## 3 Results and Discussion
### Test on Phantoms  
We subsequently tested the system on a solid switchable phantom that is capable of mimicking localized absorption perturbations in an otherwise homogeneous medium, as described in Section 2.3.  
Figure 6 displays slices of three-dimensional (3D) reconstructions along horizontal \(x\)-\(y\) planes at different depths \(z\) (rows) for different choices of the effective absorption perturbation \(\Delta\mu_{a}\) referred to a volume of 1 cm\({}^{3}\) (columns). We chose the \(y\) position of the inhomogeneity yielding the maximum contrast, which roughly corresponds to a location equidistant from each source-detector couple. The inhomogeneity can be reconstructed for all of the tested perturbations (down to \(\Delta\mu_{a}=0.005\) mm\({}^{-1}\)) in the expected \(x\)-\(y\) location (white circle), but with a shallower depth, in particular, for the weaker \(\Delta\mu_{a}\). This issue is probably due to the strong decrease of the Jacobian coefficients with depth combined to the noise increase of signal coming from deep photons, as also observed in simulations. This combination tends to give more weight to superficial voxels in the reconstruction.  
Figure 6: Slices of three-dimensional (3D) reconstructions along horizontal \(x\)-\(y\) planes at different depths \(z\) (rows) for different choices of the effective absorption perturbation \(\Delta\mu_{a}\) referred to a volume of 1 cm\({}^{3}\) (columns).  
The upper slices confirm the presence of the superficial artifacts also in these phantom measurements, but this does not affect the retrieval of the real perturbation, since the darkest and widest shadows belong to deeper \(x\)-\(y\) planes. These results are quite consistent with findings on simulations, as reported in Figure 3. We can observe a good agreement for the first two inclusions, while the second two are slightly more visible than in the simulated ones (see Figure 3). This difference is a consequence of the less than ideal performance of a real measurement, whose additional noise and imprecision could make the inclusion appear bigger than it actually is, and the reduced number of optodes, which also determines less accurate identification of the object. Figure 7 (right) displays the reconstructed \(x\)-\(y\) slices, extracted at \(z=16\) mm for different \(\Delta\mu_{a}\) values of the inhomogeneity (rows) and positions along the \(y\) axis (columns), as depicted in the scheme illustrated on the left panel.  
We observe an acceptable capability of the system to track the object location as far as it is within the detector's circle (with a radius of 30 mm, which stays between \(10<y<65\) mm), in agreement with the outcome of simulations (see Figure 5). Outside that region, the reconstructed perturbation is off-placed in the y direction. For the weakest perturbation, the detection gets worse, and it is basically lost at the edge of the detectors' circle.  
For what concerns the repeatability of the measurements, we tested it for two different configurations and we acknowledged that the absorption reconstructions were always in the same positions, with an uncertainty of at maximum 1 voxel (2.5 \(\times\) 2.5 \(\times\) 2.5 mm\({}^{3}\)), probably due to the intrinsic noise and/or regularizations during the reconstructions.  
Figure 7: **(Left)** Scheme of the measurement condition: the inclusion, represented by the black dot, is placed in different positions along \(y\) axis as shown by the bold dotted line, while the simple dotted lines qualitatively specify the superficial disposition of the probe. **(Right)** Tomographies showing horizontal \(x\)-\(y\) slices taken at \(z=16\) mm for different \(\Delta\mu_{a}\) of the inhomogeneity (rows) and y positions (columns). The white circle identifies the expected “true” location, while the black dashed lines correspond to the boundary of the detectors’ circle.