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### 随机有限集模型中的粒子方法概览 #### 引言本文档提供了一个关于粒子方法在随机有限集（RFS）模型背景下应用的综述。粒子方法是处理非线性、非高斯系统状态估计问题的有效工具，特别适用于多目标跟踪问题。文章主要面向对标准贝叶斯滤波器中的粒子方法有一定了解的研究人员。 #### 粒子方法在标准贝叶斯滤波器中的应用 - **问题表述与标准贝叶斯滤波器**：首先介绍了粒子方法的基本原理及其在处理非线性动态系统状态估计问题中的应用。 - **粒子方法简介**：粒子滤波器是一种基于蒙特卡罗模拟的方法，通过一组加权样本（即“粒子”）来近似后验概率分布。 - **系统参数校准**：为了提高粒子滤波器的性能，需要对模型参数进行适当调整。 - **示例：仅方位角跟踪**：通过对仅使用方位角测量的单目标跟踪问题的分析，展示了粒子滤波器的应用效果。 #### 随机有限集理论背景 - **随机有限集概述**：定义了随机有限集的概念，并讨论了其在多目标跟踪中的应用。 - **常见的RFS变量**：介绍了一些在多目标跟踪中经常使用的RFS变量，如伯努利随机有限集等。 #### 随机有限集贝叶斯最优滤波器的粒子方法 - **随机有限集贝叶斯最优滤波器的公式化**：详细阐述了在随机有限集框架下如何建立贝叶斯最优滤波器。 - **粒子方法近似**：探讨了如何利用粒子方法来逼近随机有限集贝叶斯滤波器的解。 - **伯努利粒子滤波器**：伯努利粒子滤波器是一种特殊的粒子滤波器，适用于存在单个目标或不存在目标的情况。 - **示例：仅方位角检测与跟踪**：通过一个仅使用方位角测量进行目标检测和跟踪的例子，展示了伯努利粒子滤波器的实际应用效果。 #### PHD粒子滤波器 - **PHD滤波器的公式化**：PHD滤波器是一种针对多目标跟踪问题的有效滤波器，能够估计多个目标的存在概率密度。 - **粒子方法应用于PHD滤波**：介绍了一种基于粒子的PHD滤波方法，该方法可以处理更复杂的多目标跟踪场景。 - **跟踪算法校准**：讨论了如何校准PHD滤波器中的参数以优化跟踪性能。 - **示例：仅方位角多目标滤波**：通过一个仅使用方位角测量的多目标跟踪案例，验证了PHD粒子滤波器的有效性和鲁棒性。 #### 带标签的随机有限集贝叶斯跟踪滤波器 - **带标签的随机有限集**：带标签的随机有限集（Labeled RFS）是一种特殊的RFS，它为每个目标分配唯一的标签，从而允许精确跟踪每个目标的轨迹。 - **带标签的多目标转移和似然模型**：定义了带标签的多目标跟踪问题中的转移模型和观测模型。 - **δ-GLMB粒子滤波器**：δ-广义标记的多伯努利（δ-GLMB）粒子滤波器是一种先进的跟踪滤波器，它结合了带标签的RFS理论和粒子滤波技术，能够在复杂环境中准确地跟踪多个移动目标。通过这些章节的详细介绍，本文档不仅涵盖了粒子方法在随机有限集模型中的基本原理和技术细节，还通过具体的示例展示了这些方法在实际应用中的效果。这对于理解现代多目标跟踪技术以及开发新型跟踪算法具有重要的参考价值。

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arXiv:1602.03945v1 [cs.SY] 12 Feb 2016

An Overview of Particle Methods for Random Finite

Set Models

Branko Ristic

a,∗

, Michael Beard

, Claudio Fantacci

RMIT University, Australia

Curtin University, Australia

Universit`a di Firenze, DINFO, Florence, Italy

Abstract

This overview paper describes the particle metho ds developed for the imple-

mentation of the a class of Bayes ﬁlters formulated using the random ﬁnite set

formalism. It is primar ily intended for the readership already familiar with the

particle methods in the context of the standard Bayes ﬁlter. The focus in on the

Bernoulli particle ﬁlter, the proba bility hypothesis density (PHD) particle ﬁlter

and the generalise d labelled multi-Bernoulli (GLMB) particle ﬁlter. The per-

formance of the described ﬁlters is demo nstrated in the context of bearings-only

target tra cking application.

Keywords: Stochastic nonlinear ﬁltering, Monte Carlo estimation, particle

methods, random set models, target tracking, bearings- only measurements

∗

Corresponding author: B. Ristic, RMIT University, School of Electrical and Computer

Engineering, Rm 10.8.14, 376-392 Swanston str eet, Melbourne, VIC 3000, Australia; email:

branko.ristic@rmit.edu.au; tel: +61 3 9925 3768

Preprint submitted to Information Fusion Fe bruary 15, 2016

Notation

X the single object state space

F(X ) the multiple o bject state space

L the s pace of labels

x the state of a single object (a random vector)

X the state of multiple-objects (a random ﬁnite set, RFS)

X the state of multiple labelled objects (a labelled RFS)

Z measurement space

z a measurement of a single object (a random vector)

Z a detector output measurement (a RFS)

k discrete-time index

p(x) a probability density function (PDF) of x ∈ X

˜p(x, ℓ) a P DF of a labelled random vector (x, ℓ) ∈ X × L

f(X) a P DF of an RFS variable X ∈ F(X )

f(X) a P DF of a labelled RFS variable X ∈ (F(X ) × L)

1. Introduction

In many areas of science and engineering there is a need to infer the be haviour

of a stochastic dynamic system, using its pa rtial and indirect observations. By

combining (typically nonlinear) mathematical models of system evolution and

sensor measurements, one can formulate the optimal sequential estimator in

the Bayesian framework. This estimator, commonly referred to as the Baye s-

optimal (or simply Bayes) ﬁlter, provides a recursive formula for the complete

probabilistic characterisation of the dynamic system in the form of a time-

varying posterior probability density of its state [1].

For most nonlinear/non-Gaussian formulations, analytic closed-from solu-

tions of the Bayes ﬁlter are intractable. Practica l solutions, therefore, need

to be ba sed on approximations. Particle ﬁlters are a class of Monte Carlo

simulation ba sed methods which can provide very accurate approximations of

the Bayes ﬁlter. Despite being computationally expensive, particle ﬁlters have

become universally popular, primarily due to their accuracy, relatively simple

implementation and the ever increasing speed of computers. As a result of their

widespread application, a few good tutorials and books have been published on

the subject of particle ﬁlters [2], [3], [4], [5], [6], [7].

Particle ﬁlters have been introduced and traditionally applied as the ap-

proximate solutions of the standard Bayes ﬁlter, fo rmulated during the 1960s

[1] under the following assumptions: (i) the s tochastic dynamic system (object,

phenomenon) is permanently active (or present); (ii) observations are noisy, but

collected with perfect detectio n (i.e. there are no false or missed detections). All

the aforementioned tutorials and books discuss the particle ﬁlters in this context

only. However, in many practical applications, one may have to deal with mul-

tiple sto chastic dynamic systems (objects), which can be simultaneously active

(present), and which can randomly switch on and oﬀ (appear/disappear). In

addition, perfect detectio n using sur veillance sensors (e.g. radar, sonar, v ideo

cameras) is rarely possible [8]. Until r e cently, particle ﬁlters have been applied

to this class of problems using a cle ver combination of Bayesian estimation the-

ory with ad-hoc log ic. However, the recent advances in Bayesian estimation

using random ﬁnite set (RFS) models [9] resulted in elegant and rigoro us math-

ematical formulations of the Bayes-optimal and principled Bayes-suboptimal

ﬁlters, a pplicable to multiple interacting on/oﬀ switching systems w ith possibly

imperfect detection of measure ments.

This overview paper describes the particle methods developed for the im-

plementation of the new class of RFS-Bayes ﬁlters. It is primarily intended for

the readership alrea dy familiar with the particle methods in the context of the

standard Bayes ﬁlter. One of the most popular and convincing applications of

particle ﬁlters, versus standard approximation methods, such as the Extended

Kalman ﬁlter (EKF) [10] and unscented Kalman ﬁlter (UKF) [11], has been

for bearings-only tracking problems [5, Ch.6]. Hence, this applica tion has been

chosen to demonstrate throughout the paper diﬀerent RFS-Bayes particle ﬁlters

and their performance. The paper is organised as follows. Sec. 2 reviews the

particle method for the standard Bayes ﬁlter. The elements of mathematics for

random ﬁnite set models are presented in Sec. 3. The particle method for the

RFS Bayes-optimal ﬁlter and its special case, the Bernoulli ﬁlter, are discussed

in Sec. 4. A multi-target particle ﬁlter, referred to as the PHD particle ﬁlter,

is presented in Sec. 5. The labelled RFS Bayes tracking ﬁlters implemented

using the particle method are discussed in Sec. 6. T he summary and pointers

to advanced research topics are given in Se c. 7.

2. Particle methods for the standard Bayes ﬁlter

In order to familia rise with the notation, let us start with a quick review of

the standard Bayes ﬁlter and the corresponding particle methods. The problem

is by no means simple and it is still an active and highly relevant rese arch topic.

2.1. Problem formulation and the standard Bayes ﬁlter

Suppose the state vector x

∈ X provides the complete speciﬁcation of the

state of a dynamic s ystem (object, phenomenon) a t time t

. Here X ⊆ R

the state spa ce, while k is the dis crete-time index corresponding to t

. Let us

adopt the discrete-time additive-noise formulation, speciﬁed by two equations:

= f

k−1

) + v

k−1

, (1)

= h

) + w

, (2)

referred to as the dynamics equation and the measurement equation, respe ctively.

Function f

k−1

: R

→ R

in (1) is a nonlinear transition function deﬁning the

temporal evolution of the state vector as a ﬁrst-o rder Markov process. Random

disturbances v

∈ R

, also known as process noise, are assumed to be indepen-

dent identically distributed (IID) according to the probability density function

(PDF) p

. Functio n h

: R

→ R

in (2) deﬁnes the relationship between

the state x

and the meas urement z

∈ Z, where Z ⊆ R

is the measurement

space. Rando m disturbances w

∈ R

, also known as measurement noise, are

assumed independent of v

, and modelled as an IID process with the PDF p

Typically n

< n

, giving rise to the term partial observations of the system.

In the formulation speciﬁed by (1)-(2), the functions f

and h

, the proba-

bility distributions p

and p

, and the PDF of the state vector at initial time

k = 0, (i.e. p

)), are all assumed known. Equations (1) and (2) eﬀectively

deﬁne two probability functions: the transitional density π

k|k−1

k−1

) =

− f

k−1

)) a nd the likelihood function g

) = p

− h

)).

The pro ble m is to compute recurs ively the posterior PDF of the state, denoted

as p

k|k

1:k

) at discrete-time k, whe re the notation z

1:k

stands for the se-

quence z

, z

, · · · , z

The solution is usually presented as a two step procedure. Let p

k−1|k−1

k−1

1:k−1

)

denote the posterior PDF at k − 1. The ﬁrst step predicts the density of the

state to time k via the Chapman-Kolmogo rov equation [1]:

k|k−1

1:k−1

k|k−1

′

)p(x

′

1:k−1

)dx

′

. (3)

The second step applies Bayes rule to update p(x

1:k−1

) using measurement

k|k

1:k

) =

) p

k|k−1

1:k−1

)

|x)p

k|k−1

(x|z

1:k−1

)dx

. (4)

Knowing the posterior p

k|k

1:k

), one can compute a point estimate of the

state

(e.g. as the mean or the mode of the posterior) and a conﬁdence (or

credible) interval.

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