小波分析是上个世纪八十年代初发展起来的新兴数学分支，它无论是对数学，还是对其他应用学科都产生了深远的影响。小波分析的出现，是不同学科、不同领域的交流与交叉学科发展的结晶。图像压缩是小波分析中十分重要的一个应用，目前小波分析在图像压缩中的应用已经十分广泛。本文所作的主要工作具体如下：
⑴ 在第二章中主要讨论了有关连续小波变换的几种不同定义，分析其联系，然后举了一些常用的连续小波例子，并通过定义验证了Haar和Mexican帽为基小波；给出了一种基于卷积的基小波的构造方法，并证明之；证明了尺度函数和多分辨分析(MRA)产生的半正交小波是基小波。详细的介绍了双正交多分辨分析；研究了图形显示算法，推导了二文情形的图形显示算法，并实现了一文情形的图形显示算法。
⑵ 在第三章中探讨了在图像压缩中小波滤波器选取的原则；研究了矩阵法构造小波滤波器的方法，对其前提条件进行总结，研究了如何确定滤波器长度与消失矩的阶数，以此构造出几种小波；对各种滤波器进行小波编码的仿真实验，实验表明新小波的性能很好。
⑶ 在第四章中讨论了有关图像压缩的一些问题，讨论了JPEG、WSQ、EZW等算法；最后采用小波分解与矢量量化结合的压缩方法，在分裂法产生初始码书的过程中充分考虑小波分解的特性，采用合适的参数进行分裂，并在编码搜索码字的时候考虑人眼对低频部分敏感而对高频部分不敏感，从而对误差采取加权的形式，这样在一定程度上可以保证重要的低频部分误差得到控制。
关键词：容许小波；连续小波变换；小波滤波器；小波编码；图像压缩。
Wavelet analysis is a new branch of mathematics developed from earlier 1980s, it has far-reaching influence not only on mathematics but many other application fields. The emergence of wavelet analysis is the result of a multidisciplinary effort that brought together many intersect fields. Image compression is an important application of wavelet analysis, now the application of wavelet analysis on image compression is very popular. The main work is as follows:
⑴ In chapter 2, the author discusses several definitions of CWT, tells the difference and relation between them, then gives some common example, and prove Haar and Mexihat to be basic wavelet. One method for constructing basic wavelet based on convolution is put forward, and is proved; the paper proves that wavelet produced by scaling function and MRA is basic wavelet. Detailed knowledge of biorthogonal multiresolution analysis is introduced; and studies Interpolation graphical display algorithm(IDGA), derived the 2-deminision case of IDGA, and then give an implemented example of IDGA.
⑵ In chapter 3, principia for choosing wavelet filter in image compression is discussed; the paper studies the matrix method of constructing wavelet filters, summarize some precondition, make how to confirm the length and vanish moments of filters, and construct some new filters with it; at last experiments for wavelet coding is done using kinds of filters, and the result show that the new wavelet filters have good performance.
⑶ In chapter 4, some issue of image compression as well as JPEG, WSQ, EZW are discussed; and then the author use the wavelet decomposition and vector quantization for image coding, considering characteristic of wavelet decomposition, use an appropriate parameter to split when producing initial codebook. When searching codeword we use weighted error to control the error of low frequency part for human eyes are sensitive to the low frequency part and not sensitive to high frequency part.
Keywords: admissible wavelet; continuous wavelet transform; wavelet filters; wavelet coding; image compression
摘 要...................................................................................................................................... I
Abstract.................................................................................................................................. II
目 录................................................................................................................................... III
第一章 绪 论........................................................................................................................... 1
§1.1引言........................................................................................................................ 1
§1.2图像压缩................................................................................................................. 1
§1.3小波变换编码的优越性............................................................................................ 2
§1.4本文的主要工作....................................................................................................... 2
第二章 小波分析的基本理论.................................................................................................... 4
§2.1连续小波变换.......................................................................................................... 4
§2.2离散小波变换........................................................................................................ 10
§2.3多分辨分析............................................................................................................ 11
§2.4 双正交多分辨分析................................................................................................ 13
§2.5图形显示算法及其实现.......................................................................................... 16
§2.6小结............................................................................................. 19
第三章 小波基的选取及构造................................................................................... 20
§3.1小波基选取原则................................................................................ 20
§3.2构造小波滤波器的矩阵方法................................................................................... 22
§3.3矩阵法构造滤波器的一些条件................................................................................ 25
§3.4具体小波的构造................................................................................ 26
§3.5小波编码中滤波器选取仿真................................................................................... 29
§3.6小结........................................................................................... 32
第四章 小波变换在图像压缩中的应用.................................................................................... 33
§4.1小波编码的基本框架.............................................................................................. 33
§4.2标量量化与矢量量化.............................................................................................. 34
§4.3误差的度量.................................................................. 35
§4.4常见的图像压缩算法.............................................................................................. 35
§4.5基于小波树结构的矢量量化压缩算法..................................................................... 43
§4.6小结............................................................... 46
第五章 总结与展望................................................................................. 47
参考文献..................................................................................... 21
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