## 4 Discussion  
We validated the feasibility of direct annihilation position classification based on DNNs from photon information measured by paired Cherenkov detectors. We used Monte Carlo simulation data of 125 point sources for validation. In addition, we evaluated the influence of the SPTR and readout pitch on the classification accuracy and spatial resolution. The classification accuracy was 96% at ideal conditions where the SPTR was 0 ps and the readout pitch was 0 mm (Table 2). This indicates that the proposed method is feasible under ideal conditions. As the classification accuracies of the readout pitch of 0 to 3 mm were almost the same (Figure 4), a readout pitch of 3 mm was considered to be adequate. In contrast, the classification accuracies dropped rapidly as the SPTR worsened. To maintain a classification accuracy of 80%, the SPTR must be 10 ps or less (Figure 4). In addition, we evaluated the confusion matrix to determine whether the MLP made major mistakes, because classification accuracy as the figure of merit does not distinguish between major and minor mistakes. When the SPTR was 10 ps and the readout pitch was 3 mm, misclassification was concentrated on the adjacent positions (Figure 5a). This can be interpreted as evidence that the MLP rarely made major mistakes in these conditions. In contrast, as the SPTR worsened to 100 ps, the misclassification was broadly distributed in the \(z\)-direction (Figure 5b). These results indicate that an SPTR of 10 ps or less is preferable in view of the magnitude of misclassification. To better understand the effects of the SPTR and readout pitch, we visualized the PSF by estimating the continuous-valued annihilation position as the center of gravity of the probability output by the MLP (Figure 6). When the SPTR was 10 ps and the readout pitch was 3 mm, the PSF was isotropic (Figure 6a). The magnitudes of misclassification in the \(x\)-, \(y\)-and \(z\)-directions were almost equal and concentrated in adjacent positions. In contrast, as the SPTR worsened to 100 ps, the PSF was elongated in a bar shape along the \(z\)-direction (Figure 6b). This can be interpreted as an indication that the SPTR has a greater effect in the \(z\)-direction than in the \(x\)- and \(y\)-directions. The projections of the PSF on the \(x\)- and \(z\)-axis show that the shape of the PSF was sharp and had side peaks (Figure 6c). This means that Gaussian fitting is not adequate for analysis of the spatial resolution. Hence, we calculated the RMSE to determine spatial resolution as in [PERSON] et al. [38]. The RMSE of the \(x\)-position estimation was slightly affected by the readout pitch (Figure 7a). This was consistent with the results of 3-D interaction position estimation based on deep learning using the Cherenkov detector [14]. The RMSE of the \(z\)-position estimation was almost the same for all readout pitches and rapidly degraded as the SPTR worsened (Figure 7b). These results may indicate that the \(z\)-position was estimated using only time information. When the SPTR was 10 ps and the readout pitch was 3 mm, the proposed method obtained a \(\sigma_{x}\) of 1.12 mm and \(\sigma_{z}\) of 1.42 mm (Table 3). The sensitivity of the proposed system was low (Figure 8). As we focused on whether direct annihilation position classification is possible in this study, we did not optimize the system to increase the sensitivity. In future studies, we aim to optimize the system parameters, for example, the distance between detectors, detector size, thickness, and surface treatment of the crystal, and the number of paired detectors, to improve the resolution and sensitivity. For example, [PERSON] et al. optimized the thickness and surface treatment of PbF2 crystals of whole-body PbF2 TOF-PET scanners, using a figure of merit which was defined as the ratio between detection efficiency and CTR [39]. In addition, the photon detection threshold can be optimized to control the tradeoff between resolution and sensitivity.