doi: 10.3934/ipi.2022002
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Direct regularized reconstruction for the three-dimensional Calderón problem

Technical University of Denmark, Department of Applied Mathematics and Computer Science, DK-2800 Kgs. Lyngby, Denmark

* Corresponding author

Received  June 2021 Revised  November 2021 Early access January 2022

Electrical Impedance Tomography gives rise to the severely ill-posed Calderón problem of determining the electrical conductivity distribution in a bounded domain from knowledge of the associated Dirichlet-to-Neumann map for the governing equation. The uniqueness and stability questions for the three-dimensional problem were largely answered in the affirmative in the 1980's using complex geometrical optics solutions, and this led further to a direct reconstruction method relying on a non-physical scattering transform. In this paper, the reconstruction problem is taken one step further towards practical applications by considering data contaminated by noise. Indeed, a regularization strategy for the three-dimensional Calderón problem is presented based on a suitable and explicit truncation of the scattering transform. This gives a certified, stable and direct reconstruction method that is robust to small perturbations of the data. Numerical tests on simulated noisy data illustrate the feasibility and regularizing effect of the method, and suggest that the numerical implementation performs better than predicted by theory.

Citation: Kim Knudsen, Aksel Kaastrup Rasmussen. Direct regularized reconstruction for the three-dimensional Calderón problem. Inverse Problems and Imaging, doi: 10.3934/ipi.2022002
References:
[1]

K. Abraham and R. Nickl, On statistical Calderón problems, Math. Stat. Learn., 2 (2019), 165-216. 

[2]

A. AdlerR. AmyotR. GuardoJ. Bates and Y. Berthiaume, Monitoring changes in lung air and liquid volumes with electrical impedance tomography, Journal of Applied Physiology, 83 (1997), 1762-1767.  doi: 10.1152/jappl.1997.83.5.1762.

[3]

G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal., 27 (1988), 153-172.  doi: 10.1080/00036818808839730.

[4]

G. Alessandrini, Singular solutions of elliptic equations and the determination of conductivity by boundary measurements, J. Differential Equations, 84 (1990), 252-272.  doi: 10.1016/0022-0396(90)90078-4.

[5]

M. Alsaker and J. L. Mueller, A D-bar algorithm with a priori information for 2-dimensional electrical impedance tomography, SIAM J. Imaging Sci., 9 (2016), 1619-1654.  doi: 10.1137/15M1020137.

[6]

K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane, Ann. of Math., 163 (2006), 265-299.  doi: 10.4007/annals.2006.163.265.

[7]

J. Bikowski, K. Knudsen and J. L. Mueller, Direct numerical reconstruction of conductivities in three dimensions using scattering transforms, Inverse Problems, 27 (2011), 015002, 19 pp. doi: 10.1088/0266-5611/27/1/015002.

[8]

G. BovermanT.-J. KaoD. Isaacson and G. J. Saulnier, An implementation of Calderón's method for 3-D Limited-View EIT, IEEE Transactions on Medical Imaging, 28 (2009), 1073-82.  doi: 10.1109/TMI.2009.2012892.

[9]

R. M. Brown, Global uniqueness in the impedance-imaging problem for less regular conductivities, SIAM J. Math. Anal., 27 (1996), 1049-1056.  doi: 10.1137/S0036141094271132.

[10]

A.-P. Calderón, On an inverse boundary value problem, in Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), Soc. Brasil. Mat., Rio de Janeiro, 1980, 65–73.

[11]

P. Caro and A. Garcia, The Calderón problem with corrupted data, Inverse Problems, 33 (2017), 085001, 17 pp. doi: 10.1088/1361-6420/aa7425.

[12]

P. Caro and K. M. Rogers, Global uniqueness for the Calderón problem with Lipschitz conductivities, Forum. Math. Pi, 4 (2016), e2, 28 pp. doi: 10.1017/fmp.2015.9.

[13]

V. CherepeninA. KarpovA. KorjenevskyV. KornienkoY. KultiasovM. OchapkinO. Trochanova and J. Meister, Three-dimensional EIT imaging of breast tissues: System design and clinical testing, IEEE Transactions on Medical Imaging, 21 (2002), 662-667.  doi: 10.1109/TMI.2002.800602.

[14]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93 of Applied Mathematical Sciences, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-02835-3.

[15]

H. CorneanK. Knudsen and S. Siltanen, Towards a $d$-bar reconstruction method for three-dimensional EIT, J. Inverse Ill-Posed Probl., 14 (2006), 111-134.  doi: 10.1515/156939406777571102.

[16]

F. DelbaryP. C. Hansen and K. Knudsen, Electrical impedance tomography: 3D reconstructions using scattering transforms, Appl. Anal., 91 (2012), 737-755.  doi: 10.1080/00036811.2011.598863.

[17]

F. Delbary and K. Knudsen, Numerical nonlinear complex geometrical optics algorithm for the 3D Calderón problem, Inverse Probl. Imaging, 8 (2014), 991-1012.  doi: 10.3934/ipi.2014.8.991.

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D. C. Dobson, Convergence of a reconstruction method for the inverse conductivity problem, SIAM J. Appl. Math., 52 (1992), 442-458.  doi: 10.1137/0152025.

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M. Dunlop and A. Stuart, The Bayesian formulation of EIT: Analysis and algorithms, Inverse Probl. Imaging, 10 (2016), 1007-1036.  doi: 10.3934/ipi.2016030.

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H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, vol. 375 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 1996.

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I. Frerichs and T. Becher, Chest electrical impedance tomography measures in neonatology and paediatrics - a survey on clinical usefulness, Physiological Measurement, 40 (2019), 054001.  doi: 10.1088/1361-6579/ab1946.

[23]

M. Frigo and S. G. Johnson, The design and implementation of FFTW3, Proceedings of the IEEE, 93 (2005), 216-231.  doi: 10.1109/JPROC.2004.840301.

[24]

N. Goren, J. Avery, T. Dowrick, E. Mackle, A. Witkowska-Wrobel, D. Werring and D. Holder, Multi-frequency electrical impedance tomography and neuroimaging data in stroke patients, Scientific Data, 5 (2018), 180112, 10 pp. doi: 10.1038/sdata.2018.112.

[25]

M. HallajiA. Seppänen and M. Pour-Ghaz, Electrical impedance tomography-based sensing skin for quantitative imaging of damage in concrete, Smart Materials and Structures, 23 (2014), 085001.  doi: 10.1088/0964-1726/23/8/085001.

[26]

S. J. HamiltonD. IsaacsonV. KolehmainenP. A. MullerJ. Toivainen and P. F. Bray, 3D electrical impedance tomography reconstructions from simulated electrode data using direct inversion $\rm t^{\exp}$ and Calderón methods, Inverse Probl. Imaging, 15 (2021), 1135-1169.  doi: 10.3934/ipi.2021032.

[27]

A. Hauptmann, M. Santacesaria and S. Siltanen, Direct inversion from partial-boundary data in electrical impedance tomography, Inverse Problems, 33 (2017), 025009, 26 pp. doi: 10.1088/1361-6420/33/2/025009.

[28]

T. C. Hou and J. P. Lynch, Electrical impedance tomographic methods for sensing strain fields and crack damage in cementitious structures, Journal of Intelligent Material Systems and Structures, 20 (2009), 1363-1379.  doi: 10.1177/1045389X08096052.

[29]

D. IsaacsonJ. MuellerJ. Newell and S. Siltanen, Reconstructions of chest phantoms by the d-bar method for electrical impedance tomography, IEEE Transactions on Medical Imaging, 23 (2004), 821-828.  doi: 10.1109/TMI.2004.827482.

[30]

B. Jin and P. Maass, An analysis of electrical impedance tomography with applications to Tikhonov regularization, ESAIM Control Optim. Calc. Var., 18 (2012), 1027-1048.  doi: 10.1051/cocv/2011193.

[31]

J. P. KaipioV. KolehmainenE. Somersalo and M. Vauhkonen, Statistical inversion and Monte Carlo sampling methods in electrical impedance tomography, Inverse Problems, 16 (2000), 1487-1522.  doi: 10.1088/0266-5611/16/5/321.

[32]

A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, vol. 120 of Applied Mathematical Sciences, 2nd edition, Springer, New York, 2011. doi: 10.1007/978-1-4419-8474-6.

[33]

K. Knudsen, A new direct method for reconstructing isotropic conductivities in the plane, Physiological Measurement, 24 (2003), 391-401.  doi: 10.1088/0967-3334/24/2/351.

[34]

K. KnudsenM. LassasJ. 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.

[35]

K. KnudsenM. LassasJ. 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.

[36]

K. Knudsen and J. L. Mueller, The Born approximation and Calderón's method for reconstruction of conductivities in 3-D, Discrete Contin. Dyn. Syst., 8th AIMS Conference. Suppl. Vol. Ⅱ, 2011,844–853.

[37]

A. Lechleiter and A. Rieder, Newton regularizations for impedance tomography: Convergence by local injectivity, Inverse Problems, 24 (2008), 065009, 18 pp. doi: 10.1088/0266-5611/24/6/065009.

[38]

S. Leonhardt and B. Lachmann, Electrical impedance tomography: The holy grail of ventilation and perfusion monitoring?, Intensive Care Medicine, 38 (2012), 1917-1929.  doi: 10.1007/s00134-012-2684-z.

[39]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. Ⅰ, Springer-Verlag, New York-Heidelberg, 1972, Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181.

[40]

E. MaloneM. JehlS. ArridgeT. Betcke and D. Holder, Stroke type differentiation using spectrally constrained multifrequency EIT: Evaluation of feasibility in a realistic head model, Physiological Measurement, 35 (2014), 1051-1066.  doi: 10.1088/0967-3334/35/6/1051.

[41]

N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation, Inverse Problems, 17 (2001), 1435-1444.  doi: 10.1088/0266-5611/17/5/313.

[42]

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.

[43]

J. L. MuellerS. Siltanen and D. Isaacson, A direct reconstruction algorithm for electrical impedance tomography, IEEE Transactions on Medical Imaging, 21 (2002), 555-559.  doi: 10.1109/TMI.2002.800574.

[44]

A. I. Nachman, Reconstructions from boundary measurements, Ann. of Math., 128 (1988), 531-576.  doi: 10.2307/1971435.

[45]

A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math., 143 (1996), 71-96.  doi: 10.2307/2118653.

[46]

A. NachmanJ. Sylvester and G. Uhlmann, An $n$-dimensional Borg-Levinson theorem, Comm. Math. Phys., 115 (1988), 595-605.  doi: 10.1007/BF01224129.

[47]

R. G. Novikov, Multidimensional inverse spectral problem for the equation $-\Delta \psi + (v(x)-Eu(x))\psi = 0$, Functional Analysis and its Applications, 22 (1988), 263-272.  doi: 10.1007/BF01077418.

[48] M. Reed and B. Simon, Methods of Modern Mathematical Physics 1, Functional Analysis, Academic Press, 1980. 
[49]

L. Rondi, On the regularization of the inverse conductivity problem with discontinuous conductivities, Inverse Probl. Imaging, 2 (2008), 397-409.  doi: 10.3934/ipi.2008.2.397.

[50]

L. Rondi, Discrete approximation and regularisation for the inverse conductivity problem, Rend. Istit. Mat. Univ. Trieste, 48 (2016), 315-352.  doi: 10.13137/2464-8728/13162.

[51]

M. Salo, Semiclassical pseudodifferential calculus and the reconstruction of a magnetic field, Comm. Partial Differential Equations, 31 (2006), 1639-1666.  doi: 10.1080/03605300500530420.

[52]

C. SchmidtS. WagnerM. BurgerU. V. Rienen and C. H. Wolters, Impact of uncertain head tissue conductivity in the optimization of transcranial direct current stimulation for an auditory target, Journal of Neural Engineering, 12 (2015), 046028.  doi: 10.1088/1741-2560/12/4/046028.

[53]

S. SiltanenJ. Mueller and D. Isaacson, Erratum: "An implementation of the reconstruction algorithm of A. Nachman for the 2D inverse conductivity problem" [Inverse Problems 16 (2000), 681–699], Inverse Problems, 17 (2001), 1561-1563.  doi: 10.1088/0266-5611/17/5/501.

[54]

S. SiltanenJ. Mueller and D. Isaacson, An implementation of the reconstruction algorithm of A. Nachman for the 2D inverse conductivity problem, Inverse Problems, 16 (2000), 681-699.  doi: 10.1088/0266-5611/16/3/310.

[55]

P. Sołtan, A Primer on Hilbert Space Operators, Springer International Publishing, 2018.

[56]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169.  doi: 10.2307/1971291.

[57]

Y. ZhaoE. ZimmermannJ. A. HuismanA. TreichelB. WoltersS. Van Waasen and A. Kemna, Broadband EIT borehole measurements with high phase accuracy using numerical corrections of electromagnetic coupling effects, Measurement Science and Technology, 24 (2013), 085005.  doi: 10.1088/0957-0233/24/8/085005.

[58]

Y. Zou and Z. Guo, A review of electrical impedance techniques for breast cancer detection, Medical Engineering and Physics, 25 (2003), 79-90.  doi: 10.1016/S1350-4533(02)00194-7.

show all references

References:
[1]

K. Abraham and R. Nickl, On statistical Calderón problems, Math. Stat. Learn., 2 (2019), 165-216. 

[2]

A. AdlerR. AmyotR. GuardoJ. Bates and Y. Berthiaume, Monitoring changes in lung air and liquid volumes with electrical impedance tomography, Journal of Applied Physiology, 83 (1997), 1762-1767.  doi: 10.1152/jappl.1997.83.5.1762.

[3]

G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal., 27 (1988), 153-172.  doi: 10.1080/00036818808839730.

[4]

G. Alessandrini, Singular solutions of elliptic equations and the determination of conductivity by boundary measurements, J. Differential Equations, 84 (1990), 252-272.  doi: 10.1016/0022-0396(90)90078-4.

[5]

M. Alsaker and J. L. Mueller, A D-bar algorithm with a priori information for 2-dimensional electrical impedance tomography, SIAM J. Imaging Sci., 9 (2016), 1619-1654.  doi: 10.1137/15M1020137.

[6]

K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane, Ann. of Math., 163 (2006), 265-299.  doi: 10.4007/annals.2006.163.265.

[7]

J. Bikowski, K. Knudsen and J. L. Mueller, Direct numerical reconstruction of conductivities in three dimensions using scattering transforms, Inverse Problems, 27 (2011), 015002, 19 pp. doi: 10.1088/0266-5611/27/1/015002.

[8]

G. BovermanT.-J. KaoD. Isaacson and G. J. Saulnier, An implementation of Calderón's method for 3-D Limited-View EIT, IEEE Transactions on Medical Imaging, 28 (2009), 1073-82.  doi: 10.1109/TMI.2009.2012892.

[9]

R. M. Brown, Global uniqueness in the impedance-imaging problem for less regular conductivities, SIAM J. Math. Anal., 27 (1996), 1049-1056.  doi: 10.1137/S0036141094271132.

[10]

A.-P. Calderón, On an inverse boundary value problem, in Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), Soc. Brasil. Mat., Rio de Janeiro, 1980, 65–73.

[11]

P. Caro and A. Garcia, The Calderón problem with corrupted data, Inverse Problems, 33 (2017), 085001, 17 pp. doi: 10.1088/1361-6420/aa7425.

[12]

P. Caro and K. M. Rogers, Global uniqueness for the Calderón problem with Lipschitz conductivities, Forum. Math. Pi, 4 (2016), e2, 28 pp. doi: 10.1017/fmp.2015.9.

[13]

V. CherepeninA. KarpovA. KorjenevskyV. KornienkoY. KultiasovM. OchapkinO. Trochanova and J. Meister, Three-dimensional EIT imaging of breast tissues: System design and clinical testing, IEEE Transactions on Medical Imaging, 21 (2002), 662-667.  doi: 10.1109/TMI.2002.800602.

[14]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93 of Applied Mathematical Sciences, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-02835-3.

[15]

H. CorneanK. Knudsen and S. Siltanen, Towards a $d$-bar reconstruction method for three-dimensional EIT, J. Inverse Ill-Posed Probl., 14 (2006), 111-134.  doi: 10.1515/156939406777571102.

[16]

F. DelbaryP. C. Hansen and K. Knudsen, Electrical impedance tomography: 3D reconstructions using scattering transforms, Appl. Anal., 91 (2012), 737-755.  doi: 10.1080/00036811.2011.598863.

[17]

F. Delbary and K. Knudsen, Numerical nonlinear complex geometrical optics algorithm for the 3D Calderón problem, Inverse Probl. Imaging, 8 (2014), 991-1012.  doi: 10.3934/ipi.2014.8.991.

[18]

D. C. Dobson, Convergence of a reconstruction method for the inverse conductivity problem, SIAM J. Appl. Math., 52 (1992), 442-458.  doi: 10.1137/0152025.

[19]

M. Dunlop and A. Stuart, The Bayesian formulation of EIT: Analysis and algorithms, Inverse Probl. Imaging, 10 (2016), 1007-1036.  doi: 10.3934/ipi.2016030.

[20]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, vol. 375 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 1996.

[21]

L. C. Evans, Partial Differential Equations, vol. 19, American Mathematical Society, 2010. doi: 10.1090/gsm/019.

[22]

I. Frerichs and T. Becher, Chest electrical impedance tomography measures in neonatology and paediatrics - a survey on clinical usefulness, Physiological Measurement, 40 (2019), 054001.  doi: 10.1088/1361-6579/ab1946.

[23]

M. Frigo and S. G. Johnson, The design and implementation of FFTW3, Proceedings of the IEEE, 93 (2005), 216-231.  doi: 10.1109/JPROC.2004.840301.

[24]

N. Goren, J. Avery, T. Dowrick, E. Mackle, A. Witkowska-Wrobel, D. Werring and D. Holder, Multi-frequency electrical impedance tomography and neuroimaging data in stroke patients, Scientific Data, 5 (2018), 180112, 10 pp. doi: 10.1038/sdata.2018.112.

[25]

M. HallajiA. Seppänen and M. Pour-Ghaz, Electrical impedance tomography-based sensing skin for quantitative imaging of damage in concrete, Smart Materials and Structures, 23 (2014), 085001.  doi: 10.1088/0964-1726/23/8/085001.

[26]

S. J. HamiltonD. IsaacsonV. KolehmainenP. A. MullerJ. Toivainen and P. F. Bray, 3D electrical impedance tomography reconstructions from simulated electrode data using direct inversion $\rm t^{\exp}$ and Calderón methods, Inverse Probl. Imaging, 15 (2021), 1135-1169.  doi: 10.3934/ipi.2021032.

[27]

A. Hauptmann, M. Santacesaria and S. Siltanen, Direct inversion from partial-boundary data in electrical impedance tomography, Inverse Problems, 33 (2017), 025009, 26 pp. doi: 10.1088/1361-6420/33/2/025009.

[28]

T. C. Hou and J. P. Lynch, Electrical impedance tomographic methods for sensing strain fields and crack damage in cementitious structures, Journal of Intelligent Material Systems and Structures, 20 (2009), 1363-1379.  doi: 10.1177/1045389X08096052.

[29]

D. IsaacsonJ. MuellerJ. Newell and S. Siltanen, Reconstructions of chest phantoms by the d-bar method for electrical impedance tomography, IEEE Transactions on Medical Imaging, 23 (2004), 821-828.  doi: 10.1109/TMI.2004.827482.

[30]

B. Jin and P. Maass, An analysis of electrical impedance tomography with applications to Tikhonov regularization, ESAIM Control Optim. Calc. Var., 18 (2012), 1027-1048.  doi: 10.1051/cocv/2011193.

[31]

J. P. KaipioV. KolehmainenE. Somersalo and M. Vauhkonen, Statistical inversion and Monte Carlo sampling methods in electrical impedance tomography, Inverse Problems, 16 (2000), 1487-1522.  doi: 10.1088/0266-5611/16/5/321.

[32]

A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, vol. 120 of Applied Mathematical Sciences, 2nd edition, Springer, New York, 2011. doi: 10.1007/978-1-4419-8474-6.

[33]

K. Knudsen, A new direct method for reconstructing isotropic conductivities in the plane, Physiological Measurement, 24 (2003), 391-401.  doi: 10.1088/0967-3334/24/2/351.

[34]

K. KnudsenM. LassasJ. 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.

[35]

K. KnudsenM. LassasJ. 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.

[36]

K. Knudsen and J. L. Mueller, The Born approximation and Calderón's method for reconstruction of conductivities in 3-D, Discrete Contin. Dyn. Syst., 8th AIMS Conference. Suppl. Vol. Ⅱ, 2011,844–853.

[37]

A. Lechleiter and A. Rieder, Newton regularizations for impedance tomography: Convergence by local injectivity, Inverse Problems, 24 (2008), 065009, 18 pp. doi: 10.1088/0266-5611/24/6/065009.

[38]

S. Leonhardt and B. Lachmann, Electrical impedance tomography: The holy grail of ventilation and perfusion monitoring?, Intensive Care Medicine, 38 (2012), 1917-1929.  doi: 10.1007/s00134-012-2684-z.

[39]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. Ⅰ, Springer-Verlag, New York-Heidelberg, 1972, Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181.

[40]

E. MaloneM. JehlS. ArridgeT. Betcke and D. Holder, Stroke type differentiation using spectrally constrained multifrequency EIT: Evaluation of feasibility in a realistic head model, Physiological Measurement, 35 (2014), 1051-1066.  doi: 10.1088/0967-3334/35/6/1051.

[41]

N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation, Inverse Problems, 17 (2001), 1435-1444.  doi: 10.1088/0266-5611/17/5/313.

[42]

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.

[43]

J. L. MuellerS. Siltanen and D. Isaacson, A direct reconstruction algorithm for electrical impedance tomography, IEEE Transactions on Medical Imaging, 21 (2002), 555-559.  doi: 10.1109/TMI.2002.800574.

[44]

A. I. Nachman, Reconstructions from boundary measurements, Ann. of Math., 128 (1988), 531-576.  doi: 10.2307/1971435.

[45]

A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math., 143 (1996), 71-96.  doi: 10.2307/2118653.

[46]

A. NachmanJ. Sylvester and G. Uhlmann, An $n$-dimensional Borg-Levinson theorem, Comm. Math. Phys., 115 (1988), 595-605.  doi: 10.1007/BF01224129.

[47]

R. G. Novikov, Multidimensional inverse spectral problem for the equation $-\Delta \psi + (v(x)-Eu(x))\psi = 0$, Functional Analysis and its Applications, 22 (1988), 263-272.  doi: 10.1007/BF01077418.

[48] M. Reed and B. Simon, Methods of Modern Mathematical Physics 1, Functional Analysis, Academic Press, 1980. 
[49]

L. Rondi, On the regularization of the inverse conductivity problem with discontinuous conductivities, Inverse Probl. Imaging, 2 (2008), 397-409.  doi: 10.3934/ipi.2008.2.397.

[50]

L. Rondi, Discrete approximation and regularisation for the inverse conductivity problem, Rend. Istit. Mat. Univ. Trieste, 48 (2016), 315-352.  doi: 10.13137/2464-8728/13162.

[51]

M. Salo, Semiclassical pseudodifferential calculus and the reconstruction of a magnetic field, Comm. Partial Differential Equations, 31 (2006), 1639-1666.  doi: 10.1080/03605300500530420.

[52]

C. SchmidtS. WagnerM. BurgerU. V. Rienen and C. H. Wolters, Impact of uncertain head tissue conductivity in the optimization of transcranial direct current stimulation for an auditory target, Journal of Neural Engineering, 12 (2015), 046028.  doi: 10.1088/1741-2560/12/4/046028.

[53]

S. SiltanenJ. Mueller and D. Isaacson, Erratum: "An implementation of the reconstruction algorithm of A. Nachman for the 2D inverse conductivity problem" [Inverse Problems 16 (2000), 681–699], Inverse Problems, 17 (2001), 1561-1563.  doi: 10.1088/0266-5611/17/5/501.

[54]

S. SiltanenJ. Mueller and D. Isaacson, An implementation of the reconstruction algorithm of A. Nachman for the 2D inverse conductivity problem, Inverse Problems, 16 (2000), 681-699.  doi: 10.1088/0266-5611/16/3/310.

[55]

P. Sołtan, A Primer on Hilbert Space Operators, Springer International Publishing, 2018.

[56]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169.  doi: 10.2307/1971291.

[57]

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Figure 1.  The piecewise constant heart-lungs phantom in a threedimensional view (A), and in the planar cross section x3 = 0 (B)
Figure 2.  Cross sections $ (x^3 = 0) $ of reconstructions using the regularized reconstruction algorithm with different choices of truncation radius $ M $, $ K = 12 $ and $ |\zeta(\xi)| = \frac{1}{4}M^{3/2} $. There is no added noise
Figure 3.  Cross sections $ (x^3 = 0) $ of reconstructions using the regularized reconstruction algorithm on noisy Dirichlet-to-Neumann maps. The noise levels correspond to relative noise levels $ \varepsilon \approx 0.1\% $ with $ \mathrm{SNR} = 12\cdot 10^3 $ (top left), $ \varepsilon \approx 0.01\% $ with $ \mathrm{SNR} = 123\cdot 10^3 $ (top right), $ \varepsilon \approx 0.001\% $ with $ \mathrm{SNR} = 1172\cdot 10^3 $ (bottom left) and $ \varepsilon \approx 0.0001\% $ with $ \mathrm{SNR} = 11299\cdot 10^3 $ (bottom right). The parameters used are $ K = 11 $ and $ |\zeta(\xi)| = \frac{1}{3\sqrt{2}}M^{3/2} $
Figure 4.  Regularized reconstruction using noisy Dirichlet-to-Neumann maps with $ \varepsilon = 10^{-2} $, which corresponds to approximately $ 1\% $ relative noise and $ \mathrm{SNR} = 1.17\cdot 10^3 $. Plot (A) shows the cross sections $ x^3 = 0 $, $ x^2 = -0.6 $, $ x^2 = {-0.05} $ and $ x^2 = 0.6 $, whereas plot (B) shows the plane corresponding to $ x^3 = 0 $. The parameters used are $ M = 9 $, $ K = 11 $ and $ |\zeta(\xi)| = \frac{1}{3\sqrt{2}}M^{3/2} $
Figure 5.  The truncation radii as predicted by theory $ M = (-1/11\log(\varepsilon))^{-1/p} $ for $ p = 3/2 $, and the chosen truncation radii for the noisy reconstructions of Figure 3 and 4
Table 1.  Summary of piecewise constant heart-lungs phantom consisting of three inclusions
Inclusion Center Radii Axes Conductivity
Ball $ (-0.09,-0.55,0) $ $ r = 0.273 $ 2
Left spheroid $ 0.55(-\sin(\frac{5\pi}{12}), \cos(\frac{5\pi}{12}), 0) $ $ r_1 = 0.468 $,
$ r_2 = 0.234 $,
$ r_3 = 0.234 $
$(\cos(\frac{5\pi}{12}),\sin(\frac{5\pi}{12}),0)$,
$(-\sin(\frac{5\pi}{12}),\cos\frac{5\pi}{12}),0)$,
$(0,0,1)$
0.5
Right spheroid $0.45(\sin(\frac{5\pi}{12}), \cos(\frac{5\pi}{12}), 0)$ $r_1 = 0.546$,
$r_2 = 0.273$,
$r_3 = 0.273$
$(\cos(\frac{5\pi}{12}),-\sin(\frac{5\pi}{12}),0)$,
$(\sin(\frac{5\pi}{12}), \cos(\frac{5\pi}{12}), 0)$,
$(0,0,1)$
0.5
Inclusion Center Radii Axes Conductivity
Ball $ (-0.09,-0.55,0) $ $ r = 0.273 $ 2
Left spheroid $ 0.55(-\sin(\frac{5\pi}{12}), \cos(\frac{5\pi}{12}), 0) $ $ r_1 = 0.468 $,
$ r_2 = 0.234 $,
$ r_3 = 0.234 $
$(\cos(\frac{5\pi}{12}),\sin(\frac{5\pi}{12}),0)$,
$(-\sin(\frac{5\pi}{12}),\cos\frac{5\pi}{12}),0)$,
$(0,0,1)$
0.5
Right spheroid $0.45(\sin(\frac{5\pi}{12}), \cos(\frac{5\pi}{12}), 0)$ $r_1 = 0.546$,
$r_2 = 0.273$,
$r_3 = 0.273$
$(\cos(\frac{5\pi}{12}),-\sin(\frac{5\pi}{12}),0)$,
$(\sin(\frac{5\pi}{12}), \cos(\frac{5\pi}{12}), 0)$,
$(0,0,1)$
0.5
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