Recursive Orthgonal Wavelet Function

本文提出了一种一维和二维递归小波函数的设计方法，并探讨了其在数字信号处理中的应用。在这种方法中，为了满足正交性，我们使用了由一些延迟和一维和二维全通滤波器组成的并行连接。此外，在低通滤波器的混叠频率处放置最大数量的零以获得规则性。通过求解联立方程，可以很容易地得到具有一定规律性和任意阶的IIR传递函数。建立了基于迭代滤波器组的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。其次，我们考虑了它在数字信号处理、奇点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割成多个分辨率空间。每个多分辨率空间使用IFS（迭代函数系统）编码。在这个时候，IFS代码是根据每个空间的特点来制定的。该方法可获得较好的重建图像。

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.