Biorthogonal Multiwavelets with Sampling Property and Application in Image Compression

This paper discusses biorthogonal multiwavelets with sampling property. In such systems, vector-valued refinable functions act as the sinc function in the Shannon sampling theorem, and their corresponding matrix-valued masks possess a special structure. In particular, for the multiplicity \(r=2\), a biorthogonal multifilter bank can be reduced to two scalar-valued filters. Moreover, if the vector-valued scaling functions are interpolating, three different concepts: balancing order, approximation order and analysis-ready order, will be shown to be equivalent. Based on this result, we introduce the transferring armlet order for constructing biorthogonal balanced multiwavelets with sampling property. Also, some balanced biorthogonal multiwavelets will be obtained. Finally, application of biorthogonal interpolating multiwavelets in image compression is discussed. Experiments show that for the same length, the biorthogonal multifilter bank is superior to the orthogonal case. Moreover, certain biorthogonal interpolating multiwavelets are also better than the classical Daubechies wavelets.     

The Fusion of Image Based on OPTFR-Multiwavelets

1　Introduction　In recent years, the study of multiwavelets as an extensionfrom scalar wavelets has received considerable of attentionfrom the wavelets research communities both in theory[1~4]as well as in applications such as signal compression and de noising[5~7]. It was shown in papers[1,8] that symmetry, or thogonality, compact support and approximation order r>1can be simultaneously achieved for multiwavelets althoughthis is not possible for scalar wavelets. But in papers[12~13],i…     

Multiwavelet moments and projection prefilters

An efficient projection procedure is derived for use of orthogonal multiwavelets in the analysis of discrete data sequences. A family of simple prefilters corresponding to numerical quadrature evaluation of the projection integrals provides exact results for locally polynomial data. The full approximation order of the multiwavelet basis can thus always be enabled. For nonpolynomial signals, the prefilters provide approximations to the coefficients of the multiwavelet series whose convergence accelerates quickly with increase in sampling rate. Comparison is also made with previous time-invariant multiwavelet prefilters     

Residue-division multiplexing for discrete-time signals

Multiplexing is a technique for dividing a single transmission channel into a number of virtual subchannels. The present paper introduces a new multiplexing system for discrete-time signals based on a polynomial factorization. In this multiplexing system, a linear-filter additive-noise channel is decomposed into independent subchannels that are also modeled as linear-filter additive-noise channels. A configuration and an analysis of the multiplexing system as applied to mobile communications are described in detail, based on a specific selection of polynomial factorization. Multicarrier code-division multiple access (MC-CDMA) is receiving much attention in the field of mobile communications because of its time and frequency diversity property. As is the case with MC-CDMA, in the proposed multiplexing system, the subchannel information is dispersed uniformly in both frequency and time so that degradation localized in frequency or time average out over the subchannels. The averaging has the effect of decreasing the total error probability of transmission. Unlike MC-CDMA however, the multiplexing system creates multiple carriers simply by up-sampling a single complex sinusoidal carrier for each subchannel user. Because of the simple mechanism, the implementation cost is less than that of MC-CDMA     

Vector-valued wavelets and vector filter banks

In this paper, we introduce vector-valued multiresolution analysis and vector-valued wavelets for vector-valued signal spaces. We construct vector-valued wavelets by using paraunitary vector filter bank theory. In particular, we construct vector-valued Meyer wavelets that are band-limited. We classify and construct vector-valued wavelets with sampling property. As an application of vector-valued wavelets, multiwavelets can be constructed from vector-valued wavelets. We show that certain linear combinations of known scalar-valued wavelets may yield multiwavelets. We then present discrete vector wavelet transforms for discrete-time vector-valued (or blocked) signals, which can be thought of as a family of unitary vector transforms     

The application of multiwavelet filterbanks to image processing

Multiwavelets are a new addition to the body of wavelet theory. Realizable as matrix-valued filterbanks leading to wavelet bases, multiwavelets offer simultaneous orthogonality, symmetry, and short support, which is not possible with scalar two-channel wavelet systems. After reviewing this theory, we examine the use of multiwavelets in a filterbank setting for discrete-time signal and image processing. Multiwavelets differ from scalar wavelet systems in requiring two or more input streams to the multiwavelet filterbank. We describe two methods (repeated row and approximation/deapproximation) for obtaining such a vector input stream from a one-dimensional (1-D) signal. Algorithms for symmetric extension of signals at boundaries are then developed, and naturally integrated with approximation-based preprocessing. We describe an additional algorithm for multiwavelet processing of two-dimensional (2-D) signals, two rows at a time, and develop a new family of multiwavelets (the constrained pairs) that is well-suited to this approach. This suite of novel techniques is then applied to two basic signal processing problems, denoising via wavelet-shrinkage, and data compression. After developing the approach via model problems in one dimension, we apply multiwavelet processing to images, frequently obtaining performance superior to the comparable scalar wavelet transform.     

Optimal differentiation based on stochastic signal models

The problem of estimating the time derivative of a signal from sampled measurements is addressed. The measurements may be corrupted by colored noise. A key idea is to use stochastic models of the signal to be differentiated and of the measurement noise. Two approaches are suggested. The first is based on a continuous-time stochastic process as a model of the signal. The second uses a discrete-time ARMA model of the signal and a discrete-time approximation of the derivative operator. Digital differentiators are presented in a shift operator polynomial form. They minimize the mean-square estimation error, and are calculated from a linear polynomial equation and a polynomial spectral factorization. The three obstacles to perfect differentiation, namely a finite smoothing lag, measurement noise, and aliasing effects due to sampling, are discussed     

Generalized Discrete Multiwavelet Transform With Embedded Orthogonal Symmetric Prefilter Bank

Prefilters are generally applied to the discrete multiwavelet transform (DMWT) for processing scalar signals. To fully utilize the benefit offered by DMWT, it is important to have the prefilter designed appropriately so as to preserve the important properties of multiwavelets. To this end, we have recently shown that it is possible to have the prefilter designed to be maximally decimated, yet preserve the linear phase and orthogonal properties as well as the approximation power of multiwavelets. However, such design requires the point of symmetry of each channel of the prefilter to match with the scaling functions of the target multiwavelet system. It can be very difficult to find a compatible filter bank structure; and in some cases, such filter structure simply does not exist, e.g., for multiwavelets of multiplicity 2. In this paper, we suggest a new DMWT structure in which the prefilter is combined with the first stage of DMWT. The advantage of the new structure is twofold. First, since the prefiltering stage is embedded into DMWT, the computational complexity can be greatly reduced. Experimental results show that an over 20% saving in arithmetic operations can be achieved comparing with the traditional DMWT realizations. Second, the new structure provides additional design freedom that allows the resulting prefilters to be maximally decimated, orthogonal and symmetric even for multiwavelets of low multiplicity. We evaluated the new DMWT structure in terms of computational complexity, energy compaction ratio as well as the compression performance when applying to a VQ based image coding system. Satisfactory results are obtained in all cases comparing with the traditional approaches.     

Design of prefilters for discrete multiwavelet transforms

The pyramid algorithm for computing single wavelet transform coefficients is well known. The pyramid algorithm can be implemented by using tree-structured multirate filter banks. The authors propose a general algorithm to compute multiwavelet transform coefficients by adding proper premultirate filter banks before the vector filter banks that generate multiwavelets. The proposed algorithm can be thought of as a discrete vector-valued wavelet transform for certain discrete-time vector-valued signals. The proposed algorithm can be also thought of as a discrete multiwavelet transform for discrete-time signals. The authors then present some numerical experiments to illustrate the performance of the algorithm, which indicates that the energy compaction for discrete multiwavelet transforms may be better than the one for conventional discrete wavelet transforms     

GHM类正交多小波滤波器组的因子化和参数化

本文提出了GHM类正交多小波滤波器组的完备的因子化形式和参数化方法。可用于这类多小波的优化设计和有效实现，同时给出了时频最优准则下的设计实例。     

Multiwavelet prefilters. II. Optimal orthogonal prefilters

For pt.I see IEEE Trans. Circuits Syst. II, vol.45, p.1106-12 (1998). Prefiltering a given discrete signal has been shown to be an essential and necessary step in applications using unbalanced multiwavelets. In this paper, we develop two methods to obtain optimal second-order approximation preserving prefilters for a given orthogonal multiwavelet basis. These procedures use the prefilter construction introduced in Hardin and Roach (1998). The first prefilter optimization scheme exploits the Taylor series expansion of the prefilter combined with the multiwavelet. The second one is achieved by minimizing the energy compaction ratio (ECR) of the wavelet coefficients for an experimentally determined average input spectrum. We use both methods to find prefilters for the cases of the DGHM and Chui-Lian (CL) multiwavelets. We then compare experimental results using these filters in an image compression scheme. Additionally, using the DGHM multiwavelet with the optimal prefilters from the first scheme, we find that quadratic input signals are annihilated by the high-pass portion of the filter bank at the first level of decomposition.     

Rate of Convergence in Approximating the Spectral Factor of Regular Stochastic Sequences

Common methods for the calculation of the spectral factorization rely on an approximation of the given spectral density by a polynomial and a subsequent factorization of this polynomial. It is known that the regularity of the stochastic sequence determines the achievable approximation rate of its spectrum. However, since the approximative polynomial should be factorized, it has to be positive. It is shown that this restriction on the approximation polynomial implies a limitation on the approximation rate for linear methods whereas for nonlinear methods the optimal approximation rate can still be achieved. This has also consequences for the rate of convergence of the spectral factor, which is investigated in the second part. There, a lower and an upper bound for the error in the spectral factor is derived, which shows the dependency on the approximation degree and on the regularity of the stochastic sequence. Finally, if the spectral density is given only on a finite set of sampling points, no linear approximation method exists such that the error in the spectral factor can be controlled by the approximation degree.      

Approximation power of biorthogonal wavelet expansions

This paper looks at the effect of the number of vanishing moments on the approximation power of wavelet expansions. The Strang-Fix conditions imply that the error for an orthogonal wavelet approximation at scale a=2^-i globally decays as a^N, where N is the order of the transform. This is why, for a given number of scales, higher order wavelet transforms usually result in better signal approximations. We prove that this result carries over for the general biorthogonal case and that the rate of decay of the error is determined by the order properties of the synthesis scaling function alone. We also derive asymptotic error formulas and show that biorthogonal wavelet transforms are equivalent to their corresponding orthogonal projector as the scale goes to zero. These results strengthen Sweldens earlier analysis and confirm that the approximation power of biorthogonal and (semi-)orthogonal wavelet expansions is essentially the same. Finally, we compare the asymptotic performance of various wavelet transforms and briefly discuss the advantages of splines. We also indicate how the smoothness of the basis functions is beneficial in reducing the approximation error     

Sorting continuous-time signals: analog median and median-typefilters

There are two natural orderings in signals: temporal order and rank order. There is no compelling reason to explore only one of these orderings, either in the discrete-time or in the continuous-time case. Nevertheless, the concept of rank order for continuous-time signals remains virtually neglected, which is in striking contrast to the discrete-time case: ranked order discrete-time filters, of which the running median is the most common example, have been intensively studied for three decades. The dependence of these nonlinear systems on the order statistics of the input samples stands in contrast with the tapped delay line filter, which depends on temporal order only. However, continuous-time signals can also be meaningfully sorted: a fact that is explored in this paper to define and study the analog median filter and other ranked-order filters. The paper introduces the basic tools needed to analyze and understand these continuous-time nonlinear filters (the distribution function and the sorting) and presents some of their properties in a tutorial way. The analog median filter is defined in terms of the (unique) nonincreasing left-continuous sorting. More general filters can also be defined, including filters similar to α-trimmed mean filters and L filters. These include filters that depend on one parameter and contain the running average and running median as special cases. The rate of convergence of the digital median filter to the analog median filter is discussed and related to the signal sampling period, the duration of the filter window, and the smoothness of the input signal. The paper introduces the concept of noise width and studies the effect of additive and multiplicative noise at the output of the analog median filter in terms of the noise width and the smoothness of the input signal     

一类具有线性相位的二重多小波的构造

给出由实单一紧支撑正交的小波构造二重正交多小波的方法。具体地,首先由实单一的紧支撑尺度函数构造出单一紧支撑正交对称的复尺度函数,再由构造出的复尺度函数去构造二重正交紧支撑多尺度函数,然后给出由二重尺度函数构造二重小波的显式公式。紧支撑正交的单一小波除Haar小波外不具有任何对称性,它用作滤波器不可能有线性相位,而由实单一紧支撑正交的尺度函数构造出的二重尺度函数却是对称的,对应的二重小波可以是对称或反对称的,从而使得这种小波在信号处理的过程中具有线性相位。最后给出相应的构造算例。     

Factorizations fornD polynomial matrices

In this paper, a constructive general matrix factorization scheme is developed for extracting a nontrivial factor from a givennD (n>2) polynomial matrix whose maximal order minors satisfy certain conditions. It is shown that three classes ofnD polynomial matrices admit this kind of general matrix factorization. It turns out that minor prime factorization and determinantal factorization are two interesting special cases of the proposal general factorization. As a consequence, the paper provides a partial solution to an open problem of minor prime factorization as well as to a recent conjecture on minor prime factorizability fornD polynomial matrices. Three illustrative examples are worked out in detail.     

Balanced multiwavelets theory and design

This article deals with multiwavelets, which are a generalization of wavelets in the context of time-varying filter banks and with their applications to signal processing and especially compression. By their inherent structure, multiwavelets are fit for processing multichannel signals. This is the main issue in which we are interested. First, we review material on multiwavelets and their links with multifilter banks and, especially, time-varying filter banks. Then, we have a close look at the problems encountered when using multiwavelets in applications, and we propose new solutions for the design of multiwavelets filter banks by introducing the so-called balanced multiwavelets     

A centrosymmetric kernel decomposition for time-frequency distribution computation

Time-frequency distributions (TFDs) are bilinear transforms of the signal and, as such, suffer from a high computational complexity. Previous work has shown that one can decompose any TFD in Cohen's class into a weighted sum of spectrograms. This is accomplished by decomposing the kernel of the distribution in terms of an orthogonal set of windows. In this paper, we introduce a mathematical framework for kernel decomposition such that the windows in the decomposition algorithm are not arbitrary and that the resulting decomposition provides a fast algorithm to compute TFDs. Using the centrosymmetric structure of the time-frequency kernels, we introduce a decomposition algorithm such that any TFD associated with a bounded kernel can be written as a weighted sum of cross-spectrograms. The decomposition for several different discrete-time kernels are given, and the performance of the approximation algorithm is illustrated for different types of signals.     

Nonparametric estimation of instantaneous frequency

The local polynomial approximation of time-varying phase is used in order to estimate the instantaneous frequency and its derivatives for a complex-valued harmonic signal given by discrete-time observations with a noise. The considered estimators are high-order nonparametric generalizations of the short-time Fourier transform and the Wigner-Ville distribution. The asymptotic variance and bias of the estimates are obtained     

Unified approach for constructing multiwavelets with approximation order using refinable super-functions

A unified approach for constructing a large class of multiwavelets is presented. This class includes Geronimo-Hardin-Massopust (1994), Alpert (1993), finite element and Daubechies-like multiwavelets. The approach is based on the characterisation of approximation order of r multiscaling functions using a known compactly supported refinable super-function. The characterisation is formulated as a generalised eigenvalue equation. The generalised left eigenvectors of the finite down-sampled convolution matrix L/sub f/ give the coefficients in the finite linear combination of multiscaling functions that produce the desired super-function. The unified approach based on the super-function theory can be used to construct new multiwavelets with short support, high approximation order and symmetry.