## 3 Results and Discussion
### Time-Based Markov Chain Multiple Scattering Inversion
However, the proposed reconstruction method reduced the cost function evaluations to only \(11\) (number of candidates) \(\times\) 10 (number of layers) = 110, which substantially reduced the computational cost and made the reconstruction feasible. Figure 5: Schematic of the time-based reconstruction methodology. The reconstruction performance of the time-based Markov inversion method is presented in Figure 7. The ground truth data was obtained using the forwarding time-based Markov model, and we also added 5% relative noise to the ground truth data to investigate the robustness of the algorithm under noise. Since the time-based inversion scheme is a deterministic method, only one reconstruction was performed, in comparison with the 10 tests using optimization algorithms in the previous work [18]. The computational cost to perform each evaluation was around 25 minutes with an Intel i7-8700 CPU using MATLAB software, which could potentially be improved by using other programming platforms. In the two evaluations performed, both reconstructions faithfully rebuilt the synthetic phantom, so we will not be showing the reconstruction here. Instead, we show the relative error of each diameter in each iteration in Figure 7. In each iteration, the transmitted angular distribution was obtained with an assumed diameter size. Then, the relative error for each angle was found against the ground truth value. In this reconstruction, only backscattered photons were used, and we chose a range within 100-170 deg, where the signal strength was higher. Then, we summarized all the absolute values of the relative error and found the droplet diameter with the lowest relative error summation. As seen, the cost function values in the first few layers were distinctively lower at 1.3 \(\upmu\)m (\(\sim\)0), compared with the rest of the candidates (\(\sim\)70). As the sought layer moves further (away from the entrance plane), the relative error using false diameter candidates also decreases and reduces the sensitivity of the reconstruction. It is worth noting that this issue is not only associated with the proposed Markov chain method, but also impacts all algorithms utilizing backscattered photon angular distributions. The reason for this is that the sensitivities of different phase functions will be reduced, as the photons have been scattered many times, making it so that the spectral features of the angular distribution are not apparent anymore, which is also evident in Figure 5e-h in the range from 90-180 deg, where the angular distribution was a quite smooth single-peak distribution. Furthermore, although the true particle diameter has been found with a 5% relative error, as shown in Figure 6b, the summarized error in layers 81-90 for the true diameter (1.3 \(\upmu\)m) was 0.82,  
Figure 6: Numerical scheme for the reconstruction effort. Figure 7: Relative error summation under different droplet diameters for each layer set: (**a**) Ground truth data from the time-based Markov Chain model without noise; (**b**) Ground truth data from the time-based Markov Chain model with 5% relative noise. compared with 0.89 for a diameter of 1.4 \(\upmu\)m. Such an observation means the capability to distinguish different phase functions is deteriorated with a higher measurement error. Therefore, an \(OD\) threshold exists beyond which the droplet diameter cannot be resolved by backscattered photons, and the time-based Markov method is also limited by such a constraint. To address the issues mentioned above, photons transmitted from the exit plane (forward scattering photons) can also be utilized to examine the fidelity of the reconstruction with additional computational costs. Such efforts will be examined and implemented in future work. Furthermore, with the forward scattering process converted into a fast matrix evaluation problem, it is also possible to utilize end-to-end deep learning algorithms for reconstruction purposes, for which the training dataset will be provided by the time-based Markov model.