# Super-Resolution Remote Imaging Using Time Encoded Remote Apertures
## 2 Time Encoded Remote Apertures (TERA)
### Tera Imaging Overview
Below, we show that the reconstruction of a point-cloud with \(n\) objects is possible under some assumptions. In this section, we use the results of [PERSON] et al. [19], which shows that, when the measured time-response of the scene contains a sufficiently rich set of first and second bounces, we can reconstruct the point-cloud up to Euclidean congruence. Here, we follow the same notation as [19]. Now, consider a scenario where the scene consist of multiple point objects (Figure 2a). 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 our imaging system (a laser and a detector are co-located). See Figure 2b. A graph \(\mathbf{S}=(p_{1}\),..., \(p_{n})\) is a scene configuration and \(\mathbf{K}=(p_{0},p_{1}\),..., \(p_{n})\) is a total configuration. The scene is illuminated with a pulsed laser and the detector observes returning signal from the scene. We define light path \(\alpha_{k}=[p_{0},p_{1}\),..., \(p_{z},p_{0}](z\in\mathbb{N})\) as a finite sequence of points where light has traveled. \(z\) is some integer. Note that any sequence \(\alpha\) starts from \(p_{0}\) and ends at \(p_{0}\), since our light source and detectors are co-located. First bounce, ping, is a light path \(\alpha_{k}=[p_{0},p_{i},p_{0}]\) for \(i=1\),..., \(n\). Second bounce, loop, is a light path \(\alpha_{k}=[p_{0},p_{i},p_{j},p_{0}]\) for \(i\
eq j\). Let \(v_{k}\) be a length of \(\alpha_{k}\). Subsequently, our measurement contains ensemble \(\beta=[v_{1},v_{2},...,v_{k}]\), where \(v_{k}\) are returned first or second bounce. Figure 2: **(a)** Illustration of the imaging system (a pulsed laser and time-of-flight camera) and the point-cloud scene. **(b)** We consider the imaging system as part of the point cloud. \(p_{0}\) is the location of the imaging system. \(p_{1},p_{2}\),..., \(p_{n}\) are point objects in the scene. **(c)** Dashed green lines are pings, and solid red lines are loop paths. If the measurement ensemble \(\beta\) contains four pings and six loops then we say that the sub-graph is contained in the measurement ensemble. **(d)** If “x” are known points and the measurement ensemble \(\beta\) contains one ping and three loops, then we say that the measurement allows for trilateration. Next, let \(\mathbf{K}_{5}\) be a sub-graph of \(\mathbf{K}\) that has 5 vertices, including \(p_{0}\). If the measurement ensemble \(\beta\) contains all pings and triangles of \(\mathbf{K}_{5}\) that starts and ends at \(p_{0}\), and then we say that \(\mathbf{K}_{5}\) is contained within \(\beta\). See Figure 2c. In this example, the measurement ensemble \(\beta\) contains four pings (\(\alpha=[p_{0},p_{i},p_{0}]\) for \(i=1\),..., 4) and six loops (\(\alpha=[p_{0},p_{i},p_{j},p_{0}]\) for \(i,j\in(1,2,3,4)\) and \(i\
eq j\)). Next, if the measurement ensemble \(\beta\) contains a ping (\(\alpha=[p_{0},p_{4},p_{0}]\)) and three loops (\(\alpha=[p_{0},p_{i},p_{4},p_{0}]\) for \(i=1\),..., 3), then we say that \(\beta\) allows for trilateration. See Figure 2d. Finally, we can apply the theorem from [PERSON] et al. [19], which says that if \(\mathbf{K}=(p_{0},p_{1}\),..., \(p_{n})\) is unknown configuration and \(\beta\) is the measurement ensemble of \(\mathbf{K}\) that allows for trilateration, then one can find trilateration based process to reconstruct \(\mathbf{K}\) up to Euclidean congruence. Readers can find detailed mathematical proof of the statement in [19]. Note that there exists a generic (or trivial) point configuration \(\mathbf{K}\) for which unique reconstruction is not possible. However, these special configurations occur very rarely [19].