# 臺灣博碩士論文加值系統

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 本文將擴散方程式應用在影像處理去雜訊的功能上，並達成邊緣保留之目的。灰階數位影像可以視為矩陣的整數資料型態，我們利用傅立葉級數(Fourier series)，將矩陣中每一列的整數資料以傅立葉級數展開，當作擴散方程式的起始條件。亦將擴散方程式的全解以傅立葉級數展開，因此擴散方程式的全解即成為不同時間所代表灰階數位影像中的灰階值，隨著時間的增加，代表雜訊的高頻部份將會逐漸衰減，因此達成去除雜訊的目的，但同時，物件的邊緣輪廓也將同時逐漸衰減，為了克服此問題，我們加入由Delta函數所組成的熱源。Delta函數由影像中物件的邊緣位置決定，其效用只影響了全解中穩態解的部份，並不會隨著時間而與暫態解一起衰減，因此加入熱源後的擴散方程式就能將灰階數位影像平滑化(smoothing)的同時，也可以將物件的銳利邊緣保留而避免被模糊的缺點。
 The paper presents the diffusion equation applied on the image denoising process with the edge preservation.The gray digital images can be regarded as matrixes with the integer data type. In each row of matrix, the integer data sequence can be represented by the Fourier series expansion and it can be regarded as the initial condition of the diffusion equation. The complete solutions of diffusion equation can be expanded by Fourier series; therefore, the gray scale of the gray level digital image is equivalent to the complete solutions of the diffusion equation. The high frequency part of the transition solution is going to decrease in the complete solution as the time increasing which achieve the aim of image denoising purpose. But simultaneously, the edge of accessory also at the same time gradually will attenuation, in order to overcome this question, we accede to the heat source which is composed by the Delta function. The location of Delta functions are decided by the sharp edge positions of the image. It only influences the steady states solution which is the part of the complete solution and does not attenuate together with the transition condition solution as the time increased. The diffusion equation can be used to smooth the gray digital image and to preserve the sharp edge avoiding the shortness of edge blurred.
 目錄誌謝辭 I摘要 IIAbstract III目錄 IV表目錄 VI圖目錄 VII第一章 緒論 11.1 研究背景 11.2 研究動機與目的 31.3 論文架構 4第二章 數學基礎理論 52.1 單位階梯函數(Heaviside Function) 52.2 單位衝量函數(Dirac Delta Function) 62.3 單位階梯函數與單位衝量函數之間的關係 8第三章 邊緣保持之影像去雜訊理論 103.1 數位影像之訊號表示 103.2 擴散方程式 133.3 Edge Preserve的非齊性擴散方程式(EPNDE) 173.4 離散數位影像之傅立葉級數表示法 193.5 Canny Edge Detection 21第四章 實驗結果 254.1 測試簡單函數 254.2 參數 284.3 DF、EPNDE及小波轉換之間的去雜訊比較 344.4 實際影像的實例 37第五章 討論與展望 45附錄一 方程式之推導 48參考文獻 50
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 1 小波函數在影像壓縮和編輯之應用 2 利用ECG訊號之離散小波轉換於個人身份辨識研究

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