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### 可标签的多目标跟踪技术详解 #### 核心概念与背景介绍在现代信息技术领域，特别是信号处理与智能系统研究中，多目标跟踪（Multi-Target Tracking, MTT）是一个极其重要的课题。它涉及从一系列观测数据中估计未知且随时间变化的目标数量及其各自的状态。本文旨在深入探讨一种名为“可标签的随机有限集”（Labeled Random Finite Sets, LRFS）的方法，并基于此方法提出了一种高效的多目标跟踪滤波器——δ-广义标记多伯努利（δ-Generalized Labeled Multi-Bernoulli, δ-GLMB）滤波器。 #### 技术要点解析 ##### 1. 基础理论：随机有限集（Random Finite Sets, RFS） - **定义**：随机有限集是一种数学模型，用于描述集合中的元素数量不确定的情况。这种模型非常适合描述多目标跟踪问题中目标数量未知的情况。 - **应用**：通过将目标视为一个随机有限集，可以有效地处理检测不确定性、杂波干扰以及数据关联不确定性等问题。 ##### 2. 标签的重要性 - **目标轨迹追踪**：在多目标跟踪场景中，不仅需要估计目标的位置等状态参数，还需要追踪每个目标的历史轨迹。为了实现这一目的，引入了标签机制。 - **实现方式**：每个目标被赋予一个唯一的标签，这些标签可以是简单的整数或更复杂的编码。通过这种方式，即使多个目标之间发生交错移动，也能够准确地追踪到每个目标的运动轨迹。 ##### 3. δ-GLMB滤波器的工作原理 - **迭代过程**：δ-GLMB滤波器的每一次迭代包括更新操作和预测操作两个步骤。 - **更新操作**：利用观测数据更新目标状态的概率分布。该过程会生成大量加权的目标指数函数，其中包含难以计算的大量项。 - **预测操作**：根据现有信息预测下一时刻目标的状态。同样会产生大量项的加权目标指数函数。 - **简化策略**： - 使用排名分配算法（Ranked Assignment Algorithm）来截断更新操作中的项，从而保留最重要的项而避免计算所有项。 - 使用第k短路径算法（K-th Shortest Path Algorithm）来简化预测操作中的项。 - 通过概率假设密度滤波（Probability Hypothesis Density Filtering, PHD）等工具进一步减少计算量。 ##### 4. 错误分析 - **误差评估**：由于截断操作会导致一定的误差，因此需要对多目标密度中的L1误差进行量化分析。 - **性能优化**：通过对误差来源的深入理解，可以采取措施进一步提高跟踪精度和效率。 #### 实际应用场景与挑战 - **分布式系统**：在分布式环境下，多个传感器协同工作，共同完成多目标跟踪任务。此时，如何有效地融合来自不同节点的数据成为关键。 - **集群协同**：对于大规模的多目标跟踪任务，需要设计合理的集群架构来支持高效的信息交换和处理。 - **信息融合**：在处理来自多个源的数据时，必须采用先进的信息融合技术来提高整体系统的鲁棒性和准确性。 #### 结论本文通过对可标签的多目标跟踪技术的研究，特别是对δ-GLMB滤波器的详细介绍，展示了其在解决复杂多目标跟踪问题方面的强大能力。随着未来技术的发展，该领域的研究将继续深化，为实际应用提供更多有力的支持。

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arXiv:1312.2372v2 [stat.CO] 28 Feb 2017

PREPRINT: IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 62, NO. 24, PP. 6554–6567, 2014 1

Labeled Random Finite Sets and the Bayes

Multi-Target Tracking Filter

Ba-Ngu Vo

∗

, Ba-Tuong Vo

†

, Dinh Phung

‡

Abstract—An analytic solution to the multi-target Bayes re-

cursion known as the δ-Generalized Lab eled Mul ti-Bernoulli (δ-

GLMB) ﬁlter has been recently proposed in [1]. As a sequel

to [1], this paper details efﬁcient implementations of the δ-

GLMB multi-target tracking ﬁlter. Each iteration of this ﬁlter

involves an update operation and a prediction operation, both of

which result in weighted sums of multi -target exponentials with

intractably large number of terms. To truncate these sums, the

ranked assignment and K-th shortest path algorithms are used

in the update and prediction, respectively, to determine the most

signiﬁcant terms without exhaustively computing all of the terms.

In addition, using tools derived from the same framework, such

as probability hypothesis density ﬁltering, we present inexpensive

(relative to the δ- GLMB ﬁlter) look-ahead strategies to reduce

the number of computations. Characterization of the L

-error in

the multi-target density arising from the truncation is presented.

Index Terms—Random ﬁnite set, marked point process, con-

jugate prior, Bayesian estimation, target tracking.

I. INTRODUCTION

Multi-target ﬁltering involves the on-line estimation of an

unknown and time-varying number of targets and their individ-

ual states from a sequence of observations [2]–[4]. While the

term multi-target ﬁltering is often used interchangeably with

multi-target tracking , there is a subtle difference. In multi-

target tracking we are a lso interested in the trajectories of

the targets (indeed, real multi-target tracking systems require

track labels). This work is concerne d with a Bayesian multi-

target ﬁltering solution that also provides estimates of target

trajectories, hence the nam e multi-target tracking ﬁlter.

The key challenges in multi-target ﬁltering/tracking include

detection uncertainty, clutter, and data association uncer-

tainty. To da te , three m a jor approaches to multi-target track-

ing/ﬁltering have em erged as the main solution paradigms.

These a re, Multiple H ypotheses Tracking (MHT), [2], [5]–

[7], Joint Probabilistic Data Association (JPDA) [2], [4], and

Random Finite Set (RFS) [3].

The ran dom ﬁnite set (RFS) approach pr ovides an ele-

gant Bayesian formulation of the mu lti- ta rget ﬁltering/tracking

problem in which the collection of target states, refer red to as

Acknowledgement: This work is supported by the Australian Research

Council under the Future Fellowship FT0991854 and Discovery Early Career

Researcher Award DE120102388

B.-N. Vo and B.-T. Vo are with the Department of Electrical and Com-

puter Engineering, Curtin University, Bentley, WA 6102, Australia (email:

{ba-ngu,ba-tuong}.vo@curtin.edu.au). D. Phung is with the Faculty of Sci-

ence, Engineering & Built Environment, Deakin University, Geelong, VIC

3220, Australia (email: dinh.phung@deakin.edu.au)

the multi-target state, is tr eated as a ﬁnite set [3], [8]. The ra-

tionale beh ind this representatio n traces back to a fundamental

consideration in estimation theory–estimation error [9]. This

mathematical framework subsequently became a very popular

multi-target estimation method with applications in sonar [10],

computer vision [11], [12], [13], [14], ﬁeld rob otics [15], [16],

[17], [18], [19] trafﬁc monitoring [20], [21], [22], cell biology

[23], [13], [24], sensor network and distributed estimation [ 25],

[26], [27], [28] etc.

The centerpiece o f the RFS ap proach is the Bayes m ulti-

target ﬁlter [3], which recursively pr opagates the ﬁltering

density of the multi-target state forward in time. This ﬁlter

is also a (multi-target) tracker when target identities or labe ls

are incor porated into individual ta rget states. Due to the nu-

merical complexity of Bayes multi-target ﬁlter, the Probability

Hypothesis Density (PHD) [8], Cardinalized PHD (CPHD)

[29], and multi-Bernoulli ﬁlters [30], [9] have be e n developed

as approximations. These ﬁlters, in principle, are not multi-

target trackers b ecause they rest on the premise that targets

are indistinguishable.

In [1], [31], the notion of labeled RFSs is introduc ed to

address target trajectories and their uniqueness. The key results

include conjugate priors that are closed under the Chapman-

Kolmogorov equation, and an analytic solution to the Bayes

multi-target trackin g ﬁlter known as the δ-generalized labeled

multi-Berno ulli (δ-GLMB) ﬁlter. Although a simulation result

was presented to verify the solution, speciﬁc implementation

details were not given.

As a sequel to [1], the aim of this paper is to complement

its theoretical co ntributions with practical algorithms that will

facilitate the development of applications in signal processing

and related ﬁelds. In pa rticular, we p resent an efﬁcient and

highly parallelizable implementation of the δ-GLMB ﬁlter.

Each iteration o f the δ-GLMB ﬁlter involves multi-target ﬁlter-

ing and predictio n densities that are weighted sums of multi-

target exponentials. While these sums are expre ssible in closed

forms, the number of terms g rows super-exponentially in time.

Furthermore, it is not tractable to exhau stively compute all the

terms of the multi- ta rget densities ﬁrst and then truncate by

discarding those deemed insig niﬁcant.

The key innovation is the truncation of the multi-ta rget

densities without exhaustively computing all th eir components.

The multi-target ﬁltering and p rediction densities are truncated

using the ranked assignment and K-shortest paths algorithms,

respectively. Techniques such as PHD ﬁltering are used as

inexpensive look-ahead strategies to drastically r educe the

number of calls to ranked assignment and K-shortest paths

algorithm s. Moreover, w e establish that truncation by discard-

PREPRINT: IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 62, NO. 24, PP. 6554–6567, 2014 2

ing δ-GLMB components with small weights minimizes the

error in the multi-target density. To our best knowledge,

this is the ﬁrst result r egar ding the effect of truncation on the

multi-target p robability law.

The paper is organize d as follows. Backgrou nd on labeled

RFS and the δ-GLMB ﬁlter is provided in section II. Section

III establishes the L

-distance between a δ-GLMB density

and its truncated version. Sections IV and V present efﬁcient

implementations δ-GLMB ﬁlter update and prediction respec-

tively. Section VI details the multi-target state estima tion, and

discusses loo k-ahead strategies to reduce the computational

load. Numer ical results are presented in Section VII and

conclud ing remarks are given in Sectio n VIII.

II. BACK GROUND

This section summarizes the lab e le d RFS formulation of

the multi-target tracking problem and the δ-GLMB ﬁlter pro-

posed in [1]. Labeled RFS is summarize d in subsection II-A,

followed by Bayes multi-target ﬁlter ing basics in subsections

II-B, II-C and II-D. T he δ-GLMB multi-target density and

recursion are summarized in subsections II-E and II-F.

Throu ghout the paper, we use the standard inner product

notation hf, gi ,

f(x)g(x)dx, and the following multi-

object exponential notation h

x∈X

h(x), where h is

a real-valued function, with h

∅

= 1 b y convention. We denote

a genera liza tion of the Kro neker delta that takes arbitrary

arguments such as sets, vectors, etc., by

(X) ,



1, if X = Y

0, otherwise

and the inclusion function, a generalization of the indicator

function, by

(X) ,



1, if X ⊆ Y

0, otherwise

We also write 1

(x) in place of 1

({x}) when X = {x}.

A. Labeled RFS

An RFS is simply a ﬁnite-set-valued random variable [32],

[33]. In this paper we use the Finite Set Statistics (FISST)

notion of integration/density to charac terize RFSs [3], [8].

Treatments o f RFS in the context of multi-target ﬁltering can

be found in [3], [34], [35].

To in corporate target identity, each state x ∈ X is augmented

with a unique label ℓ ∈ L= {α

: i ∈ N}, where N denotes

the set of positive integers and the α

’s are distinct.

Let L : X×L → L be the projection L((x, ℓ)) = ℓ, then a

ﬁnite sub set X o f X×L has distinct labels if a nd only if X and

its labels L(X) = {L(x) : x ∈ X} have the same cardinality,

i.e. δ

|X|

(|L(X)|) = 1. The function ∆(X) , δ

|X|

(|L(X)|) is

called the distinct la bel indicator.

Deﬁnition 1 A labeled RFS with state space X and (discrete)

label space L is an RFS on X×L such that each realization

has distinct labels.

The u nlabeled version of a labeled RFS is obta ined by

simply discarding the labels. Consequently, the cardinality

distribution (the distribution of the number of ob je c ts) of a

labeled RFS is the same as its unlabeled version.

For the rest of the paper, single-ob je ct states are represented

by lowercase letters (e.g. x, x), while multi-object states are

represented by uppercase letters (e.g. X, X), symb ols for

labeled states and their distributions are bolded to distinguish

them from unlabeled ones (e.g. x, X, π, etc.) , spaces are

represented by blackboard bold (e.g. X, Z, L, N, etc.), and

the class of ﬁnite subsets of a space X is denoted by F(X).

The integral of a fun ction f :X×L → R is given by

f(x)dx =

ℓ∈L

f((x, ℓ))dx.

B. Bayesian Multi-target Filtering

To incorporate target tracks in the Bayes multi-target ﬁl-

tering framework, targets are identiﬁed b y an ordered pair

of integers ℓ = (k, i), where k is the time of birth, and

i ∈ N is a unique ind ex to distinguish objects born at the

same time. Figure 1 illustrate s the assignment of labels to

target trajectories. The label space for objects born at time

k, denoted as L

, is then {k} × N. An object born at time

k, has state x ∈ X×L

. The label space for targets at

time k (including those born prior to k), denoted as L

0:k

is construc ted recursively by L

0:k

= L

0:k−1

∪ L

. A multi-

object state X at time k, is a ﬁnite subset of X×L

0:k

. Note

that L

0:k−1

and L

are disjoint.

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WLPH

WUDFNV

  «







PXOWLWDUJHWVWDWHV

Fig. 1. An example of label assignments. The two tracks born at time 1 are

given labels (1,1) and (1,2), while the only track born at time 4 is given label

(4,1). Notice also the difference between multi-target states and target tracks.

Suppose that at time k, there are N(k) target states

k,1

, . . . , x

k,N (k)

, each taking values in the (labeled) state

space X × L, and M (k) m easurements z

k,1

, . . . , z

k,M (k)

each

taking values in an observation space Z. In the random ﬁnite

set formulation, the set of targets and observations, at tim e k,

[3], [8] are treated as the multi-target state and multi-target

observation, respectively

= {x

k,1

, . . . , x

k,N (k)

= {z

k,1

, . . . , z

k,M (k)

The multi-ta rget posterior density captures all information

on the set of target trajectories conditioned on the measure-

ment histo ry Z

0:k

= (Z

, ..., Z

), and is given recursively for

k ≥ 1 by

0:k

)

∝ g

k|k−1

k−1

)π

0:k−1

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