Construction theory of dyadic scale function and dyadic wavelet function

We derive the conditions for the existence of the unique solution of the two scale integral equation and the form of the solution, according to the method of the construction of the dyadic scale function. We give the construction of the dyadic wavelet and its necessary and sufficient condition. As an application, we also develop a pyramid algorithm of the dyadic wavelet decomposition.

Cylindrical Guided Wave Signals for Underground Pipe Inspection using Different Continuous Wavelet Mother Functions

Over the past decade, wavelets provided a powerful and flexible set of tools for handling fundamental problems in science and engineering. Wavelet analyses are being used for solving problems in different engineering areas like audio de-noising, signal compression, object detection, image decomposition, speech recognition etc. Wavelet analysis employs orthonormal as well as non-orthonornal functions. This research investigates the effectiveness of wavelet analysis in detecting defects in underground steel pipe networks. Continuous Wavelet Transforms （CWT） has been performed on the received signals of cylindrical guided waves. Cylindrical Guided waves are generated and propagated through the pipe wall boundaries in a pitch-catch system. Piezo-electric transducers are used to generate as well as receive guided waves. Several mother wavelet functions such as Daubechies, Symlet, Coiflet and Meyer have been used for the Continuous Wavelet Transform to investigate the most suitable function for defect detection. This research also investigates the effect of surrounding soil on wavelet transforms for different mother wavelet functions.

Wavelet-Based Fractal Function Approximation

In this paper, we study on the application of radical B-spline wavelet scaling function in fractal function approximation system. The paper proposes a wavelet-based fractal function approximation algorithm in which the coefficients can be determined by solving a convex quadraticprogramming problem. And the experiment result shows that the approximation error of this algorithm is smaller than that of the polynomial-based fractal function approximation. This newalgorithm exploits the consistency between fractal and scaling function in multi-scale and multiresolution, has a better approximation effect and high potential in data compression, especially inimage compression.

Local Extrema of Periodic Function’s Wavelet Transform

The theory of detecling ridges in the modulus of the continuous wavelet transform is presented as well as reconstructing signal by using information on ridges,To periodic signal we suppose Morlet wavelet as basic wavelet, and research the local extreme point and extrema of the wavelet transform on periodic function for the collection of signal＇ s instantaneous amplitude and period.

Phase spectrum estimation of the seismic wavelet based on a criterion function

Phase spectrum estimation of the seismic wavelet is an important issue in high-resolution seismic data processing and interpretation. On the basis of two patterns of constant-phase rotation and root transform for wavelet phase spectrum variation, we introduce six sparse criteria, including Lu’s improved kurtosis criterion, the parsimony criterion, exponential transform criterion, Sech criterion, Cauchy criterion, and the modified Cauchy criterion, to phase spectrum estimation of the seismic wavelet, obtaining an equivalent effect to the kurtosis criterion. Through numerical experiments, we find that when the reflectivity is not a sparse sequence, the estimated phase spectrum of the seismic wavelet based on the criterion function will deviate from the true value. In order to eliminate the influence of non-sparse reflectivity series in a single trace, we apply the method to the multi-trace seismogram, improving the accuracy of seismic wavelet phase spectrum estimation.

Steganography based on wavelet transform and modulus function

In order to provide larger capacity of the hidden secret data while maintaining a good visual quality of stego-image, in accordance with the visual property that human eyes are less sensitive to strong texture, a novel steganographic method based on wavelet and modulus function is presented. First, an image is divided into blocks of prescribed size, and every block is decomposed into one-level wavelet. Then, the capacity of the hidden secret data is decided with the number of wavelet coefficients of larger magnitude. Finally, secret information is embedded by steganography based on modulus function. From the experimental results, the proposed method hides much more information and maintains a good visual quality of stego-image. Besides, the embedded data can be extracted from the stego-image without referencing the original image.

Gaussian Functions,Γ-Functions and Wavelets

The relations between Gaussian function and Γ function is revealed first at one dimensional situation. Then, the Fourier transformation of n dimensional Gaussian function is deduced by a lemma. Following the train of thought in one dimensional situation, the relation between n dimensional Gaussian function and Γ function is given. By these, the possibility of arbitrary derivative of an n dimensional Gaussian function being a mother wavelet is indicated. The result will take some enlightening role in exploring the internal relations between Gaussian function and Γ function as well as in finding high dimensional mother wavelets.

Nonstationary Wavelets Related to the Walsh Functions

Using the Walsh-Fourier transform, we give a construction of compactly supported nonstationary dyadic wavelets on the positive half-line. The masks of these wavelets are the Walsh polynomials defined by finite sets of parameters. Application to compression of fractal functions are also discussed.

UNIFORM ANALYTIC CONSTRUCTION OF WAVELET ANALYSIS FILTERS BASED ON SINE AND COSINE TRIGONOMETRIC FUNCTIONS

Based on sine and cosine functions, the compactly supported orthogonal wavelet filter coefficients with arbitrary length are constructed for the first time. When N = 2(k-1) and N = 2k, the unified analytic constructions of orthogonal wavelet filters are put forward, respectively. The famous Daubechies filter and some other well-known wavelet filters are tested by the proposed novel method which is very useful for wavelet theory research and many application areas such as pattern recognition.

APPLICATION OF WAVELET THEORY IN RESEARCHON WEIGHT FUNCTION OF MESHLESS METHOD

Multiresolution analysis of wavelet theory can give an effective way to describe the information at various levels of approximations or different resolutions, based on spline wavelet analysis,so weight function is orthonormally projected onto a sequence of closed spline subspaces, and is viewed at various levels of approximations or different resolutions. Now, the useful new way to research weight function is found, and the numerical result is given.

CHARACTERIZATIONS OF FUNDAMENTAL SCALING FUNCTIONS AND WAVELETS

The objective of Ibis paper is to establish precise characterizations of scaling functions which are orthonormal or fundamental.A criterion for the corresponding wavelets is also given.

Inversion of receiver function by wavelet transformation

A new method for receiver function inversion by wavelet transformation is presented in this paper. Receiver func-tion is expanded to different scales with different resolution by wavelet transformation. After an initial model be-ing taken, a generalized least-squares inversion procedure is gradually carried out for receiver function from low to high scale, with the inversion result for low order receiver function as the initial model for high order. A neighborhood containing the global minimum is firstly searched from low scale receiver function, and will gradu-ally focus at the global minimum by introducing high scale information of receiver function. With the gradual ad-dition of high wave-number to smooth background velocity structure, wavelet transformation can keep the inver-sion result converge to the global minimum, reduce to certain extent the dependence of inversion result on the initial model, overcome the nonuniqueness of generalized least-squares inversion, and obtain reliable crustal and upper mantle velocity with high resolution.

THE WAVELET TRANSFORM OF PERIODIC FUNCTION AND NONSTATIONARY PERIODIC FUNCTION

Some properties of the wavelet transform of trigonometric Junction, periodic function and nonstationary periodic function have been investigated. The results show that the peak height and width in wavelet energy spectrum of a periodic function are in proportion to its period. At the same time, a new equation, which can truly reconstruct a trigonometric function with only one scale wavelet coefficient, is presented. The reconstructed wave shape of a periodic function with the equation is better than any term of its Fourier series. And the reconstructed wave shape of a class of nonstationary periodic function with this equation agrees well with the function.

THREE-DIMENSIONAL ANALYSIS OF FUNCTIONALLY GRADED PLATE BASED ON THE HAAR WAVELET METHOD

A three-dimensional analysis of a simply-supported functionally graded rectangular plate with an arbitrary distribution of material properties is made using a simple and effective method based on the Haar wavelet. With good features in treating singularities, Haar series solution converges rapidly for arbitrary distributions, especially for the case where the material properties change rapidly in some regions. Through numerical examples the influences of the ratio of material constants on the top and bottom surfaces and different material gradient distributions on the structural response of the plate to mechanical stimuli are studied.

Wavelet function estimation involving time series

Nonparametric regression models are considered. The nonparametric estimators for a regression function are given by wavelets. It is shown that under certain conditions, the wavelet estimators attain the optimal or nearly optimal convergence rate in a ball of Besov spaces.

Face Recognition Based on Wavelet Packet Coefficients and Radial Basis Function Neural Networks

An efficient face recognition system with face image representation using averaged wavelet packet coefficients, compact and meaningful feature vectors dimensional reduction and recognition using radial basis function (RBF) neural network is presented. The face images are decomposed by 2-level two-dimensional (2-D) wavelet packet transformation. The wavelet packet coefficients obtained from the wavelet packet transformation are averaged using two different proposed methods. In the first method, wavelet packet coefficients of individual samples of a class are averaged then decomposed. The wavelet packet coefficients of all the samples of a class are averaged in the second method. The averaged wavelet packet coefficients are recognized by a RBF network. The proposed work tested on three face databases such as Olivetti-Oracle Research Lab (ORL), Japanese Female Facial Expression (JAFFE) and Essexface database. The proposed methods result in dimensionality reduction, low computational complexity and provide better recognition rates. The computational complexity is low as the dimensionality of the input pattern is reduced.

Identification of linear continuous-time system using wavelet modulating filters

An approach to identification of linear continuous-time system is studied with modulating functions. Based on wavelet analysis theory, the multi-resolution modulating functions are designed, and the corresponding filters have been analyzed. Using linear modulating filters, we can obtain an identification model that is parameterized directly in continuous-time model parameters. By applying the results from discrete-time model identification to the obtained identification model, a continuous-time estimation method is developed. Considering the accuracy of parameter estimates, an instrumental variable (Ⅳ) method is proposed, and the design of modulating integral filter is discussed. The relationship between the accuracy of identification and the parameter of modulating filter is investigated, and some points about designing Gaussian wavelet modulating function are outlined. Finally, a simulation study is also included to verify the theoretical results.

Denoising Method for Partial Discharge Signal of Switchgear Based on Continuous Adaptive Wavelet Threshold

Partial discharge(PD)is an important reason for the insulation failure of the switchgear.In the process of PD detection,PD signal is often annihilated in strong noise.In order to improve the accuracy of PD detection in power plant switchgear,a method based on continuous adaptive wavelet threshold switchgear PD signals denoising is proposed in this paper.By constructing a continuous adaptive threshold function and introducing adjustment parameters,the problems of over⁃processing of traditional hard threshold functions and incomplete denoising of soft threshold functions can be improved.The analysis results of simulated signals and measured signals show that the continuous adaptive wavelet threshold denoising method is significantly better than the traditional denoising method for the PD signal.The proposed method in this paper retains the characteristics of the original signal.Compared with the traditional denoising methods,after denoising the simulated signals,the signal⁃to⁃noise ratio(SNR)is increased by more than 30%,and the root⁃mean⁃square error(RMSE)is reduced by more than 30%.After denoising the real signal,the noise suppression ratio(NRR)is increased by more than 40%.The recognition accuracy rate of PD signal has also been improved to a certain extent,which proves that the method has a certain practicability.

Application of wavelet support vector regression on SAR data de-noising

A new filtering method for SAR data de-noising using wavelet support vector regression （WSVR） is developed. On the basis of the grey scale distribution character of SAR imagery, the logarithmic SAR image as a noise polluted signal is taken and the noise model assumption in logarithmic domain with Gaussian noise and impact noise is proposed. Based on the better per- formance of support vector regression （SVR） for complex signal approximation and the wavelet for signal detail expression, the wavelet kernel function is chosen as support vector kernel func- tion. Then the logarithmic SAR image is regressed with WSVR. Furthermore the regression distance is used as a judgment index of the noise type. According to the judgment of noise type every pixel can be adaptively de-noised with different filters. Through an approximation experiment for a one-dimensional complex signal, the feasibility of SAR data regression based on WSVR is con- firmed. Afterward the SAR image is treated as a two-dimensional continuous signal and filtered by an SVR with wavelet kernel function. The results show that the method proposed here reduces the radar speckle noise effectively while maintaining edge features and details well.