## 4 Results
### Two-layered Head Model
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{**Recovery Model**\(\rightarrow\)}} & \multicolumn{2}{c}{**Semi-infinite Recovery Model**} & \multicolumn{2}{c}{**Head-based Recovery Model**} \\ \hline \multirow{3}{*}{\%\(\Delta\) HbO} & Homogenous & 22.0 \(\pm\) 13.3 & \(-\)27.2 \(\pm\) 18.4 \\ \cline{2-4}  & Layer-1 & 21.9 \(\pm\) 2.9 & 5.2 \(\pm\) 4.4 \\ \cline{2-4}  & Layer-2 & 19.0 \(\pm\) 7.5 & \(-\)23.3 \(\pm\) 7.6 \\ \hline \multirow{3}{*}{\%\(\Delta\) Hb} & Homogenous & 48.3 \(\pm\) 9.3 & 15.3 \(\pm\) 22.5 \\ \cline{2-4}  & Layer-1 & \(-\)5.5 \(\pm\) 1.2 & \(-\)20.0 \(\pm\) 3.7 \\ \cline{2-4}  & Layer-2 & 45.4 \(\pm\) 6.1 & \(-\)1.36 \(\pm\) 7.6 \\ \hline \multirow{3}{*}{\(\Delta\) TOI} & Homogenous & 9.3 \(\pm\) 8.0 & 9.8 \(\pm\) 9.0 \\ \cline{2-4}  & Layer-1 & \(-\)5.1 \(\pm\) 0.6 & \(-\)4.1 \(\pm\) 0.3 \\ \cline{2-4}  & Layer-2 & 8.4 \(\pm\) 2.6 & 4.2 \(\pm\) 1.9 \\ \hline \multirow{3}{*}{\%\(\Delta\) Sa} & Homogenous & 12.3 \(\pm\) 2.2 & 37.8 \(\pm\) 0.4 \\ \cline{2-4}  & Layer-1 & \(-\)34.0 \(\pm\) 2.2 & \(-\)6.2 \(\pm\) 2.7 \\ \cline{2-4}  & Layer-2 & 4.4 \(\pm\) 14.4 & 16.2 \(\pm\) 2.7 \\ \hline \multirow{3}{*}{\%\(\Delta\) Sp} & Homogenous & \(-\)41.2 \(\pm\) 4.0 & \(-\)22.1 \(\pm\) 0.4 \\ \cline{2-4}  & Layer-1 & 17.8 \(\pm\) 0.8 & 26.8 \(\pm\) 1.1 \\ \cline{2-4}  & Layer-2 & \(-\)87.8 \(\pm\) 58.3 & \(-\)44.2 \(\pm\) 7.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Average percentage error (%) in the homogenous and two-layered recovery of parameters corresponding to a semi-infinite recovery model and a head-based recovery model. Figure 9: Estimated values of layer-1 thickness with semi-infinite recovery model corresponding to extra-cerebral tissue for 10 subject head models, along with the ground-truth probability density function of extra-cerebral tissue thickness in the region of interest. Subjects 2, 8 are not shown as the algorithm did not converge to a stable solution. boundary surface, therefore unlike the original head model which has a heterogeneous and spatially varying thickness of layer-1, the recovery model consists of uniform layer-1 thickness. So, although this model is better than the semi-infinite model by considering the external boundary structure, its internal structure is not exactly the same as the true head model. The corresponding parameter recovery is shown in Figure 8b, Figure 10, and Table 3.  
These results demonstrate that for the data corresponding to a head model, the proposed algorithm with a semi-infinite recovery-model estimates the TOI with an average recovery error of 8.4% with the two-layered recovery approach as compared to the error of 9.3% with the homogenous approach. Recovery of other tissue parameters (except for scattering power) can be achieved within 45.4% error with the two-layered approach as compared to the error of 48.3% with the homogenous recovery approach. However, using an appropriate recovery model (head-based model) the parameter recovery can be greatly improved by reducing the average recovery error of TOI to 4.2% with the two-layered approach as compared to the error of 9.8% with the homogenous approach, and the recovery of other tissue parameters (except for scattering power) can be achieved within 23.3% error with the two-layered approach as compared to the error of 37.8% with the homogenous recovery approach. However, the scattering power parameter exhibits relatively higher recovery errors due to its low sensitivity to light-intensity. In regard of the extra-cerebral tissue thickness, as the ground-truth is not a single value but rather varies spatially, we can quantify this variation in terms of a probability density functions as shown in Figure 9 & Figure 10, and by considering the tissue thickness at the peak probability as the ground-truth, the semi-infinite recovery model estimates the tissue thickness with an average error of 30.1%, whereas using an appropriate head based recovery model the error in estimating the tissue thickness is reduced to 11.8%.