Multibang regularization for electrical impedance tomography

Christina Brandt; Thi Bich Tram Do; Patrick Horn

doi:10.3934/ammc.2024007

Article Contents

2024, Volume 2, Issue 2: 208-227. Doi: 10.3934/ammc.2024007

This issue Previous Article An efficient hierarchical Bayesian method for the Kuopio tomography challenge 2023 Next Article A boundary integral equations approach to electrical impedance tomography: Experiments on the KTC2023 data

Multibang regularization for electrical impedance tomography

1.
University of Greifswald, Institute of Mathematics and Computer Science, Walther-Rathenau-Straße 47, 17487 Greifswald, Germany
2.
Institute of Biological and Medical Imaging, Helmholtz Zentrum München, Ingolstädter Landstraße 1, 85764 Neuherberg, Germany
3.
Chair of Biological Imaging at the Central Institute for Translational Cancer Research (TranslaTUM), School of Medicine and Health, Technical University of Munich, Ismaninger Str. 22, 81675 Munich, Germany
4.
Department of Mathematics, Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany

^*Corresponding author: Thi Bich Tram Do
^*Corresponding author: Thi Bich Tram Do

Received: May 2024

Revised: May 2024

Early access: June 2024

Published: June 2024

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Abstract

This study is concerned with a regularization approach for electrical impedance tomography in the Kuopio Tomography Challenge 2023. As the unknown conductivity in the electrical impedance tomography setting is known to take values on a given discrete set, we propose multibang regularization as a suitable method to recover this parameter from measurement. The numerical solution for the data set provided by the organizers of the challenge will be derived by a primal-dual splitting method. Additionally, we will have a discussion about the reconstruction when adding total variation in the regularizer.

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Mathematics Subject Classification: Primary: 49N45, 65J22; Secondary: 65J20, 65K99.

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Figure 1. Structure of the pointwise multibang penalty for $ (\sigma_1,\sigma_2,\sigma_3) = (0,1,2) $, its subdifferential and the Fenchel conjugate of its subdifferential

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Figure 2. Phantoms of the training data (first row) as well as the reconstruction for level 1 of our 3 approaches: reconstruction with multibang penalty (second row), segmentation with multibang regularization (third row) and segmentation with multibang & TV regularization (4th row)

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Figure 3. Reconstruction of phantom 4: Tikhonov reconstruction, TV regularized reconstruction and multibang reconstruction

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Figure 4. Segmentation of the training data with multibang regularization of a base reconstruction with Tikhonov regularization (12)

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Figure 5. Reconstruction and ground truth for phantom A, B, C of the validation data from difficulty level 1 (first row) till level 7 (7th row) for the segmentation approach with multibang regularization

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Figure 6. Reconstruction and ground truth for phantom A, B, C of the validation data from difficulty level 1 (first row) till level 7 (7th row) for the segmentation approach with multibang & TV regularization

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Table 1. Dual functions and proximal operators for the operator splitting. We denote with $ P_C $ the orthogonal projection operator onto the convex and closed set $ C $ and with $ ＄_\beta $ the soft shrinkage operator. With $ B_{\| {\cdot} \|_\infty,\beta}(0) $ we denote the closed ball around $ 0 $ with radius $ \beta $ with respect to the $ \| {\cdot} \|_\infty $-norm

$ f(u) $	$ f^*(p) $	$ {\rm{prox}}_{\tau f} (z) $	$ {\rm{prox}}_{\tau f^*}(z) $
$ \beta \\| u \\| _1 $	$ \delta_{B_{\\| {\cdot} \\|_\infty, \beta}(0)}(p) $	$ ＄_\beta (z) $	$ \mathrm{P}_{B_{\\| {\cdot} \\|_\infty, \beta}(0)}(z) $
$ \frac 1 2 \\| {u-y} \\|_{\Gamma_{\Delta n}^{-1}}^2 $	$ \frac 1 2 \\| {p} \\|_{\Gamma_{\Delta n}}^2 + \langle p, y \rangle $	$ (I + \tau \Gamma_{\Delta n}^{-1})^{-1}(z + \tau \Gamma_{\Delta n}^{-1} y) $	$ (I + \tau \Gamma_{\Delta n})^{-1}(z - \tau y) $
$ \frac 1 2 \\| {u-y} \\|_2^2 $	$ \frac 1 2 \\| {p} \\|_2^2 + \langle p, y \rangle $	$ \frac{z + \tau y}{1 + \tau} $	$ \frac{z-\tau y}{1 + \tau} $

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Table 2. SSIM values for the validation data set of our three methods. The method 3 was overall the best rated approach, however, we underlined those values which are lower than at least one of those of the two other methods. Moreover, we overlined those SSIM values of method 1 which are better than those of method 3

method	sample	level 1	level 2	level 3	level 4	level 5	level 6	level 7
1	$ S_{j}^{A} $	$ 0.60 $	$ 0.53 $	$ 0.52 $	$ 0.22 $	$ 0.04 $	$ \overline{0.54} $	$ 0.18 $
	$ S_{j}^{B} $	$ 0.60 $	$ 0.52 $	$ 0.60 $	$ 0.09 $	$ 0.53 $	$ 0.11 $	$ 0.07 $
	$ S_{j}^{C} $	$ 0.55 $	$ 0.55 $	$ 0.50 $	$ 0.22 $	$ 0.02 $	$ \overline{0.19} $	$ 0.52 $
2	$ S_{j}^{A} $	$ 0.87 $	$ 0.92 $	$ 0.07 $	$ 0.54 $	$ 0.34 $	$ 0.27 $	$ 0.02 $
	$ S_{j}^{B} $	$ 0.90 $	$ 0.28 $	$ 0.84 $	$ 0.39 $	$ 0.07 $	$ 0.03 $	$ 0.09 $
	$ S_{j}^{C} $	$ 0.84 $	$ 0.91 $	$ 0.59 $	$ 0.27 $	$ 0.35 $	$ -0.01 $	$ 0.68 $
3	$ S_{j}^{A} $	$ 0.91 $	$ \underline{0.22} $	$ 0.66 $	$ 0.43 $	$ 0.24 $	$ \underline{0.18} $	$ 0.22 $
	$ S_{j}^{B} $	$ 0.84 $	$ 0.61 $	$ \underline{0.39} $	$ \underline{0.24} $	$ 0.60 $	$ 0.26 $	$ 0.22 $
	$ S_{j}^{C} $	$ \underline{0.24} $	$ 0.70 $	$ 0.51 $	$ 0.45 $	$ \underline{0.13} $	$ \underline{0.16} $	$ 0.69 $

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