An implementation of the reconstruction algorithm of A Nachman for the 2D inverse conductivity problem

The 2D inverse conductivity problem requires one to determine the unknown electrical conductivity distribution inside a bounded domain ??2 from knowledge of the Dirichlet-to-Neumann map. The problem has geophysical, industrial, and medical imaging (electrical impedance tomography) applications. In 1996 A Nachman proved that the Dirichlet-to-Neumann map uniquely determines C2 conductivities. The proof, which is constructive, outlines a direct method for reconstructing the conductivity. In this paper we present an implementation of the algorithm in Nachman's proof. The paper includes numerical results obtained by applying the general algorithms described to two radially symmetric cases of small and large contrast.

[1]  D. Isaacson,et al.  A reconstruction algorithm for electrical impedance tomography data collected on rectangular electrode arrays , 1999, IEEE Transactions on Biomedical Engineering.

[2]  M. Krasnosel’skiǐ,et al.  Small solutions of operator equations , 1972 .

[3]  J. Leon,et al.  On a spectral transform of a KDV-like equation related to the Schrodinger operator in the plane , 1987 .

[4]  A. Nachman,et al.  Global uniqueness for a two-dimensional inverse boundary value problem , 1996 .

[5]  A. Nachman,et al.  Reconstructions from boundary measurements , 1988 .

[6]  John Sylvester,et al.  A convergent layer stripping algorithm for the radially symmetric impedence tomography problem , 1992 .

[7]  R. Kress,et al.  Integral equation methods in scattering theory , 1983 .

[8]  V. Isakov.  Appendix -- Function Spaces , 2017 .

[9]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[10]  G. Papanicolaou,et al.  High-contrast impedance tomography , 1996 .

[11]  R. Kohn,et al.  Determining conductivity by boundary measurements II. Interior results , 1985 .

[12]  Gunther Uhlmann,et al.  Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions , 1997 .

[13]  Willis J. Tompkins,et al.  Comparing Reconstruction Algorithms for Electrical Impedance Tomography , 1987, IEEE Transactions on Biomedical Engineering.

[14]  David Isaacson,et al.  Issues in electrical impedance imaging , 1995 .

[15]  Begnaud Francis Hildebrand,et al.  Introduction to numerical analysis: 2nd edition , 1987 .

[16]  R. Kress.  Linear Integral Equations , 1989 .

[17]  Ian Knowles.  A variational algorithm for electrical impedance tomography , 1998 .

[18]  David Isaacson,et al.  Effects of measurement precision and finite numbers of electrodes on linear impedance imaging algorithms , 1991 .

[19]  James G. Berryman,et al.  Matching pursuit for imaging high-contrast conductivity , 1999 .

[20]  Guanrong Chen,et al.  Approximate Solutions of Operator Equations , 1997 .

[21]  Kendall E. Atkinson.  An introduction to numerical analysis , 1978 .

[22]  J. Sylvester,et al.  A global uniqueness theorem for an inverse boundary value problem , 1987 .

[23]  John Sylvester,et al.  The Dirichlet to Neumann map and applications , 1989 .

[24]  David Isaacson,et al.  Electrical Impedance Tomography , 1999, SIAM Rev..

[25]  David Isaacson,et al.  Electric current computed tomography eigenvalues , 1990 .