The Supplementary Materials are uploaded, which include the Noise robust analysis, Resolution analysis, and more visualization results.
If the PDF cannot be viewed directly, the Baidu cloud link of Supplementary Materials is:https://pan.baidu.com/s/1mSp8xj9ai-u3MY6TOHQaHw?pwd=prsm Key: prsm
Electrical Impedance Tomography (EIT) reconstruction is a typical inverse problem with the characteristic of ill-posedness and nonlinearity. The reconstructed image based on sensitivity prior suffers from significant shape distortions due to the 'soft-field' nature and the limited measurements. Moreover, the existing learning-based methods still have drawbacks in expressing the complicated boundary and the multiphase distributions because of the insufficient features representation by spatial convolution. In this paper, we proposed a novel dual-domain architecture, called Spatial plus Wavelet Dual-domain Empowering (SpW-Net), to achieve high-quality EIT reconstruction, which learns the shape features and conductivity distributions in both the spatial and wavelet domains. First, the residual fast Fourier convolution is proposed to construct the spatial latent features globally. Second, a (shifted) window-based wavelet vision Transformer is introduced to not only capture multiscale and multifrequency information but also reduces the complexity efficiently. Finally, a dual-domain feature fusion block is proposed to fuse the latent representations in the spatial and wavelet domain adaptively with the attention-wise manner. Extensive experiments on both simulated and real-world data illustrate that SpW-Net performs better regarding visual results and quantitative evaluations (the Root Mean Squared Error (RMSE) is 3.972±1.931 and the Structural Similarity Index (SSIM) is 0.982±0.010). Further, the reconstructions of our method are more consistent with the ground truth compared with state-of-the-art methods on the public benchmark Kuopio Tomography Challenge 2023 (KTC2023). The SpW-Net could obtain better EIT reconstruction results, especially in complicated shapes and multiphase distributions.
The performances of our SpW-Net and other comparisons are listed as follows:
The performance of visualizations and quantitative evaluations. Left: the representative results (visualized reconstructions and quantitative metrics) of KTC-2023 test data, including multiphase inclusion distributions and complicated shapes. The compared methods include the proposed SpW-Net and CNN/Transformer-based architectures. The best results are marked as 'red.' Right: The SSIM v.s. Computational complexity (GFLOPs). The up-left area indicates better performance with higher SSIM and fewer GFLOPs. Overall, we can observe that SpW-Net offers improving visualizations and metrics and less GPU consumption with faster imaging speed.
The overall Framework of SpW-Net is:
Overview of our SpW-Net framework with hierarchical dual-branch Encoder and Decoder structure. The measured voltages from the sensors are reshaped to VFM (16×16) and expanded to the HR-VFM (256×256) in a spatial-domain branch. Four layers of up-sampling obtain the initial imaging results in the spatial domain, and the ConvNeXt layer (shown in the green dot box) and the wavelet domain are obtained by the ISTA solver. The focuses of our design are residual fast Fourier convolution (RFFC), shifted/window-based wavelet vision Transformer (W2Former), and dual-domain feature fusion block (DFFB), where the details of these three core designs are illustrated in Section III-B, Section III-C, and Section III-D in the main paper. Finally, the output image is obtained by the initial reconstruction in the spatial domain and the results of the Decoder.
The code will be released once this paper is accepted.
In the following two Figures, the more representative reconstructions are shown, where the visual results and absolute error maps are shown. The absolute error maps between reconstructions and ground truth to could highlight the differences in visual perceptions and display the texture details. As shown in these two groups of visualizations and error maps, the proposed SpW-Net is capable of reconstructing clearer and sharper boundaries and accurate electrical distributions.
In the following figure, a circular tank with 16 electrodes is used for real-world experiments. The multiphase conductivity distributions are performed utilizing food materials (cucumber, carrot, and sugarcane and injected with different concentrations of NaCl solution to change the conductivity of the materials), as well as the healthy/injured medical reconstructions are performed using lung-shaped phantoms made from agar with different conductivities. The water with a conductivity value of 0.6 S/m is used as a homogeneous background, and different inclusions are placed in the tank to set as the inhomogeneous states for obtaining the measured voltages.
(1) Measurement noise: To evaluate the robustness of SpW-Net under different levels of noise, a series of Gaussian white noise is added to the simulation test dataset, in which the SNR levels for these additive noises are set in the range from 70 to 20 dB. The robustness of our SpW-Net and four learning-based models (SWISTA-Net, IDT, DnSwin, and EWT) is compared separately. We evaluate the performance of SpW-Net and methods using average PSNR and SSIM on various noise levels, and the results are shown. It can be seen that the performance of all methods decreases as the SNR level decreases, SpW-Net outperforms the other methods in terms of both PSNR and SSIM for all levels of noise. In detail, the RMSE is lower than 5.0 as well as SSIM is higher than 0.95 of SpW-Net when the SNR is lower than 40 dB. Due to the test model being trained under the noise-free datasets, these results indicate that our SpW-Net with spatial and wavelet hybrid domains could effectively suppress the artifacts due to systemic perturbations. Moreover, the multi-scale and low/high-frequency sub-band fusion strategies in SpW-Net could enhance the valuable information describing the conductivities and shape features simultaneously.
(2) Electrode movements: Some other perturbations have to be considered for addressing practical noise issues, such as the electrode movement that may occur in medical applications. In this paper, the 1, 2, 4, and 8 electrodes random movements are considered, respectively. To better illustrate the results, our SpW-Net is compared with two SOTA imaging methods including PDCISTA-Net and EWT. The visual reconstructions are shown. The visual perceptions in Figure S2 show that our SpW-Net has more robust performances under different electrode movements. The major reasons include (1) the learnable pre-reconstruction is introduced in wavelet-domain, which couples the VFM features with spatial locations of sensor placements, (2) the spatial and wavelet domains could enhance the focused inclusions as well as suppress the artifacts, which highlight the high-frequency sub-bands and the final results have sharper boundary features.
