## 2 Prototype System and Methods
### Estimating the Motion Model with Volume Minimization  
The proposed algorithm, called volume minimization (VM), adjusts the motion parameters to find the optimal reconstruction that occupies the minimum volume of the reconstructed point cloud of the aircraft. In this way, the volume of the reconstruction is used as a metric for measuring the point cloud consistency:  
\[\text{minimize (w.r.t }\mathbf{x})\;Vol\big{[}\mathbf{U}_{\mathbf{p}t\in\mathbb{P }}\mathbf{T}(\mathbf{p}_{t},t,\mathbf{x})\big{]}, \tag{2}\]  
where \(\mathbf{p}_{t}=[x,y,z,t]\in\mathbb{P}\) is a point from \(\mathbb{P}\in\mathbb{C}=2^{\mathbb{P}}\) point cloud at \(t\in\mathbb{R}\) time, \(\mathbb{T}\in(\mathbb{P},\,\mathbb{R},\,\mathbb{R}^{n})\to\mathbb{P}\) depicts the motion model that transforms a point to epoch \(t\) based on the underlying motion model with given \(\mathbf{x}\in\mathbb{R}^{n}\) motion parameters, and finally, \(Vol\in\mathbb{C}\to\mathbb{N}\) is a function that measures the volume of a point cloud. During volume minimization, the space is decimated into cubes, and the volume is measured by the number of cubes occupied by the reconstructed point cloud, as shown in Figure 5. Figure 4 shows the cubes from top view, and clearly illustrates that as the velocity approaches the real value, \begin{table}
\begin{tabular}{c c c} \hline \hline
**Constant Velocity** & **Constant Acceleration** & **General or Polynomial** \\ \hline \(s=s_{0}+v_{s}t\) & \(s=s_{0}+v_{s}t+a_{s}\frac{t}{2}\) & \(s=s_{0}+\sum\limits_{i=1}^{n}a_{s}t^{i}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Motion models used in this study; more details can be found in Appendix B.  
and the reconstruction becomes more realistic, the cube numbers also decrease, demonstrating that the proposed metric is consistent with the reconstruction quality. It is noteworthy that instead of volume, one can use entropy similar to [21; 22]. For our dataset, we found that the volume as metric requires less computation and has better convergence properties as opposed to entropy. An objective function of Equation (2) for the CV model is presented in Figure 6 in order to analyze the problem space. Note that the figure shows the function at different scales. Although the objective function at large scale, shown in Figure 6a, seems quasi-convex, the medium scale in Figure 6b, reveals that this function is highly non-convex. The complexity of the objective function depends on the cube size, the number of points captured from the aircraft, and the applied motion model. Obviously, the objective function is non-smooth due to the volume calculation presented above. This can be clearly seen at small scale in Figure 6c, which shows that the objective function is a multivariate step function. Consequently, one possible solution can be to approximate the objective function with a smooth function to solve the problem with one of the smooth optimization techniques, which solution is very likely to be sub-optimal. Alternatively, non-smooth optimization techniques might be applicable. In this study, we implemented a simulated annealing (SA) for solving Equation (2) [23]. The input of the algorithm is the initial search boundaries for all motion parameters and the cube size. The algorithm is relatively robust against the searching space due to the favorable characteristics of the objective function at large scale. This allows for finding a value close to the optimum in a couple of iterations, even if the search space is large. The cube size, however, has to be chosen carefully. If the size is too large, then it is not adequate to describe the volume of the aircraft, and the reconstruction will be \"blurry\". If it is too small, then not enough points will be located in the cubes. In other words, one can easily imagine a situation, when the cube size is so small that each point only occupies one cube at almost all chosen motion parameter values, which clearly results in the same objective (cost) values.