# Super-Resolution Remote Imaging Using Time Encoded Remote Apertures
## 2 Time Encoded Remote Apertures (TERA)
### Tera Imaging Overview  
Consider a situation when one can detect not only direct (first bounce) back-scattered lights, but also the lights that bounced between the objects(second bounce). Such a signal can be obtained by illuminating the scene with a short-pulsed laser and detecting the returning light with a single time-of-flight detector. The temporal signal of these multi-bounce lights directly encodes the information about the distance between two object points in the scene. Figure 1b shows the simplified example of how the information about the scene is encoded in the temporal signal. Suppose the scene contains two small point objects \(p_{1}\) and \(p_{2}\) with a negligibly small diameter \(\delta\). The distances from \(p_{1}\) and \(p_{2}\) to the imaging system \(C\) are \(d_{1}\) and \(d_{2}\) correspondingly. The distance between objects \(p_{1}\) and \(p_{2}\) is \(d_{3}\). The scene is being illuminated by a pulsed laser, and a time-of-flight detector measures the returning light from the scene. The time-resolved measurement or time response of the scene is illustrated on the left side of Figure 1b. The first impulse is due to the light directly reflected from object \(p_{1}\), and it appears at \(t_{1}=\frac{2d_{1}}{c}\), where \(c\) is the speed of light. Similarly, the second impulse appearing at \(t_{2}=\frac{2d_{2}}{c}\), corresponds to the light reflected from the object \(p_{2}\). The last impulse at \(t_{3}\) is due to the light traveled following two paths: (\(C\to p_{1}\to p_{2}\to C\)) and (\(C\to p_{2}\to p_{1}\to C\)). Clearly, these two paths have the same travel distances and, thus, in the time response, they both appear at \(t_{3}=\frac{d1+d2+d3}{c}\). By using these three values, \(t_{1},t_{2},t_{3}\), one can find \(d1,d2\), and \(d3\). In other words, one can completely reconstruct relative positions of objects \(p_{1}\) and \(p_{2}\) up to Euclidean congruence. Now, we fix the distance \(d_{3}\) and keep increasing the distances \(d_{1}\) and \(d_{2}\). Naturally, at some point, the distance \(d_{3}\) will be below the Rayleigh diffraction limit for the conventional imaging system, i.e., one can not visually separate two points (\(p_{1}\) and \(p_{2}\)). However, for the proposed imaging system, the increase of the distances \(d_{1}\) and \(d_{2}\) only results in the time shift of the entire signal. The difference between three time tags, \(t_{1},t_{2},t_{3}\), is conserved. Therefore, one can still recover the scene(\(d_{1},d_{2}\) and \(d_{3}\)) up to Euclidean congruence. We note that we can find exact relative distance (\(d3\)) between the  
Figure 1: Comparison between conventional and Time Encoded Remote Apertures (TERA) imaging system. **(a)** Conventional imaging system. To infer the position of an object within the focal plane, an imaging system needs to determine the length difference between two rays. The figure shows a fundamental relation between distance, resolution, and aperture. The resolution of the system is determined by the [PERSON]’s criterion. **(b)** TERA imaging system. A pulsed laser illuminates the scene, and the SPAD detects the light reflected back from the scene. The collected time response contains information about the distance between the two targets. The resolution of the system is determined by the time resolution of the SPAD. objects and distances \((d1,d2)\) from the imaging system. However, the orientation (rotation) of the point-cloud remains ambiguous. Because we are using single-pixel detector, one can rotate the entire point-cloud along the depth direction or translate the entire point-cloud along the perpendicular direction to the depth and record the same time response. We call it reconstructing point-cloud up to Euclidean congruence. Naturally, the arising question is whether it is possible to reconstruct the scene when the number of objects is larger. Is the reconstruction unique, i.e., is it possible that two or more configurations generate the same time response?