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在现代监控和防御系统中，对于多目标的联合检测与估计（Joint Detection and Estimation，简称JDE）是一个重要的研究领域。本文讨论了如何在一个具有杂波和漏检的传感器环境下，通过单个传感器实现多目标的联合检测与状态估计，并提出了一个性能评估的误差范围。在随机有限集合（Random Finite Set，简称RFS）框架下，通过使用多伯努利或泊松近似来处理多目标贝叶斯递归，进而发展了对多目标进行联合检测与状态估计的误差范围。文章所涉及的关键知识点包括以下几个方面： 1. **多目标跟踪**：在复杂环境中，多个目标同时存在，且目标数量可能未知，如何准确跟踪这些目标的运动状态是多目标跟踪领域的重要研究课题。在本文中，多目标跟踪指的是同时估计目标数量和状态的过程。 2. **传感器性能评估**：对传感器性能进行评估是为了确定其在特定测量设置下的性能极限，这对于设计和管理传感器，以改善目标跟踪性能至关重要。文章提出了一个典型的性能评估指标——误差范围，它是所有滤波器性能极限的一个度量。 3. **杂波和漏检的影响**：在传感器测量中，除了目标的真实反射信号外，还可能存在由环境杂波或噪声引起的额外信号，这被称为杂波。漏检是指传感器未能检测到实际存在的目标。杂波密度和检测概率对误差范围有显著影响，这是通过仿真结果验证的。 4. **最大后验概率（MAP）检测器和无偏估计器**：在本研究中，所有检测器和估计器都被限制为最大后验概率检测器和无偏估计器。MAP检测器是在给定测量数据的条件下，计算目标存在概率最大的状态；而无偏估计器则是在估计目标状态时尽可能减少估计误差。 5. **最优子模式分配（OSPA）距离**：为了衡量真实和估计状态集合之间的误差度量，本研究使用了二阶最优子模式分配距离。这是用于比较两个目标集合相似度的一种度量方法，能够同时考虑目标数量和状态的差异。 6. **概率假设密度（PHD）和基数化PHD（CPHD）滤波器**：PHD滤波器是一种用于多目标跟踪的算法，它能够提供目标密度随时间的演变；而CPHD滤波器则是一种改进的PHD滤波器，它在PHD基础上考虑了目标数量的不确定性。通过研究这些滤波器在不同杂波密度和检测概率下的性能，验证了所提出的误差范围的有效性。 7. **多伯努利近似和泊松近似**：文章中采用了多伯努利近似或泊松近似来处理多目标贝叶斯递归。这种近似方法能够简化复杂的目标跟踪模型，使之在计算上更为可行，同时保持足够的准确性。 8. **随机有限集合框架（RFS框架）**：在多目标跟踪领域，RFS框架提供了一种处理不确定数量目标的方法。该框架允许在贝叶斯递归更新中考虑目标可能出现和消失的动态特性。综合以上各点，该研究工作在理论上具有重要的意义，为多目标跟踪系统的设计提供了新的评估方法，并为解决实际应用中的传感器管理和滤波器性能评估问题提供了有力的工具。同时，这些研究成果不仅在理论上具有创新性，而且在实际中具有广泛的应用前景。

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sensors

Article

Multi-Target Joint Detection and Estimation

Error Bound for the Sensor with Clutter and

Missed Detection

Feng Lian

∗

, Guang-Hua Zhang, Zhan-Sheng Duan and Chong-Zhao Han

Ministry of Education Key Laboratory for Intelligent Networks and Network Security (MOE KLINNS),

College of Electronics and Information Engineering, Xi’an Jiaotong University, Xi’an 710049, China;

MichaelZgh@stu.xjtu.edu.cn (G.-H.Z.); zsduan@xjtu.edu.cn (Z.-S.D.); czhan@xjtu.edu.cn (C.-Z.H.)

* Correspondence: lianfeng1981@xjtu.edu.cn; Tel.: +86-158-2907-9318

Academic Editor: Fabrizio Lamberti

Received: 18 November 2015; Accepted: 21 January 2016; Published: 28 January 2016

Abstract:

The error bound is a typical measure of the limiting performance of all ﬁlters for the given

sensor measurement setting. This is of practical importance in guiding the design and management

of sensors to improve target tracking performance. Within the random ﬁnite set (RFS) framework, an

error bound for joint detection and estimation (JDE) of multiple targets using a single sensor with

clutter and missed detection is developed by using multi-Bernoulli or Poisson approximation to

multi-target Bayes recursion. Here, JDE refers to jointly estimating the number and states of targets

from a sequence of sensor measurements. In order to obtain the results of this paper, all detectors

and estimators are restricted to maximum a posteriori (MAP) detectors and unbiased estimators,

and the second-order optimal sub-pattern assignment (OSPA) distance is used to measure the error

metric between the true and estimated state sets. The simulation results show that clutter density

and detection probability have signiﬁcant impact on the error bound, and the effectiveness of the

proposed bound is veriﬁed by indicating the performance limitations of the single-sensor probability

hypothesis density (PHD) and cardinalized PHD (CPHD) ﬁlters for various clutter densities and

detection probabilities.

Keywords:

performance evaluation; error bound; multi-target tracking; joint detection and

estimation; random ﬁnite set

1. Introduction

The problem of joint detection and estimation (JDE) of multiple targets arises from many

applications in surveillance and defense [

], where the number of targets is unknown and the sensor

may receive measurements generated randomly from either targets or clutters. There is no information

about which are the measurements of interest or which are the clutters. The aim of multi-target JDE is

to determine the number of targets and to estimate their states if exist using prior information, as well

as a sequence of the sensor measurements. In recent years, multi-target JDE has attracted extensive

attention, and many approaches for it have been proposed [2–10].

Obviously, it is very necessary to ﬁnd an error (lower) bound to assess the achievable performance

of the multi-target JDE algorithms for the given sensor measurements. It is well known that

Tichavsky et al. [11]

proposed a recursive posterior Cramér-Rao lower bound (CRLB) for evaluating

the performance of nonlinear ﬁlters when a target was asserted and observed by a sensor. Then, the

CRLB was extended to the cases in which clutter or missed detection was present in the sensor [

–

Nevertheless, these CRLBs [

–

] could barely be applied to such a JDE problem, since CRLB only

considers the estimation error of a target state, but not the detection error of the target number (or

Sensors 2016, 16, 169; doi:10.3390/s16020169 www.mdpi.com/journal/sensors

Sensors 2016, 16, 169 2 of 18

existence/non-existence of a target). Within random ﬁnite set (RFS) [

] framework, Rezaeian and

Vo [

] derived the static error bounds for JDE of a single target observed by a single sensor with clutter

and missed detection. Tong et al. presented a recursive form of a single-sensor single-target error bound

based on CRLB when only missed detection, but not clutter, exists [

] and then extended the result

of [

] to the single-sensor multi-target case with the more rigorous restriction that neither clutter nor

missed detection exists [

]. Note that the bounds in [

] actually do not include the detection error

generated by the uncertainty of target number, since the target number can be completely determined

by the measurement number by restricting the sensor observation model to the one in [17,18].

This paper proposes an RFS-based single-sensor multi-target JDE error bound when clutter

and missed detection may simultaneously exist in the sensor. In order to obtain the results of this

paper, the multi-target Bayes recursion is approximated as a multi-Bernoulli process [

] or a Poisson

process [

], and all detectors and estimators are restricted to maximum a posterior (MAP) detectors

and unbiased estimators. Since the JDE error is the average distance between true and estimated state

sets, the second-order optimal sub-pattern assignment (OSPA) distance [

] rather than the Euclidean

distance is used as the error metric. Finally, the simulation results show that clutter density and

detection probability have signiﬁcant impacts on the proposed bound, and the effectiveness of the

proposed bound is veriﬁed by indicating the performance limitations of the single-sensor probability

hypothesis density (PHD) [

] and cardinalized PHD (CPHD) [

] ﬁlters for various clutter densities and

detection probabilities.

The rest of the paper is organized as follows. Section 2 presents the background for deriving our

results. In Section 3, we derive the proposed bound by using multi-Bernoulli or Poisson approximation.

A numerical example is presented in Section 4. The conclusions and future work are given in Section 5.

Relevant mathematical proofs are provided in Appendices A and B.

2. Background

• Set integral: For any real-valued function ϕ(X) of a ﬁnite-set variable X, its set integral is [4]:

ϕ(X)δX=

∞

∑

n=0

(

)

(1)

· · ·dx

(n)

= ϕ(∅) +

∞

∑

n=1

(

)

(1)

· · ·dx

(n)

(1)

where

(i)

i=1

⊆ X

denotes a

-points set (that is, the cardinality of the set

) and

denotes the space of X

. In this paper, we note X

= ∅.

•

Multi-Bernoulli RFS: A multi-Bernoulli RFS

is a union of

independent Bernoulli RFSs

(i)

X =

i=1

(i)

. Its density is completely described by parameter Υ =

n

(i)

, p

(i)

o

i=1

as [6]:

f (X) = π(∅)

∑

1≤j

6=···6=j

|X|

≤M

|X|

∏

i=1

)

1 − r

)



(i)



, with π(∅) =

∏

i=1



1 − r

(i)



(2)

where

| · |

denotes the cardinality of a set,

(i)

∈ (

0, 1

)

denotes the probability of

(i)

6= ∅

and

(i)



(i)



denotes the density of x

(i)

• Poisson RFS: An RFS X is Poisson if its density f (X) is:

f (X) = e

−η

∏

x∈X

υ(x), with η =

υ(x)dx and υ(x) = η f (x) (3)

where

υ(x)

denotes the intensity function of the Poisson RFS

is the average number of elements

in X and f (x) is the density of single element x ∈ X.

Sensors 2016, 16, 169 3 of 18

•

Second-order OSPA distance: The OSPA distance of order

p =

2 between set

and its estimate

is [19]:















0, |

X| = |X| = 0

min

τ∈Π

max(|

X|,|X|)

min(|

X|,|X|)

∑

t=1

min



(t)

− ˆx

(τ(t))



X|−|X|

max(|

X|,|X|)

, |

X| + |X| > 0

(4)

where

denotes the set of permutations on

{

1, 2, ...,

c >

0 denotes the cut-off parameter,

max(·)

min(·)

denotes the maximization or minimization operation and

|| · ||

denotes the

two-norm. The OSPA metric is comprised of two components, each separately accounting for

“localization” and “cardinality” errors between two sets. The localization error arises from the

estimates paired with the nearest truths, while the cardinality error arises from the unpaired

estimates. Schuhmacher et al. [

] have proven that the OSPA distance with

p ∈ [

∞)

and

c >

0 is

indeed a metric, so it can be used as a principled performance measure.

•

Information inequality and CRLB: Given a joint probability density

f (x

X × Z

, under

regularity conditions and the existence of

∂

log f (x, z)/∂x

∂x

, the information inequality states

that [20,21]:

f (x, z)

(

−

(z)

)

dxdz ≥



∂

∂x

[

(z)

]



−1

l,l

(5)

where

ˆx(z)

denotes an estimate of

dimensional vector

based on

and

(z)

are, respectively,

the

-th components of

and

ˆx(z)

l =

1, ...,

, the notation

means the expectation with respect

to density f and J is known as the L × L Fisher information matrix:

[J]

i,j

= −E

∂

log f (x,z)

∂x

= −

f (x, z)

∂

log f (x,z)

∂x

dxdz, i, j = 1, 2..., L

(6)

where [J]

i,j

denotes the element on the i-th row and j-th column of matrix J.

For the particular case in which the estimator

ˆx(z)

is unbiased (that is,

[

ˆx(z)

]

= x

), the

information inequality of Equation (5) reduces to:

f (x, z)

(

−

(z)

)

dxdz ≥

−1

l,l

(7)

which is a result known as the CRLB. The Fisher information matrix

in Equation (7) is also

computed by Equation (6).

Note that the ordinary information inequality of Equation (5) holds without the unbiasedness

requirement on the estimator

x(z). However, unbiasedness is critical in the CRLB of Equation (7).

Explanation: In the current set up of this paper, our attention is restricted to the unbiased estimator

of multi-target states. Our future work will study the extension of the proposed bound to the

biased estimator by using the ordinary information inequality of Equation (5).

Moreover, Equation (5) or Equation (7) is satisﬁed with equality depending on a very restricted

condition. In [

], Poor concludes that, within regularity, the information lower bound is achieved

(that is, the “=” in Equation (5) or Equation (7) holds) by

ˆx(z)

if and only if

ˆx(z)

is in a one-parameter

exponential family (e.g., the linear Gaussian models for target dynamics and sensor observation

described in [11] for achieving the CRLB). More details about this can be found in [21].

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