A boundary integral equations approach to electrical impedance tomography: Experiments on the KTC2023 data

Figure 1.  Comparison of the available experimental voltages (black circles) and the numerically estimated voltages (red crosses) which correspond to the voltages one would have, assuming the found solution from the inverse solver. The first figure shows the measured voltages corresponding to the 3rd target of Level 0 (32 electrodes; example data given to the participants). Here the full 2356 measurements are available. Below, in the third plot we show the 513 available voltages in the case of the first target at Level 7 (12 electrodes disabled; challenge data unavailable to participants). The second and fourth plots show the respective relative errors $ |V_{\exp}-V_{\mathrm{num}}|/\max(|V_{\exp}|) $ for these two cases. Here $ V_{\exp} $ and $ V_\mathrm{num} $ are the experimentally measured and numerically found voltages, respectively. As most measured voltages are small, for visual clarity we display all voltages above the threshold of 0.3 V in absolute value, but only every 5^th voltage below 0.3 V

Figure 2.  Level 0: Full data available. The ground truth's yellow inclusions are more and teal inclusions less conductive compared to the background water, respectively. Numerically found location of inclusions in solid red line and the initial guess in grey dotted line. Score above each image is the objective function value at the end and $ i $ denotes the number of iterations the optimizer took. We emphasize that in case of Target A we start with an initial guess on the left that contains five vertices, whose convex hull is a square. The optimizer then finds a better fit to the data by employing all of the five vertices, by finding a pentagonal diamond shape

Figure 3.  Full 32 electrode data at Level 1 and reducing the number of voltage measurements by 2 electrodes at each increment of level of difficulty. The active electrode locations at the outer boundary are marked with red color, while the electrodes not used for measurement are marked with black. The ground truth's yellow inclusions are more and teal inclusions less conductive compared to the background water. Numerically found location of inclusions in solid red line and the initial guess in grey dotted line. Score above each image is the objective function value at the end and $ i $ denotes the number of iterations the optimizer took

Figure 4.  The active electrode locations at the outer boundary are marked with red color, while the electrodes not used for measurement are marked with black. The ground truth's yellow inclusions are more and teal inclusions less conductive compared to the background water. Numerically found location of inclusions in solid red line and the initial guess in grey dotted line. Score above each image is the objective function value at the end and $ i $ denotes the number of iterations the optimizer took

Figure 5.  Level 5: Targets A and B, 8 removed electrodes, 1012 voltage measurements. In these examples we are forcing an initial guess with incorrect number of bodies. The ground truth's yellow inclusions are more and teal inclusions less conductive compared to the background water. Numerically found location of inclusions in solid red line, and the initial guess in grey dotted line. Score above each image is the objective function value at the end and $ i $ denotes the number of iterations the optimizer took

Figure 6.  Top: Figure 2 of [20]. Photograph of the water tank (left) with two target phantoms inside and the original ground truth picture provided to the contestants (right). Visually, it appears the boundary of the domain is mismatched in the ground truth compared to the inner boundary of the water chamber. This is confirmed by overlapping the two images in a picture editing software. Bottom: A reconstruction against the original ground truth (left), and same reconstruction plotted over the photograph of the water chamber while matching the inner boundary of the water tank with the domain of reconstruction (right). Visually, the inclusions are more accurately located in the right picture. We refer to Figure 2 for more detailed view of the reconstruction against the updated ground truth figures