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 CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 59, 2017 

A publication of 

 

The Italian Association 
of Chemical Engineering 
Online at www.aidic.it/cet 

Guest Editors: Zhuo Yang, Junjie Ba, Jing Pan 
Copyright © 2017, AIDIC Servizi S.r.l. 
ISBN 978-88-95608- 49-5; ISSN 2283-9216 

Extended Target Tracking Algorithm Based on Random 

Hypersurface Model with Glint Noise 

Yawen Li 

Electronic information and electrical college of engineering, Shangluo University, Shangluo 726000, China 

YawenLi@126.com 

Due to the fact that the unmatching of extended targets’ measurements noise is also assumed as Gaussian 

distribution and conventional algorithm cannot estimate target’s extent under the circumstance of unknown 

measurement noise covariance, a new multiple extended target tracking algorithm based on variational 

bayesian random hypersurface model (VB-RHM) is proposed, which is embedded into CPHD filter frame. 

Measurement noise is modeled by glint noise with t-distribution, its parameters are assumed to have a 

Gamma prior distribution so that the predicted and updated PHDs can have mixture of Gaussians 

representations. A variational bayesian procedure is applied to iteratively estimate parameters of the mixture 

distributions through random hypersurface model CPHD prediction and update steps. The simulation results 

show that the proposed algorithm VB-RHM-CPHD can track multiple extended targets’ kinematic state and 

object extension under the condition of unknown numbers and measurement noise covariance adaptively. In 

addition, it has an improved precision compared with conventional RHM-GGM-CPHD algorithm. 

1. Introduction 

Conventional extended target algorithms are mainly based on that measurement noise model is known, and 

the noise is also assumed as Gaussian distribution (Gilholm and Salmond, 2005; Gilholm et al., 2005). But in 

many conditions, the parameters of noise, such as inverse covariance of measurement noise, are always 

unknown, particularly for radar tracking systems, changes in the target aspect toward the radar may cause 

irregular electromagnetic wave reflections and this gives rise to outliers or glint noise (Li et al., 2014). It was 

found that glint noise has a heavy-tailed probability density function and conventional filtering algorithms are 

known to show unsatisfactory performance in the presence of glint noise. So it is eagerly to introduce another 

measurement noise model. It is proved that when measurement noise model is matched with true noise, effect 

tracking result can be reached, while the accuracy of target tracking will drop dramatically or obtained a poor 

result once model unmatched. 

In order to solve the estimation problem of unknown measurement noise, a number of approximation 

algorithms are proposed to deal with errors. In Oussalah et al. 2000), weight least square method is used to 

identify noise. In Zhu (Zhu, 1999), recursive least square filtering method with forgetting factor is used to make 

up for the lack of statistical noise problem. Recently, variationally Bayesian (VB) method is applied to estimate 

joint probability density of the target state and unknown measurement noise covariance (Zhu et al., 2013). 

Because the performance of the conventional extended objects tracking degrades significantly under the 

condition of unknown measurement noise covariance, a new multiple extended object-tracking algorithm 

based on variationally Bayesian cardinality-balanced multi-object multi-Bernoulli was proposed in Li et al. (Li et 

al., 2015) without taking target extension into consideration. In that case, an extended object tracking 

algorithm based on Variationally Bayesian Random Hypersurface Model (VB-RHM) is proposed, which is in 

the situation of glint noise, and it can also estimate target extension (Li, 2016). 

Random Hypersurface Model (RHM) is a new object modeling method proposed by Baum in 2009 (Baum and 

Hanebeck, 2009), in the model, object’s measurements are produced by measurement sources and the noise 

from sensor device, while the modeling of measurement source stands for target extent, but when using this 

method to tracking object, Baum doesn’t take clutter and missing decection into consideration, which against 

application in practical engineering. In 2013, Zhang Hui and Han Yu Lan etc (Zhang et al., 2013; Han et al., 

                               
 
 

 

 
   

                                                  
DOI: 10.3303/CET1759115

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Yawen Li, 2017, Extended target tracking algorithm based on random hypersurface model with glint noise, 
Chemical Engineering Transactions, 59, 685-690  DOI:10.3303/CET1759115   

685



2013) propose a novel filter called RHM-PHD which combined RHM with ET-PHD. On this basis, we present 

an algorithm called VB-RHM-CPHD that embedded RHM into CPHD filter, it’s worth noting that the algorithm 

can also estimate object extension besides kinematic state, also, it is under the assumption of a glint 

measurement noise model with unknown inverse covariance. 

2. Extended object tracking modeling 

2.1 Extended target tracking model  

We denote the dynamics (motion equation) of the kth time step as: 

1 1 1k k k k  
 x F x w  (1) 

where 𝑥𝑘 = [𝑚𝑥,𝑘 , 𝑚𝑦,𝑘 , 𝑣𝑥,𝑘 , 𝑣𝑦,𝑘 ]
T is the kinematic state of an extended object, (mx, k, my, k) is the position of 

the centroid of the object, (vx, k, vy, k) is the velocity of the object, and 𝑊𝑘−1 is a zero-mean Gaussian process 

noise with the covariance matrix  𝑄𝑘−1. The state transition matrix  𝐹𝑘−1 is assumed known. 
An elliptic random hypersurface model (RHM) (Zhang et al., 2013) is used to model the spread of the 

extended objects in 2-D space. With the RHM model, the measurement vector z must lie within an ellipse 

defined as follows: 

    2 1| , 1
T

k k k


   z z z m A z m

  (2) 

where 𝑚𝑥,𝑘 , 𝑚𝑦,𝑘
𝑇. The positive-definite inverse covariance matrix 𝐴𝐾

−1 may be factorized as: 𝐴𝐾
−1 = 𝐿𝐾 𝐿𝐾

𝑇  

where𝑙𝑘 = [
𝑙𝑘

(1)
0

𝑙𝑘
(3)

𝑙𝑘
(2)

] is a lower triangular matrix with positive diagonal elements. The kinematic state vector 

of an extended object thus should be extended to a 1  7 vector: 

     1 2 3
, , , y,

, , , , , ,
T

k x k y k x k k k k k
m m v v l l l 
 

x  (3) 

We assume this extended target is associated with a set of nT measurements: 

 , 1,...,jk k Tj n z z . 

The measurement source model and relevant measurement model of the jth measurement can be expressed 

as: 

   ( ) ( ) ( ) ( ); , ,
T

j j j j

k k k k k k k k
s R a b   y m e    (4) 

     
  1,...,

j j j

k k k T
j n  z y c  (5) 

where 𝑐𝑘
(𝑗)

 denotes the measurement noise with unknown covariance, it is modelled by a Student’s-distribution 

with unknown parameters. The scaling factor 𝑠𝑘
(𝑗)

 is a random variable uniformly distributed over (0, 1] with 

mean s = 0.5 and variance s2 = 1/12. 𝜃𝑘
(𝑗)

is unknown, but it can be substituted by a proper estimation given 

by the angel between the positive x-axis and the vector from the current center to the measurement 𝑧𝑘
(𝑗)

. 𝑎𝑘 , 

𝑏𝑘  are the semi-major and semi-minor axis of the ellipse, respectively. ∅𝑘 ∈ [0,2𝜋] is the angle between the 
positive x-axis and semi-major axis which is positive in the clock-wise direction. Where 

       ( ) ; , , 0j j jjk k k k k k k kR a b s s    y m  and      ( ) j jjk k k k k  e y m y m  

is a uniceptor along the direction of yk(j)-mk.  

   
2

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) 2 ( ) 2

0 , , , = ( ; , , )

2 ( ; , , )

j j j j j

k k k k k k k k k

j j j j j

k k k k k k k k k k

h s s R a b

s R a b

 

 



    

Χ z c

e c c z m

 (6) 

h(𝑋𝑘 , 𝑠𝑘
(𝑗)

, 𝑧𝑘
(𝑗)

, 𝑐𝑘
(𝑗)

) maps the state vector 𝑋𝑘, the measurement noise vector 𝑐𝑘
(𝑗)

, the scaling factor 𝑠𝑘
(𝑗)

 and 

the physical measurement vector 𝑧𝑘
(𝑗)

 into a pseudo-measurement 0. Because the measurement model is 

nonlinear, an Unscented Transform (Julier andUhlmann, 2004) is applied to deal with this tracking situation. 

686



With the Unscented Transform, we augment 𝑋𝑘 as x𝑘
𝑎 = [x𝑘

𝑇  𝑠𝑘  c𝑘
𝑇 ]T. The corresponding mean and covariance 

can be expressed as 

2

2

0 0

,    and  0 0

0 0

 

   
   

    
     

T

k x

a a

k s k s

k

μ P

μ C

0 R

. (7) 

Where, k is the mean of 𝑋𝑘. 𝑠𝑘 is a Gaussian distribution with mean s and covariance 𝜎𝑠
2. 𝑅𝑘  denotes the 

unknown covariance of measurement noise.  

2.2 Variational bayesian method 

The goal of the Bayesian filtering is to estimate the posterior density given the measurement set at time k:  

1

1

( | , , ) ( , , | )
( , , | )

( | , , ) ( , , | )

k k k k k k k k
k k k k

k k k k k k k k k k k

p v p v
p v

p v p v d d dv









z x R x R Z
x R Z

z x R x R Z x R

 (8) 

This posterior density is in general rather complicated. We employ the Variational Bayesian approach to 

approximate the posterior distribution with a simpler distribution (Zhu et al., 2013) 

( , , | ) ( ) ( ) ( )
k k k k x k R k v k

p v Q Q Q vx R Z x R  (9) 

This approach promises significant reduction of computational complexity. Qx(xk), QR(Rk), and Qv(vk) are 

found by minimizing the KL divergence between 𝑄𝑥(𝑋𝑘 )𝑄𝑟 (𝑅𝑘 )𝑄𝑣(𝑉𝑘 ) and p(𝑋𝑘 , 𝑅𝑘 , 𝑉𝑘 |𝑍𝑘 ): 

 ( ) ( ) ( ) || ( , , | )

( ) ( ) ( )
( ) ( ) ( ) ln

( , , | )

x k R k v k k k k k

x k R k v k
x k R k v k k k k

k k k k

KL Q Q Q v p v

Q Q Q v
Q Q Q v d d dv

p v

 
  

 


x R x R Z

x R
x R x R

x R Z

 (10) 

Minimizing eq. (10), one has: 

| |
( ) ( ; , )

x k k k k k k
Q Nx x x P

  (11) 

, | , | ,

1

( ) ( ; , )
M

R k k l k k l k k l

l

Q G r α β


R
, (12)  

| |
( ) ( ; , )

v k k k k k k
Q v G v γ η

. (13) 

where M is the dimensions of the measurement noise inverse covarianceRk, 𝛼𝑘|𝑘,𝑙 , 𝛽𝑘|𝑘,𝑙 , 𝛾𝑘|𝑘  𝑎𝑛𝑑 𝜂𝑘|𝑘  are 

parameters with the Gamma distribution.  

3. Extended object tracking algorithm based on VB 

Based on eq.(11) to eq.(13), the complex task of joint estimation of the object state, measurement noise 

inverse covariance and degrees of freedom is simplified into parameter estimations of the product of three 

density functions, and these parameters can be predicted and updated in the framework of an extended object 

CPHD filter. The detailed process is as follows: 

Step 1 prediction 

The prediction step of the extended object CPHD filter propagates the joint PHD of object state, measurement 

noise inverse covariance, and degrees of freedom (hereafter called the joint PHD), this joint PHD can be 

expressed as (Li et al., 2014). 

     ,| 1 , | 1, , , , , ,k k S k k B kDD v D v v  x R x R x R  (14) 

Where 𝐷𝑘|𝑘−1(𝑥, 𝑟, 𝑣) and 𝐷𝑏,𝑘 (𝑥, 𝑟, 𝑣) denote respectively the predicted joint PHD intensity of survival object 

and birth object. The expression is below: 

      
         

1

( )

, | 1 , 1 , | 1 , | 1

1 1

, , | 1, , | 1, , | 1 , | 1

, , ; ,

; , ; ,

kJ M
i ii

S k k S k k k S k k S k k

i l

i i i i

k l S k k l S k k l k S k k S k k

D P w N

G r G v   



   

 

   





x R ν x m P  (15) 

687



Here, 𝑃𝑆,𝑘 stands for the survival probability. 𝑊𝑘−1
(𝑖)

, 𝑚
𝑠,𝑘|𝑘−1
(𝑖)

, 𝑃
𝑠,𝑘|𝑘−1
(𝑖)

 𝑎𝑛𝑑 𝐽𝑘−1 are the ith Gaussian component 

parameters of survival object, 𝛼
𝑠,𝑘|𝑘−1,𝑙
(𝑖)

 and 𝛽
𝑠,𝑘|𝑘−1,𝑙
(𝑖)

 are the inverse covariance parameters of the ith Gamma 

component, 𝛾
𝑠,𝑘|𝑘−1
(𝑖)

 and 𝜂
𝑠,𝑘|𝑘−1
(𝑖)

 are the degrees of freedom of the ith Gamma component. 

   

   

,

( ) ( ) ( )

, , ,

1 1

( ) ( ) ( ) ( )

, , , , ,

,

, ,

, , ; ,

; , ; ,

b kJ M
i i i

b k k b k b k

i l

i i i i

k l b k l b k l k b k

B k

b k

w N

G r G

D

v   

 





x R ν x m P  (16) 

Here, 𝑊𝑏,𝑘
(𝑖)

, 𝑚𝑏,𝑘
(𝑖)

, 𝑃𝑏,𝑘
(𝑖)

 𝑎𝑛𝑑 𝐽𝑏,𝑘 are the ith Gaussian component parameters of birth object, 𝛼𝑏,𝑘,𝑙
(𝑖)

 and 𝛽𝑏,𝑘,𝑙
(𝑖)

 are 

the inverse covariance parameters of the ith Gamma component, 𝛾𝑏,𝑘
(𝑖)

 and 𝜂𝑏,𝑘
(𝑖)

 the degrees of freedom of the 

ith Gamma component. Detailed prediction process can be found in [14]. 

Step 2 update 

Set the Gamma distribution parameters at first, 𝛼𝑘,𝑙
(𝑖)

= 𝛼
𝑘|𝑘−1,𝑙
(𝑖)

+ 0.5 , 𝛽
|𝑘,𝑙
(𝑖)(0)

= 𝛽
𝑘|𝑘−1,𝑙
(𝑖)

,  𝛾
𝑘|𝑘
(𝑖)

= 𝛾
𝑘|𝑘−1
(𝑖)

+ 0.5 , 

𝜂
𝑘|𝑘
(𝑖)(0)

= 𝜂
𝑘|𝑘−1
(𝑖)

, �̂�𝑘
(𝑖)(0)

= 1 , for l =1, …, M, i = 1, …, K. K is the number of predicted object Gaussian 

components in current time, M is the dimension of the measurement noise inverse covariance. The main 

update equations are below: 

( ) ( )

,1 ,( )( )

( )( ) ( )( )

,1 ,

, ... ,

i i

k k Mi n

k i n i n

k k M

diag
 

 

 
  

  

R ,  ( )( ) ( )( 1) ( )( ) ( ), ( ), ( ),,1 ,1 | | |
1

1
ˆ [ ][ ]

2

W

i n i n i n j j W j j W T j W T

k k k k k k k k k k k

j

tr s B B
W

 




      z μ z μ P
 (17) 

( )
p p

z h  , 
nsigma

( )

1

ˆ
p

nz m p

p

z W z


  ,   
nsigma

( )

1

ˆ ˆS
T

p

nz c p nz p nz

p

W z z z z


  
,   

nsigma
( )

1

ˆ
T

p a

xz c p k p nz

p

W z z


  P μ  (18) 

 
1

S
xz nz

K


 P ,  ˆ0a ak k nzK z  μ μ , S
a a T

k k nz
K K   C C , 

( ),

|
(1: 2)

j W a

k k x
μ μ  (19) 

( ),

|
(1: 4,1: 4)

j W a

k k k
P C , 

   ( )( ) ( )( ) 1ˆ ˆ( )
i n i n i n

k k k
s


R R , 

( )( )
( )( )

( )( )
ˆ

i n
i n k

k i n

k

a
s

b
 , 

( )

|( )( )

( )( )

|

+0.5
2

i

k ki n

k i n

k k

a 



 (20)  

 
( )

|( )( ) ( )( ) ( ), ( ), ( ),

| | |( )( )
1|

1
[ ][ ]

2 2

i W
k ki n i n j j W j j W T j W T

k k k k k k k k k ki n
jk k

γ
b tr B B

η W 
      R z μ z μ P

 (21) 

( )( ) ( )( 1) ( )( 1) ( )( 1)

| | 1
ˆ ˆ0.5[1 log( ) ]

i n i n i n i n

k k k k k k
s s

  


      (22) 

W refers to a measurement cell obtained currently after the distance partition (Granstrom et al., 2010), and 

|W| is the cardinal number of the set W. Equations are used to perform the Unscented Transform, while the 

parameter B in eq. The notation 𝑋𝑃 represents the pth Sigma point after the Unscented Transform, and the 

total number of Sigma points equals to n sigma. 𝑊𝑚
(𝑝)

 and 𝑊𝑐
(𝑝)

 denote the relevant weight. 

4. Simulation results 

We consider a two-dimensional simulation scenario where the ground truth is described as follows: Two 

objects moves inside a surveillance region for a duration of 40 seconds. The surveillance region is a rectangle 

area with lower left coordinate (500, 1000) meters and upper right coordinate (500, 200) meters. The initial 

positions of these targets are at (930, 210) meters and (930, 210) meters. During the first 10 seconds, both 

targets move toward east at a constant velocity (a CV model) V=28 m/sec. These yield two parallel trajectories. 

Then for the next 10 seconds, these objects turn toward each other at a constant turn rate (a CT model) of 4.5 

degrees/sec (=/40rad/sec), while maintaining the constant speed V. During the last 20 seconds, these 

objects move straight along their headings at the 20th second, namely south east (upper one) and north east 

(lower one) while maintaining the constant speed V. Their trajectories cross-over at the 25th second, and then 

separate apart. Their trajectories terminate at (400, 80) and (400, 80) respectively at the end of the 40th 

second. The sampling period is set to �̃�=1 sec. Detail parameters are shown in the following Table. 
 

 

688



Table 1: The parameters in simulation 

Survival probability 

,
0.99

S k
p 

 

Detection probability 

,
0.98

D k
p   

True measurement noise standard 

deviation 
= = 1.118

x y
  

 

Clutter mean 

 = 10 
 Pruning threshold

5
10T  Gaussian components number max

100J 
 

Empirical threshold 

convcrit = 0.01 

Merge threshold 

10U  

Spread factors 

0.75
   
      

 
Measurement noise 

covariance 

 2 2ˆ ,k x ydiag  R
 

Initial state means of birth objects
(1)

[ 930, 210, 0, 0, 20, 20, 0]
T


 x

( 2)
[ 930, 210, 0, 0, 20, 20, 0]

T


  x

 

Initial state covariance of birth objects

 ( ), 5,5,15,15,5,5,5
i

b k
diagP

 

 

So as to verify the availability of the proposed algorithm, traditional extended object tracking with RHMS 

(RHM-GGM-CPHD) under the measurement noise covariance being σ = 0.25,1.11,6.8  were done. The 

following part is the presentation of the performance of extended object’s tracking, such as the angle (deg), 

semi-major axis’s length, semi-minor axis’s length etc. 100 Monte Carlo (MC) simulations are performed. 

    

Figure 1: Estimation of object trajectory and Estimation of object number  

   

Figure 2: Centroid OSPA and Degree OSPA  

  

Figure 3: Semi-major axis OSPA and Semi-minor axis OSPA 

From Figure 1- Figure 3, it can be found that the proposed VB method can have a good estimation for object’s 

centroid and shape by combining with the Elliptic RHM. In traditional method of extended object tracking with 

-1000 -800 -600 -400 -200 0
-500

-400

-300

-200

-100

0

100

200

300

400

500

x position(m)

y
 p

o
s
it
io

n
(m

)

 

 

VB-RHM-CPHD

0 5 10 15 20 25 30 35 40
1

1.2

1.4

1.6

1.8

2

2.2

2.4

Time

o
b
je

c
t 

n
u
m

b
e
r

 

 

VB-RHM-CPHD

Traditonal RHM =6.8

Traditonal RHM =0.25

Traditonal RHM =1.118

0 5 10 15 20 25 30 35 40
0

10

20

30

40

50

60

70

80

90

Time

d
e
g
 O

S
P

A

 

 
VB-RHM-CPHD

Traditonal RHM =6.8

Traditonal RHM =0.25

Traditonal RHM =1.118

0 5 10 15 20 25 30 35 40
0

10

20

30

40

50

60

70

80

90

Time

c
e
n
tr

o
id

 O
S

P
A

 

 

VB-RHM-CPHD

Traditonal RHM =6.8

Traditonal RHM =0.25

Traditonal RHM =1.118

0 5 10 15 20 25 30 35 40
0

10

20

30

40

50

60

70

Time

s
e
m

i-
m

a
jo

r 
a
x
is

 O
S

P
A

 

 

VB-RHM-CPHD

Traditonal RHM =6.8

Traditonal RHM =0.25

Traditonal RHM =1.118

0 5 10 15 20 25 30 35 40
0

10

20

30

40

50

60

70

Time

s
e
m

i-
m

in
o
r 

a
x
is

 O
S

P
A

 

 

VB-RHM-CPHD

Traditonal RHM =6.8

Traditonal RHM =0.25

Traditonal RHM =1.118

689



Elliptic RHM, improper measurement noise covariance leads to poor estimation performance of object’s 

centroid and shape especially using a bigger covariance. When the measurement noise covariance is little, 

the impact on object’s shape is little caused by measurement noise. Meanwhile, an undetected phenomenon 

occurs because of the objects’ coming across at time 25. But when two objects come close to each other, the 

estimation performance of extended object’s shape is also good. 

5. Conclusion  

Based on the parameters of the Student’s t-distribution being Gamma distribution, and enlightened by ideas of 

the GM-CPHD filtering, a new implementation form of CPHD filtering is proposed in this paper, i.e., the 

predicted and updated PHD intensities are all substituted with a mixture of Gaussian-Gamma distribution. 

Meanwhile, in order to overcome the complex computing problem with the unknown measurement noise 

inverse covariance and the joint estimation of object state and measurement noise inverse covariance, the VB 

step is used to derive an approximating distribution, and the RHM-CPHD step is used to estimate the 

parameters of the approximating distribution. Simulation results show that the proposed approach costs a 

longer time, but improves the accuracy of extended objects tracking. It is expected that the proposed algorithm 

can be utilized to the extended objects tracking under the condition of unknown clutter or unknown detection 

probability. 

Acknowledgements 

The authors acknowledge thenatural science foundation research project of Shaanxi province Foundation of 

China (Grant: 2014JM8328), the Science Technology Research Project of Shangluo University (Grant: 

16SKY002) 

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