FISST based method for multi-target tracking in the image plane of optical sensors.

Abstract

A finite set statistics (FISST)-based method is proposed for multi-target tracking in the image plane of optical sensors. The method involves using signal amplitude information in probability hypothesis density (PHD) filter which is derived from FISST to improve multi-target tracking performance. The amplitude of signals generated by the optical sensor is modeled first, from which the amplitude likelihood ratio between target and clutter is derived. An alternative approach is adopted for the situations where the signal noise ratio (SNR) of target is unknown. Then the PHD recursion equations incorporated with signal information are derived and the Gaussian mixture (GM) implementation of this filter is given. Simulation results demonstrate that the proposed method achieves significantly better performance than the generic PHD filter. Moreover, our method has much lower computational complexity in the scenario with high SNR and dense clutter.

Full text

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Sensors 2012, 12, 2920-2934; doi:10.3390/s120302920

sensors
ISSN 1424-8220
www.mdpi.com/journal/sensors
Article
FISST Based Method for Multi-Target Tracking in the Image
Plane of Optical Sensors
Yang Xu *, Hui Xu, Wei An and Dan Xu
School of Electronic Science and Engineering, National University of Defense Technology,
Changsha 410073, China; E-Mails: simon863@vip.sina.com (H.X.); nudtanwei@tom.com (W.A.);
stfan79@yahoo.com.cn (D.X.)
* Author to whom correspondence should be addressed; E-Mail: xuyang012@nudt.edu.cn;
Tel: +86-138-0846-5023; Fax: +86-0731-8457-3489.
Received: 3 February 2012; in revised form: 23 February 2012 / Accepted: 1 March 2012 /
Published: 2 March 2012

Abstract: A finite set statistics (FISST)-based method is proposed for multi-target
tracking in the image plane of optical sensors. The method involves using signal amplitude
information in probability hypothesis density (PHD) filter which is derived from FISST to
improve multi-target tracking performance. The amplitude of signals generated by the
optical sensor is modeled first, from which the amplitude likelihood ratio between target and
clutter is derived. An alternative approach is adopted for the situations where the signal noise
ratio (SNR) of target is unknown. Then the PHD recursion equations incorporated with
signal information are derived and the Gaussian mixture (GM) implementation of this filter
is given. Simulation results demonstrate that the proposed method achieves significantly
better performance than the generic PHD filter. Moreover, our method has much lower
computational complexity in the scenario with high SNR and dense clutter.
Keywords: finite set statistics; signal amplitude information; PHD filter; multi-target
tracking; Gaussian mixture

1. Introduction
Optical sensors have been widely applied both in important military and civil areas due to their
properties of long detection range, high concealment ability and large coverage area. Since there is
usually a long distance between targets and sensor which generates a low SNR and dense clutter
OPEN ACCESS

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scenario, multi-target tracking in the image plane of optical sensors is a very difficult problem. In
multi-target tracking, the aim is to estimate the number of a set of targets and the state of each target
from a set of measurements received. However, due to the variation of targets number with time in the
field of view of the sensors and the existence of miss detection and dense clutters, multi-target tracking
in image plane remains a challenging problem.
In addition to the location measurement, amplitude information has been proven to improve tracking
performance [1–5]. One of the pioneering techniques was the approach proposed by Colegrove, Lerro
and Bar-Shalom [1,2] where the probability data association (PDA) filter utilizing target amplitude
was applied in the context of single-target tracking. The target amplitude has also been incorporated in
the multiple hypothesis tracking (MHT) framework [3] and Viterbi data association scheme [4]. More
recently, the significance of target amplitude has been explored for data association of closely spaced
targets in [5]. Although significant progress has been made recently, the approaches mentioned above are
based on data association technique, which requires expensive computational cost in most circumstances.
Therefore, the traditional data association approach is not the optimal option for multi-target tracking
in image plane of optical sensor in the scenarios with time-varying targets number, low SNR and
dense clutters.
A good candidate is the emerging Bayesian approach in the framework of finite set statistics
(FISST) proposed by Mahler [6]. FISST provides a set of mathematical tools that allows direct
application of Bayesian inferencing to multi-target problems. PHD proposed by Mahler [7] is a
computation tractable approximation for the optimal multi-object Bayes filter based on Finite Set
Statistics (FISST). Operating on the single-target state space, the PHD filter avoids the combinatorial
problem that arises from data association and thus leads to superior performance in comparison with
traditional MHT algorithm in multi-target tracking [8,9]. These features render the PHD filter
extremely attractive to many researchers. The primary PHD applications can be found in [10–13].
In [14], Clark et al. incorporated target amplitude into a PHD filter for the first time in multi-target
tracking, which improves the tracking performance. However, their amplitude is modeled for radar and
sonar application and the computational complexity analysis of their method is absent.
In this paper, we propose a FISST based method which uses signal amplitude information in a PHD
filter for multi-target tracking in the image plane of optical sensors. Based on analyzing the imaging
characteristics of the optical sensor, we model the signal amplitude and incorporate it into the PHD in
the form of amplitude likelihood ratio. In the situation where the SNR of target is unknown, we then
present an alternative method based on this amplitude model. Simulation results demonstrate a
significant improvement of the proposed method in tracking performance. Furthermore, the
computational complexity of our method in the scenarios with different clutter density and SNR is
also discussed.
This paper is organized as follows: Section 2 models the signal amplitude generated by the optical
sensor. Section 3 demonstrates how to incorporate the amplitude information (for known and unknown
SNR case) into a PHD filter and then the GM implementation for this filter is given. Section 4 presents
the simulation results that validate the proposed method. The conclusions are given in Section 5.

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2. Amplitude Measurement Model
2.1. Amplitude Likelihood Ratio
In an optical multi-target tracking system, the original images taken by the sensor are usually
processed by background suppression before being further used for target tracking. Assuming the noise
is addictive, we define the SNR d of the residual image as [15–17]:

(1)
where s is the mean signal value of target, σ is the standard deviation of the residual image. Each
measurement from the image consists of the two-dimensional position vector z in the image plane and
the corresponding amplitude a ≥ 0, that is, a measurement vector has the form
simplicity, we assume that the value of the target signal has no spreading in the image plane. We
assume that the noise is Gaussian, then the probability densities of the amplitude of the false alarms
and the target p0(a) and p1(a|d) can be written as [18]:
. For
(2)

(3)
This leads to the probabilities of false alarm and detection
threshold τ,
and
with a detection
(4)

(5)
where
2
1
2
erf( )exp
2
x
y
xdy
π
+∞
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
=−
∫
is the probability error function. According to Equations (4) and (5),
we have:

(6)
In target tracking application we usually choose
can be calculated via the inverse form of Equation (4) with the parameter σ, and the probability of
detection can be calculated via Equations (5) or (6) for the target with SNR = d. Table 1 lists
the values of for different SNR values d and specified
value, i.e., σ = 1.
as a fixed value. Given , the threshold τ
with σ setting to be a normalized
s
d
σ
=
()
:,
T
Ta
=
z
%
z
( )
a
2
0
2
2
1
πσ
exp
2
2
a
σ
p
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
=−
()
()
()
2
1
2
2
2
2
2
1
πσ
|exp
2
2
1
exp
2
2
as
pa d
ad
σ
σ
σ
πσ
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
−
=−
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
−⋅
=−
FA
pτ
( )d
D
p
τ
0( ) p a daerf( )
FA
p
τ
τ
∫
τ σ
∞
==
1
( )d( | ) p a d daerf()
D
pd
τ
τ
∫
τ σ
∞
==−
()
( )
d
11
erferf
FAD
ppd
ττ−−⎡⎤
⎦
−=
⎣
FA
pτ
FA
pτ
( )d
D
p
τ
( ) d
D
p
τ
FA
pτ

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2923
Table 1. under different SNR and combinations.
d
6 4 5 7 8
5 × 10−5
1 × 10−4
5 × 10−4
1 × 10−3
0.5436
0.6106
0.7610
0.8185
0.6864
0.8999
0.9563
0.9719
0.9825
0.9887
0.9966
0.9982
0.9991
0.9995
0.9999
1.0000
1.0000
1.0000
1.0000
1.0000
When the target SNR d is known, we can get our amplitude likelihood functions for the false alarm
and the target as:
( )
a
FA
p
(7)

(8)
The amplitude likelihood ratio given a threshold τ is defined as [19]:

(9)
We use the notation ca(a) and ρa(a|d) to denote the case where τ = 0, then we have:
(10)

(11)
From Equations (5) and (9) we can see that the calculations of
specified known target SNR, however, this requirement cannot be satisfied in most practical tracking
systems. We adopt an alternative approach to circumvent this issue next.
and rely on a
2.2. Method for Unknown SNR
When the SNR of target is unknown, one straightforward approach would be to estimate the
unknown parameter d from the measurement amplitudes a. However, this approach requires a large
number of measurements from the target to achieve an accurate estimate of d. Furthermore, due to the
unknown association between measurements and targets in multi-target environment with clutter, the
approaches of estimating d usually fail. Similar to the idea introduced in [14], we adopt an alternative
approach where we do not attempt to estimate d at all. Instead we marginalize out the parameter d over
the range of possible values and find a probability of detection for
that is not conditional on d.
Consider that we always have some prior information about the targets being tracked, so we assume
that p(d), defined on the possible SNR values [d1,d2], gives the expected probability distribution of
SNR values. Since the amplitude distribution in Equations (2) and (3) are symmetric thus have no
biases in high or low SNR targets, the reasonable choice for p(d) is the uniform distribution U[d1,d2].
Then we define the probability of detection and amplitude likelihood ratio where SNR is unknown as:
and a likelihood ratio for
( )
d
D
p
τ
FA
pτ
FA
pτ
0
1
τ
( ),
p acaa
τ
τ=≥
()
1
1
( )
D
| ( | ),
p a d
a
ga da
pd
τ
τ
τ=≥
()
(
τ
)
|
|
( )
a
c a
a
a
ga d
a d
τ
τ
ρ=
0
( ) ( )
ac a p a
=
()
1
p a
0
( | )
( )
|
a
p a d
a d
ρ=
( )d
D
p
τ
()
|
aa d
ρ
τ
D
pτ
a
τ
ρ

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Sensors 2012, 12


2924
(12)

(13)
From Equations (5) and (12) we have:

(14)
Note that the
offline since it does not need to be computed at each iteration. The computation of
Equation (13) can be simplified and will be presented in Section 3.2.
over the marginalized region [d1, d2] can be computed with numerical integration
as in
3. PHD Filter with Signal Amplitude Information
Suppose that at time k there are Nk target states
; and measurements (detections)
. In PHD filter, a multi-object state and a multi-object observation are represented by RFS:
⊆
R
{
{
,1kk
L
, each taking values in a state space
, each taking values in the observation space

(15)
where F(X) and F(Z) are the finite subsets of X and Z, respectively. The state
target contains the position and velocity
defined in Section 2. We assume that each target follows a linear Gaussian dynamical model and the
sensor has a linear Gaussian measurement model, i.e.,
of each
in the image plane, while the measurement z is
(16)

(17)
where
matrix, Qk-1 is the process noise covariance, Hk is the observation matrix, and Rk is the observation
noise covariance.
denotes a Gaussian density with mean m and covariance P, Fk-1 is the state transition
3.2. The PHD Recursion with Amplitude Information
We abbreviate the PHD filter incorporated with amplitude information as AI-PHD filter. Next we
derive the prediction and update equations of AI-PHD filter based on the amplitude likelihood ratio
given by Section 2. For simplicity, we do not consider target spawning in this paper.
Step 1. Prediction: The prediction equation of AI-PHD filter is the same as generic PHD filter since
their state vector and state transition matrix are the same, i.e.,

(18)
where
the transition density and
is the birth term for new targets, is the probability of target survival,
is the PHD at time k − 1.
is
2
1
( )
ν
( )
ν
d
DD
d
pppd
ττ
ν=∫
( )
a
()
2
1
( )
ν ρ
|
d
aa
d
pad
ττ
ρνν=∫
2
1
21
1
−∫
erf( /)
d
D
d
p v d
dd
τ
τ σν=−
D
pτ
( )
aa
ρ
τ
,1 ,
,
k
kk N
xx
L
x n
⊆
XR
k
M
,1,
,
k
k k M
zz
L
zn
Z
}
}
( )
( )
,1,
,
,
,
k
k
kk k N
k M
X
Z
=∈
=∈
xx
zz
L
F X
F Z
( , , , )T
x y x y
=
x
& &
( , )T
x y ( , )T
x y
& &
()
|111
( |
x x
);,
k kkk
f
−−−
′′
=
x F x QN
()
( | );,
kk
L
=
zz xz H m RN
()
; ,
⋅ m PN
|1,|1 1|1
( )x ( )x(( )x( | x x))
′
( )x
k kkS kk kkk
DpfDd
γ
−−−−
′′′
=+∫
x
( )x
k γ
,( )
S k
p
′ x
|1( |
−
)
k k
f
′
x x
1|1( )
−
kk
D−
′ x