Hybrid Iterated Kalman Particle Filter for Object Tracking Problems

Amr M. Nagy

1

, Ali Ahmed

2

and Hala H. Zayed

1

1

Faculty of Computers and Informatics, Benha University, Banha ElGadeda, Benha, Egypt

2

Faculty of Computers and Information, Menoﬁa University, Shubin el Kom, Menuﬁa, Egypt

Keywords:

Kalman Filter, Particle Filter, Nonlinear/Non-Gaussian, Object Tracking.

Abstract:

Particle Filters (PFs), are widely used where the system is non Linear and non Gaussian. Choosing the im-

portance proposal distribution is a key issue for solving nonlinear ﬁltering problems. Practical object tracking

problems encourage researchers to design better candidate for proposal distribution in order to gain better

performance. In this correspondence, a new algorithm referred to as the hybrid iterated Kalman particle ﬁlter

(HIKPF) is proposed. The proposed algorithm is developed from unscented Kalman ﬁlter (UKF) and iterated

extended Kalman ﬁlter (IEKF) to generate the proposal distribution, which lead to an efﬁcient use of the latest

observations and generates more close approximation of the posterior probability density. Comparing with

previously suggested methods(e.g PF, PF-EKF, PF-UKF, PF-IEKF), our proposed method shows a better per-

formance and tracking accuracy. The correctness as well as validity of the algorithm is demonstrated through

numerical simulation and experiment results.

1 INTRODUCTION

The increasing interest in the object tracking is moti-

vated by a huge number of promising applications that

can now be tackled in real-time applications. These

applications include performance analysis, surveil-

lance, video-indexing, smart interfaces, teleconfer-

encing and video compression and so on.

A variety of tracking algorithms have been pro-

posed and implemented. They can be roughly clas-

siﬁed into two categories: deterministic methods and

stochastic methods. Deterministic methods typically

track the object by performing an iterative search for

a similarity between the template image and the cur-

rent one. The algorithms which utilize the determin-

istic method are background subtraction ((McIvor,

2000); (LIU et al., 2001)), inter-frame difference

((Lipton et al., 1998);(Collins et al., 2000)), opti-

cal ﬂow (Meyer et al., 1998), skin color extraction

((kyung-min Cho et al., 2001); (Phung et al., 2003))

and so on. On the other hand, the stochastic meth-

ods use the state space to model the underlying dy-

namics of the tracking system such as Kalman ﬁlter

(Broida and Chellappa, 1986) and particle ﬁlter ((Is-

ard and Blake, 1998); (Ristic et al., 2004); (Sugandi

et al., 2009); (Fen and Ming., 2010); (Zhiqiang et al.,

2011); (Zhonga et al., 2012) ).

Probabilistic methods have become popular

among many researchers. The Kalman ﬁlter is a com-

mon approach for dealing with target tracking in a

probabilistic framework, but it cannot resolve a track-

ing problem where the model is nonlinear and non-

Gaussian. The extended Kalman ﬁlter can deal with

this problem, but still has a problem when the non-

linearity and non-Gaussian cannot be approximated

accurately.

Recently, the particle ﬁlter method, a numerical

method that allows ﬁnding an approximate solution

to the sequential estimation has proven very success-

ful for nonlinear and non-Gaussian estimation prob-

lems. It approximates a posterior probability density

of the state such as the object position by using sam-

ples which are called particles. A key issue in particle

ﬁltering is the selection of the proposal distribution

function. In general, it is hard to design such pro-

posals. Now many proposed distributions have been

proposed in the literature. For example, the prior, the

EKF Gaussian approximation and the UKF proposal

are used as the proposal distribution for particle ﬁlter

((Gordon et al., 1993); (Arulampalam et al., 2002);

(R Van der Merwe, 2000)).

In this paper, a new proposal distribution gener-

ating scheme for the particle ﬁltering framework is

proposed. The algorithm obtained is named as the hy-

brid Iterated Kalman particle ﬁlter (HIKPF). This al-

gorithm uses hybrid Kalman ﬁlter (HKF) to generate

375

Nagy A., Ahmed A. and H. Zayed H..

Hybrid Iterated Kalman Particle Filter for Object Tracking Problems.

DOI: 10.5220/0004211403750381

In Proceedings of the International Conference on Computer Vision Theory and Applications (VISAPP-2013), pages 375-381

ISBN: 978-989-8565-48-8

Copyright

c

2013 SCITEPRESS (Science and Technology Publications, Lda.)

proposal distribution. In this algorithm, each parti-

cle is updated by the UKF and the IEKF sequentially.

Through this procedure, efﬁcient use of the latest ob-

servations is made, which consequently improves the

performance of particle ﬁlters.

2 PARTICLE FILTER

Considering the following nonlinear system (Arulam-

palam et al., 2002):,

x

k

= f

k

(x

k−1

, v

k−1

) (1)

y

k

= h

k

(x

k

, u

k

) (2)

Where x

k

denotes the system state, and y

k

denotes the

observation at time k. The functions f(·) and h(·) rep-

resent the system transition model and the measure-

ment model respectively. The process noise v

k

and the

measurement noise u

k

are assumed independent with

known distributions. The prior knowledge of the ini-

tial state is given by the probability distribution P(x

0

).

2.1 Recursive Bayesian Estimation

The objective of the recursive Bayesian state estima-

tion problem is to ﬁnd the mean and variance of a

random variable x

k

using the conditional probability

density function P(x

k

|y

k

), using Bayes’ formula

under following assumptions (Arulampalam et al.,

2002):

• The states follow a ﬁrst-order Markov process.

• The observation are conditionally independent

given the state variables.

Y

k

denotes the set of all the available measure-

ments, i.e The posterior density P(x

k

|y

k

) is estimated

in two steps: (a) Prediction step, which is computed

before obtaining an observation.

P(x

k

|y

k−1

) =

Z

P(x

k

|x

k−1

)P(x

k−1

|y

k−1

)dx

k−1

(3)

(b) Update step, which is computed after obtaining an

observation

P(x

k

|y

k

) =

P(y

k

|x

k

)P(x

k

|y

k−1

)

P(y

k

|y

k−1

)

(4)

where

P(y

k

|y

k−1

) =

Z

P(y

k

|x

k

)P(x

k

|y

k−1

)dx

k

(5)

By substituting Eqs. (3) and (5) in Eq. (4) we can get

obtain the ﬁnal equation:

P(x

k

|y

k

) =

P(y

k

|x

k

)

R

P(x

k

|x

k−1

)P(x

k−1

|y

k−1

)dx

k−1

R

P(y

k

|x

k

)P(x

k

|y

k−1

)dx

k

(6)

The prediction and update strategy provides an op-

timal solution to the state estimation problem,which,

unfortunately, involves high-dimensional integration.

The solution is extremely general and aspects such as

multimodality, asymmetries and discontinuities can

be incorporated.

2.2 Solution through Monte Carlo

Sampling

The exact analytical solution to the recursive prop-

agation of the posterior density is difﬁcult to obtain

for a general nonlinear system, because it involves

high-dimensional integration of unknown density

functions (refer to Eqs. (3) and (6)). However, when

the process model is linear and noise sequences

are zero mean Gaussian white noise sequences,

the Kalman ﬁlter describes the optimal recursive

solution to the sequential state estimation problem

(Soderstorm, 2002) While dealing with nonlinear

systems, it becomes necessary to develop approx-

imate and computationally tractable sub-optimal

solutions to the above sequential Bayesian estimation

problem. The particle ﬁlter is a numerical method

for implementing an optimal recursive Bayesian

ﬁlter through Monte Carlo simulation. Classical

particle ﬁlters approximate the distribution P(x

k

|y

k

)

, using a set of random samples x

i

k

: i = 1, · · · , N

together with associated weights ω

i

k

: i = 1, · · · , N

and x

k

= x

j

, j = 0, ··· , k is the set of all states up

to time k. The weights are normalised such that

∑

i

ω

i

k

= 1. Then, the posterior density at k can be

approximated as

P(x

k

|y

k

) ≈

N

∑

i=1

ω

i

k

δ(x

k

− x

i

k

) (7)

where δ(x

k

− x

i

k

) denotes the Dirac delta function.

The weights ω

i

k

can be viewed as approximations to

the relative posterior probabilities of the particles. It

should be noted that the posterior density P(x

k

|y

k

)

is seldom known. Therefore, it is not possible to

draw samples from this distribution. For this reason,

q[(x

i

k

|x

i

k−1

), y

k

], a proposal density or importance

density, is used. At each sampling instant, a sample

is drawn from the proposal distribution generated

around each particle. To compensate for the dif-

ference between the proposal density and the true

posterior density, the weights are then computed as

follows:

˜

ω

i

k

=

P(x

i

k

|y

k

)

q(x

i

k

|y

k

)

(8)

VISAPP2013-InternationalConferenceonComputerVisionTheoryandApplications

376

and the updated weight equation is:

˜

ω

i

k

=

P(y

k

|x

i

k

)P(x

i

k

|x

i

k−1

)

q(x

i

k

|x

i

k−1

, y

k

)

˜

ω

i

k−1

(9)

as a result the normalized weight is given by:

ω

i

k

=

˜

ω

i

k

∑

j=1

˜

ω

j

k

(10)

2.3 Selection of Proposal Distributions

The selection of a suitable form of importance func-

tion to represent the true posterior density is a cru-

cial step in the particle ﬁlter ((Arulampalam et al.,

2002); (Rawlings and Bakshi, 2006)). The conven-

tional approach is to use the state transition den-

sity as the proposal distribution/importance function,

i.e.q[(x

i

k

|x

i

k−1

), y

k

] ≈ P[x

i

k

|x

i

k−1

], and draw particles

from the above importance function. Because the

state transition function (being used as importance

function) does not take in to account the most re-

cent observation, y

k

, the particles drawn from tran-

sition density may have very low likelihood, and their

contributions to the posterior estimation become neg-

ligible. It may be noted that the use of appropriate

importance function can signiﬁcantly reduce the num-

ber of particles required for generating accurate esti-

mates, as compared to the conventional particle ﬁlter

(Arulampalam et al., 2002). In general, it is difﬁcult

to design such a proposal and the choice of proposal

distribution is highly problem dependent.

The computational steps involved are as follows

(Arulampalam et al., 2002):

2.3.1 Initialization

At k = 0, M samples are drawn from the given distri-

bution of initial the state, ˆx

0|0

.

2.3.2 Importance Sampling

At the k

′

th time step, after obtaining measurement

y

k

, M observers (EKF or UKF or IEKF) are used in

parallel to compute means and covariances of the pro-

posal distributions, i.e. ¯x

i

k|k

,

¯

P

i

k|k

for each propagated

particle ˆx

i

k−1|k−1

. The importance density is then ap-

proximated as q[(x

i

k

|x

i

k−1

), y

k

] ≈ N[ ¯x

i

k|k

,

¯

P

i

k|k

] and used

to draw a sample around each particle.

2.3.3 Computation of Weights

The weights associated with each particle are now

computed by Eq. (9),and These

˜

ω

i

k

weights are then

normalized to obtain ω

i

k

as given by Eq. (10).

2.3.4 Re-sampling

This step involves discarding samples that have low

importance and reassigning weights to the remaining

particles. Various approaches have been suggested in

the literature for carrying out this step.

In our proposed algorithm we used the residual re-

sampling algorithm.

3 HYBRID ITERATED KALMAN

PARTICLE FILTER

Before talking about our proposed algorithm (Hybrid

Iterated Kalman Particle Filter), ﬁrstly the unscented

Kalman ﬁlter and the iterated extended kalman ﬁlter

are introduced.

3.1 Unscented Kalman Filter

The Unscented Kalman Filter belongs to a big-

ger class of ﬁlters called Sigma-Point Kalman Fil-

ters or Linear Regression Kalman Filters, which are

using the statistical linearization technique ((Gelb,

1974); (Julier, 2002); (Julier et al., 2002);(Julier and

Uhlmann, 2004); (Lefebvre and Bruyninckx, 2004)) .

This technique is used to linearize a nonlinear func-

tion of a random variable through a linear regression

between n points drawn from the prior distribution of

the random variable. The UKF is founded on the intu-

ition that it is easier to approximate a probability dis-

tribution that it is to approximate an arbitrary nonlin-

ear function or transformation (Julier and Uhlmann,

2004). The sigma points are chosen so that their

mean and covariance to be exactly x

a

k−1

and P

k−1

.

Each sigma point is then propagated through the non-

linearity yielding in the end a cloud of transformed

points. The new estimated mean and covariance are

then computed based on their statistics. This process

is called unscented transformation. The unscented

transformation is a method for calculating the statis-

tics of a random variable which undergoes a nonlinear

transformation (Wan and van der Merwe, 2001).

3.2 Iterated Extended Kalman Filter

The extended Kalman ﬁlter (EKF) is a minimum

mean-square-error (MMSE) estimator based on the

Taylor series expansions of the nonlinear functions

f(·) and h(·) around the current estimates.

In the EKF, the state distribution is represented

by using a Gaussian random variable. It only uses the

linear expansion terms.

HybridIteratedKalmanParticleFilterforObjectTrackingProblems

377

Model and Observation:

x

k

= f(x

k−1

) + v

k−1

y

k

= h(y

k

+ u

k

)

Initialization:

x

a

0

= µ

0

with error covariance P

0

Model Forecast Step/Predictor:

x

f

k

≈ f(x

a

k−1

)

P

f

k

= J

f

(x

a

k−1

)P

k−1

J

T

f

(x

a

k−1

) + Q

k−1

Data Assimilation Step/Corrector:

x

a

k

≈ x

f

k

+ K

k

(y

k

− h(x

a

k

))

K

k

= P

f

k

J

T

k

( ˆx

k

)(J

h

(x

a

k,i

)P

f

k

J

T

k

(x

a

k

) + R

k

)

−1

P

k

= (I − K

k

J

h

(x

a

k

))P

f

k

In the EKF, h(∆) is linearized about the predicted

state estimate x

f

k

. The IEKF (Liang-qun et al., 2005)

tries to linearize it about the most recent estimate, im-

proving this way the accuracy ((Lefebvre and Bruyn-

inckx, 2004) ; (Gelb, 1974) ). This is achieved by

calculating x

a

k

, K

k

, P

k

at each iteration. Denote x

a

k,i

the

estimate at time k and ith iteration. The iteration pro-

cess is initialized with x

a

k,0

= x

f

k

. Then the measure-

ment update step becomes for each i:

x

a

k,i

≈ x

f

k

+ K

k

(y

k

− h(x

a

k,i

))

K

k,i

= P

f

k

J

T

k

( ˆx

k,i

)(J

h

(x

a

k,i

)P

f

k

J

T

k

(x

a

k,i

) + R

k

)

−1

P

k,i

= (I − K

k,i

J

h

(x

a

k,i

))P

f

k

3.3 Hybrid Iterated Kalman Particle

Filter

Our proposed algorithm named hybrid iterated

Kalman particle ﬁlter is a combination of the UKF

and the IEKF . The HIKPF inherits the excellent prop-

erties of the UKF and IEKF and can make efﬁcient

use of the latest observations, which make it very

attractive for the generation of proposal distribution

within the particle ﬁltering framework.

At time k, the UKF is ﬁrstly used to update the

particles, and to obtain the state estimate ˜x

k,uf

, and

the corresponding covariance estimate P

i

k,uf

, then the

particles are updated using the IEKF with ¯x

k,uf

, and

P

i

k,uf

. After the IEKF-update, the ﬁnal state and co-

variance estimates ¯x

i

k

and

ˆ

P

i

k

j

of time step k are ob-

tained.

Using the estimates, the required proposal distri-

bution N( ¯x

i

k

j

,

ˆ

P

i

k

j

) is formed. Here, samples can be

drawn from the approximated distribution N( ¯x

i

k

j

,

ˆ

P

i

k

j

).

The following ﬁgure shows the ﬂow chart of our

proposed algorithm.

Initialize Particles

New Observations

Particles

Generations

Update Particles

Using UKF

Update Particles

Using IEKF

Normalize

Resampling

Output Estimations

More Ob-

servations

Output

yes

no

Figure 1: The schematic description of the proposed

algrithm (HIKPF).

HIKPF Algorithm Steps

—————————————————

Step 1.Initialization: k = 0

FOR i = 1, ....., N

p

Draw the particles x

i

0

from the prior P(x

0

) and set:

¯x

i

0

= E(x

i

0

)

P

i

0

= E[(x

i

0

− ¯x

i

0

)(x

i

0

− ¯x

i

0

)

T

]

¯x

i

0,a

= E[(x

i

0,a

)] = [(x

i

0

)

T

, 0, 0]

T

P

i

0,a

= E[(x

i

0,a

− ¯x

i

0,a

)(x

i

0,a

− ¯x

i

0,a

)

T

] = diag(P

i

0

QR)

END FOR

Step 2. FOR k = 1, 2, ...

(1)FOR i = 1, ....., N

p

VISAPP2013-InternationalConferenceonComputerVisionTheoryandApplications

378

(a) Update the particles using the UKF

Calculate the sigma points

χ

i

k−1,a

= [ ¯x

i

k−1,a

¯x

i

k−1,a

±

q

(n

a

+ λ)P

i

k−1,a

]

Propagate samples into future and compute

the one-step-ahead estimates:

χ

i

k|k−1,x

= f(χ

i

k−1,x

, χ

i

k−1,v

)

Y

i

k|k−1,u f

= h(χ

i

k|k−1,x

, χ

i

k−1,u

)

¯x

i

k|k−1,u f

=

2n

a

∑

j=0

W

( j)

m

χ

( j)i

k|k−1,x

P

i

k|k−1,u f

=

2n

a

∑

j=0

W

( j)

c

[χ

( j)i

k|k−1,x

− ¯x

i

k|k−1,u f

][χ

( j)i

k|k−1,x

−

¯x

i

k|k−1,u f

]

T

¯y

i

k|k−1,u f

=

∑

2n

a

j=0

W

( j)

m

Y

( j)i

k|k−1,u f

Incorporate the new observation y

k

, and update

the one-step-ahead estimates to obtain ¯x

i

k,uf

P

y

k

y

k

=

2n

a

∑

j=0

W

( j)

c

[Y

( j)i

k|k−1,u f

− ¯y

i

k|k−1,u f

][Y

( j)i

k|k−1,u f

− ¯y

i

k|k−1,u f

]

T

P

x

k

y

k

=

2n

a

∑

j=0

W

( j)

c

[χ

( j)i

k|k−1

− ¯x

i

k|k−1,u f

][Y

( j)i

k|k−1,u f

− ¯y

i

k|k−1,u f

]

T

K

k,uf

= P

x

k

y

k

P

−1

y

k

y

k

¯x

i

k,uf

= ¯x

i

k|k−1,u f

+ K

k,uf

(y

k

− ¯y

i

k|k−1,u f

)

P

i

k,uf

= P

i

k|k−1,u f

− K

k,uf

P

y

k

y

k

K

T

k,uf

(b) Use the IEKF to update estimations obtained

through UKF update process

FOR i = 1, ....., N

p

Compute the Jacobians F

i

k

&G

i

k

of the process

model

Update the particles with the IEKF

¯x

i

k|k−1,ie f

= f( ¯x

i

k,uf

)

P

i

k|k−1,ie f

= F

i

k

P

i

k,uf

(F

i

k

)

T

+ G

i

k

FOR j = 1 : c (c is the number of iteration)

Compute the Jacobians H

i

k

j

&U

i

k

j

of the

measurement model

Update the covariance and the state estimate

from the following equations obtained

from IEKF respectively .

K

k

j

,ief

= P

i

k

j

|k

j

−1,ief

(H

i

k

j

)[U

i

k

j

R

k

j

(U

i

k

j

)

T

+

H

i

k

j

P

i

k

j

|k

j

−1,ief

(H

i

k

j

)

T

]

−1

P

i

k

j

,ief

= P

i

k

j

|k

j

−1,ief

− K

k

j

,ief

H

i

k

j

P

i

k

j

|k

j

−1,ief

¯x

i

k

j

,ief

= ¯x

i

k

j

|k

j

−1,ief

+ K

k

j

,ief

(y

k

j

− h( ¯x

i

k

j

,ief

))

let ¯x

i

k

j

= ¯x

i

k

j

,ief

,

ˆ

P

i

k

j

= P

i

k

j

,ief

END FOR

Draw x

i

k

∼ q(x

i

k

|x

i

k−1

, z

k

) = N( ¯x

i

k

j

,

ˆ

P

i

k

j

)

Assign the particle a weight, w

i

k

, according to the

equation below obtained from PF

w

i

k

∝ w

i

k−1

P(z

k

|x

i

k

)P(x

i

k

|P(x

i

k−1

)

q(x

i

k

|x

i

k−1

,z

k

)

END FOR

(2) Normalize the weights

FOR i = 1, ...., N

w

i

k

=

w

i

k

∑

j=1:N

w

i

k

.

END FOR.

(3) Resample

(4) Output: calculate the required estimations

using the particle set.

END FOR.

Step 3. k = k+ 1, go to Step2 or end the algorithm

———————————————————–

4 SIMULATION AND

EXPERIMENTAL RESULTS

The simulation results of the HIKPF algorithm is

presented and discussed in this section as well as

the comparison between HIKPF and the previously

proposed algorithms including PF, PF-EKF, PF-UPF

and PF-IEKF . The system models were taken from

(R Van der Merwe, 2000) as following.

x

k

= 1 + sin(0.4πk) + 0.5x

k−1

+ v

k−1

y

k

=

0.2x

2

k

+ e

k

, if k ≤ 30

0.5x

k

− 2+ e

k

, if k > 30

Where v

k

is a Gamma ς

a

(3, 2) random variable

modelling the process noise, and the measurement

noise u

k

is drawn from a Gaussian distribution

N(0, 0.00001). In this experiment 200 particles are

used and the program is repeated 100 times for time-

steps k = 1, .., 60. The unscented transformation pa-

rameters are set to be α = 1, β = 0, and κ = 2. The

HybridIteratedKalmanParticleFilterforObjectTrackingProblems

379

output of the algorithm is the mean of samples set that

can be computed ˆx =

1

N

∑

N

s

j=1

x

j

t

. The mean square er-

rors of each run is deﬁned as

MSE =

1

T

T

∑

k=1

( ˆx

k

− x

k

)

!

1

2

(11)

Figure 2 Shows the true and the estimated state of the

system HIKPF and the other methods. It is clear from

the ﬁgure that particle ﬁlter (PF) and extended kalman

particle ﬁlter ( PF-EKF) deviate from the true states

at some time steps. The unscented Kalman particle

ﬁlter(PF-UKF) and iterated extended kalman particle

ﬁlter (PF-IEKF) gives better performance than PF and

PF-EKF but less than our proposed system (HIKPF).

5 10 15 20 25 30 35 40 45 50 55 60

−2

0

2

4

6

8

10

12

14

Time

Estimated states and true states

True x

PF

PF−EKF

PF−UKF

PF−IEKF

HIKPF

Figure 2: Estimation of the system state generated by dif-

ferent particles ﬁlters.

The performance evaluation of our system com-

pared with other methods is shown in table 1. In

this table, our proposed system (HIKPF) gives the

best performance with the lowest mean and variance

with mean value 0.015607 and variance value (

Var

)

0.00000395.

Table 1: Estimation of means and variances of

MSE

of dif-

ferent particles ﬁlters over 100 independent runs.

Algorithm

MSE

mean Var

PF 0.25881 0.057151

PF-EKF 0.32392 0.021656

PF-UKF 0.077684 0.006589

PF-IEKF 0.049368 0.0015238

HIKPF 0.015607 0.00000395

Estimation of mean squars errors (

MSEs

) of differ-

ent particle ﬁlters are shown in Figure 3 In this ﬁg-

ure, it is clear that the bottom real line (Blue Line) is

the HIKPF performance line. The proposed algorithm

10 20 30 40 50 60 70 80 90 100

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Run(s)

MSE

PF PF−EKF PF−UKF PF−IEKF HIKPF

Figure 3:

MSEs

estimation of different particles ﬁlters at

each run.

give the lowest mean square error at every indepen-

dent run.

5 CONCLUSIONS

A new algorithm referred to as the hybrid iterated

Kalman particle ﬁlter (HIKPF) is proposed. The pro-

posed algorithm is developed from unscented Kalman

ﬁlter (UKF) and iterated extended Kalman ﬁlter

(IEKF) to generate the proposal distribution leading

to efﬁcientuse of the latest observations and generates

more close approximation of the posterior probability

density. Numerical simulation and experiment results

show that HIKPF algorithm is much robust than the

previously proposed algorithms such as (PF, PF-EKF,

PF-UPF and PF-IEKF).

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