# Super-Resolution Remote Imaging Using Time Encoded Remote Apertures
## 4 Simulations  
Here, we test the modified TRIBOND algorithm using simulated data. We generate simulated data using a simplified version of the transient light transport renderer [22,23]. The renderer is successfully used in many times of flight applications [24,25]. Similar to the previous chapter, we consider the following scenario. Let \(p_{1}\),..., \(p_{n}\in\mathbb{R}^{3}\) be n point objects in the scene and \(p_{0}\in\mathbb{R}^{3}\) be a position of the TERA imaging system. Let \(d_{i}\) be the distance from \(p_{i}\) to \(p_{0}\) for \(i=1,...,n\) and \(d_{ij}\) be the distance from \(p_{i}\) to \(p_{j}\) for all \(i\
eq j\). See Figure 3. In this simulation, we assume the exact distance list and do not consider the noise. The simulated scene contains \(n=10\) point objects that are located far away from the imaging system. The Figure 3 shows the simulated time response, which contains a mixture of first and second bounce. The first and second bounces signals are also shown separately. Locations of the peaks in the signal are marked with blue triangles. We transform these locations of the peaks into a distance list and use it as an input to the modified TRIBOND algorithm. On the right side of the Figure 3, the result of reconstruction is shown. The reconstruction matches the original point-cloud configuration up to Euclidean congruence; in other words, all of the pair-wise distances were recovered; however, the rotation and transpose of the point-cloud are remaining unknown.  
We conduct the following simulation in order to address more general scenarios and numerically evaluate the algorithm. Our sensor's dark count rate is 10 photons per second. The temporal impulse response of the system is 80 ps. See Section 5 for more details on the experimental hardware. Additionally, Poisson noise was added to the simulated data. We assume that, after pulsing, noise can be avoided by using a fast gated SPAD detector, and we, therefore, do not add after pulsing noise in the simulation. Laser power is assumed to be 10 W. We generate sets of simulated scenes with five to twenty diffuse targets with varying sizes from 1 to 8 cm. Reconstructing all these scenes would not be computationally feasible. In order to evaluate whether we have sufficient data to reconstruct the scene, we note that, in order to reconstruct a vertex of a point cloud, it is sufficient to have one first bounce and two second bounce signals from that vertex (see Section 3.2 and [PERSON] et al. [19] for details). By counting number peaks and comparing to ground truth pairwise distances, we evaluate which vertexes of the scene remain recoverable given the correctly identified peaks. Figure 4 shows the results of statistical analysis. A large number and large diameter sizes of objects may occlude some possible light paths and can create overlapped peaks, which reduces the recoverability rate. However, a high recoverability rate is achieved for a small number of points with small diameter sizes, since most of the peaks are available. Note that an additional complication to the reconstruction is provided by false peaks in the data. Section 5 discusses this issue.  
Figure 3: Time response and reconstruction (**a**) Total time response of a scene with 10 randomly generated points. The time response includes the first and second bounces. The peak extraction algorithm is applied to find peaks. The peaks are marked by blue triangles. (**b**) Second bounce time response; (**c**) Reconstruction of the point-cloud. After applying a peak extraction algorithm, modified TRIBOND is used to reconstruct the point-cloud. The red dotted line corresponds to the first bounce path connecting point in the scene and imaging system. The black line is the second bounce path.