A direct D-bar method for partial boundary data electrical impedance tomography with a priori information

Figure 1.  Example simulating a patient with a pneumothorax in the left lung. The simulated noisy measurement is collected from 75% ventral data. The first image displays the true conductivity with the position of electrodes indicated. Using a partial data D-bar approach alone results in a reconstruction with low spatial resolution, where the pathology can be hardly seen (second). Incorporating a priori data corresponding to a healthy patient directly into the reconstruction method significantly improves the spatial resolution (third). Refining the prior improves the reconstruction further, allowing even sharper visualization of the pathology (fourth)

Figure 2.  Illustration of mappings involved in the measurement modeling. Top row: Neumann data with the basis function $\varphi(\theta)=\cos(\theta)/\sqrt{\pi}$ on the left and the nonorthogonal projection $J\varphi$ on the right. Bottom row: Dirichlet data where $g=u|_{\partial\Omega}$ on the left is the solution of the partial differential equation (1) and on the right the orthogonal projection to the extended electrodes

Figure 3.  The A Priori D-bar Method with Partial Data

Figure 4.  Phantoms used in numerical examples with the corresponding boundaries of the priors outlined by white dots. Note that for each example, the prior does not assume a pathology/defect. Left: A simulated pneumothorax occurring near the heart in the left lung. Middle: A simulated pleural effusion occurring away from the heart in the left lung. Right: An enclosed diamond with an ovular defect

Figure 5.  Blind priors used for the thoracic (top) and industrial (bottom) imaging examples. Take particular note that the priors do not assume any pathology/defect

Figure 6.  The real part of the ${\mu ^{{\rm{int}}}}$ data (shown in the $z$ plane for $z\in\mathcal{D}$) corresponding to the blind thoracic prior given in Figure 5(top) computed from extended radii $R_2=4.0$, 6.5, and 9.0 in the $k$ plane. Note that as the radius increases, the integral term approaches its asymptotic behavior of ${\mu ^{{\rm{int}}}}\sim 1$

Figure 7.  Scattering data corresponding to the pneumothorax example using the blind prior given in Figure 5(top). The original radius is $R=4$ and extended radius $R_2=9$. All scattering data is plotted on the same scale (real and imaginary, respectively)

Figure 8.  Pneumothorax example for 62.5% ventral data. TOP: The partial data ND D-bar reconstruction ${\sigma ^{{\rm{ND}}}}$. BOTTOM: The recovered conductivity ${\sigma _{{R_2},\alpha }}$, using the blind thoracic prior. The maximum value is 2.25, occurring of $R=4$, $\alpha=0$

Figure 9.  Left: Original prior. Right: Updated Pneumothorax prior. The left lung in the updated prior was segmented into two regions

Figure 10.  Pneumothorax example with 75% Ventral data and segmented prior. The corresponding partial data ND D-bar reconstruction ${\sigma ^{{\rm{ND}}}}$ is shown in Figure 8. Here we display the recovered conductivity ${\sigma _{{R_2},\alpha }}$ for $R_2=4,6.5$ and various $\alpha$ using the SEG AVG or SEG MIN segmented thoracic priors. The maximum value is 2.70 and occurs in the $R_2=4$, $\alpha=0$ recon using the SEG MIN prior.

Figure 11.  Pleural effusion example for 75% ventral data. TOP: The partial data ND D-bar reconstruction ${\sigma ^{{\rm{ND}}}}$. BOTTOM: The recovered conductivity ${\sigma _{{R_2},\alpha }}$ using the blind thoracic prior. The maximum value is 2.90, occurring of $R=4$, $\alpha=0$

Figure 12.  Pneumothorax Example. Results for $R_2=9.0$ and $\alpha=0.67$. The maximum is 2.71 and occurs in the 100% boundary data, BLIND prior reconstruction

Figure 13.  Pleural effusion example for 62.5% ventral data. The partial data ND D-bar reconstruction ${\sigma ^{{\rm{ND}}}}$ is shown at the top. Below, the recovered conductivity ${\sigma _{{R_2},\alpha }}$ is shown using the blind thoracic prior. The maximum value is 2.74, occurring of $R=4$, $\alpha=0$

Figure 14.  Left: Original prior. Right: Updated Pleural Effusion prior with the left lung segmented into two regions

Figure 15.  Pleural effusion example for 75% ventral data and segmented prior. The corresponding partial data ND D-bar reconstruction ${\sigma ^{{\rm{ND}}}}$ is shown in Figure 13. Here we display the recovered conductivity ${\sigma _{{R_2},\alpha }}$ for $R_2=4$, 6.5 and various $\alpha$ using the SEG AVG or SEG MAX segmented thoracic prior. The maximum value is 2.83 and occurs in the $R_2=4$, $\alpha=0$ reconstruction using the SEG MAX prior

Figure 16.  Pleural effusion example for 62.5% ventral data. The partial data ND D-bar reconstruction ${\sigma ^{{\rm{ND}}}}$ is shown at the top. Below, the recovered conductivity ${\sigma _{{R_2},\alpha }}$ is shown using the blind thoracic prior. The maximum value is 2.74, occurring of $R=4$, $\alpha=0$

Figure 17.  Pleural Effusion Example. Results for $R_2=6.5$ and $\alpha=0.67$. The maximum is 2.65 and occurs in the 100% boundary data, BLIND prior reconstruction

Figure 18.  Industrial Example: From top to bottom, conductivity reconstructions ${\sigma _{{R_2},\alpha }}$ for 100%, 75%, 62.5%, and 50% boundary data are presented with scattering radius $R_2=4$ and various weights $\alpha$. The first column displays the ${\sigma ^{{\rm{ND}}}}$ reconstructions that do not include any a priori information. The maximum value (3.12) occurs for the 50% data reconstruction with strongest weight $\alpha=0$

Figure 19.  Industrial Example: From top to bottom, conductivity reconstructions ${\sigma _{{R_2},\alpha }}$ for 100%, 75%, 62.5%, and 50% boundary data are presented with extended scattering radius $R_2=6.5$ and various weights $\alpha$. The first column displays the ${\sigma ^{{\rm{ND}}}}$ reconstructions that do not include any a priori information. The maximum value (3.13) occurs for the 50% data reconstruction with strongest weight $\alpha=0$

Figure 20.  Relative $\ell_2$-error of reconstructions from 75% ventral data of the pneumothorax example. The horizontal axis represents $\alpha$-values for increasing regularization radii $R_2$. Recall that $\alpha=0$ corresponds to the heaviest weighting of the ${\mu ^{{\rm{int}}}}$ term, while $\alpha=1$ to the weakest expression of the prior. Errors from ${\sigma ^{{\rm{ND}}}}$ are compared to the new reconstructions ${\sigma _{{R_2},\alpha }}$ for the blind and segmented priors

Figure 21.  Relative $\ell_2$-error in the lung region within the boundary of the pathology, for 75% ventral data for the pneumothorax example. The horizontal axis represents $\alpha$-values for increasing regularization radii $R_2$