Wavelet Multi-Resolution Interpolation Galerkin Method for Linear Singularly Perturbed Boundary Value Problems

In this study,a wavelet multi-resolution interpolation Galerkin method(WMIGM)is proposed to solve linear singularly perturbed boundary value problems.Unlike conventional wavelet schemes,the proposed algorithm can be readily extended to special node generation techniques,such as the Shishkin node.Such a wavelet method allows a high degree of local refinement of the nodal distribution to efficiently capture localized steep gradients.All the shape functions possess the Kronecker delta property,making the imposition of boundary conditions as easy as that in the finite element method.Four numerical examples are studied to demonstrate the validity and accuracy of the proposedwavelet method.The results showthat the use ofmodified Shishkin nodes can significantly reduce numerical oscillation near the boundary layer.Compared with many other methods,the proposed method possesses satisfactory accuracy and efficiency.The theoretical and numerical results demonstrate that the order of theε-uniform convergence of this wavelet method can reach 5.

Interpolation of Images Using Discrete Wavelet Transform to Simulate Image Resizing as in Human Vision

This paper presents discrete wavelet transform (DWT) and its inverse (IDWT) with Haar wavelets as tools to compute the variable size interpolated versions of an image at optimum computational load. As a human observer moves closer to or farther from a scene, the retinal image of the scene zooms in or out, respectively. This zooming in or out can be modeled using variable scale interpolation. The paper proposes a novel way of applying DWT and IDWT in a piecewise manner by non-uniform down- or up-sampling of the images to achieve partially sampled versions of the images. The partially sampled versions are then aggregated to achieve the final variable scale interpolated images. The non-uniform down-or up-sampling here is a function of the required scale of interpolation. Appropriate zero padding is used to make the images suitable for the required non-uniform sampling and the subsequent interpolation to the required scale. The concept of zeroeth level DWT is introduced here, which works as the basis for interpolating the images to achieve bigger size than the original one. The main emphasis here is on the computation of variable size images at less computational load, without compromise of quality of images. The interpolated images to different sizes and the reconstructed images are benchmarked using the statistical parameters and visual comparison. It has been found that the proposed approach performs better as compared to bilinear and bicubic interpolation techniques.

Statistical Modeling with a Hidden Markov Tree and High-resolution Interpolation for Spaceborne Radar Reflectivity in the Wavelet Domain

With the increasing availability of precipitation radar data from space,enhancement of the resolution of spaceborne precipitation observations is important,particularly for hazard prediction and climate modeling at local scales relevant to extreme precipitation intensities and gradients.In this paper,the statistical characteristics of radar precipitation reflectivity data are studied and modeled using a hidden Markov tree(HMT)in the wavelet domain.Then,a high-resolution interpolation algorithm is proposed for spaceborne radar reflectivity using the HMT model as prior information.Owing to the small and transient storm elements embedded in the larger and slowly varying elements,the radar precipitation data exhibit distinct multiscale statistical properties,including a non-Gaussian structure and scale-to-scale dependency.An HMT model can capture well the statistical properties of radar precipitation,where the wavelet coefficients in each sub-band are characterized as a Gaussian mixture model(GMM),and the wavelet coefficients from the coarse scale to fine scale are described using a multiscale Markov process.The state probabilities of the GMM are determined using the expectation maximization method,and other parameters,for instance,the variance decay parameters in the HMT model are learned and estimated from high-resolution ground radar reflectivity images.Using the prior model,the wavelet coefficients at finer scales are estimated using local Wiener filtering.The interpolation algorithm is validated using data from the precipitation radar onboard the Tropical Rainfall Measurement Mission satellite,and the reconstructed results are found to be able to enhance the spatial resolution while optimally reproducing the local extremes and gradients.

Integer Wavelet-Based Image Interpolation in Lifting Structure for Image Resolution Enhancement

A floating-point wavelet-based and an integer wavelet-based image interpolations in lifting structures and polynomial curve fitting for image resolution enhancement are proposed in this paper. The proposed prediction methods estimate high-frequency wavelet coefficients of the original image based on the available low-frequency wavelet coefficients, so that the original image can be reconstructed by using the proposed prediction method. To further improve the reconstruction performance, we use polynomial curve fitting to build relationships between actual high-frequency wavelet coefficients and estimated high-frequency wavelet coefficients. Results of the proposed prediction algorithm for different wavelet transforms are compared to show the proposed prediction algorithm outperforms other methods.

Wavelet multiresolution interpolation Galerkin method for nonlinear boundary value problems with localized steep gradients

The wavelet multiresolution interpolation for continuous functions defined on a finite interval is developed in this study by using a simple alternative of transformation matrix.The wavelet multiresolution interpolation Galerkin method that applies this interpolation to represent the unknown function and nonlinear terms independently is proposed to solve the boundary value problems with the mixed Dirichlet-Robin boundary conditions and various nonlinearities,including transcendental ones,in which the discretization process is as simple as that in solving linear problems,and only common two-term connection coefficients are needed.All matrices are independent of unknown node values and lead to high efficiency in the calculation of the residual and Jacobian matrices needed in Newton’s method,which does not require numerical integration in the resulting nonlinear discrete system.The validity of the proposed method is examined through several nonlinear problems with interior or boundary layers.The results demonstrate that the proposed wavelet method shows excellent accuracy and stability against nonuniform grids,and high resolution of localized steep gradients can be achieved by using local refined multiresolution grids.In addition,Newton’s method converges rapidly in solving the nonlinear discrete system created by the proposed wavelet method,including the initial guess far from real solutions.

Power System Harmonic Detection Using Frequency-Domain Interpolation Wavelet Transform

Aiming at harmonic detection, fast Fourier transform can only detect integer harmonics precisely, short time Fourier transform can detect non-integer harmonics with low resolution, and some former wavelet based methods have no aliasing-reduction scheme which result in low measurement precision and poor robustness. A frequency-domain interpolation algorithm to detect harmonics is proposed by choosing Shannon wavelet. Shannon wavelet is an orthogonal wavelet possessing best ideal frequency domain localization ability, it can restrict wavelet aliasing but bring about Gibbs oscillation phenomenon simultaneously. An interpolation algorithm is developed to overcome this problem. Simulation reveals that the proposed method can effectively cancel aliasing, spectral leakage and Gibbs phenomenon, so it provides an effective means for power system harmonic analysis.

The collision-free trajectory planning for the space robot to capture a target based on the wavelet interpolation algorithm

In the research of path planning for manipulators with many DOF, generally there is a problem in most traditional methods, which is that their computational cost (time and memory space) increases exponentially as DOF or resolution of the discrete configuration space increases. So this paper presents the collision-free trajectory planning for the space robot to capture a target based on the wavelet interpolation algorithm. We made wavelet sample on the desired trajectory of the manipulator’s end-effector to do trajectory planning by use of the proposed wavelet interpolation formula, and then derived joint vectors from the trajectory information of the end-effector based on the fixed-attitude-restrained generalized Jacobian matrix of multi-arm coordinated motion, so as to control the manipulator to capture a static body along the desired collision-free trajectory. The method overcomes the shortcomings of the typical methods, and the desired trajectory of the end-effector can be any kind of complex nonlinear curve. The algorithm is simple and highly effective and the real trajectory is close to the desired trajectory. In simulation, the planar dual-arm three DOF space robot is used to demonstrate the proposed method, and it shows that the algorithm is feasible.

2-D NONSEPARABLE SCALING FUNCTIONINTERPOLATION AND APPROXIMATION

The authors introduce nonseparable scaling function interpolation and show that its approximation can provide similar convergence properties as scalar wavelet system. Several equivalent statements of accuracy of nonseparable scaling function are also given. In the numerical experiments, it appears that nonseparable scaling function interpolation has better convergence results than scalar wavelet systems in some cases.

HAAR EXPANSIONS OF A CLASS OF FRACTAL INTERPOLATION FUNCTIONS AND THEIR LOGICAL DERIVATIVES

In this paper,we study a special class of fractal interpolation functions,and give theirHaar-wavelet expansions.On the basis of the expansions,we investigate the H(o|¨)ldersmoothness of such functions and their logical derivatives of order α.

New Edge-Directed Interpolation Based-Lifting DWT and MSPIHT Algorithm for Image Compression

The amount of image data generated in multimedia applications is ever increasing. The image compression plays vital role in multimedia applications. The ultimate aim of image compression is to reduce storage space without degrading image quality. Compression is required whenever the data handled is huge they may be required to sent or transmitted and also stored. The New Edge Directed Interpolation (NEDI)-based lifting Discrete Wavelet Transfrom (DWT) scheme with modified Set Partitioning In Hierarchical Trees (MSPIHT) algorithm is proposed in this paper. The NEDI algorithm gives good visual quality image particularly at edges. The main objective of this paper is to be preserving the edges while performing image compression which is a challenging task. The NEDI with lifting DWT has achieved 99.18% energy level in the low frequency ranges which has 1.07% higher than 5/3 Wavelet decomposition and 0.94% higher than traditional DWT. To implement this NEDI with Lifting DWT along with MSPIHT algorithm which gives higher Peak Signal to Noise Ratio (PSNR) value and minimum Mean Square Error (MSE) and hence better image quality. The experimental results proved that the proposed method gives better PSNR value (39.40 dB for rate 0.9 bpp without arithmetic coding) and minimum MSE value is 7.4.

Numerical Solution of Freholm-Volterra Integral Equations by Using Scaling Function Interpolation Method

Wavelet methods are a very useful tool in solving integral equations. Both scaling functions and wavelet functions are the key elements of wavelet methods. In this article, we use scaling function interpolation method to solve Volterra integral equations of the first kind, and Fredholm-Volterra integral equations. Moreover, we prove convergence theorem for the numerical solution of Volterra integral equations and Freholm-Volterra integral equations. We also present three examples of solving Volterra integral equation and one example of solving Fredholm-Volterra integral equation. Comparisons of the results with other methods are included in the examples.

CURVE AND SURFACE INTERPOLATIONBY SUBDIVISION ALGORITHMS

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京津冀地区近50 a降水时空变化分析

为了明确京津冀地区近年来降水时空变化特征,使用线性回归、空间插值以及M-K显著性检验、相关性分析、小波分析等方法,结合地区DEM数据,对近50 a降水逐日、逐月数据进行处理分析。结果显示,京津冀地区年降水量呈不显著增加趋势,且存在平均周期约20 a的周期性变化;年降水的不稳定性在增强,除春季外,其余季节降水的不稳定性均在增强;夏季降水与年降水具有最强的相关性;年降水量呈现自东南沿海向西北内陆递减规律,春季、秋季、冬季降水呈现两侧多、中间少的特征,夏季降水与年降水分布特征一致;绝大部分地区年降水量呈增加趋势,增幅的空间分布与年降水量的空间分布趋势相反,夏季降水与年降水在空间上仍然具有最高的一致性。

Anti-aliasing wavelet packet method for harmonic detection

A novel anti-aliasing wavelet packet transform method for harmonic detection is proposed. Aiming at the low measurement precision and poor robustness which exists in the former traditional wavelet methods for lack of the aliasing reduction scheme, an optimal interpolation wavelet packet filter is designed according to new optimal criteria. First, the limitation of anti-aliasing on the traditional wavelet filter bank is analyzed. Second, the designed optimal interpolation filters are denoted, and then the solution algorithm is given. This devised wavelet packet filter can seek a reasonable balance between signal preservation and aliasing reduction; it overcomes the inherent bug of traditional wavelet transforms, which rooted from just only concerning total aliasing cancellation but not aliasing-reduction in decomposition. Simulation and several comparative results indicate that the proposed method can effectively eliminate aliasing and precisely extract harmonic information.

Wavelet Optimized Adaptive Mesh for MHD Flow Problems

There are many problems in science and engineering where the solution shows a boundary layer character. Near the boundary the gradient is large in contrast with the smooth behaviour in the central core. A uniform grid is, therefore, not suitable for a numerical solution. MHD flow problems belong to this category where a velocity and induced magnetic field profiles get flattened in a transverse flow. In the present paper an optimized grid has been generated using interpo-lating wavelets. The results are compared with those obtained using uniform grid, the finite element method and also from the analytical solution.

COMPACTLY SUPPORTED NON-TENSOR PRODUCT FORM TWO-DIMENSION WAVELET FINITE ELEMENT

Some theorems of compactly supported non-tensor product form two-dimension Daubechies wavelet were analysed carefully. Compactly supported non-tensor product form two-dimension wavelet was constructed, then non-tensor product form two dimension wavelet finite element was used to solve the deflection problem of elastic thin plate. The error order was researched. A numerical example was given at last.

A Wavelet Based Method for the Solution of Fredholm Integral Equations

In this article, we use scaling function interpolation method to solve linear Fredholm integral equations, and we prove a convergence theorem for the solution of Fredholm integral equations. We present two examples which have better results than others.

Analysis of a Gyroscope′s Rotor Nonlinear Supported Magnetic Field Based on the B-Spline Wavelet-FEM

A supported framework of a gyroscope′s rotor is designed and the B-Spline wavelet finite element model of nonlinear supported magnetic field is worked out. A new finite element space is studied in which the scaling function of the B-spline wavelet is considered as the shape function of a tetrahedron. The magnetic field is spited by an artificial absorbing body which used the condition of field radiating, so the solution is unique. The resolution is improved via the varying gradient of the B-spline function under the condition of unchanging gridding. So there are some advantages in dealing with the focus flux and a high varying gradient result from a nonlinear magnetic field. The result is more practical. Plots of flux and in the space is studied via simulating the supported system model. The results of the study are useful in the research of the supported magnetic system for the gyroscope rotor.

中国大陆连续GPS时间序列垂向周期信号空间分布

利用中国大陆构造环境监测网络连续GPS基准站数据,揭示垂向周期振幅及相位的空间分布特征。结果表明,各站垂向周期信号表现出明显的季节变化,半年周期信号的平均振幅只有年周期信号的1/4,二者呈现出不同的空间分布。年周期信号的振幅在空间上自西向东逐渐递减,东南沿海地区振幅较小,川滇地区年周期振幅较大;半年周期振幅自西南向东北逐渐减小,华北地区和川滇地区的半年周期现象较为明显。年周期与半年周期信号的相位有较大差异,年周期信号的峰值出现在2月到8月之间,半年周期信号峰值大多出现在1月和5月,并且有明显的分区特征。年周期信号峰值出现的时间会随纬度的增大逐渐变晚,同纬度下沿海地区的年周期信号峰值出现月份晚于内陆;半年周期信号峰值在西南、东北、华北地区多出现在5月,在华南、华中、西北北部地区多出现在1月。