【作者】 王慧；

【导师】 曾理；

【作者基本信息】 重庆大学 ， 运筹学与控制论， 2003， 硕士

【摘要】 小波分析是近年来出现的崭新而有力的数学工具，由于其良好的局部化特性和弹性的时-频窗特点，而被认为是调和分析这以纯数学重要领域半个世纪以来工作之结晶，也是信号与图像分析、量子物理等科学和工程技术近十年来在数学方法上的重大突破，从而被认为是应用数学的新趋势。到目前为止，小波分析已经被广泛的应用在信号分析、图像处理、量子力学、雷达、计算机识别、模式识别、边缘检测等诸多领域。本文的研究工作分为三部分。首先，讨论了对于尺度函数，相应的母小波构成空间的标准正交基的充要条件，提出了构造a尺度母小波的算法，从理论上研究a尺度Haar小波基的构造，提出了分解与重构公式，并对如何构造具有对称性的a尺度Haar小波基进行了探讨。其次，讨论了多小波尺度函数与其生成的多小波正交的充分必要条件。并着重研究了a尺度r重正交多小波和因果有限脉冲响应多滤子库的构造，并在Jiang[9]的基础上利用矩阵因式分解理论给出了4尺度2重对称/反对称多小波的完备化参数表示，我们发现构造对称/反对称的尺度函数和小波函数，可以通过对参数的选取来控制它们的正交性和支集长度。最后，研究了多小波离散小波变换系数逼近的误差。由于多小波变换系数的计算是直接与其预滤波方式相联系的，而在单小波变换系数计算中之所以能由近似表示就是因为尺度函数具有低通特性和平移正交性，由此我们研究了能满足以上要求的预滤波，并对离散小波变换系数和连续小波变换系数之间的差异作了分析，从误差分析结果我们可以预见：为了减少误差我们可以通过对预滤波的进一步设计来控制。更多还原

【Abstract】 Wavelet transform, which is a new powerful mathematical tool appeared in recent years, is regarded as the crystal work of pure mathematical field and the great breakthrough in signal-picture analysis and quantum physics, and is thought as the new trend of applied mathematics. Because of its good time-frequency localization and flexible time-frequency windows, wavelet transform has been widely applied in such as signal study, picture process, quantum mechanics, radar, computer identification, earthquake exploration, so on and so forth. Multiwavelets have been studied recently. Where several mother wavelet functions were used to expand a function, it also can be seen as vector-valued wavelets that satisfy conditions in which matrics are involved.The main works of this paper are as follows:Firstly, although Haar function has bad vanish property in frequency domain, it is the only normal orthonormal basis with symmetry and real short-support property. We study the design of Haar wavelet for scale=a (a2) and present a decomposition and reconstruction algorithm in Chapter 3.Secondly, In Chapter 4 we study the design of orthonormal mutiwavelets of multiplicity r with scale=a (a2) .By the factorization theory ,we give parametric expressions for orthonormal causal FIR multifilter banks of r=2 and scale=4,and we found the length of scaling function can be controlled by the parameters.Finally, we provide the error analysis between discrete multiwavelet transform coefficients and continue multiwavelet transform coefficients.更多还原