The regularized monotonicity method: Detecting irregular indefinite inclusions

Henrik Garde; Stratos Staboulis

doi:10.3934/ipi.2019006

Article Contents

2019, Volume 13, Issue 1: 93-116. Doi: 10.3934/ipi.2019006

This issue Previous Article Recovering two coefficients in an elliptic equation via phaseless information Next Article Nonconvex TGV regularization model for multiplicative noise removal with spatially varying parameters

The regularized monotonicity method: Detecting irregular indefinite inclusions

Henrik Garde^1, and
Stratos Staboulis^2,

1.
Department of Mathematical Sciences, Aalborg University, Skjernvej 4A, 9220 Aalborg, Denmark
2.
Eniram Oy (A Wärtsilä company), Itälahdenkatu 22a, 00210 Helsinki, Finland

Received: December 31, 2017

Revised: July 31, 2018

Published: December 2018

This research is funded by grant 4002-00123 Improved Impedance Tomography with Hybrid Data from The Danish Council for Independent Research | Natural Sciences.

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Abstract

In inclusion detection in electrical impedance tomography, the support of perturbations (inclusion) from a known background conductivity is typically reconstructed from idealized continuum data modelled by a Neumann-to-Dirichlet map. Only few reconstruction methods apply when detecting indefinite inclusions, where the conductivity distribution has both more and less conductive parts relative to the background conductivity; one such method is the monotonicity method of Harrach, Seo, and Ullrich [17,15]. We formulate the method for irregular indefinite inclusions, meaning that we make no regularity assumptions on the conductivity perturbations nor on the inclusion boundaries. We show, provided that the perturbations are bounded away from zero, that the outer support of the positive and negative parts of the inclusions can be reconstructed independently. Moreover, we formulate a regularization scheme that applies to a class of approximative measurement models, including the Complete Electrode Model, hence making the method robust against modelling error and noise. In particular, we demonstrate that for a convergent family of approximative models there exists a sequence of regularization parameters such that the outer shape of the inclusions is asymptotically exactly characterized. Finally, a peeling-type reconstruction algorithm is presented and, for the first time in literature, numerical examples of monotonicity reconstructions for indefinite inclusions are presented.

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Mathematics Subject Classification: Primary: 35R30, 35Q60, 35R05; Secondary: 65N21.

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Figure 1. Illustration of case (a) in the proof of Theorem 2.3.(ⅱ).

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Figure 2. Profile of a conductivity distribution $\gamma$ which does not satisfy the assumption $\sup({{\kappa _ - }})<\inf(\gamma_0)$

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Figure 3. (a) Two dimensional numerical phantom with positive part ${{D_ + }}$ (square and pentagon) and negative part ${{D_ - }}$ (ball). (b) Reconstruction of ${{D_ + }}$ from noiseless datum with $\alpha_0 = 6.56\times 10^{-4}$. (c) Reconstruction of ${{D_ - }}$ from noiseless datum with $\alpha_0 = 2.36\times 10^{-3}$. (d) Reconstruction of ${{D_ + }}$ with $0.5\%$ noise and $\alpha_0 = 5.00\times 10^{-3}$. (e) Reconstruction of ${{D_ - }}$ with $0.5\%$ noise and $\alpha_0 = 2.30\times 10^{-3}$

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Figure 4. (a) Two dimensional numerical phantom with positive part ${{D_ + }}$ (wedge) and negative part ${{D_ - }}$ (ball). (b) Reconstruction of ${{D_ + }}$ from noiseless datum with $\alpha_0 = 9.00\times 10^{-4}$. (c) Reconstruction of ${{D_ - }}$ from noiseless datum with $\alpha_0 = 6.72\times 10^{-4}$. (d) Reconstruction of ${{D_ + }}$ with $0.5\%$ noise and $\alpha_0 = 4.20\times 10^{-3}$. (e) Reconstruction of ${{D_ - }}$ with $0.5\%$ noise and $\alpha_0 = 3.26\times 10^{-3}$

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Figure 5. (a) Three dimensional numerical phantom with positive part ${{D_ + }}$ (ball) and negative part ${{D_ - }}$ (L-shape). (b) Reconstruction of ${{D_ + }}$ from noiseless datum with $\alpha_0 = 7.50\times 10^{-5}$. (c) Reconstruction of ${{D_ - }}$ from noiseless datum with $\alpha_0 = 2.90\times 10^{-4}$. (d) Reconstruction of ${{D_ + }}$ with $0.5\%$ noise and $\alpha_0 = 2.40\times 10^{-4}$. (e) Reconstruction of ${{D_ - }}$ with $0.5\%$ noise and $\alpha_0 = 2.50\times 10^{-4}$

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Figure 6. For $k = 8$, $16$, and $32$ electrodes of size $\pi/k$, upper bounds of reconstructions of ${{D_ + }}$ are shown, using regularization parameter $\alpha_0 = 0$. ${{D_ + }}$ is the ball outlined on the right side of the domain and ${{D_ - }}$ is on the left

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Figure 7. For $k = 4, 5, \dots, 64$ electrodes of size $\pi/k$, the distance $d_k$ from $\partial\Omega$ to upper bounds of reconstructions of ${{D_ + }}$ is plotted (cf. Figure 6). $h = 2\pi/k$ is the corresponding maximal extended electrode diameter from [20] and [9,Theorems 2 and 3]

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References

[1]	A. Abubakar, T. M. Habashy, M. Li and J. Liu, Inversion algorithms for large-scale geophysical electromagnetic measurements, Inverse Problems, 25 (2009), Article ID 123012, 30pp. doi: 10.1088/0266-5611/25/12/123012.
[2]	L. Borcea, Electrical impedance tomography, Inverse Problems, 18 (2002), 99-136. doi: 10.1088/0266-5611/18/6/201.
[3]	T. Brander, M. Kar and M. Salo, Enclosure method for the p-Laplace equation, Inverse Problems, 31 (2015), Article ID 045001, 16pp. doi: 10.1088/0266-5611/31/4/045001.
[4]	M. Brühl, Explicit characterization of inclusions in electrical impedance tomography, SIAM Journal on Mathematical Analysis, 32 (2001), 1327-1341. doi: 10.1137/S003614100036656X.
[5]	M. Brühl and M. Hanke, Numerical implementation of two noniterative methods for locating inclusions by impedance tomography, Inverse Problems, 16 (2000), 1029-1042. doi: 10.1088/0266-5611/16/4/310.
[6]	M. Cheney, D. Isaacson and J. C. Newell, Electrical impedance tomography, SIAM Review, 41 (1999), 85-101. doi: 10.1137/S0036144598333613.
[7]	H. Garde, Comparison of linear and non-linear monotonicity-based shape reconstruction using exact matrix characterizations, Inverse Problems in Science and Engineering, 26 (2018), 33-50. doi: 10.1080/17415977.2017.1290088.
[8]	H. Garde and K. Knudsen, Distinguishability revisited: Depth dependent bounds on reconstruction quality in electrical impedance tomography, SIAM Journal on Applied Mathematics, 77 (2017), 697-720. doi: 10.1137/16M1072991.
[9]	H. Garde and S. Staboulis, Convergence and regularization for monotonicity-based shape reconstruction in electrical impedance tomography, Numerische Mathematik, 135 (2017), 1221-1251. doi: 10.1007/s00211-016-0830-1.
[10]	B. Gebauer, Localized potentials in electrical impedance tomography, Inverse Problems and Imaging, 2 (2008), 251-269. doi: 10.3934/ipi.2008.2.251.
[11]	N. Grinberg and A. Kirsch, The factorization method for obstacles with a-priori separated sound-soft and sound-hard parts, Mathematics and Computers in Simulation, 66 (2004), 267-279. doi: 10.1016/j.matcom.2004.02.011.
[12]	M. Hanke and M. Brühl, Recent progress in electrical impedance tomography, Inverse Problems, 19 (2003), S65-S90, Special section on imaging. doi: 10.1088/0266-5611/19/6/055.
[13]	M. Hanke-Bourgeois and A. Kirsch, Sampling methods, in Handbook of Mathematical Methods in Imaging, Springer, 2015,591-647.
[14]	B. Harrach and M. N. Minh, Enhancing residual-based techniques with shape reconstruction features in electrical impedance tomography Inverse Problems, 32 (2016), Article ID 125002, 21pp. doi: 10.1088/0266-5611/32/12/125002.
[15]	B. Harrach and J. K. Seo, Exact shape-reconstruction by one-step linearization in electrical impedance tomography, SIAM Journal on Mathematical Analysis, 42 (2010), 1505-1518. doi: 10.1137/090773970.
[16]	B. Harrach, Recent progress on the factorization method for electrical impedance tomography, Computational and Mathematical Methods in Medicine, 2013 (2013), Article ID 425184, 8pp. doi: 10.1155/2013/425184.
[17]	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.
[18]	B. Harrach and M. Ullrich, Resolution guarantees in electrical impedance tomography, IEEE Transactions on Medical Imaging, 34 (2015), 1513-1521. doi: 10.1109/tmi.2015.2404133.
[19]	D. S. Holder (ed.), Electrical Impedance Tomography: Methods, History, and Applications, IOP publishing Ltd., 2005.
[20]	N. Hyvönen, Approximating idealized boundary data of electric impedance tomography by electrode measurements, Mathematical Models and Methods in Applied Sciences, 19 (2009), 1185-1202. doi: 10.1142/S0218202509003759.
[21]	N. Hyvönen and L. Mustonen, Smoothened complete electrode model, SIAM Journal on Applied Mathematics, 77 (2017), 2250-2271. doi: 10.1137/17M1124292.
[22]	M. Ikehata, Size estimation of inclusion, Journal of Inverse and Ill-Posed Problems, 6 (1998), 127-140. doi: 10.1515/jiip.1998.6.2.127.
[23]	M. Ikehata, How to draw a picture of an unknown inclusion from boundary measurements. Two mathematical inversion algorithms, Journal of Inverse and Ill-Posed Problems, 7 (1999), 255-271. doi: 10.1515/jiip.1999.7.3.255.
[24]	M. Ikehata, Reconstruction of the support function for inclusion from boundary measurements, Journal of Inverse and Ill-Posed Problems, 8 (2000), 367-378. doi: 10.1515/jiip.2000.8.4.367.
[25]	M. Ikehata, A regularized extraction formula in the enclosure method, Inverse Problems, 18 (2002), 435-440. doi: 10.1088/0266-5611/18/2/309.
[26]	H. Kang, J. K. Seo and D. Sheen, The inverse conductivity problem with one measurement: Stability and estimation of size, SIAM Journal on Mathematical Analysis, 28 (1997), 1389-1405. doi: 10.1137/S0036141096299375.
[27]	K. Karhunen, A. Seppänen, A. Lehikoinen, J. Blunt, J. P. Kaipio and P. J. M. Monteiro, Electrical resistance tomography for assessment of cracks in concrete, Materials Journal, 107 (2010), 523-531. doi: 10.14359/51663973.
[28]	K. Karhunen, A. Seppänen, A. Lehikoinen, P. J. M. Monteiro and J. P. Kaipio, Electrical resistance tomography imaging of concrete, Cement and Concrete Research, 40 (2010), 137-145.
[29]	A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, USA, 2008.
[30]	A. Lechleiter, A regularization technique for the factorization method, Inverse problems, 22 (2006), 1605-1625. doi: 10.1088/0266-5611/22/5/006.
[31]	A. Lechleiter, N. Hyvönen and H. Hakula, The factorization method applied to the complete electrode model of impedance tomography, SIAM Journal on Applied Mathematics, 68 (2008), 1097-1121. doi: 10.1137/070683295.
[32]	C. Miranda, Partial Differential Equations of Elliptic Type, Springer Berlin Heidelberg, second edition, 1970. doi: 10.1007/978-3-662-35147-5.
[33]	W. F. Osgood, A Jordan curve of positive area, Transactions of the American Mathematical Society, 4 (1903), 107-112. doi: 10.1090/S0002-9947-1903-1500628-5.
[34]	S. Schmitt, The factorization method for EIT in the case of mixed inclusions, Inverse Problems, 25 (2009), Article ID 065012, 20pp. doi: 10.1088/0266-5611/25/6/065012.
[35]	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.
[36]	A. Tamburrino, Monotonicity based imaging methods for elliptic and parabolic inverse problems, Journal of Inverse and Ill-posed Problems, 14 (2006), 633-642. doi: 10.1515/156939406778474578.
[37]	A. Tamburrino and G. Rubinacci, A new non-iterative inversion method for electrical resistance tomography, Inverse Problems, 18 (2002), 1809-1829. doi: 10.1088/0266-5611/18/6/323.
[38]	L. N. Trefethen and M. Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, Princeton University Press, Princeton, NJ, 2005.
[39]	G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse Problems, 25 (2009), Article ID 123011, 39pp. doi: 10.1088/0266-5611/25/12/123011.
[40]	T. York, Status of electrical tomography in industrial applications, Journal of Electronic Imaging, 10 (2001), 608-619. doi: 10.1117/1.1377308.