A novel density-based clustering method for effective removal of spurious intersections in bearings-only localization

Abstract

In bearings-only localization, clustering-based methods have been widely used to remove spurious intersections by fusing multiple bearing measurements from different observation stations. Existing clustering methods, including fuzzy C-mean (FCM) clustering and density-based spatial clustering of applications with noise (DBSCAN), must specify the number of clusters and the threshold for defining the neighborhood density, respectively, which are always unknown and difficult to estimate. Moreover, in dense radiation source scenes, existing clustering methods for removal of spurious intersections all deteriorate significantly. Therefore, we propose a novel density-based clustering method called K-M-DBSCAN, which combines the minimum K distance algorithm with Mahalanobis distance-based DBSCAN clustering. Firstly, K-M-DBSCAN uses minimum K distance algorithm for preprocessing to remove most of the spurious intersections and reduce the computational complexity of clustering. Mahalanobis distance-based DBSCAN is used for clustering and spurious intersections recognition. In order to adapt the large variations of sample density in clustering, we use Mahalanobis distance to define an explicit neighborhood of DBSCAN instead of traditional Euclidean distance. Simulation results show that the proposed K-M-DBSCAN performs better than FCM and DBSCAN in removing of spurious intersections.

1 Introduction

Radar radiation source localization technology based on artificial intelligence and data mining has attracted more and more attentions [1,2,3,4]. Bearing only localization (BOL) is the earliest and most mature localization method in passive localization. It estimates the position of radar target by fusing multiple bearing measurements from different observation stations [5]. In passive localization system, due to the lack of distance information, BOL plays an important role in both military and civil applications, such as underwater surveillance [6,7,8], 3-D passive target tracking [9,10,11] and unmanned aerial vehicle path planning [12,13,14,15]. For multiple radar targets within the surveillance area and two or more observation stations involved in BOL, different bearing intersections without correct data association will cause serious spurious intersections problem [16,17,18], and the number of spurious intersections will exponentially grow with the number of the observation stations and radar radiation sources. Recent advances have demonstrated that the clustering method can be utilized to remove the spurious target in BOL system.

Existing approaches for removal of the spurious intersections can be mainly divided into two kinds, including data association and clustering-based approaches. One of the most widely used data association approaches is using m-best assignment formulation [19] to solve a S dimension (S-D) assignment problem. By making appropriate modifications to the cost matrix and solving a series of modified copies of the initial problem, this method can be used to find the m-best (ranked) solutions to the data association problem [20, 21]. Michael Beard et al. [22] proposed two data association algorithms: Classical Bayesian Thread Association (CBTA) and Monte Carlo Thread Association (MCTA) to solve the problem of thread association in bearings-only tracking of multiple targets. These algorithms used single target bearings-only state estimators to calculate measurement likelihood. Simulation results showed that MCTA algorithm is superior to CBTA algorithm. Some researchers proposed joint probabilistic data association method and multiple hypothesis tracking method, Kirubarajan et al. [23] combined probabilistic data association (PDA) and maximum likelihood method to estimate target motion parameters through batch processing to track low observable targets in passive sonar measurement. However, this method requires a lot of calculation and is only suitable for a small number of radiation sources. Therefore, data association methods [24,25,26] are not robust and efficient in dealing with complex environments containing multiple targets in real time. They always suffer from the NP-hard problem and are unsuitable for real-time applications.

Another approach for removal of the spurious intersections is the clustering-based method. Two classical clustering methods including FCM clustering and DBSCAN clustering are mostly widely used. Recent related research shows that FCM method combing with a correlation objective function performs well in removal of spurious intersections [27], which allows for class overlaps based on the objective functions. A correlation-based FCM method assigns each sample with degrees of membership to multiple clusters. In order to circumvent the very high complexity of a brute-force solution to the data association problem of multiple-source localization, Reed et al. [28] used the clustering of line of bearing (LOB) to estimate target position estimates and recursively updated the position estimates by matching the LOB measurements between sensors corresponding to the same target. Campello et al. [29] proposed an improved density-based hierarchical clustering method for obtaining a “flat” partition consisting of only the most significant clusters (possibly corresponding to different density thresholds). The method provided a hierarchy of clusters from which can construct a simplified tree of significant clusters The main drawback of FCM is the number of clusters should be specified as a priori. In order to overcome the drawback of FCM, some researchers have proposed some density-based clustering methods, such as DBSCAN. Duan et al. [30] proposed a local density-based clustering algorithm to overcome the serious defects of DBSCAN in processing the cluster with different local densities. The algorithm took the advantages of local outlier factor (LOF) to detect noise. Experiments showed that their method had better results than Ordering Points To Identify The Clustering Structure (OPTICS). Tran et al. [31] proposed an improved algorithm to overcome the problem that the DBSCAN algorithm became unstable when detecting border objects of adjacent clusters. The improved DBSCAN has better robustness to data sets with connected clusters, and its clustering results have nothing to do with the order of processing objects. In density-based clustering, Euclidean distance is widely used as a measure of correlation between different samples. DBSCAN is capable of handling and identifying noise, finding clusters with arbitrary shapes, and automatically discovering the number of clusters. However, there are also drawbacks for DBSCAN to remove spurious intersections. Firstly, if the spurious intersections are not preprocessed before DBSCAN, it will increase the computational complexity and introduce much interference. More importantly, the performance may degrade significantly when large variations in sample density occur in real applications.

In order to overcome the drawbacks of DBSCAN method, we propose K-M-DBSCAN method for removal of spurious intersections. The K-M-DBSCAN has two advantages over DBSCAN. Firstly, we deploy the minimum K distance preprocessing method to reduce the interference caused by spurious intersections and also reduce the computation complexity of clustering. Secondly, M-DBSCAN uses Mahalanobis distance to define density of intersections instead of the traditional Euclidean distance. More specifically, two main parameters, including neighborhood radius and minimum number of neighborhood points, are determined based on Mahalanobis distance. Since Mahalanobis distance considers the probability distribution of intersections, M-DBSCAN has the ability to adapt large variations in intersection sample density, which is critical for effective removal of spurious intersections in a dense radiation source environment.

The remainder of this paper is organized as following. Section 2 gives the principle of BOL and problem formulation for the removal of spurious intersections and then describes the proposed clustering method, called K-M-DBSCAN. Section 3 gives the results and discussion to verify the effectiveness of the proposed method. Section 4 introduces the time complexity of the three algorithms. Section 5 includes the conclusions.

2 Methods

An overview of the proposed K-M-DBSCAN method is shown in Fig. 1. Solutions for the removal of spurious intersections using K-M-DBSCAN method can be divided into five steps after the intersections are generated. We first propose the minimum K distance algorithm to filter out some spurious intersections and thus reduce the interference and the computational complexity of clustering. The second step is to define neighborhood radius (eps) and calculate the density of each intersection based on Mahalanobis distance. Besides, we discuss another important parameter, minimum number of neighbor points ($mi{n_{pts}}$) required. Thirdly, determine the core and accessibility of the given intersections. The core intersections are obtained if there are at least $mi{n_{pts}}$ intersections within its eps neighborhood. Intersection accessibility is defined as being within an observed eps range, or through some dense intersections chains that are within eps distance from each other. In the fourth step, we regard the clusters formed by the core intersections and accessible intersections as the real target clusters, and the rest intersections as spurious clusters. Finally, use the measurement error of angle to update the cluster center, and achieve the real target localization.

Fig. 1

Overview of the proposed K-M-DBSCAN method for removal of spurious intersections

In this section, we first formulate the removal of spurious intersections problem in two-dimensional BOL and then describe the details of the proposed K-M-DBSCAN method.

2.1 Problem formulation

For simplicity, we describe target in two-dimensional plane with the bearing measurements received by two observing stations, as shown in Fig. 2.

Fig. 2

Geometry of BOL

The position of the radiation target is described by its two-dimensional Cartesian coordinates. The bearing angles measured by observation stations ${S_i}\left( {{x_i},{y_i}} \right)$ and ${S_j}\left( {{x_j},{y_j}} \right)$ corresponding to the target are $\theta _n^{\left( i \right) }$ and $\theta _n^{\left( j \right) },$ respectively, which can determine the n-th intersection $\left( x_n, y_n\right)$. The target localization [32, 33] can be given as

$$\begin{aligned} \tan \theta _n^{(i)}=\frac{y_n-y_i}{x_n-x_i} \end{aligned}$$

(1)

$$\begin{aligned} \tan \theta _n^{(j)}=\frac{y_n-y_j}{x_n-x_j} \end{aligned}$$

(2)

Equations (3) and (4) can be obtained through simplification:

$$\begin{aligned} x_n=\frac{y_j-y_i+x_i \tan \theta _n^{(i)}-x_j \tan \theta _n^{(j)}}{\tan \theta _n^{(i)}-\tan \theta _n^{(j)}} \end{aligned}$$

(3)

$$\begin{aligned} {\hat{y}}_{n}=\frac{\left( x_{i}-x_{j}\right) \tan \theta _{n}^{(i)} \tan \theta _{n}^{(j)}+y_{j} \tan \theta _{n}^{(i)}-y_{i} \tan \theta _{n}^{(j)}}{\tan \theta _{n}^{(i)}-\tan \theta _{n}^{(j)}}. \end{aligned}$$

(4)

In fact, the actual BOL system involves multiple observation stations, whose bearing measurements will produce a large number of spurious intersections, as shown in Fig. 3. Assuming that there are s observing stations and t radiation sources, there are a total number of $C_s^2 * {t^2}$ intersections ($C_s^2 = s * \left( {s - 1} \right) /2$), of which $C_s^2 * t$ intersections are real targets and the rest are spurious intersections. It clearly shows that the density of real intersections near the real target is higher than those of the spurious targets. Therefore, the method based on density clustering is feasible to remove spurious intersections.

Fig. 3

Distribution of intersections for three observation stations and two radiation sources

2.2 Minimum K distance algorithm for preprocessing

In order to delete some spurious intersections initially and reduce the computational complexity of clustering, we deploy the minimum K distance algorithm before density-based clustering. The minimum K distance algorithm is inspired by the baseline least distance method [34]. The baseline least distance is a popular method to remove spurious intersections, which has low computational complexity and is insensitive to measurement error of angle. However, the baseline least distance method only keeps one intersection pair for each bearing based on the minimum Euclidean distance. Though a large number of spurious intersections are deleted, it also deletes many real intersections. In contrast, the minimum K distance algorithm used in proposed K-M-DBSCAN algorithm retains K pairs of intersections with the least distance for each bearing. As a result, the minimum K distance algorithm filters out a considerable proportion of spurious intersections, while little real intersections are deleted.

The bearing line of the i-th observation station to the j-th target is recorded as $L_{i j}$, and the bearing line of the m-th observation station to the n-th target is recorded as $L_{m n}$, and the intersection of $L_{i j}$ and $L_{m n}$ is recorded as $X_{m n, i j}$.

Randomly choose a bearing line $L_{i j}$ as the reference line, all intersections on the $L_{i j}$ form a set $\Lambda _{i j}$. The intersections of all bearing lines of the m-th ($m \ne i$) observation station and $L_{i j}$ form a set $S_{m n, i j} \left( S_{m n, i j} \subseteq \Lambda _{i j}\right)$, with fixed i, j, m and $n=1,2, \ldots , N T$. Then traverse all intersections $X_{m n, i j}(n=1,2, \ldots, N T)$ in the set $S_{m n, i j}$, and calculate the minimum distance between $X_{m n, i j}$ and the remaining points X in the set $\Lambda _{i j}\left( X \in \Lambda _{i j}, X \notin S_{m n, i j}\right)$. Each $X_{m n, i j}$ will generate a minimum distance. Next sort all the minimum distances generated by $S_{m n, i j}$, and select K minimum distance intersections to join the set P. After traversing all bearing lines, repeating above steps, and removing the repeated intersections, the generated P is the set of intersections after preprocessing. Implementation details of minimum K distance algorithm are given in Table 1.

Table 1 Minimum K distance algorithm

2.3 M-DBSCAN for removal of spurious intersections

2.3.1 Defining neighborhood radius based on Mahalanobis distance

There are two main parameters for density-based clustering method, including neighborhood radius and minimum number of neighborhood points. The former represents the area of the neighborhood elliptic region, while the latter determines the density of the intersections. For density-based clustering, setting of neighborhood radius eps is critical for ultimate clustering results. If the value is too small, intersections of the same cluster will be divided into multiple clusters, while too large value will make multiple clusters merged into one.

For neighborhood radius setting, the distance measurement criterion is a key affecting factor. Mahalanobis distance criterion is effective to measure the similarity of two unknown sample sets and it can eliminate the interference of correlation between variables. Compared with traditional Euclidean distance criterion, we construct Mahalanobis distance criterion based on the probability distribution of intersections, which adapt the large variations in sample density. Since the measurement error of angle, positions of the observation stations and radiation sources are all considered, so the constructed Mahalanobis distance to calculate eps is more robust and effective to adapt the variations in the environment and measurement error of angle.

In order to define the neighborhood radius, we firstly give the derivation of the covariance matrix of localization error. Secondly, we use the covariance matrix to calculate the Mahalanobis distance and define an explicit neighborhood. Finally, we derive neighborhood radius based on the spatial probability density distribution of the intersections.

In passive localization systems, the localization error covariance matrix is an important indicator of localization accuracy. Assuming that the measurement error of angle following a Gaussian distribution with zero mean, the localization error equation can be derived from Equations (1) and (2),

$$\begin{aligned} \left[ \begin{array}{l} \textrm{d} \theta _n^{(i)} \\ \textrm{d} \theta _n^{(j)} \end{array}\right] =\left[ \begin{array}{cc} -\frac{\sin ^2 \theta _n^{(i)}}{y_n-y_i} &{} \frac{\cos ^2 \theta _n^{(i)}}{x_n-x_i} \\ -\frac{\sin ^2 \theta _n^{(j)}}{y_n-y_j} &{} \frac{\cos ^2 \theta _n^{(j)}}{x_n-x_j} \end{array}\right] \left[ \begin{array}{l} \textrm{d} x_n \\ \mathrm {~d} y_n \end{array}\right] +\left[ \begin{array}{l} k_{\theta _n^{(i)}} \\ k_{\theta _n^{(j)}} \end{array}\right] \end{aligned}$$

(5)

Among them, $\theta _n^{\left( i \right) }$ and $\theta _n^{\left( j \right) }$ are the bearings of the observation station ${S_i}\left( {{x_i},{y_i}} \right)$ and station ${S_j}\left( {{x_j},{y_j}} \right)$ to the n target, respectively, and $\mathrm{{d}}\theta _n^{\left( i \right) }$, $\mathrm{{d}}\theta _n^{\left( j \right) }$ represent the angle measurement error of the observation station ${S_i}$ and ${S_j},$ respectively. In addition,

$$\begin{aligned} k_{\theta _n^{(i)}}=\frac{\sin ^2 \theta _n^{(i)}}{y_n-y_i} \mathrm {~d} x_i-\frac{\cos ^2 \theta _n^{(i)}}{x_n-x_i} d y_i \end{aligned}$$

(6)

$$\begin{aligned} k_{\theta _n^{(j)}}=\frac{\sin ^2 \theta _n^{(j)}}{y_n-y_j} d x_j-\frac{\cos ^2 \theta _n^{(j)}}{x_n-x_j} d y_j \end{aligned}$$

(7)

The localization error equation is further expressed as

$$\begin{aligned} \textrm{d} V={\varvec{C}} \textrm{d} X_t+\textrm{d} X_s \end{aligned}$$

(8)

where $\mathrm{{d}}V\mathrm{{ = [d}}\theta _n^{\left( i \right) }\mathrm{{,d}}\theta _n^{\left( j \right) }{]^T}$ is the observation error vector, $\textrm{d} X_t=\left[ \textrm{d} x_n, \mathrm {~d} y_n\right] ^T$ is the target localization error vector, and $\mathrm{{d}}{X_s}\mathrm{{ = [}}{k_{\theta _n^{\left( i \right) }}},{k_{\theta _n^{\left( j \right) }}}{]^T}$ is the site error vector. The coefficient matrix ${\varvec{C}}$ can be expressed as

$$\begin{aligned} {\varvec{C}}=\left[ \begin{array}{cc} -\frac{\sin ^2 \theta _n^{(i)}}{y_n-y_i} &{} \frac{\cos ^2 \theta _n^{(i)}}{x_n-x_i} \\ -\frac{\sin ^2 \theta _n^{(j)}}{y_n-y_j} &{} \frac{\cos ^2 \theta _n^{(j)}}{x_n-x_j} \end{array}\right] \end{aligned}$$

(9)

The covariance matrix of localization error is

$$\begin{aligned} \varvec{\Sigma }=E\left[ \mathrm {~d} X_t \mathrm {~d} X_t^T\right] ={\varvec{C}}^{-1}\left\{ E\left[ \mathrm {~d} V \mathrm {~d} V^T\right] +E\left[ \mathrm {~d} X_s \mathrm {~d} X_s^T\right] \right\} {\varvec{C}}^{-T} \end{aligned}$$

(10)

Among them,

$$\begin{aligned} E\left[ \mathrm {~d} V \mathrm {~d} V^{T}\right] ={\text {diag}}\left[ e_{\theta _{n}^{(i)}}^{2}, e_{\theta _{n}^{(j)}}^{2}\right] \end{aligned}$$

(11)

$$\begin{aligned} E\left[ \mathrm {~d} X_s \mathrm {~d} X_s^T\right] ={\text {diag}}\left[ \frac{e_s^2}{\sqrt{\left( x_n-x_i\right) ^2+\left( y_n-y_i\right) ^2}} \frac{e_s^2}{\sqrt{\left( x_n-x_j\right) ^2+\left( y_n-y_j\right) ^2}}\right] \end{aligned}$$

(12)

where ${e_s}$ represents the standard deviation of the site error. The covariance matrix of localization error is

$$\begin{aligned} \varvec{\Sigma }=\left[ \begin{array}{ll} e_{x}^{2} &{} e_{x y} \\ e_{x y} &{} e_{y}^{2} \end{array}\right] \end{aligned}$$

(13)

The Mahalanobis distance is defined according to the covariance matrix of localization error, which can be expressed as the following formula:

$$\begin{aligned} r=\sqrt{[{\varvec{v}}-\varvec{\mu }]^{T} \varvec{\Sigma }^{-1}[{\varvec{v}}-\varvec{\mu }]} \end{aligned}$$

(14)

where ${\varvec{v}}$ represents two-dimensional coordination vector of a certain intersection, the mean vector $\varvec{\mu }$ is the coordination vector of the assumed real intersection, and ${\varvec{\Sigma }}$ represents the covariance matrix of the localization error.

Assuming that a certain intersection is the real target, Mahalanobis distance can define an elliptical neighborhood of the intersection. We construct the spatial probability density distribution function of intersections by Mahalanobis distance. The distribution function of spatial probability density can be expressed as

$$\begin{aligned} p(x)=(2 \pi )^{-D / 2}|\varvec{\Sigma }|^{-1 / 2} \exp \left\{ -\frac{1}{2[{\textbf{v}}-\varvec{\mu }]^T \varvec{\Sigma }^{-1}[{\textbf{v}}-\varvec{\mu }]}\right\} \end{aligned}$$

(15)

where D is the Gaussian distribution dimension, the two-dimensional vector $D=2$. And the probability density distribution function can be rewritten as

$$\begin{aligned} p(r)=\frac{1}{2 \pi }|\Sigma |^{-1 / 2} \exp \left\{ -\frac{1}{2 r^2}\right\} \end{aligned}$$

(16)

Then, we can define the neighborhood radius by the probability density function. If the probability that a certain intersection falls outside the ellipse neighborhood is $\beta$, the neighborhood radius ${r_0}$ can be calculated through the spatial probability distribution. According to the distribution function of probability density, the probability P is equivalent to doing a double integral in the ellipse neighborhood ($\frac{x^2}{a^2}+\frac{y^2}{b^2}=r^2$),

$$\begin{aligned} P=\int _0^{2 \pi } \int _0^{r_0} \frac{1}{2 \pi }|\varvec{\Sigma }|^{-1 / 2} \exp \left\{ -\frac{1}{2 r^2}\right\} (a b r) d \theta d r \end{aligned}$$

(17)

where the neighborhood radius is ${r_0} = \sqrt{ - 2\log \left( {1 - \beta } \right) }$ when $\beta (0<\beta <1)$ is fixed.

Minimum number of neighborhood points $mi{n_{pts}}$ is another crucial parameter of density clustering, which has a great influence on clustering results. We deploy the mathematical expectation based on self-decay term [35, 36] to automatically determine the $mi{n_{pts}}$ parameter, and the expression is given as

$$\begin{aligned} {\text {MinPts}}=\left[ \frac{1-\lambda }{n}\right] \sum _{i=1}^n P_i \end{aligned}$$

(18)

where $\lambda \left( {0 \le \lambda \le 1} \right)$ is the self-attenuation coefficient. In this paper, the value of $\lambda$ is between 0.1 and 0.3, n is the total number of intersections (including all real and spurious intersections remained after preprocessing), and ${P_i}$ is the number of intersections in ellipse neighborhood.

2.3.2 Implementation of M-DBSCAN

M-DBSCAN algorithm does not require prior information about the number of clusters. Moreover, two important parameters including $mi{n_{pts}}$ and eps can be well obtained according to the algorithm mentioned above. We construct the Mahalanobis distance criterion to measure the distance between intersections, which enables the adaption of large variations in cluster density and measurement error of angle.

M-DBSCAN can be described with respect to the directed eps neighborhood graph, which is designed to discover clusters of arbitrary shape with a fixed eps and a density threshold $mi{n_{pts}}$. Some concepts and terms to explain M-DBSCAN can be defined as follows:

Definition 1

An intersection p is a core intersection which is defined as intersections whose eps neighborhood size is greater than or equal to $mi{n_{pts}}$.

Definition 2

An intersection p is accessible intersection which is defined as being within an observed eps range, or through some dense intersections chains that are within eps distance from each other.

Definition 3

An intersection p is noise if it is neither a core intersection nor an accessible intersection. This means that noise is not part of any clusters.

Implementation of proposed M-DBSCAN algorithm is given in Table 2. First, we randomly choose an intersection p and traverse all points in its eps neighborhood. If p is a core intersections, it will be marked as a new cluster. This cluster is extended by retrieving all accessible intersections corresponding to p and further merges these intersections into the same cluster. This process is repeated until no clusters are found which means all intersections are observed. Another possibility is that p is an accessible intersection grouped into other clusters. If the density of p is less than $mi{n_{pts}}$, it will be marked as noise.

Table 2 M-DBSCAN algorithm

2.4 Target localization

Based on the previous section, we can distinguish real target clusters and spurious target clusters. Target localization estimation is based on real target clusters. We first calculate the cluster center, but it cannot be directly used as the localization result due to some intersections with large errors. Hence, the measurement error of angle is used to update the cluster center and achieve the real target localization.

Suppose that there are a total of w real target clusters. That is, the M intersections are divided into w categories $\left\{ {{C_1},{C_2}, \ldots ,{C_w}} \right\}$, and each type ${C_i}$ contains a lot of intersections. The cluster center can be given as follows [37, 38]

$$\begin{aligned} \left( {\tilde{x}}_{i}, {\tilde{y}}_{i}\right) =1 / {\text {card}}\left( C_{i}\right) \sum _{e=1}^{M} u_{i e}\left( x_{e}, y_{e}\right) , i=1,2, \ldots , w \end{aligned}$$

(19)

where card($C_i$) represents the total number of samples in the cluster, and ${u_{ie}}$ is used to represent whether the intersection $\left( {{x_e},{y_e}} \right)$ is in ${C_i}$. And ${u_{ie}}$ is given by

$$\begin{aligned} u_{i e}=\left\{ \begin{array}{l} 1,\left( x_e, y_e\right) \in C_i \\ 0,\left( x_e, y_e\right) \notin C_i \end{array}\right. \end{aligned}$$

(20)

However, cluster $C_i$ may contain noisy intersections with large errors. Measurement error of angle is used to eliminate the noisy intersections and then update the cluster center. The implementation process is given as follows.

First, according to the localization equation, the angle $\tilde{\theta }$ between the estimated localization $\left( {{x_i},{y_i}} \right)$ and the observation station $\left( {{{\tilde{x}}_i},{{{{\tilde{y}}}}_i}} \right)$ can be calculated with

$$\begin{aligned} {\tilde{\theta }}=\arctan \left( \frac{{\tilde{y}}_i-y}{{\tilde{x}}_i-x}\right) \end{aligned}$$

(21)

So ${\theta _{ie}}$ between the intersection $\left( {{x_i}_e,{y_i}_e} \right)$ and the observation station $\left( {{{{{\tilde{x}}}}_i},{{{{\tilde{y}}}}_i}} \right)$ in the cluster can be expressed as

$$\begin{aligned} \theta _{i e}=\arctan \left( \frac{{\tilde{y}}_i-y_{i e}}{{\tilde{x}}_i-x_{i e}}\right) , i=1,2, \ldots , k; e=1,2, \ldots , M \end{aligned}$$

(22)

Secondly, compare the calculated ${{\tilde{\theta }}}$ with ${\theta _{ie}}$, if the absolute value of the difference is greater than the measurement error of angle $\sigma$, then remove $\left( {{x_{ie}},{y_{ie}}} \right)$ from this cluster. We can use the formula to express as

$$\begin{aligned} \left\{ \begin{array}{l} \left| {\tilde{\theta }}-\theta _{i e}\right| >\sigma , {\text {remove}}\left( x_{i e}, y_{i e}\right) \\ \left| {\tilde{\theta }}-\theta _{i e}\right| \le \sigma , {\text {retain}}\left( x_{i e}, y_{i e}\right) \end{array}\right. \end{aligned}$$

(23)

Finally, update the intersections in the cluster, and calculate the cluster center again according to formula (17), and use it as the target localization result. This algorithm can recognize and delete the noisy intersection samples of the cluster and improve the localization accuracy.

3 Results and discussion

In this section, performance of the proposed K-M-DBSCAN is compared with DBSCAN and FCM algorithm via Monte Carlo (MC) simulations. The localization precision threshold mentioned in this section is defined as a localization error distance threshold. The localization error distance is defined as Euclidean distance between the estimated target localization results and the true target positions. We consider target localization result of a cluster is a real intersection if localization error distance is less than the localization precision threshold. Otherwise, the target localization result of a cluster is a spurious intersection. We adopt the recall and precision as the performance metrics. Precision is the fraction of relevant instances among the retrieved instances, while recall is the fraction of relevant instances that were retrieved. When clustering returns $N_i$ clusters that we estimate via i th MC simulation, only $R_i$ of which are relevant, while failing to return $T_i$ additional relevant clusters. We can define the recall and precision obtained from M = 10000 MC simulations as follows:

$$\begin{aligned} \text{ recall } =\frac{\sum _{i=1}^M R_i}{\sum _{i=1}^M\left( T_i+R_i\right) }, \text{ precision } =\frac{\sum _{i=1}^M R_i}{\sum _{i=1}^M N_i} \end{aligned}$$

(24)

Four scenarios are carried and discussed to demonstrate the superiority of the proposed algorithm. Scenario 1 discusses dense radiation source; Scenario 2 proves that our algorithm adapts variations in sample density caused by coexistence of far and near radiation sources due to the contribution of Mahalanobis distance criterion. Variation in measurement error of angle is also taken into account in scenario 3. In scenario 4, variation in observation station layout is discussed to prove the robustness and adaptability of the proposed algorithm.

3.1 Scenario 1: dense radiation source

The distribution of radiation sources and observation stations on the two-dimensional plane is shown in Fig. 4. The coordinates of five radiation source are (86,74), (83,77), (80,80), (77,83), (74,86) with the unit km. The coordinates of three observation stations are (15,0), (30,0), (45,0), and assume that each station measure five bearing samples each time.

Fig. 4

Coordinates of observation stations and radiation sources

Figure 5 compares recall and precision performance obtained by K-M-DBSCAN, DBSCAN and FCM, respectively. When the measurement error of angle is 0.2 degrees, the recall and precision of spurious intersections in K-M-DBSCAN can still reach more than 98.5%, while recall and precision of real intersections can reach more than 95%. In contrast, the precision of real intersections in DBSCAN is less than 70%, while FCM is less than 75%. When the measurement error of angle is 0.14 degrees, the proposed algorithm can delete all spurious intersections and identify all real targets. In contrast, the precision in K-M-DBSCAN of real intersections is 0.5% and 15% higher than those of DBSCAN and FCM, respectively. The performance of the K-M-DBSCAN in removal of spurious intersections is significantly better than the other two compared algorithms.

Fig. 5

Comparison of recall and precision versus the measurement error of angle

Figure 6 compares localization precision threshold of different clustering when measurement error of angle is 0.15 degrees. As the localization precision threshold decreases, the performance of all the three algorithms gradually increases. When the localization precision threshold is 0.6 km, the four performance indicators of the K-M-DBSCAN are all close to 100%. Taking precision of real intersections as an example, precision of K-M-DBSCAN is higher than DBSCAN by nearly 10% and higher than FCM by about 30%.

Fig. 6

Comparison of recall and precision versus the localization precision threshold

In order to evaluate effects of the radiation source spacing on the removal of spurious intersections and localization performance, we set the coordinates of five radiation source to $(80+2*d,80-2*d)$, $(80+d,80-d)$, $(80-d,80+d)$ and (80, 80), respectively, with $d=1.6:0.2:3$ (km). Figure 7 plots four performance indicators of the proposed K-M-DBSCAN. When d=2.2 km, both the recall and precision can reach more than 90%; when d=2.6 km, both the recall and precision can reach more than 96%. It is worth mentioning that the distance between the observation station and the radiation source is about 100 km. When d=2.2 km, the proposed algorithm can perform well. We can find that, although the radiation source is relatively dense, the proposed algorithm can still have superior performance.

Fig. 7

Comparison of recall and precision versus radiation source spacing

3.2 Scenario 2: coexistence of far and near radiation sources

Change the position of radiation sources and observing stations, and the coexistence of far and near radiation sources is shown in Fig. 8a. In this scenario, the identification of distant targets is more difficult than nearby targets due to the scattered intersections. The intersections of nearby targets are densely distributed, which makes spurious clusters to be confused with each other. According to the clustering results in Fig. 8b–d, the K-M-DBSCAN can correctly identify all five target clusters including almost all real intersections. Although the DBSCAN algorithm also recognizes five targets, each target cluster only contains part of the real intersections, such as target 4 and target 5. Obviously, the performance of DBSCAN and FCM becomes worse when near and far radiation sources coexist, but the proposed K-M-DBSCAN almost avoids this problem. Because Mahalanobis distance takes into account the probability distribution of real intersections, the clusters generated by K-M-DBSCAN are elliptic and adaptively include almost all true intersections. In contrast, the clusters of DBSCAN are circular based on Euclidean distance and cannot adapt well the density variations caused by coexistence of far and near radiation sources.

Fig. 8

a Distribution of observation stations and radiation sources; b clustering results of K-M-DBSCAN; c clustering results of DBSCAN; d clustering results of FCM

The recall and precision of the three algorithms are given in Fig. 9. When the localization precision threshold is 0.5 km, the recall and precision of spurious intersections in the K-M-DBSCAN can reach more than 99%, while DBSCAN is lower than 97%. The recall and precision of real intersections in the K-M-DBSCAN reach about 97%, while those of DBSCAN can only reach about 87%. The performance of FCM is significantly worse than that of DBSCAN algorithm. Simulations verify that the proposed algorithm has superior performance in an environment where far and near radiation sources coexist. This can be contributed that the constructed Mahalanobis Distance-based neighborhood radius definition adapts variation in intersections density, which is caused by the coexistence of far and near radiation sources.

Fig. 9

Comparison of recall and precision versus the localization precision threshold

3.3 Scenario 3: variation in measurement error of angle

In this subsection, we deploy the radiation source and observation station according to Fig. 4, and the measurement errors of three observation stations are 0.25, 0.10 and 0.10 degrees, respectively. Variation in measurement error of observation stations will result in different intersection densities in different directions. According to the previous statement, Mahalanobis distance is more suitable for dealing with density variations by following the probability distribution of intersections. As shown in Fig. 10, when the localization precision threshold is 0.5 km, the recall and precision of spurious intersections in the K-M-DBSCAN can reach more than 98%, and the recall and precision of real intersections can reach 95%; for FCM, when the localization precision threshold is 0.5 km, the recall and precision of spurious intersections are lower than 95%, and the recall and precision of real intersections are lower than 80%. These prove the superiority of Mahalanobis distance in K-M-DBSCAN, which can better deal with large variations in cluster density. With the increase in localization precision threshold, K-M-DBSCAN curve and DBSCAN curve become close to each other, while FCM curve is different from them. It can be seen that the increase in localization precision threshold can hardly overcome the defect of FCM, whose clustering number needs to be estimated and corrected. Compared with the DBSCAN and the FCM, our algorithm overcomes large variations in cluster density caused by different measurement error of angle.

Fig. 10

Comparison of recall and precision versus the localization precision threshold

3.4 Scenario 4: variation in observation station layout

The Cartesian coordinates of the observation stations in the first three scenarios remain fixed. In order to better verify the robustness and adaptability of our algorithm, three different observation station layouts are given and compared, as shown in Fig. 11. For layout 1, the coordinates of the observation station are (15,0), (30,0) and (45,0); for layout 2, the coordinates of the observation station are (30,15), (30,0) and (45,0); for layout 3, the coordinates of the observation station are (15,15), (30,0) and (45,0), respectively.

Fig. 11

Distribution of observation stations in three ways

The shape of the intersection distribution is different for different layouts, and this is a phenomenon of large variations in cluster density. The K-M-DBSCAN algorithm is still effective in solving this problem as shown in Fig. 12. When the measurement error of angle is 0.15 degrees and the localization precision threshold is 0.5 km for K-M-DBSCAN, the precision of the three station layouts can reach more than 96%, and the difference between them is less than 5%; when the localization precision threshold is 0.4 km, the precision of layout 3 is close to 98%, and the other two layouts also exceed 93%. When the localization precision threshold of DBSCAN is 0.5 km, only the layout 3 has a precision of more than 96%, and the other two algorithms are less than 90%. The difference between layout 1 and layout 2 is close to 10%; when the localization precision threshold is 0.4 km, the precision of layout 3 is about 96%, and the other two layouts are both less than 80%. When localization precision threshold of FCM is 0.5 km, the precision is all less than 80%. These results suggest that the performance of the proposed algorithm is better than the FCM and DBSCAN in the same layout. Moreover, K-M-DBSCAN can better adapt to the variations in the layout of observation stations.

Fig. 12

Change of the precision of real intersections with localization precision threshold

4 Time complexity analysis

In Sect. 3, we compared the localization performance of three algorithms: K-M-DBSCAN, DBSCAN and FCM. The following is a comparison of the calculation time of a single location estimation of the three positioning algorithms.

Let the number of observation stations is NS and the number of targets is NT. The number of bearing lines is $NS*NT$. The total number of intersections is $C_{N S}^2 * N T^2=\frac{N S(N S-1)}{2} N T^2$.

First analyze the minimum K distance algorithm. From Table 1, we can get that the number of cycles in step 4 is $NS-1$. The maximum cycles in step 6 is NT, and the number of cycles in step 11 is $NS*NT$, so the time complexity of the algorithm is about $\textrm{O}[(N S-1) * N T * N S * N T]=\textrm{O}\left[ N S(N S-1) N T^2\right]$.

Preprocess retains K pairs of intersections for each bearing. Therefore, there are at most $K*NS*NT$ intersections in the input of M-DBSCAN algorithm. From Table 2, we can get the number of cycles in step 7 is at least ${\text {Min}}_{p t s}$ (Minimum neighbor points of the cluster). And the number of cycles in step 15 is $K*NS*NT$. The time complexity of the M-DBSCAN is about $\textrm{O}\left( K^* N S^* N T^* {\text {Min}}_{p t s}\right)$.

The input of DBSCAN has $\frac{N S(N S-1)}{2} N T^2$ intersections. And the time complexity of M-DBSCAN and DBSCAN can be regarded as the same. So the time complexity of the DBSCAN is about $\textrm{O}\left( \frac{N S(N S-1)}{2} N T^2 {\text {Min}}_{p t s}\right)$.

The input of FCM has $\frac{N S(N S-1)}{2} N T^2$ intersections. The clusters number of FCM is NT, the minimum number is ${\text {Min}}_{p t s}$, the maximum number of iterations is NI [39]. The time complexity of the FCM is about $\textrm{O}\left( \frac{N S(N S-1)}{2} N T^3 NI\right)$.

The summary of algorithm time complexity is in Table 3.

Table 3 The time complexity of algorithm

In Sect. 3, NS=3, NT=5, repeat measurement 50 times for each bearing line. Set K=3 in the minimum K distance algorithm. ${\text {Min}}_{p t s}$ can be calculated by the Mahalanobis distance criterion in the M-DBSCAN algorithm. Set ${\text {Min}}_{p t s}$=125 in the DBSCAN algorithm. Set the number of iterations NI to 15 under the premise of ensuring that Jm can converge in the FCM of algorithm. The simulation computer configuration is as follows: Intel Core i7-12700F 2.1 GHz, 48.0 GB of DDR5 RAM, Windows 10 OS. Average time required for each position estimation is shown in Table 4.

Table 4 Average time required for each position estimation

Preprocessing by the minimum K distance algorithm reduces the number of intersections, so the consumption time of the M-DBSCAN algorithm is less than that of DBSCAN.

5 Conclusion

To handle the spurious intersections problem in BOL, this paper proposes a novel clustering method called K-M-DBSCAN. Compared with existing FCM and DBSCAN clustering methods, the proposed K-M-DBSCAN performs much better for removal of spurious intersections. The proposed method can not only adapt to the distribution of dense radiation source and the coexistence of far and near radiation sources, but also solve the problem of large variations in cluster density caused by different measurement error of angle. There are two reasons for this improvement. The first one is we construct Mahalanobis Distance-based density clustering, which follows the spatial probability distribution of intersections and enables to adapt the variations in cluster sample density. The second reason is that the deployed minimum K distance algorithm for preprocessing to filter out some spurious intersections, which reduce the interference and calculation complexity to clustering to a certain extent. Simulation results show that K-M-DBSCAN can effectively remove spurious intersections and still has a high recognition for the target when the localization precision threshold is 0.5 km. The precision of spurious intersections can reach more than 98%, and the precision of real intersections can reach more than 95%. Simulation results also verify the robustness of K-M-DBSCAN with different layout of observation stations.