臺灣博碩士論文加值系統

本論文利用擬時間積分法來重建圖像。首先將原始圖像經過保群算法擴散一段時間，接著在影像重建問題我們使用擴散後的資料當作終時條件來覆蓋原始的資料，此問題為高度病態的反算非線性擴散問題，然而，在本文中利用擬時間積分法來處理高度病態的問題的結果相當良好是藉由調整虛擬時間方向的ν值來得到精確的數值結果。之後藉由幾個算例的結果探討與解析，我們可以知道擬時間積分法可以提高計算的效率以及準確性。

We propose the fictitious time integration method (FTIM) to reconstruct the image. First、the original image is diffused a period of time by using the group preserving scheme (GPS)、and then、in the image restoration problem we use the diffused data as a final time data to recover the original time data. This problem is known ad a backward in time nonlinear diffusion problem、which is highly ill-posed. However、in this thesis we can employ the fictitious time integration method to tackle this highly ill-posed image restoration problem very well、by modifying the value ν to obtain the accurate solutions. The numerical examples are examined、and we can find that the FTIM is applicable to the image restoration problems with high efficiency and accuracy.

中文摘要…………………………………………………………… i
英文摘要…………………………………………………………… ii
目錄………………………………………………………………… iii
圖目錄……………………………………………………………… v
第一章 緒論 ……………………………………………………… 1
1.1 前言…………………………………………………………… 1
1.2 研究動機與目的……………………………………………… 1
1.3 文獻回顧……………………………………………………… 2
1.4 本文架構……………………………………………………… 3
第二章 理論基礎 ………………………………………………… 4
2.1 反算問題的定義……………………………………………… 4
2.2 保群算法……………………………………………………… 4
2.3 擬時間積分法………………………………………………… 7
第三章 圖像重建的數學模型…………………………………… 10
3.1 一維圖像重建問題…………………………………………… 10
3.1.1 轉換成一組ODEs ……………………………………… 11
3.2 二維圖像重建問題…………………………………………… 12
3.2.1 轉換成一組ODEs ……………………………………… 12
第四章 數值結果與討論 ………………………………………… 14
4.1 算例1 …………………………………………………………14
4.2 算例2 …………………………………………………………14
4.3 算例3 …………………………………………………………15
4.4 算例4 …………………………………………………………16
4.5 算例5 …………………………………………………………16
第五章 結論與未來展望………………………………………… 18
5.1 結論…………………………………………………………… 18
5.2 未來展望……………………………………………………… 18
參考文獻…………………………………………………………… 19
誌謝………………………………………………………………… 50

[1] Alvarez L.; Lions P. L.; Morel J. M. 、「Image Selective Smoothing and Edge Detection by Non-linear Diffusion II」、SAIM、vol. 29、No. 3、pp. 845-866、1992
[2] Chein-Shan Liu; C.W. Chang; J.R. Chang、「Past Cone Dynamics and Backward Group Preserving Schemes for Backward Heat Conduction Problems」、CMES、vol.12、pp.67-81、2006
[3] Chein-Shan Liu、「Cone of Non-linear dynamical system and Group Preserving Schemes」、IJNLM、vol.36、pp.1047-1068、2001
[4] Chein-Shan Liu、Satya N. Atluri 「A Novel Time Integration Method for Solving a Large System of Non-Linear Algebraic Equations」、CMES、vol. 31、pp. 71-83、2008
[5] Chein-Shan Liu、「Group Preserving Scheme for Backward Heat Conduction Problem」、IJHMT、vol. 47、pp. 2567-2576、2004
[6] Chein-Shan Liu、「A Fictitious Time Integration for the Burgers Equation」、CMC、vol.9、NO.3、pp.229-252、2009
[7] Chein-Shan Liu、S. N. Atluri、「A Fictitious Time Integration Method (FTIM) for Solving Mixed Complementarity Problems with Applications to Nonlinear Optimization」、CMES、vol. 34、No. 2、pp. 155-178、2008.
[8] Chein-Shan Liu、「 A Time-Marching Algorithm for Solving Non-Linear Obstacle Problems with the Aid of an NCP-function」、CMC、vol. 8、pp. 53-65、2008
[9] Chein-Shan Liu、「 A fictitious Time Integration Method for Two-Dimensional Quasilinear Elliptic Boundary Value Problems」、CMES、vol. 33、No. 2、pp. 179-198、2008
[10] Chein-Shan Liu、S. N. Atluri、「 A New Filter Theory for the Numerical Solution of Fredholm Integral Equation and for Numerical Differentiation of Noisy Data」 CMES、vol. 41、pp. 243-261、2009.
[11] Canny J.、「A Computational Approach to Edge Detection」、IEEE Trans. Pattern Anal. Machine Intell. 、vol. PAMI-8,pp.679-698、1986
[12] Danny Barash; Ronny Kimmel、「An Accurate Operator Splitting Scheme for Nonlinear Diffusion Filtering」、HP Laboratories Israel、pp. 1-20、2000
[13] Catte F.、Lion P. L.、Morel J. M.、Coll T. 、「Image Selective Smoothing and Edge Detection by Non-linear Diffusion」、SAIM、vol. 29、No. 3、pp. 182-193,1992
[14] Joachim Weickert; Bart M. ter Haar Romeny; Member、IEEE; Max A. Viergever、「Efficient and Reliable Schemes for Nonlinear Diffusion Filtering」、IEEE Trans.、vol.7,NO.3 pp.398-410,1998
[15] Mera、N. S.、「The Method of Fundamental Solutions for the Backward Heat conduction Problem」、Inv. Prob. Sci. Eng.、vol. 13、pp. 65-78、2005
[16] Marius Lysaker; Arvid Lundervold; Xue-Cheng Tai、「Noise Removal Using Fourth-Order Partial Differential Equation With Applications to Medical Magnetic Resonance Images in Space and Time」 IEEE Trans.,vol.12,pp.1579-1590
[17] Michael Felsberg; Gerald Sommer「Scale Adaptive Filtering Derived from the Laplace Equation」,pp.1-14,2001
[18] Saint-Marc P.、Chen J.、Medioni G.、「Adaptive Smoothing: A General Tool for Early Vision」 IEEE Trans. Pattern Anal. Machine Intell.、vol.13、pp. 514–529、1991
[19] Perona P.; Malik J.、「Scale-Space and Edge Detection Using Anisotropic Diffusion」、IEEE Trans.、vol.12、No. 7、pp.629、1990
[20] Reginald L. Lagendijk、Biemond Jan、「Interative Identification and Restoration of Images」 KLUWER ACADEMIC PUBLISHERS
[21] Acton S. T.、Bovik A. C.、「Anisotropic edge detection using mean field annealing」 in Proc. IEEE Int. Conf. Acoustics、Speech、Signal Processing (ICASSP-92)、San Francisco、CA、Mar. 23–26、1992
[22] Weickert J,「Anisotropic Diffusin in Image Processing」、ECMI series、Teubner-Verlog、Stuttgart、Germany、1998
[23] Witkin A ,「Scale-space filtering」、Int. Joint Conf. Artificial Intelligence、Karlsruhe、West Germany、pp1019-1021、1983
[24] Acton S. T.、Bovik A. C.、Crawford M. M.、「Anisotropic Diffusion Pyramids for Image Segmentation,」 IEEEIP、Austin、TX、Nov. 1994
[25] You Y. L. etal、「Behavioral Analysis of Anisotropic Diffusion in Image Processing」、IEEEIP、pp.1539-1553、1996
[26] Zhouchen Lin; Qingyun Shi 、「An Anisotropic Diffusion PDE for Noise Reduction and Thin Edge Preservation」、Beijing、P.R. China
[27] 王大凱,侯榆青,彭進業，「圖像處理的偏微分方程方法」，北京：科學出版社，2008
[28] 楊新，「圖像偏微分方程的原理與應用」，上海：上海交通大學出版社，2003。