高正则二维小波的构造及在遥感图像编码中的应用

首先根据对称正交二维小波滤波器组的阶因式分解表示,结合正则性条件,建立一组以滤波器组参数为未知数的高阶多元多项式非线性方程组,并将这一组方程分解为两个子方程组,应用计算代数中的Groebner基算法分别求出解其Groebner基后获得二维小波滤波器组的全部参数,从而构造出一种集正交性、对称性和高正则性于一体的完美的"真"二维小波;其次从二维正交多分辨分析出发,推导出二维小波变换的分解和重构快速算法;最后将构造得到的3正则阶二维小波和SPIHT编码算法相结合对某地的遥感图像进行压缩编码.实验结果显示该方法具有较好的编码性能.     

用余弦调制PR—FIR方法构造正交小波基

离散小波变换将离散时间信号分解为一系列不同分辨率下的离散近似信号和离散细节．紧支的正交规范小波与完全重构正交镜象滤波器（PR-QMF)相对应。本文在“二带”正交小波基的构造条件下．利用余弦调制完全重构滤波器组的方法．实现了正交小波基的构造，计算模拟表明该方法非常简单、有效。     

用余弦调制PR－FIR方法构造正交小波基

离散小波变换将离散时间信号分解为一系列不同分辨率下的离散近似信号和离散细节，紧支的正交规范小波与完全重构正交镜象滤波器（ＰＲ－ＱＭＦ）相对应。本文在“二带”正交小波基的构造条件下，利用余弦调制完全重构滤波器组的方法，实现了正交小波基的构造，计算模拟表明该方法非常简单、有效。     

FIR完全重建滤波器组构造双正交小波基

本文论述了由双正交完全重建滤波器组构造高度正则的双正交小波基的充分条件和构造方法，系统地研究了双正交线性相位ＦＩＲ完全重建滤波器组的解的结构和已知Ｈ０（ｚ）构造完全重建滤波器组的方法，并且实现了用单一的传递函数Ａ（ｚ）构造线性相位ＦＩＲ双正交完全重建滤波器组的设计方法。这种方法的突出优点是滤波器组分析、合成部分中的滤波器可以用数值优化的方法使两者同时逼近理想低通滤波器和理想高通滤波器，即具有良好的频率选择性，并且所有滤波器都具有线性相位的特点。该滤波器组具有良好的梯形实现结构。在具体的滤波器设计中提出了基于均方误差最小准则的特征滤波器的设计方法和基于误差最大值最小准则的Ｒｅｍｅｚ交换法。而且上述方法设计的滤波器组可以构造出具有高度正则性的光滑的双正交小波基。     

航空图像压缩的双正交小波滤波器整数化设计

在航空图像压缩中,通常采用具有线性相位、正则性、消失矩和完全重构,及适于硬件实现、实时等特性的小波。根据小波滤波器设计,提出了一种基于图像压缩的构造整数双正交小波滤波器的设计方法。从选择小波基的原则为出发点,以CDF9-7小波基为参考,以压缩效果为准则来构造出更优的双正交整数小波基,并且采用航空图像为标准训练图像,以压缩比、峰值信噪比、压缩后保留能量百分比为参数,来寻找最优的小波基。试验结果证明,此方法可以实施非常简单的、无浮点乘法的运算,因而减少运算复杂性以及降低小波硬件实现的难度。     

基于PR-QMF的扩频序列

利用小波或者小波包的变换与反变换可以构造出具有完全重构性质的滤波器组,提出一种基于完全重构正交镜像滤波器组的新型扩频序列,它不同于传统的扩频序列的二值特性,是一种多值序列。分析与仿真结果表明,小波函数与尺度函数的正交性以及滤波器组的完全重构特性使的这种新型扩频序列具有更好的抗干扰能力和组网能力。     

基于双正交小波的光阑小波滤波器

提出了一种应用于4f光学系统的光学小波滤波器的设计与制作方法.结合光学信号的优势与小波变换的优点,利用双正交小波的共轭镜像特性构造出了频域形式的分解和重构滤波器.通过将滤波器放置在4f系统的频谱面上对图像成功实现了二维分解和重构.另外还根据信息处理要求和采样器件的特性,制作了振幅型和位相型两种小波滤波器.这两套滤波器都...     

一类冗余比为2的二维DFT调制滤波器组的设计

本文构建了一类冗余比为2的二维线性相位的双原型离散傅立叶变换（DFT）调制滤波器组。利用原型滤波器的多相位分解,推导出了该滤波器组的完全重构(PR)条件。基于该PR条件,我们将滤波器组的设计归结为一个关于原型滤波器的多相位分量的无约束优化问题。由于原型滤波器是线性相位的,多相位分量之间具有一定的关系,因此我们可以简化该优化问题。仿真结果验证了滤波器组PR条件的正确性。同时,仿真表明了优化算法的有效性,设计所得的滤波器组重构误差很小、频率特性较好,基本满足实际应用的需要。      

一种完全重构双正交小波基的构造方法

在有关小波的各种应用中,合适小波基的选取是一个极为重要和棘手的问题。该文利用传递函数(或滤波器)的方法建立了一种完全重构双正交小波基的构造通用方法,利用该文提供的结论,只需要适当选择系数aij(), 就可以构造出满足特定需要性质的重构小波基。因此,该文的结论对于促进小波的应用具有十分重要的理论意义和实际意义。     

最优内插的纯二维小波构造

研究了纯二维小波的构造,并根据二维图像各向同性的谱密度模型构造了一组最优内插滤波器,将其用于提升格式中,构造了具有最优内插效果的纯二维小波滤波器组.通过MSE和能量熵实验对比,证明这种纯二维小波滤波器内插效果和能量聚积方面好于或接近于著名Neville滤波器,可以用于图像的压缩和超分辨率.     

Wavelets and filter banks: theory and design

The wavelet transform is compared with the more classical short-time Fourier transform approach to signal analysis. Then the relations between wavelets, filter banks, and multiresolution signal processing are explored. A brief review is given of perfect reconstruction filter banks, which can be used both for computing the discrete wavelet transform, and for deriving continuous wavelet bases, provided that the filters meet a constraint known as regularity. Given a low-pass filter, necessary and sufficient conditions for the existence of a complementary high-pass filter that will permit perfect reconstruction are derived. The perfect reconstruction condition is posed as a Bezout identity, and it is shown how it is possible to find all higher-degree complementary filters based on an analogy with the theory of Diophantine equations. An alternative approach based on the theory of continued fractions is also given. These results are used to design highly regular filter banks, which generate biorthogonal continuous wavelet bases with symmetries     

Discrete-time wavelet extrema representation: design and consistentreconstruction

The paper studies wavelet transform extrema and zero-crossings representations within the framework of convex representations in ℒ(Z). Wavelet zero-crossings representation of two-dimensional signals is introduced as a convex multiscale edge representation as well. One appealing property of convex representations is that the reconstruction problem can be solved, at least theoretically, using the method of alternating projections onto convex sets. It turns out that in the case of the wavelet extrema and wavelet zero-crossings representations this method yields simple and practical reconstruction algorithms. Nonsubsampled filter banks that implement the wavelet transform for the two representations are also studied in the paper. Relevant classes of nonsubsampled perfect reconstruction FIR filter banks are characterized. This characterization gives a broad class of wavelets for the representations which are derived from those of the filter banks which satisfy a regularity condition     

Design of 2D oversampled linear phase DFT modulated filter banks via modified Newton's method

In this paper, the structure of the 2D oversampled DFT modulated filter banks is analyzed and a spatial-domain condition of a filter bank without transfer function distortion is derived. Based upon the spatial-domain condition, a modified Newton's method is presented for fast design of 2D oversampled linear phase (LP) DFT modulated filter banks with nearly perfect reconstruction (NPR). We formulate the design problem into an unconstrained optimization with a fourth-order objective function, which is the weighted sum of the transfer function distortion of the filter bank and the stopband energy of the prototype filter (PF). The optimization is solved by the modified Newton's method, where each of iterations updates the PF by a set of linear equations. It is proved that the iteration process fast converges to a stationary point of the objective function. Compared with the existing methods, the new method is fast in computation and can design 2D filter banks with a large number of subbands.     

Principal component filter banks for optimal multiresolutionanalysis

An important issue in multiresolution analysis is that of optimal basis selection. An optimal P-band perfect reconstruction filter bank (PRFB) is derived in this paper, which minimizes the approximation error (in the mean-square sense) between the original signal and its low-resolution version. The resulting PRFB decomposes the input signal into uncorrelated, low-resolution principal components with decreasing variance. Optimality issues are further analyzed in the special case of stationary and cyclostationary processes. By exploiting the connection between discrete-time filter banks and continuous wavelets, an optimal multiresolution decomposition of L₂(R) is obtained. Analogous results are also derived for deterministic signals. Some illustrative examples and simulations are presented     

Subband decomposition of signals with generalized sampling

The authors generalize down and up-sampling operations by proposing block sampling. Perfect reconstruction conditions for two-band subband coding with block sampling are derived. By generalizing the sampling operation, new degrees of freedom are introduced and as a result, filter banks which were not previously possible become possible. Generalized down-sampler introduces different aliasing components than that of the traditional down-sampler. This can be used to ease some the requirements of the filter bank design problem. A constructive sampling method is proposed so that coprimeness of the transfer functions of the analysis filter banks is not only a necessary but also sufficient condition for perfect reconstruction. The results are extended to the case where the filter banks are linear periodically time-varying. The multichannel case is analyzed and the relation between unimodular matrices and perfect reconstruction filter banks is discussed     

A new class of two-channel biorthogonal filter banks and waveletbases

Proposes a novel framework for a new class of two-channel biorthogonal filter banks. The framework covers two useful subclasses: i) causal stable IIR filter banks. ii) linear phase FIR filter banks. There exists a very efficient structurally perfect reconstruction implementation for such a class. Filter banks of high frequency selectivity can be achieved by using the proposed framework with low complexity. The properties of such a class are discussed in detail. The design of the analysis/synthesis systems reduces to the design of a single transfer function. Very simple design methods are given both for FIR and IIR cases. Zeros of arbitrary multiplicity at aliasing frequency can be easily imposed, for the purpose of generating wavelets with regularity property. In the IIR case, two new classes of IIR maximally flat filters different from Butterworth filters are introduced. The filter coefficients are given in closed form. The wavelet bases corresponding to the biorthogonal systems are generated. the authors also provide a novel mapping of the proposed 1-D framework into 2-D. The mapping preserves the following: i) perfect reconstruction; ii) stability in the IIR case; iii) linear phase in the FIR case; iv) zeros at aliasing frequency; v) frequency characteristic of the filters     

Orthogonal and Biorthogonal $sqrt 3$-Refinement Wavelets for Hexagonal Data Processing

The hexagonal lattice was proposed as an alternative method for image sampling. The hexagonal sampling has certain advantages over the conventionally used square sampling. Hence, the hexagonal lattice has been used in many areas. A hexagonal lattice allows radic3, dyadic and radic7 refinements, which makes it possible to use the multiresolution (multiscale) analysis method to process hexagonally sampled data. The radic3-refinement is the most appealing refinement for multiresolution data processing due to the fact that it has the slowest progression through scale, and hence, it provides more resolution levels from which one can choose. This fact is the main motivation for the study of radic3-refinement surface subdivision, and it is also the main reason for the recommendation to use the radic3-refinement for discrete global grid systems. However, there is little work on compactly supported radic3 -refinement wavelets. In this paper, we study the construction of compactly supported orthogonal and biorthogonal radic3-refinement wavelets. In particular, we present a block structure of orthogonal FIR filter banks with twofold symmetry and construct the associated orthogonal radic3-refinement wavelets. We study the sixfold axial symmetry of perfect reconstruction (biorthogonal) FIR filter banks. In addition, we obtain a block structure of sixfold symmetric radic3-refinement filter banks and construct the associated biorthogonal wavelets.     

Replacing resamplers in filter banks without losing perfect reconstruction

The condition under which the resamplers in a filter bank can be replaced without losing perfect reconstruction is presented. This is the generalisation of the common knowledge that removing resamplers and/or inserting unimodular resamplers do not destroy the perfect reconstruction of filter banks.     

自适应小波变换更新滤波器的优化研究

基于提升格式的小波变换被称为第二代小波变换,该种结构提供了一种灵活构造非线性小波分解和重构的方法.G.Piella根据提升格式小波变换的结构特点,提出了一种不需要额外附加信息就可完成精确重构的自适应更新滤波器,但算法只说明了滤波器系数需要满足精确重构的条件,没有具体说明如何确定滤波器的系数.在先选定预测滤波器的基础上,...