Post-processing electrical impedance tomography reconstructions with incomplete data using convolutional neural networks

Roberto G. Beraldo; Leonardo A. Ferreira; Fernando S. Moura; André K. Takahata; Ricardo Suyama

doi:10.3934/ammc.2024008

Article Contents

2024, Volume 2, Issue 2: 140-164. Doi: 10.3934/ammc.2024008

This issue Previous Article Data-driven approaches for electrical impedance tomography image segmentation from partial boundary data Next Article Spatial regularization and level-set methods for experimental electrical impedance tomography with partial data

Post-processing electrical impedance tomography reconstructions with incomplete data using convolutional neural networks

1.
Information Engineering, Center for Engineering, Modeling and Applied Social Sciences, Federal University of ABC - 09210-580, Av. dos Estados, 5001 - Santo André - SP, Brazil
2.
AI R&D Team, Samsung R&D Institute Brazil - 13097-160, Av. Cambacicas, 610 - Parque dos Resedás, Campinas - SP, Brazil
3.
Biomedical Engineering, Center for Engineering, Modeling and Applied Social Sciences, Federal University of ABC - 09606-045, Alameda da Universidade, s/nº - São Bernardo do Campo - SP, Brazil

^*Corresponding author: Roberto G. Beraldo
^*Corresponding author: Roberto G. Beraldo

Received: March 2024

Revised: May 2024

Early access: June 2024

Published: June 2024

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Abstract

Electrical impedance tomography (EIT) is a technique to obtain conductivity maps from electrical voltage measurements in a region of interest. In this work, we discuss the proposal we submitted to the Kuopio Tomography Challenge 2023, whose aim was to reconstruct and segment EIT images obtained from limited data, after electrode disconnection. Our proposal, denoted 01A, consisted of an initial reconstruction using the smoothness prior and post-processing steps, including denoising and deblurring with a convolutional neural network (CNN), as a way to integrate deep learning and inverse problems. The score was calculated using the structural similarity index in 21 test cases. While the score of the reconstruction using only smoothness prior was $ 9.69 $, the score of 01A was $ 12.75 $. Also, we developed an improved proposal, denoted 01A+, using hyperparameter optimization and its score was $ 13.30 $. We obtained better results using 01A and 01A+ than using the original smoothness prior, but the proposals lacked consistency when more electrodes were disconnected and when the targets were too different from the CNN training data. Even so, 01A obtained second place at KTC2023, representing a way to remove artifacts from electrode disconnection in EIT reconstructions.

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Mathematics Subject Classification: 68T07, 68U10.

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Figure 1. Level 1: Available reference voltage $ \mathbf{v}^m_{ref} $

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Figure 2. Level 7: Available reference voltage $ \mathbf{v}^m_{ref} $

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Figure 3. KTC training set: All samples available

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Figure 4. Test set samples

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Figure 5. Overview of the proposal. The ground truth image and the segmented result from Step 6 follow the color scheme defined in section 3.1.2

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Figure 6. Sensibility analysis: $ \mathbf{J}_{\ell_1} $ from Level 1

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Figure 7. Sensibility analysis: $ \mathbf{J}_{\ell_1} $ from Level 7

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Figure 8. Singular values of the Jacobian of the EIT forward operator for each level

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Figure 9. CNN training set samples: Inputs and outputs [AU]

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Figure 10. Step-by-step visualization of the proposal 01A

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Figure 11. Step-by-step visualization of the proposal 01A

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Figure 12. Step-by-step visualization of the proposal 01A

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Figure 13. Results on the test set: $ S_1^A $

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Figure 14. Results on the test set: $ S_3^B $

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Figure 15. Results on the test set: $ S_6^A $

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Figure 16. Failure cases: Ground truth (top row) and 01A reconstructions (bottom row)

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Figure 17. Example of results that were improved with the use of the best set of hyperparameter values found in the optimization

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Figure 18. Example of results that were worsened with the use of the best set of hyperparameter values found in the optimization

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Figure 19. CNN architecture visualization

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Figure 20. Influence of $ k_C $ on 01A ($ S_T^B $, level 7)

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Table 1. Rank of the Jacobian matrix for each difficulty level

Level	1	2	3	4	5	6	7
$ \mathbf{J}(\mathit{\boldsymbol{\sigma}}_0) $ rank	376	333	293	254	219	183	152

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Table 2. Results of SSIM. Values are presented as "Mean (standard deviation)" per level

Method/level	1	2	3	4	5	6	7
00X (Example code)	0.80	0.67	0.50	0.39	0.35	0.19	0.32
00X (Example code)	(0.09)	(0.06)	(0.34)	(0.07)	(0.25)	(0.03)	(0.32)
01A (2^nd place)	0.92	0.79	0.69	0.58	0.36	0.51	0.41
01A (2^nd place)	(0.11)	(0.10)	(0.19)	(0.14)	(0.24)	(0.21)	(0.28)
01B (Alternative)	0.87	0.72	0.56	0.41	0.28	0.49	0.42
01B (Alternative)	(0.10)	(0.05)	(0.24)	(0.26)	(0.25)	(0.15)	(0.26)
02G (3^rd place)	0.76	0.79	0.62	0.52	0.45	0.48	0.53
02G (3^rd place)	(0.32)	(0.02)	(0.38)	(0.08)	(0.21)	(0.17)	(0.26)
06B (1^st place)	0.92	0.85	0.85	0.57	0.69	0.64	0.56
06B (1^st place)	(0.05)	(0.04)	(0.10)	(0.03)	(0.20)	(0.05)	(0.29)

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Table 3. Results of SSIM considering all levels together

Method	Mean	Std. dev.	Sum	Max.	Min.	Improvement
00X	0.46	0.27	9.69	0.87	0.06	–
01A	0.61	0.25	12.75	0.98	0.17	$ 15\% $
01B	0.54	0.26	11.25	0.97	0.12	$ 5\% $
02G	0.59	0.24	12.45	0.95	0.21	$ 13\% $
06B	0.73	0.18	15.24	0.95	0.30	$ 27\% $

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Table 4. Range of hyperparameters considered in the automatic optimization

Hyperparameter	Reference equation	Search Space
$ \lambda $	(14)	Floats between 0.1 and 100
$ k_T $	(16)	Floats between 0 and 0.85
$ k_C $	(18)	Floats between 0.1 and 5
$ r $	(20)	Integers between 5 and 25

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Table 5. Convolutional neural network architecture

Parameter	DIP
Input and output sizes	$ 64 \times 64 \times 1 $
Max. No. of filters (Conv2D)	256^a
Min. No. of filters (Conv2D)	64^a
Filters size in each layer (Conv2D)	$ 3 \times 3 $ (all)
Activation functions	ReLu
Trainable parameters	1108097
^a except the last layer.

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