Recursive Orthgonal Wavelet Function

在本研究中，我们提出一个一、二维递归小波函数的设计方法，并探讨其在数位讯号处理上的应用。在该方法中，我们使用由若干延迟和一、二维全通滤波器组成的并联结构，以满足正交性。此外，在低通滤波器的混叠频率处设置最大数量的零点，以获得规律性。通过求解该方程组，可以方便地得到具有一定规律性的任意阶IIR传递函数。基于迭代滤波器组构造了一维和二维递归小波函数，虽然得到的小波函数与基于FIR滤波器构造的小波函数相似，但递归小波函数比非递归小波具有更高的正则性，并讨论了递归小波函数在数字信号处理、奇异性检测和图像压缩中的应用。在图像压缩中，利用小波变换将图像分解到多分辨率空间。每个多分辨空间都采用迭代函数系统（IFS）编码。在此基础上，本文提出了基于空间特征的IFS码。通过这种方法，可以得到良好的重建图像。

In this research, we propose a design method of 1 and 2 dimensional recursive wavelet functions and examine its applications for digital signal processing. In this method, we use the parallel connection consisted of some delays and 1 and 2-D allpass filter in order to satisfy the orthogonality. Furthemore a maximum number of zeros is put at aliasing frequencies in the lowpass filter to obtain the regularity. IIR transfer function with some degrees of regularity and arbitrary order can be easily obtained by solving the simultaneous equation. 1-D and 2-D recursive wavelet functions based on iterated filter banks are found. Although the obtained wavelet functions are similar to that based on FIR filters, recursive wavelet functions have higher regularity than nonrecursive wavelet.Next, we considered its application for digital signal processing, singularity detection and image compression. In image ompression, image is separated into multiresolution spaces by wavelet transform. Each multiresolution spaces is coded by using IFS (Iterated Function System) . In this time, IFS codes are make based on the characteristics of each spaces. By this method, good reconstruction image can be obtained.