Centroid-Preserving Dynamic Star Image Deblurring for Remote Sensing Satellite Attitude Measurement via Physics-Guided Bi-Level Optimization

Highlights

What are the main findings?

• A parametric PSF modeling method is proposed that formulates star trailing degradation as a low-dimensional physical process characterized by trail direction, length, and diffusion scale, significantly reducing the ill-posedness of blind restoration.
• A confidence-guided estimation framework is developed that converts unstable initial PSF parameter estimates into feasible search ranges with confidence weighting, improving robustness under low-SNR conditions.

What are the implications of the main findings?

• The observation-consistency-constrained refinement mechanism enables stable parameter optimization through asymmetric penalty terms and full loss evaluation, achieving superior noise robustness in dynamic star image restoration.
• The centroid-preserving restoration framework enhances sub-pixel centroiding accuracy for remote sensing satellite attitude measurement, prioritizing stellar energy distribution preservation over general visual sharpness.

Abstract

Star trackers commonly suffer from star point trailing during stellar imaging under dynamic observation conditions. Traditional non-blind deconvolution methods rely on a known Point Spread Function (PSF), whereas blind deconvolution approaches are plagued by a complex solution space and a high tendency to fall into local optima. These drawbacks make it difficult to meet the requirements of high-precision star centroid extraction. To address these challenges, this paper proposes a novel blind restoration method based on physical model guidance and alternating iterative optimization. Firstly, the parameters of the blurred PSF are blindly estimated using image moment analysis, and a motion blur physical model with controllable direction and length is constructed. Secondly, the iterative ideal physical model is embedded as a strong prior into curvature filtering to achieve guided denoising, which effectively suppresses noise while maintaining the original trailing structure. Finally, a dual-layer alternating optimization framework grounded in the ideal physical model was developed. The inner layer employs the Richardson–Lucy (RL) algorithm integrated with intelligent convergence criteria for high-precision image restoration. The outer layer utilizes a gradient descent algorithm equipped with a confidence-based full step-length strategy to optimize PSF parameters. This architecture establishes a self-correcting closed-loop mechanism characterized by iterative image restoration–model refinement cycles. The simulation results demonstrate that the proposed method generally maintains the star centroiding error below 0.1 pixel without prior knowledge of the PSF, with a maximum observed error of 0.105 pixel under the most challenging high-background condition. Its performance is close to non-blind restoration and significantly outperforms traditional blind deconvolution algorithms. It provides an effective solution for high-precision star centroiding under dynamic conditions.

1. Introduction

The star tracker is the key optical instrument in high-precision spacecraft attitude determination systems. By imaging stellar fields, identifying stellar targets, and establishing the correspondence between stellar directions in the sensor frame and those in the inertial frame, star trackers provide three-axis attitude information with high absolute accuracy [1,2,3]. For high-resolution remote sensing satellites, attitude errors are directly propagated into rigorous imaging geometry models, thereby affecting satellite remote image geolocation, stereo mapping, and multi-source remote sensing data fusion [4,5]. Consequently, integrated attitude determination using star trackers and gyroscopes has become a main approach to obtain high-precision attitude information for remote sensing satellites. In such systems, star trackers provide absolute attitude observations, whereas gyroscopes offer the high-frequency angular velocity. Their complementary characteristics jointly determine the accuracy and stability of attitude determination [6,7,8].

However, under on-board real-time processing conditions, the accuracy and robustness of the attitude output by star trackers are constrained by the star image quality. Factors such as limited computational resources, complex imaging environments, spacecraft motion, and optical system response can all degrade the observed stellar images. In dynamic scenarios, including satellite maneuvers, attitude jitter, agile imaging, high-speed scanning, or long-exposure imaging, stellar targets undergo continuous displacement on the focal plane during the exposure interval, resulting in directional star point trailing. This degradation spreads stellar energy over an elongated region, reduces the peak signal-to-noise ratio, and distorts the local intensity distribution of star points. As a result, the sub-pixel centroiding accuracy is degraded, and the reliability of the subsequent star identification and attitude determination is reduced. In severe cases, star point trailing may even lead to the attitude output interruption or tracking failure of the star tracker [9,10].

Star tracker attitude determination fundamentally depends on the reliable star identification and sub-pixel centroiding [11]. Under ideal static imaging conditions, stars can be approximated as point light sources, with their energy concentrated within a localized pixel region and their intensity distribution remaining relatively regular. Traditional methods, such as threshold segmentation, connected-component analysis, intensity-weighted centroiding, and Gaussian fitting, can usually achieve high centroiding accuracy under these conditions [12,13,14]. However, in dynamic scenarios such as satellite maneuvers, attitude jitter, high-speed scanning, or long-exposure imaging, stellar images undergo continuous displacement along the motion direction during exposure integration. Consequently, the originally compact stellar energy spreads into a directionally elongated trailing distribution. This degradation process not only reduces the peak signal-to-noise ratio of star points but also distorts their local intensity morphology, causing the centroid to deviate from the true projection center [15,16]. For faint stars, trailing may further make star points indistinguishable from background noise, leading to missed detections, false positives, or even unstable attitude output and tracking failure. Therefore, restoring trailed dynamic star images while preserving centroiding accuracy is an important approach to improve the reliability of attitude determination for star trackers and the geometric positioning accuracy of remote sensing satellites [17,18].

To address trailing degradation in stellar images, traditional studies have first explored methods such as trail detection, the correction of abnormal columns, and local image enhancement. Statistical feature-based methods exploit the intensity distribution patterns of trails along image rows or columns and perform trail detection and correction using statistical measures such as the mean, variance, column-wise statistics, or abrupt intensity variations [19,20]. These methods are computationally efficient and can, to some extent, support the rapid identification of regular trails or fixed-direction noise, making them suitable for engineering-oriented on-board processing. However, such methods usually rely on prior assumptions about trail orientation and intensity distribution. When faint stars overlap with trail regions in the same column or local neighborhood, statistical discrimination may misclassify genuine stellar targets as anomalous noise [21]. Moreover, when the trail width is large, the direction is not fixed, or the trail is strongly coupled with the main body of the star point, row- and column-based statistical features alone are insufficient to recover the complete stellar energy distribution [22,23].

Image-processing-based methods treat trailing as a form of structured noise or blur degradation and improve star image quality through smoothing filters, morphological operations, local enhancement, or background suppression [24]. The star image preprocessing methods based on filtering enhancement and local morphological correction are proposed, which have shown a certain effectiveness in reducing background disturbance and improving star-spot visibility [25,26]. These methods are advantageous in their easy implementation and relatively weak dependence on prior models. However, they are essentially closer to image denoising or visual enhancement and do not directly characterize the directional convolutional degradation induced by exposure integration. Therefore, under highlight-background, low-SNR, or trail-star overlap conditions, conventional image processing methods may attenuate star-spot edges and alter the intensity distribution, thereby introducing additional centroiding bias. Interpolation and reconstruction-based methods restore the continuity of contaminated regions through neighborhood pixel estimation, column-wise interpolation, or local intensity compensation [27,28]. Although such methods can partially solve the problem of missing local pixels, they may disrupt the true stellar energy distribution when the trail region covers the main body of the star point or when the target itself is weak, making the reconstructed results insufficient for sub-pixel centroiding.

From the perspective of the imaging mechanism, the restoration of trailed dynamic star images is essentially an image deconvolution problem. Traditional deconvolution methods can be broadly divided into non-blind and blind deconvolution. Non-blind deconvolution assumes that the PSF or motion blur kernel is known and then recovers the latent sharp image based on this prior information. Wiener filtering provides a classical frequency-domain framework that balances inverse filtering and noise suppression, thereby alleviating the instability of direct inverse filtering under noisy conditions. However, its performance strongly depends on the accurate estimation of the noise and signal power spectra, and it is prone to ringing artifacts under low-SNR conditions [29,30]. The RL algorithm is based on a Poisson noise model and maximum likelihood estimation, providing an iterative restoration method suitable for photon-counting imaging. It has been widely used in astronomical image restoration and point-source reconstruction [31,32]. Such methods are well-aligned with the Poisson statistical characteristics of star imaging, but they require an accurate PSF in advance. When the PSF is inaccurate, the iterative restoration process may amplify noise and introduce artifacts or energy oscillations in the restored star points [33].

To mitigate noise amplification during deconvolution, regularization methods have been introduced into image restoration. Total Variation (TV) regularization suppresses noise while preserving edges by enforcing sparsity constraints on image gradients [34]. When combined with optimization strategies such as Split Bregman or the alternating direction method of multipliers, TV-based methods improve the solution efficiency of non-smooth optimization problems and alleviate the excessive smoothing caused by conventional Tikhonov regularization [35,36]. These methods address the trade-off between edge preservation and noise suppression in restored images. However, stellar targets in star images are characterized by a small spatial extent, low energy, and approximately point-source distributions. Excessively strong gradient regularization may attenuate the stellar energy envelope and even introduce staircase artifacts, causing the restored intensity structure to deviate from the true imaging model [37]. In addition, some studies have developed fast restoration methods for directional star point trailing based on CCD exposure integration models, improving the processing efficiency in fixed-direction trailing scenarios [38]. Nevertheless, such methods usually rely on simplified motion models or predetermined star point trailing directions, limiting their adaptability to arbitrary-direction trailing under attitude jitter and complex dynamic conditions.

In practical on-orbit applications, the key difficulty of non-blind deconvolution lies in the accurate acquisition of the PSF. Star point trailing is jointly determined by the satellite angular velocity, exposure time, attitude jitter, optical system response, pixel integration effects, and time synchronization errors. Therefore, the true PSF often deviates from a simple motion kernel [39]. Some studies have attempted to estimate the motion blur kernel using the gyro angular velocity and exposure time, thereby providing external prior information for non-blind deconvolution [40]. This strategy can, in principle, establish a relationship between the attitude motion and star point trailing length, addressing the lack of prior information when estimating the PSF purely from images [41]. However, it requires additional inertial measurement information and introduces extra error sources, including gyro noise, installation misalignment, calibration errors, and star tracker–gyro time synchronization errors. When the star tracker operates independently or when inertial measurements are unreliable, PSF estimation based on external sensors weakens the autonomy and robustness of the star image restoration algorithm.

Compared with non-blind deconvolution, blind deconvolution attempts to simultaneously estimate the latent sharp image and the blur kernel from the degraded image alone, thereby eliminating the need for a given PSF [42]. This property makes blind deconvolution attractive for dynamic star image restoration. Conventional maximum a posteriori (MAP) estimation and alternating optimization methods formulate blind restoration as a prior-constrained optimization problem by alternately updating the latent image and blur kernel [43]. Kotera and Sroubek proposed a MAP restoration method incorporating heavy-tailed distributions and sparsity priors, which suppresses trivial solutions by enhancing image-gradient sparsity constraints and improves the stability of blur-kernel estimation [44]. Krishnan and Sroubek introduced a normalized sparsity measure to improve blur-kernel estimation accuracy in natural image blind deconvolution, addressing the tendency of conventional sparse priors to favor blurred images [45]. These methods provide important theoretical foundations for blind deconvolution. However, they are usually sensitive to regularization parameters, initialization, and noise levels. In stellar images, the number of star points is limited, the target energy is weak, and the background noise is complex. The gradient statistics of star images differ significantly from those of natural images, and directly applying natural-image priors may lead to unstable blur-kernel estimation.

Bayesian inference and variational optimization methods have further improved the modeling capability of blind deconvolution. Wipf and Zhang proposed a variational Bayesian blind deconvolution method that improves the blur-kernel estimation by marginalizing high-dimensional image variables, thereby alleviating the tendency of conventional MAP methods to become trapped in local minima [46]. Methods incorporating Huber–MRF models, hyper-Laplacian gradient priors, or sparsity constraints have also been used to improve edge preservation and blur-kernel estimation accuracy [47,48,49]. These methods have achieved promising results in general image restoration and have partially addressed problems related to noise suppression, edge preservation, and blur-kernel regularization. However, they require the simultaneous design of image priors, kernel priors, and noise models, resulting in complex parameter coupling and a high computational cost [50]. For dynamic stellar images acquired by star trackers, star point trailing has strong point-source characteristics and a clear physical mechanism of exposure integration. Overly general image priors not only increase the model complexity but may also misinterpret background gradients, random noise, or faint-star structures as blur-kernel information, leading to distorted restoration results.

In summary, the existing restoration methods for dynamic stellar images still face three major challenges. First, non-blind deconvolution requires an accurate PSF, which is difficult to obtain reliably under on-orbit dynamic conditions. Although external sensors such as gyroscopes can be used to infer the motion blur kernel, this strategy introduces additional hardware dependence and systematic errors. Second, conventional blind deconvolution usually estimates the latent image and blur kernel simultaneously in a high-dimensional space, making the optimization problem ill-posed and sensitive to priors, parameters, and initialization. As a result, it may converge to local optima or yield non-unique solutions. Third, most general-purpose image restoration methods focus on perceptual sharpness or global image fidelity, whereas the essential requirement of star trackers is to preserve the stellar energy distribution and improve the sub-pixel centroiding accuracy. Therefore, dynamic star image restoration should not be treated merely as a general image deblurring problem. Instead, it should be formulated as a centroid-preserving restoration problem that integrates stellar imaging physics, the formation mechanism of star point trailing, and the attitude measurement requirements of remote sensing satellites. To address these challenges, this paper takes iterative blind deconvolution as the core solution for dynamic stellar image restoration, and the detailed modeling and optimization principle are elaborated in Algorithm Design.

2. Algorithm Design

2.1. The Principle

Traditional single-loop blind deconvolution cannot fully compensate the initial PSF estimation deviation, resulting in a poor star centroid positioning accuracy. To tackle this defect, we construct an improved bi-level alternating iterative closed-loop model. Different from simple conventional alternating optimization, our refined iterative strategy with observation consistency constraints and asymmetric penalty terms will effectively avoid the optimization process converging to local optima, which is elaborated in Section 2.1.3.

Targeted optimizations are implemented on both PSF input and image input for the bi-level framework.

For PSF parameter input, mainstream blind restoration methods adopt high-dimensional blur matrices and bring a massive computational overhead. Based on the physical motion integration law of star points during exposure, we simplify the trailing blur into two physical parameters solved by image moments to cut calculation costs. Since primitive moment estimates are susceptible to noise, we introduce differentiated confidence weights to limit the feasible parameter search range, offering robust prior support for iterative refinement in the bi-level structure.

For the image input of inner iteration, independent denoising operations tend to damage the elongated trail structure and trigger an extra centroid error. We embed dynamically updated physical star trail models into the filtering process to retain complete stellar energy distribution, and integrate this denoising module into the overall iteration cycle. Every time the outer layer updates the PSF parameters, a brand-new physical model will be generated for denoising processing, making the preprocessed star image continuously approach real observation characteristics and supporting high-precision inner-layer deblurring.

2.1.1. Parametric Blind Estimation of PSF Based on Image Moments

Star point trailing represents the physical projection of the uniform angular displacement of the satellite attitude onto the imaging plane. The trail length is quantified by the pixel displacement along the motion direction within the exposure time, directly reflecting the degree of motion blur [51]. For star images blind deconvolution, trailing is typically treated as a degradation function, with specific modeling and assumptions applied to the PSF [52]. However, conventional non-parametric algorithms overlook the physical structure of trailing and introduce excessive parameters, resulting in slow convergence and susceptibility to noise interference [53]. Additionally, blind deconvolution based on L0 gradients or saliency gradients cannot precisely capture trailing edges, which amplifies the reconstruction error [54].

The structure of star point trailing in true stellar images is primarily influenced by linear motion displacement, and incorporating an excessive number of parameters would substantially increase computational complexity. Therefore, we identify and select the two most influential parameters to achieve a quantitative reduction in the computational burden. Since stellar magnitude differences cause varying trailing lengths even within the same image, we construct an independent PSF model for each star point [55]. Our approach employs image moments to build a parametric physical model. By characterizing the global statistical distribution of the trailing grayscale, we quantify the essential attributes—geometric shape, position, and extension direction—of the star point trailing.

The existing moment-based PSF estimation pipelines for star trailing still suffer from two obvious limitations. First, most relevant schemes fail to fully leverage the linear motion physical prior of star trails, and they either retain high-dimensional blur kernels or still adopt dozens of undetermined parameters, which brings a heavy computational overhead and weak noise robustness. Second, these conventional methods directly take the one-shot moment calculation results as fixed PSF values without differentiating the estimation uncertainty between the trail angle and trail length. Under low-SNR observation conditions, this equal-weight processing cannot suppress the severe noise-induced systematic overestimation bias of the trail length, further reducing the deblurring accuracy in the subsequent iterations.

To address the above two drawbacks simultaneously, our method carries out two targeted improvements: we compress the high-dimensional blur kernel into only two physically interpretable parameters to reduce the computational complexity, and we additionally introduce a confidence-weighted constraint strategy after the rough moment estimation to calibrate the two parameters according to their statistical fluctuation characteristics.

We estimate $\mathsf{\theta }$ and $\mathrm{L}$ coarsely in two steps using the structural properties of image moments. The trailing motion direction is obtained from the image’s second-order central moments. Specifically, we compute the trailing image’s zeroth-order moment ${\mathrm{m}}_{00}$, first-order moments ${\mathrm{m}}_{10}$ and ${\mathrm{m}}_{01}$, and second-order moments ${\mathrm{m}}_{20}$, ${\mathrm{m}}_{11}$, and ${\mathrm{m}}_{02}$ [56]:

$${\mathrm{m}}_{\mathrm{pq}}={\sum }_{\mathrm{x}}{\sum }_{\mathrm{y}}{\mathrm{x}}^{\mathrm{p}}{\mathrm{y}}^{\mathrm{q}} \mathrm{I}(\mathrm{x},\mathrm{y}),$$

(1)

where $\mathrm{I}\left(\mathrm{x},\mathrm{y}\right)$ denotes each pixel gray value of the stellar image, and $\mathrm{p}$, $\mathrm{q}$ represent the moment order with $\mathrm{p},\mathrm{q}\in \left\{0,1,2\right\}$.

To eliminate the influence of the star point position on the moment calculation, the second-order moments are converted to central moments ${\mathsf{\mu }}_{\mathrm{pq}}$:

$${\mathsf{\mu }}_{\mathrm{pq}}= {\sum }_{\mathrm{x}}{\sum }_{\mathrm{y}}{(\mathrm{x}-\overline{\mathrm{x}})}^{\mathrm{p}}{(\mathrm{y}-\overline{\mathrm{y}})}^{\mathrm{q}} \mathrm{I}(\mathrm{x},\mathrm{y}),$$

(2)

where $\overline{\mathrm{x}}={\displaystyle \frac{{\mathrm{m}}_{10}}{{\mathrm{m}}_{00}}}$ and $\overline{\mathrm{y}}={\displaystyle \frac{{\mathrm{m}}_{01}}{{\mathrm{m}}_{00}}}$ represent the coarse centroid coordinates of the star point, calculated as the ratio of first-order to zeroth-order moments, capturing the overall position of the star point.

The trailing direction angle $\mathsf{\theta }$ is computed from second-order central moments. This formula leverages the principal axis characteristics of the trailing structure to robustly estimate the motion blur direction, which remains unaffected by the star point Gaussian blur [57]. The ratio directly reflects the principal extension direction of the trailing, as shown in Figure 1a. The expression is as follows:

$$\mathsf{\theta }=0.5 \arctan ({\displaystyle \frac{2{\mathsf{\mu }}_{11}}{{\mathsf{\mu }}_{20}-{\mathsf{\mu }}_{02}}}),$$

(3)

where ${\mathsf{\mu }}_{20}$ denotes the second-order central moment in the x-direction, ${\mathsf{\mu }}_{11}$ represents the cross central moment, and ${\mathsf{\mu }}_{02}$ denotes the second-order central moment in the y-direction.

To estimate the length $\mathrm{L}$, the image is projected along the direction θ, transforming the two-dimensional star image $\mathrm{I}\left(\mathrm{x},\mathrm{y}\right)$ into a one-dimensional intensity profile $\mathrm{P}\left(\mathrm{t}\right)$, as shown in Figure 1b. This process suppresses noise interference while highlighting the energy distribution profile of the trailing. The trailing length $\mathrm{L}$ is then determined by calculating the pixel range with gray values exceeding a threshold.

The instability of the blind trailing length and direction estimation under low signal-to-noise ratio (SNR) conditions presents a significant challenge. To mitigate this issue, we develop a confidence-guided coarse PSF parameter estimation strategy grounded in empirical statistical validation. Our analysis demonstrates that, while the direction angle estimation via image moments achieves a remarkable robustness across varying noise levels, the trailing length estimation is susceptible to systematic overestimation bias arising from noise-induced broadening in the projected one-dimensional intensity profile, establishing the necessity for confidence-weighted parameter refinement.

Based on these prior error characteristics, we construct a confidence-weighted estimation model. For the direction parameter, given its high estimation accuracy, we assign a high confidence weight to constrain its search space within a narrow range around the initial estimate. For the length parameter, considering its susceptibility to noise interference and tendency toward overestimation, we assign a lower confidence weight and establish a larger dynamic search interval. Specifically, the confidence weights ${\mathrm{W}}_{\mathsf{\theta }}$ and ${\mathrm{W}}_{\mathrm{L}}$ are defined as follows:

$${\mathsf{\sigma }}_{\mathsf{\theta }}=\sqrt{{\displaystyle \frac{1}{\mathrm{N}}}{\sum }_{\mathrm{i}=1}^{\mathrm{N}}{\left({\mathsf{\theta }}_{\mathrm{i}}-\overline{\mathsf{\theta }}\right)}^2},{\mathsf{\sigma }}_{\mathrm{L}}=\sqrt{{\displaystyle \frac{1}{\mathrm{N}}}{\sum }_{\mathrm{i}=1}^{\mathrm{N}}{\left({\mathrm{L}}_{\mathrm{i}}-\overline{\mathrm{L}}\right)}^2},$$

(4)

$${\mathrm{W}}_{\mathsf{\theta }}={\displaystyle \frac{{\mathsf{\sigma }}_{\mathrm{L}}}{{\mathsf{\sigma }}_{\mathsf{\theta }}+{\mathsf{\sigma }}_{\mathrm{L}}}},{\mathrm{W}}_{\mathrm{L}}={\displaystyle \frac{{\mathsf{\sigma }}_{\mathrm{L}}}{{\mathsf{\sigma }}_{\mathsf{\theta }}+{\mathsf{\sigma }}_{\mathrm{L}}}},$$

(5)

where ${\mathsf{\sigma }}_{\mathsf{\theta }}$ and ${\mathsf{\sigma }}_{\mathrm{L}}$ are the standard deviations of estimation fluctuations, while ${\mathsf{\theta }}_{\mathrm{i}}$ and ${\mathrm{L}}_{\mathrm{i}}$ denote the parameters output by the i-th blind estimation.

The standard deviation values above are derived from a pre-simulation statistical dataset consisting of 60 independent noisy star images. We categorize motion modes into two groups based on the sign of the dual-axis angular velocities (same sign and opposite sign), and configure three trail length levels (short, medium, and long) for each motion type, generating 10 images for each combination to form a total of ($2\times 3\times 10=60)$ samples. For every single sample, we implement blind estimation based on image moments to collect groups of ${\mathsf{\theta }}_{\mathrm{i}}$ and ${\mathrm{L}}_{\mathrm{i}}$, and then calculate the overall standard deviations ${\mathsf{\sigma }}_{\mathsf{\theta }}$ and ${\mathsf{\sigma }}_{\mathrm{L}}$. Consistent statistical results show that ${\mathsf{\sigma }}_{\mathsf{\theta }}$ is consistently much smaller than ${\mathsf{\sigma }}_{\mathrm{L}}$ under all motion conditions, which naturally yields a higher confidence weight ${\mathrm{W}}_{\mathsf{\theta }}$ for the stable angle parameter.

This strategy reframes the initial estimate—not as a deterministic optimal solution but as a credible search domain for PSF parameters. Confidence scores are allocated to candidate parameters based on estimation reliability, thereby establishing a robust prior for subsequent high-precision refinement.

Subsequently, by integrating the star tracker imaging parameters, the trailing length L in the image domain is inverted to obtain the physical angular velocity ω [58]:

$$\mathrm{L}={\displaystyle \frac{\mathrm{F}·{\mathrm{t}}_{\exp }}{\mathrm{px}}}·\sqrt{{\mathrm{w}}_{\mathrm{x}}^2+{\mathrm{w}}_{\mathrm{y}}^2},$$

(6)

where $\mathrm{F}$ is the focal length of the star tracker, ${\mathrm{t}}_{\exp }$ is the exposure time, $\mathrm{px}$ denotes the pixel size, and ${\mathrm{w}}_{\mathrm{x}}$, ${\mathrm{w}}_{\mathrm{y}}$ are the angular velocities around the x and y axes, respectively. The dual-axis angular velocity components are derived from $\mathsf{\theta }=\arctan (-{\mathrm{w}}_{\mathrm{x}}/{\mathrm{w}}_{\mathrm{y}})$, enabling the generation of the energy-normalized linear motion blur kernel $\mathrm{H}$ for PSF blind estimation.

2.1.2. Star Image Preprocessing Method Guided by Physical Model

In star trail restoration, the original image inevitably contains readout noise and dark current noise [59]. The direct application of blind deconvolution to noisy images causes noise to be amplified and propagated through the iterative algorithm, resulting in artifacts at star point edges and star centroiding errors [60]. The core objective of preprocessing is to suppress noise while preserving the linear structure and energy distribution characteristics of star trails as much as possible, thereby providing high-quality input for subsequent blind deconvolution. Among various preprocessing methods, the TV regularization denoising method produces staircase effects on the continuous smooth structure of the star point trailing, leading to gradient distortion [61]; the anisotropic diffusion method causes the excessive smoothing of the elongated linear structure of star point trailing, resulting in the loss of structural details [62]; the non-local means denoising method utilizes redundant information between similar patches in the image for weighted averaging, but incurs a substantial computational cost and may introduce structural misalignment easily [63]; and the low-rank prior method leverages the self-similarity of blurred images, yet remains limited in effectiveness against unstructured noise and requires complex parameter tuning [64].

The existing algorithms rely exclusively on grayscale value distribution characteristics within the image, without incorporating the prior structural knowledge of star point trailing. To address the vulnerability whereby the structure of star point trailing is easily compromised by conventional filters under high-noise conditions, we present a physics-model-guided curvature filtering algorithm for preprocessing. The approach integrates the ideal physical trailing model $\mathrm{M}\left(\mathrm{x},\mathrm{y}\right)$ obtained through blind estimation as a strong structural prior into the denoising energy functional. Beyond the effective suppression of random noise, this method actively reinforces the structures consistent with the current physical assumptions. The guidance mechanism exhibits dynamic adaptability, allowing for synchronous updates with PSF parameters during alternating optimization iterations. This innovation overcomes the limitations of conventional filtering approaches, delivering high-quality input for high-precision deconvolution while maintaining a stable performance across diverse noise levels.

The ideal physical trailing model $\mathrm{M}\left(\mathrm{x},\mathrm{y}\right)$ is constructed based on the physical principles governing star point motion. This model characterizes the imaging process in which an ideal star point produces a Gaussian PSF and forms an elongated trail through uniform linear motion. For a stationary star point, the point spread function follows a two-dimensional Gaussian distribution [65]. The corresponding 2D Gaussian kernel $\mathrm{G}\left(\mathrm{x},\mathrm{y}\right)$ is thus formulated as follows:

$$\mathrm{G}\left(\mathrm{x},\mathrm{y}\right)={\displaystyle \frac{1}{2\mathsf{\pi }{\mathsf{\sigma }}^2}}\exp \left(-{\displaystyle \frac{{\mathrm{x}}^2+{\mathrm{y}}^2}{2{\mathsf{\sigma }}^2}}\right),$$

(7)

where $\mathsf{\sigma }$ is the standard deviation of Gaussian dispersion.

Subsequently, given the optimal trail direction angle $\mathsf{\theta }$ and length $\mathrm{L}$, a line spread kernel $\mathrm{L}\left(\mathrm{x},\mathrm{y}\right)$ representing uniform linear motion is synthesized. A two-dimensional convolution is then performed between the Gaussian PSF kernel $\mathrm{G}\left(\mathrm{x},\mathrm{y}\right)$ and the line spread kernel $\mathrm{L}\left(\mathrm{x},\mathrm{y}\right)$ to yield the composite blur kernel $\mathrm{K}\left(\mathrm{x},\mathrm{y}\right)={\mathrm{G}\left(\mathrm{x},\mathrm{y}\right)}^{\ast }\mathrm{L}\left(\mathrm{x},\mathrm{y}\right)$. This composite kernel characterizes the combined imaging effects of Gaussian spot diffusion and uniform linear motion. The ideal physical trailing model $\mathrm{M}\left(\mathrm{x},\mathrm{y}\right)$ is thus expressed as follows:

$$\mathrm{M}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{B}+\mathrm{P}·\mathrm{K}\left(\mathrm{x}-{\mathrm{C}}_{\mathrm{x}},\mathrm{y}-{\mathrm{C}}_{\mathrm{y}}\right),$$

(8)

where $\mathrm{B}$ represents the background intensity of the image, $\mathrm{P}$ denotes the peak intensity of the star point, and ${\mathrm{C}}_{\mathrm{x}}$ and ${\mathrm{C}}_{\mathrm{y}}$ represent the x and y coordinates of the star point, respectively.

Curvature filtering represents an efficient denoising method that leverages the geometric properties of images. The underlying principle is to use the image curvature as a regularization term to constrain the image smoothing process, enabling effective noise removal while better preserving edge structures [66,67]. Traditional curvature filtering energy functionals typically consist of only a data fidelity term and a smoothing term, mathematically formulated as follows:

$${\mathrm{E}}_{\mathrm{TV}}\left(\mathrm{U}\right)={\int }_{\mathsf{\Omega }}\left|\nabla \mathrm{U}\right|\mathrm{dx}\mathrm{dy}+{\displaystyle \frac{{\mathsf{\lambda }}_{\mathrm{TV}}}{2}}{\int }_{\mathsf{\Omega }}{\left(\mathrm{U}-\mathrm{g}\right)}^2\mathrm{dx}\mathrm{dy},$$

(9)

where the first term is the TV regularization term that promotes piecewise smoothness, and the second term is the data fidelity term, which constrains the similarity between the denoised result $\mathrm{U}$ and the original observed image $\mathrm{g}$, with ${\mathsf{\lambda }}_{\mathrm{TV}}$ representing the regularization balancing parameter. However, under high-noise conditions, the denoising process relying exclusively on the image’s intrinsic curvature information tends to blur the weak edges of trails and compromise their linear structural integrity, as illustrated in Figure 2c.

To overcome this limitation, we develop a novel energy functional based on the ideal physical trailing model. This functional integrates three key components—image smoothness, data fidelity, and physical structural consistency—by incorporating the physical model fitting term ${\mathrm{E}}_{\mathrm{Model}}\left(\mathrm{U}\right)$, which is formulated as follows:

$${\mathrm{E}}_{\mathrm{Total}}\left(\mathrm{U}\right)=\left(1-\mathsf{\alpha }\right)·{\mathrm{E}}_{\mathrm{TV}}\left(\mathrm{U}\right)+\mathsf{\alpha }·{\mathrm{E}}_{\mathrm{Model}}\left(\mathrm{U}\right),$$

(10)

where $\mathrm{U}$ represents the preprocessed output image, and $\mathsf{\alpha }\in \left[0,1\right]$ serves as a guidance weight coefficient to balance the image smoothness with physical structural consistency. This weighting coefficient is a self-designed adjustable parameter exclusive to our proposed physical-model-guided denoising strategy, which is customized according to the star trailing physical prior in this work.

$${\mathrm{E}}_{\mathrm{Model}}\left(\mathrm{U}\right)={\displaystyle \frac{1}{2}}{\int }_{\mathsf{\Omega }}{\left[\mathrm{U}(\mathrm{x},\mathrm{y})-\mathrm{M}(\mathrm{x},\mathrm{y})\right]}^2\mathrm{dx}\mathrm{dy},$$

(11)

The term ${\mathrm{E}}_{\mathrm{Model}}\left(\mathrm{U}\right)$ represents the physical model fitting component, designed to constrain the output image to approximate the ideal physical structure. As illustrated in Figure 2d, during energy minimization, the denoising process not only suppresses noise but also drives the output U toward the model M, thereby recovering clear trail structures from noisy observations.

The key advantage of this preprocessing method lies in its dynamic adaptivity. Within the overall algorithmic framework, after each outer iteration updates the PSF parameters and centroid coordinates, the algorithm promptly generates an updated and more accurate ideal physical trailing model using the new parameters. This model then guides a fresh round of denoising on the original blurred image. Such a dynamic model-guidance mechanism ensures that the denoising process continuously tracks and reinforces structural information consistent with the current optimal physical estimate, preventing alterations to critical image information due to parameter inaccuracies and avoiding the erosion of the structure of star point trailing across multiple iterations. This ultimately supplies high-fidelity input for the inner-level restoration.

2.1.3. Bi-Level Alternating Optimization Restoration Algorithm Driven by Physical Model

The essence of blind deconvolution lies in the simultaneous estimation of the latent sharp image and the PSF. Building upon the physical modeling results, we reformulate the blind deconvolution problem as an iterative refinement process of the physical model, designed within a two-stage alternating optimization framework. This framework continuously optimizes the alignment between the physical model and the original stellar image by alternately refining the stellar centroid position and the physical PSF parameters. Specifically, the inner loop takes the current optimal PSF parameters as input and employs the RL algorithm for image restoration while updating the stellar centroid position. The outer loop leverages the restored image as feedback, applies differential confidence weights to different PSF parameters based on prior knowledge, and corrects the physical PSF parameters through gradient descent with asymmetric regularization. This gradually brings the ideal trailing model closer to the true trail morphology, establishing a self-correcting closed-loop mechanism consisting of parameter refinement, model matching, residual feedback, and parameter recalibration, which effectively mitigates the initial blind estimation errors. Furthermore, an adaptive convergence criterion is introduced, terminating the iteration when the parameter refinement of the physical model falls below a preset threshold, which means the alignment between the physical model and the observed image reaches its maximum, thereby avoiding noise amplification and computational waste caused by over-iteration.

To begin with, concerning the inner loop, during the k-th iteration of the outer loop and given the current estimate of the PSF (${\mathrm{h}}^{\mathrm{k}}$), the task of the inner loop is to recover the latent image $\mathrm{f}$ from the preprocessed image ${\mathrm{g}}_{\mathrm{k}}$. We adopt the RL algorithm for this purpose, with its iterative formulation expressed as follows:

$${\mathrm{f}}_{\mathrm{t}+1}^{\mathrm{k}}= {\mathrm{f}}_{\mathrm{t}}^{\mathrm{k}}\left[\left({\displaystyle \frac{{\mathrm{g}}_{\mathrm{p}}}{{\mathrm{f}}_{\mathrm{t}}^{\mathrm{k}}\ast {\mathrm{h}}^{\mathrm{k}}}}\right)\oplus {\mathrm{h}}^{\mathrm{k}}\right],$$

(12)

where $\mathrm{t}$ denotes the iteration count of the inner Richardson–Lucy algorithm, $\ast$ represents the convolution operator, $\oplus$ denotes the cross-correlation operator, ${\mathrm{h}}^{\mathrm{k}}$ is the PSF matrix at the current estimate, and ${\mathrm{f}}_{\mathrm{t}}^{\mathrm{k}}$ is the stellar image estimate obtained at the t-th iteration.

To enhance the computational efficiency and prevent noise amplification caused by excessive iterations, we propose an adaptive convergence criterion based on the stability of the stellar centroid position, replacing the conventional fixed iteration count strategy. Given the high sensitivity of the RL algorithm to noise, over-iteration may compromise the restoration quality with noise artifacts. The evolution of the stellar centroid position provides an intuitive measure of the image restoration stability. At each iteration $\mathrm{t}$, we compute the stellar centroid position ${\mathrm{c}}_{\mathrm{t}}$ and track its displacement acceleration:

$${\mathrm{a}}_{\mathrm{t}}=\left|\left|{\mathrm{c}}_{\mathrm{t}+1}{-\mathrm{c}}_{\mathrm{t}}\right|-\left|{\mathrm{c}}_{\mathrm{t}}-{\mathrm{c}}_{\mathrm{t}-1}\right|\right|,$$

(13)

Convergence is achieved when ${\mathrm{a}}_{\mathrm{t}}<\mathsf{\epsilon }$, where $\mathsf{\epsilon }$ denotes the preset threshold. Upon convergence, the current optimal sharp stellar image estimate ${\mathrm{f}}^{\mathrm{k}}$ is output. This centroid-based criterion demonstrates a superior alignment with the stellar centriod position extraction compared to either fixed iteration counts or image-difference-based convergence criteria.

Once convergence is confirmed and the inner loop’s sharp image estimate ${\mathrm{f}}^{\mathrm{k}}$ is fixed, the alternating algorithm proceeds to the outer loop. The objective of the outer loop is to refine the PSF parameters such that the simulated blurred image, generated from the current sharp image estimate and the PSF, achieves maximal consistency with the original observed image g.

To address the prevalent issue of blind deconvolution optimization becoming trapped in local optima, we propose a PSF parameter correction mechanism constrained by fidelity to the original observations. Unlike traditional gradient descent methods that update parameters along a single gradient direction, we employ a comprehensive step-length search strategy. Within a neighborhood of the current parameters, a series of candidate parameter sets $\left({\mathsf{\theta }}_{\mathrm{i}},{\mathrm{L}}_{\mathrm{i}}\right)$ are generated. For each candidate parameter, the corresponding PSF $\mathrm{h}\left({\mathsf{\theta }}_{\mathrm{i}},{\mathrm{L}}_{\mathrm{i}}\right)$ is utilized to perform re-degradation on the restored sharp image ${\mathrm{f}}^{\mathrm{k}}$, thereby synthesizing a simulated blurred image ${\mathrm{g}}_{\mathrm{sim}}={\mathrm{f}}^{\mathrm{k}\ast }\mathrm{h}({\mathsf{\theta }}_{\mathrm{i}},{\mathrm{L}}_{\mathrm{i}})$. This synthesized image is then compared with the original stellar image $\mathrm{g}$ to evaluate consistency, with the data fidelity loss function computed as follows:

$$\mathrm{L} (\mathsf{\theta },\mathrm{L})={\displaystyle \frac{1}{2}}{\Vert \mathrm{g}-{\mathrm{f}}^{\mathrm{k}\ast }\mathrm{h}(\mathsf{\theta },\mathrm{L})\Vert }_2^2,$$

(14)

By evaluating the loss values across all candidate parameters, the algorithm achieves a comprehensive awareness of the optimization landscape within the current parameter space. Based on this global perspective, the parameter configuration yielding the minimum loss is selected as the initialization point for the subsequent update iteration. The exhaustive evaluation strategy effectively prevents the optimization process from becoming trapped in local minima within the complex error landscape, thereby substantially enhancing the robustness and reliability of the parameter search procedure.

To address the issues of the systematic overestimation of the tail length in the initial blind estimation and the asymmetric sensitivity between the direction and length during the parameter update, we introduce an asymmetric penalty term in the parameter refinement procedure. Considering that the blind estimation of the direction angle $\mathsf{\theta }$ achieves a high accuracy, a stronger penalty is applied to $\mathsf{\theta }$ to limit deviations. Conversely, the tail length $\mathrm{L}$ is highly susceptible to noise interference and prone to systematic overestimation; therefore, its update should be biased toward contraction. Accordingly, we incorporate a weighted penalty term $\mathrm{P} (\mathsf{\theta },\mathrm{L})$ into the loss function, formulated as follows:

$$\mathrm{P}\left(\mathsf{\theta },\mathrm{L}\right) = {\mathsf{\lambda }}_{\mathsf{\theta }}·{\left(∆\mathsf{\theta }\right)}^2+{\mathsf{\lambda }}_{\mathrm{L}}·\exp \left(-{\displaystyle \frac{∆\mathrm{L}}{\mathsf{\gamma }}}\right)·{\left(∆\mathrm{L}\right)}^2,$$

(15)

where $∆\mathsf{\theta }={\mathsf{\theta }}^{\mathrm{new}}-{\mathsf{\theta }}^{\mathrm{old}}$ and $∆\mathrm{L}={\mathrm{L}}^{\mathrm{new}}-{\mathrm{L}}^{\mathrm{old}}$ represent the parameter increments, with ${\mathsf{\lambda }}_{\mathsf{\theta }}$ and ${\mathsf{\lambda }}_{\mathrm{L}}$ denoting the respective penalty coefficients satisfying ${\mathsf{\lambda }}_{\mathsf{\theta }} {< \mathsf{\lambda }}_{\mathrm{L}}$ to reflect the confidence discrepancy between the direction and length estimation. The parameter $\mathsf{\gamma }$ governs the asymmetry of the length penalty mechanism. Specifically, when $∆\mathrm{L} > 0$ (indicating length expansion), the penalty term assumes a larger value to suppress length inflation; when $∆\mathrm{L}<0$ (indicating length reduction), the penalty term becomes smaller to facilitate convergence toward the ground truth. Integrated with the confidence-guided strategy, the approach imposes differentiated constraints on different parameters, ensuring that the optimization trajectory aligns more closely with physical priors while simultaneously enhancing the convergence efficiency and estimation accuracy.

Subsequently, integrating the comprehensive step-length search with the asymmetric penalty mechanism, the PSF parameters are updated through gradient descent as follows:

$${\mathsf{\theta }}^{(\mathrm{k}+1)} = {\mathsf{\theta }}^{\left(\mathrm{k}\right)}-{\mathsf{\eta }}_{\mathsf{\theta }}\cdot {\displaystyle \frac{\partial \mathrm{L}}{\partial \mathsf{\theta }}},$$

(16)

$${\mathrm{L}}^{(\mathrm{k}+1)}={\mathrm{L}}^{\left(\mathrm{k}\right)}-{\mathsf{\eta }}_{\mathrm{L}}\cdot {\displaystyle \frac{\partial \mathrm{L}}{\partial \mathrm{L}}},$$

(17)

where ${\mathsf{\eta }}_{\mathsf{\theta }}$ and ${\mathsf{\eta }}_{\mathrm{L}}$ denote the learning rates for the direction angle and tail length, respectively, with ${{\mathsf{\eta }}_{\mathsf{\theta }} > \mathsf{\eta }}_{\mathrm{L}}$ to reflect the differential confidence in their initial estimates. The updated parameters are then employed to synthesize a new PSF, which is subsequently passed to the inner stellar image restoration step in the following outer loop iteration. This iterative refinement ensures a progressive convergence toward the optimal PSF configuration while respecting the asymmetric uncertainty inherent in the initial blind estimation.

To prevent artifacts such as excessive smoothing and centroid drift resulting from prolonged iterative refinement, we establish the convergence criteria for the outer alternating optimization loop based on the relative variations in PSF parameters and the loss function. Specifically, convergence is deemed to be achieved when the relative changes between consecutive iterations satisfy the following:

$${\displaystyle \frac{\left|{\mathsf{\theta }}^{(\mathrm{k}+1)}-{\mathsf{\theta }}^{\left(\mathrm{k}\right)}\right|}{\left|{\mathsf{\theta }}^{\left(\mathrm{k}\right)}\right|}}<{\mathsf{ϵ}}_{\mathsf{\theta }}, {\displaystyle \frac{\left|{\mathrm{L}}^{(\mathrm{k}+1)}-{\mathrm{L}}^{\left(\mathrm{k}\right)}\right|}{\left|{\mathrm{L}}^{\left(\mathrm{k}\right)}\right|}}<{\mathsf{ϵ}}_{\mathrm{L}}, {\displaystyle \frac{\left|{\mathrm{L}}^{(\mathrm{k}+1)}-{\mathrm{L}}^{\left(\mathrm{k}\right)}\right|}{\left|{\mathrm{L}}^{\left(\mathrm{k}\right)}\right|}}<{\mathsf{ϵ}}_{\mathrm{L}},$$

(18)

The algorithm is considered converged when all three conditions are simultaneously satisfied. The final outputs include the restored stellar image estimate ${\mathrm{f}}^{\left(\mathrm{final}\right)}$ and the optimized PSF parameters ${\mathsf{\theta }}^{\left(\mathrm{final}\right)}$ and ${\mathrm{L}}^{\left(\mathrm{final}\right)}$. The proposed algorithm establishes a self-correcting closed-loop framework that commences from an initial blind estimate, progresses through nested inner–outer alternating optimization with denoising, and incorporates physically informed constraints, ultimately achieving high-precision star point recovery.

2.2. The Workflow

The overall self-correcting bi-level iterative pipeline proposed in Section 2.1 is fully visualized in Figure 3. This subsection provides a step-by-step graphical interpretation of every functional block and data transmission path in the flowchart, to build an explicit mapping between the theoretical design and concrete execution steps.

(1)

The original image is imported as the initial input at the top block of Figure 3 and enters the standard curvature filtering module. Different from the dynamic model-guided denoising applied in subsequent iterations, this step only executes basic noise suppression to remove random background interference. The primary purpose of this preliminary filtering is to output a low-noise image for the following blind PSF estimation via image moments, which effectively avoids severe initial parameter deviation caused by heavy noise.

(2)

After denoising, the processed image is transmitted into the parameter calculation module. We adopt image moments to preliminarily compute coarse star centroid coordinates and initial PSF parameters. Meanwhile, the global grayscale distribution of star regions is counted to confirm the peak intensity value of each star spot. After that, a sliding matching operation is carried out to further calibrate the centroid positions. All calibrated coordinate and PSF variables required for constructing the ideal physical trail model are obtained through this module.

(3)

The calibrated centroid coordinates and PSF parameters are convolved to generate an ideal physical trailing model, which is embedded into the curvature filtering function to reprocess the original blurred image and complete secondary denoising. The denoised result serves as the input of the inner RL deblurring module. Moreover, this physical model supports dynamic iteration updates: after each full round of alternating optimization and outer parameter updating, a new physical model will be reconstructed to match the latest parameter state for the next round of denoising operation.

(4)

The RL algorithm equipped with centroid-adaptive stopping criteria is applied to process the denoised input image, and optimal centroiding coordinates are obtained after the iteration. The optimized centroid values are delivered to the outer iteration module as core calculation parameters, and they also serve as one of the criteria to judge whether the whole algorithm converges.

(5)

Multi-step search is carried out on the current PSF parameters to generate groups of candidate parameter combinations. Each candidate set is used to re-blur the sharp star image obtained from inner restoration, and the observation-consistency loss is calculated within the parameter correction module. An asymmetric penalty term is introduced to constrain the searching range of the trail length, and the updated, corrected PSF parameters are finally solved.

(6)

The convergence judgment module evaluates the relative variation in the updated PSF parameters: if the parameter variation exceeds the preset threshold, the updated parameters are fed back to reconstruct a new physical model, and the full cycle restarts; if all parameter variations satisfy the convergence threshold, optimized PSF parameters and sub-pixel centroid coordinates are output as the final results.

In summary, the flowchart in Figure 3 perfectly reflects the bi-level alternating architecture introduced in Section 2.1. The outer loop undertakes confidence-constrained PSF parameter optimization, while the inner loop completes physical-model-guided star image deblurring. The cyclic feedback path forms a self-calibrating closed loop that eliminates the mutual interference between denoising and blind PSF estimation existing in traditional linear single-pass algorithms.

3. Experimental Verification and Result Analysis

To comprehensively and systematically validate the performance of the proposed algorithm, we designed a multi-level experiment including an internal mechanism analysis, external robustness testing across diverse scenarios, and complete system-level application verification. First, by selecting typical blurred star spots, we comprehensively demonstrate the complete processing procedure from initial blind estimation to convergence through alternating optimization, thereby revealing the underlying working mechanisms. Second, adhering to the principle of controlled variables, we systematically evaluate the centroid recovery accuracy under different conditions, such as the noise levels, motion conditions (e.g., angular velocity), stellar magnitudes, and background to assess the algorithm’s robustness and generalization capability. Finally, the algorithm is applied to the star image containing multiple star points to evaluate its practical value in enhancing the star identification rates and attitude determination usability.

3.1. Simulation Dataset and Experimental Parameter Settings

The simulation test is performed based on typical parameters of star trackers as listed in

Table 1. Key parameter settings for simulation experiments.

Parameter Name	Value
Image Size	1024 × 1024 pixels
Field of View	5.0°
Exposure Time	60 ms
Detection Sensitivity	6.5 Mv
Gaussian σ of Static Star Spot	1.2 pixels

. The image size was 1024 × 1024 pixels [68], and the stellar magnitude range was configured from 4.5 to 8.0 mag, covering the typical detection dynamic range of star trackers [69]. Motion blur kernels were generated by specifying angular velocity vectors, and noise-free blurred star spots were produced through the convolution of the blur kernels with two-dimensional Gaussian star point models, in accordance with physical modeling standards for authentic image blurring processes. For noise simulation, noise variances σ were set to 30, 50, and 70, corresponding to signal-to-noise ratios (SNRs) of approximately 13, 9, and 7, respectively, simulating observation conditions ranging from favorable to severe environments. Specifically, σ = 30 corresponds to typical star tracker readout noise levels, while σ = 50 and σ = 70 simulate comprehensive noise environments resulting from the superposition of dark current noise [70]. The simulation platform was implemented in Visual Studio 2022, with a hardware configuration comprising an Intel Core i9-13900K processor (24 cores, 5.8 GHz), 64 GB DDR5 memory, and an NVIDIA RTX 3080 GPU (24 GB VRAM) to accelerate convolution operations and iterative computations. Ultimately, a 1024 × 1024 blurred star image containing seven randomly distributed star spots was generated, as shown in Figure 4a. A star spot located at coordinates (252.901, 214.902) was selected, and a 64 × 64 pixel window surrounding this star spot was cropped for detailed analysis, as illustrated in Figure 4b, serving as the core case study for subsequent algorithmic workflow examination.

In addition to the hardware and simulation setup parameters described above, all hyperparameters adopted by the proposed restoration algorithm are fixed uniformly across all experiments. The specific values of these preset variables are summarized in

Table 2. Fixed hyperparameters adopted in all experiments.

Parameter Name	Value
Physical model guidance balance weight	0.6
RL iteration centroid convergence threshold	0.01 pixels
Convergence threshold of angle	0.5°
Convergence threshold of length	0.2 pixels
Effective uncertainty weight for angle	0.62
Effective uncertainty weight for length	0.38
Direction angle and trail length learning rate	0.05
Direction angle and trail length candidate step size	{0.5, 0.2, 0.1, 0.05, 0.02, 0.01}
Direction angle penalty coefficient	0.1
Trail length penalty coefficient	1.3
Maximum RL iterations	100
Maximum outer-loop iterations	10
Grayscale projection judgment threshold	0.3

. All hyperparameters listed in Table 2 are kept identical across all comparative and multi-scene robustness experiments to ensure a fair experimental comparison.

Preliminary sensitivity checks were carried out on three core hyperparameters under the test condition of noise variance ${\mathsf{\sigma }}^2=30$ and medium background brightness to validate the basic robustness of the proposed algorithm.

First, we perturbed the physical model guidance balance weight $\alpha$ within the range [0.4, 0.8]. The centroid localization error stayed below 0.075 px for the reasonable interval [0.5, 0.7], reaching a minimum value of 0.0594 px at our default setting $\alpha =0.6$. The error only rose sharply above 0.09 px when $\alpha$ shifted to extreme values (0.4 or 0.8), which proves that the physical prior can stabilize the denoising preprocessing as long as $\alpha$ falls within physically reasonable bounds.

Second, we tested the two independent learning rates for the angle and trail length correction (${\mathsf{\eta }}_{\mathsf{\theta }}=0.05,{\mathsf{\eta }}_{\mathrm{L}}=0.05$) over the interval [0.03, 0.07]. The resulting fluctuation of the centroid error was less than 0.01 px across the whole range. This insensitivity to moderate learning rate changes originates from the full step-length search strategy adopted in the outer PSF optimization loop.

Third, we relaxed the outer-loop convergence thresholds (${\mathsf{\epsilon }}_{\mathsf{\theta }}=0.5°,{\mathsf{\epsilon }}_{\mathrm{L}}=0.2 \mathrm{p}\mathrm{x})$ by 50%. The maximum centroid error only increased by 0.014 px, and no overfitting artifacts appeared even under looser termination criteria.

Overall, these sensitivity test outcomes sufficiently validate the robustness of the adopted hyperparameter set. Perturbation experiments conducted on the three selected hyperparameters yield only minor fluctuations in the centroid localization error within reasonable value ranges. Such stable performance across varied parameter settings verifies that the fixed hyperparameters listed in Table 2 are reliable, and the proposed algorithm can maintain consistent and accurate restoration results without a manual parameter adjustment.

3.2. Preliminary Algorithm Results

3.2.1. Verification of Initial PSF Estimation

The accuracy of image moments in blind initial PSF estimation was validated by selecting a representative blurred star spot from the known synthetic star image illustrated in Figure 4a. Under the conditions of the angular velocities, rotation angle, and blur length pixels, the initial blind PSF estimation results were successfully obtained.

It is revealed from

Table 3. Initial blind PSF estimation results for target star point.

Parameter	True Value	Image Moment Initial Estimation	Absolute Error
Direction Angle θ	−18.08°	−18.28°	0.2°
Trailing Length L	14.38 px	17.0 px	2.62 px
Initial Centroid	(252.901, 214.902)	(252.010, 214.600)	0.94 px

that the blind estimation method based on image moments can effectively extract the main characteristics of the PSF from blurred star points. The preliminary estimation error for the direction angle is only 0.20°. However, the length estimation exhibits an approximately 18.22% overestimation bias due to noise-induced broadening effects in the one-dimensional projection. These results validate that the initial blind estimation module can provide a physically meaningful and fundamentally correct input for optimization.

3.2.2. Analysis of Model-Guided Denoising Effect

To demonstrate the correction effect of each step of the algorithm under noise variance ${\mathsf{\sigma }}^2=30$, we generated the first ideal physical model using the initial parameters from Table 3 ($\mathsf{\theta } = -18.08°, \mathrm{L} = 14.38 \mathrm{px})$. Subsequently, both standard curvature filtering and physical-model-guided curvature filtering were applied to Figure 4b, and the results are shown in Figure 5a–c.

Figure 5a–c clearly show that, while standard filtering smooths noise, it severely blurs both ends of the trailing structure. In contrast, the model-guided filtering result exhibits a clear linear structure of the trailing with sharp boundaries. However, the brightness values at both ends of the trailing are significantly higher than those in the ideal blurred image. This is because model-guided denoising is not intended to precisely reconstruct the ideal blurred image, but rather to generate an intermediate image with a clear structure, low noise, and consistency with the current physical hypothesis. It sacrifices absolute fidelity to the ideal blurred image in exchange for the structural integrity and high signal-to-noise ratio that are crucial for subsequent restoration steps. More importantly, this deviation is knowable and correctable—it originates from PSF parameter errors, which the alternating optimization outer loop is designed to correct.

3.2.3. Verification of Alternating Optimization Iteration Process and Correction Effect

To demonstrate how the algorithm progressively corrects the initial errors, we recorded all of the five-iteration outer loop process. Figure 6 shows the correction value of the PSF parameters, Figure 7 displays the iterative improvement of the denoised images, and

Table 4. Convergence record of centroid coordinates during the alternating optimization iteration process.

Iteration k	Reconstructed Centroid (x, y)	Centroid Error (px)
0	(252.807, 214.399)	0.5117
1	(253.074, 214.922)	0.1742
2	(253.026, 214.931)	0.1283
3	(252.963, 214.944)	0.0749
4	(252.963, 214.946)	0.0760
5	(252.963, 214.946)	0.0760

records the convergence process of the centroid position.

As shown in Figure 6, the optimization of PSF parameters is not achieved in a single step. After the first iteration, the parameters remained unchanged as the algorithm first utilized the initial model to improve the image quality. The second iteration represents a critical correction step: based on the clearer image obtained from the first iteration, the algorithm successfully corrected the motion direction angle $\mathsf{\theta }$ to −18.3° and significantly adjusted the length $\mathrm{L}$ from the initial 17.0 px to 14.2 px, effectively overcoming the overestimation bias in the initial length. Ultimately, the PSF parameter errors were constrained within $∆\mathsf{\theta } < 0.5°$ and $∆\mathrm{L} < 0.5 \mathrm{px}$, achieving sub-pixel estimation accuracy.

Figure 7 demonstrates the dynamic model-guided mechanism. As the iterations proceed, the morphology of the ideal physical model changes accordingly. The structure of the guided denoising images closely follows the evolution of the ideal physical model, becoming increasingly accurate. With continued iterations, the tail length of the blurred image shows significant shortening, gradually restoring to normal. The overall tail structure progressively converges toward the standard noise-free blurred image in Figure 7a. This indicates that the denoising process and parameter optimization process are deeply coupled, where each parameter update immediately produces more accurate denoising results, providing better input for the subsequent image restoration.

Table 4 records the convergence process of the centroid position through iterative optimization. At the initial iteration (k = 0), the centroid coordinates are (252.807, 214.399) with an error of 0.512 px. After four iterations, the coordinates gradually adjust to (252.963, 214.946), with the error decreasing from 0.174 px to 0.076 px. The iteration exhibits the characteristic of the rapid decrease and slow convergence, with the final error stabilizing at 0.076 px, which meets and exceeds the 0.1-pixel design specification. This validates the effectiveness of alternating optimization framework between the image restoration and model correction, achieving high-precision star point localization.

3.3. Comparative Verification of Algorithms

To objectively evaluate the restoration performance of the proposed algorithm under non-PSF prior conditions, this section selects the classic RL non-blind deconvolution algorithm and the traditional blind deconvolution algorithm based on sparse priors as control groups [16,71]. The RL algorithm is based on maximum likelihood estimation under the Poisson noise model, representing a classic method in the field of astronomical image restoration; however, it requires precise PSF information as a prerequisite. The traditional blind deconvolution algorithm based on sparse priors, as a typical blind deconvolution method, estimates the blur kernel, relying solely on natural image statistical priors without dependence on PSF priors. Three comparative experimental scenarios were set up: Under the premise of providing the non-blind RL algorithm with precise PSF estimation, we use RL non-blind restoration (with noise variance σ² = 0) as the ideal upper-bound benchmark. We comprehensively compare the centroid localization errors among RL non-blind restoration, the sparse-prior-based blind deconvolution as a conventional control benchmark, and the proposed algorithm under identical noise levels (σ² = 30), thereby validating the accuracy and robustness of the proposed algorithm. The experimental results are shown in Figure 8.

The experimental result in Figure 8 reveals that, under noise-free ideal conditions, the RL non-blind restoration algorithm achieves an extremely high centroid localization accuracy with precise PSF information, with centroid positioning errors generally around 0.01 px, representing the theoretical performance upper bound for the restoration task. However, when typical sensor noise with σ² = 30 is introduced, the restoration accuracy of the non-blind RL algorithm exhibits significant degradation, with the centroid localization errors for some star points increasing to 0.025 px or higher. This indicates that classic algorithms exhibit limitations in noise suppression and show a strong dependence on PSF priors. In contrast, the traditional blind deconvolution method based on sparse priors exhibits localization errors exceeding 0.69 px with a substantially degraded performance. This deterioration occurs because such algorithms rely on general image priors that are mismatched with the star point target characteristics. Under strong noise and point target scenarios, these methods cannot stably estimate the blur kernel and are, therefore, unsuitable for high-precision star centroiding.

In contrast, the physical-model-based dual-layer iterative optimization algorithm proposed in this paper demonstrates exceptional performance without requiring any PSF prior input. Under a noise environment of σ² = 30, the centroid localization errors of the proposed algorithm are consistently controlled within the range of 0.02 px to 0.07 px. The restoration accuracy of the vast majority of star points is comparable to that of the non-blind RL restoration (σ² = 30), with certain star points even outperforming the latter. This result fully demonstrates that the proposed algorithm, through a closed-loop framework including parametric modeling, iterative guidance, and alternating optimization, successfully overcomes the challenges of noise sensitivity and non-unique solutions inherent in blind deconvolution problems, achieving a restoration performance directly approaching that of non-blind methods. These findings validate the engineering application value of the proposed method for high-precision star centroiding in dynamic star images.

3.4. Multi-Scene Robustness Tests

To validate the robustness and generalization capability of the proposed algorithm in complex star image observation scenarios, this paper designs a multi-dimensional testing scheme. Tests are conducted under different conditions such as noise levels, angular velocities, stellar magnitudes, and background brightness to verify the centroid restoration accuracy of individual star points. This approach aims to simulate the complex observation environments that star trackers may encounter during actual operation. By controlling a single variable while keeping other experimental conditions fixed, the centroid localization accuracy of the algorithm is systematically verified under different interference factors, clarifying the applicable range and performance boundaries of the algorithm. For each test scenario, multiple groups of random star points are selected for repeated verification, ensuring the reliability and representatives of the experimental results.

3.4.1. Accuracy Verification Under Different Noise Conditions

Sensor noise and environmental interference distort star point signals during image acquisition, degrading the centroid localization accuracy. We tested the algorithm’s noise robustness under three noise variance levels (σ² = 30, 50, and 70) [72] while varying the angular velocities (ωx, ωy). The centroid restoration results appear in Figure 9.

The results demonstrate that the proposed algorithm maintains a high centroid localization accuracy across all noise conditions. Under low-noise conditions (σ² = 30), the algorithm achieves a sub-pixel accuracy averaging 0.018 px through iterative optimization, confirming its positioning advantage under ideal observation conditions. As the noise intensity increases to σ² = 70, the centroid localization error shows a gradual upward trend, with the mean error remaining within 0.054 px and the peak error not exceeding 0.068 px. Notably, centroid error distributions for star points at different spatial positions and initial brightness levels exhibit a uniform dispersion, with the variance consistently ranging from 0.005 to 0.068 px and no significant systematic bias. This confirms the strong adaptability of the algorithm to complex star point distributions and its broad scene applicability.

3.4.2. Accuracy Verification Under Different Angular Velocities

The star trailing blur on the detector caused by carrier attitude motion directly compromises the centroid localization accuracy. The trail length and orientation depend on the angular velocity, introducing motion-induced distortion. To evaluate the algorithm’s adaptability to varying motion conditions, we tested the centroid restoration performance across multiple angular velocity scenarios (corresponding to different trail lengths and orientations) at a fixed noise level of σ² = 30 [73]. The results are summarized in Figure 10.

As shown in Figure 10, at a fixed noise level (σ² = 30), the centroid restoration errors exhibit a uniform random distribution without a directional offset, which verifies that the proposed algorithm is insensitive to the orientation of star trails. The algorithm achieves the optimal localization accuracy under medium angular velocity, and all positioning errors are maintained at a low level, demonstrating favorable robustness against varying trail lengths. In addition, the insignificant linear correlation between the centroid error and angular velocity further validates the algorithm’s strong adaptability to variations in angular velocity.

3.4.3. Accuracy Verification Under Different Stellar Magnitudes

The stellar magnitude determines the star point signal intensity. Higher magnitudes correspond to weaker signals with lower signal-to-noise ratios, making them more susceptible to noise interference and thus degrading the centroid localization accuracy. To evaluate the algorithm’s adaptability to star points of varying magnitudes, we conducted centroid restoration tests at a fixed noise level of σ² = 30, covering the typical observational range of 6 to 12 mag. The results are presented in Figure 11 [74].

Across all stellar magnitude conditions, the centroid localization accuracy remained stable, with errors bounded within 0.1 px for all magnitude ranges. The 7–8 mag stars exhibited the minimum errors, with a minimum value of 0.020 px and a mean error of approximately 0.041 px. As the stellar magnitude increased, the errors showed a gradual upward trend, reaching maximum values of 0.092 px and 0.091 px in the 10–11 mag and 11–12 mag intervals, respectively. Nevertheless, the proposed noise suppression and signal enhancement modules effectively constrain the growth rate of positioning deviation, and a favorable centroid extraction precision is preserved across the full magnitude test range. These observations verify that our algorithm can well compensate for the signal attenuation induced by varying the stellar brightness, showing a strong adaptability to targets with diverse magnitudes and reliable anti-noise robustness for faint star imaging.

3.4.4. Accuracy Verification Under Different Background Conditions

In star tracker observation environments, variations in background brightness affect the contrast between star points and the background. A higher background brightness causes star point signals to be increasingly obscured, thereby degrading the centroid localization accuracy [75]. To evaluate the algorithm’s adaptability to different background brightness conditions, centroid restoration tests were conducted under a fixed noise level (σ² = 30), covering four typical background categories: low, low–medium, medium–high, and high brightness. The experimental results are presented in Figure 12.

The experimental results in Figure 12 reveals that the algorithm maintains a satisfactory centroid localization accuracy across all background brightness conditions, demonstrating a strong adaptability to diverse observation scenarios and robust resistance to background interference. Under low-brightness conditions, the mean centroid restoration error is merely 0.036 px, enabling high-precision centroid localization. For low–medium-and medium–high-brightness conditions, the errors exhibit a gradual upward trend with increasing background brightness, while the accuracy remains stable. Under high-brightness conditions, background interference is the most severe and the star point signal distinguishability diminishes, resulting in an error of 0.105 px. However, the overall error remains within 0.10 px, with no significant accuracy degradation or localization failure observed. In summary, regardless of the background brightness variations, the algorithm achieves stable centroid restoration by accurately extracting star point features through signal enhancement and background suppression mechanisms, validating its robust adaptability to diverse background conditions.

3.5. Simulation Dataset and Experimental Setup

3.5.1. Multi-Star Blurred Star Image Noise Validation

The single star point test has validated the centroid localization accuracy of the proposed algorithm under various interference conditions. To further evaluate the algorithm’s engineering practicality and batch processing capability, the proposed algorithm was applied to the 1024 × 1024 star multiple-star-point blurred image shown in Figure 5a (with noise variance σ² = 50) for batch processing testing. During the testing process, the algorithm automatically performed star point detection, signal extraction, and centroid restoration without manual intervention. Ultimately, all seven star points were successfully detected and individually restored. The centroid coordinate changes and corrections before and after processing for all star points are summarized in

Table 5. Summary of centroid coordinate changes and corrections for all star points after processing (σ² = 50).

Original Centroid (x, y)/px	Restored Centroid (x, y)/px	Centroid Correction (Δx, Δy)/px
(181.012, 366.988)	(181.040, 366.998)	(0.028, 0.010)
(717.432, 528.877)	(717.460, 528.912)	(0.028, 0.035)
(284.612, 147.588)	(284.640, 147.650)	(0.028, 0.062)
(757.946, 527.531)	(757.977, 527.555)	(0.031, 0.024)
(889.712, 546.077)	(889.743, 546.099)	(0.031, 0.022)
(351.188, 470.301)	(351.220, 470.335)	(0.032, 0.034)
(259.210, 712.523)	(259.240, 712.557)	(0.030, 0.034)

.

It is revealed in Table 5 that, under noise interference conditions with variance σ² = 50, the proposed algorithm achieved precise detection and centroid restoration for all seven blurred star points in the star field image as shown in Figure 5a. Significant and consistent positive corrections were applied to the centroids of all star points, effectively compensating for positioning errors caused by noise interference and star point blur. The advantage of this consistent positive correction lies in its ability to simultaneously improve the positioning accuracy of all star points in the star image, avoiding the imbalance where some star points receive an accuracy improvement while others experience an accuracy degradation. This further validates the algorithm’s stability and reliability in batch processing the multi-star-point blurred star image, supporting the algorithm’s engineering applications.

3.5.2. Multi-Star Robustness Validation

To further validate the robustness and anti-interference stability of the proposed algorithm in large-scale star point processing scenarios and address the limitations of small-sample multi-star-point testing, a large-sample robustness verification experiment was conducted. The experiment maintained the noise interference condition of variance σ² = 50 as described previously, simulating the star point distribution characteristics in a true observation of the star tracker. One hundred star points were selected randomly, and the proposed algorithm was applied to perform batch centroid restoration for all star points. The centroid positioning errors for each star point were recorded and plotted as scatter plots in Figure 13.

The results of Figure 13 shows that the centroid positioning errors for all 100 star point sets fall within the range of −0.1 to 0.1 px. The error scatter points exhibit a uniform distribution pattern with no obvious clustering bias, indicating that the algorithm maintains a stable processing performance during large-scale star point operations. The algorithm demonstrates a strong adaptability to variations in star point distribution positions and signal intensities, consistently delivering high-precision centroid restoration results. These findings validate the algorithm’s robustness in batch processing scenarios with large sample sizes.

4. Conclusions

This paper proposes a novel method guided by physical models and alternating optimization iteration to address the fundamental challenges of noise sensitivity, prior deficiency, and solution non-uniqueness in the blind restoration of dynamic star images. The main conclusions drawn from this work are as follows:

First, a parametric PSF modeling framework with confidence-guided coarse estimation was developed to significantly mitigate the ill-posed nature of the problem. By transforming the high-dimensional PSF matrix estimation into low-dimensional physical parameter estimation for direction angles and lengths, and incorporating experimental error patterns for the confidence assignment to parameters, this work provides a physically interpretable search initialization for subsequent optimization processes.

Second, a physical-model-guided curvature filtering algorithm was designed for star image preprocessing, which achieves the effective preservation of trailing structures under strong noise conditions. By innovatively embedding iterative-updated ideal physical models as strong priors into the denoising process, the proposed approach successfully suppresses noise and actively reinforces linear trailing structures consistent with physical laws even at a noise variance as high as 70, thereby laying a solid foundation for high-precision deconvolution.

Third, a two-layer alternating optimization framework under raw observation consistency constraints was constructed, achieving sub-pixel level centroid localization. The inner layer employs an intelligent-converging RL algorithm for image restoration, while the outer layer corrects the PSF parameters through full-step search and asymmetric penalty mechanisms, forming a self-calibration closed-loop from image restoration to model correction. The experimental results demonstrate that this framework maintains centroid positioning errors below 0.1 px, with a maximum error of 0.105 px under the most challenging high-background condition.

In summary, the proposed method achieves high-precision star centroiding without requiring explicit PSF priors, providing a reliable software solution for the autonomous high-precision attitude determination of star trackers in dynamic environments. Future work will focus on extending the framework to handle more complex motion patterns and integrating it into real-time flight hardware for practical space mission applications.

All validation experiments in this study are based on synthetic star images generated using representative parameters of an actual star tracker. Although these simulations reproduce the principal motion-blur and noise characteristics considered in this work, they cannot fully capture all non-ideal effects present in the measured on-orbit images. Due to the current unavailability of suitable measured star-tracker frames, real-data validation was not included in this revision. Future work will evaluate the proposed method using measured on-orbit images when such data become available.