消除雜訊在影像分析中一直佔有非常重要的地位，因為在影像擷取或是傳輸的過程中，經常會受到許多雜訊的干擾，像是拍照時鏡頭的髒汙、快速移動造成的模糊等…，上述這些情況都有可能造成影像的模糊(即雜訊)，如果我們對含有雜訊的影像做進一步的分析也難以得到精確的結果，因此消除影像雜訊將會是一種必備且重要的技術。目前幾個較有名消除影像雜訊的方法：線性或非線性的雜訊濾波器、傅立葉轉換(時域轉頻域)、小波轉換等…。
近年來，多小波(multi-wavelet)轉換逐漸受到研究人員重視，因為多小波是一種時域轉時域的轉換，相對於傅立葉轉換，多小波轉換後可以在時域中得到許多傅立葉轉換後無法得到的資訊，且經過多小波轉換後的資料也含有被壓縮特性，對於大數據時代也非常有幫助。相對於小波轉換，多小波轉換可以一次擁有許多特性，例如：正交性、對稱性、反對稱性、短支撐、較大消失動量等…，不同的多小波可能含有不同特性，利用這些特性，我們可以將影像中的雜訊順利消除。
目前大多數研究都使用級數為2的Geronimo-Hardin-Masopust(GHM)多小波消除雜訊，而此篇論文主要貢獻在於利用另一種更能廣泛使用的多小波來消除雜訊－有限元素之多小波，利用有限元素之多小波來消除雜訊能有效的降低運算時間，且不管雜訊多寡都能更正確的將雜訊消除，除此之外，有限元素之多小波的級數將不受限制，可以依照需求來選擇適合的級數，並將雜訊更精確的消除。最後，我們使用四組不同的資料進行實驗，實驗內容主要分成三步驟：首先，依據級數不同做重複資料；其次，將重複後的資料通過設計好的濾波器；最後，經過資料的排列與重新整理獲得已壓縮且消除雜訊的影像。

Denoising is an important part of image process for a long time. When we are capturing or transmitting the images, they are often effected by different kinds of noise. For examples, the dirty camera lens will increase the noise of images, the blurring images caused by moving quickly, and so on. There are many cases that will make the images not clear (also called noise). If we do some further analysis on the noising images, it still hard to get the accurate conclusion. Consequently, image denoising will be an essential and important method on image process. There are some famous methods on image denoising now, such as linear or non-linear noise filter, Fourier transform (from frequency-domain to frequency-domain), wavelet transform, etc.
Multi-wavelet transform has been emphasized by researchers these years because multi-wavelet transfers the data from time-domain to time-domain. After multi-wavelet transform, we can obtain lots of information from time-domain which can not obtain by Fourier transform. Furthermore, multi-wavelet transform contains the property of compression, which is helpful for the era of big data. Compared to wavelet transform, multi-wavelet transform can contain lots properties at once, such as orthogonality, symmetric, anti-symmetric, short support, large vanishing moment, and so on. Properties varies from different multi-wavelet. Using those properties, we remove the noise on images successfully.
Geronimo-Hardin-Masopust(GHM) with multiplicity 2 has been used to relax the effect of noise for most researches recently. However, our main contribution is that introducing other multi-wavelet which is more general to denoise, called multi-wavelet with finite element. Using multi-wavelet with finite element can decrease the computation time effectively. In addition, we can remove noise more precisely when

noise is large. More importantly, multi-wavelet with finite element will not be controlled by multiplicity. We can choose proper multiplicity by our requirement to obtain well performance of denoising. Finally, we present some experiments using four kinds of different data, which can be divided into three parts. The first part is repeated data with different multiplicity. Second, passing through the repeat data by filter which is designed in advance. Third, by re-sorting and re-arranging, we can get the denoising image which is also after compressing.

1. 緒論------------------------------------1
1.1 動機與目的-------------------------------1
1.2 論文架構---------------------------------2
2. 相關研究探討-----------------------------3
3. 小波分析---------------------------------5
3.1 定義------------------------------------5
3.2 尺度函數---------------------------------5
3.3 小波函數---------------------------------5
3.4 連續小波函數-----------------------------6
3.5 離散小波函數-----------------------------8
3.6 與傅立葉轉換比較-------------------------10
4. 多小波分析-------------------------------11
4.1 定義------------------------------------11
4.2 多尺度函數-------------------------------13
4.3 多小波函數-------------------------------13
4.4 多濾波器---------------------------------14
4.5 多解析度分析-----------------------------15
4.6 離散多小波轉換---------------------------16
4.7 常見的多小波-----------------------------18
5. 有限元素之多小波-------------------------23
5.1 定義------------------------------------23
5.2 多尺度函數-------------------------------24
5.3 多小波函數-------------------------------25
5.4 低通多濾波器-----------------------------26
5.5 高通多濾波器-----------------------------28
5.6 級數為2的有限元素之多小波-----------------30
5.7 級數為3的有限元素之多小波-----------------35
5.8 級數為n的有限元素之多小波-----------------39
6. 問題定義與消除影像雜訊原理----------------43
6.1 系統架構---------------------------------43
6.2 消除影像雜訊原理-------------------------43
6.3 濾波後影像-------------------------------44
7. 實驗與結果-------------------------------47
7.1 原始影像介紹-----------------------------47
7.2 預處理係數設置---------------------------50
7.3 級數為2的多小波濾波結果比較----------------56
7.4 級數為3的多小波濾波結果比較----------------71
8. 結論-------------------------------------77
9. 未來目標---------------------------------77
參考文獻----------------------------------------78

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