# American Institute of Mathematical Sciences

May  2013, 7(2): 417-443. doi: 10.3934/ipi.2013.7.417

## Study of noise effects in electrical impedance tomography with resistor networks

 1 Computational and Applied Mathematics, Rice University, MS 134, 6100 Main St. Houston, TX 77005-1892, United States 2 Department of Mathematics, University of Utah, 155 S 1400 E RM 233, Salt Lake City, UT 84112-0090, United States 3 Institute for Computational Engineering and Sciences, University of Texas at Austin, 1 University Station C0200, Austin, TX 78712, United States

Received  December 2011 Revised  January 2013 Published  May 2013

We present a study of the numerical solution of the two dimensional electrical impedance tomography problem, with noisy measurements of the Dirichlet to Neumann map. The inversion uses parametrizations of the conductivity on optimal grids. The grids are optimal in the sense that finite volume discretizations on them give spectrally accurate approximations of the Dirichlet to Neumann map. The approximations are Dirichlet to Neumann maps of special resistor networks, that are uniquely recoverable from the measurements. Inversion on optimal grids has been proposed and analyzed recently, but the study of noise effects on the inversion has not been carried out. In this paper we present a numerical study of both the linearized and the nonlinear inverse problem. We take three different parametrizations of the unknown conductivity, with the same number of degrees of freedom. We obtain that the parametrization induced by the inversion on optimal grids is the most efficient of the three, because it gives the smallest standard deviation of the maximum a posteriori estimates of the conductivity, uniformly in the domain. For the nonlinear problem we compute the mean and variance of the maximum a posteriori estimates of the conductivity, on optimal grids. For small noise, we obtain that the estimates are unbiased and their variance is very close to the optimal one, given by the Cramér-Rao bound. For larger noise we use regularization and quantify the trade-off between reducing the variance and introducing bias in the solution. Both the full and partial measurement setups are considered.
Citation: Liliana Borcea, Fernando Guevara Vasquez, Alexander V. Mamonov. Study of noise effects in electrical impedance tomography with resistor networks. Inverse Problems and Imaging, 2013, 7 (2) : 417-443. doi: 10.3934/ipi.2013.7.417
##### References:
 [1] G. Alessandrini, Stable determination of conductivity by boundary measurements, Applicable Analysis, 27 (1988), 153-172. doi: 10.1080/00036818808839730. [2] G. Alessandrini and S. Vessella, Lipschitz stability for the inverse conductivity problem, Advances in Applied Mathematics, 35 (2005), 207-241. doi: 10.1016/j.aam.2004.12.002. [3] H. B. Ameur, G. Chavent and J. Jaffré, Refinement and coarsening indicators for adaptive parametrization: Application to the estimation of hydraulic transmissivities, Inverse Problems, 18 (2002), 775-794. doi: 10.1088/0266-5611/18/3/317. [4] H. B. Ameur and B. Kaltenbacher, Regularization of parameter estimation by adaptive discretization using refinement and coarsening indicators, Journal of Inverse and Ill Posed Problems, 10 (2002), 561-584. [5] K. Astala, L. Päivärinta and M. Lassas, Calderón's inverse problem for anisotropic conductivity in the plane, Communications in Partial Differential Equations, 30 (2005), 207-224. doi: 10.1081/PDE-200044485. [6] S. Asvadurov, V. Druskin and L. Knizhnerman, Application of the difference Gaussian rules to solution of hyperbolic problems, Journal of Computational Physics, 158 (2000), 116-135. doi: 10.1006/jcph.1999.6410. [7] J. A. Barcelo, T. Barcelo and A. Ruiz, Stability of the inverse conductivity problem in the plane for less regular conductivities, Journal of Differential Equations, 173 (2001), 231-270. doi: 10.1006/jdeq.2000.3920. [8] L. Borcea and V. Druskin, Optimal finite difference grids for direct and inverse Sturm-Liouville problems, Inverse Problems, 18 (2002), 979-1002. doi: 10.1088/0266-5611/18/4/303. [9] L. Borcea, V. Druskin and F. Guevara Vasquez, Electrical impedance tomography with resistor networks, Inverse Problems, 24 (2008), 035013, (31pp). doi: 10.1088/0266-5611/24/3/035013. [10] L. Borcea, V. Druskin, F. Guevara Vasquez and A. V. Mamonov, Resistor network approaches to electrical impedance tomography, Inverse Problems and Applications: Inside Out II, Mathematical Sciences Research Institute Publications, Cambridge University Press, 60 (2012), 55-118. [11] L. Borcea, V. Druskin and L. Knizhnerman, On the continuum limit of a discrete inverse spectral problem on optimal finite difference grids, Communications on Pure and Applied Mathematics, 58 (2005), 1231-1279. doi: 10.1002/cpa.20073. [12] L. Borcea, V. Druskin and A. V. Mamonov, Circular resistor networks for electrical impedance tomography with partial boundary measurements, Inverse Problems, 26 (2010), 045010. doi: 10.1088/0266-5611/26/4/045010. [13] L. Borcea, V. Druskin, A. V. Mamonov and F. Guevara Vasquez, Pyramidal resistor networks for electrical impedance tomography with partial boundary measurements, Inverse Problems, 26 (2010), 105009. doi: 10.1088/0266-5611/26/10/105009. [14] R. M. Brown and G. Uhlmann, Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions, Commun. Partial Diff. Eqns., 22 (1997), 1009-1027. doi: 10.1080/03605309708821292. [15] E. Curtis, E. Mooers and J. A. Morrow, Finding the conductors in circular networks from boundary measurements, RAIRO - Mathematical Modelling and Numerical Analysis, 28 (1994), 781-814. [16] E. B. Curtis, D. Ingerman and J. A. Morrow, Circular planar graphs and resistor networks, Linear Algebra and its Applications, 23 (1998), 115-150. doi: 10.1016/S0024-3795(98)10087-3. [17] E. B. Curtis and J. A. Morrow, "Inverse Problems for Electrical Networks," World Scientific, 2000. [18] Y. C. de Verdière, Reseaux electriques planaires I, Commentarii Mathematici Helvetici, 69 (1994), 351-374. doi: 10.1007/BF02564493. [19] Y. C. de Verdière, I. Gitler and D. Vertigan, Reseaux electriques planaires II, Commentarii Mathematici Helvetici, 71 (1996), 144-167. doi: 10.1007/BF02566413. [20] V. Druskin and L. Knizhnerman, Gaussian spectral rules for second order finite-difference schemes, Numerical Algorithms, 25 (2000), 139-159. doi: 10.1023/A:1016600805438. [21] V. Druskin and L. Knizhnerman, Gaussian spectral rules for the three-point second differences: I. A two-point positive definite problem in a semi-infinite domain, SIAM Journal on Numerical Analysis, 37 (2000), 403-422. doi: 10.1137/S0036142997330792. [22] B. G. Fitzpatrick, Bayesian analysis in inverse problems, Inverse problems, 7 (1991), 675-702. doi: 10.1088/0266-5611/7/5/003. [23] F. Guevara Vasquez, "On the Parametrization of Ill-posed Inverse Problems Arising from Elliptic Partial Differential Equations," PhD thesis, Rice University, Houston, TX, USA, 2006. [24] O. Y. Imanuvilov, G. Uhlmann and M. Yamamoto, Global uniqueness from partial Cauchy data in two dimensions, Arxiv preprint arXiv:0810.2286, (2008). [25] D. Ingerman, Discrete and continuous Dirichlet-to-Neumann maps in the layered case, SIAM Journal on Mathematical Analysis, 31 (2000), 1214-1234. doi: 10.1137/S0036141097326581. [26] D. Ingerman, V. Druskin and L. Knizhnerman, Optimal finite difference grids and rational approximations of the square root I. Elliptic problems, Communications on Pure and Applied Mathematics, 53 (2000), 1039-1066. doi: 10.1002/1097-0312(200008)53:8<1039::AID-CPA4>3.0.CO;2-I. [27] D. Ingerman and J. A. Morrow, On a characterization of the kernel of the Dirichlet-to-Neumann map for a planar region, SIAM Journal on Applied Mathematics, 29 (1998), 106-115. doi: 10.1137/S0036141096300483. [28] D. Isaacson, Distinguishability of conductivities by electric current computed tomography, IEEE Transactions on Medical Imaging, 5 (1986), 91-95. doi: 10.1109/TMI.1986.4307752. [29] J. P. Kaipio and E. Somersalo, "Statistical and Computational Inverse Problems," Springer Science+ Business Media, Inc., 2005. [30] K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen, Regularized d-bar method for the inverse conductivity problem, Inverse Problems and Imaging, 3 (2009), 599-624. doi: 10.3934/ipi.2009.3.599. [31] H. R. MacMillan, T. A. Manteuffel and S. F. McCormick, First-order system least squares and electrical impedance tomography, SIAM Journal on Numerical Analysis, 42 (2004), 461-483. doi: 10.1137/S0036142902412245. [32] A. V. Mamonov, "Resistor Networks and Optimal Grids for the Numerical Solution of Electrical Impedance Tomography with Partial Boundary Measurements," Ph.D thesis, Rice University, Houston, TX, USA, 2010. [33] N. Mandache, Exponential instability in an inverse problem for the Schrodinger equation, Inverse Problems, 17 (2001), 1435-1444. doi: 10.1088/0266-5611/17/5/313. [34] A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Annals of Mathematics, (1996), 71-96. doi: 10.2307/2118653. [35] Ch. Pommerenke, "Boundary Behaviour of Conformal Maps," 299 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1992. [36] M. J. Schervish, "Theory of Statistics," Springer, 1995. doi: 10.1007/978-1-4612-4250-5. [37] A. D. Seagar, "Probing with Low Frequency Electric Currents," Ph.D thesis, University of Canterbury, UK. Department of Electrical Engineering, 1983. [38] A. G. Wills and B. Ninness, "QPC - Quadratic Programming in C," Webpage, http://sigpromu.org/quadprog/.

show all references

##### References:
 [1] G. Alessandrini, Stable determination of conductivity by boundary measurements, Applicable Analysis, 27 (1988), 153-172. doi: 10.1080/00036818808839730. [2] G. Alessandrini and S. Vessella, Lipschitz stability for the inverse conductivity problem, Advances in Applied Mathematics, 35 (2005), 207-241. doi: 10.1016/j.aam.2004.12.002. [3] H. B. Ameur, G. Chavent and J. Jaffré, Refinement and coarsening indicators for adaptive parametrization: Application to the estimation of hydraulic transmissivities, Inverse Problems, 18 (2002), 775-794. doi: 10.1088/0266-5611/18/3/317. [4] H. B. Ameur and B. Kaltenbacher, Regularization of parameter estimation by adaptive discretization using refinement and coarsening indicators, Journal of Inverse and Ill Posed Problems, 10 (2002), 561-584. [5] K. Astala, L. Päivärinta and M. Lassas, Calderón's inverse problem for anisotropic conductivity in the plane, Communications in Partial Differential Equations, 30 (2005), 207-224. doi: 10.1081/PDE-200044485. [6] S. Asvadurov, V. Druskin and L. Knizhnerman, Application of the difference Gaussian rules to solution of hyperbolic problems, Journal of Computational Physics, 158 (2000), 116-135. doi: 10.1006/jcph.1999.6410. [7] J. A. Barcelo, T. Barcelo and A. Ruiz, Stability of the inverse conductivity problem in the plane for less regular conductivities, Journal of Differential Equations, 173 (2001), 231-270. doi: 10.1006/jdeq.2000.3920. [8] L. Borcea and V. Druskin, Optimal finite difference grids for direct and inverse Sturm-Liouville problems, Inverse Problems, 18 (2002), 979-1002. doi: 10.1088/0266-5611/18/4/303. [9] L. Borcea, V. Druskin and F. Guevara Vasquez, Electrical impedance tomography with resistor networks, Inverse Problems, 24 (2008), 035013, (31pp). doi: 10.1088/0266-5611/24/3/035013. [10] L. Borcea, V. Druskin, F. Guevara Vasquez and A. V. Mamonov, Resistor network approaches to electrical impedance tomography, Inverse Problems and Applications: Inside Out II, Mathematical Sciences Research Institute Publications, Cambridge University Press, 60 (2012), 55-118. [11] L. Borcea, V. Druskin and L. Knizhnerman, On the continuum limit of a discrete inverse spectral problem on optimal finite difference grids, Communications on Pure and Applied Mathematics, 58 (2005), 1231-1279. doi: 10.1002/cpa.20073. [12] L. Borcea, V. Druskin and A. V. Mamonov, Circular resistor networks for electrical impedance tomography with partial boundary measurements, Inverse Problems, 26 (2010), 045010. doi: 10.1088/0266-5611/26/4/045010. [13] L. Borcea, V. Druskin, A. V. Mamonov and F. Guevara Vasquez, Pyramidal resistor networks for electrical impedance tomography with partial boundary measurements, Inverse Problems, 26 (2010), 105009. doi: 10.1088/0266-5611/26/10/105009. [14] R. M. Brown and G. Uhlmann, Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions, Commun. Partial Diff. Eqns., 22 (1997), 1009-1027. doi: 10.1080/03605309708821292. [15] E. Curtis, E. Mooers and J. A. Morrow, Finding the conductors in circular networks from boundary measurements, RAIRO - Mathematical Modelling and Numerical Analysis, 28 (1994), 781-814. [16] E. B. Curtis, D. Ingerman and J. A. Morrow, Circular planar graphs and resistor networks, Linear Algebra and its Applications, 23 (1998), 115-150. doi: 10.1016/S0024-3795(98)10087-3. [17] E. B. Curtis and J. A. Morrow, "Inverse Problems for Electrical Networks," World Scientific, 2000. [18] Y. C. de Verdière, Reseaux electriques planaires I, Commentarii Mathematici Helvetici, 69 (1994), 351-374. doi: 10.1007/BF02564493. [19] Y. C. de Verdière, I. Gitler and D. Vertigan, Reseaux electriques planaires II, Commentarii Mathematici Helvetici, 71 (1996), 144-167. doi: 10.1007/BF02566413. [20] V. Druskin and L. Knizhnerman, Gaussian spectral rules for second order finite-difference schemes, Numerical Algorithms, 25 (2000), 139-159. doi: 10.1023/A:1016600805438. [21] V. Druskin and L. Knizhnerman, Gaussian spectral rules for the three-point second differences: I. A two-point positive definite problem in a semi-infinite domain, SIAM Journal on Numerical Analysis, 37 (2000), 403-422. doi: 10.1137/S0036142997330792. [22] B. G. Fitzpatrick, Bayesian analysis in inverse problems, Inverse problems, 7 (1991), 675-702. doi: 10.1088/0266-5611/7/5/003. [23] F. Guevara Vasquez, "On the Parametrization of Ill-posed Inverse Problems Arising from Elliptic Partial Differential Equations," PhD thesis, Rice University, Houston, TX, USA, 2006. [24] O. Y. Imanuvilov, G. Uhlmann and M. Yamamoto, Global uniqueness from partial Cauchy data in two dimensions, Arxiv preprint arXiv:0810.2286, (2008). [25] D. Ingerman, Discrete and continuous Dirichlet-to-Neumann maps in the layered case, SIAM Journal on Mathematical Analysis, 31 (2000), 1214-1234. doi: 10.1137/S0036141097326581. [26] D. Ingerman, V. Druskin and L. Knizhnerman, Optimal finite difference grids and rational approximations of the square root I. Elliptic problems, Communications on Pure and Applied Mathematics, 53 (2000), 1039-1066. doi: 10.1002/1097-0312(200008)53:8<1039::AID-CPA4>3.0.CO;2-I. [27] D. Ingerman and J. A. Morrow, On a characterization of the kernel of the Dirichlet-to-Neumann map for a planar region, SIAM Journal on Applied Mathematics, 29 (1998), 106-115. doi: 10.1137/S0036141096300483. [28] D. Isaacson, Distinguishability of conductivities by electric current computed tomography, IEEE Transactions on Medical Imaging, 5 (1986), 91-95. doi: 10.1109/TMI.1986.4307752. [29] J. P. Kaipio and E. Somersalo, "Statistical and Computational Inverse Problems," Springer Science+ Business Media, Inc., 2005. [30] K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen, Regularized d-bar method for the inverse conductivity problem, Inverse Problems and Imaging, 3 (2009), 599-624. doi: 10.3934/ipi.2009.3.599. [31] H. R. MacMillan, T. A. Manteuffel and S. F. McCormick, First-order system least squares and electrical impedance tomography, SIAM Journal on Numerical Analysis, 42 (2004), 461-483. doi: 10.1137/S0036142902412245. [32] A. V. Mamonov, "Resistor Networks and Optimal Grids for the Numerical Solution of Electrical Impedance Tomography with Partial Boundary Measurements," Ph.D thesis, Rice University, Houston, TX, USA, 2010. [33] N. Mandache, Exponential instability in an inverse problem for the Schrodinger equation, Inverse Problems, 17 (2001), 1435-1444. doi: 10.1088/0266-5611/17/5/313. [34] A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Annals of Mathematics, (1996), 71-96. doi: 10.2307/2118653. [35] Ch. Pommerenke, "Boundary Behaviour of Conformal Maps," 299 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1992. [36] M. J. Schervish, "Theory of Statistics," Springer, 1995. doi: 10.1007/978-1-4612-4250-5. [37] A. D. Seagar, "Probing with Low Frequency Electric Currents," Ph.D thesis, University of Canterbury, UK. Department of Electrical Engineering, 1983. [38] A. G. Wills and B. Ninness, "QPC - Quadratic Programming in C," Webpage, http://sigpromu.org/quadprog/.
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