淺談小波分析理論及其應用
淺談小波分析理論及其應用
摘要: 小波分析是當前數學中1個迅速發展的新領域，它同時具有理論深刻和應用10分廣泛的雙重意義。它無論對數學還是對工程上的應用都具有重要價值。然而，小波分析的理論和方法還正處于發展之中,人們正從理論上和實踐上不斷探索小波中各種問題和應用方法，所以筆者預言小波分析的前景是10分廣闊的。
本先從認識小波下筆，接著客觀描述了小波分析的理論和應用研究現狀及對它們發展趨勢的展望，既而簡單探討了小波分析和傅里葉分析在基本思想、特征和性質等方面的異同點及小波分析應用應當注意的1些問題。然后，著重利用數學工具MATLAB 7探討了小波分析在當今的應用現狀。
其中，重點結合例子探討了小波分析在圖象壓縮和數字水印技術中的應用優勢。通過兩個例子分析表明，小波分析應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用的、有效的工具；小波分析在數字水印技術中，小波變換結合信息加密技術，能夠更好地保護數字產品安全傳輸，并可以作為存取控制和征收費用的手段，最重要的是還能解決數字產品解密后的盜版問題，這是密碼學的加密技術保護數字產品所不能做到的。最后，作者還結合當前形勢大膽地指出了小波分析當前存在的問題并對小波分析的美好發展前景進行了預測和展望。
關鍵詞:小波分析;傅里葉分析;圖像壓縮；數字水印；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
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