Linearization-based direct reconstruction for EIT using triangular Zernike decompositions

Antti Autio; Henrik Garde; Markus Hirvensalo; Nuutti Hyvönen

doi:10.3934/ipi.2024040

Article Contents

2025, Volume 19, Issue 3: 456-478. Doi: 10.3934/ipi.2024040 open access

This issue Previous Article Next Article Minimizing quotient regularization model

Linearization-based direct reconstruction for EIT using triangular Zernike decompositions

1.
Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100, 00076 Helsinki, Finland
2.
Department of Mathematics, Aarhus University, Ny Munkegade 118, 8000 Aarhus C, Denmark

^*Corresponding author: Nuutti Hyvönen
^*Corresponding author: Nuutti Hyvönen

Received: April 2024

Revised: August 2024

Early access: October 2024

Published: June 2025

This work was supported by the Academy of Finland (decisions 353080, 353081, 358944) and the Aalto Science Institute (AScI). HG was supported by grant 10.46540/3120-00003B from Independent Research Fund Denmark | Natural Sciences.

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Abstract

This work implements and numerically tests the direct reconstruction algorithm introduced in [Garde & Hyvönen, SIAM J. Math. Anal., 56, 3588-3604, 2024] for two-dimensional linearized electrical impedance tomography. Although the algorithm was originally designed for a linearized setting, we numerically demonstrate its functionality when the input data is the corresponding change in the current-to-voltage boundary operator. Both idealized continuum model and practical complete electrode model measurements are considered in the numerical studies, with the examined domain being either the unit disk or a convex polygon. Special attention is paid to regularizing the algorithm and its connections to the singular value decomposition of a truncated linearized forward map, as well as to the explicit triangular structures originating from the properties of the employed Zernike polynomial basis for the conductivity.

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Mathematics Subject Classification: Primary: 65N21, 65N20; Secondary: 35R30, 35R25.

Citation:

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Figure 1. The structure of $ F^M $, for $ M = 10 $, as a block diagonal matrix with lower triangular blocks

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Figure 2. The radial components of certain right singular functions and the first hundred singular values of $ F^M $ for $ M = 32 $

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Figure 3. Reconstructions of a discoidal inclusion inside the unit disk from highly accurate data generated using a Möbius transformation with $ M = 32 $

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Figure 4. The accepted Zernike indices for the reconstructions in Fig 3

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Figure 5. Reconstructions of a wave-like conductivity perturbation in the unit disk with $ M = 32 $

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Figure 6. An SVD-based reconstruction of a smooth conductivity perturbation in a square for $ M = 32 $, $ \omega = 2 $ and $ 1\% $ of additive noise, resulting in $ p = 163 $ and the smallest accepted singular value $ 0.0099 $

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Figure 7. An SVD-based reconstruction of a smooth conductivity perturbation in a polygon for $ M = 32 $, $ \omega = 1 $ and $ 1\% $ of additive noise, resulting in $ p = 145 $ and the smallest accepted singular value $ 0.020 $

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Figure 8. An SVD-based reconstruction of a smooth conductivity perturbation from simulated CEM data with 32 electrodes, $ M = 16 $, $ \omega = 1 $ and $ 1\% $ of additive noise, resulting in $ p = 108 $ and the smallest accepted singular value $ 0.0016 $

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Figure 9. Three SVD-based reconstructions with $ M = 8 $ from real-world water tank difference data measured at 16 equiangular electrodes. For each target, the 'optimal' truncation index $ p $ has been chosen according to a visual inspection. Top: the targets. Bottom: the reconstructions

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Table 1. The real parts of the first nine right singular functions of $ F^M $ for $ M = 32 $

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Table 2. The imaginary parts of the first nine right singular functions of $ F^M $ for $ M = 32 $

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Table 3. The real parts of the first seven right singular functions of $ F^M $ that are linear combinations of Zernike polynomials for some already visited angular index $ j $ with $ M = 32 $

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Table 4. The imaginary parts of the first seven right singular functions of $ F^M $ that are linear combinations of Zernike polynomials for some already visited angular index $ j $ with $ M = 32 $

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References

[1]	A. Allers and F. Santosa, Stability and resolution analysis of a linearized problem in electrical impedance tomography, Inverse Problems, 7 (1991), 515-533. doi: 10.1088/0266-5611/7/4/003.
[2]	L. Borcea, Electrical impedance tomography, Inverse problems, 18 (2002), R99-R136. doi: 10.1088/0266-5611/18/6/201.
[3]	A. P. Calderón, On an inverse boundary value problem, in Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasil. Mat., Rio de Janeiro, 1980, 65-73.
[4]	M. Cheney, D. Isaacson and J. C. Newell, Electrical impedance tomography, SIAM Rev., 41 (1999), 85-101. doi: 10.1137/S0036144598333613.
[5]	K.-S. Cheng, D. Isaacson, J. S. Newell and D. G. Gisser, Electrode models for electric current computed tomography, IEEE Trans. Biomed. Eng., 36 (1989), 918-924. doi: 10.1109/10.35300.
[6]	T. A. Driscoll, Algorithm 756: A MATLAB toolbox for Schwarz–Christoffel mapping, ACM Trans. Math. Soft., 22 (1996), 168-186. doi: 10.1145/229473.229475.
[7]	H. Garde, Comparison of linear and non-linear monotonicity-based shape reconstruction using exact matrix characterizations, Inverse Probl. Sci. Eng., 26 (2018), 33-50. doi: 10.1080/17415977.2017.1290088.
[8]	H. Garde and N. Hyvönen, Mimicking relative continuum measurements by electrode data in two-dimensional electrical impedance tomography, Numer. Math., 147 (2021), 579-609. doi: 10.1007/s00211-020-01170-8.
[9]	H. Garde and N. Hyvönen, Series reversion in Calderón's problem, Math. Comp., 91 (2022), 1925-1953. doi: 10.1090/mcom/3729.
[10]	H. Garde and N. Hyvönen, Linearised Calderón problem: Reconstruction and Lipschitz stability for infinite-dimensional spaces of unbounded perturbations, SIAM J. Math. Anal., 56 (2024), 3588-3604. doi: 10.1137/23M1609270.
[11]	T. Gustafsson and G. D. McBain, scikit-fem: A Python package for finite element assembly, J. Open Source Softw., 5 (2020), 2369. doi: 10.21105/joss.02369.
[12]	B. Harrach and J. K. Seo, Exact shape-reconstruction by one-step linearization in electrical impedance tomography, SIAM J. Math. Anal., 42 (2010), 1505-1518. doi: 10.1137/090773970.
[13]	A. Hauptmann, V. Kolehmainen, N. M. Mach, T. Savolainen, A. Seppänen and S. Siltanen, Open 2{D} electrical impedance tomography data archive, 2017, arXiv: 1704.01178.
[14]	N. J. Higham, Accuracy and Stability of Numerical Algorithms, 2$^{nd}$ edition, SIAM, Philadelphia, 2002.
[15]	N. Hyvönen and L. Mustonen, Smoothened complete electrode model, SIAM J. Appl. Math., 77 (2017), 2250-2271. doi: 10.1137/17M1124292.
[16]	N. Hyvönen and L. Mustonen, Generalized linearization techniques in electrical impedance tomography, Numer. Math., 140 (2018), 95-120. doi: 10.1007/s00211-018-0959-1.
[17]	N. Hyvönen, L. Päivärinta and J. Tamminen, Enhancing D-bar reconstructions for electrical impedance tomography with conformal maps, Inverse Probl. Imaging, 12 (2017), 373-400. doi: 10.3934/ipi.2018017.
[18]	A. Lipponen, Se ppänen A and J. P. Kaipio, Electrical impedance tomography imaging with reduced-order model based on proper orthogonal decomposition, J. Electron. Imaging, 22 (2013), 023008. doi: 10.1117/1.JEI.22.2.023008.
[19]	Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin, 1992.
[20]	J. Saranen and G. Vainikko, Periodic Integral and Pseudodifferential Equations with Numerical Approximation, Springer-Verlag, Berlin, 2002.
[21]	E. Somersalo, M. Cheney and D. Isaacson, Existence and uniqueness for electrode models for electric current computed tomography, SIAM J. Appl. Math., 52 (1992), 1023-1040. doi: 10.1137/0152060.
[22]	L. N. Trefethen, Numerical computation of the Schwarz–Christoffel transformation, SIAM J. Sci. Stat. Comput., 1 (1980), 82-102. doi: 10.1137/0901004.
[23]	G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse Problems, 25 (2009), 123011. doi: 10.1088/0266-5611/25/12/123011.
[24]	M. Vauhkonen, Electrical Impedance Tomography with Prior Information, Ph.D thesis, University of Kuopio, 1997.
[25]	F. Zernike, Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode, Physica, 1 (1934), 689-704. doi: 10.1016/S0031-8914(34)80259-5.