A direct reconstruction method for anisotropic electrical impedance tomography

A novel computational, non-iterative and noise-robust reconstruction method is introduced for the planar anisotropic inverse conductivity problem. The method is based on bypassing the unstable step of the reconstruction of the values of the isothermal coordinates on the boundary of the domain. Non-uniqueness of the inverse problem is dealt with by recovering the unique isotropic conductivity that can be achieved as a deformation of the measured anisotropic conductivity by isothermal coordinates. The method shows how isotropic D-bar reconstruction methods have produced reasonable and informative reconstructions even when used on EIT data known to come from anisotropic media, and when the boundary shape is not known precisely. Furthermore, the results pave the way for regularized anisotropic EIT. Key aspects of the approach involve D-bar methods and inverse scattering theory, complex geometrical optics solutions and quasi-conformal mapping techniques.

[1]  Robert V. Kohn,et al.  Determining conductivity by boundary measurements , 1984 .

[2]  Samuli Siltanen,et al.  Numerical computation of complex geometrical optics solutions to the conductivity equation , 2010 .

[3]  Jutta Bikowski,et al.  Direct numerical reconstruction of conductivities in three dimensions using scattering transforms , 2010, 1003.3764.

[4]  D. Isaacson,et al.  An implementation of the reconstruction algorithm of A Nachman for the 2D inverse conductivity problem , 2000 .

[5]  S J Hamilton,et al.  A direct D-bar reconstruction algorithm for recovering a complex conductivity in 2D , 2012, Inverse problems.

[6]  Kari Astala,et al.  Calderon's inverse conductivity problem in the plane , 2006 .

[7]  A. Bukhgeǐm,et al.  Recovering a potential from Cauchy data in the two-dimensional case , 2008 .

[8]  Zhenyu Guo,et al.  A review of electrical impedance techniques for breast cancer detection. , 2003, Medical engineering & physics.

[9]  Jutta Bikowski.  Electrical Impedance Tomography reconstructions in two and three dimensions: From Calderon to direct methods , 2009 .

[10]  Jari P. Kaipio,et al.  Compensation of errors due to incorrect model geometry in electrical impedance tomography , 2010 .

[11]  Kim Knudsen,et al.  The born approximation and Calderón's method for reconstruction of conductivities in 3-D , 2011 .

[12]  John M. Lee,et al.  Determining anisotropic real-analytic conductivities by boundary measurements , 1989 .

[13]  J C Newell,et al.  Reconstruction of conductivity changes due to ventilation and perfusion from EIT data collected on a rectangular electrode array. , 2001, Physiological measurement.

[14]  Gen Nakamura,et al.  Numerical Recovery of Conductivity at the Boundary from the Localized Dirichlet to Neumann Map , 2004, Computing.

[15]  M. E. Glidewell,et al.  Anatomically constrained electrical impedance tomography for three-dimensional anisotropic bodies , 1997, IEEE Transactions on Medical Imaging.

[16]  Ville Kolehmainen,et al.  Calderón's Inverse Problem with an Imperfectly Known Boundary and Reconstruction Up to a Conformal Deformation , 2010, SIAM J. Math. Anal..

[17]  D. Djajaputra.  Electrical Impedance Tomography: Methods, History and Applications , 2005 .

[18]  K. T. Ng,et al.  Anatomically constrained electrical impedance tomography for anisotropic bodies via a two-step approach , 1995, IEEE Trans. Medical Imaging.

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

[20]  Elisa Francini.  Recovering a complex coefficient in a planar domain from the Dirichlet-to-Neumann map , 2000 .

[21]  Jennifer L. Mueller,et al.  Properties of the reconstruction algorithm and associated scattering transform for admittivities in the plane , 2009 .

[22]  David Isaacson,et al.  Reconstructions of chest phantoms by the D-bar method for electrical impedance tomography , 2004, IEEE Transactions on Medical Imaging.

[23]  J C Newell,et al.  Imaging cardiac activity by the D-bar method for electrical impedance tomography , 2006, Physiological measurement.

[24]  Jérémi Dardé,et al.  Simultaneous Reconstruction of Outer Boundary Shape and Admittivity Distribution in Electrical Impedance Tomography , 2012, SIAM J. Imaging Sci..

[25]  Ville Kolehmainen,et al.  The Inverse Conductivity Problem with an Imperfectly Known Boundary , 2005, SIAM J. Appl. Math..

[27]  I Frerichs,et al.  Electrical impedance tomography (EIT) in applications related to lung and ventilation: a review of experimental and clinical activities. , 2000, Physiological measurement.

[28]  On the Gel'fand-Calderón inverse problem in two dimensions , 2013 .

[29]  David S. Holder.  Electrical impedance tomography , 2005 .

[30]  Jari P. Kaipio,et al.  Compensation of Modelling Errors Due to Unknown Domain Boundary in Electrical Impedance Tomography , 2011, IEEE Transactions on Medical Imaging.

[31]  Matteo Santacesaria,et al.  On an inverse problem for anisotropic conductivity in the plane , 2010, 1003.1880.

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

[33]  Samuli Siltanen,et al.  Nonlinear Fourier analysis for discontinuous conductivities: Computational results , 2013, J. Comput. Phys..

[34]  Aria Abubakar,et al.  Inversion algorithms for large-scale geophysical electromagnetic measurements , 2009 .

[35]  MATTI LASSAS,et al.  Calderóns' Inverse Problem for Anisotropic Conductivity in the Plane , 2004 .

[36]  William R B Lionheart,et al.  Validation of a finite-element solution for electrical impedance tomography in an anisotropic medium , 2007, Physiological measurement.

[37]  William Breckon.  The problem of anisotropy in Electrical Impedance Tomography , 1992, 1992 14th Annual International Conference of the IEEE Engineering in Medicine and Biology Society.

[38]  William R B Lionheart,et al.  Finite elements and anisotropic EIT reconstruction , 2010 .

[39]  Matti Lassas,et al.  REGULARIZED D-BAR METHOD FOR THE INVERSE CONDUCTIVITY PROBLEM , 2009 .

[40]  David Isaacson,et al.  An implementation of the reconstruction algorithm of A Nachman for the 2D inverse conductivity problem , 2000 .

[41]  Samuli Siltanen,et al.  Linear and Nonlinear Inverse Problems with Practical Applications , 2012, Computational science and engineering.

[42]  P. Koskela.  GEOMETRIC FUNCTION THEORY AND NON-LINEAR ANALYSIS (Oxford Mathematical Monographs) By TADEUSZ IWANIEC and GAVEN MARTIN: 552 pp., £75.00, ISBN 0-19-85029-4 (Oxford University Press, 2001) , 2002 .

[43]  S. Siltanen,et al.  Reconstructing conductivities with boundary corrected D-bar method , 2011, 1109.5982.

[44]  Ville Kolehmainen,et al.  Electrical impedance tomography problem with inaccurately known boundary and contact impedances , 2006, 3rd IEEE International Symposium on Biomedical Imaging: Nano to Macro, 2006..

[45]  Matti Lassas,et al.  Anisotropic conductivities that cannot be detected by EIT. , 2003, Physiological measurement.

[46]  Gunther Uhlmann,et al.  Anisotropic inverse problems in two dimensions , 2003 .

[47]  Kim Knudsen,et al.  A new direct method for reconstructing isotropic conductivities in the plane. , 2003, Physiological measurement.

[48]  A. Calderón,et al.  On an inverse boundary value problem , 2006 .

[49]  Kari Astala,et al.  Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (Pms-48) , 2009 .

[50]  Hyeonbae Kang,et al.  Boundary Determination of Conductivities and Riemannian Metrics via Local Dirichlet-to-Neumann Operator , 2002, SIAM J. Math. Anal..

[51]  P Bertemes-Filho,et al.  A comparison of modified Howland circuits as current generators with current mirror type circuits. , 2000, Physiological measurement.

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

[53]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

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

[55]  A. Clop,et al.  INTEGRAL STABILITY OF CALDER ´ ON INVERSE CONDUCTIVITY PROBLEM IN THE PLANE , 2008, 0807.4148.

[56]  David Atkinson,et al.  Use of anisotropic modelling in electrical impedance tomography; Description of method and preliminary assessment of utility in imaging brain function in the adult human head , 2008, NeuroImage.

[57]  Samuli Siltanen,et al.  Direct Reconstructions of Conductivities from Boundary Measurements , 2002, SIAM J. Sci. Comput..

[58]  Matti Lassas,et al.  On nonuniqueness for Calderón’s inverse problem , 2003 .

[59]  J L Mueller,et al.  2D D-bar reconstructions of human chest and tank data using an improved approximation to the scattering transform. , 2010, Physiological measurement.

[60]  Jennifer L. Mueller,et al.  Direct EIT Reconstructions of Complex Admittivities on a Chest-Shaped Domain in 2-D , 2013, IEEE Transactions on Medical Imaging.

[61]  Tzu-Jen Kao,et al.  A compensated radiolucent electrode array for combined EIT and mammography , 2007, Physiological measurement.

[62]  Matti Lassas,et al.  The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary , 2003 .

[63]  V. Zvyagin,et al.  Attractors of equations of non-Newtonian fluid dynamics , 2014 .

[64]  Kari Astala,et al.  A boundary integral equation for Calderón's inverse conductivity problem , 2006 .

[65]  Per Christian Hansen,et al.  Electrical impedance tomography: 3D reconstructions using scattering transforms , 2012 .

[66]  Alexandru Tamasan,et al.  Reconstruction of Less Regular Conductivities in the Plane , 2001 .

[67]  Alberto Ruiz,et al.  Stability of Calderón's inverse conductivity problem in the plane for discontinuous conductivities , 2010 .

[68]  Adrian I. Nachman.  Multidimensional inverse scattering and nonlinear equations , 1988 .

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

[70]  Lara Herrera,et al.  Um método D-bar para estimar admitividade em 2-D através de tomografia por impedância elétrica. , 2012 .

[71]  David Isaacson,et al.  A method for analyzing electrical impedance spectroscopy data from breast cancer patients , 2007, Physiological measurement.

[72]  Alberto Ruiz,et al.  Stability of Calderón inverse conductivity problem in the plane , 2007 .

[73]  Ville Kolehmainen,et al.  Recovering boundary shape and conductivity in electrical impedance tomography , 2013 .

[74]  David S. Holder,et al.  Electrical Impedance Tomography : Methods, History and Applications , 2004 .

[75]  Matti Lassas,et al.  On determining a Riemannian manifold from the Dirichlet-to-Neumann map , 2001 .

[76]  Juan Antonio Barceló,et al.  Stability of the Inverse Conductivity Problem in the Plane for Less Regular Conductivities , 2001 .

[77]  Samuli Siltanen,et al.  Numerical solution method for the dbar-equation in the plane , 2004 .

[78]  John Sylvester,et al.  An anisotropic inverse boundary value problem , 1990 .

[79]  Nuutti Hyvönen,et al.  Simultaneous recovery of admittivity and body shape in electrical impedance tomography: An experimental evaluation , 2013 .

[80]  Samuli Siltanen,et al.  Direct electrical impedance tomography for nonsmooth conductivities , 2011 .