WAC 2014 1569945011

Noise-Estimate Particle PHD filter
Masanori Ishibashi, Yumi Iwashita, and Ryo Kurazume
Kyushu University
744, Motooka, Nishi-ku, Fokuoka, 819-0395, Japan
Email: ishibashi@irvs.ait.kyushu-u.ac.jp, yumi@ieee.org, kurazume@ait.kyushu-u.ac.jp
Abstract—This paper proposes a new radar tracking filter
named Noise-estimate Particle PHD Filter (NP-PHDF). Kalmanfilter and particle filter are popular filtering techniques for target
tracking. However, the tracking performance of the Kalman filter
severely depends on the setting of several parameters such assystem noise and observation noise. It is an open problem how
to choose proper parameters for various scenarios, and they areoften regulated in trial-and-error manner. To tackle this problem,
Noise-estimate Particle Filter (NPF) has been proposed so far.
The NPF estimates proper noise parameters of a Kalman filter
on-line based on a scheme of particle filter. In this paper, we
extend the NPF so that it enables to track multiple targets
simultaneously by combining with Probability Hypothesis Density(PHD) filter, and propose a new Noise-estimate Particle PHD
Filter (NP-PHDF). Simulation results show that the proposed
filter has higher tracking performance in various scenarios thanconventional Kalman filter, particle filter, and PHD filter for
multiple-targets tracking.

I.

the Kalman filter, too. The observation model is also difficult tobe set appropriately for various targets with a variety of shapes,
motion directions, heights, distances, etc. Currently, a target
tracking system based on Kalman filter has to be designedwith proper system parameters in a try-and-error manner tomeet the desired performance [1].
In few decades, particle filter has been attracting muchattention as a high-performance tracking filter. In particle
filter, instead of estimating the probability distribution of theobject state from the past observation and motion model in aparametric way, the probability distribution is represented by aset of particles. Each particle has a weight which represents theprobability of the state of the particle. The particle is updated
and re-sampled according to the Bayesian recursion equation.
Particle filter has been applied to various applications suchas human tracking, state estimation, etc. [2], [3], [4], [5], [6],[7], [8]. Particle filter does not depend on the assumptionof linear or Gaussian noise, and is able to be applied forvarious systems even with non-linear and non-Gaussian noise.
However, the particle filter is less effective than the optimized
Kalman filter for a target moving by a motion model, andthe improvement of the tracking performance for various
conditions is an open problem in particle filter.

I NTRODUCTION

Target tracking is a fundamental technique in not only
computer vision but also various application fields. For example, in air traffic control by a radar sensor, multiple highspeed flight vehicles must be tracked simultaneously without
lost-tracking while estimating their velocities and heights.
However, in general, radar data is severely corrupted by noisedue to atmospheric conditions or radar reflection property ofthe target, and it is still challenging to track targets stably under
severe noisy condition.

We have proposed a tracking filter named noise-estimate
particle filter (NPF) [9], which combines Kalman filter andparticle filter. As mentioned above, Kalman filter is an optimum filter in case that the motion model of targets andobservation model of sensor/radar are correctly provided. In
NPF, instead of estimating the state of the target such as position or velocity directly, the system error and the observation
error are estimated on-line by particle-filter based approach.
More correctly, the state of the target is estimated based on
Kalman filter, and its motion noise and observation noise are
optimized by particle filter. Since the critical parameters of the
Kalman filter are adjusted on-line according the observation,
the proposed filter can be applied for a variety of target motions
including not only uniform linear motion, but also sudden
motion changes such as abrupt acceleration/deceleration or
steep turn. However, the NPF is limited to a single target
tracking and cannot be applied to multiple target tracking
problem directly.

For target tracking, a time series filter is very effective tosuppress noise in sensory data and track targets smoothly andstably. In a time series filter, the current target position, whichis estimated based on past observation and a motion model,
are merged with current observation and the optimum position
is derived. Especially, the time series filter which removes thenoise component in the radar data and estimates the current
position and velocity of the target is called ”tracking filter”.
Kalman filter is the most popular and widely-used tracking
filter and has been applied to various applications so far. Basedon the assumptions on linear and Gaussian noise in sensory
data, this filter estimates the statistically-optimized state of thetarget. Uniform linear motion is usually adopted as a motion
model in simple Kalman filter. In case that target motion does
not follow the uniform linear model, a system noise parameter
which should be adjusted beforehand to absorb the systemerror affects the tracking performance severely. However, it isan open problem how to choose a proper value of the system
noise parameter beforehand for various scenarios such as nonlinear motion or rapid acceleration/deceleration. In addition, an
observation error which represents the accuracy of sensor/radarand has to be adjusted beforehand affects the performance ofIn this paper, we extend NPF so that it enables to track
multiple targets simultaneously, and propose a new tracking
filter named Noise-estimate Particle PHD Filter (NP-PHDF).
Probability Hypothesis Density (PHD) filter is a multipletarget tracking filter proposed by Vo et al.[10],[11]. In thisfilter, multiple state spaces for multiple targets are combined
to a single state space, and their probability distributions are
expressed by a set of particles. Based on PHD filter, NPF

1

is extended so that it can track multiple targets stably by
estimating optimal parameters of Kalman filter for individual
target, and a new Noise-estimate Particle PHD Filter (NPPHDF) is introduced in this paper. The performance of the
proposed NP-PHDF is verified through computer simulation
for multiple-target tracking problems.

B. Tracking processof particles at time tk is defined as Xk =
 iA setN
xk , wki , qki , rki i=1 . Here, xik is a hypnosis for a state vectorof position and velocity, wki is a weight of each particle, andqki and rki are system and observation noises at each particle,
respectively.
1) Produce N initial particles X0 = xi0 , w0i , q0i ,N
r0i i=1 . The position, velocity, and system and observation noises are set with random numbers inparticular ranges.
2) Execute step (a) to step (f) at time tk (k = 1, · · · , T )(a) Estimation
Execute prediction procedure in each particle byKalman filter and estimate the current state from theprevious state and the motion model.

Some authors proposed combination filters of particle filter
and Kalman filter[12], [13], [14], [15]. Rao-Blackwellized
Particle Filters [14] is the most popular technique in SLAM
(Simultaneous Localization and Mapping). Localization andmapping procedures are separated in this filter and implemented using particle filter and Kalman filter, individually.
Marginalized Particle Filter proposed Schon et al. [15] separates linear and nonlinear parts in the control system andassigns Kalman filter and particle filter separately. However, toour best knowledge, the proposed filter which utilizes particle
filter for estimating the parameters of optimum Kalman filter
has not been proposed so far. Satoh et al. [13] proposed a colorbased tracking technique using Kalman Particle Filter [12]. As
the noise-estimate particle filter proposed in the paper, the stateof each particle is updated by Kalman filter. In their method,
however, the observation noise is determined according to
the weight of the particle and the state noise is determinedpreviously.
II.

xik|k−1
Pik|k−1

In the noise-estimate particle filter (NPF)[9], a numberof Kalman filters with a variety of sets of parameters runs
simultaneously in a parallel way. As conventional particle filter,
particles are re-sampled according to the errors between theestimated and observed states. By evaluating the performanceof a number of Kalman filters with a variety of motion andobservation parameters estimated adaptively, optimum target
tracking which has similar high performance as optimized
Kalman filter for a fixed condition is achieved for various
motion patterns including uniform linear motion, abrupt acceleration/deceleration or steep turn.

xik|k
Pik|k
Kik

=

(5)

=

xik|k−1 + Kik [zk − Hxik|k−1 ]=(I − Kik H)Pik|k−1
(8)
i
T
i
T
i −1
Pk|k−1 H [HPk|k−1 H + Rk ] (9)=(7)

where xik|k is the estimated state of the particle attime tk , Pik|k is the estimated error covariance matrix
in each particle, and Kik is a gain matrix.
The error covariance matrix of observation noise Rikat time tk is obtained as follows.

In NPF, each particle executes a similar process as Kalmanfilter individually. Firstly, a state space model for a target
system is defined in each particle.
Φk xik + wki
Hxik + vki(4)Φk Pik−1|k−1 ΦTk + Qik(b) Smoothing
To fit the estimated state with the current observationand estimate more accurate state, smoothing processin Kalman filter is applied in each particle.

A. Model definition==

Φk xik−1|k−1

where xik|k−1 is an estimated state in each particle
and xik−1|k−1 is a previous state at time tk−1 . Pik|k−1
is an error covariance matrix in each particle andPik−1|k−1 is an previous error covariance matrix attime tk−1 .
An error covariance matrix of system noise Qik attime tk is obtained as follows.

i 2 i 2 i 2
Qik = diag qk,x(6)
, qk,y , qk,z

N OISE - ESTIMATE PARTICLE F ILTER (NPF)

xik+1
zik

=

(1)
(2)

Rik

Eq.(1) is a motion model which represents a state transition
and Eq.(2) is an observation model which shows the relationof estimated and observed states. xik is a state vector whichcontains position and velocity terms. Φk is a state transition
matrix and, in this paper, the uniform linear motion is assumed
in all particles as follows.


I ΔtI
Φk =
(3)
0
I

=

2

2

2

i
i
i
diag{rk,x, rk,y, rk,z}(10)(c) Likelihood calculation
The likelihood p(zk |xik|k ) at each particle is calculated as follows.
 2
−di
1
p(zk |xik|k ) = 
exp
(11)
2
2σs2
2πσs
where σs is a parameter to evaluate the accuracy ofthe hypnosis, di is an Euclidean distance between theposition component in xik|k and the observed position
zk . In the following experiments, we set σs = 300.
Next, the weight of each particle is updated according
to the obtained likelihood as follows.

where Δt is a sampling interval, I is an identity matrix, andH = (I 0) is an observation matrix. wki is a vector of system
noise with an average 0 and an error covariance matrix Qik ,respectively. vki is also a vector of observation noise with anaverage 0 and an error covariance matrix Rik .

wki

2

=

i
wk−1
p(zk |xik|k )(12)

a new target. In addition, the weight wki and thelabel jk is set as follows.

In addition, the sum of the weight of all particles isicalculated by Wk = ΣN
i=1 wk and the weight of eachparticle is normalized as wki = wki /Wk .
(d) State estimation
Estimated state x̂k at time tk is calculated by theweighted mean of N particles.
N

x̂k ≈
i=1

wki xik|k

(13)

2)

(e) Resampling
Particles are re-sampled according the the probability
proportional to the weight value wki . As a result,particles with lower weight are removed and ones
with higher weight are increased.
(f) Update
In contrast to updating position and/or velocity ina conventional particle filter, random offset values
obtained with a normal distribution are added tothe system noise qki and the observation noise rki ,respectively.
i
qk,s

i
rk,s

=
=

i
i
qk,s
+ Δqk,siirk,s+ Δrk,s

m=1

(14)

3)

(15)

PD (xik )L(zkm | xik )Lk−1 +Jkn
PD (xnk )L(zkm | xnk )wk−1
n=1

Where wki is the weight of the particle i at time
tk and zkm is an observed value of mth target(m = 1 ∼ Mk ). PD (xik ) is the probability for aparticle xik to detect the target.
Resampling: Particles are re-sampled according
the the probability proportional to the weightvalue wki . In PHD filter, the sum of the weights
in a certain area indicates the expectation of thenumber of targets Nk in this area. With thisexpectation, the number of particles Lk at the next
period is determined as follows.
Lk−1 +Jk

Nk =

wtn

(21)

n=1

Lk = αNk
4)
5)

6)

7)

(16)

Where Lk−1 . is the number of particles at theprevious time. In addition, new Jk particles areproduced at each time so that it can track new
targets. Jk is determined as(17)8)

α is a number of particles assigned for single
target and PB is the probability of occurrence of

3

(23)

Labeling of clusters: Calculate the correlation
with the existing clusters of trajectories.
Id(i1 ) = argi2 max{SumW gt(i1 , i2 )}Estimation: Particles at current time tk are updated by(22)

Update: Noises are added to the position and thevelocity in each particle.
Clustering: Apply clustering algorithm and determine the index of the cluster cnk which the particle
belongs to.
i1k[{xik1 , cik1 }L
i1 =1 }] = Clustring(xk )At time tk (k > 0),

PB (x)dx

(19)

(20)

A. Probability Hypothesis Density (PHD) filterLk
Let a set of particle at time tk be Xk = xik , wki , jki i=1.
Here, xik is a hypnosis for a state vector of position andvelocity, wki is a weight of each particle, and jki is thelabel indicating which trajectory the particle is assigned to at
previous time.

Jk = αjki = 0i
1 − PD (xik )wk−1
Mk

N OISE - ESTIMATE PARTICLE PHD F ILTER(NP-PHDF)

xik = Φk xik−1 , i = 1, 2, · · · , Lk−1=+

Probability Hypothesis Density (PHD) filter is a multipletarget tracking filter proposed by Vo et al.[10],[11]. In thissection, we introduce a basic procedure of PHD filter and propose a new noise-estimate particle PHD filter which combines
PHD filter and NPF mentioned above.

1)

(18)

for i = Lk−1 + 1, · · · , Lk−1 + Jk .
Likelihood calculation: The weight of each particle is calculated according to the following equation.
wkiiiand Δrk,sarewhere s ∈ {x, y, z}, and Δqk,sdetermined according to the normal random number
with an average of 0 and a variances of σq,s andσr,s , respectively. σq,s and σr,s are predeterminedparameters, and in the following experiments, we set
σq,s = 0.1 and σr,s = 5 for all s ∈ {x, y, z}.

III.

1
α

wki =

(24)

Where i2 = 1, · · · , rmax and SumW gt(i1 , i2 ) isa sum of likelihoods in particles of j = i2 among
the whole particles belonging to the cluster i1 .
Correlation calculation: For all clusters,
if Id(i1 ) = 0
New trajectoryr = rmax + 1
rmax = rmax + 1
ks = kId(i1 ) = r
else
Copy Id(i1 ) corresponding to x̂ik1 from ℵk−1
to ℵk . ℵk is a set of observed trajectories.
Delete trajectories: The trajectory which existed
at ℵk−1 but not observed at time tk is deleted.

B. Noise-Estimate Particle PHD filterof the radar as 30 [m] in distance and 0.2 [deg.] in azimuthand elevation angles.

We introduce a new noise-estimate particle PHD filter
which combines a noise-estimate particle filter and PHD filter.
In this filter, a particle holds state sik = {xik , wki , jki , qik , rik }at time tk . jki is a label indicating the trajectory which theparticle is assigned to. qik and rik are the system noise and theobservation noise, respectively.

In this simulation, the target starts to move from 70 [km]apart from the radar toward the radar with a height of 10[km]and the velocity of 306 [m/s]. Two types of trajectories
including straight and curved paths are tested as shown in
Fig.2.

In each particle, the position and the velocity of a target
are estimated through a same procedure as the noise-estimate
particle filter. At the same time, the correlation to the trajectories is also estimated in a framework of the PHD filter.
The flowchart of the proposed noise-estimate particle filter and
PHD filter is shown in Fig. 1.

For a scenario of the straight path, the target moves by the
uniform linear motion between 0 to 150 [sec.]. For a scenarioof the curved path, the target moves by the uniform linear
motion between 0 to 50 [sec.] and 100 to 150 [sec.], and withthe turning motion from 50 to 100 [sec.] during level flight. In
this scenario, the target turns 1 or 4 times on horizontal plane.

At time tk (k > 0),
1)

2)
3)
4)
5)
6)

70 km

z

Estimation: Particles are updated according toEqs. (4) and (5) by the noise-estimate particle filter. At the same time, new particles are produced
according to Eqs.(17) ∼ (19) by PHD filter.
Smoothing: Smoothing process in Kalman filter isapplied in each particle according to (7) ∼ (9).
Likelihood calculation and resampling: The
weight calculation and resampling are performed
according to Eqs.(20) ∼ (22)Update: Random noise is added to the system andobservation noises in each particle as the noiseestimate particle filter.
Clustering, labeling and correlation calculation:
Each procedure in PHD filter is applied.
Repeat 1) ∼ 5).

target10 km

Upper view
0~150 sec.

xyGroundradarxUpper view
100~150 50~100 5~50 sec.

y
radar

target(a) Scenario of straight path
Fig. 2.

yradartarget(b) Scenario of curved path

Scenario for computer simulatoin

We run Monte-Carlo simulation in 50 times and evaluates
the RMS (Root Mean Square) error. Four types of tracking
filters are compared

Set noise patameters forkalman filters randomly

1)

Estimation
Partiles are updated

2)

Smoothing
Kalman filter is applied

3)
4)

Kalman filters which are adjusted to provide the best
performance at straight path
Kalman filters which are adjusted to provide the best
performance at curved path
Conventional particle filter
Noise-estimated particle filter (NPF)

Likelihood calculation

The number of particles are 200 for conventional particle
filter and 100 for noise-estimated particle filter, which are
determined experimentally.

Resampling and update
Clustering (PHD filter)
Labeling of clusters
Correlation calculation
Delete trajectories

Fig. 1.

Tracking results for the proposed noise-estimated particle
filter which estimates system and observation noises are shownin Fig.3. In Fig.3, KF Straight is the Kalman filter for straight
trajectory, KF Curve is the Kalman filter for curved trajectory,
PF is the conventional particle filter, and NPF QR is the proposed noise-estimated particle filter. In addition, the estimatedsystem and observation noises for 4-turns trajectory are shownin Fig.4.

Flowchart of Noise-Estimate Particle PHD filter

IV.

C OMPUTER SIMULATION

We conduct several computer simulations to compare theperformance of the proposed noise-estimate particle PHD filter(NP-PHDF), noise-estimate particle filter (NPF)[9] Kalmanfilter, and conventional particle filters optimized for straightand curved paths.

The smoothing performance of the proposed noiseestimated particle filter (NPF QR) outperforms the conventional particle filter (PF), and is similar to the Kalman filter optimized for curved trajectory (KF Curve). Though the
Kalman filter optimized for straight trajectory (KF Straight)
shows the best performance in straight path, it cannot track
the target while turning at all. In addition, though the tracking
delay is occurred for the Kalman filter for curved trajectory(KF Curve) at 4-turns scenario, the accuracy of the proposed

A. Performance of single target tracking using noise-estimate
particle filter (NPF)
We assume a radar sensor is placed at the origin of thecoordinate and set the sampling interval as 0.5 [s], the accuracy

4

RMS Error [m]500

In this simulation, two targets A and B are tracked simultanously. The target A starts at the velocity of 306 [m/s] andmakes turns at the period from 50 [s] to 100 [s]. The target
B appears at 25 [s] after the target A starts the motion andmakes turns at the period from 50 [s] to 100 [s]. Both targets
are crossed at the position of 50 [km] apart from the radarwith the different of height of 300 [m], 600 [m], and 900 [m]as shown in Fig.5.

400
300
200
100
0
0

20

40

60

80
100
Time [sec.]

120

140

160

(a) StraightRMS Error [m]

600

Turn

500
400
300
200

radar

100
00

20

40

60 80 100
Time [sec.]

120

140

160

(b) 1 turn
600
500
RMS Error [m]Fig. 5.

Trajectries of targets

Turn

Table I shows the success rates of the conventional PHD
filter and the noise-estimate particle PHD filter (NP-PHDF).
We can see that the proposed noise-estimate particle PHD filter
outperforms the conventional PHD filter in all senarios.

400
300
200
100
0
0

20

40

60
80
100
Time [sec.]

120

140

TABLE I.

160

Diff. height
300 [m]
600 [m]900 [m](c) 4 turns
Tracking error (system and observation noises are estimated)

100~150 sec.
PHD filter

PHD filter

6

NP-PHD filter

NP-PHD filter

5
4

0.1

Observation noise 3
Turn

2
1

0
0

20

40

60
80 100Time [sec.]

120

140

RMS error [m]7
RMS error [m]50~100 sec.

System noise

Observation noise

5~50 sec.
0.2

System noise

NP-PHDF [%]
78
98
100

Figs.6 ∼ 8 show the RMS errors in cases that the differences of height are 900[m], 600[m], and 300[m], respectively.

System noise

Fig. 4.

PHD filter [%]
54
96
100

0
160

Fig. 6.

Estimated system and observation noises (4 turns)

Time [s](b) object B

RMS errors in case that difference of height is 900[m]PHD filter

PHD filter

NP-PHD filter

NP-PHD filter

RMS error [m]filter is similar to the conventional particle filter (PF). Consequently, it is confirmed that the proposed filter can be applied
to various scenarios more adaptively than other conventional
filters.

Time [s](a) object A

RMS error [m]Fig. 3.

S UCCESS RATE

B. Performance of multiple target tracking using noiseestimate particle PHD filter (NP-PHDF)

Time [s]Time [s](a) object A(b) object B

In this simulation, multiple targets are tracked simultaneously using the proposed noise-estimate particle PHD filter.

Fig. 7.

We evaluate the performance by the success rate of multiple
target tracking and RMS error between the actual and estimated trajectries through Monte-Carlo simulation in 50 times.
To calculate the success rate, we define the tracking is suceeded
if the same target is tracked successively until the end.

The RMS error of the proposed noise-estimate particle
PHD filter is almost same as the conventional PHD filter
in a curved trajectry, and smaller than the PHD filter in astraight trajectory. From this simulation, it is verified that the5

RMS errors in case that difference of height is 600[m]PHD filter

NP-PHD filter

NP-PHD filter

RMS error [m]Fig. 8.

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RMS error [m]PHD filter

Time [s]Time [s](a) object A(b) object B

RMS errors in case that difference of height is 300[m]proposed noise-estimate particle PHD filter has higher tracking
performance than the conventional PHD filter.
V.

C ONCLUSION

This paper proposed the new noise-estimated PHD particle
filter for a multiple target tracking system. The proposed filter
estimates the system and the observation noises in Kalman filter by using the particle filter and adapts to the abrupt changesof the target motion characteristics. In addition, multiple targets
are tracked simultaneously in a framework of PHD filter.
We examined the tracking performance of the proposednoise-estimated PHD particle filter through computer simulations for the radar tracking system, and confirmed that the
proposed filter has high smoothing performance for the tracking error and high stability performance for sudden changesof the target motions.
This paper focused on the radar-based target tracking, however, the applications of the proposed noise-estimated particle
filter is not limited to radar tracking and we can apply the
proposed filter for a variety of vision-based tracking systems.
We are going to apply the proposed filter for, for example,
pedestrian tracking using distributed cameras and laser range
finders in near future.
ACKNOWLEDGMENT
The present study was supported in part by a Grant-in-Aid
for Scientific Research (A) (26249029)
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