摘要

本文以求解精確的輻射傳遞的積微分方程式為基礎，發展在完全的散射模式下，可同時逆算多維圓柱介質之吸收及散射性質的程式。我們以修正的離散方向法求解柱狀介質受到側邊平行入射的正算問題，所得的輻射熱通量為模擬量測值，並用它與估算的輻射性質所求得的輻射熱通量之間的均方差，形成目標函數，然後再最小化此目標函數來逆算介質的性質。本文以Levenberg-Marquardt法來求解各個例子的最小均方問題。
我們首先探討重構 分佈的輻射性質。結果顯示：(一)本文的方法可以準確的重構輻射性質的分佈。(二)當散射比增大時會降低估算結果的準確性；光學薄時(光學厚度小於0.1)需更多的離散方向以求得準確的結果；逆算不平消散係數時(第二章的 )，需更多的格點數與逼近估算消散係數的展開項。(三)加上適度的模擬量測誤差，本方法仍可得到可接受的結果( 最大為8.672%)。
然後我們延伸這方法於三維的例子。我們考慮具空間可變消散係數、散射比與常相函數的重構問題。為了比較，我們分別以三維與二維的方法來重構這些例子。結果我們發現三維可以得到良好結果。然而，它比二維的方法耗掉更多的電腦運算時間。

Abstract

In this work, we develop a scheme based on solving the general integro-differential equation of radiation transport to estimate simultaneously the absorption and scattering properties of a multi-dimensional inhomogeneneous medium with less diffusive radiation. The forward problem for a cylindrical medium subjected to collimated incident radiation is solved by the modified discrete-ordinate method. The inverse radiation problem is formulated as a least square problem that minimizes the discrepancy between the measured and the calculated leaving radiative fluxes. The Levenberg-Marquardt algorithm is applied to the least square problems for a variety of cases.
First, the reconstruction of the albedo, the phase function and the two-dimensional distribution of extinction coefficient is performed. The results obtained show that this scheme can reconstruct accurate enough results for most of the cases considered. Comparisons of the results show that the accuracy of the estimated results decreases with the increase of the scattering albedo and we need more discrete ordinates to generate accurate enough estimated results for an optically thin case. The estimated results obtained from the measurement data with moderate errors are still acceptable.
Further, the above scheme is extended to three-dimensional situations. The cases with spatially variable extinction coefficients and scattering albedos and constant phase functions are considered. Both the two- and the three dimensional schemes are applied to the cases for comparison purpose. The three-dimensional scheme generates good results in most cases. However, it takes much more CPU time than the two-dimensional schemes does.

參考文獻
[1] Y. Yamada, Light-tissue interaction and optical imaging in biomedicine, in: C. L. Tien (Ed.), Annual Review of Heat Transfer, Begell House, New York, 1995, Vol. 6, Chap. 1.
[2] S.R. Arridge, Optical tomography in medical imaging, Inverse Problem 15, 1999, R41-R93.
[3] B.J. Hughey, D.A. Santavicca, A comparison of techniques for reconstructing axisymmetric reacting flow fields from absorption measurements, Combustion Science and Technology 29 (1982) 167-190.
[4] M. Ravichandran, F.C. Gouldin, Determination of temperature and concentration profiles using (a limited number of) absorption measurements, Combustion Science and Technology 45 (1986) 47-64.
[5] B.M. Agarwal, M.P. Mengüc, Forward and inverse analysis of single and multiple scattering of collimated radiation in an axisymmetric system, International Journal of Heat and Mass Transfer 34 (1991) 633-647.
[6] E.D. Torniainen, F.C. Gouldin, Tomographic reconstruction of 2-D absorption coefficient distributions from a limited set of infrared absorption data, Combustion Science and Technology 131 (1998) 85-105.
[7] W.W. Yuen, A. Ma, I.C. Hsu, G.R. Cunnington Jr., Determination of optical properties by two-dimensional scattering, Journal of Thermophysics 6 (1991) 182-185.
[8] V.V. Alpatov, Y.A. Romanovsky, Methods of optical tomography for remote sensing of the atmosphere and near-earth space, Advances in Space Research, 21 (1998) 1437-1440.
[9] C.J. Dasch, One-dimensional tomography: a comparison of Abel, onion-peeling, and filtered back projection methods, Applied Optics 31 (1992) 1146-1152.
[10] M.P. Mengüc, P. Dutta, Scattering tomography and its application to sooting diffusion flames, Journal of Heat Transfer; Transactions of the ASME 116 (1994) 144-151.
[11] E. Okada, M. Firbank, M. Schweiger, S.R. Arridge, M. Cope, D.T. Delpy, Theoretical and experimental investigation of near-infrared light propagation in a model of the adult head, Applied Optics, 36 (1997) 21-31.
[12] N.J. McCormick, Inverse radiative transfer problems: a review, Nuclear Science and Engineering 112 (1992) 185-198.
[13] R. Aronson, R.L. Barbour, J. Lubowsky, H. Graber, Application of transport theory to infra-red medical imaging, in: W. Greenberg, J. Polewczak (Eds.), Modern Mathematical Methods in Transport Theory, Birkhäuser Verlag, Basel, 1991, pp. 64-75.
[14] J. Chang, H.L. Graber, R.L. Barbour, R. Aronson, Recovery of optical cross-section perturbations in dense-scattering media by transport-theory-based imaging operators and steady-state simulated data, Applied Optics 35 (1996) 3963-3978.
[15] A.D. Klose, A.H. Hielscher, Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer, Medical Physics 26 (1999) 1698-1707.
[16] A.D. Klose, A.H. Hielscher, Optical tomography using the time-independent equation of radiative transfer-part II: inverse model, Journal of Quantitative Spectroscopy and Radiative Transfer, 72 (2002) pp. 715-732.
[17] C.-Y. Wu, B.-T. Liou, Discrete-ordinate solutions for radiative transfer in a cylindrical enclosure with Fresnel boundaries, International Journal of Heat and Mass Transfer 40 (1997) 2467-2475.
[18] B.-T. Liou, C.-Y. Wu, Ray effects in the discrete ordinate solution for surface radiation exchange, Heat and Mass Transfer 32 (1997) 271-275.
[19] M.A. Ramankutty, A.L. Crosbie, Modified discrete-ordinates solution of radiative transfer in three-dimensional rectangular enclosures, Journal of Quantitative Spectroscopy and Radiative Transfer 60 (1998) 103-134.
[20] M. Schweiger, S.R. Arridge, Comparison of two- and three-dimensional reconstruction methods in optical tomography, Applied Optics 37(1998) 7419-7428.
[21] S.-C. Hsu, C.-Y. Wu, N.-R. Ou, Azimuthally dependent radiative transfer in a nonhomogeneous cylindrical medium, Radiation Physics and Chemistry 53 (1998) 107-113.
[22] D.W. Marquardt, An algorithm for least-squares estimation of nonlinear parameters, Journal of the Society for Industrial and Applied Mathematics 11 (1963) 431-441.
[23] H.P. William, A.T. Saul, T.V. William, P.F. Brain, Numerical Recipes, 2nd ed., Cambridge University Press, New York, 1992, pp. Chap. 15.
[24] IMSL Library, Edn. 9.2, 2500 Park West Tower One, 2500 City West, Blvd., Houston, Textas (1985)
[25] J.-M. Zhang, W.H. Sutton, The source approximation method for multidimensional media using the surface-integral form of the radiative transfer equation, Journal of Quantitative Spectroscopy and Radiative Transfer, 53 (1995) 549-563