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姓名 李坤享(Kun-Hsian Lee)  查詢紙本館藏   畢業系所 電機工程學系
論文名稱 利用小波轉換於影像之系統分析
(system analysis of image compression using wavelet transform)
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摘要(中) 在過去,我們只有一種訊號的分析方式,那就是著名的傅利葉轉換,基本上在一般的日常生活中的真實訊號如聲音、影像,在我們的時間域中是無法顯現這些訊號的重要部份,既然在時間域的基底之下我們無法看清楚這個訊號的特性,我們就可以換個角度來看我們的訊號,也就是說我們透過某種的轉換關係將我們的訊號轉換到另一個角度,透過這個新的視角我們便可以看出原來的時域我們所不能看到的訊號特性,進而加以訊號處理,這就是數學上所說的基底轉換也是我們訊號分析上面所謂的轉換編碼。
傅利葉轉換就是利用弦式波為基底來對訊號作轉換,經過轉換之後訊號便從時間域被轉換到了頻率域,在頻率域中訊號的特性與重要的部份都跑了出來,因此我們便可以對訊號加以處理;然而雖然傅利葉轉換可以有效的在頻率域上分析訊號,可是對於訊號有不連續的地方如影像的邊緣,傅利葉轉換是不適合的,另外傅利葉轉換對有另外一個嚴重的缺點,就是當由時域轉換到頻域之後,在頻域之中我們便無法知道時域的資訊,換言之在頻域所發生的特性我們無法知道它是發生在時域的那個時間所發生的。在1985年之後,數學家發現了一種新的基底轉換的方式,那就是小波轉換。小波轉換不同於傅利葉轉換是用永遠持續震盪且有周期性的弦式波為基底來展開訊號,小波轉換是利用尺度函數與小波函數兩組基底的脹縮平移來展開訊號,而小波基底非持續震盪的、是沒有周期性的,這種新的轉換方式可以補救傅利葉轉換的缺陷,而且小波轉換對於訊號的區域特性具有很好的分析效果,另外小波轉換非常符合人體視覺的系統,換言之在低頻訊號的周期比較長,如果探測時間很短,可能連一個周期都看不全,反之高頻訊號的周期比較短,因此總是局部的現象,如果用太長的探測時間,可能無法補捉那些細微的變化,而小波轉換具有這樣的調適性,在相位平面上,在低頻方面其時間域上面寬,頻率域上面窄;在高頻方面其時間域窄,頻率域上面寬,因此小波轉換在壓縮、除雜訊與辨識都有很好的效率。
雖然小波轉換有許多優點,而且被廣泛的應用於數位訊號處理、地震波處理等,但是卻有各種不同性質的小波族類相繼的發表,而這些小波族類又有不同的濾波器階數,而每種不同的小波族類對於不同的品質的量測法則與不同的訊號都會有不同的效果,因此一個主要的問題就是要對一個特定的應用中如何去決定一個最有效的小波族類,而我們希望可以對一些不同的小波族類對於影像的範疇作出系統上的分析,以了解那些小波族類適合來用於設計一個影像壓縮系統,經過一連串的模擬評估後,我們發現對於各種不同的影像在壓縮評估計算上面, Biorthogonal 的小波族類有表現為最好, Daubechies 的小波族類在訊號雜訊比與像點值涵蓋率方面都有很好的表現,因此對於一個小波族類的選取只剩下實現複雜度的考量了,而基本上 db2 小波族類是一個好的選擇,因為其存在整數的實現方式;如果實數的實現方式可行的話, db4 是一個好的選擇。
摘要(英) In recent years, wavelet transform is getting more and more important in signal processing. Before 1985, we only have a tool to analyze the signal, that is "Fourier analysis". Although Fourier analysis is an efficient tool, but it has some drawbacks. Any signal can be portrayed as an overlay of sinusoidal waveform of assorted frequencies. Fourier analysis uses this sinusoidal waveform to decompose the signal. Unfortunately the Fourier analysis is ill-suited to represent signal with discontinuities such as the edges of features in images. Another serious drawback of the Fourier analysis is that in transforming to the frequency domain, time information is lost. It is impossible to tell when a particular event took place.
After 1985, another powerful concept has found in mathematics and engineering researches, that is "wavelet analysis". Compare wavelets with sinusoidal waves, which are the basis of Fourier analysis, sinusoids do not have limited duration, and sinusoids are smooth and predictable. While wavelets tend to be irregular and asymmetric. Wavelet transform comes in many shapes and sizes, and new ones are invented almost daily. And any given decomposition of signal involves a pair of parts, that is approximation parts (lowpass) and the details parts (highpass). Wavelet analysis allows the use of long time intervals where we want more precise low frequency information, and shorter regions where we want high frequency information. One advantage afforded by wavelets is the ability to perform local analysis. Wavelet analysis is capable of revealing aspects of data that Fourier analysis techniques miss, aspects like trends, breakdown points, discontinuities in higher derivatives, and self-similarity. And wavelet analysis can often compress or de-noise a signal without appreciable degradation.
關鍵字(中) ★ 小波轉變
★ 影像壓縮
關鍵字(英) ★ wavelet transform
★ image compression
論文目次 Chapter 1 Introduction……………………………………………………1
1.1 The importance of image processing...………………… 1
1.2 Motivation………………………………………………….2
Chapter 2 Wavelet transform………………………………...8
2.1 Introduction…….…………………………………………...8
2.2 Some mathematical foundation….…………………………9
2.3 Wavelet transform…………………………………………13
2.3.1 Haar function approach……………………………...13
2.3.2 Discrete wavelet transform………………………….15
2.4 The theory of the orthonormal bases of compactly supports
wavelet…………………………………………………….18
2.4.1 Scaling function……………………………………..18
2.4.2 Wavelet function…………………………………….19
2.5 Wavelet coefficients decay theory…………………………………..21
Chapter 3 Assess a suitable wavelet for image compression
system…………………………………………….23
3.1 Introduction……...……………………………………… ..23
3.2 Wavelet transform using in the image coding…………….24
3.2.1 The model of the wavelet transform……………….25
3.2.2 Some families of the wavelet………………………31
3.3 Simulation and the results………...……………………….39
Chapter 4 Conclusions………………………………………65
Reference……………………………………………………..67
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[16] Nidhal Abdulaziz, J. Godfrey Lucas, "Representation and coding of image using wavelet transform", IEEE Tencon Speech and Image Technologies for Computing and Telecommunications, pp. 719-722, 1997
[17] E. A. B. da Silva, M. Ghanbari, "On the performance of linear phase wavelet transforms in low bit-rate image coding", IEEE Transactions on image processing, pp. 689-704, May 1996
[18] Gajetan M. Akujuobi, "The effects of different wavelets on image reconstruction", IEEE pp. 293-296, 1996
[19] G. Strang, T. Nguyen, "Wavelets and filter banks", Wellesley-Cambridge Press, 1996
[20] J. W. Woods, S. D. O'Neil, "Subband coding of images", IEEE Trans, On Acoustics, Speech, and Signal Processing, ASSP-34, NO. 5, PP. 1278-1288, Oct. 1986
指導教授 蔡木金(Mu-King Tsay) 審核日期 2000-6-20
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