Kullback-Leibler散度在多目标跟踪中的高效联合概率数据关联滤波器

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"基于Kullback-Leibler散度的高效联合概率数据关联过滤器用于多目标跟踪" 在多目标跟踪领域，数据关联是关键问题之一，它涉及到如何正确地将传感器观测到的数据与已知的目标匹配。传统的联合概率数据关联（Joint Probabilistic Data Association, JPDA）过滤器在处理跟踪目标合并问题时可能存在不足。本文提出了一种新的基于Kullback-Leibler散度（Kullback-Leibler Divergence, KLD）的高效方法，以解决JPDA过滤器中的跟踪目标混淆问题。 Kullback-Leibler散度是一种衡量两个概率分布之间差异的无量纲量，常用于信息论和统计学。在JPDA中，通常假设后验概率密度函数（Posterior PDF）在每个时间步可以用单峰高斯分布来近似。然而，这种方法可能会导致跟踪性能下降，特别是在目标密度高或者观测噪声大的情况下。本文作者提出了一种新颖的优化后验PDF的方法，以获取一个新的单峰高斯分布，该分布能最小化与实际后验PDF之间的KLD。由于实际后验PDF通常表现为高斯混合模型，KLD的计算变得复杂且不可行。因此，作者引入了KLD的近似值作为成本函数，这简化了问题的求解。这个成本函数是多个不冲突的子目标函数的线性组合，这样可以同时优化多个目标，使得问题的求解更加高效。通过这种方式，该方法能够在保持跟踪精度的同时，减少计算复杂性，提高了多目标跟踪的效率。此外，这种方法还有助于避免目标丢失和误关联，这对于实时的多目标跟踪系统至关重要。在实际应用中，这种基于KLD的优化策略有望在雷达信号处理、自动驾驶、空中交通管理等场景下提升多目标跟踪的性能。这篇研究论文提供了一个新的框架，通过利用KLD优化联合概率数据关联，解决了多目标跟踪中的关键挑战，即如何有效地处理目标合并和分离，以及如何以较低的计算成本保持高精度的跟踪。这一贡献对于推动多目标跟踪技术的发展具有重要意义。

measurement

(| 1)

kkz

. Second, the valid measurements

are selected using the gating technique and the probability of

the target measurement being selected is

,Gt

[20]. Then, the

data association hypotheses are formulated and the

conditional densities of all targets as well as the probabilities

of all hypotheses are calculated. Finally, the marginalized

posterior densities of all targets are approximated using

Gaussian densities.

A hypothesis is a target-measurement association

event. In each hypothesis: 1) each target can be assigned

zero measurement or one of the measurements which falls in

the selection window of the target, 2) each measurement can

be allocated to zero or one of the target. Let



and



denote the hypothesis h and the total number of hypotheses,

respectively. The probability for



is computed as



()

() 1 ()

( ) ( ); ( | 1), (k)

() (1 )





 

















th th

htjj

PNkkk

zz S

(8)

where



( ); ( | 1), (k)

tj j

Nkkkzz S

is a Gaussian PDF with

mean

(| 1)

kkz

and covariance matrix

(k)

evaluated at

()

() ( | 1)

kkkSHP HR

is the residual error

covariance,



is the number of false measurements, V is the

volume of the valid space,

()



is the measurement

indicator (if measurement

()

is associated with a target



()1





; otherwise,

()0





()





is the

target indicator (if target t is associated with a measurement



()1







; otherwise,

()0







), and

is the

normalization factor.

Using the probabilities of hypotheses, the updated state

of target t is described as

() ( ) ()







kP kxx



(9)

where

()

is the conditionally updated state of target t in

hypothesis



() ( | 1)1 ( ) () ( )







 







tt h th

h jth j jth

kkk w kwxx x

(10)

with

1if

()

0other

















jt h

(11)

denoting that

()

can be associated with target t in



For the updated covariance of target t

() ( ) () () () () ()













tttttt

hh h h

kP kkkkkPPxxxx



(12)

Here,

()

is the conditionally updated covariance of

target t in hypothesis



() ( | 1)1 ( )

( | 1) () () () ( )







 













tt h

hjth

tttth

kkk w

kk k k k w

PKSK

(13)

and

() ( | 1) ( ())





tt t

kkk kKP HS

is the filter gain.

From (9) and (12), one can see that the posterior PDF

is approximated by a single Gaussian PDF at each time step.

So, the performance of JPDA relies heavily on Gaussian

approximations. Nevertheless, JPDA is often described in an

alternative fashion, where each track is updated by a

weighted average of the measurements.

2.3. The RFS family

In this section, we show the relation between the

PDF of ordered (labelled) targets and the PDF of unordered

set of targets [18]. This relation provides the possibility to

switch the posterior PDF of JPDA, when target identity is

irrelevant. To formulate the problem, the joint state vector is

expressed as

 

TT T

,,,

















Xx x x

(14)

where

is the state vector of target t and n is the number of

targets. In this paper, the targets are unordered and the RFS

is used to describe them. Since an RFS is without ordering,

the targets in (14) can be reordered without affecting the

RFS. The relation between the ordered PDF

()p X

and the

unordered RFS PDF

()R X

is as follows [18]



()









RpXX

(15)

where



is the set of all possible permutations of the joint

target state and the notation



describes the joint state

vector in permutation of



. That is, to go from an ordered

PDF to an RFS PDF, we sum the ordered PDF over all

possible permutations of the joint state vector.

There are several ordered PDFs corresponding to the

same RFS PDF. An ordered PDF

()p X

belonging to the

same RFS family of

()p X

can be obtained by multiplying

each







p X

with a constant weight

()





() ()









pqpXX

(16)

and the weight

()



must satisfy

() 1,0 () 1













(17)

In this way, the ordered PDF

()p X

is switched into

()p X

For the JPDA, it is possible to switch between the ordered

PDFs in the same RFS family to obtain an PDF which can

provide a more accurate Gaussian approximation than others.

3. Proposed method

In this paper, a novel and efficient approach is

proposed to improve the tracking accuracy of the JPDA.

First, the posterior PDF of the JPDA is described as a GMM.

Then, the posterior PDF is switched into another ordered

PDF in the same RFS family. To obtain a good performance,

the switching criterion is important. Therefore, we first

develop the cost function in Section 3.1. In Section 3.2, the

iterative optimization of the posterior PDF is conducted

based on the cost function. Then, an indicative example is

shown in Section 3.3. In Section 3.4, the computational

complexity of the proposed method is analyzed.

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