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Electrical impedance tomography using level set representation and total variational regularization. (English) Zbl 1072.65143

Summary: We propose a numerical scheme for the identification of piecewise constant conductivity coefficient for a problem arising from electrical impedance tomography. The key feature of the scheme is the use of level set method for the representation of interface between domains with different values of coefficients. Numerical tests show that our method can recover sharp interfaces and can tolerate a relatively high level of noise in the observation data. Results concerning the effects of number of measurements, noise level in the data as well as the regularization parameters on the accuracy of the scheme are also given.

MSC:

65N21 Numerical methods for inverse problems for boundary value problems involving PDEs
35R30 Inverse problems for PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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[1] Allers, A.; Santosa, F., Stability and resolution analysis of a linearized problem in electrical impedance tomography, Inverse problems, 7, 4, 515-533, (1991) · Zbl 0736.35141
[2] H.B. Ameur, M. Burger, B. Hackl, On some geometric inverse problems in linear elasticity, UCLA, Mathematics Department, CAM-Report 03-35, 2003 · Zbl 1086.35117
[3] Brühl, M., Explicit characterization of inclusions in electrical impedance tomography, SIAM J. math. anal., 32, 1327-1341, (2001) · Zbl 0980.35170
[4] Brühl, M.; Hanke, M., Numerical implementation of two noniterative methods for locating inclusions by impedance tomography, Inverse problems, 16, 1029-1042, (2000) · Zbl 0955.35076
[5] Burger, M., A level set method for inverse problems, Inverse problems, 17, 1327-1356, (2001) · Zbl 0985.35106
[6] M. Burger, Levenberg-Marquardt level set methods for inverse obstacle problems, UCLA, Mathematics Department, CAM-Report 03-45, 2003 · Zbl 1059.35162
[7] Burger, M., A framework for the construction of level set methods for shape optimization and reconstruction, Interfaces free boundaries, 5, 301-329, (2003) · Zbl 1081.35134
[8] Chan, T.F.; Tai, X.-C., Level set and total variation regularization for elliptic inverse problems with discontinuous coefficients, J. comput. phys., 193, 40-66, (2003) · Zbl 1036.65086
[9] Chan, T.F.; Tai, X.-C., Identification of discontinuous coefficients from elliptic problems using total variation regularization, SIAM J. sci. comp., 25, 881-904, (2003) · Zbl 1046.65090
[10] Cheney, M.; Isaacson, D.; Newell, J.C., Electrical impedance tomography, SIAM rev., 41, 85-101, (1999), (electronic) · Zbl 0927.35130
[11] Cheney, M.; Isaacson, D.; Newell, J.; Goble, J.; Simske, S., NOSER: an algorithm for solving the inverse conductivity problem, Int. J. imaging syst. technol., 2, 66-75, (1990)
[12] Dey, A.; Morrison, H.F., Resistivity modeling for arbitrarily shaped three dimensional structures, Geophysics, 44, 753-780, (1979)
[13] Dobson, D.C.; Santosa, F., An image-enhancement technique for electrical impedance tomography, Inverse problems, 10, 2, 317-334, (1994) · Zbl 0805.35149
[14] Dobson, D.C.; Santosa, F., Resolution and stability analysis of an inverse problem in electrical impedance tomography: dependence on the input current patterns, SIAM J. appl. math., 54, 6, 1542-1560, (1994) · Zbl 0813.35131
[15] Dorn, O.; Miller, E.L.; Rappaport, C.M., A shape reconstruction method for eletromagnetic tomography using adjoint and level sets, Inverse problems, 16, 1119-1156, (2000) · Zbl 0983.35150
[16] Ikehata, M.; Siltanen, S., Numerical method for finding the convex hull of an inclusion in conductivity from boundary measurements, Inverse probl., 16, 4, 1043-1052, (2000) · Zbl 0956.35133
[17] Ito, K.; Kunisch, K.; Li, Z., Level-set function approach to an inverse interface problem, Inverse probl., 17, 1225-1242, (2001) · Zbl 0986.35130
[18] Jarvenpaa, S.; Somersalo, E., Impedance imaging and electrode models, (), 65-74 · Zbl 0880.65111
[19] Kaipio, J.P.; Kolehmainen, V.; Somersalo, E.; Vauhkonen, M., Statistical inversion and Monte Carlo sampling methods in electrical impedance tomography, Inverse probl., 16, 5, 1487-1522, (2000) · Zbl 1044.78513
[20] Kolehmainen, V.; Arridge, S.R.; Lionheart, W.R.B.; Vauhkonen, M.; Kaipio, J.P., Recovery of region boundaries of piecewise constant coefficients of an elliptic PDE from boundary data, Inverse probl., 15, 5, 1375-1391, (1999) · Zbl 0936.35195
[21] Li, Y.; Oldenburg, D.W., Inversion of 3-d dc resistivity data using an approximate inverse mapping, Geophys. J. int., 116, 3, 557-569, (1994)
[22] J. Lie, M. Lysaker, X.-C. Tai, A variant of the level set method and applications to image segmentation, UCLA CAM Report 03-50, 2003. Available from: <http://www.math.ucla.edu/applied/cam> · Zbl 1096.35034
[23] Lionheart, W.R.B., Conformal uniqueness results in anisotropic electrical impedance imaging, Inverse probl., 13, 1, 125-134, (1997) · Zbl 0868.35140
[24] Lionheart, W.R.B., Boundary shape and electrical impedance tomography, Inverse probl., 14, 1, 139-147, (1998) · Zbl 0894.35129
[25] W.R.B. Lionheart, EIT reconstruction algorithms: pitfalls, challenges and recent developments, physiological measurement, to appear. Available from: <http://www.arxiv.org/abs/physics/0310151>
[26] P.R. McGillivray, Forward modelling and inversion of DC resistivity and MMR data, Ph.D. Thesis, University of British Columbia, 1992
[27] S.J. Osher, M. Burger, E. Yablonovitch, Inverse problem techniques for the design of photonic crystals, UCLA, Mathematics Department, CAM-Report 03-31, 2003
[28] Osher, S.J.; Fedkiw, R.R., Level set methods: an overview and some recent results, J. comput. phys., 169, 2, 463-502, (2001) · Zbl 0988.65093
[29] Osher, S.J.; Santosa, F., Level set methods for optimization problems involving geometry and constraints. I. frequencies of a two-density inhomogeneous drum, J. comput. phys., 171, 272-288, (2001) · Zbl 1056.74061
[30] Osher, S.J.; Sethian, J.A., Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, J. comput. phys., 79, 12-49, (1988) · Zbl 0659.65132
[31] L. Rondi, F. Santosa, Enhanced electrical impedance tomography via the Mumford-Shah functional, Control, optimisation, and calculus of variations, to appear. Available from: <http://www.math.umn.edu/ · Zbl 0989.35136
[32] Santosa, F.; Vogelius, M., A backprojection algorithm for electrical impedance imaging, SIAM J. appl. math., 50, 1, 216-243, (1990) · Zbl 0691.65087
[33] Siltanen, S.; Mueller, J.; Isaacson, D., An implementation of the reconstruction algorithm of A. nachman for the 2D inverse conductivity problem, Inverse probl., 16, 3, 681-699, (2000) · Zbl 0962.35193
[34] Smith, N.C.; Vozoff, K., Two dimensional DC resistivity inversion for dipole dipole data, IEEE trans. geosci. remote sensing, GE 22, 21-28, (1984), (special issue on electromagnetic methods in applied geophysics)
[35] Somersalo, E.; Cheney, M.; Isaacson, D., Existence and uniqueness for electrode models for electric current computed tomography, SIAM J. appl. math., 52, 4, 1023-1040, (1992) · Zbl 0759.35055
[36] B. Song, T.F. Chan, Fast algorithm for level set segmentation, UCLA CAM Report 02-68, 2002
[37] Tamburrino, A.; Rubinacci, G., A new non-iterative inversion method for electrical resistance tomography, Inverse probl., 18, 6, 1809-1829, (2002) · Zbl 1034.35154
[38] Tamburrino, A.; Ventre, S.; Rubinacci, G., Electrical resistance tomography: complementarity and quadratic models, Inverse probl., 16, 5, 1585-1618, (2000) · Zbl 0970.35169
[39] Vese, L.A.; Chan, T.F., A new multiphase level set framework for image segmentation via the Mumford and Shah model, Int. J. comput. vision, 50, 3, 271-293, (2002) · Zbl 1012.68782
[40] L. Vese, S. Osher, The level set method links active contours, Mumford-Shah segmentation, and total variation restoration. CAM Report 02-05, UCLA Mathematics Department, 2002
[41] Vogel, C., Computational methods for inverse problem, (2001), SIAM Philadelphia
[42] Ziemer, W.P., Weakly differentiable functions, (1989), Springer Berlin · Zbl 0177.08006
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