浅谈小波分析理论及其应用

浅谈小波分析理论及其应用

摘要: 小波分析是当前数学中1个迅速发展的新领域，它同时具有理论深刻和应用10分广泛的双重意义。它无论对数学还是对工程上的应用都具有重要价值。然而，小波分析的理论和方法还正处于发展之中,人们正从理论上和实践上不断探索小波中各种问题和应用方法，所以笔者预言小波分析的前景是10分广阔的。

本论文先从认识小波下笔，接着客观描述了小波分析的理论和应用研究现状及对它们发展趋势的展望，既而简单探讨了小波分析和傅里叶分析在基本思想、特征和性质等方面的异同点及小波分析应用应当注意的1些问题。然后，着重利用数学工具MATLAB 7探讨了小波分析在当今的应用现状。

其中，重点结合例子探讨了小波分析在图象压缩和数字水印技术中的应用优势。通过两个例子分析表明，小波分析应用在图像压缩中压缩比较高，压缩效果较好，可以说是图像压缩处理的实用的、有效的工具；小波分析在数字水印技术中，小波变换结合信息加密技术，能够更好地保护数字产品安全传输，并可以作为存取控制和征收费用的手段，最重要的是还能解决数字产品解密后的盗版问题，这是密码学的加密技术保护数字产品所不能做到的。最后，作者还结合当前形势大胆地指出了小波分析当前存在的问题并对小波分析的美好发展前景进行了预测和展望。

关键词:小波分析;傅里叶分析;图像压缩；数字水印；小波变换；MATLAB 7

Theory and Application on Wavelet

Abstract : Wavelet analysis is a rapid developing new field in modern mathematics. It also has dual significance on profound theory and very wide-ranging application. It is very important for whether mathematics itself or application to engineering. However, the theory and methods on wavelet analysis are developing, and people are continuous exploring various theoretically issues and practically application methods on wavelet. Therefore, I predict that the prospects for wavelet analysis is very broad.

In this paper, we start with the understanding on wavelet to begin writing. Then describe objectively the status quo and their development trend to the theory and applied research on wavelet analysis. Subsequently explore simply the similarities and differences between wavelet analysis and Fourier analysis on the basic idea, characteristics and nature and some problems with respect to the application on wavelet analysis. Then, focus on the usage of the mathematical tool MATLAB 7 to explore the application in today on wavelet analysis by analyzing some examples.

Among them, we illustrate mainly the advantages of the application wavelet analysis to image compression and digital watermarking technology. By analyzing two routines we see that applying wavelet analysis to image compression can get the compressing image which possess high compression ratio and well compression result, so wavelet analysis can be said to be a practical and effective tool on the image compression. When use wavelet analysis to digital watermarking technology, together wavelet transform with information encryption technology make the transmission of the digital products more secure, and it could serve as access control and collection costs means The most important thing is to protect the decrypted digital products from the piracy problem. This is unable to do for the encryption technology of cryptography to protect digital products. Finally, considering the current situation, the author points out the current problems on wavelet analysis and make a prospect and forecast for the gloriously developing perspective of wavelet analysis.

Keywords : wavelet analysis; Fourier analysis; image compression; digital watermarking; wavelet transform; MATLAB 7