# American Institute of Mathematical Sciences

April  2018, 12(2): 373-400. doi: 10.3934/ipi.2018017

## Enhancing D-bar reconstructions for electrical impedance tomography with conformal maps

 1 Aalto University, Department of Mathematics and Systems Analysis, P.O. Box 11100, FI-00076 Helsinki, Finland 2 Tallinn University of Technology, Department of Mathematics, Ehitajate tee 5, 19086 Tallinn, Estonia 3 University of Helsinki, Department of Mathematics and Statistics, P.O. Box 68, FI-00014 Helsinki, Finland

* Corresponding author

Received  March 2017 Revised  September 2017 Published  February 2018

Fund Project: The work of NH and JT was supported by the Academy of Finland (decision 267789). The work of LP was supported by Estonian government grant PUT1093.

We present a few ways of using conformal maps in the reconstruction of two-dimensional conductivities in electrical impedance tomography. First, by utilizing the Riemann mapping theorem, we can transform any simply connected domain of interest to the unit disk where the D-bar method can be implemented most efficiently. In particular, this applies to the open upper half-plane. Second, in the unit disk we may choose a region of interest that is magnified using a suitable Möbius transform. To facilitate the efficient use of conformal maps, we introduce input current patterns that are named conformally transformed truncated Fourier basis; in practice, their use corresponds to positioning the available electrodes close to the region of interest. These ideas are numerically tested using simulated continuum data in bounded domains and simulated point electrode data in the half-plane. The connections to practical electrode measurements are also discussed.

Citation: Nuutti Hyvönen, Lassi Päivärinta, Janne P. Tamminen. Enhancing D-bar reconstructions for electrical impedance tomography with conformal maps. Inverse Problems & Imaging, 2018, 12 (2) : 373-400. doi: 10.3934/ipi.2018017
##### References:
 [1] M. J. Ablowitz and A. I. Nachman, Multidimensional nonlinear evolution equations and inverse scattering, Physica D, 18 (1986), 223-241.  doi: 10.1016/0167-2789(86)90183-1.  Google Scholar [2] I. Akduman and R. Kress, Electrostatic imaging via conformal mapping, Inverse Problems, 18 (2002), 1659-1672.  doi: 10.1088/0266-5611/18/6/315.  Google Scholar [3] G. Alessandrini and A. Scapin, Depth dependent resolution in electrical impedance tomography, J. Inv. Ill-Posed Problems, 25 (2017), 391-402.   Google Scholar [4] K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane, Ann. Math., 163 (2006), 265-299.  doi: 10.4007/annals.2006.163.265.  Google Scholar [5] R. Beals and R. R. Coifman, Scattering, transformations spectrales et équations d'évolution non linéaires, in Goulaouic-Meyer-Schwartz Seminar, 1980-1981, pages Exp. No. XXII, 10. École Polytech., Palaiseau, 1981.  Google Scholar [6] M. Boiti, J. P. Leon, M. Manna and F. Pempinelli, On a spectral transform of a KdV-like equation related to the Schrödinger operator in the plane, Inverse Problems, 3 (1987), 25-36.  doi: 10.1088/0266-5611/3/1/008.  Google Scholar [7] L. Borcea, Electrical impedance tomography, Inverse Problems, 18 (2002), R99-R136.  doi: 10.1088/0266-5611/18/6/201.  Google Scholar [8] R. M. Brown and G. Uhlmann, Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions, Comm. Partial Differential Equations, 22 (1997), 1009-1027.  doi: 10.1080/03605309708821292.  Google Scholar [9] A. -P. Calderón, On an inverse boundary value problem, in Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), pages 65-73. Soc. Brasil. Mat., Rio de Janeiro, 1980.  Google Scholar [10] M. Cheney and D. Isaacson, Distinguishability in impedance imaging, IEEE Trans. Biomed. Eng., 39 (1992), 852-860.   Google Scholar [11] M. Cheney, D. Isaacson and J. C. Newell, Electrical impedance tomography, SIAM Rev., 41 (1999), 85-101.  doi: 10.1137/S0036144598333613.  Google Scholar [12] K. S. Cheng, D. Isaacson, J. C. Newell and D. G. Gisser, Electrode models for electric current computed tomography, IEEE Trans. Biomed. Eng., 36 (1989), 918-924.   Google Scholar [13] M. V. de Hoop, M. Lassas, M. Santacesaria, S. Siltanen and J. P. Tamminen, D-bar method and exceptional points at positive energy: A computational study, Inverse Problems, 32 (2016), 025003, 35pp.  Google Scholar [14] T. A. Driscoll, Algorithm 756: A MATLAB toolbox for schwarz-christoffel mapping, Trans. Math. Soft., 22 (1996), 168-186.  doi: 10.1145/229473.229475.  Google Scholar [15] L. D. Faddeev, Increasing solutions of the Schrödinger equation, Dokl. Phys., 10 (1966), 1033-1035.   Google Scholar [16] H. Garde and K. Knudsen, Distinguishability revisited: Depth dependent bounds on reconstruction quality in electrical impedance tomography, SIAM J. Appl. Math, 77 (2017), 697-720.  doi: 10.1137/16M1072991.  Google Scholar [17] I. M. Gelfand, Some problems of functional analysis and algebra, in International Mathematical Congress in Amsterdam (in Russian), Nauka, Moscow, pages 49-74,1961. Google Scholar [18] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985.  Google Scholar [19] H. Haddar and R. Kress, Conformal mappings and inverse boundary value problems, Inverse Problems, 21 (2005), 935-953.  doi: 10.1088/0266-5611/21/3/009.  Google Scholar [20] H. Haddar and R. Kress, Conformal mapping and impedance tomography, Inverse Problems, 26 (2010), 074002, 18pp.  Google Scholar [21] H. Hakula, L. Harhanen and N. Hyvönen, Sweep data of electrical impedance tomography, Inverse Problems, 27 (2011), 115006, 19pp.  Google Scholar [22] H. Hakula, A. Rasila and M. Vuorinen, On moduli of rings and quadrilaterals: Algorithms and experiments, SIAM J. Sci. Comput, 33 (2011), 279-302.  doi: 10.1137/090763603.  Google Scholar [23] M. Hanke, B. Harrach and N. Hyvönen, Justification of point electrode models in electrical impedance tomography, Math. Models Methods Appl. Sci., 21 (2011), 1395-1413.  doi: 10.1142/S0218202511005362.  Google Scholar [24] M. Hanke, N. Hyvönen and S. Reusswig, Convex source support and its application to electric impedance tomography, SIAM J. Imaging Sci., 1 (2008), 364-378.  doi: 10.1137/080715640.  Google Scholar [25] L. Harhanen and N. Hyvönen, Convex source support in half-plane, Inverse Probl. Imaging, 4 (2010), 429-448.  doi: 10.3934/ipi.2010.4.429.  Google Scholar [26] N. Hyvönen, Complete electrode model of electrical impedance tomography: Approximation properties and characterization of inclusions, SIAM J. Appl. Math., 64 (2004), 902-931.  doi: 10.1137/S0036139903423303.  Google Scholar [27] N. Hyvönen, Approximating idealized boundary data of electric impedance tomography by electrode measurements, Math. Models Methods Appl. Sci., 19 (2009), 1185-1202.  doi: 10.1142/S0218202509003759.  Google Scholar [28] N. Hyvönen, N. Majander and S. Staboulis, Compensation for geometric modeling errors by electrode movement in electrical impedance tomography, Inverse Problems, 33 (2017), 035006. Google Scholar [29] D. Isaacson, Distinguishability of conductivities by electric current computed tomography, IEEE Trans. Med. Imag., 5 (1986), 91-95.  doi: 10.1109/TMI.1986.4307752.  Google Scholar [30] D. Isaacson, J. L. Mueller, J. C. Newell and S. Siltanen, Reconstructions of chest phantoms by the {D}-bar method for electrical impedance tomography, IEEE Trans. Med. Imag., 23 (2004), 821-828.  doi: 10.1109/TMI.2004.827482.  Google Scholar [31] D. Isaacson, J. L. Mueller, J. C. Newell and S. Siltanen, Imaging cardiac activity by the {D}-bar method for electrical impedance tomography, Physiol. Meas., 27 (2006), S43-S50.  doi: 10.1088/0967-3334/27/5/S04.  Google Scholar [32] K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen, D-bar method for electrical impedance tomography with discontinuous conductivities, SIAM J. Appl. Math., 67 (2007), 893-913.  doi: 10.1137/060656930.  Google Scholar [33] K. Knudsen, M. Lassas, J. Mueller and S. Siltanen, Reconstructions of piecewise constant conductivities by the D-bar method for electrical impedance tomography in Journal of Physics: Conference Series, volume 124, pages 012029, IOP Publishing, 2008. doi: 10.1088/1742-6596/124/1/012029.  Google Scholar [34] K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen, Regularized D-bar method for the inverse conductivity problem, Inverse Probl. Imaging, 3 (2009), 599-624.  doi: 10.3934/ipi.2009.3.599.  Google Scholar [35] K. Knudsen, J. L. Mueller and S. Siltanen, Numerical solution method for the dbar-equation in the plane, J. Comput. Phys., 198 (2004), 500-517.  doi: 10.1016/j.jcp.2004.01.028.  Google Scholar [36] R. Kress, Inverse problems and conformal mapping, Complex Var. Elliptic Equ., 57 (2012), 301-316.  doi: 10.1080/17476933.2011.605446.  Google Scholar [37] A. Kurganov and J. Rauch, The order of accuracy of quadrature formulae for periodic functions, in Advances in Phase Space Analysis of Partial Differential Equations, volume 78 of Progress in Nonlinear Differential Equations and Their Applications, pages 155-159. Springer, 2009.  Google Scholar [38] J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, volume 1, Springer-Verlag, 1973. Translated from French by P. Kenneth.  Google Scholar [39] J. L. Mueller and S. Siltanen, Direct reconstructions of conductivities from boundary measurements, SIAM J. Sci. Comput, 24 (2003), 1232-1266.  doi: 10.1137/S1064827501394568.  Google Scholar [40] J. L. Mueller, S. Siltanen and D. Isaacson, A direct reconstruction algorithm for electrical impedance tomography, IEEE Trans. Med. Imag., 21 (2002), 555-559.  doi: 10.1109/TMI.2002.800574.  Google Scholar [41] A. I. Nachman, Reconstructions from boundary measurements, Ann. Math., 128 (1988), 531-576.  doi: 10.2307/1971435.  Google Scholar [42] A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. Math., 143 (1996), 71-96.  doi: 10.2307/2118653.  Google Scholar [43] R. G. Novikov, A multidimensional inverse spectral problem for the equation $-δψ+(v(x)- e u(x))ψ = 0$, Funct. Anal. Appl., 22 (1988), 263-272.   Google Scholar [44] Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin, 1992.  Google Scholar [45] J. Saranen and G. Vainikko, Periodic Integral and Pseudodifferential Equations with Numerical Approximation, Springer-Verlag, Berlin, 2002.  Google Scholar [46] B. E. Seagar, T. S. Yeo and R. H. T. Bates, Full-wave computed tomography part 2: Resolution limits, IEE Proc-A, 131 (1984), 616-622.  doi: 10.1049/ip-a-1.1984.0079.  Google Scholar [47] O. Seiskari, Point electrode problems in piecewise smooth plane domains, SIAM J. Math. Anal., 46 (2014), 1204-1227.  doi: 10.1137/130928674.  Google Scholar [48] S. Siltanen, Electrical Impedance Tomography and Faddeev Green's Functions, Dissertation, Helsinki University of Technology, Espoo, 1999.  Google Scholar [49] 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), 681-699.  doi: 10.1088/0266-5611/16/3/310.  Google Scholar [50] 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), 1561-1563.  doi: 10.1088/0266-5611/17/5/501.  Google Scholar [51] S. Siltanen and J. P. Tamminen, Reconstructing conductivities with boundary corrected D-bar method, J. Inv. Ill-Posed Problems, 22 (2014), 847-870.   Google Scholar [52] E. Somersalo, M. Cheney and D. Isaacson, Existence and uniqueness for electrode models for electric current computed tomography, SIAM J. Appl. Math., 52 (1992), 1023-1040.  doi: 10.1137/0152060.  Google Scholar [53] J. P. Tamminen, T. Tarvainen and S. Siltanen, The d-bar method for diffuse optical tomography: A computational study, Exper. Math., 26 (2017), 225-240.  doi: 10.1080/10586458.2016.1157775.  Google Scholar [54] L. N. Trefethen and T. A. Driscoll, Schwarz-Christoffel Mapping, Cambridge Univ. Press, Cambridge, 2002.   Google Scholar [55] G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse Problems, 25 (2009), 123011, 39pp.  Google Scholar [56] G. Vainikko, Fast solvers of the Lippmann-Schwinger equation, In Direct and inverse problems of mathematical physics (Newark, DE, 1997), volume 5 of Int. Soc. Anal. Appl. Comput., pages 423-440. Kluwer Acad. Publ., Dordrecht, 2000.  Google Scholar [57] R. Winkler and A. Rieder, Resolution-controlled conductivity discretization in electrical impedance tomography, SIAM J. Imaging Sci., 7 (2014), 2048-2077.  doi: 10.1137/140958955.  Google Scholar

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##### References:
 [1] M. J. Ablowitz and A. I. Nachman, Multidimensional nonlinear evolution equations and inverse scattering, Physica D, 18 (1986), 223-241.  doi: 10.1016/0167-2789(86)90183-1.  Google Scholar [2] I. Akduman and R. Kress, Electrostatic imaging via conformal mapping, Inverse Problems, 18 (2002), 1659-1672.  doi: 10.1088/0266-5611/18/6/315.  Google Scholar [3] G. Alessandrini and A. Scapin, Depth dependent resolution in electrical impedance tomography, J. Inv. Ill-Posed Problems, 25 (2017), 391-402.   Google Scholar [4] K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane, Ann. Math., 163 (2006), 265-299.  doi: 10.4007/annals.2006.163.265.  Google Scholar [5] R. Beals and R. R. Coifman, Scattering, transformations spectrales et équations d'évolution non linéaires, in Goulaouic-Meyer-Schwartz Seminar, 1980-1981, pages Exp. No. XXII, 10. École Polytech., Palaiseau, 1981.  Google Scholar [6] M. Boiti, J. P. Leon, M. Manna and F. Pempinelli, On a spectral transform of a KdV-like equation related to the Schrödinger operator in the plane, Inverse Problems, 3 (1987), 25-36.  doi: 10.1088/0266-5611/3/1/008.  Google Scholar [7] L. Borcea, Electrical impedance tomography, Inverse Problems, 18 (2002), R99-R136.  doi: 10.1088/0266-5611/18/6/201.  Google Scholar [8] R. M. Brown and G. Uhlmann, Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions, Comm. Partial Differential Equations, 22 (1997), 1009-1027.  doi: 10.1080/03605309708821292.  Google Scholar [9] A. -P. Calderón, On an inverse boundary value problem, in Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), pages 65-73. Soc. Brasil. Mat., Rio de Janeiro, 1980.  Google Scholar [10] M. Cheney and D. Isaacson, Distinguishability in impedance imaging, IEEE Trans. Biomed. Eng., 39 (1992), 852-860.   Google Scholar [11] M. Cheney, D. Isaacson and J. C. Newell, Electrical impedance tomography, SIAM Rev., 41 (1999), 85-101.  doi: 10.1137/S0036144598333613.  Google Scholar [12] K. S. Cheng, D. Isaacson, J. C. Newell and D. G. Gisser, Electrode models for electric current computed tomography, IEEE Trans. Biomed. Eng., 36 (1989), 918-924.   Google Scholar [13] M. V. de Hoop, M. Lassas, M. Santacesaria, S. Siltanen and J. P. Tamminen, D-bar method and exceptional points at positive energy: A computational study, Inverse Problems, 32 (2016), 025003, 35pp.  Google Scholar [14] T. A. Driscoll, Algorithm 756: A MATLAB toolbox for schwarz-christoffel mapping, Trans. Math. Soft., 22 (1996), 168-186.  doi: 10.1145/229473.229475.  Google Scholar [15] L. D. Faddeev, Increasing solutions of the Schrödinger equation, Dokl. Phys., 10 (1966), 1033-1035.   Google Scholar [16] H. Garde and K. Knudsen, Distinguishability revisited: Depth dependent bounds on reconstruction quality in electrical impedance tomography, SIAM J. Appl. Math, 77 (2017), 697-720.  doi: 10.1137/16M1072991.  Google Scholar [17] I. M. Gelfand, Some problems of functional analysis and algebra, in International Mathematical Congress in Amsterdam (in Russian), Nauka, Moscow, pages 49-74,1961. Google Scholar [18] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985.  Google Scholar [19] H. Haddar and R. Kress, Conformal mappings and inverse boundary value problems, Inverse Problems, 21 (2005), 935-953.  doi: 10.1088/0266-5611/21/3/009.  Google Scholar [20] H. Haddar and R. Kress, Conformal mapping and impedance tomography, Inverse Problems, 26 (2010), 074002, 18pp.  Google Scholar [21] H. Hakula, L. Harhanen and N. Hyvönen, Sweep data of electrical impedance tomography, Inverse Problems, 27 (2011), 115006, 19pp.  Google Scholar [22] H. Hakula, A. Rasila and M. Vuorinen, On moduli of rings and quadrilaterals: Algorithms and experiments, SIAM J. Sci. Comput, 33 (2011), 279-302.  doi: 10.1137/090763603.  Google Scholar [23] M. Hanke, B. Harrach and N. Hyvönen, Justification of point electrode models in electrical impedance tomography, Math. Models Methods Appl. Sci., 21 (2011), 1395-1413.  doi: 10.1142/S0218202511005362.  Google Scholar [24] M. Hanke, N. Hyvönen and S. Reusswig, Convex source support and its application to electric impedance tomography, SIAM J. Imaging Sci., 1 (2008), 364-378.  doi: 10.1137/080715640.  Google Scholar [25] L. Harhanen and N. Hyvönen, Convex source support in half-plane, Inverse Probl. Imaging, 4 (2010), 429-448.  doi: 10.3934/ipi.2010.4.429.  Google Scholar [26] N. Hyvönen, Complete electrode model of electrical impedance tomography: Approximation properties and characterization of inclusions, SIAM J. Appl. Math., 64 (2004), 902-931.  doi: 10.1137/S0036139903423303.  Google Scholar [27] N. Hyvönen, Approximating idealized boundary data of electric impedance tomography by electrode measurements, Math. Models Methods Appl. Sci., 19 (2009), 1185-1202.  doi: 10.1142/S0218202509003759.  Google Scholar [28] N. Hyvönen, N. Majander and S. Staboulis, Compensation for geometric modeling errors by electrode movement in electrical impedance tomography, Inverse Problems, 33 (2017), 035006. Google Scholar [29] D. Isaacson, Distinguishability of conductivities by electric current computed tomography, IEEE Trans. Med. Imag., 5 (1986), 91-95.  doi: 10.1109/TMI.1986.4307752.  Google Scholar [30] D. Isaacson, J. L. Mueller, J. C. Newell and S. Siltanen, Reconstructions of chest phantoms by the {D}-bar method for electrical impedance tomography, IEEE Trans. Med. Imag., 23 (2004), 821-828.  doi: 10.1109/TMI.2004.827482.  Google Scholar [31] D. Isaacson, J. L. Mueller, J. C. Newell and S. Siltanen, Imaging cardiac activity by the {D}-bar method for electrical impedance tomography, Physiol. Meas., 27 (2006), S43-S50.  doi: 10.1088/0967-3334/27/5/S04.  Google Scholar [32] K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen, D-bar method for electrical impedance tomography with discontinuous conductivities, SIAM J. Appl. Math., 67 (2007), 893-913.  doi: 10.1137/060656930.  Google Scholar [33] K. Knudsen, M. Lassas, J. Mueller and S. Siltanen, Reconstructions of piecewise constant conductivities by the D-bar method for electrical impedance tomography in Journal of Physics: Conference Series, volume 124, pages 012029, IOP Publishing, 2008. doi: 10.1088/1742-6596/124/1/012029.  Google Scholar [34] K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen, Regularized D-bar method for the inverse conductivity problem, Inverse Probl. Imaging, 3 (2009), 599-624.  doi: 10.3934/ipi.2009.3.599.  Google Scholar [35] K. Knudsen, J. L. Mueller and S. Siltanen, Numerical solution method for the dbar-equation in the plane, J. Comput. Phys., 198 (2004), 500-517.  doi: 10.1016/j.jcp.2004.01.028.  Google Scholar [36] R. Kress, Inverse problems and conformal mapping, Complex Var. Elliptic Equ., 57 (2012), 301-316.  doi: 10.1080/17476933.2011.605446.  Google Scholar [37] A. Kurganov and J. Rauch, The order of accuracy of quadrature formulae for periodic functions, in Advances in Phase Space Analysis of Partial Differential Equations, volume 78 of Progress in Nonlinear Differential Equations and Their Applications, pages 155-159. Springer, 2009.  Google Scholar [38] J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, volume 1, Springer-Verlag, 1973. Translated from French by P. Kenneth.  Google Scholar [39] J. L. Mueller and S. Siltanen, Direct reconstructions of conductivities from boundary measurements, SIAM J. Sci. Comput, 24 (2003), 1232-1266.  doi: 10.1137/S1064827501394568.  Google Scholar [40] J. L. Mueller, S. Siltanen and D. Isaacson, A direct reconstruction algorithm for electrical impedance tomography, IEEE Trans. Med. Imag., 21 (2002), 555-559.  doi: 10.1109/TMI.2002.800574.  Google Scholar [41] A. I. Nachman, Reconstructions from boundary measurements, Ann. Math., 128 (1988), 531-576.  doi: 10.2307/1971435.  Google Scholar [42] A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. Math., 143 (1996), 71-96.  doi: 10.2307/2118653.  Google Scholar [43] R. G. Novikov, A multidimensional inverse spectral problem for the equation $-δψ+(v(x)- e u(x))ψ = 0$, Funct. Anal. Appl., 22 (1988), 263-272.   Google Scholar [44] Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin, 1992.  Google Scholar [45] J. Saranen and G. Vainikko, Periodic Integral and Pseudodifferential Equations with Numerical Approximation, Springer-Verlag, Berlin, 2002.  Google Scholar [46] B. E. Seagar, T. S. Yeo and R. H. T. Bates, Full-wave computed tomography part 2: Resolution limits, IEE Proc-A, 131 (1984), 616-622.  doi: 10.1049/ip-a-1.1984.0079.  Google Scholar [47] O. Seiskari, Point electrode problems in piecewise smooth plane domains, SIAM J. Math. Anal., 46 (2014), 1204-1227.  doi: 10.1137/130928674.  Google Scholar [48] S. Siltanen, Electrical Impedance Tomography and Faddeev Green's Functions, Dissertation, Helsinki University of Technology, Espoo, 1999.  Google Scholar [49] 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), 681-699.  doi: 10.1088/0266-5611/16/3/310.  Google Scholar [50] 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), 1561-1563.  doi: 10.1088/0266-5611/17/5/501.  Google Scholar [51] S. Siltanen and J. P. Tamminen, Reconstructing conductivities with boundary corrected D-bar method, J. Inv. Ill-Posed Problems, 22 (2014), 847-870.   Google Scholar [52] E. Somersalo, M. Cheney and D. Isaacson, Existence and uniqueness for electrode models for electric current computed tomography, SIAM J. Appl. Math., 52 (1992), 1023-1040.  doi: 10.1137/0152060.  Google Scholar [53] J. P. Tamminen, T. Tarvainen and S. Siltanen, The d-bar method for diffuse optical tomography: A computational study, Exper. Math., 26 (2017), 225-240.  doi: 10.1080/10586458.2016.1157775.  Google Scholar [54] L. N. Trefethen and T. A. Driscoll, Schwarz-Christoffel Mapping, Cambridge Univ. Press, Cambridge, 2002.   Google Scholar [55] G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse Problems, 25 (2009), 123011, 39pp.  Google Scholar [56] G. Vainikko, Fast solvers of the Lippmann-Schwinger equation, In Direct and inverse problems of mathematical physics (Newark, DE, 1997), volume 5 of Int. Soc. Anal. Appl. Comput., pages 423-440. Kluwer Acad. Publ., Dordrecht, 2000.  Google Scholar [57] R. Winkler and A. Rieder, Resolution-controlled conductivity discretization in electrical impedance tomography, SIAM J. Imaging Sci., 7 (2014), 2048-2077.  doi: 10.1137/140958955.  Google Scholar
Top left: The real part of a FEM approximation for the solution $u$ to (3) in a homogeneous rectangular domain $\Omega$ with the boundary current density $\varphi_8$ of (33) corresponding to a Schwarz-Christoffel map $\Phi: \Omega \to D$. Top right: The real part of $\tilde{u} = u \circ \Psi$ in $D$. Bottom left: The real part of the true current density $\varphi_8$ as a function of the arclength parameter on $\partial \Omega$. Bottom right: The real part of the virtual current density $\tilde{\phi}_8$ as a function of the polar angle on $\partial D$
Left: The true conductivity, with the blue area representing the ROI that contains two black anomalies. The red dot indicates the Möbius parameter $a = 0.6$. Right: The conformally mapped conductivity $\tilde{\sigma} = \sigma \circ \mathcal{M}_a^{-1}$, with the image of the ROI under $\mathcal{M}_a$ presented by blue color
Top row: the target conductivity in three different domains. Bottom row: the virtual conducitivities, i.e., the target conductivities transformed to the unit disk by a Schwarz-Christoffel map that fixes the origin.
with $c = 10$. The white parts correspond to $|\mathbf{t}|>c$ or $|k|>R$, i.e, $\mathbf{t}_{R, c} = 0$">Figure 4.  Numerical approximations for the real part of the truncated scattering transform $\mathbf{t}_{R, c}$ of (24) for the five configurations in Figure 3 with $c = 10$. The white parts correspond to $|\mathbf{t}|>c$ or $|k|>R$, i.e, $\mathbf{t}_{R, c} = 0$
Reconstructions of the test conductivity in three different domains and the corresponding relative $L^2(\Omega )$-errors. Top row: the target configurations. Middle row: reconstructions by the D-bar method using the current patterns (19) with $N = 16$. Bottom row: reconstructions by the D-bar method in the virtual domain (i.e., the unit disk) using the conformally transformed current patterns (33) with $N = 16$ in the true domain
Magnification of a ROI. Top row: the true target configurations. Middle row: the conformally transformed conductivities in the virtual domain with magnified ROIs. On their boundaries the point corresponding to $s = 0$ is marked with a green dot. The black dot close to the ROI is mapped to the origin. Bottom row: The real parts of the conformally transformed Fourier current pattern $\varphi_{16}$ of (33) for the respective true target domains
) with the corresponding relative $L^2$-errors over the ROIs. Top row: the ROIs in the original domains of Figure 6. Middle row: reconstructions by the D-bar method using the current patterns (19) with $N = 16$. Bottom row: reconstructions by the D-bar method in the virtual domain (i.e., the unit disk) using the conformally transformed current patterns (33) with $N = 16$ in the true domain. The left column corresponds to noiseless data, whereas the reconstructions in the right column are based on measurements corrupted by $1\%$ of additive noise">Figure 7.  Reconstructions by magnifying the ROIs (cf. Figure 6) with the corresponding relative $L^2$-errors over the ROIs. Top row: the ROIs in the original domains of Figure 6. Middle row: reconstructions by the D-bar method using the current patterns (19) with $N = 16$. Bottom row: reconstructions by the D-bar method in the virtual domain (i.e., the unit disk) using the conformally transformed current patterns (33) with $N = 16$ in the true domain. The left column corresponds to noiseless data, whereas the reconstructions in the right column are based on measurements corrupted by $1\%$ of additive noise
Conductivity inhomogeneities in the upper half-plane (left) and the corresponding virtual conductivities in the unit disk corresponding to $\mathcal{M}_b$ with five different values for $b$ (right). The black dots indicate the sampling points for the boundary potentials corresponding to the (truncated) currents (33), with $N = 5$, on the measurement interval $[-2.5+b, 2.5+b]$. These dots are mapped to an (incomplete) equidistant grid of points on the unit circle by $\mathcal{M}_b$. The gray areas on the right correspond to the exterior of the rectangular subset considered on the left
)">Figure 9.  D-bar reconstruction in the upper half-plane obtained by using five different Möbius transforms, i.e., $\mathcal{M}_b$ with $b = -3, -1.5, 0, 1.5, 3$, for mapping the half-plane onto the unit disk. The vertical line segments indicate the used values of $b$ (cf. Figure 8)
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