The Biorthogonality of Multiple Vector-valued Bivariate Wavelet Packets

The notion of a sort of biorthogonal multiple vector-valued bivariate wavelet packets,which are associated with a quantity dilation matrix,is introduced.The biorthogonality property of the multiple vector-valued wavelet packets in higher dimensions is studied by means of Fourier transform and integral transform biorthogonality formulas concerning these wavelet packets are obtained.

A biorthogonality relationship for three-dimensional couple stress problem

The duality solution for elasticity and the biorthogonality relationship have been well researched. Now the couple stress theory becomes a new research spot but there is few research for the biorthogonality relationship for couple stress theory comparing to classical elasticity. A new state vector is presented for three dimensional couple stress problems of prismatic structures. A new biorthogonality relation- ship of couple stress is discovered. The dual partial differential equations of couple stress problem are derived by the new state vector. By two important identical equations the new biorthogonality rela- tionship is proved based on the method of separation of variables. The symplectic orthogonality rela- tionship to three dimensional couple stress theory may be decomposed into two independently and symmetrically orthogonality relationships. The new biorthogonality relationship includes the symplec- tic orthogonality relationship. The biorthogonality relationship of couple stress may also be degener- ated into the theory of elasticity. The new state vector and biorthogonality relationship provide theo- retic foundation for the research on the schemes of separation of variables and eigenfunction expan- sion of couple stress theory.

Representation of three-dimensional mass distribution of the Earth's interior by biorthogonal series and its use for studying internal structure of the planet

This paper utilizes the mathematical concept of approximation within an ellipsoid from a single viewpoint to present the spatial mass distribution function of the Earth's interior and its internal potential.The primary focus lies in constructing the volume distribution of masses in the planet's interior, with the expansion coefficients being linear combinations of the Stokes constants. Several possible approaches are suggested for determining accurately these coefficients employing three-dimensional(biorthogonal)polynomials. By expressing the mass distribution function of the Earth's interior and its internal potential as a series, an algorithm is introduced for the calculation of gravitational energy. It allows us to estimate fluctuations in gravitational energy. The implementation of this algorithm offers the means of establishing the extent to which the Earth deviates from a state of hydrostatic equilibrium as a celestial body.Due to the aforementioned method, calculations have been conducted to validate its effectiveness and reliability. This example is given as an illustration of a given method for studying the internal structure of planets.

Super-Fast Approximation Algorithms Using Classical Fourier Tools

In the author’s recent publications, a parametric system biorthogonal to the corresponding segment of the exponential Fourier system was unusually effective. On its basis, it was discovered that knowledge of a finite number of Fourier coefficients of function f from an infinite-dimensional set of elementary functions allows f to be accurately restored (the phenomenon of over-convergence). Below, parametric biorthogonal systems are constructed for classical trigonometric Fourier series, and the corresponding phenomena of over-convergence are discovered. The decisive role here was played by representing the space L2 as an orthogonal sum of two corresponding subspaces. As a result, fast parallel algorithms for reconstructing a function from its truncated trigonometric Fourier series are proposed. The presented numerical experiments confirm the high efficiency of these convergence accelerations for smooth functions. In conclusion, the main results of the work are summarized, and some prospects for the development and generalization of the proposed approaches are discussed.

小波基在低压电力线信道有色背景噪声建模中的应用研究

小波马尔科夫链法可用于低压电力线信道中有色背景噪声的建模,但小波基函数的不同会对噪声的建模精度产生较大影响。基于常用的几种小波基函数对同一组有色背景噪声分别开展小波马尔科夫链仿真建模,并计算了建模前后的功率谱密度及其均方根误差。研究结果表明,Daubecies、Biorthogonal和Haar小波基函数中Daubecies小波基函数的建模精度较高,Haar小波基次之,Biorthogonal小波的建模效果较差。在这3种小波基函数中,Daubecies小波可选为有色背景噪声进行小波马尔科夫链建模的最佳小波基函数。

Seismic data compression based on integer wavelet transform

Due to the particularity of the seismic data, they must be treated by lossless compression algorithm in some cases. In the paper, based on the integer wavelet transform, the lossless compression algorithm is studied. Comparing with the traditional algorithm, it can better improve the compression rate. CDF (2, n) biorthogonal wavelet family can lead to better compression ratio than other CDF family, SWE and CRF, which is owe to its capability in can- celing data redundancies and focusing data characteristics. CDF (2, n) family is suitable as the wavelet function of the lossless compression seismic data.

Wavelet Adaptive Bitzerotree Coding for Video

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The Boundary Processing of Wavelet Based Image Compression

When an image, which is decomposed by bi-orthogonal wavelet bases, is reconstructed, some information will be lost at the four edges of the image. At the same time, artificial discontinuities will be introduced. We use a method called symmetric extension to solve the problem. We only consider the case of the two-band filter banks, and the results can be applied to M-band filter banks. There are only two types of symmetric extension in analysis phrase, namely the whole-sample symmetry (WS), the half-sample symmetry (HS), while there are four types of symmetric extension in synthesis phrase, namely the WS, HS, the whole-sample anti-symmetry (WA), and the half-sample anti-symmetry (HA) respectively. We can select the exact type according to the image length and the filter length, and we will show how to do these. The image can be perfectly reconstructed without any edge effects in this way. Finally, simulation results are reported. Key words edge effect - image compression - wavelet - biorthogonal bases - symmetric extension CLC number TP 37 Foundation item: Supported by the National 863 Project (20021111901010)Biography: Yu Sheng-sheng (1944-), male, Professor, research direction: multimedia information processing, SAN.

HYPERCYCLIC OPERATORS ON QUASI-MAZUR SPACES

Modifying the method of Ansari, we give some criteria for hypercyclicity of quasi-Mazur spaces. They can be applied to judging hypercyclicity of non-complete and non-metrizable locally convex spaces. For some special locally convex spaces, for example, KSthe （LF）-sequence spaces and countable inductive limits of quasi-Mazur spaces, we investigate their hypercyclicity. As we see, bounded biorthogonal systems play an important role in the construction of Ansari. Moreover, we obtain characteristic conditions respectively for locally convex spaces having bounded sequences with dense linear spans and for locally convex spaces having bounded absorbing sets, which are useful in judging the existence of bounded biorthogonal systems.

Singularity Detecting with the SecondGeneration Wavelet

Adopting the lifting scheme, introduced by Sweldens, a new wavelet is constructed. It overcomes the shortcoming of the classical wavelets that has no flexibility in the number of vanishing moment. Two examples are given to prove the second-generation wavelets's potentialities in the singularity detection of signal: a wavelet with vanishing moment and symmetry can be constructed according to the problem. Key words biorthogonal wavelet - lifting scheme - secondgeneration wavelets - singularity CLC number O 174.2 Foundation item: Supported by the National Natural Science Foundation of China (19602014)Biography: XU Bing-lian (1977-), female, Master candidate, research direction: the wavelets application.

FREQUENCY OPTIMIZATION APPROACH FOR DESIGNING M-BAND BIORTHOGONAL SYMMETRIC WAVELETS

In this paper,a new method is presented for designing M-band biorthogonal symmetric wavelets.The design problem of biorthogonal linear-phase scaling filters and wavelet filters as a quadratic programming problem with the linear constraints is formulated.The closed-form solution is given and a design example is presented.

Methods of Constructing Optimal Wavelet Filters Based on Genetic Algorithm

In this paper, algorithms of constructing wavelet filters based on genetic algorithm are studied with emphasis on how to construct the optimal wavelet filters used to compress a given image, due to efficient coding of the chromosome and the fitness function, and due to the global optimization algorithm, this method turns out to be perfect for the compression of the images.

P wave onset time picking with the B-spline biorthogonal wavelet

The seismic wave consists of many seismic phases, which contain rich geophysical information from the hypocenter, medium of seismic wave passing through and so on. It is very important to detect and pick these seismic phases for understanding the mechanism of earthquake, the Earth structure and property of seismic waves. In order to reduce or avoid the loss resulted from the earthquake, one of the important goals of seismic event detecting is to obtain its related information before and after it occurs. Because of the particularity of P wave and S wave the seismic event detecting focuses on distinguishing P and S waves and picking their onset time, it has been becoming one of the research hotspots for many geophysicists to pick the P and S wave arrival accurately and effectively.

Performance Evaluation of Wavelet Based on Human Visual System

We have constructed a compactly supported biorthogonal wavelet that approximates the modulation transfer function (MTF) of human visual system in the frequency domain. In this paper, we evaluate performance of the constructed wavelet, and compare it with the widely used Daubechies 9 7, Daubechies 9 3 and GBCW 9 7 wavelets. The result shows that coding performance of the constructed wavelet is better than Daubechies 9 3, and is competitive with Daubechies 9 7 and GBCW 9 7 wavelets. Like Daubechies 9 3 wavelet, the filter coefficients of the constructed wavelet are all dyadic fractions, and the tap is less than Daubechies 9 7 and GBCW 9 7. It has an attractive feature in the realization of discrete wavelet transform.

On the Zeros of Daubechies Orthogonal and Biorthogonal Wavelets

In the last decade, Daubechies’ wavelets have been successfully used in many signal processing paradigms. The construction of these wavelets via two channel perfect reconstruction filter bank requires the identification of necessary conditions that the coefficients of the filters and the roots of binomial polynomials associated with them should exhibit. In this paper, orthogonal and Biorthogonal Daubechies families of wavelets are considered and their filters are derived. In particular, the Biorthogonal wavelets Bior3.5, Bior3.9 and Bior6.8 are examined and the zeros distribution of their polynomials associated filters are located. We also examine the locations of these zeros of the filters associated with the two orthogonal wavelets db6 and db8.

Adaptive Biorthogonal Local Discrete Cosine Transform for Interference Excision in Direct Sequence Spread Spectrum Communications

A novel time-frequency domain interference excision technique is proposed. The technique is based on adaptive biorthogonal local discrete cosine trans form (BLDCT). It uses a redundant library of biorthogonal local discrete cosine bases and an efficient concave cost function to match the transform basis to the interfering signal. The main advantage of the algorithm over conventional trans form domain excision algorithms is that the basis functions are not fixed but ca n be adapted to the time-frequency structure of the interfering signal. It is w e ll suited to transform domain compression and suppression of various types of in terference. Compared to the discrete wavelet transform (DWT) that provides logar ithmic division of the frequency bands, the adaptive BLDCT can provide more flex ible frequency resolution. Thus it is more insensitive to variations of jamming frequency. Simulation results demonstrate the improved bit error rate (BER) perf ormance and the increased robustness of the receiver.

Modulated Lapped Biorthogonal Transform for Non-orthogonal Narrowband Interference Excision in Spread Spectrum Communications

Traditional lapped transform domain excision techniques obtain good performance at the expense of increased processing delay. Extension of transform domain filtering techniques to the lapped biorthogonal transform domain can help solve the problem. By incorporating biorthogonality into the lapped transforms, more flexibility is obtained in the design of windows. Thus transform bases with better stopband attenuation can be generated by designing windows, but not by increasing the overlapping factor. In this paper, a new modulated lapped biorthogonal transform (MLBT) with optimized windows is introduced for efficient compression of multi-tone interfering signal energy. The bit error rate (BER) performance of the receiver employing the proposed MLBT excision technique is analyzed and compared with that of the lapped transform domain excision-based receivers. Simulation results demonstrate the improved performance and increased robustness of the proposed technique.

A General Algorithm for Biorthogonal Functions and Performance Analysis of Biorthogonal Scramble Modulation System

Applying the theorems of Mobius inverse and Dirichlet inverse, a general algorithm to obtain biorthogonal functions based on generalized Fourier series analysis is introduced. In the algorithm, the orthogonal function can be not only Fourier or Legendre series, but also can be any one of all orthogonal function systems. These kinds of biorthogonal function sets are used as scramble signals to construct biorthogonal scramble modulation (BOSM) wireless transmission systems. In a BOSM system, the transmitted signal has significant security performance. Several different BOSM and orthogonal systems are compared on aspects of BER performance and spectrum efficiency, simulation results show that the BOSM systems based on Chebyshev polynomial and Legendre polynomial are better than BOSM system based on Fourier series, also better than orthogonal MCM and OFDM systems.

Tracking of Non-Rigid Object in Complex Wavelet Domain

In this paper we have proposed an object tracking method using Dual Tree Complex Wavelet Transform (DTCxWT). The proposed method is capable of tracking the moving object in video sequences. The object is assumed to be deform-able under limit i.e. it may change its shape from one frame to another. The basic idea in the proposed method is to decompose the image into two components: a two dimensional motion and a two dimensional shape change. The motion component is factored out while the shape is explicitly represented by storing a sequence of two dimensional models. Each model corresponds to each image frame. The proposed method performs well when the change in the shape in the consecutive frames is small however the 2-D motion in consecutive frames may be large. The proposed algorithm is capable of handling the partial as well as full occlusion of the object.

A Wavelet Multigrid Method Using Symmetric Biorthogonal Wavelets

In [1], the author introduced a wavelet multigrid method that used the wavelet transform to define the coarse grid, interpolation, and restriction operators for the multigrid method. In this paper, we modify the method by using symmetric biorthogonal wavelet transforms to define the requisite operators. Numerical examples are presented to demonstrate the effectiveness of the modified wavelet multigrid method for diffusion problems with highly oscillatory coefficients, as well as for advection-diffusion equations in which the advection is moderately dominant.