February  2019, 13(1): 93-116. doi: 10.3934/ipi.2019006

The regularized monotonicity method: Detecting irregular indefinite inclusions

1. 

Department of Mathematical Sciences, Aalborg University, Skjernvej 4A, 9220 Aalborg, Denmark

2. 

Eniram Oy (A Wärtsilä company), Itälahdenkatu 22a, 00210 Helsinki, Finland

Received  January 2018 Revised  August 2018 Published  December 2018

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

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.

Citation: Henrik Garde, Stratos Staboulis. The regularized monotonicity method: Detecting irregular indefinite inclusions. Inverse Problems & Imaging, 2019, 13 (1) : 93-116. doi: 10.3934/ipi.2019006
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.  Google Scholar

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[6]

M. CheneyD. Isaacson and J. C. Newell, Electrical impedance tomography, SIAM Review, 41 (1999), 85-101.  doi: 10.1137/S0036144598333613.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

B. Gebauer, Localized potentials in electrical impedance tomography, Inverse Problems and Imaging, 2 (2008), 251-269.  doi: 10.3934/ipi.2008.2.251.  Google Scholar

[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.  Google Scholar

[12]

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

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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.  Google Scholar

[21]

N. Hyvönen and L. Mustonen, Smoothened complete electrode model, SIAM Journal on Applied Mathematics, 77 (2017), 2250-2271.  doi: 10.1137/17M1124292.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

H. KangJ. 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.  Google Scholar

[27]

K. KarhunenA. SeppänenA. LehikoinenJ. BluntJ. 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.  Google Scholar

[28]

K. KarhunenA. SeppänenA. LehikoinenP. J. M. Monteiro and J. P. Kaipio, Electrical resistance tomography imaging of concrete, Cement and Concrete Research, 40 (2010), 137-145.   Google Scholar

[29]

A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, USA, 2008.  Google Scholar

[30]

A. Lechleiter, A regularization technique for the factorization method, Inverse problems, 22 (2006), 1605-1625.  doi: 10.1088/0266-5611/22/5/006.  Google Scholar

[31]

A. LechleiterN. 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.  Google Scholar

[32]

C. Miranda, Partial Differential Equations of Elliptic Type, Springer Berlin Heidelberg, second edition, 1970. doi: 10.1007/978-3-662-35147-5.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

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

[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.  Google Scholar

[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.  Google Scholar

[38]

L. N. Trefethen and M. Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, Princeton University Press, Princeton, NJ, 2005.  Google Scholar

[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.  Google Scholar

[40]

T. York, Status of electrical tomography in industrial applications, Journal of Electronic Imaging, 10 (2001), 608-619.  doi: 10.1117/1.1377308.  Google Scholar

show all references

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.  Google Scholar

[2]

L. Borcea, Electrical impedance tomography, Inverse Problems, 18 (2002), 99-136.  doi: 10.1088/0266-5611/18/6/201.  Google Scholar

[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.  Google Scholar

[4]

M. Brühl, Explicit characterization of inclusions in electrical impedance tomography, SIAM Journal on Mathematical Analysis, 32 (2001), 1327-1341.  doi: 10.1137/S003614100036656X.  Google Scholar

[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.  Google Scholar

[6]

M. CheneyD. Isaacson and J. C. Newell, Electrical impedance tomography, SIAM Review, 41 (1999), 85-101.  doi: 10.1137/S0036144598333613.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

B. Gebauer, Localized potentials in electrical impedance tomography, Inverse Problems and Imaging, 2 (2008), 251-269.  doi: 10.3934/ipi.2008.2.251.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

M. Hanke-Bourgeois and A. Kirsch, Sampling methods, in Handbook of Mathematical Methods in Imaging, Springer, 2015,591-647.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[21]

N. Hyvönen and L. Mustonen, Smoothened complete electrode model, SIAM Journal on Applied Mathematics, 77 (2017), 2250-2271.  doi: 10.1137/17M1124292.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

H. KangJ. 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.  Google Scholar

[27]

K. KarhunenA. SeppänenA. LehikoinenJ. BluntJ. 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.  Google Scholar

[28]

K. KarhunenA. SeppänenA. LehikoinenP. J. M. Monteiro and J. P. Kaipio, Electrical resistance tomography imaging of concrete, Cement and Concrete Research, 40 (2010), 137-145.   Google Scholar

[29]

A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, USA, 2008.  Google Scholar

[30]

A. Lechleiter, A regularization technique for the factorization method, Inverse problems, 22 (2006), 1605-1625.  doi: 10.1088/0266-5611/22/5/006.  Google Scholar

[31]

A. LechleiterN. 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.  Google Scholar

[32]

C. Miranda, Partial Differential Equations of Elliptic Type, Springer Berlin Heidelberg, second edition, 1970. doi: 10.1007/978-3-662-35147-5.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

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

[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.  Google Scholar

[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.  Google Scholar

[38]

L. N. Trefethen and M. Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, Princeton University Press, Princeton, NJ, 2005.  Google Scholar

[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.  Google Scholar

[40]

T. York, Status of electrical tomography in industrial applications, Journal of Electronic Imaging, 10 (2001), 608-619.  doi: 10.1117/1.1377308.  Google Scholar

Figure 1.  Illustration of case (a) in the proof of Theorem 2.3.(ⅱ).
Figure 2.  Profile of a conductivity distribution $\gamma$ which does not satisfy the assumption $\sup({{\kappa _ - }})<\inf(\gamma_0)$
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 $\alpha_0 = 2.30\times 10^{-3}$
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 $\alpha_0 = 3.26\times 10^{-3}$
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 $\alpha_0 = 2.50\times 10^{-4}$
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
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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