Electrical Impedance Tomography (EIT) aims at reconstructing the electric conductivity distribution in a body from electro-static boundary measurements. The inverse problem is severely ill-posed, especially when only partial data is considered. In this work, we propose three methods for the combined reconstruction and segmentation in EIT with partial data. Firstly, we introduce a regularization that takes spatial information into account and corrects for limited coverage. Secondly, we exploit the Chan–Vese method for improving the segmentation step. Finally, we utilize an optimization framework with a level-set approach to simultaneously reconstruct and segment inclusions. The work is done in the context of the Kuopio Tomography Challenge 2023. We demonstrate on experimental tank data that each of the three methods performs significantly better than a classical linearization approach, especially in partial-data scenarios. In particular, the level-set method drastically improves the reconstruction of inclusions with complicated boundaries; this method is superior among our contributions.
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Figure 1. (a) An EIT setup from the University of Eastern Finland [19]. This is not the exact setup used to generate the data. (b) Conceptual illustration of the computational domain $ \Omega $ with eight electrodes $ E_m $ attached to the boundary $ \partial \Omega $. In the competition, 32 electrodes in a similar layout were used. Inside $ \Omega $ are a conductive inclusion with conductivity $ \sigma_1 $ and a resistive inclusion with conductivity $ \sigma_2 $. The background conductivity is $ \sigma_0 $
Figure 2. Reconstructions from training data of a phantom, labeled Training Phantom 1 in the figure, using the Baseline method for difficulty levels 1, 4 and 7. Top row: Unsegmented reconstructions. Bottom row: Segmented reconstructions using Otsu segmentation. The numbers above the segmented reconstructions are the SSIM scores. The red dots mark the center of the electrodes. Yellow marks the conductive inclusion, dark blue marks the resistive, and teal green marks the background
Figure 3. Reconstructions from training data of a phantom, labeled Training Phantom 4 in the figure, using the Baseline method for difficulty levels 1, 4 and 7. Top row: Unsegmented reconstructions. Bottom row: Segmented reconstructions using Otsu segmentation. The numbers above the segmented reconstructions are the SSIM scores. The red dots mark the center of the electrodes. Yellow marks the conductive inclusion, dark blue marks the resistive, and teal green marks the background
Figure 5. The three-step process used to segment the reconstruction $ \widehat{\Delta \pmb{{\sigma}}} $ into three different classes using Chan–Vese segmentation method. Left: First, Chan–Vese segmentation is applied to $ |\widehat{\Delta \pmb{{\sigma}}}| $. Middle: The type of inclusion is determined for each connected component using the values $ \widehat{\Delta \pmb{{\sigma}}}_{ij} $. Right: Final classification
Figure 6. Reconstructions from the training data of Training Phantom 1 using the SR method and the SR-CV method for difficulty levels 1, 4 and 7. Top row: Unsegmented reconstructions. Middle row: Segmented reconstructions using Otsu segmentation. Bottom row: Segmented reconstructions using Chan–Vese segmentation. The numbers above the segmented reconstructions are the SSIM scores. The red dots mark the center of the electrodes. Yellow marks the conductive inclusion, dark blue marks the resistive, and teal green marks the background
Figure 7. Reconstructions from the training data of Training Phantom 4 using the SR method and the SR-CV method for difficulty levels 1, 4 and 7. Top row: Unsegmented reconstructions. Middle row: Segmented reconstructions using Otsu segmentation. Bottom row: Segmented reconstructions using Chan–Vese segmentation. The numbers above the segmented reconstructions are the SSIM scores. The red dots mark the center of the electrodes. Dark blue marks the resistive inclusion and teal green marks the background
Figure 8. Reconstructions from the training data of Training Phantom 1 (top row) and Training Phantom 4 (bottom row) using the SR-TV-LS method for difficulty levels 1, 4 and 7. The numbers above the segmented reconstructions are the SSIM scores. The red dots mark the center of the electrodes. Yellow marks the conductive inclusion, dark blue marks the resistive, and teal green marks the background
Figure 9. Reconstructions of the highest difficulty level from the evaluation data for three different phantoms, labeled Evaluation Phantom 7A, Evaluation Phantom 7B and Evaluation Phantom 7C in the figure, using the SR method (second column), SR-CV method (third column) and SR-TV-LS method (fourth column). The numbers above the segmented reconstructions are the SSIM scores. Yellow marks the conductive inclusion, dark blue marks the resistive, and teal green marks the background
Figure 10. Performance of our proposed methods reported by the SSIM scores for the reconstructions. (Left) our computed reconstruction scores from the training data for each method averaged over 4 cases for each level. The same set of 4 phantoms are used in each level. We also show the reconstruction score for the Baseline method. (Right) computed reconstruction scores [33] from the evaluation data for each method, averaged over 3 cases for each level. In each level, a different set of 3 phantoms is used. We also show the reconstruction score for the best-performing method in the competition [33]. We note the scores on evaluation data are drawn from the distributed competition results
[1] | B. J. Adesokan, B. Jensen, B. Jin and K. Knudsen, Acousto-electric tomography with total variation regularization, Inverse Problems, 35 (2019), 035008, 25 pp. doi: 10.1088/1361-6420/aaece5. |
[2] | A. Adler and D. Holder, Electrical Impedance Tomography: Methods, History and Applications, CRC Press, 2021. doi: 10.1201/9780429399886. |
[3] | M. Aller, D. Mera, J. M. Cotos and S. Villaroya, Study and comparison of different machine learning-based approaches to solve the inverse problem in electrical impedance tomographies, Neural Computing and Applications, 35 (2023), 5465-5477. doi: 10.1007/s00521-022-07988-7. |
[4] | L. Borcea, A nonlinear multigrid for imaging electrical conductivity and permittivity at low frequency, Inverse Problems, 17 (2001), 329. doi: 10.1088/0266-5611/17/2/312. |
[5] | L. Borcea, Electrical impedance tomography, Inverse Problems, 18 (2002), R99-R136. doi: 10.1088/0266-5611/18/6/201. |
[6] | A. Borsic, B. M. Graham, A. Adler and W. R. B. Lionheart, Total variation regularization in electrical impedance tomography, https://eprints.maths.manchester.ac.uk/813/1/TVReglnEITpreprint.pdf, 2007. |
[7] | A. Borsic, W. R. B. Lionheart and C. N. McLeod, Generation of anisotropic-smoothness regularization filters for EIT, IEEE Transactions on Medical Imaging, 21 (2002), 579-587. doi: 10.1109/TMI.2002.800611. |
[8] | V. Candiani, J. Dardé, H. Garde and N. Hyvönen, Monotonicity-based reconstruction of extreme inclusions in electrical impedance tomography, SIAM Journal on Mathematical Analysis, 52 (2020), 6234-6259. doi: 10.1137/19M1299219. |
[9] | T. Chan and L. Vese, An active contour model without edges, in Mads Nielsen, Peter Johansen, Ole Fogh Olsen, and Joachim Weickert, editors, Scale-Space Theories in Computer Vision, Berlin, Heidelberg, 1999,141-151. Springer Berlin Heidelberg. doi: 10.1007/3-540-48236-9_13. |
[10] | T. F. Chan and X.-C. Tai, Level set and total variation regularization for elliptic inverse problems with discontinuous coefficients, Journal of Computational Physics, 193 (2004), 40-66. doi: 10.1016/j.jcp.2003.08.003. |
[11] | T. F. Chan and L. Vese, An active contour model without edges, in International Conference on Scale-Space Theories in Computer Vision, Springer, 1999,141-151. doi: 10.1007/3-540-48236-9_13. |
[12] | M. Cheney, D. Isaacson, J. C. Newell, S. Simske and J. Goble, NOSER: An algorithm for solving the inverse conductivity problem, International Journal of Imaging Systems and Technology, 2 (1990), 66-75. doi: 10.1002/ima.1850020203. |
[13] | E. T. Chung, T. F. Chan and X.-C. Tai, Electrical impedance tomography using level set representation and total variational regularization, Journal of Computational Physics, 205 (2005), 357-372. doi: 10.1016/j.jcp.2004.11.022. |
[14] | A. Coxson, I. Mihov, Z. Wang, V. Avramov and F. B. Barnes, S. Slizovskiy, C. Mullan, I. Timokhin, D. Sanderson, A. Kretinin, et al., Machine learning enhanced electrical impedance tomography for 2D materials, Inverse Problems, 38 (2022), 085007, 27 pp. doi: 10.1088/1361-6420/ac7743. |
[15] | O. L. Elvetun and B. F. Nielsen, A regularization operator for source identification for elliptic PDEs, Inverse Problems and Imaging, 15 (2021), 599-618. doi: 10.3934/ipi.2021006. |
[16] | H. Garde and N. Hyvönen, Series reversion in Calderón's problem, Mathematics of Computation, 91 (2022), 1925-1953. doi: 10.1090/mcom/3729. |
[17] | H. Garde and K. Knudsen, Sparsity prior for electrical impedance tomography with partial data, Inverse Problems in Science and Engineering, 24 (2016), 524-541. doi: 10.1080/17415977.2015.1047365. |
[18] | B. Harrach and M. Ullrich, Monotonicity-based shape reconstruction in electrical impedance tomography, SIAM Journal on Mathematical Analysis, 45 (2013), 3382-3403. doi: 10.1137/120886984. |
[19] | A. Hauptmann, V. Kolehmainen, N.-M. Mach, T. Savolainen, A. Seppänen and S. Siltanen, Open 2D electrical impedance tomography data archive, https://arXiv.org/pdf/1704.01178.pdf, 2017. |
[20] | D. Jiang, Y. Wu and A. Demosthenous, Hand gesture recognition using three-dimensional electrical impedance tomography, IEEE Transactions on Circuits and Systems Ⅱ: Express Briefs, 67 (2020), 1554-1558. doi: 10.1109/TCSII.2020.3006430. |
[21] | B. Jin, T. Khan and P. Maass, A reconstruction algorithm for electrical impedance tomography based on sparsity regularization, International Journal for Numerical Methods in Engineering, 89 (2012), 337-353. doi: 10.1002/nme.3247. |
[22] | N. Johansson, Implementation of a standard level set method for incompressible two-phase flow simulations, Dissertation, 2011. Available at https://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-154651. |
[23] | J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, Springer, 2005. doi: 10.1007/b138659. |
[24] | J. P. Kaipio, V. Kolehmainen, E. 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. |
[25] | K. Knudsen, M. Lassas, J. 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. |
[26] | W. R. B. Lionheart, EIT reconstruction algorithms: Pitfalls, challenges and recent developments, Physiological Measurement, 25 (2004), 125. doi: 10.1088/0967-3334/25/1/021. |
[27] | J. L. Mueller and S. Siltanen, The D-bar method for electrical impedance tomography—demystified, Inverse Problems, 36 (2020), 093001, 28 pp. doi: 10.1088/1361-6420/aba2f5. |
[28] | S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics, 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2. |
[29] | N. Otsu, A threshold selection method from gray-level histograms, IEEE Transactions on Systems, Man, and Cybernetics, 9 (1979), 62-66. doi: 10.1109/TSMC.1979.4310076. |
[30] | F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, et al., Scikit-learn: Machine learning in Python, Journal of Machine Learning Research, 12 (2011), 2825-2830. |
[31] | D. Peng, B. Merriman, S. Osher, H. Zhao and M. Kang, A PDE-based fast local level set method, Journal of Computational Physics, 155 (1999), 410-438. doi: 10.1006/jcph.1999.6345. |
[32] | M. Räsänen, P. Kuusela, J. Jauhiainen, M. Arif, K. Scheel, T. Savolainen and A. Seppänen, Kuopio Tomography Challenge 2023 open electrical impedance tomographic dataset (KTC 2023), 2024. |
[33] | M. Räsänen, P. Kuusela and J. Scheel, Kuopio Tomography Challenge 2023., https://www.fips.fi/KTC2023.php. Accessed: 2024-07-01. |
[34] | O. Ronneberger, P. Fischer and T. Brox, U-net: Convolutional networks for biomedical image segmentation, in Medical Image Computing and Computer-Assisted Intervention–MICCAI 2015: 18th International Conference, Munich, Germany, October 5-9, 2015, Proceedings, Part III 18, Springer, 2015,234-241. doi: 10.1007/978-3-319-24574-4_28. |
[35] | F. Santosa, A level-set approach for inverse problems involving obstacles, ESAIM: Control, Optimisation and Calculus of Variations, 1 (1995/96), 17-33. doi: 10.1051/cocv:1996101. |
[36] | J. A. Sethian, Level Set Methods and Fast Marching Methods, volume 3 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, second edition, 1999. Evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science. |
[37] | S. Siltanen, J. 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. |
[38] | E. Somersalo, M. 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. |
[39] | T. N. Tallman and D. J. Smyl, Structural health and condition monitoring via electrical impedance tomography in self-sensing materials: A review, Smart Materials and Structures, 29 (2020), 123001. doi: 10.1088/1361-665X/abb352. |
[40] | P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, et al., SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python, Nature Methods, 17 (2020), 261-272. doi: 10.1038/s41592-019-0686-2. |
[41] | C. R. Vogel and M. E. Oman, Iterative methods for total variation denoising, SIAM Journal on Scientific Computing, 17 (1996), 227-238. doi: 10.1137/0917016. |
[42] | H. Wang, J. A. Huisman, E. Zimmermann and H. Vereecken, Experimental design to reduce inductive coupling in spectral electrical impedance tomography (sEIT) measurements, Geophysical Journal International, 225 (2021), 222-235. doi: 10.1093/gji/ggaa594. |
[43] | Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Transactions on Image Processing, 13 (2004), 600-612. doi: 10.1109/TIP.2003.819861. |
[44] | R. Winkler and A. Rieder, Resolution-controlled conductivity discretization in electrical impedance tomography, SIAM Journal on Imaging Sciences, 7 (2014), 2048-2077. doi: 10.1137/140958955. |
(a) An EIT setup from the University of Eastern Finland [19]. This is not the exact setup used to generate the data. (b) Conceptual illustration of the computational domain
Reconstructions from training data of a phantom, labeled Training Phantom 1 in the figure, using the Baseline method for difficulty levels 1, 4 and 7. Top row: Unsegmented reconstructions. Bottom row: Segmented reconstructions using Otsu segmentation. The numbers above the segmented reconstructions are the SSIM scores. The red dots mark the center of the electrodes. Yellow marks the conductive inclusion, dark blue marks the resistive, and teal green marks the background
Reconstructions from training data of a phantom, labeled Training Phantom 4 in the figure, using the Baseline method for difficulty levels 1, 4 and 7. Top row: Unsegmented reconstructions. Bottom row: Segmented reconstructions using Otsu segmentation. The numbers above the segmented reconstructions are the SSIM scores. The red dots mark the center of the electrodes. Yellow marks the conductive inclusion, dark blue marks the resistive, and teal green marks the background
Visual representation of the spatial regularization imposed by the weights matrix
The three-step process used to segment the reconstruction
Reconstructions from the training data of Training Phantom 1 using the SR method and the SR-CV method for difficulty levels 1, 4 and 7. Top row: Unsegmented reconstructions. Middle row: Segmented reconstructions using Otsu segmentation. Bottom row: Segmented reconstructions using Chan–Vese segmentation. The numbers above the segmented reconstructions are the SSIM scores. The red dots mark the center of the electrodes. Yellow marks the conductive inclusion, dark blue marks the resistive, and teal green marks the background
Reconstructions from the training data of Training Phantom 4 using the SR method and the SR-CV method for difficulty levels 1, 4 and 7. Top row: Unsegmented reconstructions. Middle row: Segmented reconstructions using Otsu segmentation. Bottom row: Segmented reconstructions using Chan–Vese segmentation. The numbers above the segmented reconstructions are the SSIM scores. The red dots mark the center of the electrodes. Dark blue marks the resistive inclusion and teal green marks the background
Reconstructions from the training data of Training Phantom 1 (top row) and Training Phantom 4 (bottom row) using the SR-TV-LS method for difficulty levels 1, 4 and 7. The numbers above the segmented reconstructions are the SSIM scores. The red dots mark the center of the electrodes. Yellow marks the conductive inclusion, dark blue marks the resistive, and teal green marks the background
Reconstructions of the highest difficulty level from the evaluation data for three different phantoms, labeled Evaluation Phantom 7A, Evaluation Phantom 7B and Evaluation Phantom 7C in the figure, using the SR method (second column), SR-CV method (third column) and SR-TV-LS method (fourth column). The numbers above the segmented reconstructions are the SSIM scores. Yellow marks the conductive inclusion, dark blue marks the resistive, and teal green marks the background
Performance of our proposed methods reported by the SSIM scores for the reconstructions. (Left) our computed reconstruction scores from the training data for each method averaged over 4 cases for each level. The same set of 4 phantoms are used in each level. We also show the reconstruction score for the Baseline method. (Right) computed reconstruction scores [33] from the evaluation data for each method, averaged over 3 cases for each level. In each level, a different set of 3 phantoms is used. We also show the reconstruction score for the best-performing method in the competition [33]. We note the scores on evaluation data are drawn from the distributed competition results