淺談小波分析理論及其應(yīng)用

淺談小波分析理論及其應(yīng)用

摘要: 小波分析是當(dāng)前數(shù)學(xué)中1個(gè)迅速發(fā)展的新領(lǐng)域，它同時(shí)具有理論深刻和應(yīng)用10分廣泛的雙重意義。它無論對數(shù)學(xué)還是對工程上的應(yīng)用都具有重要價(jià)值。然而，小波分析的理論和方法還正處于發(fā)展之中,人們正從理論上和實(shí)踐上不斷探索小波中各種問題和應(yīng)用方法，所以筆者預(yù)言小波分析的前景是10分廣闊的。
本論文先從認(rèn)識小波下筆，接著客觀描述了小波分析的理論和應(yīng)用研究現(xiàn)狀及對它們發(fā)展趨勢的展望，既而簡單探討了小波分析和傅里葉分析在基本思想、特征和性質(zhì)等方面的.異同點(diǎn)及小波分析應(yīng)用應(yīng)當(dāng)注意的1些問題。然后，著重利用數(shù)學(xué)工具M(jìn)ATLAB 7探討了小波分析在當(dāng)今的應(yīng)用現(xiàn)狀。
其中，重點(diǎn)結(jié)合例子探討了小波分析在圖象壓縮和數(shù)字水印技術(shù)中的應(yīng)用優(yōu)勢。通過兩個(gè)例子分析表明，小波分析應(yīng)用在圖像壓縮中壓縮比較高，壓縮效果較好，可以說是圖像壓縮處理的實(shí)用的、有效的工具；小波分析在數(shù)字水印技術(shù)中，小波變換結(jié)合信息加密技術(shù)，能夠更好地保護(hù)數(shù)字產(chǎn)品安全傳輸，并可以作為存取控制和征收費(fèi)用的手段，最重要的是還能解決數(shù)字產(chǎn)品解密后的盜版問題，這是密碼學(xué)的加密技術(shù)保護(hù)數(shù)字產(chǎn)品所不能做到的。最后，作者還結(jié)合當(dāng)前形勢大膽地指出了小波分析當(dāng)前存在的問題并對小波分析的美好發(fā)展前景進(jìn)行了預(yù)測和展望。

關(guān)鍵詞:小波分析;傅里葉分析;圖像壓縮；數(shù)字水��；小波變換；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