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3D Electrical Impedance Tomography reconstructions from simulated electrode data using direct inversion $ \mathbf{t}^{\rm{{\textbf{exp}}}} $ and Calderón methods

  • * Corresponding author: Peter A. Muller

    * Corresponding author: Peter A. Muller 

The first author is supported by NIH Award R21EB028064. The third and fifth authors are supported by the Academy of Finland, the Jane and Aatos Erkko Foundation and Neurocenter Finland

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  • The first numerical implementation of a $ \mathbf{t}^{\rm{{\textbf{exp}}}} $ method in 3D using simulated electrode data is presented. Results are compared to Calderón's method as well as more common TV and smoothness regularization-based methods. The $ \mathbf{t}^{\rm{{\textbf{exp}}}} $ method for EIT is based on tailor-made non-linear Fourier transforms involving the measured current and voltage data. Low-pass filtering in the non-linear Fourier domain is used to stabilize the reconstruction process. In 2D, $ \mathbf{t}^{\rm{{\textbf{exp}}}} $ methods have shown great promise for providing robust real-time absolute and time-difference conductivity reconstructions but have yet to be used on practical electrode data in 3D, until now. Results are presented for simulated data for conductivity and permittivity with disjoint non-radially symmetric targets on spherical domains and noisy voltage data. The 3D $ \mathbf{t}^{\rm{{\textbf{exp}}}} $ and Calderón methods are demonstrated to provide comparable quality to their 2D counterparts and hold promise for real-time reconstructions due to their fast, non-optimized, computational cost.

     

    Erratum: The name of the fifth author has been corrected from Jussi Toivainen to Jussi Toivanen. We apologize for any inconvenience this may cause.

    Mathematics Subject Classification: Primary: 65N21, 94A08; Secondary: 78A46, 92C55.

    Citation:

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  • Figure 1.  Demonstration of the 3D $ \mathbf{t}^{\rm{{\textbf{exp}}}} $ Method and Calderón's Method on the 'Heart and Lungs' phantom T2-B, shown on left, using simulated electrode data. The 2D cross-sectional slices above show that the conductive heart is correctly visible in the $ x_1x_3 $ plane but absent from the $ x_2x_3 $ plane. Similarly for the lungs in the $ x_2x_3 $ plane vs the $ x_1x_3 $ plane

    Figure 2.  The simulated targets considered in this manuscript

    Figure 3.  The domains used for data simulation showing electrode locations and electrode numbering

    Figure 4.  Comparison of reconstructed conductivity (Left) and Fourier data, $ \hat{F} $, (Right) for T1 using Calderón's method (equation (16)). $ T_z = 2.7 $ for the analytic data and $ T_z = 1.3 $ for all three simulated electrode data cases. The mollifying parameter is $ t = 0.1 $ for both analytic and simulated electrode data. The vertical dashed line indicates where the Fourier domain was truncated for the simulated electrode data cases

    Figure 5.  Reconstructions of radially symmetric example T1 across algorithms using $ L = 128 $ electrodes shown in 3D and a representative $ x_2x_3 $ slice

    Figure 6.  Comparison of conductivity and susceptivity reconstructions for the complex-valued heart and lungs target T2-A

    Figure 7.  Comparison of reconstructions for the real-valued heart and lungs target T2-B using $ L = 128 $, $ 64 $, or $ 32 $ electrodes

    Figure 8.  Comparison of reconstructions for the high contrast target T3 using $ L = 128 $, $ 64 $, or $ 32 $ electrodes

    Figure 9.  Comparison of reconstructions for the real-valued heart and lungs target T2-B with increasing levels of noise added to the voltage data. Segmented 3D isosurface renderings are shown for each reconstruction as well as the $ x_1x_2 $ cross-sectional slice

    Figure 10.  Comparison of reconstructions for the high contrast target T3 using various levels of noise. Segmented 3D isosurface renderings are shown for each reconstruction as well as the $ x_1x_2 $ cross-sectional slice

    Figure 11.  Whole-image evaluation metrics for the real-valued heart and lungs target T2-B with decreasing numbers of simulated electrodes. Left: Dynamic Range, Middle: Mean Square Error, Right: Multi-Scale Structural Similarity Index

    Figure 12.  Whole-image evaluation metrics for target T3 with decreasing numbers of simulated electrodes. Left: Dynamic Range, Middle: Mean Square Error, Right: Multi-Scale Structural Similarity Index

    Figure 13.  Whole-image evaluation metrics for the real-valued heart and lungs target T2-B with increasing levels of noise added to the voltage data. Left: Dynamic Range, Middle: Mean Square Error, Right: Multi-Scale Structural Similarity Index

    Figure 14.  Whole-image evaluation metrics for the target T3 with increasing levels of noise added to the voltage data. Left: Dynamic Range, Middle: Mean Square Error, Right: Multi-Scale Structural Similarity Index

    Table 5.  T2-B evaluation metrics across all electrode configurations considered. Lung 1 is the resistive target with the larger volume

    $ \mathbf{t}^{\rm{{\textbf{exp}}}} $ Calder #243;n Smooth TV
    LE heart L=128 0.1545 0.1296 0.0086 0.0171
    L=64 0.2416 0.1899 0.0070 0.0176
    L=32 0.2424 0.1837 0.0115 0.0035
    lung 1 L=128 0.1099 0.1216 0.0130 0.0194
    L=64 0.1627 0.1628 0.0084 0.0118
    L=32 N/A N/A 0.0040 0.0098
    lung 2 L=128 0.1171 0.1531 0.0144 0.0109
    L=64 0.2221 0.2172 0.0069 0.0085
    L=32 N/A N/A 0.0135 0.0068
    RVR heart L=128 0.7526 1.1204 0.6520 0.9152
    L=64 1.0722 1.1101 0.7737 0.9869
    L=32 2.9867 1.9449 1.2085 1.2273
    lung 1 L=128 0.3977 0.6102 0.6360 0.7102
    L=64 0.3389 0.5475 0.7038 0.8359
    L=32 N/A N/A 0.9249 0.9318
    lung 2 L=128 0.3312 0.2990 0.6342 0.6600
    L=64 0.2311 0.3047 0.6908 0.8039
    L=32 N/A N/A 0.8597 0.8590
     | Show Table
    DownLoad: CSV

    Table 1.  Evaluation metrics for the high-contrast example T3 across all electrode configurations considered

    $ \mathbf{t}^{\rm{{\textbf{exp}}}} $ Calderon Smooth TV
    LE conductor L=128 0.1267 0.1181 0.0264 0.0191
    L=64 0.1522 0.1566 0.0085 0.0136
    L=32 0.1559 0.1548 0.0111 0.0093
    resistor L=128 0.0853 0.0861 0.0128 0.0074
    L=64 0.1072 0.1107 0.0050 0.0061
    L=32 0.1465 0.1062 0.0057 0.0109
    RVR conductor L=128 0.4672 0.4690 0.4122 0.6124
    L=64 0.5357 0.4343 0.4456 0.5490
    L=32 0.5883 0.5580 0.5492 0.6912
    resistor L=128 1.6747 1.6273 0.9951 1.0290
    L=64 1.9706 1.6022 1.0895 1.0479
    L=32 2.1047 2.0931 1.4676 1.1714
     | Show Table
    DownLoad: CSV

    Table 6.  Evaluation metrics for T2-B with 0.01%, 0.1% and 1% noise with $ 128 $ electrodes

    $ \mathbf{t}^{\rm{{\textbf{exp}}}} $ Calderon Smooth TV
    LE heart 0.01% noise 0.1451 0.1494 0.0096 0.0158
    0.1% noise 0.2372 0.2147 0.0184 0.0286
    1% noise 0.2061 0.1520 N/A 0.0770
    lung 1 0.01% noise 0.1294 0.1201 0.0138 0.0209
    0.1% noise 0.1541 0.1540 0.0125 0.0213
    1% noise N/A N/A N/A 0.0513
    lung 2 0.01% noise 0.1097 0.1336 0.0141 0.0111
    0.1% noise 0.1245 0.1498 0.0210 0.0220
    1% noise N/A N/A N/A 0.0444
    RVR heart 0.01% noise 1.3702 1.0072 0.6605 0.9279
    0.1% noise 2.5372 1.3738 1.0518 1.1021
    1% noise 0.6135 3.7011 N/A 1.0894
    lung 1 0.01% noise 0.5548 0.6647 0.6603 0.7192
    0.1% noise 1.0583 0.7483 0.9256 0.4780
    1% noise N/A N/A N/A 0.9519
    lung 2 0.01% noise 0.4064 0.3625 0.6504 0.6581
    0.1% noise 1.1779 0.5656 0.9098 0.4665
    1% noise N/A N/A N/A 0.7636
     | Show Table
    DownLoad: CSV

    Table 2.  Evaluation metrics for the high-contrast example T3 with 0.01%, 0.1% and 1% noise with $ 128 $ electrodes

    $ \mathbf{t}^{\rm{{\textbf{exp}}}} $ Calderon Smooth TV
    LE conductor 0.01% noise 0.1303 0.1155 0.0292 0.0204
    0.1% noise 0.1044 0.1391 0.0127 0.0262
    1% noise 0.2501 0.1854 0.0611 0.0551
    resistor 0.01% noise 0.0887 0.0814 0.0134 0.0080
    0.1% noise 0.1444 0.1046 0.0171 0.0046
    1% noise 0.2193 0.1930 0.1166 0.0529
    RVR conductor 0.01% noise 0.3992 0.4405 0.3866 0.5726
    0.1% noise 0.6590 0.5265 0.4967 0.7483
    1% noise 1.1334 0.8591 0.4805 0.7573
    resistor 0.01% noise 1.2038 1.6083 0.9931 1.0125
    0.1% noise 3.1351 2.6043 1.6627 1.4349
    1% noise 3.4953 2.2805 1.4426 2.0613
     | Show Table
    DownLoad: CSV

    Table 3.  Evaluation metrics for T1 with 128 electrodes

    $ \mathbf{t}^{\rm{{\textbf{exp}}}} $ Calderón Smooth TV
    DR 95.25% 116.89% 162.62% 143.16%
    MSE 0.0305 0.0373 0.0141 0.0124
    MS-SSIM 0.8126 0.8324 0.8984 0.8978
    LE 0.0008 0.0008 0.0021 0.0007
    RVR 0.4435 0.3734 0.5088 0.6255
     | Show Table
    DownLoad: CSV

    Table 4.  Evaluation metrics for T2-A for 128 electrode data. Lung 1 is the resistive target with the largest volume

    $ \mathbf{t}^{\rm{{\textbf{exp}}}} $ Calderón} Smooth TV
    DR Re 50.19% 59.53% 92.59% 101.22%
    Im 98.02% 62.35% 85.80% 88.88%
    MSE Re 0.0097 0.0110 0.0042 0.0038
    Im 0.0022 0.0022 0.0011 0.0010
    MS-SSIM Re 0.8235 0.8193 0.7841 0.8179
    Im 0.8956 0.8710 0.8505 0.8558
    LE heart Re 0.1302 0.1172 0.0113 0.0263
    Im 0.1751 0.2133 0.1179 0.0949
    lung 1 Re 0.1245 0.1401 0.0059 0.0105
    Im 0.1199 0.0843 0.0708 0.0412
    lung 2 Re 0.1425 0.1746 0.0141 0.0067
    Im 0.1715 0.1290 0.0934 0.0538
    RVR heart Re 0.8211 0.8112 1.0418 0.8497
    Im 0.2798 0.4819 1.0119 0.6845
    lung 1 Re 0.9114 1.0072 1.4399 1.2159
    Im 1.1566 0.6418 0.9389 0.5171
    lung 2 Re 0.8381 0.8041 1.4425 1.2289
    Im 1.3116 0.5547 0.5568 0.1976
     | Show Table
    DownLoad: CSV
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