## 3 Results
### Optical Property Extraction from MC-Generated Reflectance  
As mentioned in Section 2.3, the reflectance generated by MC simulation, after the correction, was much closer to that by the diffusion model, which was, therefore, used for optical property extraction in this study. Figure 3 shows estimated absorption and reduced scattering coefficients of each layer using different curve fitting methods. Note that the one-layered model was only used for estimating \(\mu_{a1}\) and \(\mu_{s1}\)\({}^{\prime}\) in Figure 3(a1,a2). The average absolute values of relative errors for the five curve fitting methods are summarized in Table 2. Overall, the accuracy for estimating \(\mu_{a1}\), \(\mu_{s1}\)\({}^{\prime}\), \(\mu_{a2}\) and \(\mu_{s2}\)\({}^{\prime}\) from the corrected reflectance are within 18%, which are superior to that without reflectance correction (results not presented here). The accuracy for estimating \(\mu_{s1}{}^{\prime}\) (Figure 3(a2)) was much better than that for \(\mu_{a1}\) (Figure 3(a1)) for all five curve fitting methods, which is consistent with the reflectance analyses in Section 3.1. It could be observed from Table 2 that five-variable fit performed the weakest for estimating \(\mu_{s1}{}^{\prime}\), followed by the four-variable fit, confirming that estimation accuracy improves as fewer variables are to be estimated each time [5,7]. However, for estimating \(\mu_{a1}\), four-variable fit performed the weakest, followed by the one-layered model. These results are out of expectation that one-layered model with two unknown variables could obtain higher estimation accuracy than five- and four-variable fit methods. This may be partly explained by the fact that the light samples the bottom layer as well thus seeing in part its increased absorption, especially for lower \(\mu_{a1}\), while for high \(\mu_{a1}\), the detected light does no longer go through the bottom layer, making the one-layered model a well description. Moreover, the reflectance used for curve fitting was generated by MC simulation and corrected based on the values of \(\mu_{s1}{}^{\prime}\) rather than \(\mu_{a1}\), which may also add challenges for estimating \(\mu_{a1}\) using the one-layered model; the value of \(\mu_{a1}\) is much smaller and more sensitive to the reflectance change than the \(\mu_{s1}{}^{\prime}\). Detailed analyses show that \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Optical** & **Five-Variable** & **Four-Variable** & **Two-Variable** & **One-Variable** & **One-Layered** \\
**Property** & **Fit** & **Fit** & **Fit** & **Fit** & **Model** \\ \hline \(\mu_{a1}\) (\%) & 9.84 & 17.25 & 4.78 & 8.14 & 12.23 \\ \(\mu_{s1}{}^{\prime}\) (\%) & 9.81 & 3.72 & 2.77 & 2.90 & 2.66 \\ \(\mu_{a2}\) (\%) & 10.44 & 9.86 & 4.73 & 8.30 & – \\ \(\mu_{s2}{}^{\prime}\) (\%) & 11.62 & 10.48 & 12.11 & 8.15 & – \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Average relative errors (in absolute values) of different curve fitting methods for estimating absorption (\(\mu_{a}\)) and reduced scattering (\(\mu_{s}{}^{\prime}\)) coefficients of each layer. Figure 3: Estimated absorption and reduced scattering coefficients of each layer from the corrected reflectance generated by Monte Carlo simulation using different curve fitting methods. (**a1**) \(\mu_{a1}\) varies 0.01–0.1 mm\({}^{-1}\) with \(\mu_{s1}{}^{\prime}\) = 2 mm\({}^{-1}\), \(\mu_{a2}\) = 0.02 mm\({}^{-1}\), \(\mu_{s2}{}^{\prime}\) = 1 mm\({}^{-1}\) and \(d\) = 2 mm, (**a2**) \(\mu_{s1}{}^{\prime}\) varies 0.5–4 mm\({}^{-1}\) with \(\mu_{a1}\) = 0.03 mm\({}^{-1}\), \(\mu_{a2}\) = 0.02 mm\({}^{-1}\), \(\mu_{s2}{}^{\prime}\) = 1 mm\({}^{-1}\) and \(d\) = 2 mm, (**b1**) \(\mu_{a2}\) varies 0.01–0.1 mm\({}^{-1}\) with \(\mu_{a1}\) = 0.03 mm\({}^{-1}\), \(\mu_{s1}{}^{\prime}\) = 2 mm\({}^{-1}\), \(\mu_{s2}{}^{\prime}\) = 1 mm\({}^{-1}\) and \(d\) = 2 mm, and (**b2**) \(\mu_{a2}{}^{\prime}\) varies 0.5–4 mm\({}^{-1}\) with \(\mu_{a1}\) = 0.03 mm\({}^{-1}\), \(\mu_{s1}{}^{\prime}\) = 2 mm\({}^{-1}\), \(\mu_{a2}\) = 0.02 mm\({}^{-1}\) and \(d\) = 2 mm.