## 4 Core Algorithm
### Stochastic Modified Evaluation  
For all nodes in \(S_{explorable}\), we set up a hash map \(M_{explorable}\). For each node, its location and number of required pictures for that node are given as: \(Node_{index}^{explorable}\): \((p_{index}^{explorable},\widetilde{N_{index}}^{explorable})\). The number of pictures required for nodes in \(S_{explorable}\) are calculated from the estimation function in Equation (4). The algorithm tends to select nodes located around the explored node that has the minimum number of pictures because, from Equation (4), these nodes tend to have the smallest estimated number of pictures. However, this process does not guarantee reaching the global minimum, therefore, minimizing the number of pictures. To sufficiently explore unknown nodes as well as exploit information from already explored nodes, a stochastic process is introduced to modify the picture number estimation function given in Equation (4). Assume when the locations of a camera in space deviate a smaller amount from each other, the difference of real picture number values at those locations is also smaller. Thus, among all unexplored nodes, the estimated picture number of an unexplored node, which is closer to the explored node, is more deterministic. To emphasize this feature, we make the estimated picture number follow a normal distribution with a mean \(\mu_{\widetilde{N}}\) being equal to the value from Equation (4) and standard deviation \(\sigma_{\widetilde{N}}\). As an unexplored node is further away from explored nodes, the larger value of \(\sigma_{\widetilde{N}}\) should be assumed, which means more indeterministic estimation (illustrated in Figure 8). The following Equations (14)-(16) summarize the above discussion:  
Figure 8: Both case 1 and case 2 have the same estimated value of an unexplored node (green point) from Equation (4). However, the node in case 1 has a larger distance to explored nodes (red points) compared to that in case 2. Therefore, we set different standard deviations (\(\sigma_{1}>\sigma_{2}\)) to differentiate possibility distributions. This approach distinguishes nodes that have the same estimations from Equation (4) to have different likelihoods to be explored in the next iteration. For any nodes \(Node_{index=x}^{explorable}\) in \(S_{explorable}\), its picture number estimation should follow the following normal distributions:  
\[P(Z_{x})=\mathcal{N}(\mu_{Nx}^{\sim},\sigma_{Nx}^{\sim}) \tag{14}\]  
\[\mu_{Nx}^{\sim}=\overset{\sim}{N}_{x}^{explorable} \tag{15}\]  
\[\sigma_{Nx}^{\sim}=K_{sd}*\overline{ED}_{min} \tag{16}\]  
where \(P(Z_{x})\) is the probability of a random variable \(Z\) at \(Node_{x}^{explorable}\). \(\overset{\sim}{N}_{x}^{explorable}\) is the estimated value from Equation (4) of \(Node_{x}^{explorable}\). \(K_{sd}\) is a constant parameter, and \(\overline{ED}_{min}\) is the smallest Euclidean distance among distances between that node and explored nodes. As discussed above, the algorithm ensures that the camera, at any iteration, always has enough energy to move back to the previous best node \(N_{\text{exp\_min}}\). Therefore, a new distribution can be generated, which sets values bigger than the minimum (\(N_{\text{exp\_min}}\)) in the original distribution to be the minimum value. Then, Equations (14) to (16) are modified with a new random variable \(Z_{x}^{new}\) and its expectations as follows:  
\[Z_{x}^{new}=\left\{ \begin{matrix}Z_{x},&if\ Z_{x}<N_{\text{exp\_min}}\\ N_{\text{exp\_min}},&if\ Z_{x}\ \geq N_{\text{exp\_min}}\end{matrix} \right. \tag{17}\]  
\[P(Z_{x}^{new})=P(Z_{x})=\mathcal{N}(\mu_{Nx}^{\sim},\sigma_{Nx}^{\sim}) \tag{18}\]  
\[\hat{N}_{x}^{explorable}=\sum[Z_{x}^{new}*P(Z_{x}^{new})] \tag{19}\]  
where \(\hat{N}_{x}^{explorable}\) is the stochastic modified estimated value for the node \(Node_{x}^{explorable}\). Then, a new hash map \(\hat{M}_{explorable}\) is set up with pairs as \(Node_{index}^{explorable}\): \((p_{index}^{explorable},\hat{N}_{x}^{explorable})\).