# American Institute of Mathematical Sciences

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

## Electrical impedance tomography with multiplicative regularization

 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.

Citation: Ke Zhang, Maokun Li, Fan Yang, Shenheng Xu, Aria Abubakar. Electrical impedance tomography with multiplicative regularization. Inverse Problems & Imaging, 2019, 13 (6) : 1139-1159. doi: 10.3934/ipi.2019051
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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
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 $v$
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
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 $\tau^0$
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
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
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
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
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
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)
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
Reconstructed images on the electrode plane using the multiplicative weighted $L$2-norm regularization and synthetic data from the 3D thorax model shown in Figure 11(a)
Structural similarity (SSIM) index calculated for the reconstruction results in Section 5 (Figure 6 and 9)
 2D numerical example 1 2D numerical example 2 Noise level 0% 1% 3% 5% 0% 1% 2% 3% MR-WL2$^*$ 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
 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
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 $\alpha$ $1\times10^{-3}$ $1\times10^{-3}$ $1\times10^{-3}$ $1\times10^{-3}$ $\beta$ $1\times10^{-24}$ $1\times10^{-23}$ $1\times10^{-20}$ $1\times10^{-18}$
 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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