October  2021, 15(5): 1135-1169. doi: 10.3934/ipi.2021032

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 Published  October 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 and Imaging, 2021, 15 (5) : 1135-1169. doi: 10.3934/ipi.2021032
References:
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show all references

References:
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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.

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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.

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

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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.

[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).

[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.

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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.

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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.

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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.

[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).

[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.

[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.

[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.

[14]

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

[15]

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

[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.

[17]

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

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[25]

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

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[34]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[42]

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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
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