Low-Light Micro-Vibration Sensing on Satellite Platforms via Physical Encoding Self-Supervised Learning

Abstract

Micro-vibrations of satellite platforms can reduce the pointing accuracy of space optical communication links, leading to a reduction in link margin and even link interruption. Traditional methods rely on accelerometers to obtain vibration labels, leading to hardware deployment difficulties and additional energy overhead in space or power constrained scenarios. Here, we propose a physics-encoded self-supervised vibration sensing model that directly recovers vibration signals from time-series images of the lunar surface without external sensors. The model consists of an optical flow module, a convolutional network, and a memory network, formulating vibration sensing as a physical inversion problem constrained by an image reconstruction process. This approach extracts the textural features of the lunar surface and the temporal dynamics characteristics of platform vibrations, achieving end-to-end high-precision vibration sensing. In simulation experiments, it attains excellent performance, with a coefficient of determination ($R^2$) of 0.9975, a mean absolute error (MAE) of 0.0727, and a root mean square error (RMSE) of 0.0821. Furthermore, experiments on a real vibration platform validate the engineering applicability of the model, achieving an $R^2$ of 0.9910, an MAE of 0.1180, and an RMSE of 0.1463, with predicted values closely matching the ground truth. The proposed model exhibits sub-pixel-level accuracy and excellent generalization capability in both simulated and real-world experimental scenarios, providing an effective solution for visual sensing of micro-vibrations on satellite platforms in space optical communication.

1. Introduction

In long-distance space optical communication, micro-vibrations or jitter of the satellite platform can cause severe imaging errors [1,2], degrade terminal pointing accuracy [3], and reduce received optical power or even interrupt the communication link [4]. Measured data [5,6] show that the vibration frequency of the platform is inversely related to the amplitude, i.e., the higher the frequency, the smaller the amplitude. Low-frequency vibrations have a more pronounced impact on the optical communication link. The vibration frequencies that most affect communication are concentrated within 100 Hz, with amplitudes below 100 $\mathsf{\mu }$$\mathrm{rad}$ [5]. Although platform vibrations exhibit a certain degree of randomness, their vibration characteristics are approximately periodic [7] and can be regarded as a superposition of simple harmonics [8]. Therefore, accurate analysis of micro-vibrations on satellite platforms is crucial for link acquisition and maintaining communication stability [9,10].

Most conventional predictions of satellite platform vibrations rely on accelerometers [11,12,13] to obtain vibration signals. Although these contact-based methods offer high accuracy, they require additional hardware installation, which increases system complexity and mass load. Moreover, the sensors and their cables introduce extra energy consumption and environmental interference, making them difficult to deploy in space- or mass-constrained missions. Using the onboard satellite computer to extract the centroid coordinates of the light spot can indirectly capture vibration information [14] without external sensors. Vibration prediction is then achieved by methods such as least-squares and a Back Propagation (BP) neural network [15,16]. However, the centroid method depends strongly on the intensity distribution of the light spot. Under low-light or low signal-to-noise ratio (SNR) image conditions, its localization accuracy degrades, thereby limiting the accuracy of vibration sensing and prediction. In contrast, vision-based vibration analysis [17,18] does not require external sensors and also reduces the influence of centroid localization errors on vibration measurement under low-light conditions. However, extracting vibration information from images involves complex nonlinear mappings, which conventional statistical models [19,20] cannot adequately handle. Essentially, these methods model known vibration signals or related physical quantities. Therefore, in practical scenarios such as the absence of external sensors or low-light conditions, supervised learning methods cannot be directly deployed. In addition, low-light imaging noise can disturb the image intensity and local textures, making subtle inter-frame motion more difficult to extract accurately. Therefore, how to suppress noise propagation while maintaining micro-vibration estimation accuracy under low-SNR conditions remains an important issue in vision-based vibration sensing.

For long-distance space optical communication, small line-of-sight angular jitter mainly manifests as image-plane displacement. Under the small-angle approximation, the relationship between image displacement and angular displacement is approximately linear. For the camera used in our experiments, the conversion coefficient is on the order of $\mathrm{m}\mathrm{rad}/\mathrm{pixel}$, whereas for an on-orbit camera it is on the order of $\mathsf{\mu }\mathrm{rad}/\mathrm{pixel}$. Although the angular displacements in the laboratory and on-orbit scenarios are on the order of milliradians and microradians, respectively, the corresponding pixel displacements are of comparable magnitude. Therefore, the experiments in this work validate the proposed method’s subpixel motion sensing capability under low-light conditions, rather than directly demonstrating its readiness for micro-radian-level measurement accuracy on orbit. When the angular sampling interval of the camera reaches the microradian-per-pixel level and the frame rate and exposure time meet the requirements for vibration observation, the subpixel sensing capability validated in this paper can correspond to micro-radian-level angular jitter sensing capability after pixel-to-angle calibration. In addition, on-orbit deployment would also require calibration of optical axis alignment, the camera’s intrinsic parameters and distortion, pixel-to-angle conversion, and temporal and exposure responses.

To overcome the above challenges, we propose a physics-encoded self-supervised vibration sensing model [21] that uses sequential lunar surface images for end-to-end visual processing, achieving high-precision sensing of satellite platform micro-vibrations. The lunar surface exhibits rich textural features, and its morphology remains essentially unchanged over short observation intervals, which facilitates the extraction of image motion induced by satellite platform micro-vibrations from image sequences. The model consists of optical flow (OptFlow) [22], a convolutional neural network (CNN) [23], and a long short-term memory network (LSTM) [24]. Specifically, the optical flow method first extracts displacement features from low-light lunar surface images, which are then fed into a spatiotemporal module (CNN-LSTM) for vibration sensing. In this work, physical encoding refers to mapping the estimated displacement onto the first frame of the image sequence (i.e., the reference frame) to obtain the reconstruction loss, thereby achieving self-supervised learning. Under the constraint of this physical encoding, the model requires only the image reconstruction loss to drive training in order to invert vibration signals, significantly improving its stability and generalization ability. The validation results from both the simulation and real-world experiments demonstrate the effectiveness of the model and its accuracy advantages in vibration sensing tasks.

The main contributions of this paper are as follows:

• An end-to-end low-light vibration sensing model is proposed for satellite platforms, which is capable of achieving high-precision micro-vibration sensing under low-light conditions caused by narrowband filters.
• To address the challenges of low-light environments and the absence of ground-truth labels, we develop a hybrid model that integrates optical flow, CNN, and LSTM. By leveraging an image reconstruction constraint to construct a self-supervised signal, the model simultaneously estimates vibration information in both the x- and y-directions from video sequences, thereby achieving reliable sub-pixel micro-vibration sensing.
• A non-contact vibration sensing method is presented that directly processes sequential image frames without relying on accelerometers or similar sensors. By fully exploiting image feature information, this method provides a feasible solution for vibration analysis in scenarios where external sensors are unavailable.

The remainder of this paper is organized as follows. Section 2 presents the proposed micro-vibration sensing algorithm and the simulation setup. Section 3 reports the simulation results and evaluates the algorithm performance through ablation studies and frequency-sensitivity analysis. Section 4 conducts physical experiments to further validate the effectiveness of the proposed algorithm. Section 5 discusses the experimental results and the limitations of the proposed method. Section 6 concludes the paper.

2. Methodology

2.1. Research Framework Overview

The proposed low-light satellite platform micro-vibration sensing model is shown in Figure 1a, which inverts vibration signals solely through end-to-end visual processing of sequential lunar surface images. Here, we use lunar surface images as the observation target and leverage both their short-term morphological stability and their rich texture features to achieve high-precision vibration estimation. Compared to star-based observation methods [25], our approach exhibits stronger robustness under low-light conditions. The proposed algorithm takes as input time-series images of the low-light lunar surface affected by micro-vibrations, acquired by an onboard camera, and outputs the vibration signal. The model first applies the optical flow module to estimate inter-frame motion from low-SNR lunar image sequences, converting the image sequences into an initial displacement field sequence. However, under low-light conditions, shot noise, readout noise, and dark current noise can perturb the local image intensity and spatial gradient information, thereby undermining the brightness constancy assumption on which optical flow estimation relies. For micro-vibrations with small amplitudes, the intensity variations induced by these noises may be of a comparable magnitude to those caused by actual motion, leading to random fluctuations or anomalous displacement vectors in the optical flow field. To reduce the impact of the above errors on vibration sensing, this paper employs a CNN to extract spatially consistent motion features from the optical flow maps, thereby attenuating the effect of random perturbations. Subsequently, an LSTM is used to perform temporal modeling by leveraging the temporal continuity and inter-frame correlation of vibration signals, further mitigating the influence of random disturbances on vibration sensing. It should be noted, however, that these modules can only reduce, rather than completely eliminate, the influence of noise. Specifically, the optical flow module reads T frames from the input video and applies dense optical flow method to obtain the inter-frame optical flow matrices $O_1,O_2,\dots ,O_{T-1}$, which serve as the inter-frame displacement feature maps. Each set of feature maps contains optical flow fields in both the x- and y-directions, with the same dimensions as the original image. The optical flow method is based on two fundamental assumptions: brightness constancy and small motion, from which the basic constraint equation is derived as follows [22]:

$$I_{x}u+I_{y}v+{\displaystyle \frac{\partial I}{\partial t}}=0$$

(1)

where $I_x$, $I_y$, and $\partial I/\partial t$ are the partial derivatives of the grayscale intensity along the x, y, and t directions, respectively, which can be computed from the image sequence; (u, v) are the velocity vectors along the x- and y-axes, i.e., the desired optical flow vectors. Based on the above assumptions, we accumulate the optical flow matrices of consecutive frames to obtain the displacement difference between each frame and the first frame. That is, the displacement difference in the t-th frame ($t=2,3,\dots ,T$) is the sum of the first $t-1$ optical flow matrices, thereby yielding the initial displacement feature map. A CNN processes these optical flow feature maps to extract sequences of displacement vectors. Our CNN comprises two branches: one computes the spatial mean of the optical flow matrix for a coarse displacement estimate, while the other extracts image features to produce displacement residuals as correction terms. Element-wise addition of the two branch outputs yields a displacement feature sequence. An LSTM module capable of modeling temporal information then processes this sequence. The gated architecture of this module effectively captures long-term dependencies in the sequential data, thereby enabling accurate prediction of the vibration displacement at the next time step.

Figure 1b depicts the proposed network architecture and training process. To prevent potential information leakage, the image sequences were divided chronologically into training, validation, and test sets at a ratio of 8:1:1, with the validation and test sets strictly lagging behind the training set. Based on the above partitioning, a multi-step prediction strategy is adopted for temporal modeling to capture the long-term dependencies of the vibration signals.Each sample consists of M historical segments, each containing the 2D displacement sequences corresponding to N consecutive frames (shape (M, N, 2)), and the output is the displacement sequence for the subsequent N frames (shape (N, 2)). Accordingly, training samples are generated using a sliding window of length N with a stride of 1. The sample partitioning parameters M and N can be flexibly adjusted based on different datasets. Here, we set $M=5$ and $N=10$, this configuration effectively addresses the prediction requirements of both high- and low-frequency vibrations. Given a lunar image sequence ${\left\{I_t\right\}}_{t=1}^T$, consisting of T frames, each frame is first converted to grayscale. The Farnebäck dense optical flow algorithm implemented in OpenCV is then employed to compute the optical flow field between frames $t-1$ and t ($t=2,3,\dots ,T$):

$$F_t(x,y)=\left[\begin{array}{c}{u}_t(x,y)\\ v_t(x,y)\end{array}\right]$$

(2)

where $u_t(x,y)$ and $v_t(x,y)$ denote the inter-frame displacements at pixel location $(x,y)$ in the horizontal and vertical directions, respectively. By stacking the optical flow fields of consecutive frame pairs, we obtain a sequence of two-channel inter-frame optical flow displacement maps. The Farnebäck algorithm approximates the local neighborhood of each pixel using a quadratic polynomial [26]:

$$I\left(X\right)\approx X^{T}AX+b^{T}X+c$$

(3)

where A is a symmetric matrix, b is a vector, c is a scalar, and $X={[x,y]}^{\mathsf{T}}$ denotes the pixel coordinates. The algorithm estimates pixel displacement based on the variation in polynomial coefficients between consecutive frames, and employs an image pyramid for coarse-to-fine iteration. In this work, the scale factor and the number of pyramid layers are set to $0.5$ and 3, respectively. At each pyramid level, an estimation window of size $10\times 10$ pixels is used, with three iterations performed. The polynomial expansion neighborhood size is set to 5, with the corresponding Gaussian standard deviation set to $1.1$. These parameters remain unchanged throughout all training and testing processes. The optical flow module is a fixed and non-trainable motion extraction component, and does not participate in the backpropagation or parameter updates of the network. The resulting optical flow displacement map sequence is successively passed through three convolutional layers with ReLU activation functions, a global average pooling layer (GAP), and a fully connected (FC) layer to extract spatial motion features [23,27]. The motion feature sequence is then fed into a two-layer LSTM, which models the temporal dependencies of the vibration signals via its gated memory mechanism [28]. Finally, a fully connected layer outputs the vibration sensing results in the horizontal and vertical directions. We adopt a physics-encoded strategy that combines the learned vibration variables with the first frame of the input video. An affine transformation then generates consecutive frames as self-supervised signals, addressing the challenge of unavailable ground-truth vibration labels. For a pixel position $A_1$ ($a_1$, $b_1$) in the image, the affine transformation yields $A_2$ ($a_2$, $b_2$) as follows:

$$\left[\begin{array}{c}{a}_2\\ b_2\\ 1\end{array}\right]=\left[\begin{array}{ccc}{r}_{00}& r_{01}& t_{00}\\ r_{10}& r_{11}& t_{10}\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{a}_1\\ b_1\\ 1\end{array}\right]$$

(4)

where $t_{00}$ and $t_{10}$ are the translation parameters along the x and y directions, respectively, and $r_{01}$, $r_{10}$, $r_{00}$, and $r_{11}$ are the rotation and scaling parameters. Assuming a rigid connection between the camera and the satellite platform, platform micro-vibrations induce jitter in the optical axis pointing angle. For targets at infinity, such as the Moon, this angular jitter primarily causes an overall two-dimensional translation of the target on the focal plane, while the rotation and scaling induced by micro-vibrations at the micro-radian level are negligible. Consequently, the rotation matrix reduces to the identity matrix, i.e., $r_{00}=1$, $r_{01}$ = 0, $r_{10}$ = 0, $r_{11}$ = 1. Model training aims to minimize the reconstruction loss between the reconstructed image sequence and the input image sequence at the corresponding time step, thereby driving the update of model parameters. We use Python 3.10 as the programming environment and PyTorch to build the model. The training uses the mean squared error (MSE) loss and the Adam optimizer, with a learning rate of 0.001. Furthermore, we adopt an adaptive optimization strategy based on validation loss, which halves the learning rate when the loss fails to improve for three consecutive epochs and terminates training early when no improvement occurs for five consecutive epochs. This configuration effectively mitigates both underfitting and overfitting. Given the data scale and model complexity in this paper, this configuration strikes a balance between convergence speed and computational cost.

To evaluate the model performance, we adopt the mean absolute error (MAE), mean squared error (MSE), root mean squared error (RMSE), and the coefficient of determination ($R^2$) as evaluation metrics. Their mathematical expressions are as follows:

$$\begin{array}{cc}\hfill \mathrm{MAE}& ={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}|y_i-{\widehat{y}}_i|\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill \mathrm{MSE}& ={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}{(y_i-{\widehat{y}}_i)}^2\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill \mathrm{RMSE}& =\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2}\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill R^2& =1-{\displaystyle \frac{{\sum }_{i=1}^n{(y_i-{\widehat{y}}_i)}^2}{{\sum }_{i=1}^n{(y_i-\overline{y})}^2}}\hfill \end{array}$$

(8)

where n is the number of samples, $y_i$ represents the true value, ${\widehat{y}}_i$ indicates the predicted value, and $\overline{y}$ denotes the mean of the data. The smaller the MAE, MSE, and RMSE values, the better the model performance. The value of $R^2$ ranges from 0 to 1, with values closer to 1 indicating stronger fitting ability.

2.2. Simulation Framework

We develop a simulation model based on in-orbit vibration characteristics to validate the effectiveness of the proposed algorithm. To reflect the broadband nature of on-orbit micro-vibrations, we superpose multiple single-frequency sinusoidal signals within the 1–100 Hz range to construct a composite vibration waveform containing multi-harmonic components. For simple harmonic motions along the same direction but at different frequencies, given by $s_1=A_1\cos ({\omega }_{1}t+{\phi }_1)$, $s_2=A_2\cos ({\omega }_{2}t+{\phi }_2)$, …, $s_m=A_m\cos ({\omega }_{m}t+{\phi }_m)$, the resultant motion is the linear superposition of these components:

$$s\left(t\right)={\displaystyle \sum _{j=1}^m}{s}_j={\displaystyle \sum _{j=1}^m}{A}_j\cos ({\omega }_{j}t+{\phi }_j)$$

(9)

where $A_j$ indicates the amplitude, ${\omega }_j$ represents the angular frequency, and ${\phi }_j$ denotes the initial phase.

Driven by the aforementioned vibration signal, the algorithm applies a temporal transformation to a static image, generating simulated videos affected by vibrations. This paper uses lunar surface images as the observation target, employing static lunar images captured from star chart software as the reference images. To simulate the low-light imaging conditions induced by an optical filter, we apply a global brightness scaling factor to each reference image and add random noise. In brief, the vibration signal is sampled according to the frame rate to obtain a displacement sequence. This sequence then drives a static image via physics-based encoding to generate a dynamic image sequence. Subsequently, we apply blur kernels that vary with exposure time and displacement values to each frame, thereby producing a synthetic video.

3. Simulation Analysis

3.1. Simulation Setup

This section describes the specific setup and generation results of the low-light simulation video generation algorithm. Because vibrations below 100 Hz cause the most significant interference to the communication link, we superpose single-frequency sinusoidal signals at 1, 6, 10, 20, 30, 40, 50, 60, 70, and 80 Hz to simulate the complex micro-vibrations of the satellite platform. Figure 2a(left) shows the resulting vibration curve. Using physics-based encoding, we generate a vibration-affected image sequence from a one-second vibration signal and a reference image, producing a one-second low-light video. The video simulates the degradation effects of micro-vibrations on the imaging quality of an on-orbit satellite, such as jitter and blur. To avoid frequency aliasing, the video frame rate is set to 800 Hz, and the generated low-light images have a signal-to-noise ratio below 20 $\mathrm{d}$$\mathrm{B}$. Figure 2a(right) shows the t-th frame generated by the algorithm, where the red circle marks the position of the reference image and the green circle indicates the position of the generated image. The image exhibits a translational shift of three pixels, with no rotation or scaling.

3.2. Simulation Results and Analysis

3.2.1. Algorithm Output and Analysis

Figure 2b shows the loss curves of the model. The training and validation losses exhibit consistent convergence behavior without overfitting. Figure 2c(left) shows a comparison of the predicted and true vibrations on the training set, revealing a small error between them. The scatter plot of this error is shown in Figure 2c(right), where the error values are primarily concentrated within a narrow range of $\pm 0.2$ pixels, accounting for 90.81% and 90.52% of the samples, respectively, both exceeding 90%. This indicates that the model achieves high prediction accuracy. To evaluate the generalization capability of the model, we apply it to the test set. Figure 2d(left) shows the comparison between the predicted and true vibrations. The scatter plot in the right panel of Figure 2d shows that all prediction errors fall within $\pm 0.2$ pixels, accounting for 100% of the samples. The evaluation metrics are MAE = 0.0727, MSE = 0.0067, RMSE = 0.0821, and $R^2$ = 0.9975. These results demonstrate that the model achieves high prediction accuracy, a concentrated error distribution, and few extreme outliers, indicating good stability and generalization ability. This confirms that our physics-encoded self-supervised learning model enables high-precision vibration sensing from low-light sequential images.

Frequency-domain analysis is essential in vibration signal analysis. In this work, the vibration signals in the x- and y-directions have the same amplitude and frequency but differ only in phase. We therefore analyze the frequency spectrum of the x-direction signal as an example. As shown in the upper-left panel of Figure 2e, because the signal is truncated (i.e., it does not contain an integer number of periods), performing a direct spectral analysis yields significant spectral leakage. To mitigate this, we tested several common window functions, including the Hanning window, Hamming window, Blackman window, and Flattop window. In the figure, the blue curve represents the frequency spectrum of the untruncated signal (ideal case), while the cyan, green, purplish red, and golden curves correspond to the spectra processed with the Hanning, Hamming, Blackman, and flat-top windows, respectively. Nevertheless, applying only these window functions still results in residual spectral leakage of varying severity, preventing an accurate representation of the true vibration spectrum. The Hamming window yields the best performance among them. To further improve the accuracy of spectral analysis, we adopt a combination of zero padding and the Hamming window. As shown by the red curve in the lower-right panel of Figure 2e, the resulting spectrum curve agrees almost perfectly with the true spectrum. Therefore, we select this scheme for spectral analysis in this work.

3.2.2. Ablation Studies

This section quantitatively evaluates the contribution of critical components to the overall performance through ablation studies, examining the necessity of each module for image feature and temporal feature extraction. Based on the network architecture shown in Figure 1b, the optical flow, CNN, and LSTM modules were individually removed, and each model was re-trained utilizing the same hyperparameters. The evaluation metrics of each ablation study on the test set are presented in Figure 3e. Figure 3a shows a comparison between the prediction curve of the complete algorithm described in this paper and the true curve and the two are almost perfectly aligned. Figure 3b shows the results after removing the CNN. Relative to the complete model in Figure 3a, the amplitude of the error curve fluctuations increases, with MAE and RMSE rising by 242.1% and 261.0%, respectively, and $R^2$ decreasing by 1.3%, confirming the necessity of the CNN for image-domain feature extraction. Figure 3c shows the results of removing the optical flow module. The error curve increases significantly, with MAE and RMSE rising by 497.1% and 548.7%, respectively, and $R^2$ decreasing by 4.9%. This indicates that the optical flow module can effectively extract the two-dimensional displacement field between consecutive frames, providing reliable motion information for the subsequent spatiotemporal modeling by the CNN-LSTM. Figure 3d shows the results of removing the LSTM module. The error increases substantially, and model performance deteriorated severely, with MAE and RMSE increasing by 783.9% and 868.9%, respectively, and $R^2$ decreasing by 10.6%. This indicates that the LSTM module plays a critical role in capturing temporal dependencies in time-series data. The MSE metric exhibits a similar trend. The reason for the above phenomenon is that removing any key component prevents the model from accurately extracting both spatial and temporal features simultaneously, leading to information loss. In summary, the optical flow module, CNN, and LSTM are all indispensable components of the proposed model. Removing any one of these modules leads to increased prediction error and reduced accuracy, confirming the necessity of each module for the task of inferring micro-vibrations from low-light lunar surface images.

3.2.3. Frequency Sensitivity Analysis

To investigate the model’s sensitivity to the frequency of single-frequency signals, this section quantitatively analyzes signals at 10 Hz, 40 Hz, 60 Hz, and 80 Hz, based on the characteristic that higher frequencies result in smaller amplitudes for satellite platform micro-vibrations. Figure 4a shows a comparison between the predicted vibration values and the true values. The evaluation metrics for each frequency on the test set are presented in Figure 4b. The $R^2$ values for all frequencies exceed 0.98 (reaching up to 0.9985). The MAE and MSE remain stable in order of magnitude without drastic fluctuations, indicating that the model possesses a certain degree of stability. As the frequency increases, MAE, MSE, and RMSE show a decreasing trend. At 10 Hz, the MAE is 0.1064 and RMSE is 0.1329; at 80 Hz, the MAE drops to 0.0396 and RMSE to 0.0494. However, $R^2$ does not show an upward trend with increasing frequency; instead, it decreases monotonically. Specifically, $R^2$ is 0.9985 at 10 Hz, 0.9980 at 40 Hz, 0.9974 at 60 Hz, and 0.9897 at 80 Hz. This discrepancy mainly arises from the reduced amplitude of the high-frequency components. Since MAE and RMSE are absolute error metrics, the smaller absolute errors observed at higher frequencies are associated with their smaller target amplitudes. Furthermore, sub-pixel motion at higher frequencies is more susceptible to low-light image noise and optical flow estimation errors, which leads to larger relative errors and a decrease in $R^2$. Therefore, smaller absolute errors at higher frequencies do not necessarily imply better prediction performance of the algorithm, whereas the higher R² at lower frequencies reflects more accurate waveform fitting. Although $R^2$ exhibits a decreasing trend with increasing frequency, it remains above 0.98 for all frequency components, indicating that the model maintains good waveform reconstruction capability across the frequency range examined in this study.

4. Experiments

4.1. Experimental Setup

To validate the performance of the proposed model for predicting satellite platform micro-vibrations, a vibration experimental platform was constructed to simulate the on-orbit micro-vibration environment, and low-light videos affected by vibrations were captured, as shown in Figure 5a. Computer I generates the vibration signal and drives the shaker. A high-speed camera (CMOS) is rigidly mounted on the shaker, vibrates with it, captures low-light videos, and transmits them to Computer II. In the experiment, the camera frame rate was set to 800 fps. The acquired video duration was 1s, and the image signal-to-noise ratio is below 20 $\mathrm{d}$$\mathrm{B}$. For targets at infinity, such as the Moon, when neglecting the effects of parallax, lens distortion, and rotation about the optical axis, small-amplitude pitch and yaw jitter of the satellite platform mainly manifests as an overall two-dimensional translation of the target on the focal plane. Under the small-angle approximation, the conversion relation between the focal-plane image displacement and the optical-axis pointing angular displacement can be approximated by a linear relationship:

$$\mathsf{\Delta }\theta =\arctan \left(\mathsf{\Delta }d·{\displaystyle \frac{a}{f}}\right)\approx \mathsf{\Delta }d·{\displaystyle \frac{a}{f}}$$

(10)

where $\mathsf{\Delta }d$ denotes the image displacement (in pixels), a is the pixel size (in $\mathsf{\mu }\mathrm{m}/\mathrm{pixel}$), f is the camera focal length (in $\mathsf{\mu }\mathrm{m}$), and $\mathsf{\Delta }\theta$ is expressed in radians. Thus, the conversion factor from pixel displacement to angular displacement is $a/f$, with units of $\mathrm{rad}/\mathrm{pixel}$. The conversion factor between image displacement and angular displacement for the camera used in this experiment is on the order of milliradians ($a/f$ = 0.667 $\mathrm{m}\mathrm{rad}/\mathrm{pixel}$), which is significantly higher than the microradian level of actual on-orbit cameras. Therefore, this experiment is designed to simulate the temporal and frequency variations in image displacement induced by on-orbit vibrations, rather than to reproduce the absolute angular displacement that would occur in orbit. By quantitatively analyzing the time-frequency characteristics at the image-displacement level, the algorithm’s vibration sensing capability can be effectively validated. For practical engineering applications, once the actual conversion factor is obtained, the measured image displacement can be converted into the true angular displacement, thereby enabling pointing error analysis.

To quantitatively evaluate the effectiveness of the proposed self-supervised method, two representative baseline methods are introduced: the traditional centroid tracking method and a supervised CNN-LSTM model. For centroid tracking, the displacement is estimated by computing the gray-weighted centroid between consecutive frames. The supervised CNN-LSTM adopts the same feature extraction and temporal modeling architecture as the proposed method, but is trained directly using reference displacement as labels.

4.2. Results and Analysis

Figure 5b(top) shows the time-domain curve output by the proposed algorithm, which includes a comparison between the predicted values and the actual values. The reference displacement (i.e., the actual values) is obtained from synchronized accelerometer measurements. The predicted curves generally follow the time-varying trend of the reference signal in both directions, capturing the major oscillatory trends and peak positions. Figure 5b(bottom) shows the scatter plot of errors between the actual and predicted vibrations. The error values are mainly concentrated within a narrow range of $\pm 0.2$ pixels, accounting for 70.77% and 76.79% of the samples in the x- and y-directions, respectively. When the error range is expanded to $\pm 0.3$ pixels, the proportions increase to 90.46% and 91.72%, both exceeding 90%. In satellite optical communication, spectral analysis of micro-vibrations is essential. The frequency spectra obtained using the Hamming window combined with zero padding is shown in Figure 5c. The predicted spectrum agrees well with the actual spectrum, indicating that the proposed algorithm accurately reconstructs the frequency-domain characteristics of vibration signals and captures key frequency components. This confirms the model’s high-precision sensing capability and generalization performance in the frequency domain. Experimental results demonstrate that the proposed algorithm can effectively capture the temporal variation trend of image displacement, thereby providing data support for pointing error analysis induced by on-orbit satellite micro-vibrations.

To quantitatively evaluate the effectiveness of the proposed method, it is compared with two representative baseline methods. The MAE, RMSE, and $R^2$ values for each method are provided in

Table 1. Evaluation metrics of baseline methods.

Method	MAE	RMSE	${\mathit{R}}^2$
Centroid tracking	0.2908	0.5259	0.9561
Supervised CNN-LSTM	0.1382	0.1721	0.9944
Proposed method	0.1180	0.1463	0.9910

. Compared with the centroid tracking method, the proposed method reduces the MAE and RMSE by 59.42% and 72.18%, respectively, and improves the $R^2$ by 3.65%, indicating its stronger robustness to low-light images and sub-pixel motion. Compared with the supervised CNN-LSTM, the proposed method reduces the MAE and RMSE by 14.62% and 14.99%, respectively, while the $R^2$ decreases by 0.34%. Despite not using displacement labels during training, the proposed method achieves accuracy comparable to that of the supervised approach. This indicates that the image reconstruction constraint adopted in this work can provide effective supervision for learning vibration-related temporal features from low-light image sequences.

Compared with the simulation results, the proposed algorithm exhibits a slight performance degradation on the experimental data. Specifically, the $R^2$ is 0.9910, representing a decrease of approximately 0.65%; the MAE is 0.1180, an increase of about 62.3%; and the RMSE is 0.1463, an increase of approximately 78.2%. This discrepancy is mainly attributed to the following factors: imaging noise and experimental setup errors. In real low-light environments, more complex noise interference causes fluctuations in the local intensity and texture features of the images, thereby reducing the estimation accuracy of subpixel inter-frame motion in optical flow. Although the spatial feature extraction by CNN and the temporal modeling by LSTM can alleviate random disturbances, they cannot completely eliminate their effects. In addition, the mechanical nonlinearity of the vibration shaker may introduce amplitude deviations and additional harmonic components. Nevertheless, the algorithm still achieves an $R^2$ exceeding 0.99 under real-world conditions, and the absolute error also remains within an acceptable range, demonstrating strong generalization ability and potential for engineering applications. In summary, the proposed algorithm achieves high-precision reconstruction of vibration signals under both ideal simulation conditions and real physical platform experiments, confirming its stability and reliability across different data sources. This provides a solid foundation for subsequent vibration signal analysis and prediction tasks.

5. Discussion

This paper proposes a self-supervised method for satellite micro-vibration sensing from low-light lunar-surface videos. The method first employs Farnebäck dense optical flow to extract two-dimensional motion information between adjacent frames. It then uses a CNN and an LSTM to model the spatial features and temporal dependencies, respectively, and employs an image-reconstruction constraint to enable training without vibration labels. The experimental results demonstrate that the proposed method can simultaneously estimate micro-vibration signals in the x- and y-directions from low-light lunar-surface videos and achieve good sensing performance. The ablation studies further demonstrate that both the CNN and the LSTM contribute to the performance of the proposed method. Specifically, the CNN extracts stable spatial motion features from noisy optical-flow fields, whereas the LSTM exploits the temporal correlations of the vibration signals to suppress random inter-frame disturbances.

Compared with vibration measurement methods that rely on accelerometers [11,12,13], the proposed method does not require additional vibration sensors or their labeled data during operation, and can directly estimate vibration signals from videos, which provides a complementary solution for application scenarios with limited sensor deployment. Compared with methods based on centroid coordinates [15,16], the proposed method utilizes dense motion information on the lunar surface, thereby reducing the influence of centroid localization errors caused by low-light image noise. Owing to the variations in imaging devices, vibration magnitudes, and evaluation metrics among different studies, a direct quantitative comparison of their reported errors is infeasible. To address this, this paper quantitatively evaluates the centroid method and the supervised CNN-LSTM under identical experimental conditions. Compared with the centroid method, the proposed method exhibits stronger stability under low-light images and sub-pixel motion. Compared with the supervised CNN-LSTM, the proposed method can still achieve comparable accuracy to the supervised approach without using displacement labels. This indicates that the image reconstruction constraint provides effective supervision for the model to learn temporal vibration features from low-light image sequences.

Frequency–sensitivity analysis shows that as the vibration frequency increases, the MAE and RMSE decrease progressively, whereas the $R^2$ exhibits a downward trend. This discrepancy is primarily attributed to the amplitude–frequency characteristics of the vibration signals in our study, where higher-frequency components exhibit smaller amplitudes. Since MAE and RMSE are absolute error metrics, the smaller errors for high-frequency components do not necessarily indicate better relative prediction performance. The high-frequency image motion approaches the sub-pixel level, making it more susceptible to noise in low-light images and estimation errors in the Farnebäck optical flow. Consequently, its relative error increases and the goodness of fit decreases. Despite this, the $R^2$ values for all frequency components examined in this paper exceed 0.98, demonstrating that the model maintains reliable waveform reconstruction capability.

It should be noted that this study still has certain limitations. On the one hand, although the model has been evaluated using simulated videos and laboratory-captured low-light videos, its applicability to real on-orbit satellite platforms remains to be verified. On the other hand, the Farnebäck optical flow algorithm and the CNN components involve substantial computational cost, potentially limiting the model’s feasibility for onboard real-time processing. Furthermore, according to relevant on-orbit measurement studies [5], this work primarily focuses on vibrations below 100 Hz, which have a significant impact on optical communication, while the performance at higher frequencies has not yet been verified. Future work will involve validation using real on-orbit data and the development of vibration analysis models that balance accuracy and computational efficiency. Overall, this paper provides a vision-based, self-supervised approach for satellite micro-vibration sensing in the absence of vibration labels and external sensors.

6. Conclusions

To address the pointing accuracy degradation in space optical communication caused by satellite platform micro-vibrations, we propose a physically encoded self-supervised vibration sensing model. Using only time-series lunar images, the model retrieves vibration information without external sensors, thereby providing data support for system vibration analysis. Following the physical principle that higher frequencies correspond to smaller amplitudes, we constructed multi-frequency synthetic vibration signals and their corresponding simulated videos to emulate on-orbit micro-vibrations. The proposed model achieved excellent prediction performance on the test set, with an $R^2$ of 0.9975, an MAE of 0.0727, and an RMSE of 0.0821, demonstrating its ability to reconstruct complex vibration signals with high precision. Quantitative analysis via ablation studies demonstrates that the optical flow module, CNN, and LSTM are all indispensable and removing any one of them leads to increased prediction error and reduced accuracy. The sensitivity of the model to different frequencies was investigated. For single-frequency sinusoidal signals, the model achieved $R^2>0.98$ at all tested frequencies, with lower prediction errors at higher frequencies and a better fit at lower frequencies. Low-light micro-vibration experiments validate the engineering applicability of the proposed algorithm, achieving an $R^2$ of 0.9910, an MAE of 0.1180, and an RMSE of 0.1463 on the test set. Although the performance is slightly inferior to the simulation results, the accuracy remains high, indicating that the model possesses good generalization ability. The proposed model exhibits excellent prediction accuracy and generalization performance in both simulation and laboratory scenarios, providing a reliable technical solution for satellite platform micro-vibration sensing in space optical communication. However, the generalization ability of the current model on real on-orbit platforms has not yet been validated, and its high computational overhead limits applicability to high-real-time scenarios. Future work will conduct on-orbit experiments, and explore vibration analysis models that balance accuracy and efficiency.