CN104794735A

CN104794735A - Extended target tracking method based on variational Bayesian expectation maximization

Info

Publication number: CN104794735A
Application number: CN201510152626.9A
Authority: CN
Inventors: 李翠芸; 王晋斌; 姬红兵; 王荣
Original assignee: Xidian University
Current assignee: Xidian University
Priority date: 2015-04-02
Filing date: 2015-04-02
Publication date: 2015-07-22
Anticipated expiration: 2035-04-02
Also published as: CN104794735B

Abstract

The invention discloses an extended target tracking method based on variational Bayesian expectation maximization (VBEM) and mainly solves the problem that tracking performance of a target is weakened dramatically under the condition that measurement noise covariance is unknown in the conventional extended target tracking field. The extended target tracking method includes firstly predicting relevant parameters of Gaussian inverse gamma components in joint probability hypothesis density of a target state and the measurement noise covariance; updating the parameters of the Gaussian inverse gamma components; finally acquiring the extended target state and the number by construction and combination. It is proved by simulation experiment that multiple extended targets can be well tracked under the unknown number and the unknown measurement noise covariance, and the extended target tracking method is high in tracking accuracy and can be used for tracking aircrafts and submarine targets.

Description

Extended target tracking method based on variational Bayesian expectation maximization

Technical Field

The invention belongs to the technical field of information processing, and particularly relates to a target tracking method which can be used for tracking multiple extended targets.

Background

In the conventional target tracking field, due to the limited resolution of radar, targets are usually regarded as point targets, i.e. each target can only produce one measurement at a time. In recent years, with the development of radar detection technology and the requirement of practical application, targets are more regarded as extended targets, that is, each target can generate a plurality of measurements at each moment.

The actual purposeIn the target tracking scene, the number of targets cannot be predicted in advance, so that the requirement of the target tracking theory is greatly met by the random set theory. Among a plurality of model assumptions about targets, the proposal of the extended target theory is more close to the requirement of the current tracking theory, and is widely applied in real life, and becomes a research hotspot in the field of target tracking in recent decades. In 2003, Mahler applies a random finite set theory to the multi-target tracking problem and provides Probability Hypothesis Density (PHD) filtering. In 2005, Gilholm and Salmond proposed an extended object model whose spatial distribution obeyed the poisson distribution. In 2009, Mahler derived extended target PHD filtering, that is, a target random set is predicted and updated by using a measured random set at each moment, so that the motion state of a target and the number of estimated targets can be accurately extracted. In the year 2010, the operation of the mobile phone is carried out,etc. give a gaussian mixture implementation of extended target PHD. In 2011, Orguner ET al also proposed potential distribution extended target PHD (ET-CPHD) filtering, and well solved the defects when ET-PHD estimates the number of targets. However, the conventional extended target tracking algorithm deals with the situation that the measured noise covariance is known, and in practice, when the measured noise covariance is unknown, the tracking performance of the extended target will be drastically reduced.

Disclosure of Invention

The invention aims to provide an extended target tracking method based on variational Bayesian expectation maximization to improve the tracking performance under the condition that the covariance of the measured noise is unknown.

The key technology for realizing the invention is as follows: under the framework of potential probability hypothesis density filtering, a Variational Bayesian Expectation Maximization (VBEM) technology is introduced, the joint probability hypothesis density of the target state and the measurement noise covariance is estimated, and the target tracking problem under the unknown measurement noise covariance is achieved.

VB is a method for approximating and calculating complex integrals in the field of Bayes estimation and machine learning, wherein the VB is used for approximating the joint probability hypothesis density of an extended target state and a measured noise covariance in a linear Gaussian system, and the main idea of the algorithm is to carry out parametric approximation on the joint probability hypothesis density of the extended target state and the measured noise covariance and give a parametric expression form of the joint probability hypothesis density. In the process of approximating the joint probability hypothesis density of the extended target state and the measurement noise covariance by using a VB theory, in order to judge the performance of the estimated Gaussian inverse gamma component related parameters, an expectation maximization EM algorithm is introduced on the basis of the VB. In the expectation step E, estimating an expectation value of the unknown parameter and giving a current parameter estimation; at the step of maximizing M, the distribution parameters are re-estimated to maximize the likelihood function. The parameter estimation values obtained in the M steps are used for the next E step calculation, and the process is continuously and alternately carried out. The method can be widely applied to the condition that data are defective, and has the advantages of simplicity, stability and the like.

The technical steps of the invention for carrying out extended target tracking by utilizing the VBEM technology comprise the following steps:

(1) when the time k is 0, initializing the joint probability hypothesis density of the extended target state and the measurement noise covariance as v₀(x,R)；

(2) When k is more than or equal to 1, the joint probability hypothesis density v of the extended target state and the measured noise covariance_k-1|k-1(x, R) and potential distribution P for calculating the number of extended targets_k-1|k-1(num) predicting to obtain predicted extended target joint probability hypothesis density v_k|k-1(x, R) and the predicted potential distribution P_k|k-1(num)；

(3) Joint probability hypothesis density v for predicted extended target states and metrology noise covariance_k|k-1(x, R) and potential distribution P for calculating the number of extended targets_k|k-1(num) update:

(3a) joint probability hypothesis density v using variational Bayes VB method_k|k-1(x, R) to obtain a probability hypothesis density Q of the extended target state expressed in the form of a sum of Gaussian distributions_x,k|k-1(x) And a probability hypothesis density Q of the covariance of the metrology noise expressed in the form of a summation of inverse gamma distributions_R,k|k-1(R)；

(3b) Probability hypothesis density Q of extended target states by utilizing Variational Bayesian Expectation Maximization (VBEM) method_x,k|k-1(x) Gaussian component in (1) and probability hypothesis density of the covariance of the measurement noise Q_R,k|k-1Iteratively updating the inverse gamma component in the (R) to obtain a Gaussian component representing the motion state x of the extended target and an inverse gamma component representing the covariance R of the measured noise;

(3c) for the potential distribution P obtained by prediction in the step (2)_k|k-1(num) updating to obtain updated potential distribution P_k|k(num)；

(4) Pruning and combining the updated Gaussian component and inverse gamma component, and extracting the position and speed of the combined Gaussian component and inverse gamma component as the state of the extended target;

(5) potential distribution P obtained by updating in the step (3)_k|k(num) weighted averaging, resulting in the number of extended targets:

(6) and (5) repeating the steps (2) to (5) and continuing to track the extension target.

The invention has the following advantages:

firstly, the invention introduces the variational Bayesian EM technology, effectively estimates the real measurement noise of each target at different moments by estimating the joint probability hypothesis density of the extended target state and the measurement noise covariance, provides help for the analysis of a multi-extended target tracking scene in a complex environment, and ensures that the CPHD filtering algorithm can effectively realize the extended target tracking in an unknown measurement noise covariance environment.

Secondly, the invention adopts the process of predicting relevant parameters of the inverse Gaussian gamma component, updating the inverse Gaussian gamma component parameters and finally obtaining the states and the number of the extended targets by pruning and merging, thereby improving the tracking precision compared with the traditional GM-CPHD filtering algorithm.

Drawings

FIG. 1 is a general flow chart of the present invention;

FIG. 2 is a graph of simulation results for tracking an extended target using the method of the present invention under a single experimental condition;

FIG. 3 is a graph comparing the simulation results of estimating the number of targets using the method of the present invention and the conventional GM-CPHD method under 100 experimental conditions;

FIG. 4 is a comparison graph of simulation results of determining target tracking accuracy by OSPA distance using the method of the present invention and the conventional GM-CPHD method under 100 experimental conditions.

Detailed Description

The invention is further described below with reference to the accompanying drawings.

Referring to fig. 1, the specific implementation steps of the present invention include the following:

step 1, when the time k is 0, initializing a joint probability hypothesis density of a target state and a measured noise covariance:

wherein, J₀The number of the gaussian components is represented,represents the weight of the ith Gaussian component, N (-) represents a Gaussian distribution,represents the mean of the ith gaussian component,represents the covariance of the ith Gaussian component; IG (-) represents an inverse gamma distribution,represents the covariance of the ith inverse gamma component,is a constant factor of the ith inverse gamma component,an iteration factor for the ith inverse gamma component, where l represents the ith dimension of the measured noise covariance and d represents the dimension of the measured noise covariance.

Step 2, when k is larger than or equal to 1, the joint probability hypothesis density v of the extended target state and the measured noise covariance_k-1|k-1(x, R) predicting to obtain predicted extended target joint probability hypothesis density v_k|k-1(x,R)。

2a) Joint probability hypothesis density v for surviving target states and metrology noise covariance_S,k|k-1Mean value of Gaussian components in (x, R)Sum covariancePredicting to obtain the average value of Gaussian components of the predicted survival targetSum covariance

m_{S, k | k - 1}^{(i)} = F_{S, k - 1}^{(i)} m_{S, k - 1}^{(i)}

P_{S, k | k - 1}^{(i)} = Q_{k - 1} + F_{S, k - 1}^{(i)} P_{S, k - 1}^{(i)} {(F_{S, k - 1}^{(i)})}^{T}

Wherein,representing the survival target state transition matrix, Q_k-1Representing survival target Process noise covariance, (-)^TRepresenting a transpose;

2b) joint probability hypothesis density v for surviving target states and metrology noise covariance_S,k|k-1Constant factors of inverse gamma components in (x, R)And an iteration factorPerforming prediction to obtain constant factor of inverse gamma component of the survival targetAnd the iteration factor of the inverse gamma component

Where ρ is_lRepresents a forgetting factor, and ρ_l∈(0,1]；

2c) Under the Gaussian mixture framework, the mean value of Gaussian components of the predicted survival target is utilizedCovarianceAnd constant factor of inverse gamma componentIteration factorCalculating joint probability hypothesis density of surviving target state and measured noise covariance_S,k|k-1(x,R)：

Wherein, P_S,kRepresenting extended target survival probability, J_k-1Representing the number of gaussian components at time k-1,represents the weight of the ith Gaussian component at the time k-1, N (-) represents the Gaussian distribution,joint probability hypothesis density v representing predicted surviving target state and metrology noise covariance_S,k|k-1(x, R) is the average of the ith Gaussian component,joint probability hypothesis density v representing predicted surviving target state and metrology noise covariance_S,k|k-1Covariance of ith gaussian component in (x, R); IG (-) denotes the inverse gamma distribution, d denotes the dimension of the covariance of the measurement noise,joint probability hypothesis density v representing predicted surviving target state and metrology noise covariance_S,k|k-1Constant factors of the ith inverse gamma component in (x, R),joint probability hypothesis density v representing predicted surviving target state and metrology noise covariance_S,k|k-1An iteration factor of the ith inverse gamma component in (x, R);

2d) joint probability hypothesis density b for derived target states and metrology noise covariance_k|k-1Mean value of Gaussian components in (x, R)Sum covariancePredicting to obtain the average value of the predicted derived target Gaussian componentsSum covariance

m_{b, k | k - 1}^{(i, j)} = F_{b, k - 1}^{(j)} m_{b, k - 1}^{(i, j)} + d_{b, k - 1}^{(j)}

P_{b, k | k - 1}^{(i, j)} = Q_{b, k - 1}^{(j)} + F_{b, k - 1}^{(j)} P_{b, k - 1}^{(i)} {(F_{b, k - 1}^{(j)})}^{T}

Wherein i represents the ith of the inverse Gaussian gamma component at the time k-1, j represents the jth of the inverse Gaussian gamma component derived from the time k-1 to the time k,a state transition matrix representing the derived target,a state correction amount indicating a derived target,representing derived target process noise co-ordinationVariance;

2e) joint probability hypothesis density b for derived target states and metrology noise covariance_k|k-1Constant factors of inverse gamma components in (x, R)And an iteration factorPredicting to obtain constant factor of the predicted derivative target inverse gamma componentAnd an iteration factor

2f) Under the Gaussian mixture framework, the mean value of the predicted derived target Gaussian components is utilizedCovarianceAnd constant factor of inverse gamma componentIteration factorCalculating a joint probability hypothesis density b of the derived target states and the metrology noise covariance_k|k-1(x,R)：

Wherein, J_b,kRepresenting the number of derived target gaussian components from time k-1 to time k,represents the weight of the j-th derived target gaussian component at the time k,means that the average value of the jth Gaussian component is derived from the ith Gaussian component from the k-1 time to the k time,representing time k-1 to time kDeriving the covariance of the jth Gaussian component from the ith Gaussian component;a constant factor representing the j (th) inverse gamma component derived from the i (th) inverse gamma component from the k-1 to the k (th) time,expressing that the ith inverse gamma component is derived from the ith inverse gamma component from the k-1 moment to the k moment to obtain an iteration factor of the jth inverse gamma component;

2g) calculating joint probability density gamma of new object state and measured noise covariance_k(x,R)：

Wherein, J_γ,kThe number of newly generated target Gaussian components at the moment k,the weight of the ith gaussian component of the new target at the moment k,the state mean of the ith gaussian component of the new object,motion state covariance for the ith gaussian component of the new target;a constant factor for the ith inverse gamma component of the new target,the iteration factor of the ith inverse gamma component of the new object.

2h) Calculating the joint probability hypothesis density v of the extended target state and the measured noise covariance by using the parameters obtained from the step 2a) to the step 2g)_k|k-1(x,R)：

v_k|k-1(x,R)＝v_S,k|k-1(x,R)+b_k|k-1(x,R)+γ_k(x,R)。

Step 3, when k is more than or equal to 1, aligning potential distribution P_k-1|k-1(num) to obtain a predicted potential distribution P_k|k-1(num)：

<math> <mrow> <msub> <mi>P</mi> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>num</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mi>Σ</mi> <mrow> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>num</mi> </munderover> <msub> <mi>P</mi> <mrow> <mi>birth</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>num</mi> <mo>-</mo> <mi>j</mi> <mo>)</mo> </mrow> <munderover> <mi>Σ</mi> <mrow> <mi>h</mi> <mo>=</mo> <mi>j</mi> </mrow> <mo>∞</mo> </munderover> <mfrac> <mrow> <mi>h</mi> <mo>!</mo> </mrow> <mrow> <mi>j</mi> <mo>!</mo> <mrow> <mo>(</mo> <mi>h</mi> <mo>-</mo> <mi>j</mi> <mo>)</mo> </mrow> <mo>!</mo> </mrow> </mfrac> <msub> <mi>P</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>h</mi> <mo>)</mo> </mrow> <msubsup> <mi>p</mi> <mrow> <mi>S</mi> <mo>,</mo> <mi>k</mi> </mrow> <mi>j</mi> </msubsup> <msup> <mrow> <mo>(</mo> <msub> <mrow> <mn>1</mn> <mo>-</mo> <mi>p</mi> </mrow> <mrow> <mi>S</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>)</mo> </mrow> <mrow> <mi>h</mi> <mo>-</mo> <mi>j</mi> </mrow> </msup> </mrow> </math>

Wherein, P_birth,k(num-j) represents the probability of new num-j extended targets from time k-1 to time k, p_S,kRepresenting the survival probability of the extended target at time k, P_k-1|k-1(h) Represents the probability that k-1 is now with h expansion targets! It is shown that the order of the multiplication,represents the probability that j extended targets survive from time k-1 to time k, (1-p)_S,k)^h-jThe probability that h-j extended targets disappear from the moment k-1 to the moment k is shown.

Step 4. Joint probability hypothesis density v for extended target state and measured noise covariance_k|k-1(x, R) and the predicted potential distribution P_k|k-1(num) update:

4a) assuming the density v by using a variational Bayes VB method_k|k-1(x, R) is nearThe method is as follows:

in the formula, Q_x,k|k-1(x) In the form of a sum of a gaussian distribution,

Q_R,k|k-1(R) is a summed version of the inverse gamma distribution,

wherein,denotes the prediction weight of the ith gaussian component at the kth instant, i ═ 1_k，J_kRepresents the number of the extended target Gaussian components at the kth time, N (-) represents the Gaussian distribution,the average of the ith gaussian component predicted for the kth time instant,the covariance of the ith Gaussian component predicted for the kth moment; IG (-) represents an inverse gamma distribution,the constant factor of the ith inverse gamma component predicted for the kth time instant,the iteration factor of the ith inverse gamma component predicted for the kth moment, i 1., d, represents the dimension of the covariance of the measurement noise;

4b) expanding target shape by utilizing Variational Bayesian Expectation Maximization (VBEM) methodProbability hypothesis density of states Q_x,k|k-1(x) Gaussian component in (1) and probability hypothesis density of the covariance of the measurement noise Q_R,k|k-1Iteratively updating the inverse gamma component in (R):

(4b1) setting a constant factor for the inverse gamma componentAnd an iteration factor Where l 1., d, d is the dimension of the measurement noise covariance R;

(4b2) calculating the covariance of the measured noise according to two factors of the set inverse gamma component:

where N1, N is the maximum number of iterations, diag.]Representing elements in diagonalization;

(4b3) using measured noise covarianceCalculating an update factor

S_{W}^{(n)} = H_{W} P_{k | k - 1}^{(i)} {H_{W}}^{T} + R_{W}^{(n)}

Wherein,representing a covariance matrix for metrology noiseCarrying out matrix of diagonal connection on the measured number of the current unit W, wherein blkdiag (·) represents diagonal connection on elements in the matrix, and | W | represents the measured number of the current unit W; h_WRepresentation pair observation matrix H_kThe matrix after the measured number of the current cell W is vertically connected,an observation matrix H representing the k instants_kTransposing;representing the extended target gaussian component state of motion covariance predicted at time k-1 to k,representation matrix H_WTransposing;

(4b4) using update factorsCalculating a gain matrix

K_{k}^{(i) (n)} = P_{k | k - 1}^{(i)} H_{W}^{T} {[S_{W}^{(n)}]}^{- 1}

Wherein [. ]]^-1Represents inverting the matrix;

(4b5) using a gain matrixCalculating extended target Gauss component motion stateAnd extended target gaussian component motion state covariance

m_{k | k}^{(i) (n)} = m_{k | k - 1}^{(i)} + K_{k}^{(i) (n)} (z_{W} - H_{W} m_{k | k - 1}^{(i)})

P_{k | k}^{(i) (n)} = [I - K_{k}^{(i) (n)} H_{W}] P_{k | k - 1}^{(i)}

Wherein I represents an identity matrix, z_WRepresents all measurements in a certain partition unit W;

(4b6) extracting extended target Gauss component motion stateUsing the position informationRepresents;

(4b7) using the Gaussian component position informationAnd measure noise covarianceCalculating and measuring Y_n′From locationProbability of gaussian component generation gamma of_n′i：

Wherein, J_kRepresenting the number of extended target Gaussian components, Y_n′Represents the nth 'measurement of the current unit W, n' ═ 1., | W |; n (-) represents a Gaussian distribution; pi_iWhich represents the mixing coefficient(s) of the mixture,N_iis represented by positionThe number of valid measurements resulting from the gaussian component of (a),

(4b8) using measurement of Y_n′From locationProbability of gaussian component generation gamma of_n′iAnd obtaining the position information of the extended target Gaussian component motion state by iterative update

(4b9) Position information using extended target gaussian component motion stateCoefficient of mixing pi_iMeasuring the covariance of noiseCalculating a maximum likelihood function L⁽ⁱ⁾⁽ⁿ⁾：

Wherein J_kRepresenting the number of the extended target Gaussian components;

(4b10) determine | L⁽ⁱ⁾⁽ⁿ⁾-L^(i)(n-1)If | is smaller than the constant 0.01, judging whether the current iteration number N is smaller than the maximum iteration number N100, and if so, stopping iteration; otherwise, returning to step (4b2), updating the inverse gamma component iteration factor:

wherein,represents the relative quantityAll of the elements in (a) are added up,

represents the square of the jth dimension element of the pair vector, (. DEG)_jjThe representation takes the diagonal elements of the matrix,representation pair iteration factorThe vector after the measurement number of the current cell W is vertically connected,z_Wrepresents a measurement of the current cell;

(4b11) extracting an extended target state componentExtending target motion state covarianceIteration factorThat is to say that the first and second electrodes,

wherein the target state component is extendedThe position information in (4b8) is the position information of the motion state of the extended target component obtained by the iterative update in step (4b8)

4c) Hedonic distribution P_k|k-1(num) updating to obtain updated potential distribution P_k|k(num) is as follows:

<math> <mrow> <msub> <mi>P</mi> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>num</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <mfrac> <mrow> <msub> <mi>Σ</mi> <mrow> <mi>p</mi> <mo>&angle;</mo> <mi>Z</mi> </mrow> </msub> <msub> <mi>Σ</mi> <mrow> <mi>W</mi> <mo>&Element;</mo> <mi>p</mi> </mrow> </msub> <msub> <mi>ψ</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>W</mi> </mrow> </msub> <msubsup> <mi>G</mi> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mrow> <mo>(</mo> <mi>num</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mfenced open='(' close=')' separators=' '> <mi></mi> <mtable> <mtr> <mtd> <mi></mi> <msub> <mi>G</mi> <mi>FA</mi> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mfrac> <mi>ηW</mi> <mrow> <mo>|</mo> <mi>p</mi> <mo>|</mo> </mrow> </mfrac> <mfrac> <msup> <mi>ρ</mi> <mrow> <mi>num</mi> <mo>-</mo> <mo>|</mo> <mi>P</mi> <mo>|</mo> </mrow> </msup> <mrow> <mrow> <mo>(</mo> <mi>num</mi> <mo>-</mo> <mo>|</mo> <mi>p</mi> <mo>|</mo> <mo>)</mo> </mrow> <mo>!</mo> </mrow> </mfrac> <msub> <mi>δ</mi> <mrow> <mi>num</mi> <mo>&GreaterEqual;</mo> <mo>|</mo> <mi>p</mi> <mo>|</mo> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>+</mo> <msubsup> <mi>G</mi> <mi>FA</mi> <mrow> <mo>(</mo> <mo>|</mo> <mi>W</mi> <mo>|</mo> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mfrac> <msup> <mi>ρ</mi> <mrow> <mi>num</mi> <mo>-</mo> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mrow> <mrow> <mo>(</mo> <mi>num</mi> <mo>-</mo> <mo>|</mo> <mi>p</mi> <mo>|</mo> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>!</mo> </mrow> </mfrac> <msub> <mi>δ</mi> <mrow> <mrow> <mi>num</mi> <mo>&GreaterEqual;</mo> <mo>|</mo> <mi>p</mi> <mo>|</mo> </mrow> <mo>-</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mfenced> </mrow> <mrow> <msub> <mi>Σ</mi> <mrow> <mi>p</mi> <mo>&angle;</mo> <mi>Z</mi> </mrow> </msub> <msub> <mi>Σ</mi> <mrow> <mi>W</mi> <mo>&Element;</mo> <mi>p</mi> </mrow> </msub> <msub> <mi>Ψ</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>W</mi> </mrow> </msub> <msub> <mi>l</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>W</mi> </mrow> </msub> </mrow> </mfrac> <mo>,</mo> </mtd> <mtd> <mo>|</mo> <mi>Z</mi> <mo>|</mo> <mo>&NotEqual;</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <msup> <mi>ρ</mi> <mi>num</mi> </msup> <msubsup> <mi>G</mi> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mrow> <mo>(</mo> <mi>num</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> <mrow> <msub> <mi>G</mi> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>ρ</mi> <mo>)</mo> </mrow> </mrow> </mfrac> </mtd> <mtd> <mo>|</mo> <mi>Z</mi> <mo>|</mo> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> </mrow> </math>

wherein p < Z represents that the measurement set Z is divided into p non-empty subsets, W ∈ p represents a certain unit under the p-th non-empty subset, G_k|k-1(p) represents a state prediction probability generating function,representation state predictionNum order partial derivatives, G, of the probability generating function_FA(0) Representing the false alarm probability generating function in the absence of measurement, η W representing the measurement probability generated by the extended target, | p | representing the number of all non-empty cells in the p-th partition,the | W | order partial derivative of the false alarm probability generation function,_num≥|p|indicating that the value is 1 when the number of targets num is greater than the partition unit | p |, otherwise, 0, | Z | ═ 0 indicates that no measurement is generated by the extended target, | W | indicates the number of measurements in each non-empty unit W, | W |, l_p,WRepresents the false alarm constant coefficient, psi, when the measurement partition unit is | p | -1_p,WRepresents the product of the target production metrology probabilities, ρ represents the probability that the extended target component is not detected:

<math> <mrow> <mi>ρ</mi> <mo>=</mo> <mi>Σ</mi> <msubsup> <mover> <mi>ω</mi> <mo>&OverBar;</mo> </mover> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>j</mi> </msubsup> <mo>[</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>P</mi> <mi>D</mi> </msub> <mrow> <mo>(</mo> <mo>·</mo> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>P</mi> <mi>D</mi> </msub> <mrow> <mo>(</mo> <mo>·</mo> <mo>)</mo> </mrow> <msub> <mi>G</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>|</mo> <mo>·</mo> <mo>)</mo> </mrow> <mo>]</mo> </mrow> </math>

<math> <mrow> <mi>ηW</mi> <mo>=</mo> <msub> <mi>p</mi> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>[</mo> <msub> <mi>P</mi> <mi>D</mi> </msub> <mrow> <mo>(</mo> <mo>·</mo> <mo>)</mo> </mrow> <msubsup> <mi>G</mi> <mi>z</mi> <mrow> <mo>(</mo> <mo>|</mo> <mi>W</mi> <mo>|</mo> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mn>0</mn> <mo>|</mo> <mo>·</mo> <mo>)</mo> </mrow> <munder> <mi>Π</mi> <mrow> <msup> <mi>z</mi> <mo>′</mo> </msup> <mo>&Element;</mo> <mi>W</mi> </mrow> </munder> <mfrac> <mrow> <msub> <mi>p</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>z</mi> <mo>′</mo> </msup> <mo>|</mo> <mo>·</mo> <mo>)</mo> </mrow> </mrow> <mrow> <msub> <mi>p</mi> <mi>FA</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>z</mi> <mo>′</mo> </msup> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>]</mo> </mrow> </math>

<math> <mrow> <msub> <mi>ψ</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>W</mi> </mrow> </msub> <mo>=</mo> <munder> <mi>Π</mi> <mrow> <msup> <mi>W</mi> <mo>′</mo> </msup> <mo>&Element;</mo> <mi>p</mi> <mo>-</mo> <mi>W</mi> </mrow> </munder> <msup> <mi>ηW</mi> <mo>′</mo> </msup> </mrow> </math>

wherein pi is a successive multiplication symbol,represents the proportion of the j-th inverse gamma component in all the current inverse gamma components, p_k|k-1Representing a single extended target state transition probability density function, p_z(z' | -) represents the extended target metric likelihood, p_FA(z') represents the likelihood of false alarm measurement, G_z(0 |. represents a measurement probability generating function,the | W | order partial derivative of the measurement probability generating function is expressed, z 'belongs to W and represents that the measurement z' belongs to W, P_D(. cndot.) represents the detection probability, W 'is the unit left after removing the unit W in all the units under p division, eta W' represents the generation probability of the extended target false alarm measurement,representing the | p | order partial derivative of the state prediction probability generation function,represents the | p | -1 order partial derivative of the state prediction probability generating function.

And 5, pruning and combining the updated Gaussian component and the inverse gamma component, wherein the steps are as follows:

(5a) two pruning thresholds T1 and T2 are set, one merging threshold U: t1 ═ 10^-5T2 ═ 120, U ═ 10; setting the maximum number of inverse gamma components of gaussian: j. the design is a square_max＝100；

(5b) Calculating the covariance of the measured noise corresponding to each extended target component:

(5c) setting the variable l' to be 0, pruning the updated extension target component, and obtaining a sequence number set I corresponding to the pruned extension target component as follows:

I = {i = 1, . . ., J_{k} | w_{k}^{(i)} > T 1, {| | R_{k}^{(i)} | |}_{2} < T 2};

(5d) making l '═ l' +1, takingRepresents taking the maximum weightExtracting the component meeting the merging threshold U in the trimmed extended target components to obtain a serial number set corresponding to the extended target components suitable for mergingComprises the following steps:

<math> <mrow> <mover> <mi>Q</mi> <mo>&OverBar;</mo> </mover> <mo>=</mo> <mo>{</mo> <mi>i</mi> <mo>&Element;</mo> <mi>I</mi> <mo>|</mo> <msup> <mrow> <mo>(</mo> <msubsup> <mi>m</mi> <mi>k</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </msubsup> <mo>-</mo> <msubsup> <mi>m</mi> <mi>k</mi> <mrow> <mo>(</mo> <msup> <mi>i</mi> <mo>′</mo> </msup> <mo>)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> <mi>T</mi> </msup> <msup> <mrow> <mo>(</mo> <msubsup> <mi>P</mi> <mi>k</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msubsup> <mi>m</mi> <mi>k</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </msubsup> <mo>-</mo> <msubsup> <mi>m</mi> <mi>k</mi> <mrow> <mo>(</mo> <msup> <mi>i</mi> <mo>′</mo> </msup> <mo>)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>≤</mo> <mi>U</mi> <mo>}</mo> <mo>;</mo> </mrow> </math>

(5e) respectively align sequence number setsThe weight of the corresponding extended target component in (1)State of motionConstant factorIteration factorCovarianceMerging to obtain the weight of the merged extended target componentState of motionConstant factorIteration factorCovarianceThe following were used:

<math> <mrow> <msubsup> <mover> <mi>w</mi> <mo>~</mo> </mover> <mi>k</mi> <mrow> <mo>(</mo> <msup> <mi>l</mi> <mo>′</mo> </msup> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <munder> <mi>Σ</mi> <mrow> <mi>i</mi> <mo>&Element;</mo> <mover> <mi>Q</mi> <mo>&OverBar;</mo> </mover> </mrow> </munder> <msubsup> <mi>w</mi> <mi>k</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </msubsup> <mo>,</mo> </mrow> </math>

<math> <mrow> <msubsup> <mover> <mi>m</mi> <mo>~</mo> </mover> <mi>k</mi> <mrow> <mo>(</mo> <msup> <mi>l</mi> <mo>′</mo> </msup> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <mfrac> <mn>1</mn> <msubsup> <mover> <mi>w</mi> <mo>~</mo> </mover> <mi>k</mi> <mrow> <mo>(</mo> <msup> <mi>l</mi> <mo>′</mo> </msup> <mo>)</mo> </mrow> </msubsup> </mfrac> <munder> <mi>Σ</mi> <mrow> <mi>i</mi> <mo>&Element;</mo> <mover> <mi>Q</mi> <mo>&OverBar;</mo> </mover> </mrow> </munder> <msubsup> <mi>w</mi> <mi>k</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </msubsup> <msubsup> <mi>m</mi> <mi>k</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </msubsup> <mo>,</mo> </mrow> </math>

<math> <mrow> <msubsup> <mover> <mi>α</mi> <mo>~</mo> </mover> <mi>k</mi> <mrow> <mo>(</mo> <msup> <mi>l</mi> <mo>′</mo> </msup> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <mfrac> <mn>1</mn> <msubsup> <mover> <mi>w</mi> <mo>~</mo> </mover> <mi>k</mi> <mrow> <mo>(</mo> <msup> <mi>l</mi> <mo>′</mo> </msup> <mo>)</mo> </mrow> </msubsup> </mfrac> <munder> <mi>Σ</mi> <mrow> <mi>i</mi> <mo>&Element;</mo> <mover> <mi>Q</mi> <mo>&OverBar;</mo> </mover> </mrow> </munder> <msubsup> <mi>w</mi> <mi>k</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </msubsup> <msubsup> <mi>α</mi> <mi>k</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </msubsup> <mo>,</mo> </mrow> </math>

<math> <mrow> <msubsup> <mover> <mi>β</mi> <mo>~</mo> </mover> <mi>k</mi> <mrow> <mo>(</mo> <msup> <mi>l</mi> <mo>′</mo> </msup> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <mfrac> <mn>1</mn> <msubsup> <mover> <mi>w</mi> <mo>~</mo> </mover> <mi>k</mi> <mrow> <mo>(</mo> <msup> <mi>l</mi> <mo>′</mo> </msup> <mo>)</mo> </mrow> </msubsup> </mfrac> <munder> <mi>Σ</mi> <mrow> <mi>i</mi> <mo>&Element;</mo> <mover> <mi>Q</mi> <mo>&OverBar;</mo> </mover> </mrow> </munder> <msubsup> <mi>w</mi> <mi>k</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </msubsup> <msubsup> <mi>β</mi> <mi>k</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </msubsup> <mo>,</mo> </mrow> </math>

<math> <mrow> <msubsup> <mover> <mi>P</mi> <mo>~</mo> </mover> <mi>k</mi> <mrow> <mo>(</mo> <msup> <mi>l</mi> <mo>′</mo> </msup> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <mfrac> <mn>1</mn> <msubsup> <mover> <mi>w</mi> <mo>~</mo> </mover> <mi>k</mi> <mrow> <mo>(</mo> <msup> <mi>l</mi> <mo>′</mo> </msup> <mo>)</mo> </mrow> </msubsup> </mfrac> <munder> <mi>Σ</mi> <mrow> <mi>i</mi> <mo>&Element;</mo> <mover> <mi>Q</mi> <mo>&OverBar;</mo> </mover> </mrow> </munder> <msubsup> <mi>w</mi> <mi>k</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msubsup> <mi>P</mi> <mi>k</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </msubsup> <mo>+</mo> <mrow> <mo>(</mo> <msubsup> <mover> <mi>m</mi> <mo>~</mo> </mover> <mi>k</mi> <mrow> <mo>(</mo> <msup> <mi>l</mi> <mo>′</mo> </msup> <mo>)</mo> </mrow> </msubsup> <mo>-</mo> <msubsup> <mi>m</mi> <mi>k</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <msubsup> <mover> <mi>m</mi> <mo>~</mo> </mover> <mi>k</mi> <mrow> <mo>(</mo> <msup> <mi>l</mi> <mo>′</mo> </msup> <mo>)</mo> </mrow> </msubsup> <mo>-</mo> <msubsup> <mi>m</mi> <mi>k</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>)</mo> </mrow> <mo>;</mo> </mrow> </math>

(5f) the sequence number set I corresponding to the extension target component which is suitable for combination and obtained in the step (5d) in the sequence number set I corresponding to the extension target component obtained in the step (5c) after pruningRemoving the same elements, and judging after trimmingIf the sequence number set I corresponding to the extended target component is not an empty set, returning to the step (5d), otherwise executing the step (5 g);

(5g) judging whether the variable l' is greater than the maximum Gauss inverse gamma component number J_maxIf l' > J_maxThen the weight value will beArranging corresponding inverse Gauss components from big to small, and taking the first J_maxIndividual weight valueThe position and the speed of the inverse gamma component of the gauss larger than 0.5 are taken as the state of the extended target; if l' < J_maxThen all the weights are calculatedThe position and velocity of the corresponding inverse gamma component of gaussian greater than 0.5 are taken as the state of the extended target.

Step 6. updating the obtained potential distribution P according to the step 4c)_k|k(num) weighted averaging, resulting in the number of extended targets:

and completing the tracking of the extended target after the state and the number of the extended target are obtained.

The effect of the invention for tracking the extended target can be further illustrated by the following simulation experiment:

1. simulation conditions

Considering the situation that 4 targets do uniform linear motion in a two-dimensional plane and the motion tracks at two positions are crossed, the sampling period isThe whole observation process lasts for 40 moments, and the motion equation and the measurement equation of the target are respectively as follows:

x_k＝Fx_k-1+Gw_k

y_k＝Hx_k+v_k

wherein the Gaussian noise w_kStandard deviation of (a)_w2. Extended target state transition matrix F, input matrix G, Gaussian noise w_kThe observation matrix H is set to:

measurement noise v in actual target tracking scene_kThe covariance of (a) is unknown, and the actual standard deviation sigma of the measured noise is set in the experiment_v＝1；

Setting three standard deviations of measurement noise adopted in the traditional GM-CPHD algorithm as 0.05,1 and 50 respectively;

setting the measurement number generated by the target to obey Poisson distribution, wherein a Poisson distribution parameter beta is 20, and the measurement position generated by the target obeys Gaussian distribution;

set target survival probability p_S,k0.99, detection probability p_D,k＝0.98；

And (3) setting the clutter number to obey Poisson distribution with the average value of 5, and enabling the clutter to obey uniform distribution in the whole observation area.

Setting the state of the newborn target to

Covariance of newborn target is P_γ＝diag[10,10,10,10]；

Setting the pruning threshold T1 to 10^-5The pruning threshold T2 is 120, the combining threshold U is 10, and the maximum inverse gamma component number J is gaussian_maxThe forgetting factor ρ is 0.9, and the number of experimental simulations is 100.

2. Simulation content and results

Simulation 1, adopting the method of the present invention to track a single experiment of expanding the target motion trajectory under unknown measured noise covariance, the result is shown in fig. 2. As can be seen from FIG. 2, the method of the present invention can track the motion trail of the extended target well.

Simulation 2, comparing the target number estimation of 100 experiments using the method of the present invention with the conventional three GM-CPHD methods using different measured noise covariances, respectively, the result is shown in FIG. 3. It can be seen from fig. 3 that the method of the present invention works as well for estimating the target number as the conventional GM-CPHD method at a given true metrology noise covariance. However, in the conventional GM-CPHD method, when the measured noise covariance is unknown and the deviation between the measured noise covariance and the true value is large, the estimated target number may have a large deviation, and at the same time, when two target motion trajectories, k being 20 and k being 35, cross, the target number has missing detection;

simulation 3, comparing the tracking accuracy of 100 experiments by using the method of the present invention with that of the conventional three GM-CPHD methods respectively using different measured noise covariance by means of OSPA distance, and the result is shown in FIG. 4. It can be seen from fig. 4 that the method of the present invention tracks as well as the conventional GM-CPHD algorithm given the true metrology noise covariance. However, the conventional GM-CPHD method may have poor tracking accuracy when the measurement noise covariance used is greatly deviated from the true measurement noise covariance.

Experiments show that when the method disclosed by the invention is used for processing an extended target tracking scene under the condition that the measured noise covariance is unknown, the tracking effect of the method is superior to that of the traditional GM-CPHD extended target tracking method.

Claims

1. An extended target tracking method based on variational Bayesian expectation maximization comprises the following steps:

2. The extended target tracking method based on variational Bayes expectation maximization as claimed in claim 1, wherein the joint probability hypothesis density v using variational Bayes VB method in step (3a)_k|k-1(x, R) is approximated as follows:

wherein Q is_x,k|k-1(x) Is in the form of a sum of Gaussian distributions, expressed as

Q_R,k|k-1(R) is the summed form of the inverse gamma distribution, expressed as

Denotes the weight of the ith gaussian component at the kth instant, i ═ 1_k，J_kRepresents the number of the extended target Gaussian components at the kth time, N (-) represents the Gaussian distribution,the average of the ith gaussian component predicted for the kth time instant,the covariance of the ith Gaussian component predicted for the kth moment; IG (-) represents an inverse gamma distribution,the constant factor of the ith inverse gamma component predicted for the kth time instant,the iteration factor of the i-th inverse gamma component predicted for the k-th time instant, i 1.. d, represents the dimensionality of the covariance of the metrology noise.

3. The extended target tracking method based on variational Bayesian expectation maximization as claimed in claim 1, wherein the probability hypothesis density Q of the extended target state using variational Bayesian expectation maximization VBEM method in step (3b) is_x,k|k-1(x) Gaussian component in (1) and probability hypothesis density of the covariance of the measurement noise Q_R,k|k-1And (R) performing iterative updating on the inverse gamma component in the (R) according to the following steps:

(3b1) setting a constant factor for the inverse gamma componentAnd an iteration factor Where l 1., d, d is the dimension of the measurement noise covariance R;

(3b2) and calculating the covariance of the measurement noise according to two set factors of the inverse gamma component:where N1, N is the maximum number of iterations, diag.]Representing elements in diagonalization;

(3b3) using measured noise covarianceCalculating an update factor

S_{W}^{(n)} = H_{W} P_{k | k - 1}^{(i)} {H_{W}}^{T} + R_{W}^{(n)}

Wherein,representing a covariance matrix for metrology noiseCarrying out diagonal connection on the matrix of the current unit W after measuring the number, wherein blkdiag (·) represents carrying out diagonal connection on elements in the matrix, and | W | represents the measuring number of the current unit W; h_WRepresentation pair observation matrix H_kThe matrix after the measured number of the current cell W is vertically connected, an observation matrix H representing the k instants_kTransposing;representing the extended target gaussian component state of motion covariance predicted at time k-1 to k,representation matrix H_WTransposing;

(3b4) using update factorsCalculating a gain matrix

K_{k}^{(i) (n)} = P_{k | k - 1}^{(i)} H_{W}^{T} {[S_{W}^{(n)}]}^{- 1},

Wherein [. ]]^-1Represents inverting the matrix;

(3b5) using a gain matrixCalculating extended target Gauss component motion stateAnd extended target gaussian component motion state covariance

m_{k | k}^{(i) (n)} = m_{k | k - 1}^{(i)} + K_{k}^{(i) (n)} (z_{w} - H_{w} m_{k | k - 1}^{(i)})

P_{k | k}^{(i) (n)} = [I - K_{k}^{(i) (n)} H_{W}] P_{k | k - 1}^{(i)}

(3b6) extracting extended target Gauss component motion stateUsing the position informationRepresents;

(3b7) using the Gaussian component position informationAnd measure noise covarianceCalculating and measuring Y_n′Is from the positionProbability of gaussian component generation gamma of_n′i：

(3b8) using measurement of Y_n′From locationProbability of gaussian component generation gamma of_n′iAnd obtaining the position information of the extended target Gaussian component motion state by iterative update

(3b9) Position information using extended target gaussian component motion stateCoefficient of mixing pi_iMeasuring the covariance of noiseCalculate the bestLarge likelihood function L⁽ⁱ⁾⁽ⁿ⁾：

Wherein J_kRepresenting the number of the extended target Gaussian components;

(3b10) determine | L⁽ⁱ⁾⁽ⁿ⁾-L^(i)(n-1)If |, is smaller than the constant 0.01, and meanwhile, whether the current iteration number N is smaller than the maximum iteration number N is judged to be 100, if yes, the iteration is stopped, otherwise, the step (3b2) is returned, and the inverse gamma component iteration factor is updated:

represents the square of the jth dimension element of the pair vector, (. DEG)_jjThe representation takes the diagonal elements of the matrix,representation pair iteration factorThe vectors after the measurement number of the current cell W are vertically connected,z_Wrepresents a measurement of the current cell;

(3b11) extracting an extended target state componentExtending target motion state covarianceIteration factorThat is to say that the first and second electrodes,

wherein the target shape is expandedComponent of stateThe position information in (3b8) is the position information of the motion state of the extended target component obtained by iterative update in step (3b8)

4. The extended target tracking method based on variational bayes expectation maximization as claimed in claim 1, wherein said step (3c) is on the potential distribution P_k|k-1(num) updating to obtain updated potential distribution P_k|k(num) is as follows:

<math> <mrow> <msub> <mi>P</mi> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>num</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <mfrac> <mrow> <msub> <mi>Σ</mi> <mrow> <mi>p</mi> <mo>&angle;</mo> <mi>Z</mi> </mrow> </msub> <msub> <mi>Σ</mi> <mrow> <mi>W</mi> <mo>&Element;</mo> <mi>p</mi> </mrow> </msub> <msub> <mi>ψ</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>W</mi> </mrow> </msub> <msubsup> <mi>G</mi> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mrow> <mo>(</mo> <mi>num</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mfenced open='(' close=')'> <mtable> <mtr> <mtd> <msub> <mi>G</mi> <mi>FA</mi> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mfrac> <mi>ηW</mi> <mrow> <mo>|</mo> <mi>p</mi> <mo>|</mo> </mrow> </mfrac> <mfrac> <msup> <mi>ρ</mi> <mrow> <mi>num</mi> <mo>-</mo> <mo>|</mo> <mi>P</mi> <mo>|</mo> </mrow> </msup> <mrow> <mrow> <mo>(</mo> <mi>num</mi> <mo>-</mo> <mo>|</mo> <mi>p</mi> <mo>|</mo> <mo>)</mo> </mrow> <mo>!</mo> </mrow> </mfrac> </mtd> <mtd> <msub> <mi>δ</mi> <mrow> <mi>num</mi> <mo>&GreaterEqual;</mo> <mo>|</mo> <mi>p</mi> <mo>|</mo> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi></mi> <mo>+</mo> <msubsup> <mi>G</mi> <mi>FA</mi> <mrow> <mo>(</mo> <mo>|</mo> <mi>W</mi> <mo>|</mo> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mfrac> <msup> <mi>ρ</mi> <mrow> <mi>num</mi> <mo>-</mo> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mrow> <mrow> <mo>(</mo> <mi>num</mi> <mo>-</mo> <mo>|</mo> <mi>p</mi> <mo>|</mo> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>!</mo> </mrow> </mfrac> </mtd> <mtd> <msub> <mi>δ</mi> <mrow> <mi>num</mi> <mo>&GreaterEqual;</mo> <mo>|</mo> <mi>p</mi> <mo>|</mo> <mo>-</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mfenced> </mrow> <mrow> <msub> <mi>Σ</mi> <mrow> <mi>p</mi> <mo>&angle;</mo> <mi>Z</mi> </mrow> </msub> <msub> <mi>Σ</mi> <mrow> <mi>W</mi> <mo>&Element;</mo> <mi>p</mi> </mrow> </msub> <msub> <mi>ψ</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>W</mi> </mrow> </msub> <msub> <mi>l</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>W</mi> </mrow> </msub> </mrow> </mfrac> <mo>,</mo> </mtd> <mtd> <mo>|</mo> <mi>Z</mi> <mo>|</mo> <mo>&NotEqual;</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <msup> <mi>ρ</mi> <mi>num</mi> </msup> <msubsup> <mi>G</mi> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mrow> <mo>(</mo> <mi>num</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> <mrow> <msub> <mi>G</mi> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>ρ</mi> <mo>)</mo> </mrow> </mrow> </mfrac> </mtd> <mtd> <mi></mi> <mo>|</mo> <mi>Z</mi> <mo>|</mo> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> </mrow> </math>

wherein p < Z represents that the measurement set Z is divided into p non-empty subsets, W ∈ p represents a certain unit under the p-th non-empty subset, G_k|k-1(p) represents a state prediction probability generating function,num order partial derivatives, G, representing state prediction probability generating functions_FA(0) Representing a false alarm probability generating function in the absence of measurement, eta W representing the measurement probability generated by the extended target, and | p | representing the p-th divisionThe number of all non-empty cells of (c),the | W | order partial derivative of the false alarm probability generation function,_num≥pindicating that the value is 1 when the number of targets num is greater than the partition unit | p |, otherwise, 0, | Z | ═ 0 indicates that no measurement is generated by the extended target, | W | indicates the number of measurements in each non-empty unit W, | W |, l_p,WRepresents the false alarm constant coefficient, psi, when the measurement partition unit is | p | -1_p,WRepresents the product of the target production metrology probabilities, ρ represents the probability that the extended target component is not detected:

<math> <mrow> <msub> <mi>ψ</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>W</mi> </mrow> </msub> <mo>=</mo> <munder> <mi>Π</mi> <mrow> <msup> <mi>W</mi> <mo>′</mo> </msup> <mo>&Element;</mo> <mi>p</mi> <mo>-</mo> <mi>W</mi> </mrow> </munder> <mi>η</mi> <msup> <mi>W</mi> <mo>′</mo> </msup> </mrow> </math>

wherein pi is a successive multiplication symbol,represents the proportion of the j-th inverse gamma component in all the current inverse gamma components, p_k|k-1Representing a single extended target state transition probability density function, p_z(z' | -) represents the extended target metric likelihood, p_FA(z') represents the likelihood of false alarm measurement, G_z(0 |. represents a measurement probability generating function,the | W | order partial derivative of the measurement probability generating function is expressed, z 'belongs to W and represents that the measurement z' belongs to W, P_D(. represents)Detecting probability, wherein W 'belongs to p-W and represents the unit left after removing the unit W in all the units under p division, eta W' represents the probability of generating the extended target false alarm measurement,representing the | p | order partial derivative of the state prediction probability generation function,represents the | p | -1 order partial derivative of the state prediction probability generating function.

5. The extended target tracking method based on variational bayes expectation maximization according to claim 1, wherein the step (4) of pruning and merging the updated gaussian component and inverse gamma component is performed according to the following steps:

I = {i = 1, . . ., J_{k} | w_{k}^{(i)} > T 1, {| | R_{k}^{(i)} | |}_{2} < T 2};

<math> <mrow> <msubsup> <mover> <mi>w</mi> <mo>~</mo> </mover> <mi>k</mi> <mrow> <mo>(</mo> <msup> <mi>l</mi> <mo>′</mo> </msup> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <munder> <mi>Σ</mi> <mrow> <mi>i</mi> <mo>&Element;</mo> <mover> <mi>Q</mi> <mo>&OverBar;</mo> </mover> </mrow> </munder> <msubsup> <mi>w</mi> <mi>k</mi> <mrow> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </msubsup> </mrow> </math>

(5f) the sequence number set I corresponding to the extension target component which is suitable for combination and obtained in the step (5d) in the sequence number set I corresponding to the extension target component obtained in the step (5c) after pruningRemoving the same elements, and then judging whether the sequence number set I corresponding to the pruned extended target component is empty or notIf not, returning to the step (5d), otherwise, executing (5 g);