4362
IEEE SENSORS JOURNAL, VOL. 16, NO. 11, JUNE 1, 2016
A Bernoulli Filter for Extended Target Tracking
Using Random Matrices in a UWB Sensor Network
Abdulkadir Eryildirim and Mehmet Burak Guldogan, Member, IEEE
Abstract— In this paper, we propose a new tractable Bernoulli
filter based on the random matrix framework to track an
extended target in an ultra-wideband (UWB) sensor network. The
resulting filter jointly tracks the kinematic and shape parameters
of the target and is called the extended target Gaussian inverse
Wishart Bernoulli (ET-GIW-Ber) filter. Closed form expressions
for the ET-GIW-Ber filter recursions are presented. A clustering
step is inserted into the measurement update stage in order to
have a computationally tractable filter. In addition, a new method
that is consistent with the applied clustering method is embedded
into the filter recursions in order to adaptively estimate the
time-varying number of measurements of the extended target.
The simulation results demonstrate the robust and effective
performance of the proposed filter. Furthermore, real data
collected from a UWB sensor network are used to assess the
performance of the proposed filter. It is shown that the proposed
filter yields a very promising performance in estimation of the
kinematic and shape parameters of the target.
Index Terms— Random finite set, extended target tracking,
Bernoulli filter, Gaussian inverse Wishart, inverse Wishart
distribution, random matrix, ultra-wideband, sensor network,
localization.
I. I NTRODUCTION
ITH the advent of modern sensors, targets are generally treated as extended targets which can produce
multiple measurements per time step [1]–[3]. From a target, a
high resolution sensor usually collects multiple measurements
per time-step across the target’s surface and/or over target’s
boundaries such as edges [4]. For example, Fig. 1 shows an
extended target tracking scenario, where a camera is used as a
sensor. The dataset (PETS 2000, [5]) includes the video frames
of the red car which enters and leaves the scene. By applying
a corner detection method, it is possible to collect multiple
detections of the red car at each frame. Also note that, as can
be seen from the two different frames given in the figure, pixelwise size (i.e. number of possible detections) and orientation
of the red car changes over time. Therefore, the car can be
considered as an extended target and not only the kinematic
parameters but also the shape parameters can be estimated.
Recent developments in sensor technologies along
with advances in processor speeds have made extended
target tracking feasible and this has attracted researchers’
W
Manuscript received February 14, 2016; accepted March 13, 2016. Date
of publication March 22, 2016; date of current version April 26, 2016. This
work was supported by the Scientific and Technological Research Council of
Turkey under Grant 114E214. The associate editor coordinating the review of
this paper and approving it for publication was Dr. Amitava Chatterjee.
The authors are with the Department of Electrical and Electronics Engineering, Turgut Özal University, Ankara 06010, Turkey (e-mail:
aeryildirim@turgutozal.edu.tr; bguldogan@turgutozal.edu.tr).
Digital Object Identifier 10.1109/JSEN.2016.2544807
Fig. 1. Two images (frame 134 and frame 172) corresponding to two different
time steps demonstrating the detections gathered in tracking of the red car
using a camera; yellow dots are extended target detections which can be
obtained by a corner detection method [5]. (a) Frame 134. (b) Frame 172.
attention [1]–[3], [6]–[15]. In the concept of extended target
tracking, several different applications are studied in the
literature. In [13], a group of closely spaced targets is
tracked. Hammarstrand et al. [2], Fortin et al. [3], and
Lundquist et al. [7] concern with the tracking using
automotive radars. In [16], target tracking is achieved using
data sets acquired with a laser range sensor. In [14], a moving
car is tracked using the data obtained from a surveillance
camera. Baum [1] investigates tracking of a single moving
object on a table with a camera that provides RGB and depth
images. In [17], tracking using an incoherent X-band radar is
considered. Ultra-Wide Band (UWB) is a very promising technology and provides important advantages: high resolution,
immunity to multi-paths, small size, and low-power [18]–[20].
These advantages make UWB technology very crucial
especially for in-door positioning [18], [21]–[25]. Currently,
UWB sensor systems are employed in many applications [24], [26]–[30]. In literature, there exist some works
regarding the detection, surveillance and tracking of persons
or objects by using UWB sensor network (UWB-SN).
1558-1748 © 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
ERYILDIRIM AND GULDOGAN: BERNOULLI FILTER FOR EXTENDED TARGET TRACKING USING RANDOM MATRICES
However, all of these previous works use point-target
assumption [24], [28], [31]–[35]. Therefore, the use of an
effective extended target tracking filter can play a significant
role to enhance the detection, tracking and classification
tasks by estimating the target shape jointly with the target
kinematics. Consequently, an extended target filter can
overcome the challenges associated with the raw UWB sensor
measurements and enhance the feasibility of similar systems
for further advanced tasks.
In this work, we present a new extended target tracking
approach to track size and shape parameters along with
the kinematic parameters from the raw UWB sensor network measurements. The core of the proposed approach is a
Bernoulli filter which can jointly detect and track an extended
target in the presence of data association uncertainty, targetmeasurement rate uncertainty, detection uncertainty, noise and
false alarms [36], [37]. The Bernoulli filter is the exact Bayes
filter which propagates the parameters of a Bernoulli Random
Finite Set (RFS) for a single dynamic system which can
randomly switch on and off [37]–[41]. The Bernoulli filter
provides reliable and computationally inexpensive extraction
of state estimates and joint detection and state estimation [42].
Considering extended target tracking using the probability
hypothesis density filters [6], [16], [43]–[45], there exists
almost no work on extended target tracking using the Bernoulli
filter.
The novelty of this work is two-fold. Firstly, we present
the Gaussian Inverse Wishart (GIW) implementation of the
Bernoulli filter, which is capable of estimating target kinematics and target shape, for extended target tracking. The
proposed extended target GIW Bernoulli filter (ET-GIW-Ber)
assumes random matrix framework for target extent and binomial RFS model of target-originated measurements [4], [37].
Closed form formulas for filter recursions are obtained. Since
the proposed tracking filter provides an analytical solution, this
makes it a good candidate to easily satisfy real-time demands
of a practical system. Furthermore, another contribution is the
adaptive estimation of the time varying number of scatters
of the extended target, which corresponds to an adaptive
target-originated measurement rate estimation scheme. Thus,
it increases the feasibility of the filter for realistic scenarios.
The proposed scheme uses the output of a clustering step
integrated into the measurement update stage of the filter. The
proposed filter makes use of the clustering approach in [16]
in order to have a tractable implementation. To the best of
our knowledge, such a filter has not been presented before.
In literature, the research regarding practical extended target
tracking using Bernoulli filter is very limited, hence indicating
that it is a premature research area. In [14], a Bernoulli
filter is presented for extended target tracking that models the
extended target state by a vector and the part of the state
vector corresponding to the target extent evolves according to
random walk. Different than [14], the proposed ET-GIW-Ber
filter provides a closed form solution and estimates both the
kinematic state and the extent of a target by exploiting the
random matrix framework.
Secondly, the performance of the proposed filter is evaluated over real-data acquired by an UWB sensor network
4363
system by using an accurate ground-truth data established
under controlled experiments. In most extended target tracking
works, ground-truth data is not available or its quality is very
low. To the best of our knowledge, accurate ground truth
data, especially for target extent, are not available in previous
works regarding extended target tracking [3], [7], [14]–[16],
[43], [46]. Target tracking based on the point-target concept
via UWB sensors has been considered in some studies in
the literature [27], [28], [30], [47]–[49]. However, this paper
presents extended target tracking with UWB sensors for the
first time.
II. T HE R ANDOM M ATRIX M ODEL
In the framework of the random matrix model based
tracking, the target extent is modeled by symmetric, positive
definite random matrix, Xk of size d × d, which approximates
the target extent as an ellipsoid, at each time k [4]. The target
extent is assumed to be a part of the target state such that
ξk = (xk , Xk ) where ξk represents the augmented state, xk and
Xk correspond to the kinematical state vector whose dimension
is s × d and target extent state respectively at time k. In the
remaining parts of the paper, ξ is used to denote the extended
target state which is the augmentation of the kinematical state
and target extent state. In this work, the target dynamics is
modeled as in [4]:
(i)
(i)
= (Fk+1|k ⊗ Id )xk(i) + wk+1
,
xk+1
(i)
(i)
(1)
(i)
(i)
where wk+1 ∼ N (0, k+1|k ) and k+1|k = Qk+1|k ⊗ Xk+1 .
The notation ⊗ defines the Kronecker product of two matrices.
(i)
(i)
The kinematical state xk has the following form: xk =
(rkT , ṙkT , r̈kT )T where rk , ṙk and r̈k correspond to the spatial position, velocity and acceleration respectively. Id is a
d × d identity matrix, N (m, P) corresponds to the Gaussian
probability density function (pdf) with mean vector m and
covariance matrix P. In this work, we have s = 3 and d = 2.
Thus, the matrix Fk+1|k and Qk+1|k are given by:
⎡
1 2⎤
T
1 Ts
⎢
2 s⎥
⎢
(2)
Fk+1|k = ⎣0 1
Ts ⎥
⎦,
0
0
e
−Ts
θ
Qk+1|k = 2 (1 − e−2Ts /θ )diag([0 0 1]),
(3)
where Ts is the sampling time, is the acceleration width
and θ is the maneuver correlation time constant. The sensor
measurement model is given by the following linear equation:
( j)
(i)
(i)
(4)
zk = H k x k + ek ,
(i)
where Hk = Hk ⊗ Id , Hk = [1 0 0] and ek is white
Gaussian noise whose covariance is equal to the target extent
matrix Xk . This model assumes that the sensor measurements
are generated by the target centroid and they are spread over
the target extent. Thus, the sensor noise is neglected compared
with the target extension. This model leads to analytical
Bayesian filter recursions for the estimation of the extension
from sensor measurements [4]. Once the target extent is estimated, the target size, shape and orientation can be extracted.
4364
IEEE SENSORS JOURNAL, VOL. 16, NO. 11, JUNE 1, 2016
TABLE I
L IST OF M ATHEMATICAL S YMBOLS
The Bernoulli filter prediction equations are given as [37]:
(6)
qk|k−1 = pb (1 − qk−1|k−1 ) + ps qk−1|k−1 ,
pb (1 − qk−1|k−1 )bk|k−1 (ξ )
sk|k−1 (ξ ) =
qk|k−1
ps qk−1|k−1 πk|k−1 (ξ |ξ ′ )sk−1|k−1 (ξ ′ )dξ ′
,
+
qk|k−1
(7)
where pb and ps are the probability of target birth and
probability of target survival, respectively. The target state
evolves from ξ ′ to ξ with the transitional density πk|k−1 (ξ |ξ ′ ).
The PDF bk|k−1 (ξ ) models the target births.
In this work, the binomial RFS model of measurements originating from the extended target, which is presented in [37],
is used. It is assumed that at time k, the target consists of
L k scatters (measurement generating scattering points) with
the probability of detection, pd . The corresponding Bernoulli
filter update equations are given as [37]:
qk|k =
1 − k
qk|k−1 ,
1 − qk|k−1 k
(8)
(1 − pd ) L k +
ψk
W ∈P1:L k (Zk )
sk|k (ξ ) =
z∈W
1 − k
gk (z|ξ )
λc(z)
sk|k−1 (ξ ),
(9)
where
For the sake of clarity and readability, Table I summarizes the
notation used in this work.
III. T HE B ERNOULLI F ILTER FOR AN E XTENDED TARGET
In the RFS framework, both targets and measurements take
values as unordered finite sets [50]. Thus, in this work, the
target state at time k is modeled by the Bernoulli RFS.
The sensor measurements, Zk , at time k are also defined
( j ) Nz,k
{zk } j =1
as the RFS Zk =
where Nz,k is the number of
sensor measurements at time k. ξ and z denote the extended
target state and sensor measurement respectively. The finite set
statistics probability density function (FISST PDF) [37], [50]
statistically characterizes an RFS. The ’FISST PDF’ is abbreviated by the term PDF in this work for simplicity. For the
Bernoulli filter considered in this work, the Bernoulli RFS,
which models the target state, assumes that at most one target
can be present at time k. The Bernoulli RFS can either be
empty set, denoted by ∅, with probability 1 − q or it can
have one element with probability q distributed over the state
space X with PDF s(ξ ). The PDF of a Bernoulli RFS X is
given by [50]:
1 − q if X = ∅,
f (X) =
(5)
qs(ξ ) if X = {ξ }.
For tracking an extended target, the Bernoulli filter propagates the following two quantities recursively [37]: the
posterior probability of target existence qk|k = P(|X| =
1|Z1:k ) and the posterior spatial PDF sk|k (ξ ) = p(ξ |Z1:k ).
k = 1 − (1 − pd ) L k
−
ψk
gk (z|ξ )sk|k−1 (ξ )dξ
z∈W
W ∈P1:L k (Zk )
λc(z)
,
(10a)
z∈W
|W |
pd
Lk !
ψk =
.
(L k − |W |)! (1 − pd )|W |−L k
(10b)
The notation P1:L k (Zk ) corresponds to the set of all subsets
of Zk with cardinalities equal to 1, 2, . . . , L k . gk (z|ξ ) is the
likelihood function and c(z) denotes the spatial distribution of
the clutter, which is assumed to be known. λ is the average rate
of the clutter measurements modeled by a Poisson distribution
at time k. The parameter L k must be set in order to update the
filter. In most practical applications, L k is unknown. Therefore,
we will present a method for adaptive estimation of L k in the
following section.
IV. T HE E XTENDED TARGET G AUSSIAN I NVERSE
W ISHART B ERNOULLI (ET-GIW-BER) F ILTER
In [4], under certain assumptions which are justified for
most scenarios and ellipsoidal target extent model, a Bayesian
recursive filter which describes the posterior density as a
product of Gaussian and Wishart-related densities is presented.
Thus, considering that the Bernoulli filter is a Bayesian
optimal filter, the filter prediction and update results given
in [4] are also preserved for the extended target Bernoulli
filter [43]. Therefore, in this work, the results presented in [4]
are adapted into the extended target Bernoulli filter framework
accordingly.
ERYILDIRIM AND GULDOGAN: BERNOULLI FILTER FOR EXTENDED TARGET TRACKING USING RANDOM MATRICES
( j)
ηk−1|k−1 and
( j)
k−1|k−1
4365
are the inverse Wishart degrees of
freedom and inverse scale matrix of the j -th component. The
notation IW(X; η, ) corresponds to an inverse Wishart pdf
defined over the variable X with degrees of freedom η and
inverse scale matrix [51]. In order to shorten the equations,
here we define the following variable:
( j) ( j)
( j)
( j)
( j) △
(12)
,
P
,
η
,
=
m
k|k
k|k k|k
k|k .
k|k
Fig. 2. Extended target measurements with the ellipsoidal extension model:
Measurements correspond to the x-y coordinates of an extended target
at two different time instances. Gray squares represent the measurements
collected by a sensor, red squares correspond to the centroid of the extended
target and red ellipses represent the true shape of the target. Bearing,
semi-major and semi-minor axes lengths are seen for two time instances.
An extended target and measurements generated at different
times according to the models used in this work are illustrated
in Fig. 2. From an extended target, a high-resolution sensor in
general collects multiple measurements originating from the
scatters of the target at each time. Assuming that sensor measurements consist of position coordinates x and y of the target,
an extended target might generate multiple measurements at
each time as shown in Fig. 2. Both cardinality and spatial
spread of extended target measurements at each time are
assumed to be random. The measurements originated from the
scattering points vary with time depending on the properties
of the target extent, sensor-to-target geometry and properties
of sensor. Therefore, binomial RFS model for extended target
measurements is a very appropriate model.
In this work, the target shape is approximated by an ellipsoid
represented by the symmetric positive definite matrices as
in [4]. As shown in Fig. 2, target shape parameters including
the bearing φ, the semi-major axes length A1 and semiminor axes length a1 can be extracted from the random
matrix model used in this work. Ellipsoidal shape representation is appropriate in many practical target tracking
scenarios [4], [43]. This is due to the fact that in many practical
cases, the target-to-sensor range is neither high enough to
generate only a single measurement nor is it low to allow
target features to be distinguished in detail.
It is assumed that the spatial PDF sk−1|k−1 (·) at time k −1 is
given by a GIW mixture of the form,
Assuming the birth PDF has a mixture of GIW distribution,
the form of the prediction equation (7) results in a mixture of
GIW distribution with the parameters Mk|k−1 , wk|k−1 , k|k−1
according to the model and assumptions for prediction step
in [4]. Thus, applying the equations (24), (25), (28), (29) in [4],
we obtain the predicted components corresponding to existing
targets as:
ps qk−1|k−1 ( j )
( j)
wk−1|k−1 ,
(13a)
wk|k−1 =
qk|k−1
( j)
=
j =1
( j)
k−1|k−1
,
(11)
( j)
where Mk−1|k−1 is the number of components, wk−1|k−1 is
( j)
=
(13c)
( j)
k|k−1
=
=
( j)
T
Qk|k−1 + Fk|k−1 Pk−1|k−1 Fk|k−1
,
−Ts /τ ( j )
e
ηk−1|k−1 ,
( j)
ηk|k−1 − d − 1 ( j )
k−1|k−1 ,
( j)
ηk−1|k−1 − d − 1
( j)
k|k−1
( j)
the weight of j -th component; mk−1|k−1 and Pk−1|k−1 are
the Gaussian mean and covariance of the j -th component;
(13d)
(13e)
=
( j)
b,k .
(15)
The probability of existence is predicted according to (6).
The updated spatial distribution is again a GIW mixture;
D
sk|k
(ξ, W ),
ND
sk|k (ξ ) = sk|k
(ξ ) +
(16)
W ∈P1:L k (Zk )
where ND and N stand for no detection and detection
cases, respectively. No detection part of the updated spatial
N D (ξ ), is updated by
distribution, sk|k
( j)
wk|k =
( j)
k|k
( j)
( j)
( j)
wk−1|k−1 N x; mk−1|k−1 , Pk−1|k−1 ⊗ X
( j)
×IW X; ηk−1|k−1 ,
(13b)
( j)
Pk|k−1
( j)
ηk|k−1
where τ is a temporal decay constant. The parameter, τ , is
used to accommodate the variation of the target extent over
time. We assume that the birth model, bk|k−1 (ξ ), has exactly
the same form as sk−1|k−1 (·) with parameters Mb,k , wb,k , b,k .
This assumption can be validated for many realistic scenarios.
Then, the contribution of birth model to the prediction is
given by:
pb (1 − qk−1|k−1 ) ( j )
( j)
wb,k ,
(14)
wk|k−1 (ξ ) =
qk|k−1
sk−1|k−1 (ξ )
Mk−1|k−1
( j)
mk|k−1 = (Fk|k−1 ⊗ Id )mk−1|k−1 ,
=
(1 − pd ) L k ( j )
wk|k−1 ,
1 − k
( j)
k|k−1 .
(17)
(18)
In order to obtain the detection part of the updated distribution,
D (ξ, W ), which is also a GIW mixture, the likelihood of the
sk|k
measurements in each subset W is to be multiplied with the
predicted spatial PDF components. Assuming uniform clutter
distribution, c(z k ) = V1 where V is the surveillance volume,
we have the following structure for the likelihood part:
gk (zk |ξ )
−|W |
= αc,k
N (zk ; (Hk ⊗ Id )x, X), (19)
λc(z k )
zk ∈W
zk ∈W
4366
IEEE SENSORS JOURNAL, VOL. 16, NO. 11, JUNE 1, 2016
where αc,k = Vλ is the average number of clutter measurements
per surveillance volume per time. Therefore, we obtain the
following form for the product of sk|k−1 (ξ ) and (19):
⎞
⎛
−|W |
αc,k ⎝
N (zk ; (Hk ⊗ Id )x, X)⎠ sk|k−1 (ξ )
zk ∈W
=
( j,W )
( j,W )
x; mk|k , Pk|k ⊗ X
( j,W )
( j,W )
,
×IW X; ηk|k , k|k
−|W | ( j,W )
αc,k Lk
N
(20)
( j,W )
where Lk
represents a kind of likelihood. Then, using the
filter update results derived in [4], the updated components are
obtained as,
( j,W )
( j,W )
( j)
( j,W )
(21a)
mk|k = mk|k−1 + K k|k−1 ⊗ Id E k|k−1 ,
( j,W )
Pk|k
( j,W )
ηk|k
( j,W )
k|k
( j)
( j,W ) T
( j,W ) ( j,W )
= Pk|k−1 − K k|k−1 Sk|k−1 (K k|k−1 ) ,
=
=
( j)
ηk|k−1 + |W |,
( j)
( j,W )
k|k−1 + Nk|k−1
(21b)
(21c)
+ Z kW .
(21d)
The respective intermediate variables in the above equations
are given by,
z̄kW =
1
|W |
z(i)
k ,
(22a)
(i)
zk ∈W
T
(i)
W
W
(z(i)
k − z̄k )(zk − z̄k ) ,
Z kW =
(22b)
(i)
zk ∈W
1
( j,W )
( j)
Sk|k−1 = Hk Pk|k−1 Hk T +
,
|W |
( j,W )
( j)
( j,W ) −1
K k|k−1 = Pk|k−1 Hk T Sk|k−1
,
( j,W )
E k|k−1
( j,W )
=
z̄kW
( j,W )
( j,W )
Lk
(22d)
( j)
− (Hk ⊗ Id )mk|k−1 ,
( j,W ) −1 ( j,W )
( j,W ) T
Nk|k−1 = Sk|k−1
The likelihood term Lk
(22c)
E k|k−1 E k|k−1
(22e)
(22f)
.
in (20) has the following form [43]:
1
=
( j,W ) d/2
π |W | |W |Sk|k−1
|
( j)
k|k−1 |
|
( j,W )
k|k |
( j)
ηk|k−1
2
η
( j,W )
k|k
2
( j,W )
(ηk|k )
)
2
.
( j)
ηk|k−1
Ŵd ( 2 )
Ŵd (
(23)
where | | corresponds to the determinant of the matrix
and |W | is the number of measurements in the subset W . The
weights of the updated GIW mixture are obtained as:
( j)
wk|k =
1
( j,W ) −|W | ( j )
αc,k wk|k−1 ,
ψk Lk
1 − k
(24)
where k given in (10a) has the following form:
k = 1 − (1 − pd ) L k
Mk|k−1
−|W |
−
( j,W )
Lk
ψk αc,k
W ∈P1:L k (Zk )
j =1
( j)
wk|k−1 .
(25)
The estimate of the target extent is obtained from the
inverse Wishart parameters by using the following formula:
X̂ = η−2d−2 [4]. In order to have a tractable number of
components, pruning and merging of GIW components is
achieved similarly as in [43].
The computational complexity and required memory size
for the ET-GIW-Ber filter are both proportional to the total
number of subsets, P1:L k (Zk ), generated from the measurements Zk . Thus, total number of subsets, P1:L k (Zk ), exponentially increases as cardinality of measurements and L k
corresponding to the true number of scattering points of
extended target at time step k increases. Even if L k is
small, increased clutter rate leads to huge number of subsets.
Considering augmented vector and matrices used in the measurement update step, required memory size might become
infeasible to do the computation for even powerful computers.
Therefore, gating of measurements is commonly applied
to alleviate the computational burden for extended target
tracking [37]. However, target-originated measurements must
be preserved in order to have a good extended target tracking
performance. Thus, high number of target-originated measurements leads to intractable solution even if gating is used to
eliminate the clutter. Considering this problem, we insert a
clustering step into the measurement update stage of the proposed filter in order to have a computationally tractable filter.
The clustering method approximates the set of all subsets,
P1:L k (Zk ), generated in the measurement update stage with a
subset of partitions. Assume that the output of the clustering
is denoted by P Zk . The notation P Zk means that P
partitions the measurement set Zk into non-empty clusters .
Then, P Zk replaces P1:L k (Zk ). Thus, partitions or clusters,
P Zk , are processed in the measurement update instead of
using all subsets, P1:L k (Zk ). Therefore, the proposed filter can
handle large number of target-originated measurements per
time, which is often the case in many real world scenarios.
Having said that, we note that the parameter L k still
influences the tracking performance of the proposed filter as
it is embedded in the measurement update equations. Thus,
we still need to set the value of L k in the measurement
update stage of the proposed filter. In [14], a fixed formula
which uses the cardinality of current measurements, probability of detection and known clutter rate is given for the
estimation of L k . In order to estimate L k , we propose a new
method which is consistent with the applied clustering method.
Consequently, we cluster the measurement set received at time
step k spatially with the distance partitioning method presented
in [43] by exploiting the knowledge of the spatial distribution
of the extended target measurements. At the output of this
clustering process, typically, one large cluster corresponding
to the extended target and other clusters are obtained. Also, the
cluster corresponding to the extended target in general contains
more measurements than the other clusters. To sum up, in
our method, the parameter, L k , is adaptively estimated using
the output of the applied clustering method, which is distance
partitioning.
Assume that the clustering step which employs the distance
partitioning method generates n number of clusters, P Zk ,
at time step k such that each cluster and the corresponding
ERYILDIRIM AND GULDOGAN: BERNOULLI FILTER FOR EXTENDED TARGET TRACKING USING RANDOM MATRICES
4367
cardinality is represented by (i) and |(i) | where i =
1, . . . , n. Then, the parameter L k is estimated such that:
Mk,max = max {|(1)|, |(2) |, . . . , |(n) |},
Mk,max
⌉.
Lk = ⌊
pd
(26)
(27)
where ⌊·⌉ corresponds to the nearest integer operation and
Lk
represents the adaptive estimate of L k .
V. S IMULATION R ESULTS
We demonstrate the ET-GIW-Ber filter performance on a
scenario involving time-varying target extent and time-varying
number of target scatters. For nearly all kinds of sensors
from laser range finders to cameras, quality and statistical characteristics of target-originated measurements strongly
depend on target-to-sensor distance, which is also called as
target range. In general, as target range increases over time,
target occupies fewer number of resolution cells and signal-tonoise ratio decreases. Hence, the number of target-originated
measurements (which is also called as detections) as well as
the detected target size reduce. However, it is desired that
a tracking filter maintains its track while the target range
increases. Therefore, the considered scenario is frequently
encountered in real-world applications.
The surveillance area is taken as 600 m2 . The simulation
duration is 100 s where each time k corresponds to 1 s
(i.e. the sampling time is T = 1 s). The target is born
at time k = 5 and dies at k = 72. The target-originated
measurements consisting of Cartesian x-y coordinates are
generated according to (4). The true target extents are given by
Xk = Mk diag([ A1,k a1,k ])(Mk )T where Mk is the rotation matrix which determines the bearing (orientation),
A1,k and a1,k are the semi-major and semi-minor axes lengths
of the extended target at time k respectively.
In the considered scenario, the target extent is time-varying
such that the lengths of the principal axes reduce to their
halves with constant rate over time assuming initial values
(A1,birt h , a1,birt h ) = (30 m, 10 m). Besides that, the bearing of the target linearly increases up to 75 degrees from
15 degrees over time. The true number of scatters of the
target L k,true linearly reduces to 10 over time starting from
the value 20 such that L k,true = ⌊dk + e⌉ for 5 ≤ k ≤ 72
where d = −0.1493 and e = 20.7463. Thus, the target extent
undergoes rotation and scaling which often occur in practical
applications. The scenario might correspond to a target moving
far away from a possibly radar sensor such that as the target-tosensor distance increases, its size detected by the radar sensor
reduces and fewer number of measurements are received by
the radar.
The true target trajectory, which is shown in Fig. 3a, is
generated according to the dynamic model in [4]. The average
clutter rate is taken as λ = 10. The clutter is uniformly
distributed over the surveillance area. For most practical applications, there is not a-priori knowledge on the locations where
the target is likely to appear. Thus, assuming that we lack
a-priori information about target birth, we use the following birth model: mb,k = [250 250 3 3 0 0],
Fig. 3. A single run of the ET-GIW-Ber filter: (a) All measurements generated, true target (centroid) trajectory (blue line), estimated target (centroid)
trajectory by the ET-GIW-Ber filter (red dots). (b) The corresponding targetoriginated measurements and tracking results at k = 7, 11, 20, 30, 35, 40, 46,
50, 54, 64, 71: the target-originated measurements (gray dots), blue ellipses
representing true extended target shape, red ellipses representing estimated
extended target shape.
Pb,k = diag([1502 502 502 ]) ηb,k = 7, b,k = diag([1 1]).
The birth model that we use here is generally the case in
most real applications and therefore makes the scenario more
realistic [39], [52]. The main parameters of the ET-GIWBer filter used are: ps = 0.99, pb = 0.01, pd = 0.9,
= 0.15 m2 /s, θ = 0.5 s, and τ = 2 s. The pruning/merging
parameters are selected as Ttrunc = 10−4 , U = 20 and
Jmax = 100 where Ttrunc is the truncation threshold for
weights, U is the merging threshold, and Jmax is the maximum
allowable number of GIW mixture components. Clustering of
measurements is achieved by using the distance partitioning
method. The threshold τ for the target detection is chosen
as 0.5 (i.e. a target is declared as detected if qk|k ≥ τ ).
The performance of the ET-GIW-Ber filter is evaluated via
Monte Carlo simulations. For each Monte Carlo run, random
sensor measurements including the ones originating from the
target and clutter are generated accordingly. For performance
evaluation, optimal subpattern assignment (OSPA) error is
4368
IEEE SENSORS JOURNAL, VOL. 16, NO. 11, JUNE 1, 2016
Fig. 4.
The average OSPA errors of the ET-GIW-Ber filter.
used with cut-off parameter, c = A1,birt h = 30 m and
p = 1 [53]. The sum of the localization error and the
cardinality error is both involved in OSPA error. The relative
weighting of the costs assigned to localization and cardinality
errors depends on the selection of c. We define four different
OSPA errors: OSPA position, bearing, size, and shape error.
The OSPA position error corresponds to the OSPA between the
true target centroid coordinates and the estimated ones. The
OSPA bearing, size and shape error are computed according to
the true shape parameter vectors [φ], [ A1 a1 ] and [ A1 a1 φ]
respectively where A1 , a1 , and φ are the semi-major axes
length, semi-minor axes length (in meters) and the bearing
angle of the target (in degrees) respectively. All OSPA errors
are averaged over 1000 Monte Carlo runs. The results of a
single Monte Carlo run of the proposed ET-GIW-Ber filter
are shown in Fig.3. As seen in Fig.3a, the ET-GIW-Ber filter
provides accurate tracking of the extended target position. The
estimates of the extended target shape as well as the bearing
and size variation of the target over time are both shown in
Fig.3b at certain time steps.
Fig.4 shows the OSPA error performances of the proposed
filter. According to Fig.4, the ET-GIW-Ber filter provides
a very promising accuracy in estimating both the centroid
position and the size/bearing of the target. At around k = 5
when the target is born, both the average OSPA position and
shape/size errors rise up to some value which is lower than
the cut-off value c. This is a desired result indicating that
the ET-GIW-Ber filter can initiate the track accurately and
immediately following the target birth. Similar to the target
birth, the ET-GIW-Ber filter is sensitive to the death of the
target at k = 72 as the corresponding OSPA errors very
quickly reduce to zero after k = 72. In [4], it is claimed
that time-varying target extents, whose shape and number of
scatters change over time, are tractable within the presented
framework. Consequently, the simulation results suggest that
target with time-varying extent can be estimated with low error
by using the proposed ET-GIW-Ber filter. Furthermore, the
proposed filter performs well with unknown target-originated
measurement rate and lack of a-priori information about target
births.
VI. E XPERIMENT R ESULTS W ITH THE UWB S YSTEM
We used UWB Real-time Location System (RTLS) provided by Ubisense to perform extended target tracking experiments [54]. Ubisense RTLS is an in-door UWB sensor
Fig. 5. The UWB-SN experimental setup. (a) Indoor environment where
the experiments take place, a person moves inside the area whose corners
are assumed to be the UWB sensors shown by the red circles. (b) UWB
tags mounted on a person: The tags are shown as numbered from one to
six corresponding to their IDs. (c) The UWB sensor (Ubisense).
network (UWB-SN) which uses active tags mounted on moving objects for positioning. Therefore, the UWB-SN system
installation is composed of a network of UWB sensors that
are installed in a room and small radio-emitting tags that can
be mounted on a person as shown in Fig. 5 [54].
The measurement data collected by the UWB-SN are (x,y,z)
spatial coordinates of the tags and their related acquisition
time slot. The UWB-SN allocates time slots to each tag for
emitting its signal in order to eliminate interference between
tags. Position refreshment frequency, i.e. measurement rate,
differs for each tag. The localization accuracy of the system
is claimed to be 0.15 m with 99% of errors being within
0.30 m [55] in an ideal setting. However, in reality, due to
the environmental effects, its positioning accuracy degrades.
In [56], it is claimed that for dynamic positioning where the
UWB tag is moving, they obtain a horizontal positioning mean
error of 0.37 m and a standard deviation of 0.24 m for the
accuracy of the UWB-SN system.
For the physical set-up of the UWB-SN system, the sensors
are fixed at a height of 1.74 m from the ground. Four sensors
are placed in the four vertices of the bounding rectangle
ERYILDIRIM AND GULDOGAN: BERNOULLI FILTER FOR EXTENDED TARGET TRACKING USING RANDOM MATRICES
covering the area where experiments take place as seen in
Fig. 5a. Different scenarios including different trajectories and
human movements are achieved inside this area.
In all experimental scenarios, the elevation (height) of the
tag remains fixed. Hence, the tags attached to the humans
keep a constant distance to the floor and tag movement is
considered to be only horizontal. Therefore, we have a 2D
tracking problem.
Data collected by the UWB-SN system bring several challenges [35], [56], [57]. First, due to lost packets or low signalto-noise ratio (SNR), missing values occur at certain time
steps. This means that detection uncertainty exists for the
UWB-SN system. Second, a certain percentage of the collected
data has high error - which can be called outliers or clutter.
Lastly, mobility rate of the tag, non-line-of-sight conditions as
well as obstruction due to the human body and other factors
further degrade the accuracy of the UWB-SN system compared
to the ideal accuracy values.
In [56], Ubisense tags are mounted on wheel chairs of the
athletes and a Leica TS-30 robotic total station is employed
for establishing the ground-truth. In this work, an average
horizontal positioning error of 0.37 m (with standard deviation
0.24 m) is reported for athletes tracking. For our experiments,
the conditions of our environment do not favor the Ubisense
system. Furthermore, the geometry and placement of Ubisense
sensors cause non-line-of-sight for some sensors at certain
time steps during the movement of the person. Consequently,
in this work, considering the quality of the acquired data, it is
assumed that the Ubisense positioning accuracy is 0.4 m. This
assumption also means that assuming identical conditions for
both axes, the standard deviation of the measurement noise
is 0.4 m.
6 UWB tags are mounted on the body of a person as
shown in Fig. 5b. During the experiments, the person with
the UWB tags follows pre-defined trajectories with relatively
constant velocity. The parameters of the proposed filter as
well as the values of the parameters in the target dynamics
and sensor model are adjusted according to the motion of the
human, measurement statistics of the UWB-SN system and
other conditions of the experiments.
Since the person can start its movement at any location
inside the surveillance area, it is assumed that we do not
know the target birth locations a-priori. Thus, the spontaneous
birth density is defined in the center of the surveillance area
with an appropriate covariance such that target appearance can
occur in any location of the area. The birth density used in
the experiments is given by:
bk|k−1 = 0.2N (x; mb , Pb ),
(28)
mb = [3, 6, 0, 0, 0.0], Pb = di ag([32, 52 , 52 ]).
(29)
where
Consequently, accommodating all scenarios, this approach is
realistic for practical considerations. The clutter is assumed
to follow a uniform spatial distribution over the rectangular
surveillance area whose corners are determined by four UWB
sensors.
4369
Fig. 6.
The UWB-SN measurements and ground truth trajectory for
Scenario I: The UWB-SN measurements corresponding to different UWB tags
are denotes with different color coded markers. Solid blue line represents the
ground truth.
Pruning stage of the proposed filter is achieved with the
same parameter values used in the simulations such that
Ttrunc = 10−5 , U = 4 and Jmax = 100. The threshold τ
for the target detection of the ET-GIW-Ber is chosen as 0.5.
In our tracking problem, due to the non-line-of-sight (NLOS)
and signal attenuation problems, unreliable measurements are
collected at certain time instants. Also, no measurement might
be collected at some time instants for a corresponding tag.
Thus, clutter rate is taken as λ = 1 with clutter intensity
distributed uniformly over the surveillance area.
We present the results from the experiments achieved with
the UWB-SN system. Two different scenarios, which we call
as Scenario I and II respectively, were considered in order to
evaluate the performance of the proposed filter.
In order to obtain a definite measure of target tracking
performance, ground truth data for the state of the target and
its extent are required. Since the experiments were achieved
under our control, the ground-truth for the trajectory of the
person were obtained in a reasonable and relatively accurate
manner. The ground-truth trajectories for both scenarios are
shown in Fig. 7a and 9a respectively. While it is relatively
an easy task to build the ground-truth accurately and reliably
for the target position, it was relatively difficult to obtain the
ground truth data in a very accurate manner for the target
extent, especially for the bearing. However, by examining the
geometric configuration of the tags mounted on the person,
true trajectory and accuracy of the UWB-SN system, the true
target size and bearing were established.
Scenario I involves a person walking over a straight line
along the y direction. The person follows a trajectory starting
from coordinates (3.0, 3.5) m and ending in coordinates
(3.0, 7.2) m along a straight line. Besides that, the person
moves in the middle of the surveillance area. This enhances
the line-of-sight condition for all sensors. As the number of
sensors seeing the UWB tag directly increases, the probability
of detection of the system increases and the accuracy of
the localization performance improves [55], [56]. Moreover,
due to the target trajectory, the bearing of the person, which
4370
Fig. 7.
Scenario I, extended target tracking results for the UWB-SN
measurements: (a) Solid blue line represents the ground-truth for the target
trajectory and red-diamonds represent the estimates of the ET-GIW-Ber filter.
(b) Extended target size tracking results: Estimates of the major and minor
axes lengths, A1 and a1 and the corresponding true values, respectively.
(c) Extended target bearing tracking results: Solid blue lines represent the
lower and upper bounds for the ground-truth and blue-crosses correspond to
the bearing estimates.
is considered as an extended target, is expected to remain
approximately constant with some amount of fluctuations. This
is due to the placement of the tags. There exist two tags
mounted close to the knees of the persons. The movement of
legs of the person causes some amount of bearing change over
time. Thus, while it is difficult to obtain an accurate groundtruth for the bearing of the person, a reasonable assumption is
made. Considering the spatial spread of the leg movement is
IEEE SENSORS JOURNAL, VOL. 16, NO. 11, JUNE 1, 2016
limited, the variation of the true bearing should be within some
bounds. An investigation of the trajectory, which is along y
direction only, and geometric configuration of the tags leads
to the conclusion that the true bearing is around zero degree.
The random matrix model ignores the contribution of the
measurement noise [4]. Thus, it assumes that all targetoriginated measurements are contained inside the target extent.
Therefore, the proposed filter estimates the target extension
plus sensor accuracy [58]. Thus, in order to properly define
the true target size, i.e. the ground-truth for the semi-major
and minor axes lengths, the accuracy of the UWB-SN system
is involved to obtain the ground-truth for the target size.
Considering the geometric configuration of the tags and the
trajectory in Scenario I, it is assumed that the extent of
the target along y direction is approximately determined by
the average accuracy (error) of the UWB-SN. Based on the
collected data for our mobile measurements, the standard
deviation of the measurement noise in both directions is
assumed to be 0.4 m, as mentioned previously. Consequently,
assuming that the width between the two shoulders of the
person is 0.4 m, we have the following true values for the
semi-major and minor axes lengths: A1,true = 0.6 m and
a1,true = 0.4 m.
The collected measurements are shown in Fig. 6. The corresponding target position estimates provided by the proposed
filter are shown in Fig. 7a. The differences between the
blue line representing the ground-truth positions and the red
diamonds corresponding to the filter estimates are very small
in this scenario. Thus, the results show that the proposed filter
can estimate the position of the person accurately in this case.
The tracking performance for the target extent is shown
in Fig. 7. The accuracy of target size tracking appears very
good as shown in Fig. 7b. As seen in Fig. 6, the UWB-SN
measurements contain more distortions at the beginning of the
trajectory. This also affects the size estimation performance of
the filter for the initial measurements. In general, the proposed
filter provides smooth and accurate estimates which are close
to the ground-truth. Besides that, the proposed filter provides
reasonable bearing estimates consistent with the true values
as shown in Fig. 7c. We think that the movement of person
includes some small gait changes as it is not easy for the person to preserve the same posture during the whole trajectory.
Due to this and other factors, the fluctuations of the bearing
estimates as well as the true bearing values have different
behavior during different portions of the trajectory. However,
over-all, the bearing estimates are good and consistent with
the expected true bearing values.
The second scenario (Scenario II) corresponds to a more
challenging target trajectory with target maneuvers in a small
area, as shown in Fig. 8. The trajectory of the person starts at
the position (2.3, 3.5) m and ends at the position (3.5, 3.5) m.
As seen in Fig. 8, the trajectory is in the shape of a rectangular
with one missing edge. Therefore, the trajectory followed by
the person in Scenario II can be divided into three segments,
Segment 1, 2 and 3 respectively. The person first moves along
Segment 1 which is a straight line segment along y direction.
Then, the person does a maneuvering-like action in order to
enter Segment 2 which is a shorter straight line segment
ERYILDIRIM AND GULDOGAN: BERNOULLI FILTER FOR EXTENDED TARGET TRACKING USING RANDOM MATRICES
4371
Fig. 8.
The UWB-SN measurements and ground truth trajectory for
Scenario II: The UWB-SN measurements corresponding to different UWB
tags are denotes with different color coded markers. Solid blue line represents
the ground truth.
along x direction. Finally, the person follows Segment 3
in reverse direction with respect to the one in Segment 1
and finishes its trajectory. At certain locations on the considered trajectory, non-line-of-sight conditions occur due to
the orientation of the sensors and their limited field of view.
Furthermore, the human body parts might obstruct and weaken
the signals of some tags depending on the geometry. Therefore,
at certain locations, the UWB-SN measurements degrade more
severely as exhibited by the plot shown in Fig. 8. However,
it is desired that the proposed filter maintains track during
these unfavorable and challenging conditions. Similar to the
previous experiment, the red diamonds in Fig. 9a are very
close to the real positions of the person and the proposed filter
performs very well for this difficult scenario.
Similar effective performances are also exhibited for the size
and bearing tracking provided by the proposed filter as shown
in Fig. 9b and 9c. The filter successfully captures the sudden
bearing changes which occurs when the person moves from
Segment 1 to 2 and from Segment 2 to 3. This is exhibited by
jumps in the bearing estimates in the interval between k = 30
and k = 50 approximately as shown in Fig. 9c. This time
interval corresponds to the movement of person in Segment 2
including the maneuvering actions while entering and leaving
Segment 2. Thus, the results indicate that the proposed filter
can track the bearing of an extended target. The size estimation
performance of the proposed filter for this scenario is also good
as indicated by the small differences between the ground-truth
and estimates as shown in Fig. 9b.
The two scenarios above are promising evaluations of the
ET-GIW-Ber filter and they present a proof of concept study
for indoor person tracking using an UWB sensor network
system. In [35], it is claimed that although the accuracy of
Ubisense system is about 0.15 m in ideal setting, however,
in practice, the quality of Ubisense raw measurements makes
human action recognition challenging. Being an extended
target tracking filter capable of the estimating the shape of
target, the proposed filter provides extra information and
inherently can handle the noise, occlusions, distortions and
Fig. 9.
Scenario II, extended target tracking results for the UWB-SN
measurements: (a) Solid blue line represents the ground-truth for the target
trajectory and red-diamonds represent the estimates of the ET-GIW-Ber filter.
(b) Extended target size tracking results: Estimates of the major and minor
axes lengths, A1 and a1 and the corresponding true values, respectively.
(c) Extended target bearing tracking results: Solid blue lines represent the
lower and upper bounds for the ground-truth and blue-crosses correspond to
the bearing estimates.
anomalies associated with UWB systems. Thus, we think that
the proposed filter can be useful for tasks like human action
recognition or behavior analysis of humans.
VII. C ONCLUSION
In this work, a tractable extended target Bernoulli filter
using random matrix model is presented. The proposed filter is
4372
IEEE SENSORS JOURNAL, VOL. 16, NO. 11, JUNE 1, 2016
capable of estimating both the target kinematics and the nonstationary target extent parameters. Analytical formulas are
presented for filter recursions based on GIW approximation.
A method which uses the output of the integrated clustering
method is also applied in order to adaptively estimate the timevarying number of scatters of the extended target. Simulation
results show that the proposed filter provides accurate and
robust performance in extended target tracking. Moreover,
performance of the proposed filter is tested in an indoor person
tracking scenario using real UWB data. For future work, we
intend to improve the performance of the proposed filter by
considering different models developed in the random matrix
framework for different scenarios. For instance, we plan to
achieve experiments which involves the scaling of the target
size. Furthermore, we are planning test the performance of
the proposed filter in a tag-less experiment using UWB radar
sensors.
R EFERENCES
[1] M. Baum, “Simultaneous tracking and shape estimation of extended
objects,” Ph.D. dissertation, Dept. Fakultät für Informatik, Karlsruhe
Inst. Technol., Karlsruhe, Germany, 2013.
[2] L. Hammarstrand, L. Svensson, F. Sandblom, and J. Sorstedt, “Extended
object tracking using a radar resolution model,” IEEE Trans. Aerosp.
Electron. Syst., vol. 48, no. 3, pp. 2371–2386, Jul. 2012.
[3] B. Fortin, R. Lherbier, and J. C. Noyer, “A labeled PHD filter
for extended target tracking in lidar data using geometric invariance properties: Vehicular application,” in Proc. 16th Int. Conf. Inf.
Fusion (FUSION), Jul. 2013, pp. 1744–1751.
[4] J. W. Koch, “Bayesian approach to extended object and cluster tracking
using random matrices,” IEEE Trans. Aerosp. Electron. Syst., vol. 44,
no. 3, pp. 1042–1059, Jul. 2008.
[5] PETS: Performance Evaluation of Tracking and Surveillance,
accessed on Jan. 31, 2016. [Online]. Available: http://www.cvg.
reading.ac.uk/slides/pets.html
[6] R. Mahler, “PHD filters for nonstandard targets, I: Extended targets,” in
Proc. 12th Int. Conf. Inf. Fusion, 2009, pp. 915–921.
[7] C. Lundquist, U. Orguner, and F. Gustafsson, “Extended target tracking
using polynomials with applications to road-map estimation,” IEEE
Trans. Signal Process., vol. 59, no. 1, pp. 15–26, Jan. 2011.
[8] F. Lian, C. Han, J. Liu, and H. Chen, “Convergence results for the
Gaussian mixture implementation of the extended-target PHD filter and
its extended Kalman filtering approximation,” J. Appl. Math., vol. 2012,
May 2012, Art. no. 141727.
[9] L. Mihaylova, A. Y. Carmi, F. Septier, A. Gning, S. K. Pang, and
S. Godsill, “Overview of Bayesian sequential Monte Carlo methods for
group and extended object tracking,” Digit. Signal Process., vol. 25,
pp. 1–16, Feb. 2014.
[10] M. Baum and U. D. Hanebeck, “Extended object tracking with random
hypersurface models,” IEEE Trans. Aerosp. Electron. Syst., vol. 50, no. 1,
pp. 149–159, Jan. 2014.
[11] M. Baum and U. D. Hanebeck, “Extended object tracking based on settheoretic and stochastic fusion,” IEEE Trans. Aerosp. Electron. Syst.,
vol. 48, no. 4, pp. 3103–3115, Oct. 2012.
[12] U. Orguner, “A variational measurement update for extended target
tracking with random matrices,” IEEE Trans. Signal Process., vol. 60,
no. 7, pp. 3827–3834, Jul. 2012.
[13] M. Feldmann and D. Franken, “Advances on tracking of extended objects
and group targets using random matrices,” in Proc. 12th Int. Conf. Inf.
Fusion, Jul. 2009, pp. 1029–1036.
[14] B. Ristic and J. Sherrah, “Bernoulli filter for joint detection and tracking
of an extended object in clutter,” IET Radar, Sonar Navigat., vol. 7, no. 1,
pp. 26–35, Jan. 2013.
[15] K. Granström, A. Natale, P. Braca, G. Ludeno, and F. Serafino, “Gamma
Gaussian inverse Wishart probability hypothesis density for extended
target tracking using X-band marine radar data,” IEEE Trans. Geosci.
Remote Sens., vol. 53, no. 12, pp. 6617–6631, Dec. 2015.
[16] K. Granström, C. Lundquist, and U. Orguner, “Extended target tracking
using a Gaussian-mixture PHD filter,” IEEE Trans. Aerosp. Electron.
Syst., vol. 48, no. 4, pp. 3268–3286, Oct. 2012.
[17] K. Granstrom, A. Natale, P. Braca, G. Ludeno, and F. Serafino, “PHD
extended target tracking using an incoherent X-band radar: Preliminary
real-world experimental results,” in Proc. 17th Int. Conf. Inf. Fusion
(FUSION), Jul. 2014, pp. 1–8.
[18] Z. Sahinoglu, S. Gezici, and I. Guvenc, Ultra-Wideband Positioning
Systems: Theoretical Limits, Ranging Algorithms, and Protocols. Cambridge, U.K.: Cambridge Univ. Press, 2006.
[19] S. Bartoletti, A. Conti, and A. Giorgetti, “Analysis of UWB radar sensor
networks,” in Proc. IEEE Int. Conf. Commun. (ICC), May 2010, pp. 1–6.
[20] Y. Rahayu, T. A. Rahman, R. Ngah, and P. Hall, “Ultra wideband
technology and its applications,” in Proc. 5th IFIP Int. Conf. Wireless
Opt. Commun. Netw. (WOCN), May 2008, pp. 1–5.
[21] S. Gezici et al., “Localization via ultra-wideband radios: A look at
positioning aspects for future sensor networks,” IEEE Signal Process.
Mag., vol. 22, no. 4, pp. 70–84, Jul. 2005.
[22] S. Gezici and H. V. Poor, “Position estimation via ultra-wide-band
signals,” Proc. IEEE, vol. 97, no. 2, pp. 386–403, Feb. 2009.
[23] S. Canovic, “Application of UWB technology for positioning, a feasibility sudy,” M.S. thesis, Dept. Electron. Telecommun., Norwegian Univ.
Sci. Technol., Trondheim, Norway, 2007.
[24] M. A. Stelios, A. D. Nick, M. T. Effie, K. M. Dimitris, and
S. C. A. Thomopoulos, “An indoor localization platform for ambient
assisted living using UWB,” in Proc. 6th Int. Conf. Adv. Mobile Comput.
Multimedia (MoMM), 2008, pp. 178–182.
[25] D.-H. Kim, D.-W. Lim, L. Shen, H.-M. Kim, S.-C. Woo, and H. K. Yu,
“Localization methods of multi-targets for UWB radar sensor networks,”
in Proc. 3rd Int. Asia-Pacific Conf. Synth. Aperture Radar (APSAR),
Sep. 2011, pp. 1–4.
[26] Y. K. Cho and J.-H. Youn, “Wireless sensor-driven intelligent navigation
robots for indoor construction site security and safety,” in Proc. 23rd Int.
Symp. Autom. Robot. Construct. (ISARC), 2006, pp. 493–498.
[27] K. C. Cheok, M. Radovnikovich, P. Vempaty, G. R. Hudas,
J. L. Overholt, and P. Fleck, “UWB tracking of mobile robots,” in Proc.
IEEE 21st Int. Symp. Pers. Indoor Mobile Radio Commun. (PIMRC),
Sep. 2010, pp. 2615–2620.
[28] B. Gulmezoglu, M. B. Guldogan, and S. Gezici, “Multiperson tracking
with a network of ultrawideband radar sensors based on Gaussian
mixture PHD filters,” IEEE Sensors J., vol. 15, no. 4, pp. 2227–2237,
Apr. 2015.
[29] A. Aryan, “Evaluation of ultra-wideband sensing technology for position
location in indoor construction environments,” M.S. thesis, Dept. Civil
Eng., Univ. Waterloo, Waterloo, ON, Canada, 2011.
[30] E. Nasr, T. Shehab, and A. Vlad, “Tracking systems in construction:
Applications and comparisons,” in Proc. 49th ASC Annu. Int. Conf.,
2013, pp. 1–9.
[31] M. E. Bowen, J. Craighead, C. A. Wingrave, and W. D. Kearns,
“Real-time locating systems (RTLS) to improve fall detection,”
Gerontechnology, vol. 9, no. 4, pp. 464–471, 2010.
[32] H. Ravikrishnan, “Ultra-wideband position tracking on an assembly
line,” M.S. thesis, Dept. Elect. Eng., Clemson Univ., Clemson, SC, USA,
2014.
[33] L. Mucchi, F. Trippi, and A. Carpini, “Ultra Wide band real-time location
system for cinematic survey in sports,” in Proc. 3rd Int. Symp. Appl. Sci.
Biomed. Commun. Technol. (ISABEL), Nov. 2010, pp. 1–6.
[34] T. H. Riehle, P. Lichter, and N. A. Giudice, “An indoor navigation system
to support the visually impaired,” in Proc. 30th Annu. IEEE Eng. Med.
Biol. Conf., Aug. 2008, pp. 4435–4438.
[35] R. Piltaver, B. Cvetković, and B. Kaluža, “Denoising human-motion
trajectories captured with Ultra-Wideband real-time location system,”
Informatica, vol. 39, no. 3, pp. 311–322, 2015.
[36] R. Mahler, “An introduction to multisource-multitarget statistics and
its applications,” Lockheed Martin, Bethesda, MD, USA, Tech.
Rep., 2000.
[37] B. Ristic, B.-T. Vo, B.-N. Vo, and A. Farina, “A tutorial on Bernoulli
filters: Theory, implementation and applications,” IEEE Trans. Signal
Process., vol. 61, no. 13, pp. 3406–3430, Jul. 2013.
[38] B. N. Vo, B.-T. Vo, N. T. Pham, and D. Suter, “Bayesian multi-object
estimation from image observations,” in Proc. 12th Int. Conf. Inf. Fusion,
Jul. 2009, pp. 890–898.
[39] M. B. Guldogan, “Consensus Bernoulli filter for distributed detection
and tracking using multi-static Doppler shifts,” IEEE Signal Process.
Lett., vol. 21, no. 6, pp. 671–675, Jun. 2014.
[40] A. Gunes and M. B. Guldogan, “Joint underwater target detection and
tracking with the Bernoulli filter using an acoustic vector sensor,” Digit.
Signal Process., vol. 48, pp. 246–258, Jan. 2016.
ERYILDIRIM AND GULDOGAN: BERNOULLI FILTER FOR EXTENDED TARGET TRACKING USING RANDOM MATRICES
[41] A. Gunes and M. B. Guldogan, “Multi-target bearing tracking with a
single acoustic vector sensor based on multi-Bernoulli filter,” in Proc.
OCEANS, Genova, Italy, May 2015, pp. 1–5.
[42] B.-T. Vo, B.-N. Vo, and A. Cantoni, “The cardinality balanced multitarget multi-Bernoulli filter and its implementations,” IEEE Trans. Signal
Process., vol. 57, no. 2, pp. 409–423, Feb. 2009.
[43] K. Granstrom and U. Orguner, “A PHD filter for tracking multiple
extended targets using random matrices,” IEEE Trans. Signal Process.,
vol. 60, no. 11, pp. 5657–5671, Nov. 2012.
[44] R. Mahler, “PHD filters for nonstandard targets, II: Unresolved targets,”
in Proc. 12th Int. Conf. Inf. Fusion, Jul. 2009, pp. 922–929.
[45] C. Lundquist, K. Granström, and U. Orguner, “An extended target CPHD
filter and a gamma Gaussian inverse Wishart implementation,” IEEE
J. Sel. Topics Signal Process., vol. 7, no. 3, pp. 472–483, Jun. 2013.
[46] M. Beard, S. Reuter, K. Granström, B.-T. Vo, B.-N. Vo, and A. Scheel,
“Multiple extended target tracking with labeled random finite sets,” IEEE
Trans. Signal Process., vol. 64, no. 7, pp. 1638–1653, Apr. 2016.
[47] S. Jovanoska and R. Thoma, “Multiple target tracking by a distributed
UWB sensor network based on the phd filter,” in Proc. 15th Int. Conf.
Inf. Fusion (FUSION), Jul. 2012, pp. 1095–1102.
[48] B. Sobhani, E. Paolini, A. Giorgetti, M. Mazzotti, and M. Chiani, “Target
tracking for UWB multistatic radar sensor networks,” IEEE J. Sel. Topics
Signal Process., vol. 8, no. 1, pp. 125–136, Feb. 2014.
[49] X. W. Zhao, A. Gaugue, C. Lièbe, J. Khamlichi, and M. Mé́nard,
“Through the wall detection and localization of a moving target with
a bistatic UWB radar system,” in Proc. Eur. Radar Conf. (EuRAD),
Sep./Oct. 2010, pp. 204–207.
[50] R. P. S. Mahler, Statistical Multisource-Multitarget Information Fusion.
Norwood, MA, USA: Artech House, 2007.
[51] A. K. Gupta and D. K. Nagar, Matrix Variate Distributions. London,
U.K.: CRC Press, 1999.
[52] M. B. Guldogan, D. Lindgren, F. Gustafsson, H. Habberstad, and
U. Orguner, “Multi-target tracking with PHD filter using Doppler-only
measurements,” Digit. Signal Process., vol. 27, pp. 1–11, Apr. 2014.
[53] D. Schuhmacher, B. T. Vo, and B. N. Vo, “A consistent metric for
performance evaluation of multi-object filters,” IEEE Trans. Signal
Process., vol. 56, no. 8, pp. 3447–3457, Aug. 2008.
[54] Ubisense Real-Time Location System (RTLS), accessed on Dec. 12, 2015.
[Online]. Available: http://ubisense.net/en/products/rtls-platform
[55] A. D. Angelis, P. Handel, and J. Rantakokko, “Measurement report.
laser total station campaign in KTH R1 for Ubisense system accuracy
evaluation,” KTH Royal Inst. Technol., Stockholm, Sweden, Tech. Rep.
QC 20120618, 2012.
4373
[56] B. Perrat, M. J. Smith, B. S. Mason, J. M. Rhodes, and
V. L. Goosey-Tolfrey, “Quality assessment of an ultra-wide band positioning system for indoor wheelchair court sports,” in Proc. Inst. Mech.
Eng., Jun. 2015, pp. 81–91.
[57] P. H. Abreu, J. Xavier, D. C. Silva, L. P. Reis, and M. Petry, “Using
Kalman filters to reduce noise from RFID location system,” Sci. World J.,
vol. 2014, Jan. 2014, Art. no. 796279.
[58] M. Feldmann, D. Fränken, and W. Koch, “Tracking of extended objects
and group targets using random matrices,” IEEE Trans. Signal Process.,
vol. 59, no. 4, pp. 1409–1420, Apr. 2011.
Abdulkadir Eryildirim was born in Turkey in 1984.
He received the B.S. degree in electrical and electronics engineering from Middle East Technical University, Turkey, in 2007, the M.S. degree in electrical
and electronics engineering from İ.D. Bilkent University, Turkey, in 2009, the master’s degree in space
communication systems from ISAE (SUPAÉRO),
France, in 2012. He is currently pursuing the
Ph.D. degree with Turgut Özal University, Turkey.
His research interests are signal processing, detection and estimation, UWB systems, radar systems,
localization, and target tracking.
Mehmet Burak Guldogan received the B.S., M.S.,
and Ph.D. degrees from Bilkent University, Turkey,
in 2003, 2006, and 2010, respectively, all in electrical and electronics engineering. From 2010 and
2012, he was a Post-Doctoral Fellow with the Automatic Control Group, Linkoping University, Sweden.
He joined Turgut Özal University, Ankara, Turkey, in
2012, where he is currently an Associate Professor
with the Department of Electrical and Electronics
Engineering. His current research interests are in statistical signal processing, sensor fusion, estimation
theory, and target tracking. He has been an Associate Editor of Digital Signal
Processing (Elsevier) since 2012.