## 3 Results and Discussion
### Numerical Verification against Monte Carlo Simulations
Even for different methods which have been proposed to reduce the computational cost of Monte Carlo simulations, such as using cloud computing [8; 9] or cell processors [20], hardware improvement could also accelerate the computation of the Markov chain model, especially when no computational optimization was done to the Markov chain time-based model at the time. In this work, we just demonstrated the capability of the proposed Markov time-based model and its inversion method. We will focus on the computational cost and accuracy in our future work. As can be seen in Figure 2, the newly developed time-based model approximated the results given by the scattering-based model and Monte Carlo simulations well, although the resultant angular distribution could be complex and wavy. Furthermore, as we have discussed, the accuracy of the time-based algorithm suffered from a higher \(OD\), and it is notable that the results under \(OD=10\) and \(OD=20\) cases were a bit off from the ground truth when compared with the \(OD=2\) and \(OD=5\) cases. Nevertheless, the proposed time-based model could still achieve a relative error of less than 1% compared with the Monte Carlo simulation results, which indicates that the proposed time-based model can satisfy the need of approximating Monte Carlo simulations. To further demonstrate the error from the Markov chain approximation, we incorporated the relative error of each model, as shown in Figure 3. As we introduced in our previous work [17], the Markov chain model had the worst performance near 90\({}^{\circ}\), around which the signal was hard to capture by the sensors in the experiments. Therefore, in Figure 3, we just compared the results at 135\({}^{\circ}\) (corresponding to the maximal point in 90\({}^{\circ}\)-180\({}^{\circ}\)) to demonstrate the error of the time-based model, and we also confirmed that this error was representative of the overall error of the Markov chain models. As can be seen, the relative error of the scattering-based model increased as the \(OD\) increased, but in general, the relative error was less than 0.5%, which was in agreement with our previous research [16]. For the time-based model, the error was around 0.7% when \(OD=2\). Then, the error increased steadily and reached \(\sim\)1.6% when \(OD=10\). The error increased less remarkably afterward with higher \(OD\)s. Thus, although the time-based model would incur error from the two assumptions mentioned above, the overall accuracy was still acceptable for calculating the transmitted angular distribution. In order to examine the capacity of the time-based Markov chain model for distinguishing transmitted photons at different timings (i.e., propagating different path lengths inside the turbid media), Figure 4 compares the results obtained by Markov chain approximation and the Monte Carlo simulations, respectively. In Markov chain realization, the photon transmitted can be obtained via Equation (6). For instance, the transmitted angular distribution within \(t=100\Delta z/c-t=125\Delta z/c\) can be achieved by adding Equation (6), with \(n\) ranging from 100 to 125. For Monte Carlo simulation, as each p  
Figure 3: Relative error as a function of the \(OD\). The transmitted value at the bin of 135\({}^{\circ}\) (the maximal value for backpropagated, transmitted photons) was used for comparison. Monte Carlo data was used as the ground truth. pathlength that this photon has propagated for can be recorded. Therefore, when all the photons have been simulated, the sought distribution within a range from \(t=100\Delta z/c\) to \(t=125\Delta z/c\) can be obtained by counting all photons that propagate a total path length of 100-125\(\Delta z\) before being transmitted. In this investigation, a total of 100 turbid layers were used, and the results in Figure 4 were obtained under a total \(OD\) of 5.0, suggesting that the \(OD\) in each layer was 0.05. The thickness of each layer was noted as \(\Delta z\). Figure 4a-h presents the photons transmitted during different periods of time.