Gaussian Mixture initialization in passive tracking applications

Abstract

This paper describes the approximation of a nonlinear posterior density by a Gaussian Mixture (GM). The GM is used to initialize a bank of Kalman filters. For each Gaussian term, a Kalman filter is started. The basic conditions and the quality of the approximation are discussed. Examples from different tracking applications, the multistatic tracking and passive emitter localization using TDOA measurements, are investigated. The results are discussed and compared with existing approaches. The RMS error of the estimate is used as an evaluation criterion. The performance of the Gaussian Mixture approach is analyzed in Monte Carlo simulations.

Full text

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Gaussian Mixture Initialization in Passive Tracking
Applications
Martina Daun
Dept. Sensor Data and Information Fusion
Fraunhofer FKIE
Wachtberg, Germany
martina.daun@fkie.fraunhofer.de
Regina Kaune
Dept. Sensor Data and Information Fusion
Fraunhofer FKIE
Wachtberg, Germany
regina.kaune@fkie.fraunhofer.de
Abstract – This paper describes the approximation of
a nonlinear posterior density by a Gaussian Mixture
(GM). The GM is used to initialize a bank of Kalman
filters.For each Gaussian term, a Kalman filter is
started.The basic conditions and the quality of the
approximation are discussed. Examples from different
tracking applications, the multistatic tracking and pas-
sive emitter localization using TDOA measurements,
are investigated. The results are discussed and com-
pared with existing approaches. The RMS error of the
estimate is used as an evaluation criterion. The per-
formance of the Gaussian Mixture approach is analyzed
in Monte Carlo simulations.
Keywords: Gaussian Mixture, tracking, multistatic,
TDOA, Kalman filter.
1
This paper investigates two different types of passive
tracking applications.First, in Multistatic Passive
Radar the signal of illuminators of opportunity (like
terrestrial television or radio antennas) are exploited to
measure the reflected signal of an air target at a sepa-
rate receiver, see for example [1]. Secondly, in passive
emitter localization the target itself is emitting a signal
which is obtained by a pair of sensors. In both appli-
cations we measure the time when the signal arrives at
the receiver(s).
In passive radar applications, we are often faced with
incomplete measurement information, which means
that we are not able to estimate our full target state
from measurements of a single source and receiver.
Even though the Kalman Filter algorithm has been
shown to provide a quite a robust solution for fusing
measurements of different time steps in many applica-
tions, the quality of the resulting estimate is strongly
dependent on a good initialization [2].
Difficulties appear if the approximation of the ini-
tial Cartesian covariance matrix by a single Gaussian
fails.A similar effect has also been noted for other
Introduction
tracking scenarios (see for example [3]) if the sensor
provides only poor azimuth measurements. With in-
creasing azimuth inaccuracy and increasing distance to
the sensor, the Cartesian uncertainty region develops
more and more to a crescent shape. If we approximate
this by a single ellipse (approximation by single Gaus-
sian), the ellipse must either be very large or ignores
possible target states.
This paper discusses the initialization problems of
the two introduced tracking applications and analyzes
the use of a GM approach.
results analyzing the Cartesian estimation performance
in terms of the root mean squared error and comparison
to the Cram´ er Rao Lower Bound (CRLB).
The paper is organized as follows: after this intro-
duction, we will describe the passive tracking scenarios
in section 2. Section 3 provides some fundamentals on
GM. Applications to passive tracking and results are
given in 4. Section 5 concludes this paper.
We will give numerical
2Scenario Description
In this section the two types passive radar are de-
scribed and the application of GM approximation is
motivated.
2.1
Without loss of generality, the observer will be posi-
tioned at the origin. The position of the ith stationary
illuminator is given in Cartesian coordinates by xs,i.
Time Difference of Arrival (TDoA) and the bistatic
Doppler shift are measured, which are directly related
to the bistatic range riand bistatic range-rate ˙ ri, i.e.
Multistatic Passive Radar
TDoA =ri− ||xs,i||
c
=||p|| + ||p − xs,i|| − ||xs,i||
?
c
,
Doppler = −˙ ri
λ= −1
λ
p
||p||+
p − xs,i
||p − xs,i||
?T
· v,
(1)
where c is the speed of light, λ the wavelength of the
RF signal and p and v denote the target’s position

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and velocity in Cartesian coordinates.
range equation describes ellipsoids in three-dimensional
Cartesian space with foci at the observer and illumina-
tor position. The bistatic Doppler depends on the tar-
get velocity component normal to the surface of the
ellipsoid.Considering the target state x = (p,v)T
and the measurement vector zi = (ri, ˙ ri)T, the mea-
surement equation is in the following abbreviated by
zi= h(x,xs,i).
For reason of association complexity [1], we estimate
the target position and velocity from a minimum num-
ber of two synchronous measurements using two dif-
ferent illuminators (xs,1 and xs,2). In a simplified 2D
consideration, this would be intersecting two ellipses
with foci at the origin and xs,i. This basically means
solving two quadratic equations successively, which can
result in up to four solutions; but since in this case the
two ellipses share one focal point, there will be only two
solutions. Thus, in the 2D scenario and, even more gen-
erally, if the target height is exactly known, the target
position can be derived analytically.
Considering airborne targets the assumption of 2D
problem is generally not fulfilled. The unknown target
height has influence on the position estimate, which
was discussed in more detail in [4]. The calculation
of CRLB [4] characterizes the impact of the unknown
height on the position estimate. In figure 1 the accuracy
of the position estimate as indicated by the CRLB is
plotted for each point in a 80 km×80 km region and a
fixed constellation of two illuminators and one receiver.
Already for the optimistic case of known target height
(figure 1(a)), we notice that there are regions of poor
estimation performance that enlarge with decreasing a-
priori knowledge on target height (see figure 1(b) for a
standard deviation (std) of 2km in target height).
In 3D, intersecting two ellipsoids yields as a solution
an ellipse in 3D. Assuming some directional information
and neglecting heights below zero we can constrain the
locations of potential targets (uncertainty region) to an
ellipse sector that is plotted as an example in figure 2
for three different targets locations in y-z view. If we
additionally constrain the target height to values be-
low, for example, 10 km, we can draw conclusions from
the curvature of the ellipse sector on the magnitude of
position estimation error. If the target is in the far field
of the sensors1, the influence of the height on the es-
timate is insignificant. We deduce that, for targets in
the far field of the illuminators and receiver, the bistatic
range equations holds no information about the target
height. For near field considerations it is quite obvi-
ous that approximating by a single Gaussian fails; thus
we propose a GM approximation for more robust filter
initialization.
The bistatic
1where the range is measured orthogonal to the line between
the illuminators, which corresponds to the y-component of the
range in the scenario considered here
−40−200 2040
−40
−20
0
20
40


0
500
1000
1500
2000
2500
3000
3500
x - range in km
y - range in km
(a) CRLB (POS), z known
−40−20 02040
−40
−20
0
20
40


0
500
1000
1500
2000
2500
3000
3500
x - range in km
y - range in km
(b) CRLB (POS), z ∼ N(z;5km,(2km)2)
Figure 1: CRLB by exploiting bistatic range and range-
rate information generated by two illuminators (trian-
gles) and one receiver (circle) at one single time scan.
Units of the colorbar are in m.
2.2
We consider the situation of tracking a non-cooperative
emitter using TDOA measurements from a pair of
sensors. The emitter is assumed to be stationary, while
sensors si, i = 1,2 are moving and collecting TDOA
measurements over time.
the emitter and sensors are located in the same
plane. Therefore a two-dimensional TDOA scenario is
studied. One noise free TDOA measurement localizes
the emitter on a branch of a hyperbola with the two
sensors as foci. The measurement equation
Passive emitter localization
For reasons of simplicity,
h(x) =r1− r2
c
.
(2)
is nonlinear, where riis the distance of the emitter to
sensor i,i = 1,2. The TDoA measurement equation
depends only on the emitter position, denoted by x,
the sensor positions enter as parameters. The measure-

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05 10
y - range in km
1520 25 30
0
5
10
15


target 1
target 2
target 3
z - range in km
Figure 2: Impact of height uncertainty on Position esti-
mate: estimation ambiguity for three potential targets
is plotted in y-z view. The uncertainty can be described
by a curve that is bent according to the distance of tar-
get, illuminators and receiver. For targets in the near
field, the curve is more bent, which means that the inac-
curacy in height has a significant impact on the position
estimate.
ment function is modeled by adding white Gaussian
noise to the measurement equation: zt= h(x) +ν. We
want to model the likelihood function after the first
measurement p(z1|x) and the posterior density p(x|z1).
Figure 3 shows the posterior density after the first mea-
surement. The approximation with a single Gaussian
Figure 3: Posterior density after one TDOA measure-
ment
fails. Thus the localization with a single Kalman Fil-
ter, EKF or UKF, depends strongly on the initializa-
tion and the KF diverges in adverse scenarios, see [5].
Therefore we approximate the density with a GM. For
each summand of the GM an EKF is started. Results
of a static GM filter are shown in [5]. In the present pa-
per, we introduce an hypothetic azimuth measurement
at a certain range, and obtain a two-dimensional mea-
surement space which we transform into the Cartesian
state to find the parameters of the GM.
The need for a GM approximation in the two pas-
sive tracking applications is cause of the incomplete
measurement information. To determine a Cartesian
estimate, prior information about target height or an
azimuth can be exploited.
3Some Fundamentals on GM
approximations
In this section we shortly repeat some fundamentals on
GMs and derive formulas for passive tracking applica-
tions. The GM approximation lemma [3] states that
every probability density p can be approximated (as
closely as desired) by a GM:
p(x) ≈
n
?
i=1
αiN(x;mi,Bi),
(3)
where n is sufficiently large and the positive scalars αi,
mean vectors mi and positive-definite covariance ma-
trices are chosen properly.
In target tracking we need to determine the likelihood
p(z|x), from that we can derive the posterior density
according to Bayes rule:
p(x|z) =
p(z|x) · p(x)
?p(z|x) · p(x)dx
(4)
Since for track initialization a priori knowledge in x is
not available, p(x) is constant, which means that every
possible state of x is equally likely, i.e.
p(x|z) =
p(z|x)
?p(z|x)dx.
(5)
Thus, approximating p(x|z) can be transferred to ap-
proximating p(z|x) by a GM. Since this is not a prob-
ability density of x, the Gaussian sum takes the form:
p(z|x) ≈
m
?
i=1
αiN(x;mi,Bi),
(6)
where?m
3.1
i=1αi=?p(z|x)dx ?= 1.
How to find the parameters of the
GM?
Approximating a given probability density by a GM
(Eq. 3) means finding the best fit of parameters ac-
cording to a pre-defined number of summands; thus a
multi-dimensional search needs to be performed. The
Expectation Maximization (EM) algorithm is quite a
popular solution to this problem, see for example [6],
due to its simple implementation and its proven con-
vergence to local maxima. However the EM requires
a good initialization and its computational complexity

Page 4

depends on the dimension of x (Eq. 3). An optimal
search is therefore often not applicable for real-time
track initialization.
In [3] an approach is presented that requires for the
known functional relationship of the measurement and
target state, i.e. z = h(x) + w, with some noise vector
w, only an approximation of the usually known pdf of
w, i.e. pw(z−h(x)). The desired Gaussian sum approx-
imation in Cartesian coordinates can then be deduced
from this by transformation. This provides two crucial
advantages: First, finding a Gaussian sum approxima-
tion of pw is generally simpler than of the Cartesian
pdf. Second, assuming identical pdfs of different mea-
surements the parameter fitting needs to be done only
once to derive different types of Cartesian sums.
In our applications the problem is reduced to fitting a
uniform density. Following the description in [7] we use
the following steps to approximate a uniform density in
the interval [a,b]:
• choose n mean values μi to be equally spaced in
the considered interval, i.e. mi= a + (2i − 1)(b −
a)/(2n)
• set the weighting factors αi= 1/n
• select the standard deviation σμi= σμ such that
the L1distance is minimized, i.e.
?????
Thus, only a one-dimensional search is required.
σμ= argmin
σ
?∞
−∞
1
b − a−
n
?
i=1
αiN(x;μi,σ2)
?????dx
(7)
A given GM approximation of pw, results in the follow-
ing approximation of p(z|x):
p(z|x) = pw(z − h(x))
N
?
N
?
To approximate equation (6) by means of a GM de-
pendent on the Cartesian state vector x we follow the
derivation in [3]:
≈
i=1
˜ αiN(z − h(x); ˜ mi,˜Bi)
=
i=1
˜ αiN(h(x); ¯ mi,˜Bi)
(8)
N(h(x); ¯ mi,˜Bi)
≈
1
|2π˜Bi|1/2e−(1/2)(x−mi)TB−1
where mi = h−1( ¯ mi), Bi = [Jh(mi)T˜B−1
and Jh(mi) is the Jacobi matrix with respect to h. Cal-
culation holds only if h is invertible, which also implies
that dim(x) = dim(z).
1
|2π˜Bi|1/2e−(1/2)( ¯ mi−h(x))T˜B−1
i
( ¯ mi−h(x))
=
i
(x−mi),
(9)
iJh(mi)]−1
Thus, the Gaussian pdf can be approximated by a
pdf on x and an adequate weight factor:
N(h(x); ¯ mi,˜Bi)
≈|2πBi|1/2
= |Jh(mi)|N(x;mi,Bi),
|2π˜Bi|1/2N(x;mi,Bi)
(10)
The inverse function theorem states that
Jh(mi) = Jh(h−1( ¯ mi)) = (Jh−1( ¯ mi))−1
(11)
Thus:
Bi= [Jh(mi)T˜B−1
= [Jh−1( ¯ mi)˜BiJh−1( ¯ mi)T]
iJh(mi)]−1
(12)
In summary:
The desired GM approximation in Cartesian coordi-
nates is given by
p(z|x) =
N
?
=
i=1
αiN(x;mi,Bi),
(13)
with
means mi
[Jh−1( ¯ mi)˜BiJh−1( ¯ mi)T].
covariance of the transformed density match with the
results of transforming N(z; ¯ mi,˜Bi) by linearization
according to h−1.
weights
αi
h−1( ¯ mi) and covariances Bi
Note that the mean and
˜ αi|(Jh−1( ¯ mi))−1|, the
==
4Application of the GM for
Track Initialization
This section provides GM approximation results.
4.1 Multistatic Passive Radar: GM Ap-
proach for Unknown Target Heights
We model the target height to be uniformly distributed
in a suitable large interval [a,b], for example [0 km,10
km], and approximate this probability density by a GM.
Following the steps for approximating a uniform density
described above we calculate the Gaussian summands
for n = 6,10,20 and 49, results are shown in figure 4(a).
Obviously, the GM approximation converges to the true
density for increasing N. The approximation is worst
in vicinity of the boundaries of the interval; this seems
to be admissible in our application and could be coun-
teracted by enlarging the interval.
The std (chosen by the one-dimensional search pro-
cedure) corresponding to N
799,522,282 and 124 respectively. Choosing 10 mix-
tures seem to give an appropriate approximation of the
true density and correspond to a height error about
500 m of each component of the sum. Utilizing the GM
∈{6,10,20,49} are

Page 5

−5 05 10 15
0
0.2
0.4
0.6
0.8
1
1.2
1.4
target height [km]
pdf [1e−4]
Figure 4: GM approximations of uniform target height
for different number of mixtures N ∈ {6,10,20,49}
approximation of the target height results in (compare
[7]):
pw(z(a)|x) ≈
N
?
N
?
(r1,r2, ˙ r1, ˙ r2,ht,˙ht)T
i=1
αiN(h(x);z(a)
i ,P(a)
i)
≈
i=1
βiN(x;y(a)
i
,Q(a)
i)
(14)
where z(a)
diag(σr,σr,σ˙ r,σ˙ r,σht,σ˙ht), βi= αi·?det(Qi) and yi
surements into Cartesians (either by Unscented Trans-
form (UT) or Linearization). It follows that:
=and P(a)
i
=
and Qiare the results of the transformation from mea-
p(x|z) ≈
N
?
i=1
˜βiN(x;y(a)
i,Q(a)
i),
(15)
where˜βi= βi/(?N
in figure 5. Two sources are placed at (−10 km,√3·10
km,0 km)Tand (10 km,√3·10 km,0 km)T; the target is
located at (0 km,20 km,1 km) and the receiver is placed
in the origin. As seen in the picture, the Cartesian co-
variances increase with increasing height; thus also the
probabilities of the respective Gaussians increase.
i=1βi). Results for 10 mixture com-
ponents and a specific multistatic geometry are shown
Simulation Results
In simulation we consider a 80 km×80 km region,
see figure 6, and for each point in a grid we run
100 Monte-Carlo Runs and simulate measurements in
bistatic range (std σr = 100 m) and bistatic range-
rate (std σ˙ r = 1m/s) for two illuminators (shown by
triangles) and a single receiver (circle).
height is sampled from a uniform distribution in [0
km, 10 km], and velocity from a Gaussian density with
˙ x, ˙ y ∼ N(0,100m/s2); velocity in height is assumed to
The target
−0.5−0.4−0.3−0.2−0.100.10.2
−2
0
2
4
6
8
10
12
target height in km
x - range in km
(a) Approximated Cartesian density (x-z view)
810121416182022
−2
0
2
4
6
8
10
12
target height in km
y - range in km
(b) Approximated Cartesian density (y-z view)
Figure 5: GM approximations of 3D Cartesian position.
Note the different scaling in (a) and (b).
be zero. At the first time scan, we generate a Gaussian
sum with 10 components as explained above and start
a bank of UKFs from these initialization points. The
Root Mean Squared Error (RMSE) is calculated for the
“best” result (according to the track score) after a fil-
tering period of 60 seconds. Results for position, and
height estimates are provided in figure 6(a), and fig-
ure 7(a). The Gaussian sum approach is compared to
the initialization by a single Gaussian, see figure 6(b),
and figure 7(b).The results show that with the GM
approximation, filter divergence due to bad initializa-
tion can be prevented in some regions. Initialization is
worse, even for the GM approach, for decreasing range,
measured orthogonal to the line of illuminator and re-