An underwater multi-target tracking trajectory smoothing method based on KD-tree

Abstract

The present invention discloses an underwater multi-target tracking trajectory smoothing method based on KD tree, belonging to the technical field of hydrodynamic experiments. During the forward filtering of MeMBer, the present invention no longer calculates the state transition probability of each target. Instead, it uses the KD tree method to classify according to different states, and improves the original algorithm by using the fast N-body Monte Carlo method to solve the problem of the current labeled multi-Bernoulli smoothing algorithm having too high computational complexity and too long computational time. By using the underwater multi-target tracking trajectory smoothing algorithm based on KD tree described in the present invention, the formed trajectory is clear, complete, continuous, and the computational complexity is greatly reduced. It can be applied to many fields such as passive sonar target tracking, submarine detection, and guided missile detection. The following gives the detailed modeling and tracking process.

Description

Technical Field

[0001] The present invention relates to a method for smoothing the underwater multi-target tracking trajectory based on a KD tree, belonging to the technical field of hydrodynamic experiments. Background Art

[0002] Various intractable interferences existing in the underwater environment have always been the key research topics. For example, in shallow water areas, the impacts brought by wind noise, ship noise, industrial noise, etc., and in deep water areas, the impacts of water flow changes, rugged underwater terrain, and activities of various large aquatic organisms, etc., all lead to the irregularity and strong interference of the actual underwater space physical characteristics. How to obtain high-precision underwater target tracking has always been the focus and difficulty of technological breakthroughs in this field.

[0003] Track-before-detect (TBD) is a powerful method for detecting "low, slow, and small" targets by modern radar systems. This method does not set a threshold, can fully exploit multi-frame raw radar scan data, improve the signal-to-noise ratio through energy accumulation, establish track tracking for moving targets based on multi-frame information, and then detect the presence of targets. Due to the theoretical advantage of multi-target tracking based on the random finite set (RFS) framework in avoiding data association, many scholars have regarded the TBD method based on RFS theory as a research hotspot in the field of track-before-detect.

[0004] Multi-target multi-Bernoulli (MeMBer) filtering is another RFS-based multi-target tracking method proposed by Mahler after probability hypothesis density (PHD) and cardinality probability hypothesis density (CPHD) filtering. For multi-target non-linear filtering, Vo et al. proposed MeMBer filtering. Compared with the above two algorithms, this algorithm has more advantages in terms of filtering accuracy and computational complexity. Vo et al. also improved the problem of over-estimation of the number of targets caused by inaccurate approximation of the probability generation functional in the update process of MeMBer filtering, that is, cardinality balanced multi-target multi-Bernoulli (CBMeMBer) filtering.

[0005] Although the smoothing operation consumes more time, it improves the estimation accuracy compared with filtering. At present, there has been theoretical verification of applying the smoothing idea to pre-detection tracking. Vo et al. used Bernoulli RFS forward and backward smoothing to model and complete the tracking of a single target. While improving the accuracy of Bernoulli filtering for target number changes and target state estimation, Bernoulli forward and backward smoothing also verified the feasibility of Bernoulli filtering forward and backward smoothing. Sun Jie et al. proposed a smoothing method based on multi-Bernoulli, which improved the tracking performance of multi-Bernoulli RFS in multi-target scenarios. Wong et al. completed smooth pre-detection tracking of road motor vehicles under high clutter background, thus demonstrating the feasibility and performance advantages of forward and backward smoothing TBD.

[0006] In order to solve the problem of high computational complexity and long computational time of multi-Bernoulli smoothing algorithm, this paper proposes an improved fast smoothing method based on KD tree method. In the process of MeMBer forward filtering, this algorithm no longer calculates the state transition probability of each target, but uses KD tree method to classify according to different states, and improves the original algorithm by fast N-body Monte Carlo method. The computational complexity of the algorithm is greatly reduced from O(N) of traditional algorithm to 2 ) is reduced to O9NlogN) and the smoothing effect is better. Summary of the invention

[0007] The present invention proposes a KD-tree-based underwater multi-target tracking track smoothing method. In the process of MeMBer forward filtering, the state transition probability of each target is no longer calculated. Instead, the KD tree method is used to classify the targets according to their states. The fast N-body Monte Carlo method is used to improve the original algorithm to solve the problems of high computational complexity and long computational time of the current label multi-Bernoulli smoothing algorithm.

[0008] A method for underwater multi-target tracking track smoothing based on a KD tree, the underwater multi-target tracking method based on an underwater multi-target tracking track smoothing algorithm of a KD tree comprises the following steps:

[0009] S100, forward filtering prediction obtains a multi-Bernoulli parameter set in LMB form at time k;

[0010] S200, after prediction, converting the multi-Bernoulli parameter set in LMB form at time k into a δ-GLMB form, that is, obtaining a δ-GLMB prediction parameter set at time k;

[0011] S300, calculating a δ-GLMB form update parameter set according to the δ-GLMB prediction parameter set at time k and converting the updated result back to the LMB form;

[0012] S400, backward smoothing, set the smoothing step size to t l> 0, when the current time k is greater than t l Start smoothing when

[0013] S500. Set the smoothing threshold r sm and perform smoothing on the target label and there is a probability for the target, and keep the states of the remaining targets unchanged;

[0014] S600. Use the KD - tree to divide the particle set at time k - 1 into different state spaces;

[0015] S700. Calculate the average state - transfer function value as the state - transfer function of all particles in the state space;

[0016] S800. Substitute the improved state - transfer function value obtained by the KD - tree algorithm into the label multi - Bernoulli smoothing formula in S500 to replace and calculate the smoothed probability density distribution p;

[0017] S900. Substitute the smoothed probability density distribution p into the k - time LMB - form multi - Bernoulli parameter set in S200, draw the final multi - target tracking trajectory represented by the label multi - Bernoulli method, and obtain the smoothed tracking trajectory.

[0018] Furthermore, in S100, specifically,

[0019] At time k, the multi - Bernoulli parameter set is where r and p represent the existence probability and probability density of the multi - target respectively, k represents the time, l is the target label, and each state distribution is composed of a series of weighted particles i represents the i - th target at this time. The predicted multi - Bernoulli random parameter set of the target at time k + 1 can be expressed by the following formula,

[0020]

[0021] This formula is divided into two parts, where is the parameter set of the surviving targets, is the parameter set of the new - born targets. The above formulas are unified and written in one formula as

[0022] Furthermore, in S200, specifically,

[0023] For δ - GLMB filtering, the form of the transmitted parameters is no longer the existence probability and probability distribution but the hypothesis weight and the probability distribution corresponding to the hypothesis h is the same as i above, representing the h - th target, and different algebraic expressions are used for distinction,

[0024] According to Bayes' theorem, the recursive form of δ-GLMB for pre-detection tracking is obtained. At time k, the distribution of the δ-GLMB of the target follows the distribution of the following formula. Define the mapping which represents the mapping from the label state space to the label space. For a target with state X (1) =(x (1) , l (1) ), its label is

[0025]

[0026] where Ξ is a discrete space, which is used to represent the association history of measurement update in tracking. ξ ∈ Ξ represents the association history at a certain moment, including the target label history of birth and death. represents the space composed of finite subsets of . I represents the set of tracking labels. The δ-generalized label multi-Bernoulli distribution is composed of multiple hypotheses. The weight w (I,ξ) represents the probability size of the corresponding hypothesis. The sum of the weights of all hypotheses is 1. And p (ξ) represents the single-target state distribution in the corresponding hypothesis. H represents the number of hypotheses. δ Y (X) is the Kronecker δ function.

[0027] Then the predicted target probability distribution at time k + 1 is shown in the following formula.

[0028]

[0029] Furthermore, in S300, specifically,

[0030] The posterior target probability distribution at time k + 1 is shown in the following formula. According to this formula for update, the updated result is converted back to the LMB form.

[0031]

[0032] The existence probability and probability distribution of the target are obtained from the following two formulas.

[0033]

[0034] where l Y (X) is the inclusion function, as shown in the following formula.

[0035]

[0036] Furthermore, in S400, specifically,

[0037] Backward smoothing, set the smoothing step size to t l > 0. When the current time k is greater than tl Start smoothing at

[0038] The set of posterior density parameters at known time k is The set of predicted density parameters at time k+1 is The posterior density parameter is Use the following two equations to calculate the smoothing parameter set at time k

[0039]

[0040] where α and β represent the target extinction index and survival index respectively, and the calculation methods are as follows, f k+1|k (Y|X) is the target transition probability. It can be found from the formula that backward smoothing is performed on the targets at previous times, and the newly born targets after that do not participate in smoothing.

[0041]

[0042] Furthermore, in S500, specifically,

[0043] Set the smoothing threshold r sm , for the target label and the existence probability of the target, perform smoothing, and keep the states of the remaining targets unchanged.

[0044] For any target that meets the conditions, assume that the target existence probability is independent of the target state and is p s , at time k, the predicted density and the posterior density only have different weights for the weight particles representing the probability distribution function, and the states represented by the particles remain unchanged. First, calculate the target existence probability and parameters α and β, and then calculate the n k particles in respectively to k+1|k the n k+1|k transfer function f of the particles in If the number of targets does not change before and after filtering, then probabilities will be obtained, and finally calculate the probability density distribution after smoothing

[0045]

[0046]

[0047] Furthermore, in S600, specifically,

[0048] For the state transition function in S500 During the calculation, it is regarded as a Gaussian distribution for calculation. For multi-object tracking, it is overcome by particle filtering, and the posterior is approximated by a set of weighted samples ("particles"). The probability distribution of the multi-Bernoulli algorithm based on this algorithm contains a series of particles with weights. Therefore, when smoothing it, the state transition probability of each particle must be calculated, which is related to the state difference between time k-1 and time k. So it is regarded as a function related to the state distance. Thus, the KD tree can be used to divide the particle set at time k-1 into different state spaces.

[0049] Furthermore, in S700, specifically,

[0050] For the particles in the same state space, take the distance d to the target state that is the closest in the state space min and the distance d to the target state that is the farthest. max Calculate the average state transition function value as the state transition function of all particles in the state space. The calculation formula is as follows:

[0051]

[0052] In practical applications, the distance can be used as the basis for dividing the state space. Different grouping intervals are specified for different distances. For the state space with a distance exceeding the set threshold, its state transition function is directly replaced by the minimum value.

[0053] Furthermore, in S800, specifically,

[0054] Substitute the improved state transition function value obtained by the KD tree algorithm into the S500 label multi-Bernoulli smoothing formula to replace and calculate the smoothed probability density distribution p.

[0055] Furthermore, in S900, specifically,

[0056] It is known from S200 that the k-time LMB-form multi-Bernoulli parameter set consists of represented. S800 has finally obtained the smoothed probability density distribution p, and the method for calculating the target existence probability r remains unchanged. The k-time LMB-form multi-Bernoulli parameter set consists of Draw the final multi-object tracking trajectory represented by the label multi-Bernoulli method to obtain the smoothed tracking trajectory.

[0057] Advantages of the present invention: A method for underwater multi-target tracking trajectory smoothing based on KD tree proposed by the present invention classifies and arranges targets according to their states using the KD tree method, and performs simulation analysis using the fast N-body Monte Carlo algorithm. It effectively improves the drawback of the traditional forward-backward smoothing algorithm that the state transition probability of each particle must be calculated due to calculating the particle weights, improves the accuracy of the tracking trajectory, reduces the computational amount, and reduces the actual running time. This method first converts the predicted multi-Bernoulli parameter set in LMB form to δ-GLMB form after the prediction step in the forward filtering stage, updates it in δ-GLMB form and then converts it back to LMB form, retaining the intuitive mathematical results and preventing the exponential growth of the components of the generalized labeled multi-Bernoulli filter. In the backward smoothing step, to avoid unnecessary calculations, a smoothing threshold r sm is set to smooth the targets with target labels and existence probability , and the states of the remaining targets remain unchanged. Then, the state transition function in the labeled multi-Bernoulli smoothing formula is improved by the KD tree algorithm, and the KD tree is used to divide the particle set at time k-1 into different state spaces, as shown in Figure 3 . Next, the distance d min from the state space to the target state that is the closest and the distance d max from the state space to the target state that is the farthest are taken to calculate the average state transition function value as the state transition function of all particles in the state space, as shown in Figure 4 . Finally, the improved state transition function value obtained by the KD tree algorithm is substituted into the labeled multi-Bernoulli smoothing formula to obtain the final smoothing result. Through simulation experiments, this method proves that the research algorithm has a smoothing effect on the filtering result, and the fast smoothing algorithm realizes smoothing within the allowable accuracy range and greatly shortens the calculation time. The smoothed trajectory also significantly reduces the deviation degree of the target particles from the trajectory, proving the effectiveness of the research algorithm. Using the underwater multi-target tracking trajectory smoothing algorithm based on KD tree described in the present invention, the formed trajectory is clear, complete, continuous, and the computational complexity is greatly reduced. It can be applied to many fields such as passive sonar target tracking, submarine detection, and guided missile detection. The following gives the detailed modeling and tracking process. Brief Description of the Drawings

[0058] Figure 1 is a schematic diagram of the forward filtering backward smoothing algorithm process.

[0059] Figure 2 is a flowchart of the smoothing algorithm based on labeled multi-Bernoulli of the present invention.

[0060] Figure 3 is a schematic diagram of the KD tree dividing the particle state space of the present invention.

[0061] Figure 4It is a schematic diagram of calculating the average state transition function in the present invention.

[0062] Figure 5 It is a simulation result diagram of the fast smoothing algorithm in the present invention. Specific implementation manners

[0063] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without making creative efforts shall fall within the protection scope of the present invention.

[0064] The present invention proposes an underwater multi-target tracking trajectory smoothing method based on a KD tree. The underwater multi-target tracking method based on the underwater multi-target tracking trajectory smoothing algorithm of the KD tree includes the following steps:

[0065] S100. Forward filtering prediction to obtain the predicted multi-Bernoulli parameter set in the LMB form at time k;

[0066] S200. After prediction, convert the multi-Bernoulli parameter set in the LMB form at time k into the δ-GLMB form, that is, obtain the predicted parameter set of δ-GLMB at time k;

[0067] S300. Calculate the updated parameter set in the δ-GLMB form according to the predicted parameter set of δ-GLMB at time k and convert the updated result back to the LMB form;

[0068] S400. Backward smoothing, set the smoothing step size as t l >0, start smoothing when the current time k is greater than t l ;

[0069] S500. Set the smoothing threshold r sm , and smooth the target with the target label and the existence probability , and keep the states of the remaining targets unchanged;

[0070] S600. Use the KD tree to divide the particle set at time k-1 into different state spaces;

[0071] S700. Calculate the average state transition function value as the state transition function of all particles in the state space;

[0072] S800. Substitute the improved state transition function value obtained by the KD tree algorithm into the label multi-Bernoulli smoothing formula in S500 to replace to calculate the smoothed probability density distribution p;

[0073] S900. Substitute the smoothed probability density distribution p into the multi-Bernoulli parameter set in the LMB form at time k in S200, plot the final multi-target tracking trajectory represented by the labeled multi-Bernoulli method, and obtain the smoothed tracking trajectory.

[0074] Furthermore, in S100, specifically,

[0075] At time k, the multi-Bernoulli parameter set is where r and p represent the existence probability and probability density of multi-targets respectively, k represents the time, l is the target label, and each state distribution is composed of a series of weighted particles i is the i-th target at this time. The predicted multi-Bernoulli random parameter set of the target at time k + 1 can be expressed by the following formula

[0076]

[0077] This formula is divided into two parts, where is the parameter set of surviving targets, is the parameter set of newly born targets. The above formulas are uniformly written in one formula as

[0078] Furthermore, in S200, specifically,

[0079] For δ-GLMB filtering, the transmitted parameter form is no longer the existence probability and probability distribution but the assumed weight and the probability distribution corresponding to the assumption h is the same as i above, representing the h-th target, and different algebraic expressions are used for distinction.

[0080] According to Bayes' theorem, the recursive form of δ-GLMB for tracking before detection is obtained. When at time k, the distribution of δ-GLMB of the target follows the distribution of the following formula. Define the mapping representing the mapping from the label state space to the label space. For the target with state X (1) =(x (1) , l (1) ), its label is

[0081]

[0082] where Ξ is the discrete space, which is used to represent the association history of measurement update in tracking. ξ ∈ Ξ represents the association history at a certain time, including the target label history of birth and death. represents the space composed of finite subsets of I represents the set of tracking labels. The δ-generalized labeled multi-Bernoulli distribution is composed of multiple assumptions, and the weight w(I,ξ) represents the probability magnitude corresponding to the hypothesis. The sum of all hypothesis weights is 1, and p (ξ) represents the single-target state distribution in the corresponding hypothesis. H represents the number of hypotheses, and δ Y (X) is the Kronecker delta function,

[0083] Then the predicted target probability distribution at time k + 1 is shown as follows,

[0084]

[0085] Furthermore, in S300, specifically,

[0086] The posterior target probability distribution at time k + 1 is shown as follows. Update according to this formula and convert the updated result back to the LMB form,

[0087]

[0088] The existence probability and probability distribution of the target are obtained from the following two formulas,

[0089]

[0090] where l Y (X) is the inclusion function, as shown in the following formula,

[0091]

[0092] Furthermore, in S400, specifically,

[0093] Backward smoothing, set the smoothing step size to t l > 0. When the current time k is greater than t l start smoothing,

[0094] Given that the posterior density parameter set at time k is The predicted density parameter set at time k + 1 is The posterior density parameter is Calculate the smoothing parameter set at time k using the following two formulas

[0095]

[0096]

[0097] where α and β represent the target death index and survival index respectively, and the calculation methods are as follows, f k+1|k (Y|X) is the target transition probability. It can be found from the formula that backward smoothing is performed on the targets at previous times, and the newly born targets after that do not participate in the smoothing,

[0098]

[0099] Further, in S500, specifically,

[0100] Set a smoothing threshold r sm , and smooth the target labels and there is a probability for the targets, while keeping the states of the remaining targets unchanged.

[0101] Since the probability density distribution consists of a series of weighted particles, calculating the transition probability of each particle is required during smoothing, which consumes a large amount of time. When there are many targets, the program runs extremely slowly. Therefore, to avoid unnecessary calculations, set the smoothing threshold r sm , and smooth the target labels and there is a probability for the targets, while keeping the states of the remaining targets unchanged.

[0102] For any target that meets the conditions, assume that the target existence probability is p and is independent of the target state s . At time k, the predicted density and the posterior density only differ in the weights of the weighted particles representing the probability distribution function, and the states represented by the particles remain unchanged. First, calculate the target existence probability and the parameters α and β, and then calculate the n k particles in respectively to the k+1|k n particles in k+1|k transition function f . If the number of targets does not change before and after filtering, then probabilities will be obtained. Finally, calculate the smoothed probability density distribution

[0103]

[0104]

[0105] Further, in S600, specifically,

[0106] For the state transition function in S500 regard it as a Gaussian distribution for calculation during calculation. For multi-target tracking, overcome it through particle filtering, and approximate the posterior by a set of weighted samples ("particles"). The probability distribution of the multi-Bernoulli algorithm based on this algorithm contains a series of particles with weights. Therefore, when smoothing it, the state transition probability of each particle must be calculated, which is related to the state difference between time k - 1 and time k. So regard it as a function related to the state distance Thus, the KD tree can be used to divide the particle set at time k - 1 into different state spaces.

[0107] Further, in S700, specifically,

[0108] For the particles in the same state space, take the distance d to the target state that is the closest in the state space min and the distance d to the target state that is the farthest max Calculate the average state transition function value as the state transition function of all particles in the state space. The calculation formula is as follows:

[0109]

[0110] In practical applications, the distance can be used as the basis for dividing the state space. Different grouping intervals are specified for different distances. For the state space with a distance exceeding the set threshold, its state transition function is directly replaced with the minimum value.

[0111] Further, in S800, specifically,

[0112] Substitute the improved state transition function value obtained by the KD tree algorithm into the S500 label multi-Bernoulli smoothing formula to replace and calculate the smoothed probability density distribution p.

[0113] Further, in S900, specifically,

[0114] As known from S200, the k - moment LMB - form multi - Bernoulli parameter set is composed of represented. S800 has finally obtained the smoothed probability density distribution p, and the method for calculating the target existence probability r remains unchanged. The k - moment LMB - form multi - Bernoulli parameter set is composed of Draw the final multi - target tracking trajectory represented by the label multi - Bernoulli method to obtain the smoothed tracking trajectory.

[0115] Next, smooth the low - SNR tracking trajectory. The target signal is a narrow - band signal with a frequency of 1450 Hz. The input signal - to - noise ratio of the target signal is constantly - 16 dB, and the sampling frequency is 20 kHz. The standard deviation in the target motion equation is set to 1.5° / s 2 , the target survival probability p s is set to 0.99, the newborn target survival probability r Γ is set to 0.05, the number of newborn targets per period is 10, the newborn density follows a Gaussian distribution, the mean is uniformly distributed in ten intervals evenly divided in (0°, 360°], and the variance is diag([8, 1]) 2 , the smoothing step size is set to 2, and the smoothing threshold r sm is 0.1.

[0116] From Figure 2As can be seen, the present invention is based on the cross-track tracking filtering results of the labeled multi-Bernoulli detection-before-tracking algorithm. The red straight line represents the true target azimuth trajectory, and the points of different colors above represent the tracking results. The algorithm based on the present invention can continuously and effectively track the target even at low signal-to-noise ratios, and the trajectory is clear and the number of targets is definite.

[0117] An underwater multi-target tracking trajectory smoothing algorithm based on KD-tree proposed by the present invention reduces the computational complexity while achieving a good smoothing effect. The computational complexity is reduced from O(N 2 ) of the traditional algorithm to O(NlogN). The spectral estimation of the target is based on the MUSIC algorithm and implemented by particles. The position where the target most likely exists is obtained by searching for peaks and controlling thresholds. Comparing the actual tracking result trajectory with the theoretical actual result trajectory can verify the performance of the tracking algorithm of the present invention, and then form a sustainable, accurate and complete multi-target tracking trajectory. The present invention reasonably utilizes the probability of the targets detected at each moment, estimates the position and number of the targets with high precision, reduces the number of target tracking losses and interruptions, and improves the overall integrity of the target trajectory, so as to effectively detect and track underwater multi-targets.

Claims

1. An underwater multi-target tracking trajectory smoothing method based on KD-tree, characterized in that, The underwater multi-target tracking method of the underwater multi-target tracking trajectory smoothing algorithm based on KD tree includes the following steps: S100. Forward filtering prediction to obtain the predicted multi-Bernoulli parameter set in LMB form at time k; S200. After prediction, convert the multi-Bernoulli parameter set in LMB form at time k to δ-GLMB form, that is, obtain the predicted parameter set of δ-GLMB at time k; S300. Calculate the updated parameter set in δ-GLMB form according to the predicted parameter set of δ-GLMB at time k and convert the updated result back to LMB form; S400, backward smoothing, set the smoothing step size to t l > 0, when the current time k is greater than t l start smoothing S500. Set the smoothing threshold r sm , for the target label and there is a probability smooth the target, and keep the states of the remaining targets unchanged; S600. Use the KD tree to divide the particle set at time k-1 into different state spaces; S700. Calculate the average state transition function value as the state transition function of all particles in the state space; S800. Substitute the improved state transition function value obtained by the KD tree algorithm into the label multi-Bernoulli smoothing formula in S500 for replacement to calculate the smoothed probability density distribution p; S900. Substitute the smoothed probability density distribution p into the multi-Bernoulli parameter set in LMB form at time k in S200, draw the final multi-target tracking trajectory represented by the labeled multi-Bernoulli method, and obtain the smoothed tracking trajectory.

2. The underwater multi-target tracking trajectory smoothing method based on KD-tree according to claim 1, characterized in that, Specifically, in S100 At time k, the multi-Bernoulli parameter set is where r and p represent the existence probability and probability density of multiple targets respectively, k represents the time, l is the target label, and each state distribution is composed of a series of weighted particles constitute, i is the i-th target at this time, and the predicted multi-Bernoulli random parameter set of the target at time k + 1 can be expressed by the following formula, This formula is divided into two parts, where is the set of survival target parameters, is the set of new-born target parameters, and the above formula is uniformly written in one formula as 3. The underwater multi-target tracking trajectory smoothing method based on KD-tree according to claim 2, characterized in that, Specifically, in S200 For the δ-GLMB filter, the form of the transmitted parameters is no longer the existence probability and the probability distribution but rather the hypothesized weights and the probability distributions corresponding to the hypotheses h is the same as i above, representing the h-th target, and different algebraic expressions are used for distinction According to Bayes' theorem, the recursive form of δ-GLMB for pre-detection tracking is obtained. At time k, the distribution of the δ-GLMB of the target follows the distribution of the following formula. Define the mapping which represents the mapping from the label state space to the label space. For a target with state X (1) =(x (1) ,l (1) ), its label is where Ξ is a discrete space, which is used to represent the association history of measurement updates in tracking. ξ ∈ Ξ represents the association history at a certain moment, including the target label history of birth and death. denotes the space constituted by the finite subsets of, I represents the set of tracking labels, and the δ-generalized labeled multi-Bernoulli distribution consists of multiple hypotheses, with the weight w (I,ξ) representing the probability magnitude of the corresponding hypothesis. The sum of the weights of all hypotheses is 1, while p (ξ) represents the single-target state distribution in the corresponding hypothesis, H represents the number of hypotheses, and δ Y (X) is the Kronecker delta function. Then the predicted target probability distribution at time k+1 is shown in the following formula 4. The underwater multi-target tracking trajectory smoothing method based on KD-tree according to claim 3, characterized in that, Specifically, in S300 The posterior target probability distribution at time k+1 is shown in the following formula. Update according to this formula and convert the updated result back to LMB form The existence probability and probability distribution of the target are obtained from the following two formulas where l Y (X) is an inclusion function, as shown in the following formula, 5. The underwater multi-target tracking trajectory smoothing method based on KD-tree according to claim 4, characterized in that, Specifically, in S400 Backward smoothing, setting the smoothing step size to t l > 0, when the current time k is greater than t l start smoothing when The posterior density parameter set after time k is The predicted density parameter set at time k+1 is The posterior density parameter is Calculate the smoothing parameter set at time k using the following two equations where α and β represent the extinction index and survival index of the target respectively, and the calculation methods are as follows, f k+1|k (Y|X) is the target transition probability. It can be found from the formula that backward smoothing is performed on the targets at previous times, and the newly born targets afterwards do not participate in the smoothing.

6. The underwater multi-target tracking trajectory smoothing method based on KD tree according to claim 5, wherein, Specifically, in S500 Set the smoothing threshold r sm , for the target label and there is a probability smooth the target, and keep the status of the remaining targets unchanged For any target that meets the conditions, assume that the target existence probability is \(p\) and is independent of the target state s , at time \(k\), the predicted density and the posterior density only differ in the weights of the particles representing the probability distribution function, and the states represented by the particles remain unchanged. First, calculate the target existence probability and the parameters \(\alpha\) and \(\beta\), and then calculate the \(n\) in k each of the \(n\) particles to the \(n\) in k+1|k transfer function \(f\) of each of the \(n\) particles k+1|k . If the number of targets does not change before and after filtering, then a number of probabilities will be obtained, and finally the smoothed probability density distribution The formula is as follows 7. The underwater multi-target tracking trajectory smoothing method based on KD tree according to claim 6, wherein, Specifically, in S600 For the state transition function in S500 During calculation, it is regarded as a Gaussian distribution for calculation. For multi-target tracking, it is overcome through particle filtering, approaching the posterior through a set of weighted samples ("particles"). The probability distribution of the multi-Bernoulli algorithm based on this algorithm contains a series of particles with weights. Therefore, when smoothing it, the state transition probability of each particle must be calculated, which is related to the state difference between time k-1 and time k. So it is regarded as a function related to the state distance Thus, the KD tree can be used to divide the particle set at time k-1 into different state spaces.

8. The underwater multi-target tracking trajectory smoothing method based on KD tree according to claim 7, wherein, Specifically, in S700 For particles in the same state space, take the distance d that is the closest to the target state in the state space min and the distance d that is the farthest from the target state max Calculate the average state transition function value as the state transition function of all particles in the state space. The calculation formula is as follows In practical applications, the distance can be used as the basis for dividing the state space, different grouping intervals are specified for different distances, and for the state space with a distance exceeding the set threshold, its state transition function is directly replaced with the minimum value.

9. The underwater multi-target tracking trajectory smoothing method based on KD tree according to claim 8, wherein, Specifically, in S800 Substitute the improved state transition function values obtained by the KD-tree algorithm into the S500 label multi-Bernoulli smoothing formula to replace calculate the smoothed probability density distribution p.

10. The underwater multi-target tracking trajectory smoothing method based on KD tree according to claim 9, wherein, Specifically, in S900 As known from S200, the LMB-form multi-Bernoulli parameter set at the k-th moment is composed of It is shown that S800 has finally obtained the smoothed probability density distribution p, and the method for calculating the target existence probability r remains unchanged. The LMB-form multi-Bernoulli parameter set at the k-th moment is composed of Draw the final multi-target tracking trajectory represented by the labeled multi-Bernoulli method to obtain the smoothed tracking trajectory.

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