## 2 Prototype System and Methods
### Refinement of the Volume Minimization: Motion Estimation with Cube Trajectories  
The presented volume minimization algorithm preforms well for constant velocity and constant acceleration motion models. Unfortunately, it is not stable for higher-order trajectories, supposedly,  
Figure 5: Measuring the volume of the reconstruction: the space is decimated with cubes, and the volume is represented by the number of cubes occupied by the reconstruction.  
Figure 6: Illustration of the problem space for CV motion model at various scales: (**a**) large, (**b**) medium, and (**c**) small scale.  
because the objective function is more complex. In this section, we present a solution, called cube trajectory estimation (CT) that is used for refining the trajectory provided by the volume minimization algorithm to solve for higher-order general motion models.  
After obtaining the initial reconstruction with assuming CV or CA motion models and using volume minimization, the proposed CT algorithm decimates this reconstruction into cubes similar to the space decimation presented for the volume calculation in the previous section (see Figure 7). Those points that are located in the same cube can be assumed to be captured from the same spot of the aircraft body but at different times. Thus, these points represent correspondences. The X, Y, and Z coordinate components of the cube trajectories can be calculated for each cube based on these points; these trajectories are shown in black lines in Figure 7. Next, the velocity of each cube is derived based on the time differences of their points. This allows the visualization of these velocities on a time-velocity diagram (see Figure 8). Assuming that all cubes, and thus all points of the aircraft, have the same motion, a chosen general motion model can be fitted to the velocity instances with regression (see Figure 8). The residuals from the regression provide a feedback and used later to assess the errors.  
Note that the correspondences rely on the initial reconstruction. If the initial point cloud is not close to the real solution, then the correspondences might not be correct. Thus, executing the steps presented above results in a reconstruction where the correspondences are different than they were in the initial reconstruction, requiring the repetition of the entire trajectory estimation until the  
Figure 8: Time–velocity graph of cubes from curvy linear motion: the cube velocities along the X and Y directions are depicted with blue and red dots, respectively; in this example, a constant acceleration (CA) motion model is used, thus the parameters a line are estimated with regression.  
Figure 7: Illustration of the cube trajectory (CT) algorithm: the initial reconstruction (red dots) is calculated from the data (blue dots), then this reconstruction is decimated into cubes, and then the trajectories (black lines) of all cubes are determined based on the points in the cubes.  
correspondences do not change. Since the refinement requires only a simple regression computation, the runtime with an optimized code is one or two orders of magnitude less than it was for the volume minimization algorithm.