NASA estimates there are millions of pieces of space junk in low Earth orbit (LEO) – mostly pieces of old spacecraft and defunct satellites. This debris can reach speeds up to 18,000 mph, posing a major danger to the 2,612 satellites that operate at LEO. Without effective tools for tracking space debris, parts of LEO may become too hazardous for satellites.

In a paper publishing in the SIAM Journal on Imaging Sciences, Matan Leibovich (New York University), George Papanicolaou (Stanford University), and Chrysoula Tsogka (University of California, Merced) introduce a new method for taking high-resolution images of fast-moving, rotating objects in space. They created an imaging process that first utilizes a novel algorithm to estimate the speed and angle at which an object in space is rotating and then applies those estimates to develop a high-resolution picture of the target.

Leibovich, Papanicolaou, and Tsogka used a theoretical model of a space imaging system to construct and test their imaging process. The model depicts a piece of fast-moving debris as a cluster of very small, highly reflective objects that represent the strongly reflective edges of an item in orbit, such as the solar panels on a satellite. The cluster of reflectors all move together with the same speed and direction and rotate about a common center. In the model, multiple sources of radiation on Earth’s surface emit pulses that are reflected by target pieces of space debris. A distributed set of receivers then detects and records the signals that bounce off the targets. The first step of the authors’ imaging process was thus to correlate the data taken at different receivers, which can help reduce the effects of these distortions.

Objects in LEO can spin on timescales that range from a full rotation every few seconds to every few hundred seconds, which complicates imaging. It’s thus important to know details about the rotation before developing the image. The authors needed to estimate the parameters related to the object’s rotation before synthesizing the data from different receivers. Although checking all possible parameters to see which ones yield the sharpest image is technically feasible, doing so requires a lot of computational power. Instead of employing this brute force approach, the authors developed an algorithm that can analyze the imaging data to estimate the object’s rotation speed and the direction of its axis.

After accounting for the rotation, the next step in the authors’ imaging process was to analyze the data to develop a picture of the space debris that would hopefully be as accurate and well-resolved as possible. One method that researchers often employ for this type of imaging of fast-moving objects is the single-point migration of cross correlations. Although atmospheric fluctuations do not usually significantly impair this technique, it does not have a very high resolution. A different, commonly used imaging approach called Kirchhoff migration can achieve a high resolution, as it benefits from the inverse synthetic aperture configuration; however, the trade-off is that it is degraded by atmospheric fluctuations. With the goal of creating an imaging scheme that is not too heavily affected by atmospheric fluctuations but still maintains a high resolution, the authors proposed a third approach: an algorithm whose result they call a rank-1 image. “The introduction of the rank-1 image and its resolution analysis for fast-moving and rotating objects is the most novel part of this study,” Leibovich said.

To compare the performance of the three imaging schemes, the authors gave simulated data of a rotating object in LEO to each one and compared the images that they produced. The rank-1 image was much more accurate and well-resolved than the result of single-point migration. It also had similar qualities to the output of the Kirchhoff migration technique. But this result was not entirely surprising, given the problem’s configuration. “It is important to note that the rank-1 image benefits from the rotation of the object,” Papanicolaou said. Although a rotating object generates more complex data, one can actually incorporate this additional information into the image processing technique to improve its resolution. Rotation at certain angles can also increase the size of the synthetic aperture, which significantly improves the resolution for the Kirchhoff migration and rank-1 images.

Further simulations revealed that the rank-1 image is not easily muddled by errors in the new algorithm for the estimation of rotation parameters. It is also more robust to atmospheric effects than the Kirchhoff migration image. If receivers capture data for a full rotation of the object, the rank-1 image can even achieve optimal imaging resolution.

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