臺灣博碩士論文加值系統

本論文是估測一維的電阻抗斷層成像的阻抗係數，而估測的方法是使用一種特別的方法-李群打靶法(Lie-Group Shooting Method, LGSM)。由數學模型推出一維的電阻抗斷層成像問題其實就是反算史特姆-呂維爾邊界值問題(Inverse Sturm-Liouville Boundary Value Problem)。我們所使用的方法是將反算史特姆-呂維爾問題轉化為熱傳導方程的參數識別問題。然後將熱傳導方程離散化為常微分系統，並且使用李群打靶法估測。例題數值計算結果可以知道李群打靶法對於估測一維的電阻抗斷層成像的阻抗係數是可行的。經過本文探討，李群打靶法可以提高計算的效率、準確性和穩定性；而其數值程序簡單且省時。

The purpose of this thesis we are applying the Lie-Group Shooting Method (LGSM), which is a kind of special method for estimating the impedance coefficient of one-dimensional electrical impedance tomography. The problem of the one-dimensional electrical impedance tomography by the mathematical model is an inverse Sturm-Liouville boundary value problem. The method we employ is to transform the inverse Sturm-Liouville problem into a parameter identification problem of a heat conduction equation. Then a LGSM is developed to estimate the coefficients in a system of ordinary differential equations discretized from the heat conduction equation. Numerical examples were worked out, which show that the LGSM is applicable for estimating the impedance coefficient of one-dimensional electrical impedance tomography. Through this study, it can be concluded that the LGSM is accurate, effective, and stable. Its numerical implementation is very simple and the computational speed is very fast.

誌謝.................................................... i
摘要.................................................... iii
Abstract................................................. iv
目錄.................................................... v
表目錄.................................................. viii
圖目錄.................................................. ix
符號表.................................................. xi
第一章 緒論............................................ 1
1.1 研究動機........................................ 1
1.2 文獻回顧........................................ 2
1.2.1 電阻抗斷層成像理論方面....................... 2
1.2.2 電阻抗斷層成像重建算法方面................... 2
1.2.3 李群打靶法(Lie-Group Shooting Method) ......... 3
1.3 本文架構......................................... 4
第二章 電阻抗斷層成像的數學模型........................ 5
2.1 數學模型......................................... 5
2.2 連續模型......................................... 5
2.3 一維的電阻斷層抗成像問題......................... 6
2.4 反算史特姆-呂維爾問題的歷史背景.................. 7
2.5 雙重形式轉換..................................... 7
2.5.1 轉換成PDE的型式............................. 7
2.5.2 轉換成一組ODEs.............................. 8
第三章 李群打靶法(Lie-Group Shooting Method) ............. 9
3.1 保群算法(Group-preserving scheme, GPS ) ............. 9
3.2 一步保群算法(One-step GPS )........................ 11
3.3 廣義中值定理..................................... 12
3.4 光錐上兩點之李群映射............................. 12
3.5 使用李群打靶法估測導納阻抗 ..................... 15
第四章 數值結果與討論.................................. 18
4.1 算例一............................................ 18
4.2 算例二............................................ 18
4.3 算例三............................................ 19
第五章 結論與未來展望.................................. 21
5.1 結論.............................................. 21
5.2 未來展望.......................................... 31
參考文獻................................................ 22
附錄 論文口試投影片.................................... 41

1. V. A. Cherepenin, A. Y. Karpov, A. V. Korjenevsky, V. N. Kornienko, Y. S. Kultiasov, M. B. Ochapkin, O. V. Trochanova, and J. D. Meister, “Three-dimensional EIT imaging of breast tissues : system design and clinical testing” IEEE Transactions on Medical Imaging, vol. 21, no. 6, pp. 662-667, 2002.
2. J. H. Li, C. Joppek, and U. Faust, “Fast EIT data acquisition system with active electrodes and its application to cardiac imaging” Physiological Measurement, vol. 17, no. 4A, p. A25, 1996.
3. R. H. Smallwood, A. R. Hampshire, B. H. Brown, R. A. Primhak, S. Marven, and P. Nopp, “A comparison of neonatal and adult lung impedances derived from EIT images” Physiological Measurement, vol. 20, no. 4, p. 401, 1999.
4. R. A. Erol, R. H. Smallwood, B. H. Brown, P. Cherian, and K. D. Bardhan, “Detecting oesophageal-related changes using electrical impedance tomography” Physiological Measurement, vol. 16, no. 3A, p. A143, 1995.
5. I. Jurgens, rgens, J. Rosell, and P. J. Riu, “Electrical impedance tomography of the eye: in vitro measurements of the cornea and the lens” Physiological Measurement, vol. 17, no. 4A, p. A187, 1996.
6. A. T. Tidswell, A. Gibson, R. H. Bayford, and D. S. Holder, “Electrical impedance tomography of human brain activity with a two- dimensional ring of scalp electrodes” Physiological Measurement, vol. 22, no. 1, p. 167, 2001.
7. Z. Szczepanik, and Z. Rucki, “Frequency analysis of electrical impedance tomography system”, IEEE Transactions on Instrumentation and Measurement, vol. 49, no. 4, pp. 844-851, 2000.
8. A.P. Calderon, “On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to continuum physics” Brasileira de Matematica, R?o de Janeiro, pp. 65-73, 1980.
9. J. Sylvester, and G. Uhlmann, “A global uniqueness theorem for an inverse boundary value problem” Annals of Mathematics, vol. 125, pp. 153-169, 1987.
10. A.I. Nachman, “Reconstructions from boundary measurements” Annals of Mathematics, vol. 128, no. 3, pp. 531-576, 1988.
11. A.I. Nachman, “Global uniqueness for a two-dimensional inverse boundary problem” Annals of Mathematics, vol. 143, no. 1, pp. 71-96, 1996.
12. R.M. Brown, and G.A. Uhlmann, “Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions” Communications in Partial Differential Equations, vol. 22, pp. 1009-1027, 1997.
13. Y Kim, J.G. Webster, and W.J. Tompkins, “Electrical impedance imaging of the thorax” J Microwave Power, Vol. 18, No. 3, pp. 245-257, 1983.
14. A. Wexler, B. fry, and M.R. Neuman, “Impedance-computed tomography algorithm and system” Applied Optics, Vol. 24, No. 23, pp. 3985-3992, 1985.
15. T. Murai, and Y. Kagawa, “Electrical impedance computed tomography based on a finite element model” IEEE Transactions on Biomedical Engineering, vol. 32, no. 3, pp. 177-184, 1985.
16. D.C. Barber, and A.D. Seagar, “Fast reconstruction of resistive images” Clinical Physics and Physiological Measurement, vol. 8, pp. 47-54, 1987.
17. T.J. Yorkey, J.G. Webster, and W.J. Tompkins, “Comparing Reconstruction Algorithms for Electrical Impedance Tomography” IEEE Transactions on Biomedical Engineering, vol. 34, no. 11, pp. 843-852, 1987.
18. M. Zadehkoochak, T.K. Hames, B.H. Blott, and R. F. George, “A transputer implemented algorithm for electrical impedance tomography” Clinical Physics and Physiological Measurement, vol. 11, no. 3, pp. 223-230, 1990.
19. A. Adler, and R. Guardo, “A neural network image reconstruction technique for electrical impedance tomography” IEEE Transactions on Medical Imaging, vol. 13, no. 4, pp. 594-600, 1994.
20. S. Meeson, A.L.T. Killingback, and B.H. Blott, “The dependence of EIT images on the assumed initial conductivity distribution: a study of pelvic imaging” Physics in Medicine and Biology, vol. 40, no. 5, pp. 643-657, 1995.
21. 王 超、王 化祥，「基於閾值的廣義逆電阻抗成像的重建算法」 自然科學進展, vol. 12, no. 11, pp. 1193-1196, 2001.
22. M. Wang, “Inverse solutions for electrical impedance tomography based on conjugate gradients method” Measurement science & technology, vol. 13, pp. 101-117, 2002.
23. C.S. Liu, “Cone of non-linear dynamical system and group preserving schemes” International Journal of Non-Linear Mechanics, vol. 36, pp. 1047-1068, 2001.
24. C.S. Liu, “One-step GPS for the estimation of temperature-dependent thermal conductivity” International Journal of Heat and Mass Transfer, vol. 49, pp. 3084-3093, 2006.
25. C.W. Chang, C.S. Liu, and J.R. Chang, “A group preserving scheme for inverse heat conduction problems” Computer Modeling in Engineering & Sciences, vol. 10, pp. 13-38, 2005.
26. C.S. Liu, L.W. Liu, and H.K. Hong, “Highly accurate computation of spatial-dependent heat capacity in inverse thermal problem” Computer Modeling in Engineering & Sciences, vol. 17, pp. 1-18, 2007
27. C.S. Liu, “An efficient simultaneous estimation of temperature- dependent thermophysical properties” Computer Modeling in Engineering and Sciences, vol. 14, pp. 77-90, 2006.
28. C.S. Liu, “Identification of temperature-dependent thermophysical properties in a partial differential equation subject to extra final measurement data” Numerical Methods for Partial Differential Equations, vol. 23, pp. 1083-1109, 2007.
29. C.S. Liu, L.W. Liu, and H.K. Hong, “Highly accurate computation of spatial-dependent heat conductivity and heat capacity in inverse thermal problem” Computer Modeling in Engineering and Sciences, vol. 17, pp. 1-18, 2007.
30. C.S. Liu, “The Lie-group shooting method for nonlinear two-point boundary value problems exhibiting multiple solution” Computer Modeling in Engineering & Sciences, vol. 13, pp. 149-163, 2006.
31. J.R. Chang, C.S. Liu, and C.W. Chang, “A new shooting method for quasi-boundary regularization of backward heat conduction problems” International Journal of Heat and Mass Transfer, vol. 50, pp. 2325-2332, 2006.
32. I.M. Gel’fand, and B.M. Levitan, “On the determination of a differential equation from its spectral function” American Mathematical Society Translations, vol. 1, pp. 253-304, 1955.
33. J.R. McLaughlin, “Analytical methods for recovering coefficients in differential equations from spectral data” SIAM Review, vol. 28, no. 1, pp. 53-72, 1986.
34. D. Boley, and G.H. Golub, “A survey of matrix inverse eigenvalue problem” Inverse Problems, vol. 3, no. 4, pp. 595-622, 1987.
35. C.S. Liu, “Solving an inverse Sturm-Liouville problem by a Lie-group method” Boundary Value Problems, vol. 2008, Article ID 749865, 18 pages, 2008
36. C.S. Liu, “Nonstandard Group-preserving Schemes for Very Stiff Ordinary Differential Equation” Computer Modeling in Engineering & Sciences, vol. 09, pp. 255-272, 2005.
37. Y.L. Keung, and J. Zou, “An efficient linear solver for nonlinear parameter identification problems” SIAM Journal on Scientific Computing, vol. 22, no. 5, pp. 1511-1526, 2000.