## 3 Results and Discussion
Horizontal axis of each subplot is the simulated \(x\) position of the inclusion and vertical axis is the simulated \(z\) position. Colors show an error of \(-100\%\) (blue) to \(+100\%\) (red). Figure 7 is obtained in the same condition as Figure 6, but it shows the Contrast-to-Noise Ratio defined as:  
\[CNR=\frac{\overline{\Delta\mu}_{rec,roi}-\overline{\Delta\mu}_{rec,roi}}{\sqrt{w_ {roi}\sigma_{roi}^{2}-w_{roi}^{2}}}\]  
where \(\overline{\Delta\mu}_{rec,roi}\) and \(\sigma_{roi}\) are the mean value and standard deviation of the reconstructed \(\Delta\mu_{a}^{rec}\) calculated in a region of interest; \(\overline{\Delta\mu}_{rec,roi}\) and \(\sigma_{roi}\) are calculated on the complementary region of interest (\(\overline{roi}=1-roi\)); \(w_{roi}\) and \(w_{roi}\) are weights defined as the volume rations \(w_{roi}=\frac{V_{roi}}{V_{tot}}\) and \(w_{roi}=\frac{V_{roi}}{V_{tot}}\). The region of interest is defined as the region of space in which the reconstructed \(\Delta\mu_{a}^{rec}\) is greater than half of the maximum reconstructed value. This figure of merit is an indicator of the goodness of the reconstruction for specific structures within the volume [36]. The CNR strongly decreases with the depth for any regularization value, but quicker for higher \(\tau\) values. The column \(\tau=10^{-3}\) and \(\tau=10^{-2}\) provide the best trend for the CNR, leading to a CNR close to 10 at a depth of 2 cm, even though \(\tau=10^{-2}\) seems to offer a more homogeneous distribution at all depths. Overall, Figures 6 and 7 suggest that the simulated system would be capable of detecting inclusions up to 2 cm depth with a significant accuracy and reasonable contrast. The previous figures show the spatial distribution of the figures of merit in different experimental conditions (i.e., photon count rate, regularization parameter). However, the most common method of assessing the goodness of a reconstruction is through the cross sections of the overall tomographic image. Thus, in the following Figures 8 and 9, we show an example of the simulated tomography problem and of the obtained reconstruction, respectively. Each subplot corresponds to a different depth (expressed in millimeters) and the X/Y axis is the _xyy_ coordinate in space. The color bar shows the value of the absorption coefficient in mm\({}^{-1}\) units. The count rate is fixed at \(10^{6}\) cps and three hardware gates of the detectors are simulated. The regularization parameter is set to \(10^{-2}\). We simulated the spherical inclusion (whose features are described in Section 2.3) in the center of the _x-y_ plane and at a 10 mm-depth, thus spanning from 5 mm to 15 mm. Figure 6: Map of the volumetric error on the absorption coefficient. The horizontal axis of each subplot is the simulated \(x\) position of the inclusion and the vertical axis is the simulated \(y\) position. Rows are for different \(z\) coordinates of the inclusion, while columns are for different \(\tau\). Colors show an error of \(-100\%\) (blue) to \(+100\%\) (red). Figure 8: Cross sections of the simulated tomography at different depths. The inclusion is set in the center of the \(x\)-\(y\) plane and at a 10 mm-depth. Figure 7: Map of the CNR on the absorption coefficient. Horizontal axis of each subplot is the simulated \(x\) position of the inclusion and vertical axis is the simulated \(y\) position. Rows are for different \(z\) coordinates of the inclusion while the columns for different \(\tau\). Colors show a range from 1 (blue) to 37 (red), that is the maximum CNR obtained. The reconstruction in Figure 9 shows small surface artifacts, but the localization in the \(x\)-\(y\) plane is good as the inclusion is set in the center of such plane, as expected. The \(z\) coordinate is reconstructed at a slightly shallower position, as the center of the inclusion is at 9 mm, but the overall height is properly identified. The quantification error is limited to 5%.