approximation space, and refined by adding details in the detail spaces over several levels till the equation is resolved to the desired accuracy. The scale-decoupling condition of the stiffness matrix of element is proposed by introducing wavelet vanishing moments, and the principle of constructing the scale-decoupling wavelet bases is established. The method establishes an important connection between finite element analysis and multiresolution analysis. The numerical examples have illustrated that the proposed method is powerful to analyze the field problems with changes in gradients and singularities.

The design of approximate Hilbert transform pairs of wavelet bases

Several authors have demonstrated that significant improvements can be obtained in wavelet-based signal processing by utilizing a pair of wavelet transforms where the wavelets form a Hilbert transform pair. This paper describes design procedures, based on spectral factorization, for the design of pairs of dyadic wavelet bases where the two wavelets form an approximate Hilbert transform pair. Both orthogonal and biorthogonal FIR solutions are presented, as well as IIR solutions. In each case, the solution depends on an allpass filter having a flat delay response. The design procedure allows for an arbitrary number of vanishing wavelet moments to be specified. A Matlab program for the procedure is given, and examples are also given to illustrate the results.

Dispersion of time domain wavelet Galerkin method based on Daubechies' compactly supported scaling functions with three and four vanishing moments

The wavelet-Galerkin method for time-domain electromagnetic field modeling based on Daubechies' compactly supported wavelets proposed by Cheong et al. (see ibid., vol. 9, no. 8, p. 297-299, 1999) has been extended to the use of the scaling functions with three and four vanishing wavelet moments together with the approximate shifted interpolation property. The numerical dispersion properties of the methods are precisely investigated and compared with those of other wavelet-based and finite-difference methods. It was found that Daubechies' scaling functions with larger number of vanishing moments generally give higher accuracy while maintaining the comparable computational expenditure.

Wavelet-Galerkin scheme of time-dependent inhomogeneous electromagnetic problems

A wavelet-Galerkin scheme based on the time-dependent Maxwell's equations is presented. Daubechies' wavelet with two vanishing wavelet moments is expanded for basis function in spatial domain, and Yee's leap-frog approach is applied. The shifted interpolation property of Daubechies' wavelet family leads to the simplified formulations for inhomogeneous media without the additional matrices for the integral or material operator. The storage effectiveness, execution time reduction, and accuracy of this scheme are demonstrated by calculating the resonant frequencies of the homogeneous and inhomogeneous cavities.

Rank M Wavelets with N Vanishing Moments

This work generalizes the rank 2 (scale factor of 2) orthogonal wavelet sequences of Daubechies to the case of a rank $M$ wavelet matrix. Several equivalent definitions of $N$th order vanishing moments for rank $M$ wavelets are developed. These notions are used to find an explicit formula for rank $M$ wavelet scaling sequences with $N$ vanishing wavelet moments (of degree $N$ in our terminology). A full wavelet matrix (scaling sequence and $M-1$ wavelet sequences) is constructed, with explicit examples.