A Track Segment Association Method Based on Heuristic Optimization Algorithm and Multistage Discrimination

Abstract

The fragmentation of vessel tracks represents a significant challenge in the context of high-frequency surface wave radar (HFSWR) tracking. This paper proposes a new track segment association (TSA) algorithm that integrates optimal tracklet assignment, iterative discrimination, and multi-stage association. This paper reformulates the optimal tracklet assignment task as an optimal state search problem for modeling and solution purposes. To determine whether competing old and new tracklets can be associated, we assume the existence of a public state that represents the correlation between the tracklets. However, due to track fragmentation, this public state remains unknown. We need to search for the optimal public state of all candidate tracklet pairs within a feasible parameter space, using a fitness function value as the evaluation criterion. The old and new tracklets pairs that match the optimal public state are considered optimally associated. Since the solution process involves searching for the optimal state across multiple dimensions, it constitutes a high-dimensional optimization problem. To accomplish this task, the catch fish optimization algorithm (CFOA) is employed for its ability to escape local optima and handle high-dimensional optimization, enhancing the reliability of tracklet assignment. Furthermore, we achieve precise one-to-one associations by assigning new tracklet to old tracklet through the optimal tracklet assignment method we proposed, a process we abbreviate as AN2O, and its inverse process, which assigns old tracklet to new tracklet, abbreviated as AO2N. This dual approach is further complemented by an iterative discrimination mechanism that evaluates unselected tracklets to identify potential associations that may exist. The algorithm effectiveness of the proposed is validated using field experiment data from HFSWR in the Bohai Sea region, demonstrating its capability to accurately process complex tracklet data.

1. Introduction

As a vital tool for monitoring maritime vessels, HFSWR possesses several advantages, including over-the-horizon detection, extensive coverage, all-weather functionality, and cost-effectiveness, enabling it to effectively detect non-cooperative vessels [1,2,3]. Despite the numerous advantages of HFSWR in maritime vessel tracking, its tracking accuracy remains relatively low [4,5]. The presence of various forms of clutter and noise can significantly impact the performance of target detection systems in maritime environments. These interference sources include highly maneuvering targets, ground clutter, sea clutter, ionospheric clutter, and radio-frequency interference [6], all of which can lead to missed detections and false alarms [7]. Additionally, as a non-cooperative sensor, HFSWR is unable to acquire the track identity information of the targets [8]. These limitations hinder its effectiveness in complex environments. Due to the non-cooperative nature of HFSWR and its constraints in tracking precision, the phenomenon of track interruption frequently occurs during the vessel tracking process [9]. Consequently, the same target may be fragmented into multiple short track segments (a.k.a. tracklets), preventing continuous tracking. This phenomenon significantly undermines the continuity and accuracy of vessel tracking, thereby reducing the reliability and effectiveness of monitoring systems [10]. These broken tracks can be divided into two categories: new tracklets and old tracklets. Among them, old tracklets refer to interrupted tracks that cannot be updated due to a lack of necessary measurement data. These tracklets may have failed to maintain tracking due to technical limitations or data loss. New tracklets, on the other hand, refer to newly initiated tracklets, which may be continuations of old tracklets or entirely new starting tracklets. In light of the impact of track interruption on vessel tracking, the development of an effective TSA algorithm is of paramount importance.

Tracklet association is one of the critical issues in multi-target tracking, aimed at linking tracklets generated by the same target across different times and spatial contexts, thereby facilitating long-term tracking of multiple targets. Current track association technologies can be categorized into two main types. The first category employs predictive models to perform forward prediction on old tracklets and backward prediction on new tracklets. Techniques such as optimal assignment and polynomial fitting are utilized to connect the old and new tracklets, thereby facilitating the association of interrupted track. For instance, Yeom et al. [11] employed the interacting multiple model (IMM) [12] approach for tracklet tracking and prediction, subsequently utilizing a two-dimensional optimal assignment algorithm for tracklet matching, which yielded favorable results. In a similar manner, Zhang et al. [13] employed an IMM estimator to perform the bidirectional prediction of tracklet segments, addressing the issue of track interruptions caused by the target’s maneuvering behavior during movement-stop-movement transitions. Sun et al. [14] proposed a method for track segment association using Gaussian regression analysis, which similarly employs bidirectional prediction techniques. Additionally, the Hungarian algorithm [15] is utilized to optimize tracklet assignment. Qi et al. [16] introduced a multi-hypothesis motion model incorporating prior knowledge to address the pairing and association issues arising from interrupted tracks, particularly focusing on the inaccuracies in state estimation caused by maneuvering movements of the target during the interruption period. Sun et al. [17] combined bidirectional prediction with the optimal assignment based on the Jonker–Volgenant–Castanon(JVC) [18] algorithm to facilitate tracklet association. To address the issue of track interruptions caused by strong clutter, Zhang et al. [19] proposed a method for HFSWR ship tracklet prediction that integrates multiple mechanisms and models through a multi-scale convolutional neural network (MSCNN) [20]. The second category of techniques assesses the similarity between tracklets to determine whether they belong to the same target. For instance, Jian et al. [21] introduced the concept of fuzzy tracklet membership based on fuzzy mathematics theory to associate tracklets. Cao et al. [22] employed the Holt–Winters method for the bidirectional prediction and extrapolation of tracklets, and then applied a fuzzy tracklet association algorithm for tracklet association. Bae et al. [23] leveraged the confidence of tracklets and online discriminative appearance learning to achieve robust multi-target tracking, via associating tracklets through the online learning of target appearance models. Zhang et al. [24] utilized extreme learning machines (ELM) [25] to extract tracklet information and perform tracklet association based on tracklet characteristics. Sun et al. [26] noted that the inherent uncertainties associated with HFSWR measurements pose significant challenges to the tracklet association process, particularly due to the similarities in kinematic parameters among adjacent targets. To address this issue, they proposed a tracklet association method for HFSWR based on maximum likelihood estimation (MLE) [27]. Lv et al. [28] developed a tracklet association system that tightly integrates the feature extraction and association processes of the tracklet, achieving an end-to-end tracklet association method. However, when the fragmentation of a track is caused by measurement errors, the plot information at the breakage becomes unreliable. This undoubtedly has a significant impact on the TSA algorithm that rely on the plot information at the breakage. Furthermore, existing TSA methods often rely on a singular spatiotemporal association criterion for determining associations and may fail to incorporate all tracklets within a specific region into a comprehensive evaluative framework. In complex environments or those with substantial interference, such approaches are prone to generating false associations and missed associations.

In this study, the optimal tracklet assignment is treated as an optimization task. We randomize the public state at the fragmentation and determine the optimal public state between the tracklet pairs to be associated within a given range based on the search strategy (i.e., fitness function). This optimal public state will serve as the basis for decision-making to achieve optimal assignment of the tracklets. An advanced meta-heuristic optimization algorithm inspired by human behavior, known as CFOA [29], is utilized as the foundational approach to handle the optimization task. Compared to traditional optimization algorithms, the CFOA effectively addresses challenges such as local optima and non-convex optimization in complex, high-dimensional problems, and it has demonstrated exceptional performance across various optimization scenarios. We employed the proposed optimal tracklet assignment method to perform both AN2O and AO2N processes on the tracklets, thereby achieving precise one-to-one correspondence between the tracklets. Furthermore, we have designed an iterative discrimination mechanism to evaluate unselected tracklets, assessing whether they still possess potential associations with other tracklets. If such tracklet pairs are identified, the algorithm will re-execute the AO2N and AO2N processes until all potential associations have been thoroughly evaluated. To enhance the efficiency of associations and reduce the probability of false associations, we employ a straightforward spatiotemporal discrimination mechanism at both the beginning and the end of the algorithm. In the experimental section, we validate the effectiveness of the proposed algorithm through an analysis of field experiment data collected from the HFSWR in the Bohai Sea region. The findings indicate that the algorithm effectively yields efficient and precise association results when managing intricate tracklet data.

The primary contributions of this paper are as follows:

• A Novel Optimal Tracklet Assignment Method: By reformulating the optimal tracklet assignment task as an optimal state search problem, we enable the identification of optimal association based on a public state. This method does not rely on plot information from the interruption period during the tracklet assignment process, thereby effectively reducing errors that may arise from potentially erroneous plots. Furthermore, the method provides an optimal state at the fragmentation, serving as a more reliable basis for data support;
• HFSWR Data Processing Using CFOA: The paper employs the CFOA algorithm to handle the high-dimensional optimization problem involved in the search for the optimal public state. The CFOA’s ability to escape local optima and handle high-dimensional optimization enhances the accuracy and reliability of the tracklet assignment process;
• AN2O and AO2N and the Iterative Discrimination Mechanism: We used our optimal tracklet assignment method for both AN2O and AO2N processes, ensuring accurate one-to-one matching of tracklets. Additionally, the iterative discrimination mechanism allows for a more comprehensive exploration of potential associations. These mechanisms allow for a systematic and comprehensive search for the best tracklet associations.

2. Methodology

The flowchart of the proposed algorithm is illustrated in Figure 1.

Initially, we perform a first-stage association, referred to as coarse association, on the pre-processed tracklets. During this phase, the algorithm considers the spatiotemporal relationships between tracklet pairs to determine their potential associations. Following the coarse association, we conduct a second-stage association. During this phase, we conduct optimal assignments of the tracklets to achieve precise one-to-one associations. Subsequently, we iteratively reassess those tracklets that were not selected, examining whether they still maintain coarse associations with other tracklets. If such tracklet pairs exist, the algorithm will re-execute the second-stage association until all tracklet pairs that satisfied coarse association have been thoroughly evaluated. Finally, we perform a third-stage association, which resembles the coarse association but employs stricter threshold settings to eliminate redundant associated tracklets.

2.1. Tracklets Pre-Processing

2.1.1. Smoothing

The tracklet information obtained from HFSWR measurements may exhibit significant measurement errors, which could lead to deviations from the true values. In this study, we employ the interacting multiple model extended Kalman filter (IMMEKF) [30] to smooth the tracklet data. For the tracklets under consideration, the tracklet that is temporally earlier is defined as the old tracklet i, while the tracklet that is temporally later is referred to as the new tracklet j. We perform a smoothing process on the old tracklet i from its initiation to its termination, which is termed as forward smoothing. Conversely, the new tracklet j undergoes smoothing from its termination to its initiation, a process designated as backward smoothing. This method aims to eliminate noise and emphasize the primary characteristics of the tracklet, effectively reducing errors during the association process while significantly alleviating the computational burden of the algorithm, thereby enhancing both efficiency and accuracy.

2.1.2. Error Plot Pruning

Due to the occurrence of track-plot association errors during the generation of tracks by HFSWR, subsequent tracklet association operations may exhibit significant inaccuracies. Typically, erroneous plots manifest as abrupt changes in the radial distance and azimuth angle, which, in turn, lead to corresponding abrupt variations in the smoothed value at that moment. This paper calculates the standard deviations of the radial distance and azimuth angle estimates for the last n moments of a given tracklet in order to identify and eliminate such erroneous plots.

The calculation involves determining the sample standard deviation of the radial distance and azimuth angle for the last n time instances of the tracklet:

$$\begin{array}{c}{x}_{\mathit{std}}=\sqrt{{\displaystyle \frac{1}{n-1}}{\sum }_{i=1}^n{(x_i-\overline{x})}^2}\end{array}$$

(1)

In this context, x represents the radial distance smoothing value $\widehat{r}$ or the azimuthal angle smoothing value $\widehat{a}$, where $x_i$ denotes the $i^{\mathit{th}}$ value of x, and $\overline{x}$ represents the sample mean of the variables. A threshold for outlier removal is established at $1.5$ times the standard deviation:

$$x_{\mathit{threshold}}=1.5x_{\mathit{std}}$$

(2)

The measurement noise of the smoothed tracklet can be assumed to follow a Gaussian distribution. Under this distribution, 1.5 standard deviations encompass approximately 90% of the data based on the three-sigma rule in statistics. Therefore, using 1.5 times the standard deviation as the threshold effectively identifies and eliminates outliers that significantly deviate from the normal range. If the absolute difference between the selected $i^{\mathit{th}}$ value and the mean of the selected values exceeds the predetermined threshold, we can regard the $i^{\mathit{th}}$ estimate as an outlier and exclude it from subsequent analyses.

2.2. First-Stage Association: Coarse Association

Coarse association aims to retain as many associable tracklets as possible while eliminating those that cannot be associated. This approach alleviates the computational burden associated with second-stage association, thereby enhancing the overall efficiency and reliability of the system.

2.2.1. Spatio-Temporal Discrimination Between Old and New Tracklets

In the process of tracklet association, identifying effective association features is crucial. HFSWR primarily relies on the target’s radial distance r, azimuth angle a, and radial velocity v to characterize its motion state. Consequently, time t and the smoothed values $\widehat{r}$, $\widehat{a}$, and $\widehat{v}$ have been selected as the key features for the old tracklet i and the new tracklet j. The accuracy of the association results is positively correlated with the number of association features utilized. This relationship can be expressed as follows:

$$\begin{array}{c}\left\{\begin{array}{c}{r}_{\mathit{diff}}=\left|{\widehat{r}}_{\mathit{start}}^j-{\widehat{r}}_{\mathit{end}}^i\right|\\ a_{\mathit{diff}}=\left|{\widehat{a}}_{\mathit{start}}^j-{\widehat{a}}_{\mathit{end}}^i\right|\\ v_{\mathit{diff}}=\left|{\widehat{v}}_{\mathit{start}}^j-{\widehat{v}}_{\mathit{end}}^i\right|\\ t_{\mathit{diff}}=t_{\mathit{start}}^j-t_{\mathit{end}}^i\end{array}\right.\end{array}$$

(3)

In the equation, the subscripts $\mathit{start}$ and $\mathit{end}$ denote the relevant values at the starting and ending points of the tracklet, respectively. The superscripts i and j represent the old and new tracklets, respectively. The subscript $\mathit{diff}$ indicates the difference between the corresponding values. If these differences are all below predetermined thresholds, it can be concluded that the tracklet pair satisfies the spatiotemporal discrimination criteria, indicating that the two tracklets are similar enough in both spatial and temporal dimensions to be considered potentially associable in the analysis. The thresholds are set to avoid being overly strict, ensuring that as many potentially associable tracklets as possible are retained for further processing.

2.2.2. Spatial Discrimination at the Middle Moment of Fragmentation

Spatial discrimination at the middle moment of fragmentation is applied after the coarse spatio-temporal discrimination of tracklets. Specifically, the spatial discrimination at the middle moment $t_{\mathrm{mid}}$ evaluates whether two tracklets can be associated by comparing their predicted states ${\widehat{\mathbf{X}}}_{\mathrm{mid}}^i$ and ${\widehat{\mathbf{X}}}_{\mathrm{mid}}^j$ at $t_{\mathrm{mid}}$. Compared to traditional unidirectional prediction methods, this bidirectional prediction strategy offers enhanced predictive reliability [17]. After applying the IMMEKF to perform both forward and backward predictions on the tracklets to smooth them, we extrapolate both the new and old tracklets to the middle moment $t_{\mathit{mid}}$. We then consider the spatial discrimination between the estimated values ${\widehat{\mathbf{X}}}_{mid}^i$ and ${\widehat{\mathbf{X}}}_{mid}^j$ of the tracklets i and j at $t_{\mathit{mid}}$. A schematic representation is provided below.

The black lines in Figure 2 represent the measurements of tracklets i and j, while the red and blue lines denote the results of forward and backward predictions, respectively. The dashed lines illustrate the process of extrapolating the two tracklets to the middle moment of the fracture. Due to the absence of actual measurements during the fracture period, it is necessary to perform a one-step prediction to estimate the required state values and extrapolate them to the middle moment $t_{\mathit{mid}}$. Taking the forward prediction of the old tracklet as an example,

$$\begin{array}{c}\widehat{\mathbf{X}}\left(t_{mid}\right)={\mathbf{F}}^{\mathit{CT}}·\widehat{\mathbf{X}}\left(t_{\mathit{end}}\right)\end{array}$$

(4)

where ${\mathbf{F}}^{\mathit{CT}}$ represents the state transition matrix of a constant turn (CT) motion model:

$$\begin{array}{c}{\mathbf{F}}^{\mathit{CT}}=\left[\begin{array}{cccc}1& {\displaystyle \frac{sin\omega T}{\omega }}& 0& {\displaystyle \frac{cos\omega T-1}{\omega }}\\ 0& cos\omega T& 0& -sin\omega T\\ 0& {\displaystyle \frac{1-cos\omega T}{\omega }}& 1& {\displaystyle \frac{sin\omega T}{\omega }}\\ 0& sin\omega T& 0& cos\omega T\end{array}\right]\end{array}$$

(5)

It can be observed that the ${\mathbf{F}}^{\mathit{CT}}$ is determined by the motion model’s turn rate $\omega$ and the time T. Specifically,

$$\begin{array}{c}T=t_{\mathit{mid}}\end{array}$$

(6)

The turn rate is calculated by taking the mean of $\omega \left(t_{\mathit{end}}-n\right),\omega \left(t_{\mathit{end}}-n+1\right),\dots ,\omega \left(t_{\mathit{end}}\right)$ during the time period from tracklet intervals $t_{\mathit{end}}-n$ to $t_{\mathit{end}}$. The calculation of the turn rates at each time point can be found in [31].

$$\begin{array}{c}\left\{\begin{array}{c}\overline{\omega }=\sum _{i=1}^n{\displaystyle \frac{\omega \left(t_{\mathit{end}}-i\right)}{n}}\\ \omega =\overline{\omega }\end{array}\right.\end{array}$$

(7)

Similarly, for the backward prediction of a new tracklet, we can select the first n turn rates from the tracklet’s initiating plots to calculate the average turn rate for the new tracklet to use. Thus, we can obtain the state estimates ${\widehat{\mathbf{X}}}_{mid}^i$ and ${\widehat{\mathbf{X}}}_{mid}^j$ at the moment $t_{\mathit{mid}}$. Through this method, we are able to assign a unique motion model to each tracklet, thereby significantly enhancing the reliability of the associations. Additionally, this approach provides a reference for determining the search range of the optimal state value in subsequent second-stage association. Next, we will employ a methodology consistent with that described in Section 2.2.1 to evaluate whether the tracklet pair corresponding to ${\widehat{\mathbf{X}}}_{\mathit{mid}}^i$ and ${\widehat{\mathbf{X}}}_{\mathit{mid}}^j$ satisfies the discrimination criteria.

Figure 2. Schematic diagram of forward and backward prediction.

Figure 2. Schematic diagram of forward and backward prediction.

2.2.3. Intersecting Tracklets Filtering

In tracklet association analysis, intersecting tracklets are typically considered non-associable. However, due to the relatively lenient conditions of coarse association, such intersecting tracklets may still be classified as associated under coarse association criteria. This not only increases the computational burden for subsequent second-stage association but also raises the likelihood of erroneous associations. Therefore, it is essential to exclude intersecting tracklets from consideration prior to the second-stage association process.

In summary, for the tracklet pair under consideration, if there are no intersections and they satisfy the criteria outlined in Section 2.2.1 and Section 2.2.2 they can be deemed to meet the criterion for coarse association.

After conducting the first-stage association analysis, the algorithm creates four tracklet sets and categorizes the different types of associated tracklet pairs into these sets. Let ${\mathcal{T}}_{\mathrm{old}}$ and ${\mathcal{T}}_{\mathrm{new}}$ represent the sets of old and new tracklets, respectively; $T_{\mathrm{old}}$ and $T_{\mathrm{new}}$ denote a single old and new tracklet, respectively; and k and m represent the number of associated tracklets. The four sets are defined as follows:

Set of tracklets that do not satisfy the coarse association criterion: This set contains all old tracklets that fail to meet the criterion and are excluded from further analysis.

$$S_{(1\to 0)}=\left\{\left(\{T_{\mathrm{old}}^1,\dots ,T_{\mathrm{old}}^m\},\varnothing \right)\mid T_{\mathrm{old}}^i\in {\mathcal{T}}_{\mathrm{old}},T_{\mathrm{old}}^i\mathrm{does}\mathrm{not}\mathrm{satisfy}\mathrm{the}\mathrm{criterion}\right\}$$

(8)

Set of tracklets where each old tracklet is associated with multiple new tracklets: This set contains all combinations of old tracklets and their associated sets of new tracklets, where each old tracklet satisfies the criterion and is associated with at least two new tracklets.

$$S_{(1\to n)}=\left\{\left(\{T_{\mathrm{old}}^1,\dots ,T_{\mathrm{old}}^m\},\{T_{\mathrm{new}}^1,\dots ,T_{\mathrm{new}}^k\}\right)\mid T_{\mathrm{old}}^i\in {\mathcal{T}}_{\mathrm{old}},T_{\mathrm{new}}^j\in {\mathcal{T}}_{\mathrm{new}},k\ge 2\right\}$$

(9)

Set of tracklets where each old tracklet is associated with one new tracklet: This set contains all combinations of old tracklets and their associated single new tracklets, where each old tracklet satisfies the criterion and is associated with exactly one new tracklet.

$$S_{(1\to 1)}=\left\{\left(\{T_{\mathrm{old}}^1,\dots ,T_{\mathrm{old}}^m\},\{T_{\mathrm{new}}^1,\dots ,T_{\mathrm{new}}^k\}\right)\mid T_{\mathrm{old}}^i\in {\mathcal{T}}_{\mathrm{old}},T_{\mathrm{new}}^j\in {\mathcal{T}}_{\mathrm{new}},k=m\right\}$$

(10)

Set of tracklets where each new tracklet is associated with multiple old tracklets: This set contains all combinations of new tracklets and their associated sets of old tracklets, where each new tracklet satisfies the criterion and is associated with at least two old tracklets. Currently, the set remains empty, and tracklet pairs will be integrated in subsequent processing stages.

$$S_{(n\to 1)}=\left\{\left(\{T_{\mathrm{new}}^1,\dots ,T_{\mathrm{new}}^m\},\{T_{\mathrm{old}}^1,\dots ,T_{\mathrm{old}}^k\}\right)\mid T_{\mathrm{new}}^i\in {\mathcal{T}}_{\mathrm{new}},T_{\mathrm{old}}^j\in {\mathcal{T}}_{\mathrm{old}},k\ge 2\right\}$$

(11)

2.3. Second-Stage Association: Optimal Tracklet Assignment

During the second-stage association analysis, the focus shifts to the tracklet pairs within the set $S_{(1\to n)}$. By converting the association results into one-to-one mappings, these tracklet pairs are integrated into the set $S_{(1\to 1)}$.

If the old and new tracklet pairs originate from the same target, the hypothesized tracklet from the interruption period should be consistent, meaning they should share the same states during interruption. We select one representative state, referred to as the public state ${\widehat{\mathbf{X}}}_{\mathit{pub}}$, which is intended to reflect the correlation and consistency between the old and new tracklet. In the case of competing tracklets, where multiple new tracklets are associated with a single old tracklet or vice versa, we assign such a public state ${\widehat{\mathbf{X}}}_{\mathit{pub}}$ to each candidate old and new tracklet pair. However, due to track interruption, ${\widehat{\mathbf{X}}}_{\mathit{pub}}$ is unknown, which transforms the optimal tracklet assignment task into an optimal state value search problem. For each tracklet pair under consideration, we randomize ${\widehat{\mathbf{X}}}_{\mathit{pub}}$ and search for the optimal ${\widehat{\mathbf{X}}}_{\mathit{pub}}$ within a feasible parameter space based on the search criterion, i.e., the fitness function. The tracklet pair that aligns with this optimal ${\widehat{\mathbf{X}}}_{\mathit{pub}}$ is, thus, considered to be optimally associated.

Figure 3 presents a schematic illustration of the optimal ${\widehat{\mathbf{X}}}_{\mathit{pub}}$ generation for the competitive tracklet pairs. The solid line portion is consistent in meaning with those in Figure 2, while the dashed line portion represents the possible tracklet, which is constructed based on the ${\widehat{\mathbf{X}}}_{\mathit{pub}}$. The tracklet pair $i-j$ that corresponds to this optimal ${\widehat{\mathbf{X}}}_{\mathit{pub}}$ is, therefore, regarded as being optimally associated. In contrast, the tracklet $i-l$ is excluded from consideration because it corresponds to a suboptimal state.

For the $i^{\mathit{th}}$ tracklet, there exist k associated tracklets that satisfy the coarse association criterion, forming k competitive tracklet pairs. For each k tracklet pair, we perform ℓ iterations, resulting in $\mathit{\ell }\times k$ corresponding fitness function values, represented as

$$f_{all}={\left(\begin{array}{cccc}{f}_{11}& f_{12}& \dots & f_{1k}\\ f_{21}& f_{22}& \dots & f_{2k}\\ ⋮& ⋮& \ddots & ⋮\\ f_{\mathit{\ell }1}& f_{\mathit{\ell }2}& \dots & f_{\mathit{\ell }k}\end{array}\right)}_{\begin{array}{c}\hfill \mathit{\ell }\times k\end{array}}$$

(12)

To identify the optimal tracklet pair, the minimum value of the fitness function, denoted as $f_{nm}$, is selected from all computed fitness values:

$$f_{nm}=\underset{\begin{array}{c}1\le a\le \mathit{\ell }\\ 1\le b\le k\end{array}}{min}\left(f_{ab}\right)$$

(13)

Here, n is the iteration index at which the minimum value $f_{ab}$ is achieved, and m is the index of the tracklet pair corresponding to this minimum value. Consequently, the $m^{th}$ tracklet pair, denoted as ${\mathit{trk}}_m$, is identified as the optimal tracklet pair and is referred to as ${\mathit{trk}}_{opt}$:

$${\mathit{trk}}_{opt}={\mathit{trk}}_m,\mathrm{where}m=\underset{1\le b\le k}{arg\; min}{f}_{nb}$$

(14)

During the solution process, the public time ${\widehat{t}}_{pub}$ and its corresponding radial distance, azimuth angle, and radial velocity, ${\widehat{r}}_{pub}$, ${\widehat{a}}_{pub}$, and ${\widehat{v}}_{pub}$, will be searched for. These elements collectively form the public state ${\widehat{\mathbf{X}}}_{\mathit{pub}}$, resulting in a high-dimensional optimization problem. Considering the potential performance degradation of some optimization algorithms when dealing with high-dimensional problems, we choose a novel algorithm, CFOA, for its ability to effectively avoid local optima and its advantage in handling high-dimensional issues, with the hope of achieving more accurate calculations in the tracklet association problem.

The optimal search range for ${\widehat{\mathbf{X}}}_{\mathit{pub}}$ is determined by the smoothed value ${\widehat{\mathbf{X}}}_{\mathit{end}}^i$ at the end time of tracklet i and the forward prediction value ${\widehat{\mathbf{X}}}_{\mathit{mid}}^i$ at the middle moment $t_{\mathit{mid}}$. Additionally, it incorporates the smoothed value ${\widehat{\mathbf{X}}}_{\mathit{start}}^j$ at the initiation time of tracklet j and the backward prediction value ${\widehat{\mathbf{X}}}_{\mathit{mid}}^j$ at the same time $t_{\mathit{mid}}$. The search range is, thus, defined by the maximum and minimum values of these parameters in Equation (15), expressed as

$$\left\{\begin{array}{c}{t}_{\mathit{max}}=max\left\{t_{\mathit{end}}^i,t_{\mathit{start}}^j\right\}\\ t_{\mathit{min}}=min\left\{t_{\mathit{end}}^i,t_{\mathit{start}}^j\right\}\\ r_{\mathit{max}}=max\left\{{\widehat{r}}_{\mathit{end}}^i,{\widehat{r}}_{\mathit{start}}^j,{\widehat{r}}_{\mathit{mid}}^i,{\widehat{r}}_{\mathit{mid}}^j\right\}\\ r_{\mathit{min}}=min\left\{{\widehat{r}}_{\mathit{end}}^i,{\widehat{r}}_{\mathit{start}}^j,{\widehat{r}}_{\mathit{mid}}^i,{\widehat{r}}_{\mathit{mid}}^j\right\}\\ a_{\mathit{max}}=max\left\{{\widehat{a}}_{\mathit{end}}^i,{\widehat{a}}_{\mathit{start}}^j,{\widehat{a}}_{\mathit{mid}}^i,{\widehat{a}}_{\mathit{mid}}^j\right\}\\ a_{\mathit{min}}=min\left\{{\widehat{a}}_{\mathit{end}}^i,{\widehat{a}}_{\mathit{start}}^j,{\widehat{a}}_{\mathit{mid}}^i,{\widehat{a}}_{\mathit{mid}}^j\right\}\\ v_{\mathit{max}}=max\left\{{\widehat{v}}_{\mathit{end}}^i,{\widehat{v}}_{\mathit{start}}^j,{\widehat{v}}_{\mathit{mid}}^i,{\widehat{v}}_{\mathit{mid}}^j\right\}\\ v_{\mathit{min}}=min\left\{{\widehat{v}}_{\mathit{end}}^i,{\widehat{v}}_{\mathit{start}}^j,{\widehat{v}}_{\mathit{mid}}^i,{\widehat{v}}_{\mathit{mid}}^j\right\}\end{array}\right.$$

(15)

In the equation, the subscripts $\mathit{max}$ and $\mathit{min}$ denote the maximum and minimum values of corresponding parameters. These parameters on the right-hand side will collectively define the search range for the optimal state value ${\widehat{\mathbf{X}}}_{\mathit{pub}}$. Ultimately, ${\widehat{\mathbf{X}}}_{\mathit{pub}}$ can be defined as

$${\widehat{\mathbf{X}}}_{\mathit{pub}}=\left[\begin{array}{c}{\widehat{t}}_{pub}\\ {\widehat{r}}_{pub}\\ {\widehat{a}}_{pub}\\ {\widehat{v}}_{pub}\end{array}\right]\mathrm{where}\begin{array}{cc}\hfill & t_{\min }\le {\widehat{t}}_{pub}\le t_{\max }\hfill \\ \hfill & r_{\min }\le {\widehat{r}}_{pub}\le r_{\max }\hfill \\ \hfill & a_{\min }\le {\widehat{a}}_{pub}\le a_{\max }\hfill \\ \hfill & v_{\min }\le {\widehat{v}}_{pub}\le v_{\max }\hfill \end{array}$$

(16)

2.3.1. Design of Fitness Function

The fitness function is defined as a weighted combination of four components, A, B, C, and D, which collectively aim to find an optimal public state that simultaneously minimizes the discrepancy between the optimal public state and the old tracklet, as well as between the optimal public state and the new tracklet. Each component is designed to quantify specific aspects of the relationship between the tracklets and the optimal public state, thereby establishing a comprehensive evaluation framework:

• A quantifies the discrepancy between the old tracklet and the optimal public state;
• B quantifies the discrepancy between the optimal public state and the new tracklet;
• C quantifies the geometric relationship among the old tracklet, the optimal public state, and the new tracklet;
• D quantifies the discrepancy between the old and new tracklets, independent of the optimal public state.

The selection rules for the four components of the fitness function are illustrated in Figure 4, Figure 5, Figure 6.

A, B, C, and D together play a crucial role in the formulation of the fitness function, ensuring the accurate identification of the optimal public state. The expression for the fitness function is as follows:

$$\begin{array}{c}f=w_{1}A+w_{2}B+w_{3}C+w_{4}D\end{array}$$

(17)

where the variables $w_1$, $w_2$, $w_3$, and $w_4$ represent the weights, which satisfy the following conditions:

$$\begin{array}{c}\left\{\begin{array}{c}\begin{array}{c}{w}_1+w_2+w_3+w_4=1\\ w_1=w_2\end{array}\end{array}\right.\end{array}$$

(18)

It is worth noting that A only considers the discrepancy between the old tracklet and the optimal public state, without considering the discrepancy between the new tracklet and the optimal public state, while B does the exact opposite. The same weights for A and B ensure that the discrepancies between the old tracklet and the optimal public state, as well as those between the new tracklet and the optimal public state, are balanced during the optimization process, avoiding excessive bias towards one side that could lead to extreme situations, ultimately preventing the overall discrepancy from being minimized. The same weights for A and B are helpful to find the optimal public state. The four components of the fitness function are calculated as follows.

A consists of the following factors:

• The angle ${\theta }_A$ is formed by three plots: the smoothed state ${\widehat{\mathbf{X}}}_{\mathit{end}}^i$ at the end of the old tracklet i, the smoothed state ${\widehat{\mathbf{X}}}_{\mathit{end}-1}^i$ at the preceding time step, and the public state ${\widehat{\mathbf{X}}}_{\mathit{pub}}$. As the angle ${\theta }_A$ approaches $\pi$, the line connecting tracklet i and the public state ${\widehat{\mathbf{X}}}_{\mathit{pub}}$ tends to become linear.
• The fuzzy membership degree ${\mathit{fuz}}_A$ between ${\widehat{\mathbf{X}}}_{\mathit{end}}^i$ and ${\widehat{\mathbf{X}}}_{\mathit{pub}}$ is analyzed. Such a value closer to 1 indicates a higher degree of membership between tracklet i and ${\widehat{\mathbf{X}}}_{\mathit{pub}}$.
• The difference $t_{\mathit{A}}$ between $t_{\mathit{pub}}$ and the end time $t_{\mathit{end}}^i$ of old tracklet i is considered. A lower value of $t_{\mathit{A}}$ suggests a greater likelihood that tracklet i and ${\widehat{\mathbf{X}}}_{\mathit{pub}}$ originate from the same track.

In summary, A can be expressed as

$$\begin{array}{c}A=w_{\mathit{A}\mathit{1}}\left(\pi -{\theta }_A\right)+w_{\mathit{A}\mathit{2}}\left(1-{\mathit{fuz}}_{\mathit{A}}\right)+w_{\mathit{A}\mathit{3}}{t}_{\mathit{A}}\end{array}$$

(19)

The variables r, a, and v are regarded as fuzzy factors characterizing the motion state of the vessel. Since the error distributions of these three fuzzy factors can be approximated by Gaussian distributions, this study adopts Gaussian distribution-based membership functions. The fuzzy membership degree ${\mathit{fuz}}_A$ is defined as

$$\begin{array}{c}fu{z}_A={\displaystyle \sum _{k\in \{r,a,v\}}}{\alpha }_k·exp\left(-{\tau }_k·{\displaystyle \frac{u_k^2}{{\sigma }_k^2}}\right)\end{array}$$

(20)

where ${\alpha }_k$($k\in \{r,a,v\}$) represents the fuzzy weight, ${\tau }_k$($k\in \{r,a,v\}$) denotes the adjustment degree, and ${\sigma }_k$($k\in \{r,a,v\}$) indicates the spread for the $k^{th}$ fuzzy factor, respectively. $u_k$ ($k\in \{r,a,v\}$) represents the difference between the endpoint of the old tracklet i and the public state for the $k^{th}$ factor, expressed as follows:

$$u_k=\left|k_{end}^i-k_{pub}\right|,\mathrm{where}k\in \{r,a,v\}$$

(21)

B consists of the following factors:

B composes similar to A, consisting of the smoothed values ${\widehat{\mathbf{X}}}_{\mathit{start}}^j$ and ${\widehat{\mathbf{X}}}_{\mathit{start}+1}^j$ of the new tracklet j at the initial and next moments, respectively, along with ${\widehat{\mathbf{X}}}_{\mathit{pub}}$. These elements form the angle ${\theta }_{\mathit{B}}$ and the fuzzy membership ${\mathit{fuz}}_B$, and also include the time difference $t_B$ between the initial moment $t_{\mathit{start}}^j$ and the public moment $t_{\mathit{pub}}$. The weight settings of B are consistent with those of A:

$$\begin{array}{c}B=w_{\mathit{A}\mathit{1}}\left(\pi -{\theta }_{\mathit{B}}\right)+w_{\mathit{A}\mathit{2}}\left(1-{\mathit{fuz}}_B\right)+w_{\mathit{A}\mathit{3}}{t}_B\end{array}$$

(22)

C consists of the following factor:

${\widehat{\mathbf{X}}}_{\mathit{end}}^i$, ${\widehat{\mathbf{X}}}_{\mathit{start}}^j$, and ${\widehat{\mathbf{X}}}_{\mathit{pub}}$ are fitted to form an arc. C is defined by the ratio R of the arc length $le{n}_{arc}$ to the chord length $le{n}_{chord}$ of this arc. Similar to the angle, as R approaches 1, the fitted circular arc more closely approximates a straight line, indicating a smaller discrepancy between ${\widehat{\mathbf{X}}}_{\mathit{pub}}$ and both the old and new tracklets. This relationship can be expressed as

$$\{\begin{array}{c}R={\displaystyle \frac{le{n}_{arc}}{le{n}_{chord}}}\\ C=R\end{array}$$

(23)

D consists of the following factors:

D is composed of the differences between the elements of ${\widehat{\mathbf{X}}}_{\mathit{end}}^i$ and ${\widehat{\mathbf{X}}}_{\mathit{start}}^j$, specifically, $r_{\mathit{diff}}$, $a_{\mathit{diff}}$, $v_{\mathit{diff}}$, and $t_{\mathit{diff}}$, along with the Euclidean distance d. The parameter D does not involve ${\widehat{\mathbf{X}}}_{\mathit{pub}}$, yet it effectively reflects the discrepancy between tracklets i and j. In addition, we consider $r_{\mathit{diff}}, a_{\mathit{diff}}$, and $v_{\mathit{diff}}$ as an integrated entity. The weight assignments for these three factors are consistent with the fuzzy weights in fuzzy. Therefore, D can be expressed as

$$D=w_{\mathit{D}\mathit{1}}\left({\alpha }_r{r}_{\mathit{diff}}+{\alpha }_a{a}_{\mathit{diff}}+{\alpha }_v{v}_{\mathit{diff}}\right)+w_{\mathit{D}\mathit{2}}{t}_{\mathit{diff}}+w_{\mathit{D}\mathit{3}}d$$

(24)

Data Normalization

All numerical values mentioned above require normalization, which is performed using the min-max normalization method. The specific formula is as follows:

$$\begin{array}{c}{\mathit{val}}_{\mathit{norm}}={\displaystyle \frac{\mathit{val}-\mathit{min}}{\mathit{max}-\mathit{min}}}\end{array}$$

(25)

In this equation, ${\mathit{val}}_{\mathit{norm}}$ represents the normalized value, while $\mathit{val}$ denotes the original value to be processed. The variables $\mathit{max}$ and $\mathit{min}$ correspond to the maximum and minimum values of $\mathit{val}$, respectively. This normalization technique effectively scales the data to a range of $[0,1]$, thereby mitigating the impact of differing units of measurement on subsequent analyses.

2.3.2. AN2O: Assigning New Tracklet to Old Tracklet

AN2O is designed to resolve cases where an old tracklet is associated with multiple new tracklets after first-stage association. Specifically, for each old tracklet, AN2O evaluates all new tracklets associated with this old tracklet within the set $S_{(1\to n)}$. Using the proposed optimal tracklet assignment method, AN2O identifies the most suitable new tracklet for each old tracklet.

2.3.3. AO2N: Assigning Old Tracklet to New Tracklet

After completing the AN2O process, a situation may arise where multiple old tracklets are associated with the same new tracklet. Such tracklet pairs are grouped into the set $S_{(n\to 1)}$. Similar to AN2O, AO2N utilizes the proposed optimal tracklet assignment method to assign the most suitable old tracklet to each new tracklet within $S_{(n\to 1)}$, ensuring a precise and unambiguous one-to-one correspondence between old and new tracklets.

To provide a clearer understanding of the AN2O and AO2N processes and their roles, we utilize a schematic diagram to illustrate these concepts, as depicted in Figure 7.

In the schematic diagram, the red and blue lines represent the old and new tracklets, respectively, while the black dashed arrows pointing from the new tracklet to the old tracklet indicate the assignment of the new tracklet to the old one, and vice versa. During the AN2O process, we postulate the existence of two groups of competing tracklet pairs: new tracklet 1 (denoted as N1) competes with new tracklet 2 (N2) for old tracklet 1 (O1), and N1 competes with new tracklet 3 (N3) for old tracklet 2 (O2). The AN2O process is assumed to yield O1-N1 and O2-N1 as the optimal tracklet pairs for their respective groups, resulting in multiple old tracklets (O1 and O2) being associated with the same new tracklet (N1). Consequently, the tracklet pairs O1-N1 and O2-N1 are placed into set $S_{(n\to 1)}$ for subsequent AO2N processing, where they become competing tracklet pairs. The AO2N process assigns either O1 or O2 to N1, forming the optimal tracklet pair. In this case, we assume that AO2N identifies O2-N1 as the optimal one.

It is evident that AN2O and AO2N operate sequentially: The AN2O process is applied to all tracklet pairs in set $S_{(1\to n)}$ before proceeding to AO2N. The confirmed tracklet pairs are then placed into set $S_{(1\to 1)}$, representing the final association results. The unselected tracklets undergo further discrimination, which will be elaborated in subsequent sections.

2.3.4. Iterative Discrimination

In the AO2N process, although some old tracklets are excluded due to their inability to establish associations with new tracklets (such as tracklet O1), this does not imply that they are entirely irrelevant. These discarded old tracklets may still have potential associations with other new tracklets. Therefore, if the excluded old tracklets form tracklet pairs with other new tracklets during the coarse association, the AN2O and AO2N processes should be reapplied to these tracklet pairs. This process should persist until all tracklet pairs that satisfied coarse association have been thoroughly evaluated. After each iteration is completed, the set $S_{(n\to 1)}$ will be cleared to accommodate tracklet pairs for the next iteration.

The pseudocode for the AN2O and AO2N processes, along with the iterative discrimination procedure, is provided in Algorithm 1.

Algorithm 1 AN2O and AO2N processes with iterative discrimination

1: Input $S_{(1\to n)}$, $S_{(1\to 1)}$ 2: repeat 3:    for each tracklet pair in $S_{(1\to n)}$ do 4:        Perform AN2O process 5:        Remove the successfully selected tracklet pairs from $S_{(1\to n)}$ 6:    end for 7:    Generate $S_{(n\to 1)}$ based on AN2O 8:    for each tracklet pair in $S_{(n\to 1)}$ do 9:       Perform AO2N process 10:    end for 11:    Update $S_{(1\to 1)}$ based on AN2O and AO2N 12: until$S_{(1\to n)}=\varnothing$ or $S_{(n\to 1)}=\varnothing$ 13: Clear $S_{(n\to 1)}$ 14: Output $S_{(1\to 1)}$

2.4. Third-Stage Association: Strict Association

In the third-stage association, some unreasonable tracklet pairs within the set $S_{(1\to 1)}$ will be identified and eliminated. This stage is crucial for enhancing the accuracy and reliability of the tracklet association process by removing mismatches that may have been included in previous stages.

In the coarse association stage, a portion of tracklet pairs cannot be reasonably classified as belonging to the same target. These tracklet pairs are typically excluded during the second-stage association due to their low association probability compared to the optimal tracklet pair. However, there are also some tracklet pairs that have not undergone second-stage association, where an old tracklet is only associated to a single new tracklet through coarse association. In such cases, due to the lack of competition, these tracklet pairs are not subjected to second-stage association and are, therefore, retained in $S_{(1\to 1)}$. Given the potential for erroneous associations in these tracklet pairs, it is essential to evaluate them.

The third-stage association process follows the same methodology as the coarse association detailed in Section 2.2.1, but it employs stricter thresholds to determine whether a tracklet pair satisfies the discrimination criteria. Consequently, this third-stage association is referred to as strict association. The objectives of the third-stage association are as follows:

• Identifying tracklet pairs that satisfied coarse association but did not undergo second-stage association;
• Identifying old tracklets that were excluded during the first-time AO2N but subsequently established association with other new tracklet in later iterations. Since these tracklet pairs were deemed untrustworthy during the AO2N process, they require rigorous association evaluation.

Following the removal of the tracklet pairs deemed unreasonable from the set $S_{(1\to 1)}$ during the third-stage association, the remaining tracklet pairs within the set $S_{(1\to 1)}$ constitute the final association results.

3. Results

In this study, we conducted experiments in a multi-target scenario to validate the feasibility of the proposed algorithm. The experimental data were sourced from field data collected by the HFSWR system in Bohai, China, on 30 April 2021. The radar system operates at a frequency of 4.7 MHz with a coherent integration time (CIT) of 262.144 s and a data rate of 1 frame/min. It employs an eight-element receiving antenna array with an aperture size of 105 m (interelement distance: 15 m) and utilizes a frequency-modulated interrupted continuous wave (FMICW) as the transmitting waveform, achieving a maximum detection range of 150 km, a range resolution of 2.5 km, and a Doppler velocity resolution of 0.44 km/h. These HFSWR tracklets have been confirmed using the vessel identification number for each track segment through an association algorithm that integrates HFSWR data with the Automatic Identification System (AIS). A total of 107 tracklets were obtained, with 58 of them being associable tracklets. The distribution of these tracklets is illustrated in Figure 8.

We have established three groups of thresholds: coarse spatio-temporal thresholds of tracklets in Section 2.2.1, spatial thresholds of estimation points during fragmentation in Section 2.2.2, and strict spatio-temporal thresholds of tracklets in Section 2.4. These thresholds are established determined based on practical experience, as shown in

Table 1. Thresholds for different groups.

Section	$\mathit{r}\left(\mathbf{km}\right)$	$\mathit{a}\left(\mathbf{deg}\right)$	$\mathit{v}\left(\mathbf{kph}\right)$	$\mathit{t}\left(\mathbf{min}\right)$
Section 2.2.1	40	40	8.5	80
Section 2.2.2	24	28	5.4	N/A
Section 2.4	18	17	3.9	40

.

For Equations (17), (19), (22) and (24), which are used to define the fitness function, the weights assigned in our study are based on extensive experimental tests. Among the features proposed in the equations, the fuzzy membership $fuz$, along with r, a, and v, is considered the most significant, followed by the angle $\theta$ and the arc-chord ratio R, while interruption time t and the Euclidean distance d are the least important. Based on the varying importance of these factors, the parameters we identified are presented in

Table 2. Weights for designing the fitness function.

Equation Reference	Weight Parameters	Weighted Terms	Parameter Values
(17)	$w_1$	A	0.33
	$w_2$	B	0.33
	$w_3$	C	0.2
	$w_4$	D	0.14
(19) and (22)	$w_{A1}$	$\pi -{\theta }_A$ and $\pi -{\theta }_B$	0.3
	$w_{A2}$	$1-{\mathit{fuz}}_A$ and $1-{\mathit{fuz}}_B$	0.5
	$w_{A3}$	$t_A$ and $t_B$	0.2
(24)	$w_{D1}$	${\alpha }_r{r}_{\mathit{diff}}+{\alpha }_a{a}_{\mathit{diff}}+{\alpha }_v{v}_{\mathit{diff}}$	0.55
	$w_{D2}$	$t_{\mathit{diff}}$	0.25
	$w_{D3}$	d	0.2

.

Where the fuzzy weights ${\alpha }_k$ ($k\in \{r,a,v\}$) for calculating the fuzzy membership in fuzzy are based on the radar’s error covariance, lower measurement errors correspond to higher weights of 0.22, 0.16, and 0.62, respectively. The adjustment degrees ${\tau }_k$ ($k\in \{r,a,v\}$) and spreads ${\sigma }_k$ ($k\in \{r,a,v\}$) are set to 0.027, 0.041, 0.039 and 1.45, 2, 0.51, respectively.

3.1. Statistical Results

We will conduct a comparative analysis of the proposed algorithm against the traditional fuzzy algorithm [21,22] and the improved Gale–Shapley (IGS) method [8] to evaluate its performance. True positives (TP), true negatives (TN), false positives (FP), and false negatives (FN) will be counted and analyzed to assess the performance of the algorithms. Additionally, the accuracy (Acc) is calculated as the proportion of correctly classified samples out of the total samples. This metric, along with the true positive rate (TPR), also known as recall or sensitivity, which measures the proportion of actual positive cases that are correctly identified by the algorithm, will be statistically analyzed in our study. Calculations for Acc and TPR are as follows:

$$\mathrm{Acc}={\displaystyle \frac{TP+TN}{TP+TN+FP+FN}}$$

(26)

$$\mathrm{TPR}={\displaystyle \frac{TP}{TP+FN}}$$

(27)

Higher values of TP, TN, Acc, and TPR being closer to their maximum values (58, 49, 100%, and 100%, respectively) indicate a better algorithm performance. Conversely, lower values of FP and FN reflect a better algorithmic effectiveness. The statistical results of the three algorithms are shown in

Table 3. Statistical results of the three algorithms.

	TP ↑ (Count)	TN ↑ (Count)	FP ↓ (Count)	FN ↓ (Count)	Acc ↑ (%)	TPR ↑ (%)
Ideal Case	58	49	0	0	100	100
Proposed	49	42	7	9	85.05	84.48
IGS	38	33	16	20	66.36	65.52
Fuzzy	35	32	17	23	62.62	60.34

The arrows indicate the preferred direction: “↑” indicates a higher value is better, while “↓” indicates a lower value is better.

.

The data presented in Table 3 show that the results of the two benchmark algorithms are similar, with the fuzzy method exhibiting poorer performance, which can be attributed to the potential maneuvers of vessels during track interruption. These maneuvers may reduce the similarity between tracklets. Consequently, the fuzzy algorithm, which relies heavily on similarity between tracklets, exhibits degraded performance. In contrast, the proposed algorithm demonstrates an improvement of approximately 25% in both Acc and TPR compared to the fuzzy method, and 20% in both Acc and TPR compared to the IGS method. Additionally, the FN and TN are all lower than those obtained using the traditional algorithm. It is noteworthy that FN is slightly higher than FP. This discrepancy can primarily be attributed to the significant breakage periods of some tracks, which hinder their association through preliminary coarse matching.

To evaluate the enhancement effect of CFOA on algorithm performance, we compared the particle swarm optimization (PSO) algorithm [32] with its CFOA-integrated version within the proposed algorithm framework. Both optimization algorithms are configured with a population size of 50 and 1500 iterations; the inertia weight is 0.4, individual and swarm confidence factors are set to 2, and the maximum velocity is 5% for PSO. The statistical results are shown in

Table 4. Statistical results of the algorithms based on CFOA and PSO, respectively.

	TP ↑ (Count)	TN ↑ (Count)	FP ↓ (Count)	FN ↓ (Count)	Acc ↑ (%)	TPR ↑ (%)
Proposed	49	42	7	9	85.05	84.48
PSO	41	37	12	17	72.90	70.69

.

As shown in Table 4, the proposed CFOA-based algorithm achieves approximately a 15% improvement in both Acc and TPR compared to the PSO-based algorithm, while reducing the FP and FN by nearly half. These results underscore the challenges faced by PSO in addressing such problems, including the curse of dimensionality and premature convergence. In contrast, CFOA effectively enhances algorithmic performance, significantly improving overall accuracy.

In the following section, we will further elucidate the advantages of the proposed algorithm over the fuzzy method by analyzing the association results of some representative cases.

3.2. Case Analysis

3.2.1. Case 1

This case will illustrate the performance differences between the proposed algorithm and the traditional algorithm. Through the analysis of AIS data, it has been confirmed that old tracklet 84 and new tracklet 72 belong to the same vessel. The comparative relationship between the proposed algorithm and traditional algorithm is depicted in Figure 9 and Figure 10.

Figure 9 shows the result of the AN2O after the first-stage association of old tracklet 84 using the proposed algorithm, while Figure 10 displays the result of associating old tracklet 84 using the traditional algorithm. To present these tracklets more intuitively, we converted the polar coordinate data obtained from HFSWR measurements into Cartesian coordinates. In these two figures, the red line represents the pre-processed old tracklet, while the blue lines indicate the new tracklet(s) that have been pre-processed and meet the criteria for first-stage association discrimination, with the green line representing the association between the old and new tracklets. In particular, the pentagram in Figure 9 represents the optimal state generated by the proposed algorithm. In subsequent analyses, a similar legend representation will be employed, and further elaboration on this will not be repeated.

Figure 9 and Figure 10 demonstrate that the proposed algorithm effectively associates the tracklets, whereas the traditional algorithm exhibits false association. This discrepancy stems from the higher fuzzy membership degree between the leading end of tracklet 29 and the trailing end of tracklet 84, leading the traditional algorithm to incorrectly associate tracklet 84 with tracklet 29. In contrast, the proposed algorithm incorporates a comprehensive assessment of multiple factors when designing the fitness function. Specifically, the arc-chord ratio and the angle between tracklet pair 84-29 are more favorable than those between tracklet pair 84-72. After a thorough evaluation, the fitness function value and optimal public state for tracklet pair 84-72 indicate a more optimal association.

3.2.2. Case 2

The algorithm proposed exhibited a failure in association in this case, similar to traditional algorithm. Verification by the AIS confirmed that old tracklet 77 and new tracklet 53 originated from the same vessel. However, both the proposed algorithm and the traditional algorithm erroneously associated old tracklet 77 with new tracklet 80. The results of this association are illustrated in Figure 11 and Figure 12.

The algorithm proposed in this study identifies the fundamental cause of tracklet association errors as stemming from a high degree of fuzzy membership in tracklet pair 77-80 compared to tracklet pair 77-53. Furthermore, in comparison to tracklet pair 77-53, tracklet pair 77-80 exhibits significantly greater advantages in key metrics such as arc-chord ratio, angle, and interruption time, leading to more favorable values for both the public state and the fitness function. These discrepancies cause the algorithm to incorrectly select tracklet pair 77-80 as the associated pair. The resulting association error is the type that the algorithm is unable to rectify.

3.2.3. Case 3

This case study aims to explore the primary function of the iterative discrimination mechanism, emphasizing its critical role in ensuring that no potential associations are omitted and enhancing the overall accuracy of the associations.Verification through the AIS indicates that old tracklet 31 and new tracklet 67 correspond to the same vessel, as do old tracklet 36 and new tracklet 43. Figure 13 and Figure 14 illustrate only the initial AN2O results of the proposed algorithm. It should be noted that these AN2O results are not the final association outcomes. The final association result for old tracklet 31 is depicted in Figure 15. The association results from traditional algorithms are not analyzed in this section.

In Figure 13, old tracklet 31 and new tracklet 43 were erroneously associated due to reasons similar to those observed in Case 2. Conversely, in Figure 14, old tracklet 36 successfully established a correct association with new tracklet 43. This situation led to the inclusion of old tracklets 31 and 36 and new tracklet 43 in the AO2N process. During the AO2N process, new tracklet 43 was successfully associated to old tracklet 36, due to a lower fitness function value, resulting in the exclusion of old tracklet 31. Subsequently, for old tracklet 31, the iterative discrimination mechanism was employed to evaluate whether there are any new tracklets, aside from new tracklet 43, that meet the criteria for coarse association discrimination, specifically, new tracklet 67 and new tracklet 24. Therefore, tracklet pairs 31-67 and 31-24 were re-executed with the AN2O operation, which led to the final outcome that tracklet pair 31-67 was assessed as superior to 31-24, as illustrated in Figure 15. After the three-stage association discrimination of the tracklet pair 31-67, a successful association was ultimately established, ensuring that both tracklet pair 31-67 and tracklet pair 36-43 received accurate association. The results of this case analysis demonstrate that the proposed algorithm effectively incorporates all tracklets within a specific region into a comprehensive evaluative framework, significantly reducing the likelihood of both missed associations and false associations.

4. Discussion

In the operational context of HFSWR, track fragmentation can occur at any time, especially in complex environments such as densely populated channel scenarios. The proposed TSA algorithm demonstrates exceptional performance in addressing track fragmentation. It effectively mitigates disruptions caused by interfering factors while maintaining high performance in densely populated channel environments. The algorithm exhibits remarkable robustness, enabling the long-term continuous tracking of multiple targets with high accuracy and adaptability. These characteristics underscore its feasibility as an efficient and superior TSA algorithm. However, instances of erroneous associations persist. While many erroneously associated tracklet pairs can be eliminated through subsequent discrimination processes, some are retained. Future research will focus on a a comprehensive investigation into the underlying factors responsible for association failures, with the objective of refining and enhancing the algorithm. Furthermore, we will conduct extensive validation of the algorithm’s efficacy across a wide range of scenarios. By performing comparative performance analyses in diverse contexts, we aim to elucidate its performance characteristics and establish a robust foundation for subsequent optimization efforts.

5. Conclusions

This paper addresses the issue of track fragmentation encountered in ship tracking using HFSWR. We propose a novel TSA algorithm that integrates multi-stage association, optimal tracklet assignment, and iterative discrimination. The algorithm comprises several steps: First, a coarse association of pre-processed tracklet data is conducted, focusing on the spatiotemporal relationships between tracklet pairs. This approach aims to retain all potential associated tracklet pairs. Subsequently, for the tracklet pairs selected through coarse association, we introduce an innovative optimal tracklet assignment method to facilitate both AN2O and AO2N processes. By incorporating an iterative discrimination mechanism, this method effectively achieves precise one-to-one associations of tracklets. Finally, a third-stage association process is implemented, employing stricter threshold settings to eliminate redundant associations and ensure the accuracy of the results. Experimental validation demonstrates that the proposed algorithm shows performance improvements compared to existing methods. Tt achieves an approximate 25% increase in both Acc and TPR over the fuzzy method, and a 20% increase in both Acc and TPR over the IGS method. Furthermore, the FN and TN are consistently lower than those obtained using traditional algorithms, thereby showcasing its superior performance.

In summary, the proposed TSA algorithm demonstrates strong robustness and effectiveness in interference-prone environments, enabling the accurate long-term tracking of targets, thereby underscoring its viability as a superior TSA algorithm. Future work will focus on validating the algorithm’s effectiveness across diverse scenarios through comparative analyses to gain a deeper understanding of its performance characteristics and provide a foundation for further optimization.