# 小波变换及在图像压缩中的应用

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:

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.

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.

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

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Abstract.................................................................................................................................. II

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§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

§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

§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

§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

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