Regularized full waveform inversion for low frequency ultrasound tomography with a structural similarity EIT prior

Scott Ziegler; Talles Santos; Jennifer L. Mueller

doi:10.3934/ipi.2023023

Article Contents

2024, Volume 18, Issue 1: 86-103. Doi: 10.3934/ipi.2023023

This issue Previous Article An inverse problem for a semi-linear wave equation: A numerical study Next Article A uniqueness theorem for inverse problems in quasilinear anisotropic media II

Regularized full waveform inversion for low frequency ultrasound tomography with a structural similarity EIT prior

1.
Department of Mathematics, Colorado State University, USA
2.
School of Biomedical Engineering, Colorado State University, USA

^*Corresponding author: Jennifer L. Mueller
^*Corresponding author: Jennifer L. Mueller

Received: October 2022

Revised: March 2023

Early access: May 2023

Published: February 2024

The project was supported by Award Number R21EB024683 from the National Institute of Biomedical Imaging and Bioengineering. The content is solely the responsibility of the authors and does not necessarily represent the official view of the National Institute of Biomedical Imaging and Bioengineering or the National Institutes of Health.

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Abstract

In previous work by Rueter et al., low frequency (10–750 kHz) ultrasound was shown to penetrate the lung, motivating its use as a nonionizing tomographic technique for pulmonary monitoring. Here, we present a method for regularized full waveform inversion for low-frequency tomographic ultrasound data. A novel structural similarity index metric (SSIM)-based regularization term is introduced that compares the structural correlation of the current sound speed iterate to a prior reconstruction computed by electrical impedance tomography (EIT). Full waveform reconstructions of sound speeds from numerically simulated low frequency ultrasound data with 0.1% additive Gaussian noise are computed, and the results are compared using Tikhonov regularization, total variation regularization, and both terms combined with the SSIM-EIT regularization term. The EIT reconstruction was computed from simulated voltage data on the same phantom using one step of a Newton-Raphson method with a high-pass Gaussian filter as a regularizer. Reconstructions including the SSIM-EIT regularization term converged fastest, and an adaptive regularization parameter provided the most accurate reconstructions. The method is then iterated by computing an EIT reconstruction using the USCT result as a prior, and a subsequent USCT reconstruction is computed using the SSIM-EIT regularization term with the updated EIT reconstruction.

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Mathematics Subject Classification: Primary: 35R30, 65N21; Secondary: 78A46.

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Figure 1. The transducer locations surrounding the lung phantom (left) and the filtered, transmitted ultrasonic signal sent through each transducer in succession (right)

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Figure 2. EIT phantom and conductivity reconstruction. Note that the conductivity values are negative because these represent difference images $ \sigma -\sigma_0 $

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Figure 3. Left: The Tikhonov+SSIM-EIT USCT image used to construct the prior for the OSGN EIT reconstruction. Center: The threshold segmentation with assigned conductivity values in mS/cm for the construction of the prior. Right: OSGN EIT reconstruction using equation (13)

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Figure 4. Left: The TV+SSIM-EIT USCT image used to construct the prior for the OSGN EIT reconstruction. Center: The threshold segmentation with assigned conductivity values in mS/cm for the construction of the prior. Right: OSGN EIT reconstruction using equation (13)

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Figure 5. (A)–(D): USCT sound speed reconstructions for four different regularization terms. The reconstruction with the optimal choice of regularization parameter is plotted. (E)–(F): Comparison of the error term $ E(c) $ and $ L^2 $ error for the best performing reconstructions using the four different regularization terms

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Figure 6. (A)–(B): USCT sound speed reconstructions using a second iteration of the Tikhonov+SSIM-EIT prior at iteration 14 and TV+SSIM-EIT prior at iteration 5 of Algorithm 1 with an adaptive regularization term. (C)–(D): error measures for the same reconstructions

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Table 1. USCT and EIT parameters in the numerical lung phantom

Structure Index	Tissue Type	Sound Speed (m/s)	Conductivity (mS/m)
1	Healthy Lung	1440	109.0
2	Pleural Effusion	1500	275.1
3	Heart	1560	222.1
4	Spine	1640	22.9
5	Adipose	1532	367.1
6	Background	1540	0

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Table 2. Normal distribution parameters of the conductivity region variations

Organ	Mean (S/m)	Std. (S/m)
Lungs	0.0	0.240
Heart	0.0	0.192
Domain	0.0	0.140

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Table 3. Threshold segmentation

Organ	Threshold	Region
Lungs	$ \text{Ref. Image}< {1490} $	2
Heart	$ {1580} < \text{Ref. Image} < {1610} $	3
Domain	-	1

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