Improved particle filtering schemes for target tracking

Abstract

In this paper, we propose two improved particle filtering schemes for target tracking, one based on a gradient proposal and the other based on the turbo principle. We present the basic ideas and derivations and show detailed results of three tracking applications. Favorable experimental findings have shown the efficiency of our proposed schemes and their potential in other tracking scenarios.

Full text

IMPROVED PARTICLE FILTERING SCHEMES FOR TARGET TRACKING
Zhe Chen
1
, Thia Kirubarajan
1
, Mark R. Morelande
2
1. Communications Research Lab, McMaster University, Canada
2. Center for Sensor, Signal and Information Processing, University of Melbourne, Australia
ABSTRACT
In this paper, we propose two improved particle filtering schemes
for target tracking, one based on gradient proposal and the other
based on Turbo principle. We present the basic ideas and deriva-
tions and show detailed results of three tracking applications. Fa-
vorable experimental findings have shown the efficiency of our
proposed schemes and their potential in other tracking scenarios.
1. INTRODUCTION
Recent years have witnessed ever-growing efforts in applying par-
ticle filters to signal processing, communications, and machine
learning [1]. Bearing the nature of recursive Bayesian estimation
and sequential Monte Carlo sampling, particle filtering has demon-
strated its potential in various nonlinear, non-Gaussian, non-stationary
sequential estimation problems. Among many, target tracking prob-
lem provides a testbed for particle filter. As well known, the con-
ventional Bayesian bootstrap [3] or SIR filtering (using prior pro-
posal) has a drawback of ignoring the most recent observation. In
this paper, we propose two improved particle filtering schemes to
overcome this weakness and apply them in several tracking appli-
cations. The first improvement scheme is to use gradient infor-
mation of the measurement model. The idea of gradient proposal
is very heuristic, but it is very simple to implement and turns out
to be quite efficient in practice [4]. We also propose another new
particle filtering method based on Turbo principle (motivated from
Turbo decoding in communications). Basically, we use one filter
(so-called slave filter) to produce a first-stage (rough) estimate; and
we run another filter (master filter) in parallel to yield the second-
stage (ultimate) estimate, which uses its current as well as previous
estimate for importance weights update in a recursive particle fil-
tering fashion.
The rest of the paper is organized as follows: In section 2, we
briefly discuss the Bayesian bootstrap filter and then introduce the
improved schemes for particle filtering. Section 3 is devoted to
two simulated and one real-life tracking applications, followed by
concluding remarks in Section 4.
2. PARTICLE FILTERING AND IMPROVED SCHEMES
2.1. State-Space Model and Bayesian Bootstrap Filter
Consider a generic discrete-time nonlinear state space model:
x
n+1
= f (n, x
n
, d
n
), (1a)
y
n
= g(n, x
n
, v
n
), (1b)
where d
n
and v
n
characterize the dynamic and measurement noise
processes, respectively. The state equation (1a) characterizes the
state transition probability p(x
n+1
|x
n
), whereas the measurement
equation (1b) describes the probability p(y
n
|x
n
) which is further
related to the measurement noise model.
Simply say, particle filter uses a number of independent ran-
dom variables called particles, sampled directly from the state space,
to represent the posterior probability, and update the posterior by
involving the new observations; the “particle system” is properly
located, weighted, and propagated recursively according to the
Bayesian rule. Specifically, using a sequential important sampling
(SIS) scheme, it can be shown [2] that the importance weights up-
date has the following recursive form:
W
(i)
n
= W
(i)
n−1
p(y
n
|x
(i)
n
)p(x
(i)
n
|x
(i)
n−1
)
q(x
(i)
n
|x
(i)
0:n−1
, y
0:n
)
. (2)
where W
(i)
n
= p(x
(i)
n
)/q(x
(i)
n
) denotes the importance weight,
and q(x
(i)
n
|x
(i)
0:n−1
, y
0:n
) represents the proposal distribution. Choos-
ing a proper proposal often has a crucial effect on the particle fil-
tering performance.
The well-known SIR and Bayesian bootstrap filter [3] use a
transition prior as proposal, i.e. q(x
n
|x
n−1
, y
0:n
)=p(x
n
|x
n−1
);
it then simplifies (2) to
W
(i)
n
= W
(i)
n−1
p(y
n
|x
(i)
n
), (3)
which essentially neglects the effect of recent observation y
n
. De-
spite its appealing simplicity, this proposal distribution is far from
optimal and its resulted performance can be quite poor even a large
number of particles are used. Many improved schemes (such as the
auxiliary variable) have been developed in the literature [1]. In the
following, we will describe two improved schemes in an attempt
to efficiently incorporate the observation information into the sam-
pling step.
2.2. Particle Filtering Using Gradient Proposal
In order to use the recent observation, we propose to use the gra-
dient information of (1b) to select the “informative” particles [4].
The main idea behind it is to introduce a MOVE-step to sampling
for the proposal distribution, which is plugged in before the sam-
pling step in the conventional SIR filter. This new algorithm es-
sentially calculates the gradient information from the likelihood
model and guides the particles towards the low-error region, along
the gradient-descent direction; by assuming an additive measure-
ment noise model in (1b), the MOVE-step is described by
ˆ
x
n|n−1
=
ˆ
x
n−1|n−1
− η
∂(y
n
− g(x))
2
∂x




x=
ˆ
x
n−1|n−1
(4)
IV - 1450-7803-8874-7/05/$20.00 ©2005 IEEE ICASSP 2005
➠
➡

slave filter
master filter
x
n
(i)
y
n
time n
time n+1
x
n|n
,W
n
(i)
x
n
-1
(i)
,W
n
-1
(i)
Fig. 1. A schematic diagram of Turbo particle filtering.
where η ∈ [0.001, 0.01] is a small-valued step-size parameter. As
expected from (4), the inclusion of current observation y
n
and the
calculation of gradient information will tend to push the samples to
a high-likelihood region, thereby providing more reliable predic-
tive samples for the next step. In summary, the improved particle
filtering with gradient proposal reads as follows:
1. For i =1, ··· ,N
p
, sample x
(i)
0
∼ p(x
0
), W
(i)
0
=1/N
p
.
2. For each sample {x
(i)
n−1
}, update the sample via (4).
3. Importance sampling:
ˆ
x
(i)
n
∼ p(x
n
|
ˆ
x
(i)
n|n−1
).
4. Importance weights update:
W
(i)
n
= W
(i)
n−1
p(y
n
|
ˆ
x
(i)
n
)
p(
ˆ
x
(i)
n
|
ˆ
x
(i)
n−1|n−1
)
p(
ˆ
x
(i)
n
|
ˆ
x
(i)
n|n−1
)
.
5. Calculate effective sample size
ˆ
N
ef f
,if
ˆ
N
ef f
>N
p
/2,
resampling; otherwise go to Step 2.
2.3. Turbo Particle Filtering
A schematic diagram of Turbo particle filtering is illustrated in
Fig. 1. In Fig. 1, there are two filters in parallel, to be run iter-
atively. The slave filter, being an extended Kalman filter (EKF)
here, is used to produce a rough estimate
ˆ
x
n|n
, given the current
observation y
n
and previous state estimates x
(i)
n−1
. This is done
by the typical EKF equations. Note that in the prediction step,ev-
ery particle x
(i)
n−1
is passed through the state equation, where the
predicted covariance can be estimated by the sample covariance,
ˆ
P
n|n−1
, or calculated through linearization; in the filtering step,
instead of using all of the samples {
ˆ
x
(i)
n|n−1
}, we only use its mean
value,
ˆ
x
n|n−1
=

ˆ
x
(i)
n|n−1

, to perform the EKF update:
ˆ
x
n|n
=
ˆ
x
n|n−1
+ K
n
(y
n
− g(
ˆ
x
n|n−1
)), (5)
P
n|n
=
ˆ
P
n|n−1
+ K
n
C
n
ˆ
P
n|n−1
, (6)
where K
n
=
ˆ
P
n|n−1
C
T
n
(C
n
ˆ
P
n|n−1
C
T
n
+Σ
v
)
−1
, and C
n
is the
linearized Jacobian matrix of the measurement equation, P
n|n
is
the filtered state covariance. Note that the filtered estimate
ˆ
x
n|n
is more accurate than the predicted estimate
ˆ
x
n|n−1
, since it uti-
lizes the observation y
n
. In the meantime, the master filter, given
y
n
and the previous simulated samples {x
(j)
n−1
}, as well as the
first-stage estimate
ˆ
x
n|n
, runs a particle filtering procedure with a
constructed suboptimal proposal distribution, and further produces
a second-stage posterior estimate x
(i)
n
. After a complete step, the
master filter propagates its samples to the slave filter for the next
iteration. Essentially, two filters are trying to solve the same fil-
tering problem but looking at it from different perspectives; each
one takes advantage of the result of the other at the previous step
and thereby produces the solution in a cooperative way. Due to its
similarity to Turbo decoding, we call the proposed filter structure
as Turbo particle filter (TPF). In what follows, we will derive the
update equation mathematically in detail.
Let us write the filtering posterior in a slightly different way:
p(x
n
|y
0:n
)=p(x
n
|y
n
, y
0:n−1
)
=
p(y
n
|x
n
, y
0:n−1
)p(x
n
|y
0:n−1
)
p(y
n
|y
0:n−1
)
∝ p(y
n
|x
n
)p(x
n
|y
0:n−1
). (7)
Next, suppose we can draw samples {x
(i)
n
} from a proposal dis-
tribution, we need to find the importance ratios to appropriately
weight the samples. Using the importance sampling trick, we have
W
(i)
n
=
p(y
n
|x
(i)
n
)p(x
(i)
n
|y
0:n−1
)
q(x
(i)
n
|y
n
)
, (8)
where q(x
(i)
n
|y
n
) is the proposal distribution. Assuming that at
time n, we have the simulated particles for approximating the pos-
terior of time n − 1:
p(x
n−1
|y
0:n−1
) ≈
N
p

j=1
˜
W
(j)
n−1
δ(x
n−1
− x
(j)
n−1
), (9)
where
˜
W
(j)
n−1
are the normalized importance weights with the sum
equal to unity. Hence, we can have
p(x
(i)
n
|y
0:n−1
)=

p(x
(i)
n
|x
n−1
)p(x
n−1
|y
0:n−1
)dx
n−1
≈
N
p

j=1
˜
W
(j)
n−1
p(x
(i)
n
|x
(j)
n−1
). (10)
Substituting (10) into (8) yields the importance weights update:
W
(i)
n
=
p(y
n
|x
(i)
n
)
N
p

j=1
˜
W
(j)
n−1
p(x
(i)
n
|x
(j)
n−1
)
q(x
(i)
n
|y
n
)
. (11)
Using the Gaussian approximation proposal (from the EKF), we
can draw samples {x
(i)
n
} from q = N(
ˆ
x
n|n
, P
n|n
), where P
n|n
is obtained from (6). Now equation (11) naturally combines the
prior and likelihood information, using previous weights and sam-
ples as well as observation y
n
. Finally, the master filter produces
a (second-stage) mean estimate x
n
=

N
p
i=1
˜
W
(i)
n
x
(i)
n
.
In summary, a complete-step of TPF runs as follows:
1. At time n, for j =1,...,N
p
,givenx
(j)
n−1
(obtained from
the master filter in the previous step) and y
n
, run the EKF
updates (for the slave filter) to calculate the approximated
Gaussian sufficient statistics (
ˆ
x
n|n
, P
n|n
).
2. Draw N
p
samples {x
(i)
n
} from N

ˆ
x
n|n
, P
n|n

.
3. For the master filter, for i =1,...,N
p
, calculate the im-
portance weights via (11), and normalize them to get {
˜
W
(i)
n
}.
4. Calculate the second-stage estimate: x
n
=

N
p
i=1
˜
W
(i)
n
x
(i)
n
.
5. Calculate
ˆ
N
ef f
,if
ˆ
N
ef f
<N
p
/2, perform the resampling.
6. Copy the N
p
particles to the slave filter for the next step.
IV - 146
➡
➡

Table 1. Experimental results of bearing-only target tracking
based on 50 Monte Carlo runs (excluding the divergence trials).
filter MSE NMSE diver. rate
EKF 0.0026 0.0168 0/50
UKF
0.0045 0.0232 0/50
SIR-prior (N
p
= 100) 0.0021 0.0150 3/50
SIR-gradient (N
p
= 100) 0.00014 0.0078 1/50
TPF-EKF (N
p
=30) 0.00006 0.0009 0/50
TPF-UKF (N
p
=30) 0.00006 0.0008 0/50
3. TRACKING APPLICATIONS
Bearing-Only Tracking: First, we consider a bearing-only
tracking benchmark problem [3]. Let (ν, ˆν,η, ˆη) denote the x − y
positions and velocities of a moving target. The state-space equa-
tions are described as follows:
x
n+1
= Fx
n
+ Cd
n
,
y
n
= tan
−1
(η
n
/ν
n
)+v
n
,
where x
n
=[ν
n
, ˙ν
n
,η
n
, ˙ν
n
]
T
, d
n
=[d
1,n
,d
2,n
]
T
, and
F =
⎡
⎢
⎣
1100
0100
0011
0001
⎤
⎥
⎦
, C =
⎡
⎢
⎣
10
10
01
01
⎤
⎥
⎦
.
The observation is a noisy bearing, and d
n
∼N(0, 0.001
2
I),
v
n
∼N(0, 0.005
2
). The goal is to reconstruct the trajectory
x
0:n
given the observed bearings y
0:n
and initial condition x
0
=
[−0.05, 0.001, 0.7, −0.055]
T
. The priors of particle filters are
set up as p(x
0
) ∼N(0, diag{0.05
2
, 0.005
2
, 0.03
2
, 0.01
2
}) and
E
p(x
0
)
[x
0
]=[−0.06, 0.0015, 0.65, −0.05]
T
. Note that here the
prior and likelihood are both peaked.
The experimental comparisons are conducted between (i) EKF;
(ii) unscented Kalman filter (UKF); (iii) SIR filter with a prior pro-
posal; (iv) gradient proposal particle filter; (v) TPF with an EKF
as slave filter; and (vi) TPF with a UKF as slave filter. A total of
50 Monte Carlo experiments with different random seeds are per-
formed for each filtering scheme, using different number of parti-
cles. The error metrics of interest here are the mean-squared error
(MSE), the normalized MSE (NMSE), as well as the divergence
rate.
1
Experimental results are summarized in Table 1. As seen,
the two proposed particle filtering schemes outperform the con-
ventional bootstrap filter. In this example, the TPF produces the
best tracking result; but no big difference is observed between (v)
and (vi), partially because the state equation is linear and Gaus-
sian. Using the same number of particles, the gradient proposal
obviously produces better results than the prior proposal; however,
as expected, when N
p
gradually increases, the difference between
them will be reduced; this has been confirmed in our experiments
(see Fig. 2).
Tracking with Coordinate Turn: Next, we consider a typ-
ical target tracking through a coordinated turn (CT), where the
state-space model is described by a stochastic differential equation
(SDE) and approximated by a 2nd-order weak Taylor approxima-
tion. See [5] for background and details. Let x
t
=[ξ
t
,ς
t
,s
t
,θ
t
,ω
t
]
T
1
By divergence we mean the filter deviates from the target trajectory
and is unable to come back to the true track.
0
0.2
0.4
0.6
0.8
1
1.2
x 10
Ŧ3
NMSE
SIR
SIR+gradient
100
200
500
N
p
Fig. 2. Performance (NMSE) comparison between the SIR and
gradient proposal particle filters using varying numbers of parti-
cles, each based on 20 independent Monte Carlo runs.
denote the state vector containing target position in x and y coor-
dinate, target speed, heading, as well as turn rate. Under the SDE
theory, the constant speed and turn rate (in the ideal CT model)
will be instead modified as a Wiener process.
Specifically, the 2nd-order Taylor approximation of the continuous-
time state equation is described as [5]:
x
t
= f (x
τ
)+G(x
τ
)w
t
(12)
where δ = t − τ (we use δ =1in discretization), and
f (x
τ
)=
⎛
⎜
⎜
⎜
⎝
ξ
τ
+ δs
τ
cos(θ
τ
) − δ
2
s
τ
ω
τ
sin(θ
τ
)/2
ς
τ
+ δs
τ
sin(θ
τ
)+δ
2
s
τ
ω
τ
sin(θ
τ
)/2
s
τ
θ
τ
+ δω
τ
ω
τ
⎞
⎟
⎟
⎟
⎠
,
G(x
τ
)=E(x
τ
)V
δ
, with
E(x
τ
)=
⎛
⎜
⎜
⎜
⎝
σ
s
cos(θ
τ
)0 0 0
σ
s
sin(θ
τ
)0 0 0
00σ
s
0
0 σ
ω
00
000σ
ω
⎞
⎟
⎟
⎟
⎠
,
V
δ
=


δ
3
/30
√
3δ/2
√
δ/2

⊗ I
2×2
where ⊗ denotes the Kronecker product; w
t
is a Wienner process
approximated by standardized white Gaussian noise N(0, I
4×4
).
The measurement equation consists of a “range-bearing” pair:
y
t
=


ξ
t
t
+ ς
2
t
, tan
−1
(ς
t
/ξ
t
)

T
+ v
t
(13)
where the Gaussian measurement noise v
t
∼N(0, Σ
v
) is inde-
pendent on the initial state and w
t
. The data trajectory was gen-
erated using the Euler approximation with sampling period of 1
second and 1000 intervals per sampling instant. Measurements
are collected for 200 seconds with a constant sampling period.
The noise and initial parameter setup in our experiment is as fol-
lows: σ
2
s
=1/5, σ
2
ω
=5× 10
−7
, Σ
v
= diag{100, (π/180)
2
},
x
0
∼N(µ
0
, Σ
0
) with
µ
0
= [1000, 2650, 150,π/2, −π/45]
T
,
Σ
0
= diag{400, 400, 25, (5π/180)
2
, (0.2π/180)
2
}.
IV - 147
➡
➡

0 20 40 60 80 100 120 140 160 180 200
0
5000
10000
xŦaxis
0 20 40 60 80 100 120 140 160 180 200
0
5000
yŦaxis
0 20 40 60 80 100 120 140 160 180 200
100
150
200
Target speed
0 20 40 60 80 100 120 140 160 180 200
Ŧ20
0
20
Target heading
0 20 40 60 80 100 120 140 160 180 200
Ŧ0.1
Ŧ0.08
Ŧ0.06
Turn rate
Time step
Fig. 3. One typical result in tracking with CT experiment using
gradient proposal particle filter (N
p
= 500, RMSE=87.5).
Note that here the dynamic noise is peaked whereas the measure-
ment noise is rather flat. Table 2 shows the comparison between
the bootstrap filter (with prior proposal) and the SIR filter (with
gradient proposal) and TPF, with varying number of particles. Fig.
3 shows one typical tracking result.
MIMO Wireless Channel Tracking: Finally, we study a real-
life MIMO wireless channel tracking problem (actually it is a chan-
nel/sybmol joint estimation problem but we ignore the symbol de-
coding part due to space limitation) [4]. The real-life wireless nar-
rowband MIMO channel data were recorded in midtown Manhat-
tan, New York city, January 2001. In particular, the state equa-
tion of the channel can be described by a first-order AR model
driven by non-Gaussian noise (such as the mixture of Gaussians),
whereas the measurement equation is described as [4]:
y
j,n
=
#receivers

k=1
s
k,n
x
jk,n
+ v
j,n
,
where s
k,n
is the block of encoded symbols radiated by the kth
transmitter at time n; x
jk,n
is the channel coefficient from the kth
transmitter to the jth receiver at time n; y
j,n
is the signal observed
at the input of the jth receiver; and v
j,n
is the measurement noise
at the input of the jth receiver at time n, which is modeled by a
Middleton Class A noise model. See [4] for more details.
We have compared different trackers (including Kalman filter,
mixture Kalman filters, and particle filters) [4], but here we only
highlight the efficiency of the gradient proposal in this real-life ap-
plication. As seen from Table 3, the proposed gradient proposal
particle filter produces much better results (in terms of MSE as
well as symbol error rate) than the conventional SIR filter, espe-
cially when using a small number particles. Namely, the particles
drawn from the gradient proposal are more informative. This phe-
nomenon has also been evidenced in the previous two applications.
In terms of relative complexity and performance gain, compared
to the Kalman filter (with complexity 1), the relative complexity
factors of the SIR (with 100 particles) and our gradient SIR filters
(with 20 particles) are 3.2 and 1.5, respectively; while their per-
formance gains are 2.6 and 2.7, respectively! This huge gain is
quite significant when complexity issue is concerned in industrial
practice.
Table 2. The RMSE ≡

|ξ −
ˆ
ξ|
2
+ |ς − ˆς|
2
performance com-
parison (based on 20 Monte Carlo runs) with varying N
p
.
Filter RMSE
SIR-prior (N
p
=500) 144.3
SIR-prior (N
p
= 1000) 91.6
SIR-prior (N
p
= 2000) 81.9
SIR-gradient (N
p
=200) 142.3
SIR-gradient (N
p
=500) 90.2
SIR-gradient (N
p
=1000) 74.3
TPF-EKF (N
p
=100) 158.7
Table 3. The MSE of 2-by-2 MIMO wireless channel estimate and
symbol error rate (SER) for various numbers of particles (at 10dB
SNR) based on 100 Monte Carlo runs.
Number of SIR-prior SIR-gradient
Particles, N
p
MSE SER MSE SER
10 0.0615 0.0681 0.0233 0.0353
20
0.0431 0.0460 0.0206 0.0305
40
0.0338 0.0387 0.0196 0.0291
100
0.0257 0.0301 0.0188 0.0275
200
0.0227 0.0285 0.0183 0.0272
4. CONCLUDING REMARKS
We have proposed two improved particle filtering schemes and
demonstrated their potential merits in three tracking applications.
In particular, the ad-hoc gradient proposal is quite efficient in var-
ious noise scenarios (esp. with small Σ
v
); whereas the TPF usu-
ally works well when variances Σ
d
and Σ
v
are both small (since
the deterministic EKF gradually reduces the state-error variance).
Eventually, the issue of performance and complexity tradeoff of
particle filtering is central: choosing a better proposal distribution
(with more computation per step) might ironically reduce the total
complexity (in terms of required particle number simulation). In
practice, we might design certain decision criteria for using differ-
ent particle filters in different scenarios; in other words, we have to
study the problem first and then choose a problem-specific filter.
Acknowledgements: The authors thank Kris Huber for assis-
tance in wireless channel tracking investigation. Z. C. was finan-
cially supported by a NSERC grant of Prof. Simon Haykin.
5. REFERENCES
[1] A. Doucet, N. de Freitas, and N. Gordon, Eds. Sequential
Monte Carlo Methods in Practice, Springer, 2001.
[2] A. Doucet, S. Godsill, and C. Andrieu, “On sequential Monte
Carlo sampling methods for Bayesian filtering,” Statist. Com-
put., vol. 10, pp. 197–208, 2000.
[3] N. Gordon, D. Salmond, and A. F. M. Smith, “Novel ap-
proach to nonlinear/non-gaussian Bayesian state estimation,”
IEE Proc.-F, vol. 140, pp. 107–113, 1993.
[4] S. Haykin, K. Huber, and Z. Chen, “Bayesian sequential state
estimation for MIMO wireless communication,” Proc. IEEE,
vol. 92, no. 3, pp. 439–454, March 2004.
[5] M. R. Morelande and N. J. Gordon, “Target tracking through
a coordinated turn,” manuscript submitted to ICASSP2005.
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