多未解析目标组联合检测与估计的误差界限分析
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"这篇研究论文探讨了在存在杂波和漏检的情况下，对多个未解决目标组进行联合检测和估计的误差范围问题。利用随机有限集（RFS）理论和信息不等式，作者推导出了一个联合检测和估计（JDE）的误差界限。JDE的目标是确定未解析目标组的数量并估计其状态。为了得出这些结果，首先将未解决目标组的状态建模为多伯努利RFS。"
正文:
在信息技术和信号处理领域，尤其是在雷达、无线通信和图像处理等应用中，对多个目标的检测和估计是一项核心任务。当目标过于接近，无法单独分辨时，这种情况被称为“未解决目标”。在这种情况下，联合检测和估计（JDE）技术成为了解决问题的关键，因为它可以同时处理多个未解析目标组，提供关于它们数量和状态的信息。
这篇由Lian Feng、Duan Zhan-Sheng、Yuan Xiang-Hui和Han Chong-Zhao共同撰写的论文，发表在《数字信号处理》（Digital Signal Processing）期刊上，深入研究了在存在杂波和漏检情况下的JDE误差边界。论文中的主要贡献是通过随机有限集（RFS）理论建立数学模型，这是一种处理不确定性和复杂性的强大工具，特别适用于目标跟踪和检测问题。
RFS理论允许研究人员以概率方式描述和分析一组不确定的、可能互有交叠的目标。在这里，多伯努利RFS被用来表示未解决目标组的状态，它结合了单个目标的存在概率和多个目标的统计特性。通过这种方式，论文能够考虑目标组的不确定性，如目标数量的变化和目标状态的估计误差。
接着，论文利用信息不等式来推导误差界限，这是一个关键的分析工具，可以量化在特定条件下信息处理的性能极限。在存在杂波（即无用信号）和漏检（未检测到的目标）的情况下，这些界限对于评估检测和估计算法的性能至关重要。通过这些界限，研究人员和工程师可以更好地理解算法的潜在表现，并优化他们的设计。
此外，论文还讨论了如何将这些理论成果应用于实际系统，例如雷达系统，以提高目标检测和跟踪的精度。这包括如何处理传感器数据，如何设计有效的检测和估计算法，以及如何在实际操作中减小误差。
这篇研究论文对理解和改进在复杂环境中处理多个未解析目标的问题提供了理论基础和实用指导。通过推导误差界限，它为系统设计者提供了评估和比较不同JDE方法的依据，有助于推动未来在信号处理领域的技术创新。
12 L. Feng et al. / Digital Signal Processing 51 (2016) 10–24
is n), Y
n
denotes the space of Y
n
. In this paper, we note Y
0
=∅.
The expectation of η(Y ) over density f (Y ) is
E[η]=
η(Y ) f (Y )δY . (2)
2.2. Multi-Bernoulli RFS
A multi-Bernoulli RFS Y is defined by a union of M indepen-
dent
Bernoulli RFSs Y
(i)
, Y =
M
i
=1
Y
(i)
. The density of the multi-
Bernoulli
RFS Y is given by [29],
π(Y ) = π(∅)
1≤ j
1
=···= j
|Y |
≤M
|Y |
i=1
r
j
i
1 − r
j
i
f
j
i
( y
i
), (3)
where |Y | denotes the cardinality of a set Y , r
i
∈[0, 1] denotes the
probability of Y
(i)
= ∅, f
i
( y
i
) denotes the density of y
i
,
π(∅) =
M
i=1
(1 − r
i
). (4)
Thus, a multi-Bernoulli RFS Y is completely described by the
multi-Bernoulli parameter set ϒ ={(r
i
, f
i
)}
M
i
=1
.
2.3. OSPA distance
Let e(Y
n
,
ˆ
Y
ˆ
n
) denote the error metric between the set Y
n
=
{
y
1
, ..., y
n
} and its estimate
ˆ
Y
ˆ
n
={
ˆ
y
1
, ...,
ˆ
y
ˆ
n
}. Since Y
n
and
ˆ
Y
ˆ
n
are random sets, the usual Euclidean distance error for random
vectors cannot be applied here. To overcome this, the OSPA dis-
tance
of order p = 2with cut-off c > 0 [30,32] is used to define
e(Y
n
,
ˆ
Y
ˆ
n
),
e
2
(Y
n
,
ˆ
Y
ˆ
n
) =
⎧
⎨
⎩
0,
ˆ
n = n = 0,
min
τ ∈
max(
ˆ
n,n)
min(
ˆ
n,n)
t=1
min(c
2
, y
t
−
ˆ
y
τ (t)
2
2
)+c
2
|
ˆ
n−n|
max(
ˆ
n,n)
,
ˆ
n + n > 0, (5)
where
n
denotes the set of permutations of {1, 2, ..., n}, max(·)
or min(·) denotes the maximization or minimization operation,
·
2
denotes 2-norm.
The
OSPA metric is comprised of two components each sepa-
rately
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. [30] have proved that the
OSPA distance with p ∈[1, ∞) and c > 0is indeed a metric, so it
can be used as a principled performance measure.
2.4. Information inequality
Given a joint probability density f (x, z) on X × Z, under reg-
ularity
conditions and the existence of ∂
2
log f (x, z)/∂x
i
∂x
j
, the
following inequality holds ([28], pp. 169–171, 186),
Z
X
f (x, z)
x
l
−
ˆ
x
l
(z)
2
dxdz ≥
J
−1
l,l
, (6)
where
ˆ
x(z) =[
ˆ
x
1
(z), ...,
ˆ
x
L
(z)]
T
denotes an unbiased estimate of
the L dimensional vector x =[x
1
, ..., x
L
]
T
using z; x
l
and
ˆ
x
l
(z) are,
respectively, the lth components of x and
ˆ
x(z ), l = 1, ..., L; J is the
L × L Fisher information matrix
[ J]
i, j
=−E
f
∂
2
log f (x, z)
∂x
i
∂x
j
=−
Z
X
f (x, z)
∂
2
log f (x, z)
∂x
i
∂x
j
dxdz, i, j = 1, 2,...,L. (7)
where [ J]
i, j
denotes the element on the ith row and jth column
of matrix J .
If
x is estimated by
ˆ
x(z) = E[x|z], then (6) is satisfied with
equality [24].
3. Problem statement
The state of a single unresolved target-group is modeled by an
augmented vector of form
x
=[a, x]
T
∈ X
, (8)
where the L-dimensional vector x =[x
1
, ..., x
L
]
T
∈ X ⊂ R
L
de-
notes
a conventional single-target state in the space X ; a denotes
the expected number of individual targets involved in the group
(according to the concept of continuous individual target number
proposed by Mahler ([3], pp. 437–440), a ∈ A ⊂ R
+
is a positive
real number); the L + 1-dimensional vector x
models a cluster
of a individual targets collocated at the single state x; X
denotes
the state space of a single unresolved target-group, X
= A × X ⊂
R
+
× R
L
.
Let
X
(t)
denote the state set of the tth unresolved target-group.
The tth group either exists with probability r
t
∈[0, 1] with its state
x
satisfying the density f
t
(x
t
), or does not exist with probability
1 − r
t
. Thus, it is suitable to model X
(t)
as a Bernoulli RFS with
the density
π
X
(t )
=
1 − r
t
, X
(t )
=∅,
r
t
f
t
(x
t
), X
(t )
={x
t
},
(9)
where r
t
denotes the probability of X
(t)
= ∅, f
t
(x
t
) (defined on
X
) denotes the density of x
t
.
Let
X
denote the state set of multiple unresolved target-groups.
Naturally, X
can be described by a union of N independent state
set X
(t)
of a single unresolved target-group,
X
=
N
t=1
X
(t )
, (10)
where N denotes the maximum number of unresolved target-
groups
over a surveillance region.
Since
X
(t)
(t = 1, ..., N) constituting the union in (10) has
been modeled as a Bernoulli RFS, it is reasonable to model X
as
a multi-Bernoulli RFS according to the definition in Section 2.2.
Then, the density of X
can be described by the parameter set
ϒ ={(r
t
, f
t
)}
N
t
=1
as
f
X
=
π(∅)
1≤ j
1
=···= j
|X
|
≤N
|X
|
t=1
r
j
t
1 − r
j
t
f
j
t
x
t
,
(11)
where | X
| denotes the cardinality of the set X
, and
π(∅) =
N
t=1
(1 − r
t
). (12)
In fact, the multi-Bernoulli state model has been extensively ap-
plied
in the traditional multiple (point) targets tracking problem
([3], pp. 655–658, 29).
Let
z ∈ Z denote a single measurement received by a sensor
and Z the set of sensor measurements, where Z is the measure-
ment
space. The measurement originates either from unresolved
target-group or from random clutter. For a given state set X
of
unresolved target-groups, x
t
∈ X
(t = 1, ..., |X
|) is either de-
tected
with probability p
D
(x
t
) and generates a measurement set
t
(x
t
) = ∅ with likelihood g(
t
(x
t
)|x
t
), or missed (
t
(x
t
) =∅)
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