## 3 Results and Discussion
At the same time, the localization error is  
spatially more uniform for higher regularization: At a fixed count rate, the error dispersion tends to reduce as the contour lines are straighter and zones with different error value are merged together. However, values exceeding \(10^{-1}\) could worsen the figure of merit, indicating that a high value of regularization might induce artifacts. The count rate value does not seem to affect the \(x\) coordinate localization error for \(\tau\geq 10^{-3}\), as within each column a low error (close to 0 mm) is obtained for every experimented condition. A good localization is achieved even in depth down to 25 mm for \(\tau\geq 10^{-3}\). The same considerations apply to the localization along the \(y\) coordinate (data not shown). Figure 4 shows the localization on the \(z\) coordinate, i.e., the reconstructed \(z\) coordinate of the inclusion with respect to the simulated one. Similar conclusion to Figure 3 can be drawn for Figure 4, but it is worth noticing the low localization error (around 0 mm) even in depth (25 mm) for \(\tau=10^{-3}\), which seems to be the best value for the regularization parameter. This is a key aspect as the maximum probed depth is a typical issue when dealing with DO systems, and 25 mm is considered to be a significant value. Moreover, higher regularization values cause the inclusion to be reconstructed at shallower depths, as the negative (blue) error increases for a fixed z coordinate value (same row). As an example, let us consider the last row (count rate = \(10^{8}\) cps) and Zp = 15 mm. The error increases (in absolute value) from 0 mm at \(\tau=10^{-3}\) to \(-2\) mm at \(\tau=10^{-1}\), and reaches \(-3\) mm at \(\tau=1\). Figure 5 shows the relative volumetric error on the absorption coefficient. The figure clearly highlights that high regularization values cause a negative error--i.e., an underestimation of the absorption coefficient--rapidly increasing with the regularization parameter (i.e., left to right within each row). Again, the best value of \(\tau\) seems to be \(10^{-3}\) as the volumetric error is kept as low as 20% (in absolute value), even at 2 cm-depth. Clearly, the error on the quantitation of \(\Delta\mu_{a}\) can be related to the underestimation of \(z\). The regularization tends to attract the perturbation towards the surface and consequently a lower \(\Delta\mu_{a}\) is enough to match the observed contrast. For \(\tau=10^{-3}\), a photon count rate \(\geq 10^{6}\) leads to a wide low-error region (\(<\)10%) up to 2 cm depth, which is a positive result as \(\sim\)10\({}^{6}\) counts per second (cps) is a realistic signal level. Figure 3: Localization error on the \(x\) coordinate (expressed in mm) at different values of \(\tau\) (columns) and count rates (rows). 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 \(-\)10 mm (blue) to \(+\)10 mm (red). Figure 6 sums up the previous results and corresponds to simulation F. In this case the Y axis of each subplot is the \(y\) coordinate, while rows correspond to a different \(z\) coordinate of the inclusion. Count rate is now fixed to \(10^{6}\) cps. A low error region is located within the rectangle identified by sources and detectors positions, and a 2-cm depth can be reached with a \(\sim\)80% accuracy level in the estimate of the absorption coefficient if a \(\tau=10^{-3}\) is chosen. Higher regularization values cause underestimation of the absorption coefficient as well as worse localization. Figure 4: Localization error on the \(z\) coordinate for different count rates (rows) and \(\tau\) (columns). 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 \(-10\) mm (blue) to \(+10\) mm (red). Figure 5: Volumetric error on the absorption coefficient for different count-rates (rows) and \(\tau\) (columns).