대한수학회지 (Journal of the Korean Mathematical Society)

  • 제45권1호
  • /
  • Pages.97-119
  • /
  • 2008
  • /
  • 0304-9914(pISSN)
  • /
  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

INTEGRAL REPRESENTATIONS IN ELECTRICAL IMPEDANCE TOMOGRAPHY USING BOUNDARY INTEGRAL OPERATORS

  • Kwon, Ki-Woon (Department of Biomedical Engineering Younsei University)
  • 발행 : 2008.01.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

Electrical impedance tomography (EIT) problem with anisotropic anomalous region is formulated in a few different ways using boundary integral operators. The Frechet derivative of Neumann-to-Dirichlet map is computed also by using boundary integral operators and the boundary of the anomalous region is approximated by trigonometric expansion with Lagrangian basis. The numerical reconstruction is done in case that the conductivity of the anomalous region is isotropic.

키워드

참고문헌

  1. D. C. Barber and B. H. Brown, Applied potential tomography, J. Phys. E. Sci. Instrum. 17 (1984), 723-733 https://doi.org/10.1088/0022-3735/17/9/002
  2. M. Cheney, D. Isaacson, J. Newell, J. Goble, and S. Simske, Noser: An algorithm for solving the inverse conductivity problem, Internat. J. Imaging Systems and Technology 2 (1990), 66-75 https://doi.org/10.1002/ima.1850020203
  3. M. Cheney, D. Isaacson, and J. C. Newell, Electrical impedance tomography, SIAM Rev. 41 (1999), no. 1, 85-101 https://doi.org/10.1137/S0036144598333613
  4. R. R Coifman, A. McIntosh, and Y. Meyer, L'integrale de Cauchy definit un operateur borne sur $L^2$ pour les courbes lipschitziennes, Ann. of Math. (2) 116 (1982), no. 2, 361-387 https://doi.org/10.2307/2007065
  5. D. Colton, R. Kress, and P. Monk, Inverse scattering from an orthotropic medium, J. Comput. Appl. Math. 81 (1997), no. 2, 269-298 https://doi.org/10.1016/S0377-0427(97)00065-4
  6. D. C. Dobson, Convergence of a reconstruction method for the inverse conductivity problem, SIAM J. Appl. Math. 52 (1992), no. 2, 442-458 https://doi.org/10.1137/0152025
  7. K. Erhard and R. Potthast, The point source method for reconstructing an inclusion from boundary measurements in electrical impedance tomography and acoustic scattering, Inverse Problems 19 (2003), no. 5, 1139-1157 https://doi.org/10.1088/0266-5611/19/5/308
  8. L. Escauriaza and J. K. Seo, Regularity properties of solutions to transmission problems, Trans. Amer. Math. Soc. 338 (1993), no. 1, 405-430 https://doi.org/10.2307/2154462
  9. G. B. Folland, Introduction to partial differential equations, Second edition. Princeton University Press, Princeton, NJ, 1995
  10. D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Second edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 224. Springer-Verlag, Berlin, 1983
  11. D. G. Gisser, D. Isaacson, and J. C. Newell, Current topics in impedance imaging, Clin. Phys. Phyiol. Meas. 8 (1987), 39-46
  12. F. Hettlich and W. Rundell, The determination of a discontinuity in a conductivity from a single boundary measurement, Inverse Problems 14 (1998), no. 1, 67-82 https://doi.org/10.1088/0266-5611/14/1/008
  13. B. Hofmann, Approximation of the inverse electrical impedance tomography problem by an inverse transmission problem, Inverse Problems 14 (1998), no. 5, 1171-1187 https://doi.org/10.1088/0266-5611/14/5/006
  14. B. Hofmann, A denseness theorem with an application to a two-dimensional inverse potential refraction problem, SIAM J. Math. Anal. 30 (1999), no. 4, 896-911 https://doi.org/10.1137/S0036141098336108
  15. T. Hohage and C. Schormann, A Newton-type method for a transmission problem in inverse scattering, Inverse Problems 14 (1998), no. 5, 1207-1227 https://doi.org/10.1088/0266-5611/14/5/008
  16. M. Ikehata, Identification of the curve of discontinuity of the determinant of the anisotropic conductivity, J. Inverse Ill-Posed Probl. 8 (2000), no. 3, 273-285
  17. V. Isakov, On uniqueness of recovery of a discontinuous conductivity coefficient, Comm. Pure Appl. Math. 41 (1988), no. 7, 865-877 https://doi.org/10.1002/cpa.3160410702
  18. H. Kang and J. K. Seo, The layer potential technique for the inverse conductivity problem, Inverse Problems 12 (1996), no. 3, 267-278 https://doi.org/10.1088/0266-5611/12/3/007
  19. H. Kang, J. K. Seo, and D. Sheen, Numerical identification of discontinuous conductivity coefficients, Inverse Problems 13 (1997), no. 1, 113-123 https://doi.org/10.1088/0266-5611/13/1/009
  20. H. Ki and D. Sheen, Numerical inversion of discontinuous conductivities, Inverse Problems 16 (2000), no. 1, 33-47 https://doi.org/10.1088/0266-5611/16/1/304
  21. R. Kohn and M. Vogelius, Identification of an unknown conductivity by means of measurements at the boundary, Inverse problems (New York, 1983), 113-123, SIAM-AMS Proc., 14, Amer. Math. Soc., Providence, RI, 1984
  22. R. V. Kohn and A. McKenney, Numerical implementation of a variational method for electrical impedance tomography, Inverse Problems 6 (1990), no. 3, 389-414 https://doi.org/10.1088/0266-5611/6/3/009
  23. R. Kress, Linear integral equations, Second edition. Applied Mathematical Sciences, 82. Springer-Verlag, New York, 1999
  24. R. Kress, Newton's method for inverse obstacle scattering meets the method of least squares, Special section on imaging. Inverse Problems 19 (2003), no. 6, S91-S104 https://doi.org/10.1088/0266-5611/19/6/056
  25. K. Kwon, Identification of anisotropic anomalous region in inverse problems, Inverse Problems 20 (2004), no. 4, 1117-1136 https://doi.org/10.1088/0266-5611/20/4/008
  26. K. Kwon and D. Sheen, Anisotropic inverse conductivity and scattering problems, Inverse Problems 18 (2002), no. 3, 745-756 https://doi.org/10.1088/0266-5611/18/3/315
  27. M. Lassas and G. Uhlmann, On determining a Riemannian manifold from the Dirichletto-Neumann map, Ann. Sci. Ecole Norm. Sup. (4) 34 (2001), no. 5, 771-787 https://doi.org/10.1016/S0012-9593(01)01076-X
  28. J. M. Lee and G. Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure Appl. Math. 42 (1989), no. 8, 1097-1112 https://doi.org/10.1002/cpa.3160420804
  29. W. R. B. Lionheart, Conformal uniqueness results in anisotropic electrical impedance imaging, Inverse Problems 13 (1997), no. 1, 125-134 https://doi.org/10.1088/0266-5611/13/1/010
  30. A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math. (2) 143 (1996), no. 1, 71-96 https://doi.org/10.2307/2118653
  31. R. Potthast, Frechet differentiability of boundary integral operators in inverse acoustic scattering, Inverse Problems 10 (1994), no. 2, 431-447 https://doi.org/10.1088/0266-5611/10/2/016
  32. F. Santosa and M. Vogelius, A backprojection algorithm for electrical impedance imaging, SIAM J. Appl. Math. 50 (1990), no. 1, 216-243 https://doi.org/10.1137/0150014
  33. E. Somersalo, M. Cheney, D. Isaacson, and E. L. Isaacson, Layer stripping: a direct numerical method for impedance imaging, Inverse Problems 7 (1991), no. 6, 899-926 https://doi.org/10.1088/0266-5611/7/6/011
  34. Z. Sun and G. Uhlmann, Anisotropic inverse problems in two dimensions, Inverse Problems 19 (2003), no. 5, 1001-1010 https://doi.org/10.1088/0266-5611/19/5/301
  35. J. Sylvester, An anisotropic inverse boundary value problem, Comm. Pure Appl. Math. 43 (1990), no. 2, 201-232 https://doi.org/10.1002/cpa.3160430203
  36. J. Sylvester, A convergent layer stripping algorithm for the radially symmetric impedance tomography problem, Comm. Partial Differential Equations 17 (1992), no. 11-12, 1955-1994 https://doi.org/10.1080/03605309208820910
  37. G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains, J. Funct. Anal. 59 (1984), no. 3, 572-611 https://doi.org/10.1016/0022-1236(84)90066-1
  38. E. J.Woo, J.Webster, and W. J. Tompkins, The improved Newton-Raphson method and its parallel implementation for static impedance imaging, In Proc. IEEE-EMBS Conf. Part 1, volume 5, pages 102-103, 1990
  39. T. J. Yorkey, J. G. Webster, and W. J. Tompkins, Comparing reconstruction algorithms for electrical impedance tomography, IEEE Trans. Biomed. Engr. 34 (1987), 843-852 https://doi.org/10.1109/TBME.1987.326032

피인용 문헌

  1. The method of fundamental solutions for the inverse conductivity problem vol.18, pp.4, 2010, https://doi.org/10.1080/17415971003675019