Abstract
Numerical implementation of the reconstruction formulae of Nakamura and Tanuma [Recent Development in Theories and Numerics, International Conference on Inverse Problems 2003] is presented. With the formulae, the conductivity and its normal derivative can be recovered on the boundary of a planar domain from the localized Dirichlet to Neumann map. Such reconstruction method is needed as a preliminary step before full reconstruction of conductivity inside the domain from boundary measurements, as done in electrical impedance tomography. Properties of the method are illustrated with reconstructions from simulated data.
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Astala, K., Päivärinta, L.: Calderón’s inverse conductivity problem in the plane. University of Helsinki Department of Mathematics, Preprint 370, September 2003.
Borcea, L.: Electrical impedance tomography. Inverse Problems 18, R99–R136 (2002).
Borcea, L.: Addendum to ‘‘Electrical Impedance Tomography’’. Inverse Problems 19, 997–998 (2002).
Brown, B. H., Barber, D. C., Seagar, A. D.: Applied potential tomography: possible clinical applications. Clin. Phys. Physiol. Meas. 6, 109–121 (1985).
Brown, R. M.: Recovering the conductivity at the boundary from the Dirichlet to Neumann map: a pointwise result, J. Inverse and Ill-posed Prob. 9, 567–574 (2001).
Brown, R. M., Uhlmann, G.: Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions. Communications in Partial Differential Equations 22, 1009–1027 (1997).
Calderón, A. P.: On an inverse boundary value problem. In: Seminar on Numerical Analysis and its Applications to Continuum Physics. Soc. Brasileira de Matemàtica pp. 65–73 (1980).
Cheney, M., Isaacson, D., Newell, J. C.: Electrical impedance tomography. SIAM Rev. 41, 85–101 (1999).
Cheng, K.-S., Isaacson, D., Newell, J. C., Gisser, D. G.: Electrode models for electric current computed tomography. IEEE Trans. on Biomedical Imaging, pp. 918–924 (1989).
Hecht, F., Pironneau, O., Ohtsuka, K.: Freefem++ Manual, http://www.ann.jussieu.fr/ hecht/freefem++.htm 2003.
Isaacson, D., Mueller, J. L., Newell, J. C., Siltanen, S.: Reconstructions of chest phantoms by the d-bar method for electrical impedance tomography. (submitted).
Kang, H., Yun, K.: Boundary determination of conductivities and Riemannian metrics via local Dirichlet-to-Neumann operator. SIAM J. Math. Anal. 34, 719–735 (2003).
Knudsen, K.: On the inverse conductivity problem, Ph.D. thesis, Aalborg University (2002).
Knudsen, K.: A new direct method for reconstructing isotropic conductivities in the plane. Physiol. Meas. 24, 391–401 (2003).
Knudsen, K., Tamasan, A.: Reconstruction of less regular conductivities in the plane, MSRI Preprint 2001–035 (2001).
Mueller, J. L., Siltanen, S.: Direct reconstructions of conductivities from boundary measurements. SIAM J. Sci. Comput. 24, 1232–1266 (2003).
Nachman, A. I.: Global uniqueness for a two-dimensional inverse boundary value problem. Ann Math. 143, 71–96 (1996).
Nakamura, G., Tanuma, K.: Local determination of conductivity at the boundary from the Dirichlet-to-Neumann map. Inverse Problems 17, 405–419 (2001).
Nakamura, G., Tanuma, K.: Direct determination of the derivatives of conductivity at the boundary from the localized Dirichlet to Neumann map. Comm. Korean Math. Soc. 16, 415–425 (2001).
Nakamura, G., Tanuma, K.: Formulas for reconstructing conductivity and its normal derivative at the boundary from the localized Dirichlet to Neumann map. In: Recent Development in Theories and Numerics. Int. Conf. on Inverse Problems (Yiu-Chung Hon, Masahiro Yamamoto, Jin Cheng, June-Yub Lee, eds.), pp. 192–201, World Scientific (2003).
Nakamura, G., Tanuma, K.: Formulas for reconstructing conductivity and its normal derivative at the boundary from the localized Dirichlet to Neumann map (preprint).
Siltanen, S., Mueller, J. L., Isaacson, D.: An implementation of the reconstruction algorithm of A. Nachman for the 2-D inverse conductivity problem. Inverse Problems 16, 681–699 (2000).
Somersalo, E., Cheney, M., Isaacson, D.: Existence and uniqueness for electrode models for electric current computed tomography. SIAM J. Appl. Math. 52, 1023–1040 (1992).
Sylvester, J., Uhlmann, G.: Inverse boundary value problem at the boundary-continuous dependence. Comm. Pure Appl. Math. 61, 197–219 (1988).
Vogel, C.: Computational methods for inverse problems. SIAM (2002).
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Nakamura, G., Siltanen, S., Tanuma, K. et al. Numerical Recovery of Conductivity at the Boundary from the Localized Dirichlet to Neumann Map. Computing 75, 197–213 (2005). https://doi.org/10.1007/s00607-004-0095-x
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DOI: https://doi.org/10.1007/s00607-004-0095-x