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local posterior density is introduced in the following way. By
p(x apply the proposed
exploiting k|k−1 For the number sake
x
processing given that l|k = P
Kalman l|k
p(x apply the proposed scheme directly to an arbitrary number
x l|k = P l|k P s −1
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l|k fil
In 2008 and 2009, first T2TF the product formula for Gaussians, we replace all
of of simplicity, sensors. schemeInfor we 2010, arbitrary
here the assume generalized conditions solution where was . measurement
x k; F
derived standard
(3)
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of false measurements. For notational simplicity let us assume
all sensor covariances are known.
instants of time, which is equivalent local covariances by a one:
Kalman presented to a
filtering in Kalman [2]. filter
is applicable, i.e. in the
P
case −1
l|k
S synchronized sensors produce measurements the same
= S and
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of wellseparated
Notational center, posterior combinations
P s −1
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introduce this type
Kalman presentedfiltering in [2]. is applicable, processing i.e. inallthe measurements case of wellseparated
Notational targets, Preliminaries:
assuming perfect Let
in a fusion local Convex
was density
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The proposed methodology can be directly extended to asynchronous
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all sensors
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The proposed methodology can beNotational directly extended Preliminaries: to asynchronous
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(4)
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properties of the of
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. The statistical matrices P
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time of simplicity, series produced we here by the assume measurements conditions
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produced ˜x tracks
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likelihood function, which needs to be known up to a constant ˜P
local covariances by a globalized
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we replace all
l is described
s=1
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l |xl), also called x k; Fsensor
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S synchronized sensors produce measurements at the same
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likelihood function, which needs ofsection to simplicity, be known states the we upproblem to here a constant assume addressed conditions this paper. In particular,
x
lk
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we
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introduce
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be directly extended to asynchronous
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, Ps k|kdensity is introduced (4) in the following way. By
Structure: This paper is k|k P
exploiting the product formula for Gaussians, we replace all
of T2TF. Based on the results of the cited preliminary paper,
of false measurements. For notational l up
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the approach of a globalized [6], it is likelihood not required
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function is derived to update
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(7)
section states the problem addressed in this paper. In particular,
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−1
we introduce the product representation for the fused posterior
the key defined k; ˜x
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in
local
[2] Z k s for = {Z
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instants of time t s anymore. This two-stage prediction s=1
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IV. We close the with a conclusion given in section V. (globalization
the cited preliminary of an individual
and application
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can be was
asensor directly
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.
reveal
section The statistical
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II. FORMULATION OF impact properties of the sensor
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We obtain discussed z s s=1
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of
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s anymore.
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likelihood function, which needs to be known up to a constant
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In this paper,
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= ˜P k|k P s −
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properties of an individual sites. To sensor this end, measurement section we states achieved z s k|k , ˜P
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II. FORMULATION OF THE PROBLEM
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Structure: This paper is organized as follows. The next
lthe is problem described
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l
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impact in this paper. on practical In particular, implementations
Sensor
S DISTRIBUTED KALMA
discussed in section S
the local −1 sensor index s anymore
impact on practical implementations S is discussed in section the local sensor DISTRIBUTED index s KALMAN anymore. This PROCESSING
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density which was the key element in [2] for an exact solution
k|k model) = S wasP s −1
l|k N x l; x s l|k , Ps l|k (1) First of all, we introduce a new notation. The globalized
s=1
local necessary k|k . (8)
necessary to reveal a general scheme for decorrelated tracks: posterior to reveal for thea state general x k
sche
s=1
local posterior for the state x s=1
at s
II.
of T2TF. Based on the results of the cited preliminary II. FORMULATION paper, k at sensor s will be denoted by
FORMULATION OF THE PROBLEM
OF THE PROBLEM
We obtain updates of local track We obtain updates of local track
a globalized likelihood function is derived in section III. Its Note that Sensor estimates using the global
Sensor
the globalized covariance covariance instead of the local one
In this paper, we address the problem of optimal T2TF
˜P k|k does not depend on
covariance instead of the local one. In other words, we engage
In this paper, we addressimpact the problem on practical of optimal implementations T2TF is discussed in section the local sensor index s anymore. aThis modified two-stage likelihood prediction function i
at arbitrary instants of time. at a arbitrary modified instants likelihood of function time. Asindiscussed order toin keep [2], the thistracks
can
IV. As Wediscussed close thein paper [2], with this acan
conclusion given in section V. (globalization and application of the decorrelated. evolution In model) this paper, was we d
be achieved, if all measurement error covariances are known be decorrelated. achieved, ifInallthis measurement paper, we
necessary
error derive covariances a closed formula
to reveal a
are
general
known for
scheme thisfor likelihood decorrelated function. tracks:
at the sensor sites. To this end, we II. achieved at this thelikelihood sensor sites. function. To this end, we achieved a product
FORMULATION a product OF THE PROBLEM
We obtain updates of local track estimates using the global
representation of the posterior density of the state x representation Fusion of the posterior
covariance
density of
instead
the state
of the
x l,l
local
≤ k:
one. In other III. words, GLOBALIZED we engageLIKELIH
In this paper, we address l,l ≤ k:
the problem
III. GLOBALIZED LIKELIHOOD FUNCTION FOR
of optimal T2TF
Center
S a modified DISTRIBUTED KALMA
at S DISTRIBUTED KALMAN
arbitrary instants of time. As discussed in [2],
PROCESSING likelihood function in order to keep the tracks
p(x l|Z k this can
p(x )=c l|k decorrelated. N x l; x s l|k In , Ps this l|k paper, we (1) derive
be achieved, if all measurement error covariances are known
First a closed of all, formula we introduce for
l|Z k )=c a ne
l|k N x l; x s l|k , Ps l|k (1) First of all, we introduce a new notation. The globalized
s=1 this likelihood function.
s=1 at the sensor sites. To this end, local we achieved posterior for a product the state x local posterior for the state x k at
k at sensor s will be denoted by
representation Sensor of the posterior density of the state x l,l ≤ k: III. Sensor GLOBALIZED LIKELIHOOD FUNCTION FOR
S DISTRIBUTED KALMAN PROCESSING
p(x l|Z k )=c l|k N x l; x s l|k , Ps l|k (1) First of all, we introduce a new notation. The globalized
s=1
local posterior for the state x k at sensor s will be denoted by
Sensor
Figure 3. Schematic illustration of a distributed Kalman filter. The sensor nodes process the data
to local auxiliary tracking parameters. When communication is successful, these can be fused
at the fusion center in order to obtain the estimated track. When applying exact Track-to-Track fusion,
the result is equivalent to a central Kalman filter receiving all sensor data
In the following the most common and state-of-the-art schemes for target tracking
using multiple sensors are listed. The performance of all of them will be compared
in the next section. These variable approaches can roughly be divided in the categories
Measurement-to-Track Fusion (M2TF) and Track-to-Track Fusion (T2TF).
• Central Kalman Filter (CKF): A Kalman filter at the fusion center is processing
the measurements of all sensors. This scheme results in an optimal
solution with respect to the mean squared error metric.
• Single Kalman Filter (SKF): Each sensor node in the scenario performs M2TF
using its local data. The tracking algorithm is a Kalman filter processing
the linearised measurements. At each time step, the node sends its current
estimate and estimation error covariance to the fusion center, which in turn