A wavelet frame constrained total generalized variation model for imaging conductivity distribution

Tian Zhiwei; Shi Yanyan; Wang Meng; Kong Xiaolong; Li Lei; Fu Feng

doi:10.3934/ipi.2021074

Article Contents

2022, Volume 16, Issue 4: 753-769. Doi: 10.3934/ipi.2021074

This issue Previous Article A new approach to the inverse discrete transmission eigenvalue problem Next Article

A wavelet frame constrained total generalized variation model for imaging conductivity distribution

1.
College of Electronic and Electrical Engineering, Henan Normal University, Henan Key Laboratory of Optoelectronic Sensing Integrated Application, Xinxiang, 453007, China
2.
School of Biomedical Engineering, Fourth Military Medical University, Xi'an, 710032, China

^*Corresponding author: Meng wang, Feng Fu
^*Corresponding author: Meng wang, Feng Fu

^†Y. Shi and Z. Tian contributed equally to this work

Received: June 30, 2021

Revised: July 31, 2021

Early access: December 2021

Published: August 2022

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Abstract

Electrical impedance tomography (EIT) is a sensing technique with which conductivity distribution can be reconstructed. It should be mentioned that the reconstruction is a highly ill-posed inverse problem. Currently, the regularization method has been an effective approach to deal with this problem. Especially, total variation regularization method is advantageous over Tikhonov method as the edge information can be well preserved. Nevertheless, the reconstructed image shows severe staircase effect. In this work, to enhance the quality of reconstruction, a novel hybrid regularization model which combines a total generalized variation method with a wavelet frame approach (TGV-WF) is proposed. An efficient mean doubly augmented Lagrangian algorithm has been developed to solve the TGV-WF model. To demonstrate the effectiveness of the proposed method, numerical simulation and experimental validation are conducted for imaging conductivity distribution. Furthermore, some comparisons are made with typical regularization methods. From the results, it can be found that the proposed method shows better performance in the reconstruction since the edge of the inclusion can be well preserved and the staircase effect is effectively relieved.

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Mathematics Subject Classification: Primary: 78A46.

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Figure 1. Configuration of the EIT system

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Figure 2. Comparison of the convergence

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Figure 3. Reconstructed images for inclusions with the same conductivity

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Figure 4. Reconstructed images for inclusions with different conductivities

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Figure 5. Comparison of different methods in suppressing the staircase effect

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Figure 6. Reconstructed images for inclusions under the noise level of 5%

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Figure 7. Reconstructed images for inclusions under the noise level of 10%

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Figure 8. Reconstructed images for experimental phantom

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Table 1. Comparisons of Ree and Coc for inclusions with the same conductivity


Model	TV	TGV	TGV-WF
Model	Ree/Coc	Ree/Coc	Ree/Coc
1	0.3513/0.8868	0.3428/0.8962	0.3380/0.9126
2	0.3798/0.8748	0.3698/0.8804	0.3502/0.8978
3	0.4071/0.8765	0.3814/0.8963	0.3754/0.9103

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Table 2. Comparisons of Ree and Coc for inclusions with different conductivities


Model	TV	TGV	TGV-WF
Model	Ree/Coc	Ree/Coc	Ree/Coc
4	0.3824/0.8619	0.3726/0.8735	0.3681/0.8904
5	0.4326/0.8022	0.4158/0.8202	0.4006/0.8356

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Table 3. The values of Ree and Coc under 5% noise level


Model	TV	TGV	TGV-WF
Model	Ree/Coc	Ree/Coc	Ree/Coc
1	0.3756/0.8753	0.3548/0.8850	0.3426/0.9026
2	0.3863/0.8648	0.3759/0.8688	0.3652/0.8864
3	0.4129/0.8653	0.3958/0.8806	0.3884/0.9021
4	0.3901/0.8493	0.3786/0.8652	0.3712/0.8865
5	0.4504/0.7899	0.4296/0.8057	0.4153/0.8286

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Table 4. The values of Ree and Coc under 10% noise level


Model	TV	TGV	TGV-WF
Model	Ree/Coc	Ree/Coc	Ree/Coc
1	0.3865/0.8693	0.3627/0.8768	0.3528/0.8934
2	0.3923/0.8521	0.3806/0.8602	0.3714/0.8813
3	0.4315/0.8523	0.4021/0.8712	0.3996/0.8923
4	0.4082/0.8203	0.3864/0.8486	0.3794/0.8746
5	0.4712/0.7396	0.4355/0.7863	0.4251/0.8093

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