A Resolution and Localization Algorithm for Closely-Spaced Objects Based on Improved YOLOv5 Joint Fuzzy C-Means Clustering

The presence of numerous space objects poses significant challenges to spacecraft launch, operation, and space security. To address this issue, a novel algorithm based on improved YOLOv5 joint Fuzzy C-Means (FCM) has been proposed for discerning and localizing closely-spaced objects (CSOs) in space. The algorithm employs the lightweight neural network YOLOv5 to estimate the quantity of the targets and employs the improved FCM localizing. In the first stage, it enhances the precision of CSOs quantity estimation by integrating small target layers and the Convolutional Block Attention Module (CBAM) into the YOLOv5 network. In the second stage, leveraging the estimated target quantity as input, the FCM algorithm based on Lanczos3 interpolation and particle distribution was used for the sub-pixel localization of each target cluster. The experimental results show that the algorithm's quantity estimation accuracy is more than 90%, and the average localization error is within 0.32 pixels. Moreover, the average running time of the algorithm is less than 0.034 s. Compared with other methods, the algorithm shows excellent performance, and its effectiveness is important for target tracking, 3D localization, and detailed analysis.

targets may undergo variations in posture and quantity due to the release of decoys or accompanying objects, resulting in clusters of closely-spaced targets known as closely-spaced objects (CSOs).However, due to the structural characteristics of the optical system, these CSOs exhibit overlapping clusters of spots on the image plane, making it impossible to discern crucial information such as the quantity and positions of the targets [2].Consequently, the study of super-resolution algorithms for infrared CSOs assumes significant importance in distinguishing the positional parameters of individual targets from the spot clusters.
The essence of the super-resolution positioning algorithm for infrared CSOs lies in utilizing the image plane response model and noise model to estimate parameters for point targets [3], [4].Early research on super-resolution algorithms for infrared CSOs primarily relied on the Bayesian theory, establishing objective functions based on Maximum a Posteriori (MAP) estimation [5], [6], [7], Maximum Likelihood (ML) estimation [2], [3], [8], [9], [10], [11], or Least Square (LS) estimation, and employing optimization algorithms to solve for target information.The ML estimation method degrades to the LS estimation method under the assumption of Gaussian noise.To address the optimization problem of the objective function, researchers such as Lin et al. utilized Particle Swarm Optimization (PSO) algorithm [10] and Quantum Particle Swarm Optimization (QPSO) algorithm [9] to improve the accuracy of target quantity estimation, albeit at the cost of computational speed.Subsequently, some scholars proposed a CSOs super-resolution method based on sparse reconstruction theory, leveraging the sparsity of targets on the image plane [3], [12], [13], [14], [15].This method constructs an overcomplete dictionary through discrete sampling, establishes a sparse measurement model, and transforms the norm optimization problem into a second-order cone programming problem for sol.Super-resolution algorithms based on sparse reconstruction theory are capable of accurately estimating target positions with minimal sensitivity to noise, but they are computationally intensive and require high prior knowledge.Reference [2] (pp.32-45) and [16] (pp.268-275) introduced a method that decomposes pixel clusters using dis-tributed particles and iteratively optimizes the solution using the Expectation-maximum (EM) clustering algorithm.This approach achieves a more precise estimation of the number of targets but suffers from sensitivity to noise.In addition, Lu et al. used the generalized likelihood ratio test (GLRT) based on a linearized observation model for the measurement extraction of two targets with unknown equal intensities in the focal plane [17].Tian et al. used two frames of FPA in which the targets were slightly shifted to estimate the positions and intensities of two closely-spaced objects with unknown intensities and used GLRT for performing decisionmaking [18].
In recent years, some researchers have attempted to apply the rapidly advancing neural network algorithms to CSOs superresolution, proposing a hybrid detection-classification method capable of simultaneously detecting and segmenting closely spaced targets [19], [20], [21].However, currently, such algorithms are only applicable to scenarios with simple conditions and a target quantity of 2 in CSOs situations.
Thus, the challenges in super-resolution and localization of CSOs can be summarized as the following three points: inaccurate estimation of the number of targets, the prominent contradiction between target center-of-mass localization accuracy and algorithm runtime, and algorithms are unstable, or even fail, in low signal-to-noise ratio (SNR) images of CSOs.To address the first and second challenges mentioned above, this paper proposes a joint improved YOLOv5 and fuzzy C-mean algorithm for super-resolution and localization of CSOs.The three main contributions of this work are: 1) The complete neural network structure is firstly applied to the classification and resolution of infrared CSOs, and a more accurate estimation of the number of CSOs is achieved.2) For the characteristics of infrared CSOs, the YOLOv5 network used to estimate the number of targets is improved by adding a small target detection layer and Convolutional Block Attention Module (CBAM), which makes the network more sensitive to point targets.
3) The traditional Fuzzy C-Means (FCM) clustering algorithm is improved.Based on the idea of pseudodownsampling, the FCM algorithm is combined with Lanczos3 interpolation and particle distribution to realize the sub-pixel centroid localization of CSOs.The rest of this paper is structured as follows.Section II analyzes the CSOs infra-red imaging model.In Section III, the proposed algorithm is described in detail and the image data used is illustrated.Experimental results and comparisons are given in Section IV.Finally, conclusions are given in Section V.

II. CSOS INFRARED IMAGING MODEL
Due to the long detection distance, whether it is mid-range targets, stars, or space debris, they can all be considered point targets for spaceborne infrared detection systems.During the imaging process, the diffraction effect of the optical system causes the energy of the point targets in the image plane to spread to their neighboring pixels, forming a central bright spot known as the Airy pattern, which accounts for approximately 84% of the total energy [22].This diffraction effect is typically approximated using a two-dimensional Gaussian Point Spread Function (PSF), whose expression in the image plane is as follows [4], [22].
where, (x i , y i ) is the projected position of the target in the image plane, and the standard deviation σ psf characterizes the range of energy diffusion, which is determined by the sensor focal length and detection wavelength [4], [12], [22].In this paper, σ psf =0. 5 .By integrating the PSF over the image pixel, we can obtain the unit amplitude response for object radiation as follows.
where (u, v) is the coordinate corresponding to the pixel, and d is the pixel size.
In the presence of multiple targets, the magnitude of each pixel is the result of the linear combination of the individual target pixel responses.Suppose there are K targets in CSOs and the projection coordinates of each target in the image plane are (x k , y k ), k = 1, 2, . . ., K. The response of the target in a certain pixel is represented as g k (u, v), k = 1, 2, . . ., K. Then the measurement model for the image plane can be represented as follows [2], [4].
(3) where, s is the response intensity vector of each target, n is the noise vector.In this study, it is assumed that the noise in each pixel is independent and follows a Gaussian distribution.
In optical systems, the radius of the Airy pattern has a size of 1.22λ/D (λ is the detection wavelength and D is the lens diameter), which is also the circle radius 1.9σ pf s of the two-dimensional Gaussian point spread function in the image plane [22].This is the physical discrimination limit of the sensor between two point targets, called the Rayleigh Unit (R) [12], [22].Algorithms that can distinguish point targets with a separation distance smaller than 1R are considered to possess super-resolution capability.
However, when the distance between two point targets is less than 1R, the energy of the targets overlaps significantly, causing the concealment of information regarding the number and positions of the targets, as illustrated in Fig. 1.
For CSOs with a larger number of targets, differentiating the targets and estimating parameters becomes more complex.Fig. 2 illustrates CSOs composed of seven targets with varying grayscale values and sizes.

III. METHODS AND MATERIALS
To address the issues of slow estimation speed and large estimation errors in the quantity and position estimation of neighboring targets in space, this paper proposes a CSOs quantity and position estimation algorithm called LYLF, which combines an improved version of YOLOv5 and FCM.This algorithm achieves significant improvements in the accuracy of target quantity and position estimation, as well as algorithm speed.The algorithm can be roughly divided into two modules: the quantity estimation module (LC-YOLOv5) and the centroid estimation module (LP-FCM).In the quantity estimation module, a lightweight YOLOv5 network is employed for the first time in the classification of point targets.It is enhanced with a smalllabel layer and attention mechanism to adapt to the estimation of small target clusters.In the centroid estimation module, the following improvements have been made to the traditional FCM clustering algorithm.Based on the pseudo-oversampling idea, it expands the image information using the Lanczos3 interpolation algorithm and combines particle distribution and downsampling processing to achieve sub-pixel decomposition of target clusters.This enables the FCM algorithm to extract target centroids at a sub-pixel level, effectively improving the estimation accuracy of target centroid positions.Fig. 3 illustrates the structural diagram of the proposed algorithm in this paper.

A. An Improved YOLOv5 Algorithm for Estimating the Number of CSOs
YOLOv5 is a single-stage detection algorithm with a network structure composed of a head, backbone, and neck.It can simultaneously output predicted bounding box positions and class confidences of objects [23], [24], [25].With its fast detection speed and high accuracy, YOLOv5 demonstrates significant advantages in the field of large object detection.Therefore, we consider applying the YOLOv5 network to the classification of CSOs to achieve improvements in both the accuracy of quantity estimation and the runtime speed of CSOs super-resolution algorithms.
However, for smaller objects, particularly the point targets described in this paper, the detection and classification capabilities of YOLOv5 require further enhancement.
To address these issues, we introduce a small object detection layer and CBAM to optimize the YOLOv5 network structure, referred to as LC-YOLOv5, aiming to improve its classification accuracy for small objects.
The addition of the small object detection layer aims to overcome the challenges posed by the larger downsampling multiple and the deep network architecture in extracting feature information from small objects.The new small object detection layer can capture feature maps at a larger scale, thereby addressing the difficulty of extracting information from small objects [26], [27].
Furthermore, the CBAM attention mechanism [28], [29], [30] is integrated into YOLOv5.This lightweight module enhances detection accuracy while introducing minimal computational overhead.The CBAM attention mechanism consists of both channel attention and spatial attention.The spatial attention mechanism enables the neural network to focus on the crucial pixel regions in the image, while the channel attention mechanism manages the allocation of feature map channels.CBAM allocates attention in both channel and spatial dimensions, thereby enhancing the performance improvement achieved by attention mechanisms.The specific structure of CBAM is depicted in Fig. 4.
The specific improvements to the algorithm are as follows.A connectivity layer is added to the neck and backbone portions of the YOLOv5 network, and the last feature layer of the neck is upsampled to expand the feature map.Adding the small target detection layer can improve the network's ability to extract small target features.In addition, adding CBAM to the C3 module in the backbone part increases the performance of the algorithm in small target scenarios.We call the improved YOLOv5 algorithm as LC-YOLOv5 and its network structure is shown in Fig. 5.
The improved LC-YOLOv5 algorithm significantly enhances performance in the estimation of CSOs quantity while sacrificing only a small fraction of the runtime.The performance comparison results of the algorithm will be presented in Section IV.

B. FCM Algorithm Based on Lanczos3 Interpolation and Particle Dispersion
Traditional super-resolution algorithms refer to the process of restoring images from low resolution to high resolution by using a series of algorithms to recover image details.In our study, as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.our research focus is on CSOs, which occupy a small number of pixels, we adopt the concept of image super-resolution and combine Lanczos3 interpolation with particle distribution.Firstly, based on image oversampling ideas, we utilize Lanczos3 interpolation to expand the image information.Subsequently, particles are dispersed across the expanded image, and the density of particles represents the gray-scale value of pixels.Then the image is downsampled back to its original size, enabling the sub-pixel decomposition of pixels.Finally, the FCM clustering algorithm is employed to partition the target regions and estimate their centroid positions.We refer to this FCM algorithm, based on Lanczos3 interpolation and particle dispersion, as LP-FCM.The specific implementation steps of LP-FCM are as follows: 1) The Lanczos3 Interpolation Based on Pseudo-Oversampling: Increasing the sampling frequency in imaging, also known as oversampling imaging [31], [32], [33], [34], has a noticeable effect in reducing aliasing effects and improving spatial resolution.Based on the principles of oversampling imaging, the image of the target region is initially processed using Lanczos3 interpolation.Subsequently, the over-sampling technique of the sensor is emulated by traversing pixels using a convolution template [16], aiming to expand the pixels.
The Lanczos3 interpolation algorithm is a traditional image super-resolution algorithm that calculates the discrete convolution between the input image and the Lanczos kernel to obtain the corresponding values of the expanded pixels [35], [36], [37].The Lanczos3 kernel is shown below.
Let s uv be a pixel point in the image with coordinate (u, v).The Lanczos3 interpolation function of the image is expressed as follows.

S(x, y)
Then, the target region i = α β γ δ after Lanczos3 interpolation is filled to obtain the pixel matrix I.
Sliding the convolution template R = 1 1 1 1 over the pixel matrix I to obtain the pseudo-oversampling interpolated pixel Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.matrix I p .
2) Particle Dispersion and Downsampling Processing: Particles are distributed based on the proportion of the grayscale value of each pixel in the interpolated pixel matrix to the total grayscale value of the target cluster, enabling sub-pixel decomposition of the target cluster.The density of particle distribution reflects the magnitude of the pixel grayscale value.The representation of the number of particle distributions in the mth pixel of the interpolated pixel matrix I p is as follows.
where S m is the grayscale value of the mth pixel, n is the total number of pixels in this interpolated pixel matrix I p , and N I p is the total number of particles distributed on this target cluster.Finally, the image is downsampled to map the interpolated matrix back to the original target cluster, obtaining a sub-pixel particle set.
x o = (x p + 1)/2 where (x o , y o ) denotes the coordinates of the particle in the original target region, and (x p , y p ) denotes the coordinates of the particle in the interpolation matrix.The aforementioned process, which includes oversampling and downsampling of the image, is referred to as pseudooversampling [16].Using Fig. 2 presented in Section II as an example, Fig. 6 demonstrates the procedure of the Lanc-zos3 interpolation and particle distribution algorithm based on pseudo-oversampling.
3) Fuzzy C-Mean Clustering Algorithm: By following the above steps, we obtain the sub-pixel dataset expanded with The subsequent FCM algorithm [38], [39] is utilized to perform clustering on this dataset, which leads to the identification of cluster centers representing the target centers.FCM algorithm is an iterative clustering algorithm that employs the sum of squared errors as its objective function.The membership degrees are utilized to express the probability of a sample belonging to a specific class.The objective function of the FCM algorithm is defined as follows.
where d ij denotes the spacing between the sample and the cluster center (Euclidean distance is generally used).m is the membership factor.And uij is called the membership degree, which indicates the probability that the jth sample in the data set belongs to the ith classification.
Here, we consider the particles distributed over the pixels as the set of data samples, denoted as where χ k = x k y k represents the coordinates of the kth particle.
Then the expressions of the above parameters are as follows.
v i denotes the clustering center with the following iterative formula.
During the iteration, the membership degree uij should satisfy the following constraints.
The LP-FCM algorithm takes the estimated number of targets by LC-YOLOv5 and their corresponding images as input.After iterative processing, the clustering centers vi that meet specific criteria are considered as the results of target localization in the image plane.The super-resolution and localization effectiveness of this algorithm for various CSOs will be shown in the experimental part of Section IV.

C. Experimental Data
This paper focuses on the space background, which is devoid of complex features such as landmasses and clouds.Therefore, we simulate long-wave infrared images to verify the effectiveness of the algorithm in this paper.When the infrared detection system operates outside the atmosphere, the noise mainly originates from shot noise, thermal noise, and 1/f noise.We analyze the combined effects of various noises on the detector and find that the noise distribution approximately conforms to the Gaussian distribution.In addition, the noise of each pixel is independent.Therefore, we can utilize the Gaussian noise with different strengths added to the simulated image to verify the robustness of the algorithm to noise.
On the other hand, to adapt the algorithm to the real image data, we use the data from the long-wave infrared camera that shoots stars for verification.Due to the special characteristics of the research object in this paper, it is almost difficult to capture the real scene.We propose to segment and reorganize several different stellar images to simulate CSOs scenes to validate the algorithm.
Therefore, the experimental data in this paper is divided into two parts, simulated data and real captured data.
1) Simulated Data: Simulated data is divided into training and test sets.We simulated a total of 28000 images with a size of 64 × 64 pixels containing spatially adjacent target clusters.The data was split into a 9:1 ratio for training and testing.The simulated data includes CSOs with 2 to 7 targets of various intensities and sizes.Among them, individual target sizes were randomly generated in 3 × 3, 5 × 5, 7 × 7 and 9 × 9.The centroids of adjacent targets are positioned at distances of 0.8 to 3 pixels (as shown in Fig. 7).Additionally, we simulated an additional 18000 CSOs images for performance validation, including noise-free images and images with SNR ranging from 25 to 40 dB.The SNR is calculated as follows.

SN R = 10 log 10
T − B σ n (15) where T was the minimum of peak gray values of all targets, B represents the background gray value, and σ n represents the standard deviation of noise.
2) Combined Data: To verify the algorithm's effectiveness and feasibility in real-world scenarios, we captured several sets of star images (as shown in Fig. 8(a)).Among them, the size of the infrared images is 512 × 512.Five sets of images were captured, totaling 920 images.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.It should be noted that since images of real CSOs scenarios are difficult to take, a combination of real star images and background images are used for validation in this paper.Different stars in other images were randomly combined and placed in background images, mimicking infrared spatially adjacent target clusters.We selected one of these sets of images as a background and lined up different stars on the images, combining a total of 206 images.Every combined image includes CSOs with 2 to 6 targets.The centroids of adjacent targets are positioned at distances of 1 to 3 pixels.The following Fig. 8 are examples of combined images.

A. Evaluation of Indicators
The LC-YOLOv5 algorithm is quantified by using precision, recall, mean average precision (mAP), the accuracy of CSOs quantity estimation, and runtime in terms of CSOs number estimation performance.The expressions are as follows.

P recision = T P T P + F P (16)
Recall = T P T P + F N (17) where, Precision and Recall are computed based on true positives (TP), false positives (FP), true negatives (TN), and false negatives (FN).Plot the Precision-Recall curve (P-R), defining the area under the curve as average precision (AP).The mean value of all categories of AP is called the mean average precision (mAP).When IoU = 0.5, the mAP is written as mAP@0.5.
In addition, the accuracy of CSOs quantity estimation is the ratio of the estimated target number to the true target number.
The LYLF algorithm is also quantified by using the Probability of false detection, Probability of missing target, Joint Root Mean Square Error (JRMSE) of target location estimation, and Probability of correctly distinguishing and locating the target in terms of CSOs number and location estimation performance.The equation of JRMSE is as follows.
where (x i , ŷi ) is the estimated coordinates of the ith target, (x i , y i ) is the true coordinates of the ith target, and N is the number of targets.
It should be noted that the probability of correctly distinguishing and localizing a target is the ratio of the number of correctly located targets to the number of correctly estimated targets.

B. Analysis of Algorithm Parameters 1) Selection of YOLOv5
Network Architectures: YOLOv5 offers five basic versions of network architectures: YOLOv5s, YOLOv5n, YOLOv5m, YOLOv5l, and YOLOv5x.These architectures share the same overall network structure but differ in model size and computational complexity.YOLOv5s is the smallest model with the lowest computational complexity and the poorest detection performance.On the other hand, YOLOv5x is the largest model with the best detection performance but also the highest computational complexity.In this study, performance testing was conducted on different network architectures (as shown in Table I).Due to hardware limitations and feedback on testing results, all tested epochs were set to 60, and the batch size was set to 8.  The experiments revealed that YOLOv5x, the largest model, achieved only a 7% improvement in accuracy at the cost of more than five times longer runtime compared to YOLOv5s.Considering the accuracy of CSOs classification the realtime performance of the algorithm, this study decided to use the smallest network architecture, YOLOv5s.
2) The Effects of Different Attention Mechanisms on YOLOv5: As mentioned earlier, this study addresses the drawback of the insensitivity of YOLOv5 to small objects by making two improvements.Firstly, the CBAM attention mechanism is utilized to focus the network on informative regions while disregarding irrelevant areas.The effectiveness of CBAM compared to other attention mechanisms (SE, CA, and ECA) is experimentally validated in this study, as shown in Table II.
The experimental results (shown in Table II) demonstrate that integrating the CBAM attention mechanism into the YOLOv5 network yields more significant improvements in CSOs classification compared to SE, CA, and ECA.It also shows a more noticeable enhancement in quantity estimation accuracy without a significant increase in computation time.

3) Selection of the Number of LP-FCM Particles:
The LP-FCM algorithm proposed in our study is influenced by the number of particles in terms of the accuracy of target position estimation.Additionally, different distributions of particles can affect the operating speed.Therefore, this study conducts experiments and analysis to investigate the impact of different particle distribution quantities on algorithm performance, as shown in Fig. 9.
Based on the experimental results, it is observed that the position estimation error decreases as the number of particles increases within the range of 100-1000.However, when the number of particles exceeds 1000, the position estimation error no longer decreases.Therefore, the effective range of particle quantity in the LP-FCM algorithm proposed in this study lies between 100 and 1000 particles.To control the runtime of the algorithm without significantly compromising its performance, a particle quantity of 200 is used.

C. Algorithm Performance Testing 1) Ablation Experiments of LC-YOLOv5 Algorithm:
To verify the effectiveness of LC-YOLOv5 compared to the initial network, the following ablation experiments are designed and the results are shown in Table III.
The experimental data in Table III confirms that the inclusion of the small object detection layer and CBAM attention mechanism improves the adaptability of the YOLOv5 network to small objects, resulting in a significant enhancement in CSOs quantity estimation accuracy.The improved YOLOv5 algorithm achieves an 11.1% increase in compared to the original version, and the accuracy of target number estimation improves by approximately 20.61%.Although the speed decreases to 0.0091s, it still meets the real-time requirements.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.The specific training results of the improved YOLOv5 algorithm are illustrated in Fig. 10, including precision, recall, mAP@0.5, mAP@0.5:0.95parameters, and the precision-recall curve.Among them, the precision is 0.863, recall is 0.875, and mAP@0.5 is 0.939.

2) Ablation Experiments of LP-FCM Algorithm:
To validate the effectiveness of the proposed LP-FCM algorithm compared to the original algorithm, we conducted an ablation experiment using the correct number of targets as input.The experimental results are shown in Fig. 11.
It can be observed that the LP-FCM algorithm proposed in this study effectively improves the centroid position estimation accuracy of CSOs.Moreover, the average runtime of this algorithm does not significantly increase and is approximately 50% faster than the FCM algorithm that solely utilizes particle distribution.
3) Comparative Testing of Different Algorithms: To evaluate the effectiveness of the LYLF algorithm for CSOs resolution and localization, it was compared with four other algorithms: the HK-based K-means algorithm [12], the particle distributionbased FCM algorithm, the pseudo-oversampling-based C3PC algorithm [16], and the targets extraction method based on maximum likelihood estimate (MLE) [17].Among them, the fourth algorithm takes the correct number of CSOs as the input and it is only used to compare the effect of target centroid extraction.The experiments were conducted in two groups, using simulated noise-free images and combined images by real star data as test data.After each experiment, performance metrics were calculated for each algorithm, including (1) CSOs quantity estimation accuracy, (2) false alarm probability, (3) missed detection probability, (4) JRMSE for target position estimation, (5) probability of correctly differentiating and locating targets, and (6) operating speed.The results are summarized in Table IV.
From Table IV, it can be observed that for noise-free images, the LYLF algorithm outperforms the other three algorithms in terms of CSOs quantity estimation accuracy, JRMSE for target position estimation, and the probability of correctly differentiating and locating targets.However, it slightly lags in terms of false alarm probability and missed detection probability.This is because the LC-YOLOv5 algorithm accurately estimates the number of targets, while the LP-FCM clustering algorithm struggles with locating weak targets that are occluded by other objects, leading to false alarms.Additionally, the utilization of particle distribution in the LP-FCM algorithm for sub-pixel decomposition of CSO regions increases algorithm complexity and runtime, though it still meets real-time requirements.Fig. 12 demonstrates the resolution and localization effects of LYLF for different numbers of CSOs on the simulated images.
For real image data, compared to simulated data, the LYLF algorithm exhibits a slight decrease in quantity estimation performance.However, its accuracy remains above 90, with centroid position estimation errors below 0.32 pixels.This demonstrates the potential of the proposed algorithm for practical applications.Interestingly, the LYLF algorithm even shows improvements in false alarms, missed detections, and the probability of correctly detecting and accurately locating targets.
There are several factors contributing to these star data, containing fewer types of CSOs and simpler combinations compared to the simulated images.This simplified composition makes it easier to differentiate and identify individual CSOs.Additionally, due to the lack of precise star centroid positions, the distances between two stars are not well controlled, resulting in larger inter-star spacings in the real images.As a result, it becomes relatively easier to determine the individual target information within the CSOs.These findings highlight the robustness and potential applicability of the LYLF algorithm when dealing with real images of CSOs.

4) Analysis of Factors Affecting the Performance of Proposed Algorithm:
a) Noise robustness tests of LC-YOLOv5: A noise robustness test was conducted to assess the performance of the LC-YOLOv5 algorithm under different levels of noise.The experimental results, illustrating the relationship between algorithm performance and SNR of images, are presented in Fig. 13.Note that this result is produced under the condition that noiseless simulated images are used as the training set.
As the algorithm does not incorporate denoising techniques during the super-resolution and localization process and the improved YOLOv5 was trained using noise-free images, it exhibits sensitivity to noise.Consequently, the algorithm achieves estimation accuracy above 90% only for images with an SNR greater than 28 dB.However, it is noteworthy that, thus far, the majority of tested images exhibited good SNR, with significant noise being present in only a few extreme cases.Therefore, the current algorithm effectively addresses the challenges of quantity estimation and super-resolution localization for most CSOs.However, it is important to acknowledge that its performance may be compromised in scenarios with high levels of noise.The experiment is divided into two parts, both with the correct number of targets as input.In the first part, the number of targets is fixed to 2 in the CSOs, and the distance between targets is set between 0.8 and 3 pixels.The target sizes are 3 × 3, 5 × 5, 7 × 7 and 9 × 9.A total of 4355 images are simulated for algorithm performance verification.Then noise is added to the images and experiments are conducted again to verify the algorithm Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.performance concerning SNR when the number of targets and distance are fixed.
The second part of the experiment verifies the relationship between the proposed algorithm and the number of targets and SNR.In this case, the settings of target distance and size are the same as in the first part of the experiment.The number of targets in the CSOs is between 2 and 7. A total of 18000 images are simulated.Then noise is added to verify the relationship between algorithm performance and SNR.
The experimental results show that the performance of the LP-FCM algorithm decreases with the increase in the number of targets, the decrease in target distance, and the increase in noise.The proposed algorithm is valid when SNR ≥ 10 dB, but the JRMSE increases with the decrease of SNR.When there are only two targets in CSOs and SNR≥15 dB, JRMSE can be controlled within 0.3 pixels.When the number of targets is between 2 and 7, the performance of the proposed algorithm decreases with the increase in the number of targets.When SNR ≥ 15 dB, the JRMSE of target location estimation is less than 0.3 pixels.
5) Algorithm Performance and the Cramer-Rao Lower Bound: We define the parameter vector as where, K is the number of targets in the CSOs and (x k , y k ) denotes the coordinates of the kth target.The log-likelihood function is derived from reference [11] and (1)-(3).

Λ(θ;
The Fisher information matrix (FIM) of θ is defined as The Cramer-Rao Lower Bound (CRLB), which defines a lower bound on the estimation accuracy for an unbiased estimator, is given by the inverse of J(θ) as The FIM is calculated numerically [17], [40], [41].Using the number of CSOs and the distance between targets as variables, we calculate the error variance for each parameter of θ, respectively.The first part of the experiment will run a test with the number of targets as a variable, based on 3000 Monte Carlo runs.Next, fixing the number of targets as 2, the experiment will be conducted with target distance as the variable, based on 200 Monte Carlo runs.The results can be found in Fig. 15.
It can be seen that the parameter estimation accuracy of the proposed algorithm is affected by the distance between targets and the number of CSOs, with the former being more influential.When the distance between the targets increases, the parameter variance of the proposed algorithm gradually approaches the Cramer-Rao Lower Bound.

V. DISCUSSION
The proposed method improves the accuracy of the estimation of the number of CSOs and the center position of every target, which is mainly related to three improvements.
1) First, the neural network is used to estimate the number of targets instead of a traditional clustering algorithm.The traditional algorithm is based on the Gaussian features of infrared targets to segment the multi-target region and calculate the number of targets.When two targets are highly overlapped (center distance less than 1R), the traditional algorithm easily fails.However, trained neural networks that have learned the features of various types of CSOs can classify more types of CSOs based on the number of targets, giving them an advantage in terms of accuracy.2) Second, a small target detection layer and CBAM are added to the YOLOv5 network to enhance the sensitivity of the network to small targets.These improvements enable the YOLOv5 network to better extract features from CSOs and improve the quantity estimation accuracy.3) Finally, sub-pixel localization of CSOs was performed using a modified FCM algorithm.Considering that point targets occupy very few pixels in infrared images, the scarcity of effective information has a great impact on the estimation accuracy of the target center.Based on a pseudo-downsampling idea, the target region is first upsampled using Lanczos3 interpolation to extend the pixel information of each point target.Then the pixels are decomposed using a corresponding number of particles distributed according to the pixel gray value.The extended region is then downsampled to obtain a sub-pixel particle set reflecting the target region.Finally, the FCM algorithm is used on the sub-pixel particle set to obtain the exact center position of targets.Under the condition that the number of particles is used properly, both the algorithm accuracy and running time can get better results.

VI. CONCLUSION
To address the challenges of quantity and position estimation for CSOs, the LYLF algorithm is proposed in this paper.The algorithm enhances the lightweight neural network YOLOv5 by incorporating a small object layer and the CBAM attention mechanism.This improved network is then applied to the resolution and localization algorithm for CSOs quantity estimation, significantly enhancing its estimability.Furthermore, a novel LP-FCM algorithm is introduced, which utilizes Lanczos3 interpolation and particle distribution based on the pseudo-oversampling idea.This approach enables the sub-pixel representation of target regions and leverages the sparsity of particles to capture the grayscale variations of each pixel, thereby improving the accuracy of target centroid estimation.
The comparative experiments conducted above demonstrate that the LYLF algorithm achieves superior super-resolution and localization performance.Whether tested using simulated data or real stellar composite data, the proposed algorithm consistently attains quantity estimation accuracy of over 90% and target position estimation errors within 0.32 pixels, significantly outperforming other algorithms.However, it is important to note that the algorithm does not incorporate denoising techniques Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
during the resolution and localization process, making it sensitive to noise.Therefore, future work will focus on developing CSOs resolution and localization algorithms that are robust to low SNR, as well as the application of lightweight neural networks in low SNR scenarios.

Fig. 1 .
Fig. 1.CSOs infrared plane images and two-dimensional gray-scale cloud diagrams.(a) Two targets at 3 pixels apart infrared plane image.(b) Individual PSFs of the two point-targets.(c) Two targets at 0.95 pixels (1R) apart infrared plane image.(d) The linear mixture of the two PSFs.

Fig. 2 .
Fig. 2. CSOs with seven targets infrared plane images and two-dimensional gray-scale cloud diagrams.(• indicates the center location of point targets).(a) CSOs infrared plane image.(b) Zoomed image of the CSOs area.(c) Twodimensional gray-scale cloud diagram.

Fig. 7 .
Fig. 7. Simulated CSOs images with different numbers of targets (• indicates the center location of point targets).

Fig. 8 .
Fig. 8. Star images.(a) The actual captured image of a star.(b) An image combined from four different star images.(c) Actual captured star images which were combined to form Fig. 8(b).

Fig. 12 .
Fig. 12. Resolution and localization effect of LYLF algorithm for different CSOs ( represents the target center estimated by the LYLF algorithm, Rrepresents the actual center position of the target, represents the target center incorrectly estimated by the algorithm; and • represents the center of the missed target).

Fig. 13 .
Fig. 13.The trend of algorithm performance with image SNR.

Fig. 14 .
Fig. 14.Analysis of factors affecting the performance of the LP-FCM algorithm.(a) The trend of LP-FCM algorithm performance with target distance and SNR when the number of targets is 2. (b) The trend of LP-FCM algorithm performance with the number of CSOs and SNR.

Fig. 15 .
Fig. 15.Error variance for each parameter as a function of the number and distances of CSOs.(a) (b) Error variance for each parameter as a function of the number of CSOs.(c) (d) Error variance for each parameter as a function of the distance between two targets.

TABLE I PERFORMANCE
ANALYSIS OF YOLOV5 ALGORITHMS FOR DIFFERENT NETWORK STRUCTURES Fig. 9. Variation of LP-FCM algorithm performance with the number of particles.

TABLE II PERFORMANCE
ANALYSIS OF YOLOV5 ALGORITHMS FOR DIFFERENT ATTENTION MECHANISMS

TABLE III COMPARISON
OF THE IMPROVED YOLOV5 ALGORITHM METRICS