# Reconstructing Satellites in 3D from Amateur Telescope Images

Zhiming Chang, Boyang Liu, Yifei Xia, Youming Guo, Boxin Shi and He Sun
Zhiming Chang and He Sun are with the College of Future Technology and the
National Biomedical Imaging Center, Peking University. Zhiming Chang is also
with the College of Engineering, Peking University, Beijing, China (Email:
<EMAIL_ADDRESS><EMAIL_ADDRESS>Yifei Xia, Boxin Shi are with
the National Key Laboratory for Multimedia Information Processing and National
Engineering Research Center of Visual Technology, School of Computer Science,
Peking University, Beijing, China (Email: {yfxia<EMAIL_ADDRESS>Boyang
Liu is with the Earthrise Edu&Tech (Email<EMAIL_ADDRESS>Youming Guo is
with the Institute of Optics and Electronics, Chinese Academy of Sciences,
Chengdu, China (Email<EMAIL_ADDRESS>Corresponding author: He Sun

###### Abstract

This paper proposes a framework for the 3D reconstruction of satellites in
low-Earth orbit, utilizing videos captured by small amateur telescopes. The
video data obtained from these telescopes differ significantly from data for
standard 3D reconstruction tasks, characterized by intense motion blur,
atmospheric turbulence, pervasive background light pollution, extended focal
length and constrained observational perspectives. To address these
challenges, our approach begins with a comprehensive pre-processing workflow
that encompasses deep learning-based image restoration, feature point
extraction and camera pose initialization. We proceed with the application of
an improved 3D Gaussian splatting algorithm for reconstructing the 3D model.
Our technique supports simultaneous 3D Gaussian training and pose estimation,
enabling the robust generation of intricate 3D point clouds from sparse, noisy
data. The procedure is further bolstered by a post-editing phase designed to
eliminate noise points inconsistent with our prior knowledge of a satellite’s
geometric constraints. We validate our approach using both synthetic datasets
and actual observations of China’s Space Station, showcasing its significant
advantages over existing methods in reconstructing 3D space objects from
ground-based observations.

###### Index Terms:

Computational Imaging, Astrophotography, 3D Reconstruction, Gaussian
Splatting, Imaging through Turbulence

##  1 Introduction

The monitoring of man-made objects in space is an essential endeavor in space
domain awareness (SDA), particularly critical for ensuring the safe operation
and on-orbit servicing of civilian satellites in the era of burgeoning large-
scale satellite constellations, such as SpaceX’s Starlink. Within the broad
spectrum of satellite monitoring activities, including detection,
categorization, and tracking, 3D reconstruction stands out as a promising
advancement in the field, as it uniquely provides unparalleled insights into
the real-time operational status of satellites.

To date, 3D reconstruction has predominantly been investigated in the context
of space-based space surveillance missions, aiming to derive three-dimensional
information from videos captured by neighboring satellites. Nonetheless, the
potential of 3D reconstruction from ground-based observations remains markedly
under-explored, despite its advantages in terms of cost-effectiveness and
versatility. Unlike their space-based counterparts, ground-based observations
present a series of unique challenges: 1) atmospheric turbulence significantly
distorts images captured by ground telescopes; 2) the extended focal length of
the ground telescope complicates the accurate estimation of a satellite’s
pose; and 3) the rapid motion of satellites necessitates short exposure times,
resulting in low signal-to-noise (SNR) ratios and constrained observational
perspectives.

In this paper, we propose an innovative imaging approach for the 3D
reconstruction of satellites using ground-based telescope images. Leveraging
the latest advances in imaging through turbulence and computer graphics
volumetric rendering, we have developed a cutting-edge 3D Gaussian splatting
technique. This novel approach facilitates simultaneous volume reconstruction
and pose estimation, while also incorporating sophisticated pre- and post-
processing pipelines specifically tailored for limited, distorted, and noisy
ground telescope observations. We validate our method using both simulated and
experimental datasets. Notably, we achieve the successful reconstruction of a
3D model of China’s Space Station (CSS, or Tiangong Station), a satellite in
low Earth orbit, using solely images from an amateur 35cm-aperture telescope.

Figure 1: Problem definition. Our objective is to reconstruct a 3D model of a
satellite in low-Earth orbit using limited, blurry, and noisy images captured
by an amateur ground-based telescope, which are significantly affected by
atmospheric turbulence and light pollution.

##  2 Related Work

Figure 2: Overview of the 3D satellite reconstruction pipeline. This pipeline
begins with the raw satellite images (a) obtained from ground-based
telescopes, plagued by severe jitter, blurriness, and complex noise. Our
methodology unfolds in three pivotal steps: 1) Image Pre-processing (b), which
incorporates techniques such as centering, lucky imaging, wavelet
decomposition sharpening, and deep learning denoising; 2) Joint 3D
Reconstruction and Pose Estimation, covering the annotation of feature points
(c), the initialization of the point cloud and camera poses (d), alongside
simultaneous pose refinement (e) and 3D Gaussian training (f); 3) Point Cloud
Post-editing (g), to refine and finalize the reconstructed 3D model.

###  2.1 Spaceborne 3D reconstruction and pose estimation

In space surveillance and servicing missions, such as satellite maintenance,
debris removal, and formation flying, accurately estimating the pose of non-
cooperative entities like spacecraft or debris is essential for safe and
efficient operations. Monocular cameras stand out among pose estimation
techniques for their low weight and power consumption, offering a simpler
alternative to complex sensor systems such as differential GNSS[1], LiDAR, or
stereo-vision cameras. However, applying conventional computer vision
techniques [2, 3], originally designed for terrestrial environments, to space
images poses significant challenges, given the simple textures, high noise
levels, and symmetry commonly present in such images.

Deep learning has recently emerged as the leading method for spaceborne
monocular pose estimation. This advancement has been significantly supported
by the availability of open-source datasets of synthetic spacecraft images
(e.g., SPEED[4], SPEED+[5], and SHIRT[6]), which are generated through
computer graphics techniques. A variety of CNN-based methods[4, 7, 8, 9],
developed and validated in ground-based flight testbeds simulating space-like
conditions, have shown promising results. These CNN algorithms are further
enhanced by incorporating 3D prior knowledge, such as point clouds[9] or
wireframe models[10, 11], resulting in more robust pose estimation.

Despite these advances, much of the existing research relies on prior
knowledge of the target’s structure and geometry, limiting its applicability
to monitoring unknown space objects like debris. Recent developments are
bridging vision-based pose estimation with 3D reconstruction to enable the
characterization of uncharted spacecraft and asteroids[12, 13]. Leveraging
extensive training on synthetic datasets, cutting-edge deep learning
techniques are now capable of extracting both the shape and pose of unknown
space objects from merely a single 2D image [14]. This progress marks a
significant leap towards more versatile space surveillance capabilities.

###  2.2 NeRF and 3D Gaussian splatting

3D reconstruction is a critical area in computer vision, traditionally
dominated by multi-view geometry process such as Structure from Motion
(SFM)[15]. These methods typically extract feature descriptors (e.g.,
SIFT[16]) and establish correspondences between images using the RANSAC
algorithm[17], followed by camera parameter estimation and point cloud
generation through bundle adjustment[18]. Despite their effectiveness, they
often struggle with capturing detailed textures, modeling complex lighting and
sensor noise, and rendering new viewpoints accurately.

The advent of Neural Radiance Fields (NeRF)[19] introduced a paradigm shift in
3D reconstruction and view synthesis by using deep neural networks to model
scenes’ volumetric density and color. NeRF stands out for its ability to
generate high-quality images from limited viewpoints, delivering realistic
renderings that handle complex lighting and transparency with finesse, and
capturing scene details with exceptional accuracy. However, NeRF’s reliance on
dense neural network architectures brings about substantial computational
demands, slowing down both training and rendering processes. Moreover, NeRF
does not offer an explicit scene representation, complicating straightforward
scene editing tasks. To address these issues, significant efforts have been
made to refine NeRF, aiming at accelerating the training[20] and rendering[21,
22, 23], enhancing the model’s interpretability and editability[24], and
improving the precision of pose estimation[25, 26, 27, 28, 29].

More recently, the emergence of 3D Gaussian splatting techniques[30] has
offered a promising alternative, noted for its rapid training, real-time
rendering capabilities, and user-friendly editing features. 3D Gaussian
splatting employs differentiable, splittable, and movable 3D Gaussian points,
along with spherical harmonics derived from point clouds, to explicitly
represent complex scenes. This explicit representation accelerates scene
rendering compared to fully connected neural networks while simultaneously
simplifying the process of scene editing[31, 32, 33], marking a significant
evolution in the efficient manipulation and modification of reconstructed 3D
environments.

###  2.3 Imaging through turbulence

Atmospheric turbulence significantly compromises image quality in ground-based
telescope observations. For decades, efforts to address this challenge have
led to the development of various mitigation techniques. One widely adopted
approach is Adaptive Optics (AO) [34], which utilizes deformable mirrors to
actively compensate for wavefront aberrations caused by atmospheric
turbulence. However, the implementation of AO systems requires sophisticated
and costly hardware, confining their use primarily to large astronomical
observatories.

Classical algorithmic methods [35, 36, 37] for turbulence reduction often
employ lucky imaging[38], a strategy of capturing multiple images and
selecting those least affected by turbulence according to predefined image
quality metrics. This technique capitalizes on the rare moments when
atmospheric conditions yield clearer images. Recent advancements in rapid,
physics-based turbulence simulators [39, 40, 41] have facilitated the rise of
learning-based approaches[42, 43, 44, 45] as an effective alternative. By
utilizing extensive datasets of turbulence-affected images, these approaches
leverage neural networks to decipher and learn the intricate patterns of
turbulence degradation. The ability to harness the power of deep learning and
large-scale data enables these methods to surpass classical techniques, paving
a new way for enhanced imaging through turbulence.

##  3 Method

This paper aims to accomplish the 3D reconstruction of space objects based on
monocular images captured by a ground-based telescope. In contrast to common
3D reconstruction tasks in computer vision, ground-based telescope
observations are significantly contaminated by atmospheric turbulence and
light pollution, as shown in Fig.2(a). Turbulence disrupts light paths through
the atmosphere, while light pollution from artificial sources reduces the
visibility of celestial objects. Moreover, the fixed and limited viewpoints
from the telescope introduce further complexity, resulting in a super
challenging 3D reconstruction problem from constrained, distorted, blurry, and
noisy images.

In response to these challenges, we propose a novel 3D reconstruction approach
based on the cutting-edge Gaussian splatting algorithm. This approach
encompasses three major steps: 1) pre-processing of raw telescope images
(Fig.2(b)), 2) joint pose estimation and 3D Gaussian splatting reconstruction
(Fig.2(c)-(f)), and 3) post-reconstruction editing of Gaussian point clouds
(Fig.2(g)). The implementation details of each step are explained below.

Figure 3: Post-editing significantly improves the accuracy of the
reconstructed 3D model. (a) Primitively reconstructed point cloud amidst
highly complex noise conditions, where red box indicates the region of
satellite. (b) Point cloud after spatial region filtering. (c) Point cloud
after statistical outlier filtering.

###  3.1 Pre-processing of Raw Telescope Images

Pre-processing raw images from ground telescopes is a critical first step to
mitigate distortions and enhance image quality for subsequent reconstruction,
as demonstrated in Fig.2(b). The pre-processing involves the following steps:

a) Centering and lucky imaging: To address distortion in telescope data, we
initially apply lucky imaging to sequences of short-exposure raw images using
image processing software like Autostakkert[46]. Lucky imaging consists of two
main activities. The first involves ranking images by contrast and clarity to
select a subset of high-quality frames. These selected frames represent
moments least affected by the atmospheric turbulence and light pollution, thus
identifying the sharpest images. The second involves registering and stacking
these frames to increase the signal-to-noise ratio. Given that space targets
often shift due to atmospheric dynamics and telescope instability, we center
the images on their centroids before applying lucky imaging[47].

b) Image sharpening with wavelet decomposition: Although stacking improves
signal clarity by reducing noise, it inevitably diminishes high-frequency
details. To counteract this effect, we sharpen the stacked images by
amplifying the high-frequency features through Gaussian wavelet decomposition,
accomplished by using the public software RegiStax[48]. This process also
helps diminish the background light pollution by attenuating low-frequency
features.

c) Image enhancement with video restoration transformer: Image sharpening can
introduce additional high-frequency noise or artifacts. To address this, we
apply the state-of-the-art Video Restoration Transformer (VRT)[49], a
foundational model for video processing pre-trained on extensive datasets, to
further denoise the post-sharpening images.

###  3.2 Joint pose estimation and 3D reconstruction

In astrophotography, space objects orbit in high speed, whereas telescopes
remain stationary on the Earth. For simplicity in adapting cutting-edge
computer vision techniques, we simplify this dynamic by assuming the space
object remains stationary and the telescope/camera moves around it for data
acquisition. This process involves the following key steps:

a) Annotation of video’s feature points: We begin by manually annotating
feature points on the satellite, such as the corners of a satellite’s solar
wings, highlighted in red in Fig.2(c). Atmospheric distortion and the
satellite’s geometric simplicity result in images with limited texture and
blurred edge details. Consequently, feature extraction using standard and
neural network-based detectors like SIFT[2] and SuperPoint[50, 51] proves
unreliable, and manually annotating these points becomes necessary. We leave
the automated feature point annotation for future work.

b) Initializing camera poses and 3D Gaussians with SfM: Following the
identification of feature points, we employ the Structure from Motion
(SfM)[15] algorithm to create an initial 3D point cloud of the target and
determine the camera’s pose and focal length for each frame. These initial
estimates are marked by red for poses and focal lengths and green for the
point cloud in Fig.2(d). Given the long focal lengths of ground-based
telescopes, we often encounter inaccuracies in camera-to-target distance
estimation. To counter this, we refine our pose estimates utilizing satellite
trajectory calculations informed by astrodynamics. In instances where poses
cannot be determined due to unavailable or inaccurate feature point
annotations—resulting in the failure of SfM convergence—we resort to
interpolation to fill the missing values. Quadratic interpolation is applied
for the camera’s translational vector, $T$, and spherical linear interpolation
is used for the camera’s rotation matrix, $R$. The updated initialization is
depicted in green in Fig.2(e).

c) Alternating between pose estimation and 3D reconstruction: The Gaussian
splatting algorithm is then used for reconstructing the space object in 3D. To
avoid overfitting, we carefully manage the growth of Gaussian points and
concurrently update camera poses. Initially, we train the 3D Gaussians for
1,000 epochs, applying a loss function that combines L1 and SSIM metrics. Pose
adjustments are then performed using the Branch and Bound (BnB)[52] algorithm,
updating camera’s orientations and translations while fixing camera-to-target
distances. We then alternate between updating 3D Gaussians and refining camera
poses every 500 epochs, as depicted in Fig.2(f). Such iteration increases the
number of Gaussian points progressively and gradually fits the point cloud to
the target satellite. After completing 22 cycles (12,000 epochs), we stop
point cloud growth (splitting) and pose adjustments, focusing solely on
refining the parameters of existing Gaussian points until convergence after
30,000 epochs.

TABLE I: Comparison of our method and NeRF on five simulation datasets and on-sky observations of China’s Space Station. The optimal result in each column (with training and validation views considered separately) is highlighted in bold. |  | Simulation 1 | Simulation 2 | Simulation 3 | Simulation 4 | Simulation 5 | On-sky
---|---|---|---|---|---|---|---
| Method | PSNR$\uparrow$ | SSIM$\uparrow$ | LPIPS$\downarrow$ | PSNR$\uparrow$ | SSIM$\uparrow$ | LPIPS$\downarrow$ | PSNR$\uparrow$ | SSIM$\uparrow$ | LPIPS$\downarrow$ | PSNR$\uparrow$ | SSIM$\uparrow$ | LPIPS$\downarrow$ | PSNR$\uparrow$ | SSIM$\uparrow$ | LPIPS$\downarrow$ | PSNR$\uparrow$ | SSIM$\uparrow$ | LPIPS$\downarrow$
Dirty Image |  | 23.28 | 0.07 | 0.41 | 22.23 | 0.07 | 0.36 | 23.02 | 0.07 | 0.36 | 23.16 | 0.07 | 0.36 | 24.19 | 0.13 | 0.38 | $\times$ | $\times$ | $\times$
Pre-Processed |  | 27.60 | 0.53 | 0.10 | 26.71 | 0.55 | 0.12 | 27.92 | 0.53 | 0.11 | 28.3 | 0.52 | 0.11 | 23.08 | 0.54 | 0.29 | $\times$ | $\times$ | $\times$
Train | Ours | 26.44 | 0.92 | 0.10 | 24.88 | 0.91 | 0.16 | 27.30 | 0.92 | 0.14 | 27.69 | 0.91 | 0.10 | 25.47 | 0.92 | 0.15 | 36.97 | 0.98 | 0.01
NeRF | 25.39 | 0.65 | 0.13 | 19.10 | 0.61 | 0.19 | 25.00 | 0.62 | 0.15 | 26.97 | 0.64 | 0.12 | 23.21 | 0.67 | 0.18 | 36.60 | 0.97 | 0.03
Validation | Ours | 26.32 | 0.92 | 0.10 | 24.65 | 0.90 | 0.16 | 27.12 | 0.91 | 0.14 | 27.51 | 0.91 | 0.11 | 25.38 | 0.92 | 0.16 | 36.80 | 0.98 | 0.01
NeRF | 17.42 | 0.55 | 0.36 | 17.15 | 0.56 | 0.33 | 21.40 | 0.59 | 0.20 | 22.64 | 0.60 | 0.19 | 21.40 | 0.65 | 0.21 | 23.21 | 0.67 | 0.18

###  3.3 Post-reconstruction editing of gaussian point clouds

The reconstructed 3D Gaussian point clouds often include stray points, as
illustrated in Fig.3(a). These stray points come from the residual background
noise and artifacts in the telescope images used in Gaussian splatting
training. To achieve a cleaner reconstruction, we apply geometric filters to
remove these outliers, employing two primary strategies. This step effectively
capitalizes on the strengths of 3D Gaussian splatting in editability.

a) Spatial region filter: Acknowledging the compact structure of satellites,
we begin the refinement process with a spatial region filter. This involves
defining a spherical boundary around the target and discarding points outside
this specified perimeter. Considering that solar wings usually mark the
outermost boundary of a satellite, we set the radius of our spherical region
to be 1.1 times the radius derived from the initial feature points at the
corners of the solar wings. The center is placed at the mean position of these
points. This process results in a more concentrated point cloud, as visualized
in Fig.3(b).

b) Statistical outlier filter: We then implement a statistical outlier filter
to eliminate isolated noise points—those significantly distant from their
neighbors, as visualized in Fig. 3(c). This approach identifies outliers based
on each point’s average distance to its $k$ nearest neighbors, $\bar{d_{i}}$.
Given the mean ($\mu$) and standard deviation ($\sigma$) of these distances
across the cloud, we set a threshold $\tau=\mu+\alpha\cdot\sigma$, where
$\alpha$ is a multiplier (set to 1 in our method), and flag out points with
$\bar{d_{i}}>\tau$.

After applying these filters, we resume Gaussian splatting training on the
refined point cloud for an additional 500 epochs, repairing any
discontinuities introduced by the filtering. This adjustment further enhances
the overall quality of the reconstructed point cloud.

##  4 Experiments

In this section, we report the experiments conducted to evaluate our 3D
reconstruction method, with NeRF serving as the benchmark for comparison. We
utilized three standard imaging metrics—SSIM, PSNR, and LPIPS—on 2D views of
the reconstructed volumes to gauge the reconstruction accuracy. Our method
demonstrates superior reconstruction quality and robustness on both synthetic
ground telescope satellite images and real observational data of China’s Space
Station.

###  4.1 Simulation Results

Synthetic Ground Telescope Satellite Images  For simulation, we created five
distinct 3D satellite models in Blender, an open-source 3D computer graphics
software, and produced five datasets of corresponding ground telescope images.
Each model is roughly 60 meters in size, positioned in a circular low-Earth
orbit at approximately 650 kilometers altitude, equivalent to a radius of 7100
kilometers. Since the telescope is stationary on the Earth, it captures only a
13∘ segment of the satellite’s full 360∘ trajectory. Our virtual telescope,
equipped with a 0.35-meter aperture and a 3.2-meter focal length, allowed us
to render 700 high-quality images for each model. To simulate atmospheric
conditions, 20 frames were perturbed for each original image, introducing
effects of turbulence and light pollution, thus totaling 14,000 frames with
distorted imagery. Turbulence in each frame varied, with a Fried parameter
(coherence length) between 0.07 and 0.35 meters. Light pollution contributed
an additional 5%-7% to the background intensity, measured against the
satellite’s brightness. Our dataset will be made publicly available after the
paper’s acceptance.

Implementation Details  In preprocessing the synthetic datasets, we fine-tuned
the lucky imaging parameters to distill 148 clean images from a pool of 14,000
distorted frames. To evaluate the efficacy of our preprocessing workflow, we
first assessed the improvements in image quality after the preprocessing step
based on the Ground Truth images, with the results detailed in Table I.
Subsequently, we selected one out of every ten clean frames, in total 15
frames, for Gaussian splatting or NeRF training, with the remaining images set
aside for 3D reconstruction validation. The camera poses optimized through our
combined approach of Gaussian splatting training and Branch and Bound (BnB)
pose refinement were saved and employed in NeRF training. This ensured the
elimination of pose estimation accuracy as a variable in performance
differences observed between the methodologies.

Figure 4: Results of 3D reconstruction and view synthesis for the 1st
simulation dataset. (a) Simulated raw images featuring atmospheric turbulence,
light pollution, and sensor noise. (b) Rendering quality comparison in
training views between our method and NeRF. (c) Rendering quality comparison
in validation views between our method and NeRF. (d) Imaging metrics evaluated
across various frames, encompassing both training and validation views. Our
method demonstrated outstanding precision in both training and validation
views, while NeRF encountered difficulties in rendering unseen viewpoints.
Figure 5: Results of 3D reconstruction and view synthesis for the China’s
Space Station Observation. (a) Raw images from the amateur telescope. (b)
Rendering quality comparison in training views between our method and NeRF.
(c) Rendering quality comparison in validation views between our method and
NeRF. (d) Imaging metrics evaluated across various frames, encompassing both
training and validation views. The performance discrepancy between NeRF and
our method becomes more significant.

3D Reconstruction  Figure.4 illustrates the results of 3D reconstruction for
the first simulation dataset, comparing the performance of both NeRF and our
method through selected views of the reconstructed volumes. NeRF reveals
reasonable reconstruction quality in the training views (Fig.4(b)) but
struggles when rendering novel viewpoints (Fig.4(c)) due to overfitting
issues. In contrast, our method demonstrates superior generalization ability,
excelling in both training and validation views. This indicates that our
reconstructed 3D Gaussian point cloud precisely captures the satellites’ 3D
shapes. The effectiveness of our method is also supported by the curves of
image quality metrics (SSIM, PSNR, LPIPS) across various frames, illustrated
in Fig.4(d). Extending this experiment to all five datasets, our technique
consistently surpasses NeRF in all evaluated metrics, as presented in Table I.
These findings affirm our approach’s efficiency and accuracy in 3D
reconstruction challenges. We would also like to highlight that our method
significantly outpaces NeRF in terms of training speed. It takes approximately
7 minutes to reconstruct a 3D volume with our approach on a single NVIDIA’s
RTX 4090—around 3 minutes for Gaussian splatting training and an additional 4
minutes for BnB pose search. In contrast, training a basic NeRF model requires
more than 4 hours.

###  4.2 On-Sky Results

We applied our algorithm to real observations of China’s Tiangong Space
Station, a low-Earth-orbit satellite with publicly available geometric
specifications.

Figure 6: Imaging Instrument: the Celestron C14HD telescope paired with a
QHY5III678M planetary camera.

Observation of China’s Tiangong Space Station  The observation took place on
September 15, 2023, during a pass of the Tiangong Station from 20:48:30 to
20:51:17 (UTC).

The observation site is situated near Miyun Observatory on the northern bank
of Miyun Reservoir, Beijing, China, chosen for its advantageous seeing
conditions. The Tiangong Station was 389.4 km away at its closest, reaching a
maximum elevation of 78.3 degrees. We used a Celestron C14HD telescope (0.35m
diameter, f/11) paired with a QHY5III678M planetary camera for this
observation. This setup, with a telescope focal length of 3.91 meters and a
camera pixel pitch of 2 micrometers, provided a spatial sampling rate of 0.106
arcsec/pixel, enabling us to image the Tiangong Station at 0.2 m/pixel
resolution at its nearest point.

Figure 7: Composition of the Chinese Space Station (CSS). The CSS comprises four primary modules: 1 - Tianhe Core Module, 2 - Mengtian Laboratory Module, 3 - Wentian Laboratory Module, 4 - Tianzhou-6 Cargo Spacecraft. (a) An official image of the CSS. (b) A rendering derived from our 3D reconstruction of the CSS. TABLE II: Metrology of the China’s Space Station modules and their solar wings. | Measured Value | Ground Truth[53]
---|---|---
Solar wing angle |  |
Tianhe Core Module | 0.78 | $\times$
Mengtian Laboratory Module | 63.01 | $\times$
Wentian Laboratory Module | 68.71 | $\times$
Tianzhou-6 | 3.22 | $\times$
Length |  |
Tianhe Core Module | 16.11 | 16.6
Mengtian Laboratory | 17.30 | 17.88
Wentian Laboratory | 17.30 | 17.88
Tianzhou-6 | 9.71 | 10.6
Diameter |  |
Tianhe Core Module | 5.65 | 4.2
Mengtian Module | 5.50 | 4.2
Wentian Module | 5.63 | 4.2
Tianzhou 6 | 3.30 | 3.35
The length of the solar wings |  |
Tianhe Core Module | 25.44 | $\times$
Mengtian Laboratory Module | 55.62 | 55
Wentian Laboratory Module | 55.62 | 55
Tianzhou 6 | 4.89 | $\times$
TABLE III: Ablation study on five simulated data. For clarity, the optimal result in each column (with training and validation views considered separately) is highlighted in bold, while the next best result is underlined. |  | Simulation 1 | Simulation 2 | Simulation 3 | Simulation 4 | Simulation 5
---|---|---|---|---|---|---
| Method | PSNR$\uparrow$ | SSIM$\uparrow$ | LPIPS$\downarrow$ | PSNR$\uparrow$ | SSIM$\uparrow$ | LPIPS$\downarrow$ | PSNR$\uparrow$ | SSIM$\uparrow$ | LPIPS$\downarrow$ | PSNR$\uparrow$ | SSIM$\uparrow$ | LPIPS$\downarrow$ | PSNR$\uparrow$ | SSIM$\uparrow$ | LPIPS$\downarrow$
Train | Ours | 26.44 | 0.92 | 0.10 | 24.88 | 0.91 | 0.16 | 27.30 | 0.92 | 0.14 | 27.69 | 0.91 | 0.10 | 25.47 | 0.92 | 0.15
No Sharpening | 24.22 | 0.90 | 0.15 | 24.34 | 0.54 | 0.20 | 25.91 | 0.90 | 0.16 | 27.63 | 0.90 | 0.17 | 24.42 | 0.80 | 0.15
No Gaussian initialization | 24.03 | 0.62 | 0.17 | 16.59 | 0.88 | 0.29 | 24.41 | 0.62 | 0.15 | 19.74 | 0.87 | 0.23 | 22.77 | 0.68 | 0.19
No pose refinement | 22.86 | 0.85 | 0.17 | 18.47 | 0.79 | 0.20 | 17.56 | 0.78 | 0.24 | 24.77 | 0.82 | 0.17 | 19.22 | 0.84 | 0.20
No Regulation of Gaussian | 21.78 | 0.86 | 0.15 | 18.41 | 0.82 | 0.23 | 23.55 | 0.87 | 0.14 | 23.14 | 0.69 | 0.18 | 22.66 | 0.87 | 0.16
No Point cloud Filtering | 27.86 | 0.64 | 0.11 | 25.76 | 0.63 | 0.16 | 27.38 | 0.64 | 0.14 | 28.00 | 0.62 | 0.10 | 25.82 | 0.70 | 0.15
Validation | Ours | 26.32 | 0.91 | 0.10 | 24.65 | 0.90 | 0.16 | 27.12 | 0.91 | 0.14 | 27.51 | 0.91 | 0.11 | 25.38 | 0.92 | 0.16
No Sharpening | 24.12 | 0.90 | 0.15 | 23.92 | 0.52 | 0.20 | 25.78 | 0.90 | 0.17 | 27.43 | 0.90 | 0.17 | 23.99 | 0.80 | 0.15
No Gaussian initialization | 19.48 | 0.50 | 0.23 | 16.37 | 0.78 | 0.30 | 19.91 | 0.50 | 0.23 | 18.77 | 0.77 | 0.26 | 17.34 | 0.53 | 0.27
No pose refinement | 22.85 | 0.85 | 0.17 | 18.39 | 0.78 | 0.20 | 17.39 | 0.78 | 0.24 | 24.34 | 0.81 | 0.18 | 19.09 | 0.84 | 0.20
No Regulation of Gaussian | 21.72 | 0.86 | 0.15 | 17.71 | 0.80 | 0.25 | 22.24 | 0.85 | 0.15 | 21.99 | 0.68 | 0.20 | 21.39 | 0.85 | 0.16
No Point cloud Filtering | 27.55 | 0.64 | 0.11 | 25.39 | 0.63 | 0.16 | 27.16 | 0.64 | 0.14 | 27.70 | 0.62 | 0.11 | 25.57 | 0.69 | 0.15

The observation utilized the ”Fake Polar Axis Method” with a German-type
equatorial mount (CGX-L), uniquely oriented not towards the north but towards
a precisely calculated azimuth relative to the apparent track of the space
object. This innovative method prevented the need for a meridian flip during
the observation and minimized field rotation, particularly beneficial for the
central part of the image sequence. During the 166-second observation period,
we captured images at an average rate of 87.8 fps with 5-millisecond
exposures. However, due to cache degradation, the frame rate varied, starting
higher at 93.5 fps and decreasing towards the end. This resulted in a dataset
of 13,800 raw images.

Implementation Details  Employing lucky imaging techniques, we obtained 157
frames from the inital pool of 13,800 dirty images. By further filtering out
images plagued by exceptionally low signal-to-noise ratios, we finalized a set
of 138 clean frames. Consistent with the strategy of our simulation
experiments, we selected one image for every ten frames for reconstruction
training, totaling 15 images. The remaining images were allocated to the
validation set. For each selected training image, we recorded the optimized
camera poses and conducted a parallel reconstruction with NeRF using identical
poses. Due to the lack of ground-truth images in on-sky experiments, we
assessed imaging metrics by directly comparing the reconstruction outcomes
with the pre-processed images.

3D Reconstruction and Metrology  The 3D reconstruction outcomes for the
Tiangong dataset, as shown in Fig.5, echo our simulation findings—NeRF
struggled with rendering unseen views, highlighting its challenges in
accurately capturing the satellite’s 3D structure. In contrast, our approach
achieved exceptional 3D reconstruction precision across both training and
validation views, with the performance discrepancy between NeRF and our method
being notably more pronounced in this real-world dataset than in simulation
scenarios. This contrast in image quality metrics is further elaborated in the
last column of Table I.

Additionally, we performed detailed measurements on four modules of the space
station (Tianhe, Mengtian, Wentian, and Tianzhou-6) and the angles and lengths
of their solar wings, utilizing our 3D Gaussian splatting-reconstructed
volume, as illustrated in Fig.7. To measure the modules’ sizes (length and
diameter), we densely sampled 2D views, pinpointing the frames that depicted
each module at its maximum length (from orthogonal perspectives) and
determining dimensions via pixel counts. The angles of the solar wings were
ascertained based on their relative positioning to the station’s main modules.
These metrology findings, outlined in Table II, align closely with the
officially released specifications

Figure 8: Ablation studies on the 1st simulation dataset. This figure
showcases the impact of each component in our workflow on the reconstructed
point clouds and novel view synthesis. (a) Results using our complete method.
(b) Excluding wavelet decomposition image sharpening results in a blurry
reconstruction. (c) Excluding SfM-based 3D Gaussian initialization introduces
ghosting effects. (d) Excluding simultaneous pose refinement causes
distortions and artifacts. (e) Excluding the regulation of 3D Gaussian growth
rate also leads to distortions and artifacts. (f) Without point cloud post-
editing, the background remains cluttered and noisy.

###  4.3 Ablation Study

Comprehensive ablation studies were carried out to ascertain the necessity of
each element within our 3D reconstruction framework. Fig.8 showcases the
consequences of excluding particular components from our pipeline, displaying
the reconstructed points on the left and representative rendered views on the
right for each subplot. From the top-left to the bottom-right in Fig.8(b)-(f),
the components omitted are wavelet decomposition sharpening (Sec.3.1.b), SfM-
based 3D Gaussian initialization (Sec.3.2.b), BnB search for pose refinement
(Sec.3.2.c), regulation of 3D Gaussian growth rate (Sec.3.2.c, splitting large
Gaussians every 100 epochs instead of 500 epochs), and point cloud filtering
(Sec.3.3). The exclusion of these steps visibly impairs the reconstruction
outcome, manifesting as blurriness, ghosting, fringe artifacts, and unclean
backgrounds, among other issues. Furthermore, Table III details the
quantitative results of these ablation studies, underscoring the degradation
in reconstruction quality when vital steps are skipped. The ablation results
for lucky imaging (Sec.3.1.a) and feature annotation (Sec.3.2.a) are not
presented, as omitting these steps resulted in the inability to reconstruct
anything.

##  5 Conclusion

In this paper, we introduced a novel 3D imaging framework designed for
reconstructing satellites from ground telescope observations. Central to this
approach is the utilization of the cutting-edge Gaussian splatting volume
rendering algorithm, complemented by sophisticated pre-processing,
simultaneous pose refinement, and post-editing techniques to effectively
manage the significant image distortions caused by atmospheric turbulence,
light pollution, and other factors. We validated our method using both
synthetic datasets and actual observations of China’s Tiangong Space Station.

A notable limitation of our current methodology is the need for manual
annotation of feature points, a step necessitated by the considerable
ambiguity present in raw telescope images. Future efforts will focus on
addressing this challenge by developing a feature extraction neural network
specifically designed for imaging through turbulence scenarios. We aim to
finally establish a fully automated end-to-end workflow from video capture to
3D model generation.

We believe the insights from this research can significantly benefit 3D
reconstruction tasks across various scientific fields facing complex image
distortions. Potential applications include, for example, remote sensing,
biomedical ultrasound/photoacoustic imaging and deep tissue imaging in
microscopy.

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# Reduction mod. 2 of del Pezzo lattices

Arnaud Beauville  Université Côte d’Azur
CNRS – Laboratoire J.-A. Dieudonné
Parc Valrose
F-06108 Nice cedex 2, France<EMAIL_ADDRESS>
###### Abstract.

For an even lattice $L$, the form $v\mapsto\frac{1}{2}v^{2}$ induces a
quadratic form $q$ on the $\mathbb{F}_{2}$-vector space $L/2L$. For the
lattices associated to some particular root systems, we show that reduction
mod. 2 induces a bijection between the roots of $L$ and the vectors of $L/2L$
with $q=1$, and an isomorphism $\operatorname{O}(L)/\\{\pm 1\\}\vbox{\hbox
to22.76228pt{\hfill{$\scriptstyle\sim$}\hfill} \nointerlineskip\hbox
to22.76228pt{{\hfill$\longrightarrow$\hfill}} }\operatorname{O}(L/2L)$.

##  1\. Introduction

Let $L$ be an even lattice. The form $v\mapsto\frac{1}{2}v^{2}$ on $L$ induces
a quadratic form $q$ on $L_{2}:=L/2L$; we want to investigate what information
on $L$ we get from the quadratic $\mathbb{F}_{2}$-vector space $L_{2}$. In
general not much can be said, but for the lattices associated to certain root
systems (specifically $A_{1}\times A_{2},A_{4},D_{5},E_{6},E_{7},E_{8}$), the
relation is tight: the roots correspond to the vectors of $L_{2}$ with $q=1$
(with a grain of salt for $E_{7}$), and the group $\operatorname{O}(L)$ is
very close to the orthogonal group $\operatorname{O}(L_{2})$. We call the
corresponding lattices _del Pezzo lattices_ , since they appear naturally in
the Picard group of del Pezzo surfaces [M, § 25], but this will play no role
in this note.

$\bullet$ Added in version 2: Igor Dolgachev pointed out the reference [G],
which contains much more complete results (with a much more elaborate
technology). I leave this preprint as a possible elementary introduction to
the subject.

##  2\. Reduction mod. $2$ of roots

Let $L$ be an even lattice. We endow $L_{2}:=L/2L$ with the bilinear form $(\
|\ )$ deduced from the scalar product on $L$, and the quadratic form $q$
deduced from the form $v\mapsto\frac{1}{2}v^{2}$. We have
$q(x+y)=q(x)+q(y)+(x|y)$ for $x,y$ in $L_{2}$.

######  Lemma.

Let $R$ be a root system of type $A,D,E$, and $L$ the corresponding root
lattice.

$1)$ The inclusion $R\subset L$ induces an injection $R/\\{\pm
1\\}\hookrightarrow L_{2}$.

$2)$ The reduction map
$\rho:\operatorname{O}(L)\rightarrow\operatorname{O}(L_{2})$ induces an
injection $\operatorname{O}(L)/\\{\pm
1\\}\hookrightarrow\operatorname{O}(L_{2})$.

Proof : 1) Let $\alpha,\beta\in R$ such that $\alpha-\beta=2\lambda$, with
$\lambda\in L$, $\lambda\neq 0$. Then
$8\leq(\alpha-\beta)^{2}=4-2(\alpha\cdot\beta)$, hence
$(\alpha\cdot\beta)\leq-2$; by the Cauchy-Schwarz inequality this implies
$\beta=-\alpha$.

2) Let $u\in\operatorname{O}(L)$ such that $\rho(u)=1$. For each root
$\alpha$, $u(\alpha)$ is a root, with $u(\alpha)-\alpha\in 2L$. By 1), this
implies $u(\alpha)=\varepsilon\alpha$, with $\varepsilon=\pm 1$. Let $\beta$
be another root with $(\alpha\cdot\beta)\neq 0$; since $(u(\alpha)\cdot
u(\beta))=(\alpha\cdot\beta)$, we have $u(\beta)=\varepsilon\beta$. Taking for
$\alpha$ a simple root, it follows that $u(\gamma)=\varepsilon\gamma$ for all
simple roots $\gamma$, hence $u=\varepsilon\cdot 1_{L}$.∎

From now on, we will only consider a particular type of lattices, which we
call _del Pezzo lattices_. For $3\leq n\leq 8$, let $L_{0}$ be the free
$\mathbb{Z}$-module with basis $E_{0},\ldots,E_{n}$, endowed with the
Lorentzian scalar product defined by $E_{0}^{2}=-1$, $E_{i}^{2}=1$ for $i\geq
1$, and $(E_{i}\cdot E_{j})=0$ for $i\neq j$. Let $K$ be the element
$3E_{0}-\sum\limits_{i\geq 1}E_{i}$. We put $L=K^{\perp}\subset L_{0}$; this
is an even lattice, of discriminant $9-n$. The vectors of square 2 form a root
system $R\subset L$, of type $A_{1}\times A_{2},A_{4},D_{5},E_{6},E_{7},E_{8}$
for $n=3,\ldots,8$ [M, § 25].

Let $\varphi:R\rightarrow L_{2}$ be the reduction mod. 2. By the Lemma,
$\varphi$ induces an injective map $R/\\{\pm 1\\}\hookrightarrow L_{2}$; its
image is contained in $q^{-1}(1)$.

######  Proposition 1.

$\varphi$ induces a bijection from $R/\\{\pm 1\\}$ onto $q^{-1}(1)$ if $n\neq
7$, and onto $q^{-1}(1)\smallsetminus\\{k\\}$ if $n=7$, where $k$ is the
nonzero element of the kernel of $(\ |\ )$.

Proof : Let $e_{0},\ldots,e_{n}$ be the images of $E_{0},\ldots,E_{n}$ in
$L_{0}/2L_{0}$; then $L_{2}$ is the orthogonal in $L_{0}/2L_{0}$ of the vector
$k=\sum e_{i}$. We have

$q(e_{0}+e_{i})=0\,,\quad q(e_{i}+e_{j})=1\quad\mbox{for }0<i<j\,.$

The bilinear form $(\ |\ )$ on $L_{2}$ is non-degenerate if $n$ is even; if
$n$ is odd its kernel is $\\{0,k\\}$.

Let $v\in q^{-1}(1)$. Then $v=\sum\limits_{i\in I}e_{i}$, with $\\#I=m$ even.
We put $E_{I}:=\sum\limits_{i\in I}E_{i}$.

a) Suppose $0\notin I$. If $m=2$, $v=e_{i}+e_{j}=\varphi(E_{i}-E_{j})$. If
$m=6$, $v=\varphi(2E_{0}-E_{I})$.

b) Suppose $0\in I$. If $m=4$, $v=\varphi(2E_{0}-E_{I})$. If $m=n=8$,
$v=\varphi(4E_{0}-E_{I}-2E_{\ell})$, with $\ell\notin I$.

c) It remains to show that for $n=7$, the vector $k=\sum e_{i}$ cannot be
written $\varphi(\alpha)$ for some $\alpha\in R$. Indeed we would have
$\alpha=\sum n_{i}E_{i}$ with each $n_{i}$ odd; but then $n_{i}^{2}\equiv 1\
\operatorname{mod.}\,8$, hence $\alpha^{2}\equiv 6\ \operatorname{mod.}\,8$,
contradicting $\alpha^{2}=2$.∎

It is instructive to check the equality $\frac{1}{2}\\#R=\\#q^{-1}(1)$ (or
$\\#q^{-1}(1)-1$ for $n=7$) in each case:

a) $n=2m$.

In this case the form $(\ |\ )$ is non-degenerate; the order of $q^{-1}(1)$ is
$2^{m-1}(2^{m}+(-1)^{a})$, where $a$ is the _Arf invariant_
$\operatorname{Arf}(q)$: if
$(\delta_{1},\ldots,\delta_{m};\varepsilon_{1},\ldots,\varepsilon_{m})$ is a
symplectic basis of $L_{2}$, $a=\sum q(\delta_{i})q(\varepsilon_{i})$.

We can take for instance $\delta_{k}:=\sum\limits_{i=0}^{2k-1}e_{i}$,
$\varepsilon_{k}=\sum\limits_{i=0}^{2k-2}e_{i}+e_{2k}$. Then
$q(\delta_{k})=q(\varepsilon_{k})=0$ for $k$ odd, $1$ for $k$ even, hence
$\operatorname{Arf}(q)=0$ for $n=4$ or $6$, $\operatorname{Arf}(q)=1$ for
$n=8$. Therefore we get $\\#q^{-1}(1)=10,36,120=\frac{1}{2}\\#R$ for
$n=4,6,8$.

b) $n=3$ or $7$.

For $n$ odd, the vector $k$ is in $L_{2}$, and spans the kernel of the form
$(\ |\ )$. For $n=3$ or $7$ we have $q(k)=1$. The involution $v\mapsto v+k$
exchanges $q^{-1}(1)$ and $q^{-1}(0)$, hence
$\\#q^{-1}(1)=\frac{1}{2}\\#L_{2}=2^{n-1}$. Indeed $\\#(A_{1}\times
A_{2})=2+6=8$, and $\\#E_{7}=126=2(2^{6}-1)$.

c) $n=5$.

Here $q(k)=0$. The form $q$ descends to a non-degenerate form $\bar{q}$ on
$L_{2}/\langle{k}\rangle$; we claim that $L_{2}/\langle{k}\rangle$ is
isomorphic to the quadratic $\mathbb{F}_{2}$-space $L^{\prime}_{2}$ associated
to $A_{4}$. Indeed, the subspace $N$ of $L_{2}$ spanned by $e_{i}+e_{j}$ for
$i<j<5$, with the quadratic form induced by $q$, is isomorphic to
$L^{\prime}_{2}$, and the quotient map $N\rightarrow L_{2}/\langle{k}\rangle$
is an isomorphism. Thus
$\\#q^{-1}(1)=2\\#\bar{q}^{-1}(1)=\\#A_{4}=20=\frac{1}{2}\\#D_{5}$.

##  3\. Orthogonal groups

######  Proposition 2.

Let $L$ be a del Pezzo lattice of rank $\geq 4$. The reduction map
$\rho:\operatorname{O}(L)\rightarrow\operatorname{O}(L_{2})$ induces an
isomorphism of $\operatorname{O}(L)/\\{\pm 1\\}$ onto
$\operatorname{O}(L_{2})$.

Proof : The induced map is injective by the Lemma, so we have to prove that
$\rho$ is surjective.

a) $n=4,6,8$.

In this case the form $(\ |\ )$ is non-degenerate. By [D, § 10], the group
$\operatorname{O}(L_{2})$ is generated by the reflections $r_{v}:x\mapsto
x+(x|v)v$ for $v$ in $L_{2}$ with $q(v)=1$ – except for one case that we will
consider below. By Proposition 1, we have $v=r(\alpha)$ for some root
$\alpha$; then $r_{v}$ is the reduction mod. 2 of the reflection with respect
to $\alpha$.

The exception occurs for $n=4$, if $L_{2}$ contains a 2-dimensional subspace
$N$ such that $q_{|N}=0$ (_loc. cit._). This does not happen in our case,
since the nonzero vectors $v$ with $q(v)=0$ are $f_{0}:=k+e_{0}$ and
$f_{i}:=e_{0}+e_{i}$ for $1\leq i\leq 4$, and they satisfy $(f_{i}|f_{j})=1$
for $i\neq j$. ∎

b) $n=3,7$.

In this case the nonzero element $k$ of $\operatorname{Ker}(\ |\ )$ satisfies
$q(k)=1$. Let $H$ be a hyperplane in $L_{2}$ such that
$L_{2}=H\oplus\mathbb{F}_{2}k$, and let $p_{H}$ be the projector from $L_{2}$
onto $H$. The restriction of $(\ |\ )$ to $H$ is a symplectic form, and the
map $u\mapsto p_{H}\circ u_{|H}$ is an isomorphism from
$\operatorname{O}(L_{2})$ onto $\operatorname{Sp}(H)$ [D, § 7]; a transvection
$x\mapsto(x|v)v$ in $\operatorname{Sp}(H)$ corresponds to the reflection with
respect to $v+(1+q(v))k$. Since $\operatorname{Sp}(H)$ is generated by
transvections, $\operatorname{O}(L_{2})$ is generated by reflections, and we
conclude as in a) (note that the reflection with respect to $k$ is the
identity).

c) $n=5$.

Here it is simpler to use a counting argument. The group
$\operatorname{O}(L)/\\{\pm 1\\}$ is isomorphic to the Weyl group of $D_{5}$,
that is, the semi-direct product $(\mathbb{Z}/2)^{4}\rtimes\mathfrak{S}_{5}$
[B]. As we have seen in c) above, the space $L_{2}/\\{k\\}$ with the form
induced by $q$ is isomorphic to the quadratic $\mathbb{F}_{2}$-space
$L^{\prime}_{2}$ for $n=4$; the quotient map induces a surjective map
$\pi:\operatorname{O}(L_{2})\rightarrow\operatorname{O}(L^{\prime}_{2})$. The
kernel of $\pi$ consists of all automorphisms of $L_{2}$ of the form $x\mapsto
x+\ell(x)k$, with $\ell\in L_{2}^{*}$, $\ell(k)=0$; it is isomorphic to
$(\mathbb{Z}/2)^{4}$. By a), the group $\operatorname{O}(L^{\prime}_{2})$ is
isomorphic to the Weyl group $\mathfrak{S}_{5}$ of $A_{4}$. The result
follows.∎

######  Corollary.

$\rho$ induces an isomorphism of the Weyl group $W$ of $L$ onto
$\operatorname{O}(L_{2})$ for $n=4,5$ or $6$, and of $W/\\{\pm 1\\}$ onto
$\operatorname{O}(L_{2})$ for $n=7$ or $8$.

This follows from the relation between $W$ and $\operatorname{O}(L)$, see for
instance [B].

_Remarks_ $.-\ 1)$ For $n=3$, $\rho$ is surjective, with kernel $\\{\pm
1\\}\times\\{\pm 1\\}$. More precisely, we have $L=L^{\prime}\oplus
L^{\prime\prime}$, where $L^{\prime}$ and $L^{\prime\prime}$ are lattices of
type $A_{1}$ and $A_{2}$; then
$\operatorname{O}(L)=\operatorname{O}(L^{\prime})\times\operatorname{O}(L^{\prime\prime})\allowbreak=\\{\pm
1\\}\times(\mathfrak{S}_{3}\times\\{\pm 1\\})$, and
$\operatorname{O}(L_{2})=\operatorname{O}(L_{2}^{\prime})\times\operatorname{O}(L_{2}^{\prime\prime})=\\{1\\}\times\mathfrak{S}_{3}$.

2) Propositions 1 and 2 do not hold for other root systems. For $A_{n}$ for
instance, the number of roots is $n(n+1)$, while $\\#q^{-1}(1)$ is roughly
$2^{n-1}$. Similarly the order of $\operatorname{O}(L_{2})$ is roughly
$2^{n^{2}/2}$, much larger than $n!$.

## References

  * [B] N. Bourbaki : Lie groups and Lie algebras, Ch. 6. Springer-Verlag, Berlin, 2002.
  * [D] J. Dieudonné : La géométrie des groupes classiques. Ergebnisse der Math. und ihrer Grenzgebiete 5, 3rd ed. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1971.
  * [G] R. Griess : Quotients of Infinite Reflection Groups. Math. Ann. 263 , 267-278 (1983).
  * [M] Y. Manin : Cubic forms: Algebra, geometry, arithmetic. North-Holland Math. Library 4, 2nd ed. North-Holland, Amsterdam, 1986.


# High resolution spectroscopy of thulium atoms implanted in solid noble gas
crystals.

Vinod Gaire<EMAIL_ADDRESS>Mi Y Do<EMAIL_ADDRESS>Yiting Pei
<EMAIL_ADDRESS>Anthony Semenova<EMAIL_ADDRESS>Colin V. Parker
<EMAIL_ADDRESS>School of Physics, Georgia Institute of Technology,
Atlanta, Georgia, 30332, USA

###### Abstract

Optically active defects in solid-state systems have many applications in
quantum information and sensing. However, unlike free atoms, which have fixed
optical transition frequencies, the inhomogeneous broadening of the
transitions in solid-state environments limit their use as identical
scatterers for such applications. Here we show that crystals of argon and neon
prepared in a closed-cycle cryostat doped with thulium atoms at cryogenic
temperatures are an exception. High resolution absorption and emission
spectroscopy show that the 1140 nm magnetic dipole transition is split into
multiple components. The origin of this splitting is likely a combination of
different classes of trapping sites, crystal field effects within each site,
and hyperfine interactions. The individual lines have ensemble widths as small
as $0.6\textrm{ GHz}$, which temperature dependence and pump-probe
spectroscopy indicate is likely a homogeneous effect, suggesting inhomogeneity
is well below the GHz scale.

###### pacs:

Valid PACS appear here

##  I Introduction

With the advancement of quantum technologies, new sensing applications are
emerging that offer new capabilities as well as the ability to probe new
systems and environments. Various types of quantum sensors, including clocks
Kitching  _et al._ (2011); Ushijima  _et al._ (2015); Camparo (2007),
interferometers and gyroscopes Biedermann  _et al._ (2017); Charrière  _et
al._ (2012); McGuinness  _et al._ (2012), and magnetometers, Wickenbrock  _et
al._ (2013); Patton  _et al._ (2014) are being developed from isolated atoms
in vapor phase. However, building such sensors with isolated atoms requires a
hermetically sealed environment, often within ultra high vacuum, and yields
low atomic density. Alternatively, one can use solid-state platforms,
including superconducting quantum interferometer devices (SQuIDs) Zotev  _et
al._ (2007); Matlachov  _et al._ (2004); Gatteschi and Sessoli (2003); Sawicki
_et al._ (2011) and color centers in diamond Barry  _et al._ (2020); Lesik
_et al._ (2019); Childress and Hanson (2013); Acosta  _et al._ (2009); Taylor
_et al._ (2008). However, such systems typically experience material
inhomogeneity effects that lead to variation between fabricated devices or
between individual color centers Rogers  _et al._ (2014); van Dam  _et al._
(2019). A third alternative is atoms embedded in solid noble gases, where the
inertness of the host material limits inhomogeneous effects. Compared to vapor
phase atoms, these so-called “matrix-isolated” systems offer higher density,
can be deposited arbitrarily close to the surface of any desired substrate or
device, and can be co-deposited with molecules of interest. We are interested
in such systems with narrow linewidths for detection of the environment
surrounding the target atoms. In the majority of matrix isolation work with
atoms, however, alkali atoms have been used where the optical transitions
suffer broadening at the THz level despite the inert host Kanagin  _et al._
(2013).

In our previous work, we observed that when thulium atoms are trapped in argon
and neon crystals, the magnetic dipole transition between the fine structure
levels of the ground state (${}^{2}F_{7/2}\leftrightarrow{}^{2}F_{5/2}$) was
split into at least two components with linewidth less than 1 nm, far narrower
than in alkali metals, and limited by the spectrometer itself. The transition
was studied by excitation of higher energy levels with visible light and
detection of fluorescence on the infrared transition near 1140 nm Gaire  _et
al._ (2019). Since the partially filled 4$f$ shell is submerged underneath the
fully filled outer $5s$ and $5p$ shells, this transition is isolated from the
external environment and was also observed to be narrow when trapped in solid
and liquid helium Ishikawa  _et al._ (1997). In this work, we perform both
absorption and emission spectroscopy with far better resolution using laser-
induced fluorescence at 1140 nm, revealing that this line in fact splits into
many components with linewidths down to 0.6 GHz, likely due to a combination
of multiple host trapping sites and crystal field effects at each site.
Furthermore, we use the temperature dependence of the linewidth and the lack
of spectral hole burning to argue that this linewidth is homogeneously
broadened, suggesting MHz-scale population linewidths are possible in the
solid state if this system is cooled further. If this prediction proves
correct, the Tm:Ar system could contend for the least inhomogeneously-
broadened solid-state optical emitter known. Even the currently obtained value
of 0.6 GHz full-width half-maximum (FWHM) for an ensemble width is
significantly below most other solid-state systems; for example, Eu3+ in Y2O3
thin films (5.1 GHz at temperatures $<$ 100 mK Singh  _et al._ (2020)), NV
centers in diamond (5.6 GHz van Dam  _et al._ (2019)) and Yb3+:YAG ( 3.6
GHzBöttger  _et al._ (2016)). Comparison to solid-state platforms that can
resolve individual color centers (such as SiV Rogers  _et al._ (2014)) is less
direct, but considering the distribution to be normal with a standard
deviation of 300 MHz, we estimate an ensemble line would be Gaussian with FWHM
of 0.7 GHz.

##  II Experimental Setup

Figure 1: (a) Setup for spectroscopy of thulium atoms trapped in argon or neon
crystals. (b) Mount for clamping the fiber tip, which is glued into a small
copper tube. The circular groove is for holding an annular permanent magnet.

A schematic of the cryostat chamber and the basic beam paths for the
excitation lasers used in the matrix isolation experiment is shown in Fig. 1,
similar to that used in Gaire  _et al._ (2019). However, unlike in previous
work, here the inert gas (argon or neon) is condensed directly on a multimode
optical fiber tip (50 $\mu$m core diameter, NA 0.2). The thulium atoms are co-
deposited with the matrix by ablation with a pulsed laser. The fiber tip is
glued with thermally conductive epoxy into a copper tube with outer diameter
1.6 mm. A purpose-built mount attached to the second stage of the cryostat,
shown in the inset of Fig. 1, holds the tip-tube assembly clamped with indium
metal into a V-shaped groove. This mount also can hold an annular permanent
magnet (dimensions: 25.4 mm O.D., 7.9 mm I.D., 6.4 mm thickness) if needed for
doing experiments with magnetic field near the atoms.

In the beginning, when the setpoint temperature is reached, there is a period
of 30-60 seconds when only undoped argon or neon is deposited. During this
period, the deposition rate of the noble gas can be recorded in situ using a
diode laser. A diode laser illuminates the fiber at the other end of the
sample, and the reflected beam spot is detected as it comes out from the fiber
and separated from the incident light by a beam splitter. As the thickness of
the crystal increases, the reflected light intensity undergoes Fabry-Perot
oscillations as shown in Fig.2. The thickness (t) of the sample is given by
$2tn_{rg}=m\lambda$, where $n_{rg}$ is the refractive index of the rare gas
used, and $\lambda$ the wavelength of monitoring laser. By fitting the data to
a sinusoidal curve with an exponentially decaying term, the crystal growth is
calculated and extrapolated for 30 minutes of sample growth to determine the
total thickness of the sample. Using the period of Fabry-Perot oscillation,
the deposition rate of the crystals calculated for our case is about 100-200
nm/s.

Figure 2: Oscillations in the monitor laser as the neon flow begins. The
period of oscillation is used to monitor the growth rate of the sample.

After this initial period of undoped crystal growth, which also helps in
protecting the fiber from becoming coated with thulium, the pulsed laser is
turned on to dope the remaining growth with thulium atoms. The ablation is
done for 10-30 minutes, so the total thickness of the crystal is
$60-360\textrm{ }\mu\textrm{m}$. At the end of the growth, the temperature is
5 K at the cold head and 6.9 K on the lower part of the clamp. The sample may
be hotter than either of these, as the cooling path to the sample runs through
friction mounts, adhesive, optical fiber, and the rare gas matrix itself. We
estimate the density of the thulium atoms is in parts per thousand.

The sample is excited and the fluorescence signal is collected through the
fiber and detected using an InGaAs single photon avalanche diode (SPAD). The
experiment and data acquisition sequence are synchronized by an internal clock
from a microcontroller. To perform absorption spectroscopy, a function
generator sweeps the laser frequency by modulating the cavity length via a
triangle wave voltage to the piezoelectric actuator. This scans the laser
continuously over a mode-hop-free range of about 0.01-0.02 nm. While the laser
frequency is being scanned, the laser is turned on/off by the TTL pulses
generated by the microcontroller and amplified to get the required logic
output for use with a radio frequency switch and the AOM. The light is on for
a period of 4.9 ms, and the counting begins 200 $\mu s$ after the laser is
turned off and continues for another 5.1 ms. A synchronized TTL signal
triggers the microcontroller to begin acquisition at the start of each sweep.
For detailed scans, we sweep every 22 s; for faster scans, we sweep every 2 s.
The acquisition is done while the excitation light is switched off, and the
timing sequence for absorption spectroscopy is shown in Fig. 3. One scan is
completed after a set number of pulses, which occurs slightly before the next
trigger is received. The acquired data is averaged for multiple such scans.
Usually, only the second half of the acquired data is used, thus discarding
the data where the wavelength of the laser is decreasing. Intensity for
spectroscopy is typically a few $\textrm{W}/\textrm{cm}^{2}$.

Figure 3: The timing sequence for the absorption spectroscopy.

Emission spectroscopy involves a similar sequence synchronized to the motion
of a linear translation stage, forming a Michelson interferometer and allowing
us to operate as a Fourier transform spectrometer as described in Gaire and
Parker (2022).

For performing spectral hole burning experiments, we use sequences with three
different variations, allowing us to use only a single laser. All sequences
are divided into three 5 ms time windows: pump, probe, and count. During the
pump window, the triangle wave is disconnected from the piezo and the laser is
at a fixed wavelength, while for the probe and count windows, the piezo scans
as normal. To be consistent, the piezo control always features these three
windows; however, we switch the AOM to be on for only the pump, only the
probe, or both, as shown in Fig. 4. In fact, there is a lag of 200 $\mu$s
after the laser turns off before counting so that the excitation is not
directly detected.

Figure 4: Timing sequence for the spectral hole burning spectroscopy. The
duration AOM is turned on/off is shown by the black lines for three
experiments with both pump-probe, pump-only, and probe-only light.

##  III Results and Discussion

Figure 5: (a)Absorption spectrum of thulium trapped in neon. The inset shows a
zoomed in view of the spectrum near 1140 nm, where each color represents one
of the individual scans that are stitched together. (b) Absorption spectrum of
thulium atoms trapped in an argon sample as grown (black) and in another
sample after annealing (red). The inset shows a zoomed in view of the annealed
sample showing features near 1140.02 nm. The data from the annealed sample is
scaled by a factor of 3 for better comparison.

Using absorption spectroscopy and detecting all emitted infrared light, we
obtain the spectrum shown in Fig. 5. Because the mode-hop-free tuning range of
the cateye laser is small, each of the color traces in the inset represents a
separate 0.01 to 0.02 nm scan of the laser. Since large scans must be pieced
together from many small scans, conditions vary slightly between acquisitions.
Such conditions include laser power and precise alignment but may include
other environmental factors. In order to present a more consistent spectrum,
we have manually adjusted the spectra to overlap one another, which requires
only very small corrections (by scaling up to 15%). In both argon and neon,
multiple narrow peaks are resolved. The presence of multiple peaks can be due
to a multiplicity of trapping sites or a splitting of the
${}^{2}\mathrm{F}_{7/2}$ and ${}^{2}\mathrm{F}_{5/2}$ states into sub-levels.
Such sub-levels could result from the crystal field environment breaking
rotation symmetry (crystal field splitting), from vibrational states of the
trapped atom with its host “cage”, or from hyperfine interactions with the
nucleus. For vibrational states, if we estimate the potential energy landscape
of a trapped thulium atom with the Ar-Ar Lennard-Jones potentialDobbs and
Jones (1957), we obtain an estimated vibrational frequency of 275 GHz or 1.2
nm, which is far too large to explain our results. Multiple peaks may also be
due to the presence of other species - for example, ionized $\textrm{Tm}^{+}$
or $\textrm{Tm}^{2+}$, thulium dimers, clusters, or other compounds. However,
$\textrm{Tm}^{+}$ has a different ground state fine structure due to coupling
of the 6s electron with the f shell, which splits $J$ levels so that the only
nearby line would be a dipole-forbidden $J=4\to J=2$ transition at 1140.3 nm.
We find good evidence for both trapping site multiplicity and crystal field
effects. In argon, the spectrum changes substantially as the temperature is
raised and subsequently lowered (annealing) (see Fig. 5), favoring the
existence of multiple trapping sites that reconfigure during the anneal. Some
variation of the spectrum with growth temperature is also observed, further
supporting a model of multiple sitesDargyte  _et al._ (2021). Although Fig. 5
compares data between different samples, samples grown under the same
conditions have reproducible spectra. Our emission spectrum (described below)
reveals that excitation at one wavelength can yield emission at several
others, suggesting the presence of sub-levels within the
${}^{2}\mathrm{F}_{7/2}$ and ${}^{2}\mathrm{F}_{5/2}$ states. However,
considering each of the levels should be Kramers doublets, a maximum of 12
lines could be obtained - but more than these are apparent, possibly because
each trapping site has its own set of crystal field levels. Unfortunately, due
to the dense packing of so many lines, we could not identify specific sub-
levels or measure their energy spacing.

Figure 6: (a) Detailed spectrum in argon near 1140.02 nm taken at various
temperatures (cold head temperature indicated). This sample was annealed to 25
K prior to the experiment, so the temperature dependence shown is
reproducible. Solid lines are fits to a Lorentzian model. In this dataset
only, a periodic noise source affected the detector, so a filter was applied
to remove it. (b) Linewidth (FWHM) from the Lorentzian model as a function of
temperature (both cold plate temperature and clamp temperature are shown). At
the lowest temperatures, the thermal conductivity between components is
limited and the sample temperature can’t be precisely known.

The inset to Fig. 5b shows the region near 1140.02 nm in detail. At this
scale, additional structure is seen, with four identifiable peaks spaced by
0.5-1 GHz (0.002-0.004 nm). Such separations are comparable to the 1.5 GHz
ground state hyperfine splitting in vacuum Giglberger and Penselin (1967) and
may be provisionally assigned to a hyperfine origin. In a simple picture, both
the ground and excited state Kramers doublets are split into two levels by the
coupling to the nuclear spin, and there are then four transitions between them
in a roughly symmetric pattern.

To further investigate the sources of broadening, we performed spectroscopy
near 1140.02 nm at different temperatures. The results are shown in Fig. 6. At
each temperature, the spectrum is fit to a sum of three Lorentzians and a
background (the fourth peak observed in the inset of Fig. 5b is just outside
the scan). Note that the fit is constrained to have the locations of each peak
fixed at all temperatures, and equal linewidth for all three peaks at a given
temperature. This linewidth is plotted as a function of temperature in Fig.
6b, which shows the linewidth decreasing as temperature is reduced and
supports our hypothesis that the linewidth is homogeneous thermal broadening
from phonons. At temperatures much below the Debye temperature (67 K and 93 K
for Ne and Ar, respectivelyPollack (1964)), the temperature dependence of the
linewidth is expected to be $T^{7}$ Friedrich and Haarer (1984); Könz  _et
al._ (2003). It is not clear why the narrowing appears to saturate at lower
temperatures or why this high power law dependence is not observed. While this
could be an unknown additional broadening source, it may also be related to
thermometry, as the two thermometers on the cold head and clamp bottom begin
to deviate strongly as the temperature is reduced below 15 K, so the sample
temperature may be significantly above either reading under the coldest
conditions.

We also attempted to find evidence for inhomogeneous broadening using a
spectral hole burning strategy. For this experiment, the laser was jumped to a
“pump” wavelength, then subsequently to a probe wavelength, then the light was
turned off and the fluorescence recorded. These data are shown in Fig. 7. We
focus on the suppression caused by the pump on the probe fluorescence (and
vice versa, since they are the same intensity), so we show the fluorescence of
the probe alone (minus background), compared against the pump and probe
together minus the pump alone (labeled pump/probe). For ease of comparison, we
also show a scaled-up version of the pump/probe data. In the case of dominant
inhomogeneous broadening, we expect the saturation to only affect those sites
with nearby transition frequencies, leaving a suppressed signal, or “hole,” in
the spectrum at the pump wavelength. However, no such effect is observed, and
all lines are suppressed greatly over their entire width. This is observed for
two different excitation powers. In fact, the line being pumped is actually
suppressed less than neighboring lines, suggesting the dynamics are beyond a
two-level model. This is also consistent with observation of significant
saturation effects (four times the power gives only twice the fluorescence
even without the pump) but rather small power broadening (linewidth increases
in the fit by about 15% as the power quadruples). Considering these facts, we
conclude that these particular lines are split due to sub-levels (likely
hyperfine) rather than multiplicity of trapping sites, and that the
inhomogeneous broadening is too small to detect with present methods.

Figure 7: Spectral hole burning experiment in argon, performed with two
different incident intensities. The yellow band indicates the wavelength of
the pump. Note that the pump and probe are distinguished by the pump being
applied first (each is applied for 5 ms), and are equal in power. We show
three curves for each of two power levels: probe only (with background
subtracted), pump/probe (the fluorescence from both together minus the pump-
only fluorescence), and a rescaled version of pump/probe to compare with probe
only. Solid lines are fits to a Lorentzian model. Figure 8: Emission features
in Tm:Argon obtained by excitation at wavelengths given in each panel (shaded
orange). Inset shows the emission observed near 770 nm for the same excitation
as the main panel.

Finally, we performed Fourier-transform spectroscopy on the fluorescence
light. Fig. 8 shows the emission spectra for four different excitation
wavelengths. As can be seen, excitation at one wavelength yields emission at
many others, supporting the existence of sub-levels from crystal field
splitting. Both red- and blue-shifted lines are observed because the energy
difference between these sub-levels is within the thermal spread. Common
patterns are visible as well, as emission at 1140.6 nm occurs more strongly
for excitation at 1139.88 nm or 1140.20 nm, and these wavelengths are also
detected in the emission pattern when exciting at 1140.6 nm. If the only
source of multiple lines were a multiplicity of trapping sites, then
frequency-shifted emission would not be expected. Even if the neighboring
atoms could exchange excitation by some inelastic mechanism, we would expect
such a mechanism to predominantly occur for atoms with similar energies, yet
for 1140.6 nm emission, one obtains stronger emission by exciting at 1139.88
nm than at 1140.01 nm. This pattern makes perfect sense in a combined
trapping-site/sub-level model if there are trapping sites with sub-levels
responsible for both 1140.6 nm and 1139.88 nm emission, while 1140.01 nm
emission involves another site.

Although the sample is excited by infrared light, there is emission at visible
wavelengths as well, and the sample can be seen by eye. This is likely due to
atoms in the long-lived ${}^{2}\mathrm{F}_{5/2}$ state being further excited
and decaying by an electric dipole allowed transition in the visible range.
The majority of these emission wavelengths match known transitions in the
570-597 nm range, but an unknown emission line near 770 nm is present when the
sample is excited with 1140.0 nm light. This could possibly be coming from a
transition out of the $4f^{12}({}^{3}\textrm{H}_{6})5d_{3/2}6s^{2}$
configuration with $J=9/2$ into the ground ${}^{2}\mathrm{F}_{7/2}$ state,
which would have a vacuum wavelength of $762\textrm{ nm}$.

##  IV Conclusion

Thulium atoms were embedded in crystals of argon and neon and we obtained
laser-induced fluorescence spectra by excitation with infrared light. By
exciting the magnetic dipole transition, it was observed in both the
absorption and emission spectra that the transition was split into multiple
components. These splittings likely come from the crystal field of the argon
or neon host, which breaks the rotational symmetry at the trapping site. The
splitting of the 2F7/2 and 2F5/2 levels will depend on the detail of the
symmetry breaking at the trapping site, which is not currently known. However,
by a simple counting argument, we believe at least some lines originate from
different trapping sites. Furthermore, we believe most of the linewidth is
accounted for by homogeneous broadening, and there is no evidence from pump-
probe experiments of any significant inhomogeneous broadening. This means that
it should be possible to obtain still narrower lines by lowering the sample
temperature. This, together with more sophisticated pump-probe experiments,
should enable a complete characterization of the sub-levels of matrix-isolated
thulium, paving the way for sensing and quantum information applications
requiring a high density of identical emitters.

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