doi: 10.3934/ipi.2021032
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

3D Electrical Impedance Tomography reconstructions from simulated electrode data using direct inversion $ \mathbf{t}^{\rm{{\textbf{exp}}}} $ and Calderón methods

1. 

Department of Mathematical and Statistical Sciences, Marquette University, Milwaukee, WI 53233, USA

2. 

Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180, USA

3. 

Department of Applied Physics, University of Eastern Finland, FI-70210 Kuopio, Finland

4. 

Department of Mathematics & Statistics, Villanova University, Villanova, PA 19085, USA

5. 

Department of Mathematics, Drexel University, Philadelphia, PA 19104, USA

* Corresponding author: Peter A. Muller

Received  July 2020 Revised  January 2021 Early access May 2021

Fund Project: The first author is supported by NIH Award R21EB028064. The third and fifth authors are supported by the Academy of Finland, the Jane and Aatos Erkko Foundation and Neurocenter Finland

The first numerical implementation of a $ \mathbf{t}^{\rm{{\textbf{exp}}}} $ method in 3D using simulated electrode data is presented. Results are compared to Calderón's method as well as more common TV and smoothness regularization-based methods. The $ \mathbf{t}^{\rm{{\textbf{exp}}}} $ method for EIT is based on tailor-made non-linear Fourier transforms involving the measured current and voltage data. Low-pass filtering in the non-linear Fourier domain is used to stabilize the reconstruction process. In 2D, $ \mathbf{t}^{\rm{{\textbf{exp}}}} $ methods have shown great promise for providing robust real-time absolute and time-difference conductivity reconstructions but have yet to be used on practical electrode data in 3D, until now. Results are presented for simulated data for conductivity and permittivity with disjoint non-radially symmetric targets on spherical domains and noisy voltage data. The 3D $ \mathbf{t}^{\rm{{\textbf{exp}}}} $ and Calderón methods are demonstrated to provide comparable quality to their 2D counterparts and hold promise for real-time reconstructions due to their fast, non-optimized, computational cost.

 

Erratum: The name of the fifth author has been corrected from Jussi Toivainen to Jussi Toivanen. We apologize for any inconvenience this may cause.

Citation: Sarah J. Hamilton, David Isaacson, Ville Kolehmainen, Peter A. Muller, Jussi Toivanen, Patrick F. Bray. 3D Electrical Impedance Tomography reconstructions from simulated electrode data using direct inversion $ \mathbf{t}^{\rm{{\textbf{exp}}}} $ and Calderón methods. Inverse Problems & Imaging, doi: 10.3934/ipi.2021032
References:
[1]

A. Adler, J. H. Arnold, R. Bayford, A. Borsic, B. Brown, P. Dixon, T. J. Faes, I. Frerichs, H. Gagnon, Y. Gärber and B. Grychtol, GREIT: A unified approach to 2d linear EIT reconstruction of lung images, Physiological Measurement, 30 (2009), S35–S55. doi: 10.1088/0967-3334/30/6/S03.  Google Scholar

[2]

M. AlsakerS. J. Hamilton and A. Hauptmann, A direct D-bar method for partial boundary data Electrical Impedance Tomography with a priori information, Inverse Problems and Imaging, 11 (2017), 427-454.  doi: 10.3934/ipi.2017020.  Google Scholar

[3]

G. Alessandrini, Stable determination of conductivity by boundary measurements, Applicable Analysis, 27 (1988), 153-172.  doi: 10.1080/00036818808839730.  Google Scholar

[4]

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

[5]

M. Alsaker and J. L. Mueller, EIT images of human inspiration and expiration using a D-bar method with spatial priors, Applied Computational Electromagnetics Society Journal, 34 (2019). Google Scholar

[6]

M. AlsakerJ. L. Mueller and R. Murthy, Dynamic optimized priors for D-bar reconstructions of human ventilation using electrical impedance tomography, Journal of Computational and Applied Mathematics, 362 (2019), 276-294.  doi: 10.1016/j.cam.2018.07.039.  Google Scholar

[7]

D. C. Barber and B. H. Brown, Applied potential tomography, Journal of Physics E: Scientific Instruments, 17 (1984), 723-733.  doi: 10.1088/0022-3735/17/9/002.  Google Scholar

[8]

R. Beals and R. R. Coifman, Multidimensional inverse scatterings and nonlinear partial differential equations, Pseudodifferential Operators and Applications, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 43 (1985), 45-70.  doi: 10.1090/pspum/043/812283.  Google Scholar

[9]

P. Blomgren and T. F. Chan, Modular solvers for image restoration problems using the discrepancy principle, Numerical Linear Algebra with Applications, 9 (2002), 347-358.  doi: 10.1002/nla.278.  Google Scholar

[10]

G. Boverman, D. Isaacson, T.-J. Kao, Saulnier, G. J. and J. C. Newell, Methods for direct image reconstruction for EIT in two and three dimensions, in Proceedings of the 2008 Electrical Impedance Tomography Conference, (Dartmouth College, Hanover, New Hampshire, USA), (2008). Google Scholar

[11]

G. BovermanT.-J. KaoD. Isaacson and G. J. Saulnier, An implementation of Calderón's method for 3-D limited view EIT, IEEE Trans. Med. Imaging, 28 (2009), 1073-1082.  doi: 10.1109/TMI.2009.2012892.  Google Scholar

[12]

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

[13]

J. Bikowski and J. Mueller, 2D EIT reconstructions using Calderón's method, Inverse Problems and Imaging, 2 (2008), 43-61.  doi: 10.3934/ipi.2008.2.43.  Google Scholar

[14]

L. Borcea, Addendum to "Electrical impedance tomography", Inverse Problems, 19 (2002), 997-998.  doi: 10.1088/0266-5611/19/4/501.  Google Scholar

[15]

L. Borcea, Electrical impedance tomography, Inverse Problems, 18 (2002), R99–R136. doi: 10.1088/0266-5611/18/6/201.  Google Scholar

[16]

B. H. Brown, Medical impedance tomography and process impedance tomography: A brief review, Measurement Science and Technology, 12 (2001), 991-996.  doi: 10.1088/0957-0233/12/8/301.  Google Scholar

[17]

B. H. Brown, Electrical impedance tomography (EIT): A review, J Med. Eng. & Tech., (2009), 97–108. doi: 10.1080/0309190021000059687.  Google Scholar

[18]

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

[19]

K. S. ChengD. IsaacsonJ. C. Newell and D. G. Gisser, Electrode models for electric current computed tomography, IEEE Transactions on Biomedical Engineering, 36 (1989), 918-924.   Google Scholar

[20]

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

[21]

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

[22]

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

[23]

M. DeAngelo and J. L. Mueller, 2d D-bar reconstructions of human chest and tank data using an improved approximation to the scattering transform, Physiological Measurement, 31 (2010), 221-232.  doi: 10.1088/0967-3334/31/2/008.  Google Scholar

[24]

M. Dodd and J. L. Mueller, A real-time D-bar algorithm for 2-D electrical impedance tomography data, Inverse Problems and Imaging, 8 (2014), 1013-1031.  doi: 10.3934/ipi.2014.8.1013.  Google Scholar

[25]

L. D. Faddeev, Increasing solutions of the Schrödinger equation, Soviet Physics Doklady, 10 (1966), 1033-1035.   Google Scholar

[26]

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. doi: 10.1038/sdata.2018.112.  Google Scholar

[27]

G. GonzálezJ. M. J. HuttunenV. KolehmainenA. Seppänen and M. Vauhkonen, Experimental evaluation of 3d electrical impedance tomography with total variation prior, Inverse Problems in Science and Engineering, 24 (2016), 1411-1431.   Google Scholar

[28]

P. C. Hansen, Analysis of discrete ill-posed problems by means of the l-curve, SIAM Review, 34 (1992), 561-580.  doi: 10.1137/1034115.  Google Scholar

[29]

A. Hauptmann, Approximation of full-boundary data from partial-boundary electrode measurements, Inverse Problems, 33 (2017), 125017, 22 pp. doi: 10.1088/1361-6420/aa8410.  Google Scholar

[30]

S. J. Hamilton, C. N. L. Herrera, J. L. Mueller and A. Von Herrmann, A direct D-bar reconstruction algorithm for recovering a complex conductivity in 2-D, Inverse Problems, 28 (2012), 095005, 24 pp. doi: 10.1088/0266-5611/28/9/095005.  Google Scholar

[31]

S. J. Hamilton, W. R. B. Lionheart and A. Adler, Comparing d-bar and common regularization-based methods for electrical impedance tomography, Physiological Measurement, 40 (2019), 044004. doi: 10.1088/1361-6579/ab14aa.  Google Scholar

[32]

N. Hyvönen and L. Mustonen, Generalized linearization techniques in electrical impedance tomography, Numerische Mathematik, 140 (2018), 95-120.  doi: 10.1007/s00211-018-0959-1.  Google Scholar

[33]

S. J. Hamilton, J. L. Mueller and T. R. Santos, Robust computation in 2d absolute eit (a-eit) using d-bar methods with the 'exp' approximation, Physiological Measurement, 39 (2018), 064005. doi: 10.1088/1361-6579/aac8b1.  Google Scholar

[34]

D. S. Holder (ed.), Electrical Impedance Tomography; Methods, History and Applications, IOP Publishing Ltd., 2005. Google Scholar

[35]

L. Horesh, Some Novel Approaches in Modelling and Image Reconstruction for Multi Frequency Electrical Impedance Tomography of the Human Brain, Ph.D. thesis, University of London, 2006. Google Scholar

[36]

M. Hallaji, A. 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.  Google Scholar

[37]

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.  Google Scholar

[38]

D. IsaacsonJ. L. MuellerJ. C. Newell and S. Siltanen, Reconstructions of chest phantoms by the D-bar method for electrical impedance tomography, IEEE Trans. Med. Imaging, 23 (2004), 821-828.  doi: 10.1109/TMI.2004.827482.  Google Scholar

[39]

D. Isaacson, J. L. Mueller, J. C. Newell and S. Siltanen, Imaging cardiac activity by the D-bar method for electrical impedance tomography, Physiological Measurement, 27 (2006), S43–S50. doi: 10.1088/0967-3334/27/5/S04.  Google Scholar

[40]

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.  Google Scholar

[41]

K. KnudsenM. LassasJ. 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.  Google Scholar

[42]

K. Knudsen and J. L. Mueller, The born approximation and Calderón's method for reconstructions of conductivities in 3-D, Discrete and Continuous Dynamical Systems, (2011), 884–893.  Google Scholar

[43]

C. LiebermanK. Willcox and O. Ghattas, Parameter and state model reduction for large-scale statistical inverse problems, SIAM Journal on Scientific Computing, 32 (2010), 2523-2542.  doi: 10.1137/090775622.  Google Scholar

[44]

E. Malone, M. Jehl, S. Arridge, T. 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. doi: 10.1088/0967-3334/35/6/1051.  Google Scholar

[45]

P. A. MullerJ. L. Mueller and M. M. Mellenthin, Real-time implementation of Calderón's method on subject-specific domains, IEEE Trans. Med. Imaging, 36 (2017), 1868-1875.   Google Scholar

[46]

J. L. Mueller, P. Muller, M. Mellenthin, R. Murthy, M. Alsaker M. Capps, R. Deterding, S. D. Sagel and E. DeBoer, Estimating regions of air trapping from electrical impedance tomography data, Physiological Measurement, 39 (2018), 05NT01. doi: 10.1088/1361-6579/aac295.  Google Scholar

[47]

P. A. Muller, J. L. Mueller, M. Mellenthin, M. Capps R. Murthy, B. D. Wagner, M. Alsaker, R. Deterding, S. D. Sagel and J. Hoppe, Evaluation of surrogate measures of pulmonary function derived from electrical impedance tomography data in children with cystic fibrosis, Physiological Measurement, 39 (2018), 045008. doi: 10.1088/1361-6579/aab8c4.  Google Scholar

[48]

J. L. Mueller and S. Siltanen, Linear and nonlinear Inverse Problems with Practical Applications, Computational Science & Engineering, 10. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2012. doi: 10.1137/1.9781611972344.  Google Scholar

[49]

A. I. Nachman, Reconstructions from boundary measurements, Annals of Mathematics, 128 (1988), 531-576.  doi: 10.2307/1971435.  Google Scholar

[50]

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

[51]

R. G. Novikov and G. M. Khenkin, The $\overline\partial$-equation in the multidimensional inverse scattering problem, Uspekhi Mat. Nauk., 42 (1987), 93–152,255.  Google Scholar

[52]

A. NissinenA. LehikoinenM. MononenS. Lähteenm$\ddot{\mathrm{k}}$i and M. Vauhkonen, Estimation of the bubble size and bubble loading in a flotation froth using electrical resistance tomography, Minerals Engineering, 69 (2014), 1-12.  doi: 10.1016/j.mineng.2014.07.001.  Google Scholar

[53]

R. G. Novikov, A 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.  Google Scholar

[54]

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd editon, Springer Series in Operations Research, Springer Verlag, New York, 2006. doi: 10.1007/b98874.  Google Scholar

[55]

W. Padden, 3D Simpson's Integrator, 2008, https://www.mathworks.com/matlabcentral/fileexchange/23250-3d-simpson-s-integrator. Google Scholar

[56]

C. Rodgers, S2kit mex file for matlab, https://rodgers.org.uk/software/s2kit, Accessed: 2020-04-29. Google Scholar

[57]

L. I. RudinS. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[58]

E. SomersaloM. Cheney and D. Isaacson, Existence and uniqueness for electrode models for electric current computed tomography, SIAM Journal on Applied Mathematics, 52 (1992), 1023-1040.  doi: 10.1137/0152060.  Google Scholar

[59]

S. SiltanenJ. 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

[60]

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

[61]

A. SeppänenM. VauhkonenP. J. VauhkonenE. Somersalo and J. P. Kaipio, State estimation with fluid dynamical evolution models in process tomography - an application to impedance tomography, Inverse Problems, 17 (2001), 467-483.  doi: 10.1088/0266-5611/17/3/307.  Google Scholar

[62]

H. S. TappA. J. PeytonE. K. Kemsley and R. H. Wilson, Chemical engineering applications of electrical process tomography, Sensors and Actuators B: Chemical, 92 (2003), 17-24.  doi: 10.1016/S0925-4005(03)00126-6.  Google Scholar

[63]

C. R. Vogel, Computational Methods for Inverse Problems, Frontiers in Applied Mathematics, SIAM, 2002. doi: 10.1137/1.9780898717570.  Google Scholar

[64]

M. VauhkonenD. VadászP. A. KarjalainenE. Somersalo and J. P. Kaipio, Tikhonov regularization and prior information in electrical impedance tomography, IEEE Trans. Med. Imaging, 17 (1998), 285-293.  doi: 10.1109/42.700740.  Google Scholar

[65]

Z. Wang, E. P. Simoncelli and A. C. Bovik, Multiscale structural similarity for image quality assessment, in The Thrity-Seventh Asilomar Conference on Signals, Systems & Computers, IEEE, (2003), 1398–1402. doi: 10.1109/ACSSC.2003.1292216.  Google Scholar

show all references

References:
[1]

A. Adler, J. H. Arnold, R. Bayford, A. Borsic, B. Brown, P. Dixon, T. J. Faes, I. Frerichs, H. Gagnon, Y. Gärber and B. Grychtol, GREIT: A unified approach to 2d linear EIT reconstruction of lung images, Physiological Measurement, 30 (2009), S35–S55. doi: 10.1088/0967-3334/30/6/S03.  Google Scholar

[2]

M. AlsakerS. J. Hamilton and A. Hauptmann, A direct D-bar method for partial boundary data Electrical Impedance Tomography with a priori information, Inverse Problems and Imaging, 11 (2017), 427-454.  doi: 10.3934/ipi.2017020.  Google Scholar

[3]

G. Alessandrini, Stable determination of conductivity by boundary measurements, Applicable Analysis, 27 (1988), 153-172.  doi: 10.1080/00036818808839730.  Google Scholar

[4]

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

[5]

M. Alsaker and J. L. Mueller, EIT images of human inspiration and expiration using a D-bar method with spatial priors, Applied Computational Electromagnetics Society Journal, 34 (2019). Google Scholar

[6]

M. AlsakerJ. L. Mueller and R. Murthy, Dynamic optimized priors for D-bar reconstructions of human ventilation using electrical impedance tomography, Journal of Computational and Applied Mathematics, 362 (2019), 276-294.  doi: 10.1016/j.cam.2018.07.039.  Google Scholar

[7]

D. C. Barber and B. H. Brown, Applied potential tomography, Journal of Physics E: Scientific Instruments, 17 (1984), 723-733.  doi: 10.1088/0022-3735/17/9/002.  Google Scholar

[8]

R. Beals and R. R. Coifman, Multidimensional inverse scatterings and nonlinear partial differential equations, Pseudodifferential Operators and Applications, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 43 (1985), 45-70.  doi: 10.1090/pspum/043/812283.  Google Scholar

[9]

P. Blomgren and T. F. Chan, Modular solvers for image restoration problems using the discrepancy principle, Numerical Linear Algebra with Applications, 9 (2002), 347-358.  doi: 10.1002/nla.278.  Google Scholar

[10]

G. Boverman, D. Isaacson, T.-J. Kao, Saulnier, G. J. and J. C. Newell, Methods for direct image reconstruction for EIT in two and three dimensions, in Proceedings of the 2008 Electrical Impedance Tomography Conference, (Dartmouth College, Hanover, New Hampshire, USA), (2008). Google Scholar

[11]

G. BovermanT.-J. KaoD. Isaacson and G. J. Saulnier, An implementation of Calderón's method for 3-D limited view EIT, IEEE Trans. Med. Imaging, 28 (2009), 1073-1082.  doi: 10.1109/TMI.2009.2012892.  Google Scholar

[12]

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

[13]

J. Bikowski and J. Mueller, 2D EIT reconstructions using Calderón's method, Inverse Problems and Imaging, 2 (2008), 43-61.  doi: 10.3934/ipi.2008.2.43.  Google Scholar

[14]

L. Borcea, Addendum to "Electrical impedance tomography", Inverse Problems, 19 (2002), 997-998.  doi: 10.1088/0266-5611/19/4/501.  Google Scholar

[15]

L. Borcea, Electrical impedance tomography, Inverse Problems, 18 (2002), R99–R136. doi: 10.1088/0266-5611/18/6/201.  Google Scholar

[16]

B. H. Brown, Medical impedance tomography and process impedance tomography: A brief review, Measurement Science and Technology, 12 (2001), 991-996.  doi: 10.1088/0957-0233/12/8/301.  Google Scholar

[17]

B. H. Brown, Electrical impedance tomography (EIT): A review, J Med. Eng. & Tech., (2009), 97–108. doi: 10.1080/0309190021000059687.  Google Scholar

[18]

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

[19]

K. S. ChengD. IsaacsonJ. C. Newell and D. G. Gisser, Electrode models for electric current computed tomography, IEEE Transactions on Biomedical Engineering, 36 (1989), 918-924.   Google Scholar

[20]

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

[21]

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

[22]

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

[23]

M. DeAngelo and J. L. Mueller, 2d D-bar reconstructions of human chest and tank data using an improved approximation to the scattering transform, Physiological Measurement, 31 (2010), 221-232.  doi: 10.1088/0967-3334/31/2/008.  Google Scholar

[24]

M. Dodd and J. L. Mueller, A real-time D-bar algorithm for 2-D electrical impedance tomography data, Inverse Problems and Imaging, 8 (2014), 1013-1031.  doi: 10.3934/ipi.2014.8.1013.  Google Scholar

[25]

L. D. Faddeev, Increasing solutions of the Schrödinger equation, Soviet Physics Doklady, 10 (1966), 1033-1035.   Google Scholar

[26]

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. doi: 10.1038/sdata.2018.112.  Google Scholar

[27]

G. GonzálezJ. M. J. HuttunenV. KolehmainenA. Seppänen and M. Vauhkonen, Experimental evaluation of 3d electrical impedance tomography with total variation prior, Inverse Problems in Science and Engineering, 24 (2016), 1411-1431.   Google Scholar

[28]

P. C. Hansen, Analysis of discrete ill-posed problems by means of the l-curve, SIAM Review, 34 (1992), 561-580.  doi: 10.1137/1034115.  Google Scholar

[29]

A. Hauptmann, Approximation of full-boundary data from partial-boundary electrode measurements, Inverse Problems, 33 (2017), 125017, 22 pp. doi: 10.1088/1361-6420/aa8410.  Google Scholar

[30]

S. J. Hamilton, C. N. L. Herrera, J. L. Mueller and A. Von Herrmann, A direct D-bar reconstruction algorithm for recovering a complex conductivity in 2-D, Inverse Problems, 28 (2012), 095005, 24 pp. doi: 10.1088/0266-5611/28/9/095005.  Google Scholar

[31]

S. J. Hamilton, W. R. B. Lionheart and A. Adler, Comparing d-bar and common regularization-based methods for electrical impedance tomography, Physiological Measurement, 40 (2019), 044004. doi: 10.1088/1361-6579/ab14aa.  Google Scholar

[32]

N. Hyvönen and L. Mustonen, Generalized linearization techniques in electrical impedance tomography, Numerische Mathematik, 140 (2018), 95-120.  doi: 10.1007/s00211-018-0959-1.  Google Scholar

[33]

S. J. Hamilton, J. L. Mueller and T. R. Santos, Robust computation in 2d absolute eit (a-eit) using d-bar methods with the 'exp' approximation, Physiological Measurement, 39 (2018), 064005. doi: 10.1088/1361-6579/aac8b1.  Google Scholar

[34]

D. S. Holder (ed.), Electrical Impedance Tomography; Methods, History and Applications, IOP Publishing Ltd., 2005. Google Scholar

[35]

L. Horesh, Some Novel Approaches in Modelling and Image Reconstruction for Multi Frequency Electrical Impedance Tomography of the Human Brain, Ph.D. thesis, University of London, 2006. Google Scholar

[36]

M. Hallaji, A. 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.  Google Scholar

[37]

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.  Google Scholar

[38]

D. IsaacsonJ. L. MuellerJ. C. Newell and S. Siltanen, Reconstructions of chest phantoms by the D-bar method for electrical impedance tomography, IEEE Trans. Med. Imaging, 23 (2004), 821-828.  doi: 10.1109/TMI.2004.827482.  Google Scholar

[39]

D. Isaacson, J. L. Mueller, J. C. Newell and S. Siltanen, Imaging cardiac activity by the D-bar method for electrical impedance tomography, Physiological Measurement, 27 (2006), S43–S50. doi: 10.1088/0967-3334/27/5/S04.  Google Scholar

[40]

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.  Google Scholar

[41]

K. KnudsenM. LassasJ. 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.  Google Scholar

[42]

K. Knudsen and J. L. Mueller, The born approximation and Calderón's method for reconstructions of conductivities in 3-D, Discrete and Continuous Dynamical Systems, (2011), 884–893.  Google Scholar

[43]

C. LiebermanK. Willcox and O. Ghattas, Parameter and state model reduction for large-scale statistical inverse problems, SIAM Journal on Scientific Computing, 32 (2010), 2523-2542.  doi: 10.1137/090775622.  Google Scholar

[44]

E. Malone, M. Jehl, S. Arridge, T. 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. doi: 10.1088/0967-3334/35/6/1051.  Google Scholar

[45]

P. A. MullerJ. L. Mueller and M. M. Mellenthin, Real-time implementation of Calderón's method on subject-specific domains, IEEE Trans. Med. Imaging, 36 (2017), 1868-1875.   Google Scholar

[46]

J. L. Mueller, P. Muller, M. Mellenthin, R. Murthy, M. Alsaker M. Capps, R. Deterding, S. D. Sagel and E. DeBoer, Estimating regions of air trapping from electrical impedance tomography data, Physiological Measurement, 39 (2018), 05NT01. doi: 10.1088/1361-6579/aac295.  Google Scholar

[47]

P. A. Muller, J. L. Mueller, M. Mellenthin, M. Capps R. Murthy, B. D. Wagner, M. Alsaker, R. Deterding, S. D. Sagel and J. Hoppe, Evaluation of surrogate measures of pulmonary function derived from electrical impedance tomography data in children with cystic fibrosis, Physiological Measurement, 39 (2018), 045008. doi: 10.1088/1361-6579/aab8c4.  Google Scholar

[48]

J. L. Mueller and S. Siltanen, Linear and nonlinear Inverse Problems with Practical Applications, Computational Science & Engineering, 10. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2012. doi: 10.1137/1.9781611972344.  Google Scholar

[49]

A. I. Nachman, Reconstructions from boundary measurements, Annals of Mathematics, 128 (1988), 531-576.  doi: 10.2307/1971435.  Google Scholar

[50]

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

[51]

R. G. Novikov and G. M. Khenkin, The $\overline\partial$-equation in the multidimensional inverse scattering problem, Uspekhi Mat. Nauk., 42 (1987), 93–152,255.  Google Scholar

[52]

A. NissinenA. LehikoinenM. MononenS. Lähteenm$\ddot{\mathrm{k}}$i and M. Vauhkonen, Estimation of the bubble size and bubble loading in a flotation froth using electrical resistance tomography, Minerals Engineering, 69 (2014), 1-12.  doi: 10.1016/j.mineng.2014.07.001.  Google Scholar

[53]

R. G. Novikov, A 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.  Google Scholar

[54]

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd editon, Springer Series in Operations Research, Springer Verlag, New York, 2006. doi: 10.1007/b98874.  Google Scholar

[55]

W. Padden, 3D Simpson's Integrator, 2008, https://www.mathworks.com/matlabcentral/fileexchange/23250-3d-simpson-s-integrator. Google Scholar

[56]

C. Rodgers, S2kit mex file for matlab, https://rodgers.org.uk/software/s2kit, Accessed: 2020-04-29. Google Scholar

[57]

L. I. RudinS. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[58]

E. SomersaloM. Cheney and D. Isaacson, Existence and uniqueness for electrode models for electric current computed tomography, SIAM Journal on Applied Mathematics, 52 (1992), 1023-1040.  doi: 10.1137/0152060.  Google Scholar

[59]

S. SiltanenJ. 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

[60]

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

[61]

A. SeppänenM. VauhkonenP. J. VauhkonenE. Somersalo and J. P. Kaipio, State estimation with fluid dynamical evolution models in process tomography - an application to impedance tomography, Inverse Problems, 17 (2001), 467-483.  doi: 10.1088/0266-5611/17/3/307.  Google Scholar

[62]

H. S. TappA. J. PeytonE. K. Kemsley and R. H. Wilson, Chemical engineering applications of electrical process tomography, Sensors and Actuators B: Chemical, 92 (2003), 17-24.  doi: 10.1016/S0925-4005(03)00126-6.  Google Scholar

[63]

C. R. Vogel, Computational Methods for Inverse Problems, Frontiers in Applied Mathematics, SIAM, 2002. doi: 10.1137/1.9780898717570.  Google Scholar

[64]

M. VauhkonenD. VadászP. A. KarjalainenE. Somersalo and J. P. Kaipio, Tikhonov regularization and prior information in electrical impedance tomography, IEEE Trans. Med. Imaging, 17 (1998), 285-293.  doi: 10.1109/42.700740.  Google Scholar

[65]

Z. Wang, E. P. Simoncelli and A. C. Bovik, Multiscale structural similarity for image quality assessment, in The Thrity-Seventh Asilomar Conference on Signals, Systems & Computers, IEEE, (2003), 1398–1402. doi: 10.1109/ACSSC.2003.1292216.  Google Scholar

Figure 1.  Demonstration of the 3D $ \mathbf{t}^{\rm{{\textbf{exp}}}} $ Method and Calderón's Method on the 'Heart and Lungs' phantom T2-B, shown on left, using simulated electrode data. The 2D cross-sectional slices above show that the conductive heart is correctly visible in the $ x_1x_3 $ plane but absent from the $ x_2x_3 $ plane. Similarly for the lungs in the $ x_2x_3 $ plane vs the $ x_1x_3 $ plane
Figure 2.  The simulated targets considered in this manuscript
Figure 3.  The domains used for data simulation showing electrode locations and electrode numbering
Figure 4.  Comparison of reconstructed conductivity (Left) and Fourier data, $ \hat{F} $, (Right) for T1 using Calderón's method (equation (16)). $ T_z = 2.7 $ for the analytic data and $ T_z = 1.3 $ for all three simulated electrode data cases. The mollifying parameter is $ t = 0.1 $ for both analytic and simulated electrode data. The vertical dashed line indicates where the Fourier domain was truncated for the simulated electrode data cases
Figure 5.  Reconstructions of radially symmetric example T1 across algorithms using $ L = 128 $ electrodes shown in 3D and a representative $ x_2x_3 $ slice
Figure 6.  Comparison of conductivity and susceptivity reconstructions for the complex-valued heart and lungs target T2-A
Figure 7.  Comparison of reconstructions for the real-valued heart and lungs target T2-B using $ L = 128 $, $ 64 $, or $ 32 $ electrodes
Figure 8.  Comparison of reconstructions for the high contrast target T3 using $ L = 128 $, $ 64 $, or $ 32 $ electrodes
Figure 9.  Comparison of reconstructions for the real-valued heart and lungs target T2-B with increasing levels of noise added to the voltage data. Segmented 3D isosurface renderings are shown for each reconstruction as well as the $ x_1x_2 $ cross-sectional slice
Figure 10.  Comparison of reconstructions for the high contrast target T3 using various levels of noise. Segmented 3D isosurface renderings are shown for each reconstruction as well as the $ x_1x_2 $ cross-sectional slice
Figure 11.  Whole-image evaluation metrics for the real-valued heart and lungs target T2-B with decreasing numbers of simulated electrodes. Left: Dynamic Range, Middle: Mean Square Error, Right: Multi-Scale Structural Similarity Index
Figure 12.  Whole-image evaluation metrics for target T3 with decreasing numbers of simulated electrodes. Left: Dynamic Range, Middle: Mean Square Error, Right: Multi-Scale Structural Similarity Index
Figure 13.  Whole-image evaluation metrics for the real-valued heart and lungs target T2-B with increasing levels of noise added to the voltage data. Left: Dynamic Range, Middle: Mean Square Error, Right: Multi-Scale Structural Similarity Index
Figure 14.  Whole-image evaluation metrics for the target T3 with increasing levels of noise added to the voltage data. Left: Dynamic Range, Middle: Mean Square Error, Right: Multi-Scale Structural Similarity Index
Table 5.  T2-B evaluation metrics across all electrode configurations considered. Lung 1 is the resistive target with the larger volume
$ \mathbf{t}^{\rm{{\textbf{exp}}}} $ Calder #243;n Smooth TV
LE heart L=128 0.1545 0.1296 0.0086 0.0171
L=64 0.2416 0.1899 0.0070 0.0176
L=32 0.2424 0.1837 0.0115 0.0035
lung 1 L=128 0.1099 0.1216 0.0130 0.0194
L=64 0.1627 0.1628 0.0084 0.0118
L=32 N/A N/A 0.0040 0.0098
lung 2 L=128 0.1171 0.1531 0.0144 0.0109
L=64 0.2221 0.2172 0.0069 0.0085
L=32 N/A N/A 0.0135 0.0068
RVR heart L=128 0.7526 1.1204 0.6520 0.9152
L=64 1.0722 1.1101 0.7737 0.9869
L=32 2.9867 1.9449 1.2085 1.2273
lung 1 L=128 0.3977 0.6102 0.6360 0.7102
L=64 0.3389 0.5475 0.7038 0.8359
L=32 N/A N/A 0.9249 0.9318
lung 2 L=128 0.3312 0.2990 0.6342 0.6600
L=64 0.2311 0.3047 0.6908 0.8039
L=32 N/A N/A 0.8597 0.8590
$ \mathbf{t}^{\rm{{\textbf{exp}}}} $ Calder #243;n Smooth TV
LE heart L=128 0.1545 0.1296 0.0086 0.0171
L=64 0.2416 0.1899 0.0070 0.0176
L=32 0.2424 0.1837 0.0115 0.0035
lung 1 L=128 0.1099 0.1216 0.0130 0.0194
L=64 0.1627 0.1628 0.0084 0.0118
L=32 N/A N/A 0.0040 0.0098
lung 2 L=128 0.1171 0.1531 0.0144 0.0109
L=64 0.2221 0.2172 0.0069 0.0085
L=32 N/A N/A 0.0135 0.0068
RVR heart L=128 0.7526 1.1204 0.6520 0.9152
L=64 1.0722 1.1101 0.7737 0.9869
L=32 2.9867 1.9449 1.2085 1.2273
lung 1 L=128 0.3977 0.6102 0.6360 0.7102
L=64 0.3389 0.5475 0.7038 0.8359
L=32 N/A N/A 0.9249 0.9318
lung 2 L=128 0.3312 0.2990 0.6342 0.6600
L=64 0.2311 0.3047 0.6908 0.8039
L=32 N/A N/A 0.8597 0.8590
Table 1.  Evaluation metrics for the high-contrast example T3 across all electrode configurations considered
$ \mathbf{t}^{\rm{{\textbf{exp}}}} $ Calderon Smooth TV
LE conductor L=128 0.1267 0.1181 0.0264 0.0191
L=64 0.1522 0.1566 0.0085 0.0136
L=32 0.1559 0.1548 0.0111 0.0093
resistor L=128 0.0853 0.0861 0.0128 0.0074
L=64 0.1072 0.1107 0.0050 0.0061
L=32 0.1465 0.1062 0.0057 0.0109
RVR conductor L=128 0.4672 0.4690 0.4122 0.6124
L=64 0.5357 0.4343 0.4456 0.5490
L=32 0.5883 0.5580 0.5492 0.6912
resistor L=128 1.6747 1.6273 0.9951 1.0290
L=64 1.9706 1.6022 1.0895 1.0479
L=32 2.1047 2.0931 1.4676 1.1714
$ \mathbf{t}^{\rm{{\textbf{exp}}}} $ Calderon Smooth TV
LE conductor L=128 0.1267 0.1181 0.0264 0.0191
L=64 0.1522 0.1566 0.0085 0.0136
L=32 0.1559 0.1548 0.0111 0.0093
resistor L=128 0.0853 0.0861 0.0128 0.0074
L=64 0.1072 0.1107 0.0050 0.0061
L=32 0.1465 0.1062 0.0057 0.0109
RVR conductor L=128 0.4672 0.4690 0.4122 0.6124
L=64 0.5357 0.4343 0.4456 0.5490
L=32 0.5883 0.5580 0.5492 0.6912
resistor L=128 1.6747 1.6273 0.9951 1.0290
L=64 1.9706 1.6022 1.0895 1.0479
L=32 2.1047 2.0931 1.4676 1.1714
Table 6.  Evaluation metrics for T2-B with 0.01%, 0.1% and 1% noise with $ 128 $ electrodes
$ \mathbf{t}^{\rm{{\textbf{exp}}}} $ Calderon Smooth TV
LE heart 0.01% noise 0.1451 0.1494 0.0096 0.0158
0.1% noise 0.2372 0.2147 0.0184 0.0286
1% noise 0.2061 0.1520 N/A 0.0770
lung 1 0.01% noise 0.1294 0.1201 0.0138 0.0209
0.1% noise 0.1541 0.1540 0.0125 0.0213
1% noise N/A N/A N/A 0.0513
lung 2 0.01% noise 0.1097 0.1336 0.0141 0.0111
0.1% noise 0.1245 0.1498 0.0210 0.0220
1% noise N/A N/A N/A 0.0444
RVR heart 0.01% noise 1.3702 1.0072 0.6605 0.9279
0.1% noise 2.5372 1.3738 1.0518 1.1021
1% noise 0.6135 3.7011 N/A 1.0894
lung 1 0.01% noise 0.5548 0.6647 0.6603 0.7192
0.1% noise 1.0583 0.7483 0.9256 0.4780
1% noise N/A N/A N/A 0.9519
lung 2 0.01% noise 0.4064 0.3625 0.6504 0.6581
0.1% noise 1.1779 0.5656 0.9098 0.4665
1% noise N/A N/A N/A 0.7636
$ \mathbf{t}^{\rm{{\textbf{exp}}}} $ Calderon Smooth TV
LE heart 0.01% noise 0.1451 0.1494 0.0096 0.0158
0.1% noise 0.2372 0.2147 0.0184 0.0286
1% noise 0.2061 0.1520 N/A 0.0770
lung 1 0.01% noise 0.1294 0.1201 0.0138 0.0209
0.1% noise 0.1541 0.1540 0.0125 0.0213
1% noise N/A N/A N/A 0.0513
lung 2 0.01% noise 0.1097 0.1336 0.0141 0.0111
0.1% noise 0.1245 0.1498 0.0210 0.0220
1% noise N/A N/A N/A 0.0444
RVR heart 0.01% noise 1.3702 1.0072 0.6605 0.9279
0.1% noise 2.5372 1.3738 1.0518 1.1021
1% noise 0.6135 3.7011 N/A 1.0894
lung 1 0.01% noise 0.5548 0.6647 0.6603 0.7192
0.1% noise 1.0583 0.7483 0.9256 0.4780
1% noise N/A N/A N/A 0.9519
lung 2 0.01% noise 0.4064 0.3625 0.6504 0.6581
0.1% noise 1.1779 0.5656 0.9098 0.4665
1% noise N/A N/A N/A 0.7636
Table 2.  Evaluation metrics for the high-contrast example T3 with 0.01%, 0.1% and 1% noise with $ 128 $ electrodes
$ \mathbf{t}^{\rm{{\textbf{exp}}}} $ Calderon Smooth TV
LE conductor 0.01% noise 0.1303 0.1155 0.0292 0.0204
0.1% noise 0.1044 0.1391 0.0127 0.0262
1% noise 0.2501 0.1854 0.0611 0.0551
resistor 0.01% noise 0.0887 0.0814 0.0134 0.0080
0.1% noise 0.1444 0.1046 0.0171 0.0046
1% noise 0.2193 0.1930 0.1166 0.0529
RVR conductor 0.01% noise 0.3992 0.4405 0.3866 0.5726
0.1% noise 0.6590 0.5265 0.4967 0.7483
1% noise 1.1334 0.8591 0.4805 0.7573
resistor 0.01% noise 1.2038 1.6083 0.9931 1.0125
0.1% noise 3.1351 2.6043 1.6627 1.4349
1% noise 3.4953 2.2805 1.4426 2.0613
$ \mathbf{t}^{\rm{{\textbf{exp}}}} $ Calderon Smooth TV
LE conductor 0.01% noise 0.1303 0.1155 0.0292 0.0204
0.1% noise 0.1044 0.1391 0.0127 0.0262
1% noise 0.2501 0.1854 0.0611 0.0551
resistor 0.01% noise 0.0887 0.0814 0.0134 0.0080
0.1% noise 0.1444 0.1046 0.0171 0.0046
1% noise 0.2193 0.1930 0.1166 0.0529
RVR conductor 0.01% noise 0.3992 0.4405 0.3866 0.5726
0.1% noise 0.6590 0.5265 0.4967 0.7483
1% noise 1.1334 0.8591 0.4805 0.7573
resistor 0.01% noise 1.2038 1.6083 0.9931 1.0125
0.1% noise 3.1351 2.6043 1.6627 1.4349
1% noise 3.4953 2.2805 1.4426 2.0613
Table 3.  Evaluation metrics for T1 with 128 electrodes
$ \mathbf{t}^{\rm{{\textbf{exp}}}} $ Calderón Smooth TV
DR 95.25% 116.89% 162.62% 143.16%
MSE 0.0305 0.0373 0.0141 0.0124
MS-SSIM 0.8126 0.8324 0.8984 0.8978
LE 0.0008 0.0008 0.0021 0.0007
RVR 0.4435 0.3734 0.5088 0.6255
$ \mathbf{t}^{\rm{{\textbf{exp}}}} $ Calderón Smooth TV
DR 95.25% 116.89% 162.62% 143.16%
MSE 0.0305 0.0373 0.0141 0.0124
MS-SSIM 0.8126 0.8324 0.8984 0.8978
LE 0.0008 0.0008 0.0021 0.0007
RVR 0.4435 0.3734 0.5088 0.6255
Table 4.  Evaluation metrics for T2-A for 128 electrode data. Lung 1 is the resistive target with the largest volume
$ \mathbf{t}^{\rm{{\textbf{exp}}}} $ Calderón} Smooth TV
DR Re 50.19% 59.53% 92.59% 101.22%
Im 98.02% 62.35% 85.80% 88.88%
MSE Re 0.0097 0.0110 0.0042 0.0038
Im 0.0022 0.0022 0.0011 0.0010
MS-SSIM Re 0.8235 0.8193 0.7841 0.8179
Im 0.8956 0.8710 0.8505 0.8558
LE heart Re 0.1302 0.1172 0.0113 0.0263
Im 0.1751 0.2133 0.1179 0.0949
lung 1 Re 0.1245 0.1401 0.0059 0.0105
Im 0.1199 0.0843 0.0708 0.0412
lung 2 Re 0.1425 0.1746 0.0141 0.0067
Im 0.1715 0.1290 0.0934 0.0538
RVR heart Re 0.8211 0.8112 1.0418 0.8497
Im 0.2798 0.4819 1.0119 0.6845
lung 1 Re 0.9114 1.0072 1.4399 1.2159
Im 1.1566 0.6418 0.9389 0.5171
lung 2 Re 0.8381 0.8041 1.4425 1.2289
Im 1.3116 0.5547 0.5568 0.1976
$ \mathbf{t}^{\rm{{\textbf{exp}}}} $ Calderón} Smooth TV
DR Re 50.19% 59.53% 92.59% 101.22%
Im 98.02% 62.35% 85.80% 88.88%
MSE Re 0.0097 0.0110 0.0042 0.0038
Im 0.0022 0.0022 0.0011 0.0010
MS-SSIM Re 0.8235 0.8193 0.7841 0.8179
Im 0.8956 0.8710 0.8505 0.8558
LE heart Re 0.1302 0.1172 0.0113 0.0263
Im 0.1751 0.2133 0.1179 0.0949
lung 1 Re 0.1245 0.1401 0.0059 0.0105
Im 0.1199 0.0843 0.0708 0.0412
lung 2 Re 0.1425 0.1746 0.0141 0.0067
Im 0.1715 0.1290 0.0934 0.0538
RVR heart Re 0.8211 0.8112 1.0418 0.8497
Im 0.2798 0.4819 1.0119 0.6845
lung 1 Re 0.9114 1.0072 1.4399 1.2159
Im 1.1566 0.6418 0.9389 0.5171
lung 2 Re 0.8381 0.8041 1.4425 1.2289
Im 1.3116 0.5547 0.5568 0.1976
[1]

Fabrice Delbary, Kim Knudsen. Numerical nonlinear complex geometrical optics algorithm for the 3D Calderón problem. Inverse Problems & Imaging, 2014, 8 (4) : 991-1012. doi: 10.3934/ipi.2014.8.991

[2]

Albert Clop, Daniel Faraco, Alberto Ruiz. Stability of Calderón's inverse conductivity problem in the plane for discontinuous conductivities. Inverse Problems & Imaging, 2010, 4 (1) : 49-91. doi: 10.3934/ipi.2010.4.49

[3]

Erfang Ma. Integral formulation of the complete electrode model of electrical impedance tomography. Inverse Problems & Imaging, 2020, 14 (2) : 385-398. doi: 10.3934/ipi.2020017

[4]

Nuutti Hyvönen, Harri Hakula, Sampsa Pursiainen. Numerical implementation of the factorization method within the complete electrode model of electrical impedance tomography. Inverse Problems & Imaging, 2007, 1 (2) : 299-317. doi: 10.3934/ipi.2007.1.299

[5]

Marius Ionescu, Luke G. Rogers. Complex Powers of the Laplacian on Affine Nested Fractals as Calderón-Zygmund operators. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2155-2175. doi: 10.3934/cpaa.2014.13.2155

[6]

Petteri Piiroinen, Martin Simon. Probabilistic interpretation of the Calderón problem. Inverse Problems & Imaging, 2017, 11 (3) : 553-575. doi: 10.3934/ipi.2017026

[7]

Fabrice Delbary, Rainer Kress. Electrical impedance tomography using a point electrode inverse scheme for complete electrode data. Inverse Problems & Imaging, 2011, 5 (2) : 355-369. doi: 10.3934/ipi.2011.5.355

[8]

Pedro Caro, Mikko Salo. Stability of the Calderón problem in admissible geometries. Inverse Problems & Imaging, 2014, 8 (4) : 939-957. doi: 10.3934/ipi.2014.8.939

[9]

Matteo Santacesaria. Note on Calderón's inverse problem for measurable conductivities. Inverse Problems & Imaging, 2019, 13 (1) : 149-157. doi: 10.3934/ipi.2019008

[10]

Sun-Sig Byun, Yunsoo Jang. Calderón-Zygmund estimate for homogenization of parabolic systems. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 6689-6714. doi: 10.3934/dcds.2016091

[11]

Oleg Yu. Imanuvilov, Masahiro Yamamoto. Calderón problem for Maxwell's equations in cylindrical domain. Inverse Problems & Imaging, 2014, 8 (4) : 1117-1137. doi: 10.3934/ipi.2014.8.1117

[12]

Yernat M. Assylbekov. Reconstruction in the partial data Calderón problem on admissible manifolds. Inverse Problems & Imaging, 2017, 11 (3) : 455-476. doi: 10.3934/ipi.2017021

[13]

Sun-Sig Byun, Yumi Cho, Shuang Liang. Calderón-Zygmund estimates for quasilinear elliptic double obstacle problems with variable exponent and logarithmic growth. Discrete & Continuous Dynamical Systems - B, 2020, 25 (10) : 3843-3855. doi: 10.3934/dcdsb.2020038

[14]

Angkana Rüland, Eva Sincich. Lipschitz stability for the finite dimensional fractional Calderón problem with finite Cauchy data. Inverse Problems & Imaging, 2019, 13 (5) : 1023-1044. doi: 10.3934/ipi.2019046

[15]

Kim Knudsen, Jennifer L. Mueller. The born approximation and Calderón's method for reconstruction of conductivities in 3-D. Conference Publications, 2011, 2011 (Special) : 844-853. doi: 10.3934/proc.2011.2011.844

[16]

María Ángeles García-Ferrero, Angkana Rüland, Wiktoria Zatoń. Runge approximation and stability improvement for a partial data Calderón problem for the acoustic Helmholtz equation. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021049

[17]

Jifa Jiang, Fensidi Tang. The complete classification on a model of two species competition with an inhibitor. Discrete & Continuous Dynamical Systems, 2008, 20 (3) : 659-672. doi: 10.3934/dcds.2008.20.659

[18]

Woojoo Shim. On the generic complete synchronization of the discrete Kuramoto model. Kinetic & Related Models, 2020, 13 (5) : 979-1005. doi: 10.3934/krm.2020034

[19]

David J. Aldous. A stochastic complex network model. Electronic Research Announcements, 2003, 9: 152-161.

[20]

Daniel Morales-Silva, David Yang Gao. Complete solutions and triality theory to a nonconvex optimization problem with double-well potential in $\mathbb{R}^n $. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 271-282. doi: 10.3934/naco.2013.3.271

2020 Impact Factor: 1.639

Metrics

  • PDF downloads (109)
  • HTML views (263)
  • Cited by (0)

[Back to Top]