Learned enclosure method for experimental EIT data

Figure 1.  The image on the right shows an experimental phantom with plastic shapes placed in saline (salt water). On the left is a simulated phantom resembling the experimental setting. The image shows the line determined by the support function $ {{h_D(\omega)}: = \sup_{x\in D}x\cdot\omega} $ in the direction $ \omega\in S^1 $ for the inclusions $ D\subset\Omega $, where $ \Omega $ is the unit disk

Figure 2.  An example of the slope of the least squares fit approximating the true support function of the phantom on the right in the direction $ \omega_4 $. The values of $ \frac{1}{2} \text{log} | I_{\omega_4} (\tau_j)| $, $ j = 1,\dots,10 $, are shown on the left as dots. The LS fit is plotted as a dotted and $ h_D(\omega_4)\tau+b $ as a solid line. The true support function $ h_D(\omega_4) $ corresponds to the distance of the probing half-plane from the origin, shown on the right in dark gray. The slope of the least squares fit approximates this distance but overestimates it, as can be seen from the light gray half-plane. To get the full convex hull of inclusions, this process is repeated for all directions $ \omega_i $

Figure 3.  An example of the indicator function matrix (B, top) and the input (B, bottom) given to the neural network for one conductivity phantom (A) from the training data set. The output (C) is a vector, plotted here as a line over $ \omega $, where each element corresponds to the true support function in the direction $ \omega_i $

Figure 4.  Neural network architecture, with two convolutional layers and a fully connected output layer

Figure 5.  We measure the quality of reconstructions of convex hulls using formula (22). This picture illustrates how the error can be seen as a sum of false negatives and false positives

Figure 6.  Histograms showing the distribution of the relative errors of the learned hulls (blue) and least squares hulls (red) in the simulated testing data set. The horizontal axis represents the relative error (22) and the vertical axis represents the frequency. Most of the learned hulls have a relative error that falls under 10%, whereas the relative errors of the least squares hulls range from 7% to 78%. False positives account for most of the errors, meaning that both approaches tend to overestimate the hull

Figure 7.  Comparison of the support vectors and the hulls of the simulated phantoms, computed using LS (dotted line), learned approach (dashed line), and the ground truth (solid line). The error relative to the ground truth is shown below each phantom

Figure 8.  Comparison of the ground truth, learned and least squares hulls of the experimental phantoms. The error relative to the ground truth is shown below each phantom

Figure 9.  Comparison of the ground truth, learned and least squares hulls of the experimental phantoms. The error relative to the ground truth is shown below each phantom

Figure 10.  Comparison of the ground truth, learned and least squares hulls of the experimental phantoms. The error relative to the ground truth is shown below each phantom

Figure 11.  Comparison of the ground truth, learned and least squares hulls of the experimental phantoms. The error relative to the ground truth is shown below each phantom

Figure 12.  The image on the left shows an example of the segmentation and the image on the right shows the resulting convex hull that is used as ground truth