American Institute of Mathematical Sciences

Advanced
December  2019, 13(6): 1139-1159. doi: 10.3934/ipi.2019051

Electrical impedance tomography with multiplicative regularization

Ke Zhang 1, , Maokun Li 1,, , Fan Yang 1, , Shenheng Xu 1, and  Aria Abubakar 2,

1. 

State Key Laboratory on Microwave and Digital Communications, Beijing National Research Center for Information Science and Technology (BNRist), Department of Electronic Engineering, Tsinghua University, Beijing 100084, China
2. 

Schlumberger, Houston, TX 77056, USA

* Corresponding author: Maokun Li

Received  November 2017 Revised  June 2019 Published  October 2019

Fund Project: This work was supported in part by the National Science Foundation of China under Grant 61571264, in part by the National Key R & D Program of China under Grant 2018YFC0603604, in part by the Guangzhou Science and Technology Plan under Grant 201804010266, in part by the Beijing Innovation Center for Future Chip, and in part by the Research Institute of Tsinghua, Pearl River Delta.

It is known that EIT inversion is an ill-posed problem, meaning that the solution is unstable if noise exists in the measured data. Generally, a regularization scheme is needed to alleviate the ill-posedness. In this work, a multiplicative regularization scheme is applied to EIT inversion. In this regularization scheme, a cost functional is constructed in which the data misfit functional is multiplied by a regularization factor, and no regularization parameter is needed. The regularization factor is based on the weighted $ L2 $-norm favoring 'blocky' profiles in the reconstructed images. Gauss–Newton method is used to minimize the cost functional iteratively. In the implementation of the multiplicative regularization scheme, the spatial gradient and divergence need to be computed on triangular meshes. For this purpose, the discrete exterior calculus (DEC) theory is applied to formulate the related discrete operators. Numerical and experimental results show good anti-noise performance of the multiplicative regularization scheme in EIT inverse problem.

Keywords: Discrete exterior calculus (DEC), electrical impedance tomography (EIT), finite-element meshes, Gauss–Newton inversion, multiplicative regularization, spatial derivatives, thorax imaging, total variation (TV).
Mathematics Subject Classification: Primary: 65N21.
Citation: Ke Zhang, Maokun Li, Fan Yang, Shenheng Xu, Aria Abubakar. Electrical impedance tomography with multiplicative regularization. Inverse Problems and Imaging, 2019, 13 (6) : 1139-1159. doi: 10.3934/ipi.2019051
References:
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show all references

References:
[1]

A. Abbasi, B. V. Vahdat and G. E. Fakhim, An inverse solution for 2D electrical impedance tomography based on electrical properties of material blocks, Journal of Applied Sciences, 9 (2009), 1962-1967.  doi: 10.3923/jas.2009.1962.1967.

[2]

A. Abbasi and B. V. Vahdat, A non-iterative linear inverse solution for the block approach in EIT, Journal of Computational Science, 1 (2010), 190-196.  doi: 10.1016/j.jocs.2010.09.001.

[3]

A. Adler and R. Guardo, Electrical impedance tomography: Regularized imaging and contrast detection, IEEE Trans. Med. Imag., 15 (1996), 170-179.  doi: 10.1109/42.491418.

[4]

A. Adler et al., GREIT: A unified approach to 2D linear EIT reconstruction of lung images, Physiol. Meas., 30 (2009), S35.

[5]

M. Alsaker and J. L. Mueller, A D-bar algorithm with a priori information for 2-dimensional electrical impedance tomography, SIAM J. Imaging Sci., 9 (2016), 1619-1654.  doi: 10.1137/15M1020137.

[6]

M. Alsaker, S. J. Hamilton and A. Hauptmann, A direct D-bar method for partial boundary data electrical impedance tomography with a priori information, Inverse Probl. Imag., 11 (2017), 427-454.  doi: 10.3934/ipi.2017020.

[7]

A. Borsic, B. M. Graham, A. Adler and W. R. B. Lionheart, In vivo impedance imaging with total variation regularization, IEEE Trans. Med. Imag., 29 (2010), 44-54.  doi: 10.1109/TMI.2009.2022540.

[8]

X. Chen, Ill-posed problems and regularization, Computational Methods for Electromagnetic Inverse Scattering, John Wiley & Sons, (2018), 281–289. doi: 10.1002/9781119311997.app1.

[9]

D. Isaacson, M. Cheney, J. C. Newell, S. Simske and J. Goble, NOSER: An algorithm for solving the inverse conductivity problem, Int. J. Imag. Syst. Technol., 2 (1990), 66-75. 

[10]

M. Cheney, D. Isaacson and J. C. Newell, Electrical impedance tomography, SIAM Rev., 41 (1999), 85-101.  doi: 10.1137/S0036144598333613.

[11]

K.-S. Cheng, D. Isaacson, J. C. Newell and D. G. Gisser, Electrode models for electric current computed tomography, IEEE Trans. Biomed. Eng., 36 (1989), 918-924. 

[12]

E. T. Chung, T. F. Chan and X. C. Tai, Electrical impedance tomography using level set representation and total variational regularization, J. Comput. Phy., 205 (2005), 357-372.  doi: 10.1016/j.jcp.2004.11.022.

[13]

C. Cohen-Bacrie, Y. Goussard and R. Guardo, Regularized reconstruction in electrical impedance tomography using a variance uniformization constraint, IEEE Trans. Med. Imag., 16 (1997), 562-571. 

[14]

E. L. V. Costa, J. B. Borges, A. Melo, F. Suarez-Sipmann, C. Toufen, S. H. Bohm and M. B. P. Amato, Bedside estimation of recruitable alveolar collapse and hyperdistension by electrical impedance tomography, Intens. Care Med., 2012,165–170. doi: 10.1007/978-3-642-28270-6_34.

[15]

K. Crane, Discrete Differential Geometry: An Applied Introduction, Carnegie Mellon University, Pittsburgh, USA, 2018.

[16]

D. C. Dobson and F. Santosa, An image-enhancement technique for electrical impedance tomography, Inverse Probl., 10 (1994), 317-334.  doi: 10.1088/0266-5611/10/2/008.

[17]

G. Franchineau, N. Brechot, G. Lebreton, G. Hekimian, A. Nieszkowska, J. Trouillet, P. Leprince, J. Chastre, C. Luyt, A. Combes and M. Schmidt, Bedside contribution of electrical impedance tomography to setting positive end-expiratory pressure for extracorporeal membrane oxygenation-treated patients with severe acute respiratory distress syndrome, Am. J. Respir. Crit. Care Med., 196 (2017), 447-457.  doi: 10.1164/rccm.201605-1055OC.

[18]

I. Frerichs, J. Hinz, P. Herrmann, G. Weisser, G. Hahn, M. Quintel and G. Hellige, Regional lung perfusion as determined by electrical impedance tomography in comparison with electron beam CT imaging, IEEE Trans. Med. Imag., 21 (2002), 646-652.  doi: 10.1109/TMI.2002.800585.

[19]

I. Frerichs, Z. Zhao, T. Becher, P. Zabel, N. Weiler and B. Vogt, Regional lung function determined by electrical impedance tomography during bronchodilator reversibility testing in patients with asthma, Physiol. Meas., 37 (2016), 698-712.  doi: 10.1088/0967-3334/37/6/698.

[20]

N. P. Galatsanos and A. K. Katsaggelos, Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation, IEEE Trans. Image Process., 1 (1992), 322-336.  doi: 10.1109/83.148606.

[21]

B. Gebauer and N. Hyvönen, Factorization method and irregular inclusions in electrical impedance tomography, Inverse Probl., 23 (2007), 2159-2170.  doi: 10.1088/0266-5611/23/5/020.

[22]

M. Gehre, T. Kluth, A. Lipponen, B. Jin, A. Seppänen, J. P. Kaipio and P. Maass, Sparsity reconstruction in electrical impedance tomography: An experimental evaluation, J. Comput. Appl. Math., 236 (2012), 2126-2136.  doi: 10.1016/j.cam.2011.09.035.

[23]

G. H. Golub, M. Heath and G. Wahba, Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics, 21 (1979), 215-223.  doi: 10.1080/00401706.1979.10489751.

[24]

G. González, J. M. J. Huttunen, V. Kolehmainen, A. Seppänen and M. Vauhkonen, Experimental evaluation of 3D electrical impedance tomography with total variation prior, Inverse Probl. Sci. En., 24 (2016), 1411-1431. 

[25]

G. González, V. Kolehmainen and A. Seppänen, Isotropic and anisotropic total variation regularization in electrical impedance tomography, Comput. Math. Appl., 74 (2017), 564-576.  doi: 10.1016/j.camwa.2017.05.004.

[26]

T. M. Habashy and A. Abubakar, A general framework for constraint minimization for the inversion of electromagnetic measurements, Prog. Electromagn. Res., 46 (2004), 265-312.  doi: 10.2528/PIER03100702.

[27]

G. Hahn, A. Just, T. Dudykevych, I. Frerichs, J. Hinz, M. Quintel and G. Hellige, Imaging pathologic pulmonary air and fluid accumulation by functional and absolute EIT, Physiol. Meas., 27 (2006), S187–S198. doi: 10.1088/0967-3334/27/5/S16.

[28]

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Figure 1.  Simplices and their circumcentric duals in 2D space. The Greek letter $ \tau $ is used to represent the simplex. The superscript of a simplex is its dimension. In the diagrams of the duals of the simplices, the notation '$ \star $' is the star duality operator, and the solid points inside the triangles are their circumcenters
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Figure 2.  An illustration of the computation of the discrete gradient at the dual of Triangle $ \tau^2 $. The three vertices of Triangle $ \tau^2 $ are represented by $ \tau_1^0 $, $ \tau_2^0 $, and $ \tau_3^0 $. Here we choose Vertex $ \tau_3^0 $ as the reference vertex $ v $
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Figure 3.  A 1-ring of a vertex $ \tau^0 $ to illustrate the computation of the discrete divergence at $ \tau^0 $. The shaded region is the Voronoi region of Vertex $ \tau^0 $. The edges sharing Vertex $ \tau^0 $ are pointing outwards
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Figure 4.  A 1-ring of a vertex $ \tau^0 $ to illustrate the computation of the discrete Laplacian at $ \tau^0 $. All the information needed is the conductivity values at Vertex $ \tau^0 $ and the vertices adjacent to it, as well as the geometry of the mesh. The shaded region is the Voronoi region of Vertex $ \tau^0 $
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Figure 5.  Two-dimensional thoracic numerical model and the inversion mesh. (a) The model. Conductivity settings: lungs: 0.05 S/m, heart: 0.2 S/m, background: 0.15 S/m. Sixteen electrodes are around the boundary (shown as the bolded line segments). (b) An arbitrary mesh used for inversion
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Figure 6.  Reconstruction results of the model shown in Figure 5(a) using two different kinds of regularization schemes: multiplicative weighted $ L2 $-norm regularization (MR-WL2, first row) and additive TV regularization (AR-TV, second row). The first, second, third, and fourth columns correspond to the reconstruction results using data with 0%, 1%, 3%, and 5% noise, respectively. All the reconstructions are at the 12th iterations
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Figure 7.  Comparison of convergence curves for the multiplicative weighted $ L $2-norm regularization (MR-WL2) and the additive TV regularization (AR-TV). The curves correspond to the numerical example shown in Figure 6
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Figure 8.  Two-dimensional thoracic numerical model with pleural effusion in the dorsal part of the right lung. The conductivity settings are the same as those in the model shown in Figure 5(a) except that the conductivity of the pleural effusion area is 0.2 S/m
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Figure 9.  Reconstruction results of the model shown in Figure 8 using two different kinds of regularization schemes: multiplicative weighted $ L2 $-norm regularization (MR-WL2, first row) and additive TV regularization (AR-TV, second row). The first, second, third, and fourth columns correspond to the reconstruction results using data with 0%, 1%, 2%, and 3% noise, respectively. All the reconstructions are at the 12th iterations
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Figure 10.  Reconstruction results using measured data. The first column shows the photos of the phantoms. Phantom 2.2: two plastic inclusions; Phantom 3.1: three plastic inclusions; Phantom 4.1: one metallic (circular, hollow) and one plastic (triangular) inclusions; Phantom 5.2: two metallic (circular, hollow) and one plastic (triangular) inclusions. The second column shows the reconstruction results corresponding to each phantom using the multiplicative weighted $ L $2-norm regularization (MR-WL2). The third column shows the reconstruction results corresponding to each phantom using the additive TV regularization (AR-TV)
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Figure 11.  Three-dimensional thoracic numerical model and the inversion mesh. The tetrahedral elements in the upper part of the meshes are masked in order to show the interior structures. (a) The model. Conductivity settings: lungs: 0.05 S/m, heart: 0.2 S/m, background: 0.15 S/m. Sixteen circular electrodes are around the same height of the boundary. (b) An arbitrary mesh used for inversion
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Figure 12.  Reconstructed images on the electrode plane using the multiplicative weighted $ L $2-norm regularization and synthetic data from the 3D thorax model shown in Figure 11(a)
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Table 1.  Structural similarity (SSIM) index calculated for the reconstruction results in Section 5 (Figure 6 and 9)
2D numerical example 1 2D numerical example 2
Noise level 0% 1% 3% 5% 0% 1% 2% 3%
MR-WL2$ ^* $ 0.9532 0.9007 0.8845 0.8718 0.9571 0.9107 0.8894 0.8807
AR-TV$ ^\dagger $ 0.9591 0.8954 0.8755 0.8530 0.9581 0.9005 0.8805 0.8636
* Multiplicative weighted L2-norm regularization
† Additive total variation regularization
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2D numerical example 1 2D numerical example 2
Noise level 0% 1% 3% 5% 0% 1% 2% 3%
MR-WL2$ ^* $ 0.9532 0.9007 0.8845 0.8718 0.9571 0.9107 0.8894 0.8807
AR-TV$ ^\dagger $ 0.9591 0.8954 0.8755 0.8530 0.9581 0.9005 0.8805 0.8636
* Multiplicative weighted L2-norm regularization
† Additive total variation regularization
Table 2.  Parameter values of the additive TV regularization in the reconstructions using the tank data
Phantom 2.2 Phantom 3.1 Phantom 4.1 Phantom 5.2
$ \alpha $ $ 1\times10^{-3} $ $ 1\times10^{-3} $ $ 1\times10^{-3} $ $ 1\times10^{-3} $
$ \beta $ $ 1\times10^{-24} $ $ 1\times10^{-23} $ $ 1\times10^{-20} $ $ 1\times10^{-18} $
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Phantom 2.2 Phantom 3.1 Phantom 4.1 Phantom 5.2
$ \alpha $ $ 1\times10^{-3} $ $ 1\times10^{-3} $ $ 1\times10^{-3} $ $ 1\times10^{-3} $
$ \beta $ $ 1\times10^{-24} $ $ 1\times10^{-23} $ $ 1\times10^{-20} $ $ 1\times10^{-18} $
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