Multiple Sensor Multiple Object Tracking With GMPHD Filter - ISIF

ange tracking, and multiple speaker tracking are presented in
the section V.
II. PROBLEM FORMULATION
The multi-sensor multi-object tracking problem can be
modelled by random nite set (RFS) framework. Let X be
the single object state space then multiple object state at
time k is presented by X k = fx k;1 ; x k;2 :::; x k;Nk g 2 F(X ),
where F(X ) denotes the collection of all nite subsets of the
space X . For a multi-object state X k 1 at time k 1, each
x k 1 2 X k 1 can continue to exist at time k with probability
p S;k or die at time k with probability (1 p S;k ). Let S k (x k 1 )
denote the object that is the transition from x k 1 at time k on
condition that the object is survived and let B kjk 1 (x k 1 ) be
objects spawned at time k from an object with previous state
x k 1 . Let k be RFS of spontaneous births at time k and
can be determined by using the assumption of spontaneous
birth models. Given a multi-object state X k 1 at time k 1,
the multi-object state X k at time k is given by union of the
surviving objects and new objects,
X k =
h[ h[ i
Sk (x k 1 )i
[ Bkjk 1 (x k 1 )
[ [ k] (1)
The RFS X k encapsulates all aspects of multi-object tracking
problem, such as time varying number of objects, object
motion.
Similarly, let Z i be the measurement space of single object
at sensor i then measurements collected at sensor i time k is
Zk i 2 F(Zi ). A given object state x k 2 X k is either detected
with probability p D or missed with probability (1 p D ).
Conditional on detection, the measurement from x k at sensor
i is dened by the RFS i k (x k). The sensor i also can receive
a set of clutters Ck i . So, given a multi-object state X k at time
k, the measurement set from sensor i at time k is formed by
the union of object generated measurements and clutters,
" #
[
Zk i = i k(x k ) [ Ck i (2)
x k 2X k
Assuming that we have Q sensors, the RFS of measurements
at time k is modelled by
h
i
Z k = Zk; 1 Zk; 2 : : : ; Z Q k
(3)
The RFS Z k encapsulates all sensor characteristics such as
measurement noise, sensor eld of view, clutter.
The multi-sensor multi-object tracking can be posed as
follows: given set of measurement Z 1:k collected from sensors
up to time k, the problem is to nd ^X k is expectation or
maximization of the posterior density function p(X k jZ 1:k ).
III. PROBABILITY HYPOTHESIS DENSITY APPROACH
In multiple object tracking problem, we usually need to
obtain the posterior density p(X k jZ 1:k ). When the number
of object increases, the multiple object state space become
large. Hence, it is difcult to obtain the posterior density function.
Fortunately, this density function can be approximately
recovered from the probability hypothesis density (PHD) [9].
The PHD is dened as follows. For a random nite set X on
X with probability distribution P , the PHD is the density v(x)
such that for each region S X , the integral of v over region
S gives the expected number of elements of X that are in S,
Z
Z
j X \ S j P (dX) = v(x)dx; (4)
Thus, instead of estimating states of objects from posterior
density, we can estimate them by investigating peaks of PHD.
It helps to reduce from searching in multiple object state space
to single object state space.
IV. GAUSSIAN MIXTURE PROBABILITY HYPOTHESIS
DENSITY FILTER IN MULTI-SENSOR MULTI-OBJECT
A. Assumptions
TRACKING
First, there are some assumptions. The transition function
of each object follows a linear Gaussian model, i.e.,
f kjk 1 (xj) = N(x; F k 1 ; Q k 1 ) (5)
where N(:; m; P ) denotes a Gaussian density with mean m
and covariance P , F k 1 is the state transition matrix, Q k 1
is the process noise covariance. There are Q sensors, the
likelihood function at each sensor is also a linear Gaussian
model, i.e.,
g i k(zjx) = N(z; H i kx; R i k) (6)
where H i k is the observation matrix of the sensor i, and Ri k is
the observation noise covariance of the sensor i. The survival
and detection probabilities are
S
p S;k (x) = p S;k (7)
p D;k (x) = p D;k (8)
The intensity of the spontaneous birth RFS is
J ;k
X
k (x) =
i=1
w (i)
;kN(x; m(i)
;k ; P (i)
;k ) (9)
where J ;k is the number of birth Gaussian components at
time k, and w (i)
;k
is the weight for i-th Gaussian component.
The posterior intensity at time k 1 is a Gaussian mixture of
the form
v k 1 (x) =
J
X k 1
i=1
w (i)
k 1
N(x; m(i)
k 1 ; P (i)
k 1 ) (10)
where J k 1 is the number of Gaussian components of posterior
intensity at time k 1, and w (i)
k 1
is the weight for i-th
Gaussian component.
B. GMPHD lter with one sensor
Vo [12] proposed a closed form expression of the PHD
lter for linear Gaussian multi-object tracking, called the
Gaussian mixture probability hypothesis density lter. Under
assumptions in IV-A, the initial prior intensity is a Gaussian
mixture, the posterior intensity at any subsequent time step is