Direct reconstructions of conductivities from boundary measurements

SIAM Journal on Scientific Computing

Volumn 24, Issue 4, 2003, Pages 1232-1263

  Mueller, Jennifer L ^a   Siltanen, Samuli ^b  

^a Department of Environmental and Radiological Health Sciences   (United States)

^b GE Healthcare Finland Oy   (Finland)

Author keywords

Electrical impedance tomography; Exponentially growing solution; Faddeev's Green's function; Inverse conductivity problem; Scattering transform; Equation

Indexed keywords

COMPUTER SIMULATION; ELECTRIC CONDUCTIVITY; IMAGE RECONSTRUCTION; ITERATIVE METHODS; MEDICAL IMAGING; NONDESTRUCTIVE EXAMINATION; NONLINEAR EQUATIONS; PROBLEM SOLVING;

NUMERICAL RECONSTRUCTION;

BOUNDARY CONDITIONS;

EID: 0037505387     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827501394568     Document Type: Article

References (79)

1. M. ABLOWITZ, D. BAR YAACOV, AND A. S. FOKAS, On the inverse scattering transform for the Kadomtsev-Petviashvili equation, Stud. Appl. Math., 69 (1983), pp, 135-143.
2. R. A. ADAMS, Sobolev Spaces, Academic Press, New York-London, 1975.
3. G. ALESSANDRINI, Stable determination of conductivity by boundary measurements, Appl. Anal., 27 (1988), pp. 153-172.
4. G. ALESSANDRINI, Singular solutions of elliptic equations and the determination of conductivity by boundary measurements, J. Differential Equations, 84 (1990), pp. 252-273.
5. A. ALLERS AND F. SANTOSA, Stability and resolution analysis of a linearized problem in electrical impedance tomography, Inverse Problems, 7 (1991), pp. 515-533.
6. K. ATKINSON, An Introduction to Numerical Analysis, 2nd ed., John Wiley, New York, 1989.
7. D. C. BARBER AND B. H. BROWN, Applied potential tomography, J. Phys. E. Sci. Instrum., 17 (1984), pp. 723-733.
8. D. C. BARBER AND B. H. BROWN, Progress in electrical impedance tomography, in Inverse Problems in Partial Differential Equations, SIAM, Philadelphia, 1990, pp. 151-164.
9. J. A. BARCELÓ, T. BARCELÓ, AND A. RUIZ, Stability of the inverse conductivity problem in the plane for less regular conductivities, J. Differential Equations, 173 (2001), pp. 231-270.
10. R. BEALS AND R. COIFMAN, Scattering, transformations spectrales et équations d'évolution non linéaires, Goulaouic-Meyer-Schwartz Seminar 1980-1981 22, École Polytechnique, Palaiseau, France, 1981.
11. R. BEALS AND R. COIFMAN, Scattering, transformations spectrales et équations d'évolution non linéaires. II, Goulaouic-Meyer-Schwartz Seminar 1981/1982 21, École Polytechnique, Palaiseau, France, 1982.
12. R. BEALS AND R. R. COIFMAN, Multidimensional inverse scattering and nonlinear partial differential equations, A.M.S., 43 (1985), pp. 45-70.
13. R. BEALS AND R. R. COIFMAN, The d-bar approach to inverse scattering and nonlinear evolution equations, Physica, 18 (1986), pp. 242-249.
14. C. BERENSTEIN AND E. C. TARABUSI, Inversion formulas for the k-dimensional Radon transform in real hyperbolic spaces, Duke Math. J., 62 (1991), pp. 1-19.
15. R. BLUE, Real-Time Three-Dimensional Electrical Impedance Tomography, Ph.D. thesis, R.P.I., Troy, NY, 1997.
16. L. BORCEA, Nonlinear Multigrid for Imaging Electrical Conductivity and Permittivity at Low Frequency, Inverse Problems, 17 (2001), pp. 329-359.
17. L. BORCEA, J. G. BERRYMAN, AND G. PAPANICOLAOU, High contrast impedance tomography, Inverse Problems, 12 (1996), pp. 835-858.
18. L. BORCEA, J. G. BERRYMAN, AND G. C. PAPANICOLAOU, Matching pursuit for imaging high-contrast impedance tomography, Inverse Problems, 15 (1999), pp. 811-849.
19. R. M. BROWN AND G. UHLMANN, Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions, Comm. Partial Differential Equations, 22 (1997), pp. 1009-1027.
20. A. P. CALDERÓN, On an inverse boundary value problem, in Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasil. Mat., Rio de Janeiro, 1980, pp. 65-73.
21. S. CHANILLO, A problem in electrical prospection and an n-dimensional Borg-Levinson theorem, Proc. Amer. Math. Soc., 108 (1980), pp. 761-767.
22. M. CHENEY, D. ISAACSON, AND E. L. ISAACSON, Exact solutions to a linearized inverse boundary value problem, Inverse Problems, 6 (1990), pp. 923-934.
23. M. CHENEY, D. ISAACSON, J. C. NEWELL, J. GOBLE, AND S. SIMSKE, Noser: An algorithm for solving the inverse conductivity problem, Internat. J. Imaging Systems and Technology, 2 (1990), pp. 66-75.
24. M. CHENEY, D. ISAACSON, AND J. C. NEWELL, Electrical impedance tomography, SIAM Rev., 41 (1999), pp. 85-101.
25. W. DAILY AND A. RAMIREZ, Electrical resistance tomography during in-situ tricholoethylene remediation at the savannah river site, Appl. Geophys., 33 (1995), pp. 239-249.
26. W. DAILY, A. RAMIREZ, AND R. JOHNSON, Electrical impedance tomography of a perchloroethy- lene release, JEEG, 2 (1998), pp. 189-201.
27. W. DAILY, A. RAMIREZ, D. LABRECQUE, AND J. NITAO, Electrical resistivity tomography of vadose water movement, Water Resources Res., 28 (1992), pp. 1429-1442.
28. D. C. DOBSON AND F. SANTOSA, An image-enhancement technique for electrical impedance tomography, Inverse Problems, 10 (1994), pp. 317-334.
29. O. DORN, H. BERTETA-AGUIRRE, J. G. BERRYMAN, AND G. PAPANICOLAOU, A nonlinear inversion method for 3D-electromagnetic imaging using adjoint fields, Inverse Problems, 15 (1999), pp. 1523-1558.
30. M. R. EGGLESTON, R. J. SCHWABE, D. ISAACSON, AND J. C. NEWELL, The application of electric current computed tomography to defect imaging in metals, in Review of Progress in Quantitative NDE, Vol. 9A, Plenum Press, New York, 1989, pp. 455-462.
31. L. D. FADDEEV, Increasing solutions of the Schrödinger equation, Sov. Phys. Dokl, 10 (1966), pp. 1033-1035.
32. A. S. FOKAS AND M. J. ABLOWITZ, The inverse scattering transform for multidimensional 2+1 problems, in Nonlinear Phenomena, K. B. Wolf, ed., Lecture Notes in Physics, Springer, Berlin, 1984, pp. 137-183.
33. M. IKEHATA AND S. SILTANEN, Numerical method for finding the convex hull of an inclusion in conductivity from boundary measurements, Inverse Problems, 16 (2000), pp. 1043-1052.
34. M. IKEHATA AND S. SILTANEN, Numerical Solution of the Cauchy Problem Using Faddeev's Green Function, preprint.
35. D. ISAACSON AND M. CHENEY, Effects of measurement precision and finite numbers of electrodes on linear impedance imaging algorithms, SIAM J. Appl. Math., 15 (1991), pp. 1705-1731.
36. V. ISAKOV, Completeness of products of solutions and some inverse problems for PDE, J. Differential Equations, 92 (1991), pp. 305-317.
37. V. ISAKOV, Uniqueness and stability in multi-dimesional inverse problems, Inverse Problems, 9 (1993), pp. 579-621.
38. V. ISAKOV, Inverse Problems for Partial Differential Equations, Springer, New York, 1998.
39. D. S. JERISON AND C. E. KENIG, Boundary behaviour of harmonic functions in non-tangentially accessible domains, Adv. Math., 46 (1982), pp. 80-147.
40. J. KAIPIO, V. KOLEHMAINEN, E. SOMERSALO, AND M. VAUHKONEN, Statistical inversion and Monte Carlo sampling methods in electrical impedance tomography, Inverse Problems, 16 (2000), pp. 1487-1522.
41. J. S. KALLMAN AND J. G. BERRYMAN, Weighted least-squares methods for electrical impedance tomography, IEEE Trans. Med. Imaging, 11 (1992), pp. 284-292.
42. M. KLIBANOV, A Newton-Kantorovich Method for Impedance Computed Tomography, preprint.
43. R. V. KOHN AND A. MCKENNEY, Numerical implementation of a variational method for electrical impedance imaging, Inverse Problems, 9 (1990), pp. 389-414.
44. R. V. KOHN AND M. VOGELIUS, Determining conductivity by boundary measurements, Comm. Pure Appl. Math., 37 (1984), pp. 289-298.
45. R. V. KOHN AND M. VOGELIUS, Determining conductivity by boundary measurements II. Inte- rior results, Comm. Pure Appl. Math., 38 (1985), pp. 643-667.
46. R. V. KOHN AND M. VOGELIUS, Relaxation of a variational method for impedance computed tomography, Comm. Pure Appl. Math., 40 (1987), pp. 745-777.
47. L. LIU, Stability Estimates for the Two-Dimensional Inverse Conductivity Problem, Ph.D. thesis, University of Rochester, Rochester, NY, 1997.
48. J. L. MUELLER, D. ISAACSON, AND J. C. NEWELL, A reconstruction algorithm for electrical impedance tomography data collected on rectangular electrode arrays, IEEE Trans. Biomed. Engr., 49 (1999), pp. 1379-1386.
49. J. L. MUELLER, D. ISAACSON, AND J. C. NEWELL, Reconstruction of conductivity changes due to ventilation and perfusion from EIT data collected on a rectangular electrode array, Physiol. Meas., 22 (2001), pp. 97-106.
50. A. NACHMAN, Inverse scattering at fixed energy, in Proceedings of 10th International Congress on Mathematical Physics, K. Schmüdgen, ed., Springer-Verlag, Berlin, 1992, pp. 434-441.
51. A. I. NACHMAN, Reconstructions from boundary measurements, Ann. of Math. (2), 128 (1988), pp. 531-576.
52. A. I. NACHMAN, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math. (2), 143 (1996), pp. 71-96.
53. A. NACHMAN AND M. J. ABLOWITZ, A multi-dimensional inverse scattering method, Stud. Appl. Math., 71 (1984), pp. 243-250.
54. A. I. NACHMAN, J. SYLVESTER, AND G. UHLMANN, An n-dimensional Borg-Levinson theorem, Comm. Math. Phys., 115 (1988), pp. 595-605.
55. R. G. NOVIKOV AND G. M. KHENKIN, The ∂̄-equation in the multidimensional inverse scattering problem, Uspekhi Mat. Nauk., 42 (1987), pp. 93-152.
56. A. RAMIREZ, W. DAILY, D. LABRECQUE, E. OWEN, AND D. CHESNUT, Monitoring an underground steam injection process using electrical resistance tomography, Water Resources Research, 29 (1993), pp. 73-87.
57. A. RAMIREZ, W. DAILY, A. BINLEY, D. LABRECQUE, AND D. ROELANT, Detection of leaks in underground storage tanks using electrical resistance methods, J. Envir. Eng. Geophys., 1 (1996), pp. 189-203.
58. F. SANTOSA AND M. VOGELIUS, A backprojection algorithm for electrical impedance imaging, SIAM J. Appl. Math., 50 (1990), pp. 216-243.
59. S. SILTANEN, J. MUELLER, AND D. ISAACSON, An implementation of the reconstruction algorithm of A. Nachman for the 2-D inverse conductivity problem, Inverse Problems, 16 (2000), pp. 681-699.
60. S. SILTANEN, J. MUELLER, AND D. ISAACSON, Errata: An implementation of the reconstruction algorithm of A. Nachman for the 2-D inverse conductivity problem. Inverse Problems, 17 (2001), pp. 1561-1563.
61. S. SILTANEN, J. MUELLER, AND D. ISAACSON, Reconstruction of high contrast 2-D conductivities by the algorithm of A. Nachman, in Radon Transforms and Tomography, E. T. Quinto, L. Ehrenpreis, A. Faridani, F. Gonzalez, and E. Grinberg, eds., AMS, Providence, RI, 2001, pp. 241-254.
62. S. SILTANEN, Electrical Impedance Tomography and Faddeev Green's Functions, Ph.D. thesis, Helsinki University of Technology, Helsinki, 1999.
63. E. SOMERSALO, M. CHENEY, D. ISAACSON, AND E. ISAACSON, Layer stripping: A direct numerical method for impedance imaging, Inverse Problems, 7 (1991), pp. 899-926.
64. S. STPHANESCI, C. SCHLUMBERGER, AND M. SCHLUMBERGER, Sur la distribution electrique autour d'une prise de terre ponctuelle dans un terrain a couches horizontales, homogenes et isotropes, J. Physics and Radium Ser., 7 (1930), pp. 132-140.
65. Z. SUN, On an inverse boundary value problem in two dimensions, Comm. Partial Differential Equations, 14 (1989), pp. 1101-1113.
66. Z. SUN, The inverse conductivity problem in two dimensions, J. Differential Equations, 87 (1990), pp. 227-255.
67. Z. SUN AND G. UHLMANN, Generic uniqueness for an inverse boundary value problem, Duke Math. J., 62 (1991), pp. 131-155.
68. A. J. SUROVIEC, S. S. STUCHLY, J. R. BARR, AND A. SWARUP, Dielectric properties of breast carcinoma and the surrounding tissues, IEEE Trans. Biomed. Eng., 35 (1988), pp. 257-263.
69. A. SWARUP, S. S. STUCHLY, AND A. SUROWIEC, Dielectric properties of mouse meal fibrosar- coma at different stages of development, Bioelectromagnetics, 12 (1991), p. 108.
70. J. SYLVESTER, A convergent layer stripping algorithm for the radially symmetric impedance tomography problem, Comm. Partial Differential Equations, 17 (1992), pp. 1955-1994.
71. J. SYLVESTER AND G. UHLMANN, A uniqueness problem for an inverse boundary value problem in electrical prospection, Comm. Pure Appl. Math., 39 (1986), pp. 91-112.
72. J. SYLVESTER AND G. UHLMANN, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2), 125 (1987), pp. 153-169.
73. J. SYLVESTER AND G. UHLMANN, Inverse boundary value problems at the boundary - continuous dependence, Comm. Pure Appl. Math., 41 (1988), pp. 197-219.
74. J. SYLVESTER AND G. UHLMANN, The Dirichlet to Neumann map and applications, in Inverse Problems in Partial Differential Equations, Philadelphia, SIAM, 1990, pp. 101-139.
75. G. VAINIKKO, Fast solvers of the Lippmann-Schwinger equation, in Direct and Inverse Problems of Mathematical Physics (Newark, DE), Int. Soc. Anal. Appl. Comput. 5, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000, pp. 423-440.
76. A. WEXLER, B. FRY, AND M. NEUMAN, Impedance computed tomography algorithm and system, Appl. Opt., 24 (1985), pp. 3982-3985.
77. R. A. WILLIAMS AND M. S. BECK, EDS., Process Tomography-Principles, Techniques, and Applications, Butterworth-Heinemann, Oxford, UK, 1995.
78. C. G. XIE, S. M. HUANG, AND B. S. BOYLE, Electrical capacitance tomography for flow imaging: System model for development of image reconstruction algorithms and design of primary sensors, IEE Proceedings, Part G, Circuits, Devices and Systems, 139 (1992), pp. 89-98.
79. T. J. YORKEY, J. G. WEBSTER, AND W. J. TOMPKINS, Comparing reconstruction algorithms for electrical impedance tomography, IEEE Trans. Biomed. Eng., 34 (1987), pp. 843-852.