V

Abstract—This paper proposes a fast target tracking method in which particle filter is improved using Gaussian kernel and evolutionary strategy. We use Gaussian kernel function to replace the Dirac kernel function, which can decrease the degeneracy problem of the traditional particle filter partly. To further improve the performance of particle filter, we introduce evolutionary strategy into the process of Gaussian kernel particle filtering. It uses only mutation operation, which has less computation than genetic algorithm. And it can prevent the impoverishment problem and steer the particles towards local mode of posterior probability effectively. The proposed method can track fast target robustly using fewer particles than the standard particle filter and Gaussian kernel particle filter.

I. INTRODUCTION ISUAL tracking is an important problem in computer vision, especially fast target tracking. Generally visual tracking can be described as a non-Gaussian and

nonlinear problem, in which the optimal system state is estimated using the observation of the tracked target. There are two usual tracking methods based on Bayesian estimation to obtain the optimal system state of visual target. One is the extended Kalman filter [1], and the other is particle filter [2] [3] [4]. Compared to Kalman filter, the extended Kalman filter can partly handle a non-Gaussian and nonlinear motion model. But if the target motion is heavily nonlinear, it is difficult to model the motion process of the tracked target. So the effect of the extended Kalman filter is still unsatisfactory. In contrast, particle filter does not have such limitation, which uses a set of weighting particles to express the motion probability model of the tracked target. However, the importance sampling process of the traditional particle filter can decrease the estimation performance greatly. The reason of this problem is due to the particle impoverishment phenomenon [2] [5]. After several iterations, many particle weights are to be small, even close to zero. Although they are recomputed in the importance sampling process repeatedly, their contributions for the filtering system are very small. They are regarded as the useless particles whose weight degeneracy occurs. So, to track target effectively using particle filter, the diversity of particle set must be maintained. We can employ a resampling process to deal with the weight degeneracy after the importance sampling [2]. The main purpose of the resampling process is that we can remove those particles with small weight and resample the particles with large weight. The resampling process can prevent the degeneracy of particles to some extent, but it can lead to the particle impoverishment problem. The repeated resampling for a particle with large weight can generate many duplicated

Q.C.Wang is with the Department of Computer Science, Xiamen University, Xiamen 361005 (e-mail: qcwang@xmu.edu.cn)

particles, which decreases the diversity of particle set greatly. Then it leads to the inaccurate approximation of posterior probability and finally affects the estimation performance of particle filter. Many methods have been proposed to deal with the particle impoverishment phenomenon. Increasing the number of particles is the simplest method, but the computation increases. Other improved methods, such as partitioned sampling [6] [7], resampling moving algorithm which uses Markov Chain Monte Carlo (MCMC) to redistribute particles for posterior probability [8], simulated annealing particle filter which introduces anneal importance sampling and middle distribution to improve the algorithm performance when prior tail observation exists [9], and auxiliary particle filter whose resampling process can select particles in high likelihood region [10]. These methods need complex sampling strategy or some priori knowledge about target, which can reduce particle impoverishment partly. In [11], some similar particle filters based on mean shift are proposed, which employ mean shift to each particle. In [12], Gaussian kernel based particle filter is proposed, in which Gaussian kernel is used and a resampling process is embedded at each iteration step so that fewer particles can be used to estimate posterior probability effectively.

From [13], we can see that the implementation characteristics of particle filter are similar to genetic algorithm. In [14], the sampling algorithm is considered as the fittest survive in the evolution theory. In [15], the relationship between particle filter and genetic algorithm is set up through Monte Carlo simulation. In [16], Bayesian framework and evolutionary computation are combined to improve the function optimization performance. The reasons leading to the particle impoverishment phenomenon are that only some of particles which have large weight are resampled and uniform sampling or roulette sampling is used in particle filtering. In this paper, we consider the particle impoverishment problem from the perspective of evolutionary strategy. We introduce evolutionary strategy into particle filter to maintain the diversity of particle set. Another problem of the traditional particle filter is the lack of particles in the high-probability region. In the implementation process of evolutionary algorithm, although we initially do not know the location of the high-probability region in solution space, it can automatically move towards the high-probability region using fitness function which employs mutation and selection operator to evolve [17]. In the proposed method, evolutionary strategy in particle filtering has two functions: searching sub-optimal mode and realizing implicit resampling. The first one can steer new particles towards the local mode of posterior probability, which makes the local expression of posterior probability to be appropriate. The mutation operator of evolutionary strategy is embedded into the resampling process to enrich the diversity of particle

A Novel Particle Filter for Tracking Fast Target Qicong Wang, Wenxiao Jiang, Chenhui Yang, and Yunqi Lei

V

288

Third International Workshop on Advanced Computational Intelligence August 25-27, 2010 - Suzhou, Jiangsu, China

978-1-4244-6337-4/10/$26.00 @2010 IEEE

set. As long as the iteration number and mutation probability are set appropriately, the impoverishment problem can be prevented effectively, and finally we can use fewer particles to track the fast target robustly.

II. SAMPLING FILTERING OF THE TRADITIONAL PARTICLE FILTER

Suppose ( ) ( )p x xπ∝ , i.e., ( )p x is up to

proportionality. ( )p x is a posterior probability distribution which can not be computed directly from a particle set, but

( )xπ can. If the particle set, ( ) ( ){ },i ik kx w , is produced by

the importance probability density ( )q x , where kx is

particle state, kw is particle weight, 1,2, ,i N= and N is particle number. Then the weighted approximation of posterior probability ( )p x is estimated as follows [2]:

( ) ( ) ( )( )1

Ni i

ip x w x xδ

=≈ −∑ (1)

where ( )δ • is Dirac function, ( ) 0ikw ≥ , ( )

11

Nik

iw

==∑ and

( )( )( )( )( )i

ii

xw

q x

π∝ (2)

is the normalized weight of the i th particle. But the filtering process can lead to the weight degeneracy problem. After several iterations, the weights of some particles would be close to zero. To prevent this problem, the resampling is used to update those particles which contribute rarely to the computation of posterior probability [2]. In other words, it can remove the particles with small weight and resample those particles with large weight. The resampled probability of a particle in the resampling process is according to:

( ) ( )( ) ( )Pr i j jk k kx x w′ = = (3)

After resampling, each particle weight must be normalized

by ( ) 1ikw N′ = .

Particle filter with resampling is named as sampling importance resampling particle filter. This process is depicted as follows:

When 0k =

1. Sample importance distribution ( )0 0q x y and get an

initial particle set ( ){ }0ix , 1, ,i N=

2. Compute each particle weight

( )( )( ) ( )( )( )( )

0 0 0

0

0 0

i i

i

i

p y x p xw

q x y=

3. Compute the normalized weight for each particle

( )( )

( )

00

01

ii

Nn

n

www

=

=∑

When 1k ≥

1. Sample importance density ( )( )1 1:,ik k kq x x y− and get a

particle set ( ){ }ikx , 1, ,i N=

2. Compute each particle weight

( ) ( )( )( ) ( ) ( )( )( ) ( )( )

1

1

1 1:,

i i ik k k ki i

k k i ik k k

p y x p x xw w

q x x y

−

−

−

=

3. Compute the normalized weight for each particle

( )( )

( )

1

ii kk N

nk

n

www

=

=∑

4. Compute the threshold( )( )2

1

1N

ik

i

Tw

=

=∑

. If T c≥ ,

where c is the normalized cumulative probability,

then ( ) ( )i ik kx x′ = , otherwise, resample according to each

particle weight to obtain a new particle set. 5. Calculate the current posterior probability of particle set

by ( ) ( ) ( )( )1:1

Ni i

k k k k ki

p x y w x xδ=

≈ −∑ .

We resample particles according to uniform distribution, but the absolute random number is impossible to obtain in fact. So this pseudo-random inevitably lead to the impoverishment problem. In [5], Amaud analyzed the impoverishment problem. From [19], we can see that the selection process of evolutionary algorithm employs random universal sampling strategy to decrease the selection biases and impoverishment problem.

III. TARGET TRACKING BASED ON GAUSSIAN KERNEL PARTICLE FILTER

The bandwidth of Gaussian kernel is wider than the Dirac, the particle filter based on the Gaussian kernel can improve the performance [12]. It has three advantages: not need to optimize the kernel parameters, existing an implicit resampling and using non-zero bandwidth kernel function to realize the sequential estimation of posteriori probability. The prediction equation of Gaussian kernel particle filter is

( )( ) ( ) ( ) ( )( )

1: 1

1 1 1 1 1 11

|

| ; ,

k k

N i i ik k k k k k ki

p x y

w p x x N x x dx

−

− − − − − −== Σ∑ ∫

(4)

where ( )1ikx − is Gaussian kernel function mea, Σ is

covariance matrix which can reflect the change of Gaussian

289

kernel at each iteration. The process of visual tracking based on Gaussian kernel

particle filter can be depicted as follows. When 0k = , we use Gaussian distribution to get an initial

particle set ( ) ( ) ( ){ }0 0 0, ,i i ix w ∑ , 1, ,i N= ,where ( )0ix is the

mean of the thi Gaussian kernel, ( )0

1iwN

= is the

corresponding weight and ( ) ( )0 0covi x I∑ = i is the corresponding covariance matrix.

When 1k ≥ 1. Estimate the mean of each Gaussian kernel as particle

state at time k according to the system dynamic

model ( ) ( ) ( )1

i i ik k kx Fx v−= + , where F is the state transition

matrix, 2. Get observation ky at time k

3. Compute each particle weight at time k as ( ) ( )

( ) ( )

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

12

1 1exp22

Ti i i i ik k k k k k kd

ik

w w y x y xπ

−

−⎛ ⎞= − − ∑ −⎜ ⎟⎝ ⎠∑

i

4. Compute Jacobian ( )i

kxJ

5. Calculate the corresponding covariance using ( )

( )( )

( )1i ik k

Ti ik kx x

J J−∑ = ∑ at time k .

6. Normalize particle weight ( )( )

( )

1

ii k

k Nnk

n

www

=

′ =∑

7. Resampling (1) Compute the normalized cumulative probability kc′

(2) Produce N random numbers obeyed to uniform distributed [ ]0,1nr ∈

(3) Select the minimum j from ( ){ }1j

kc −′ which

satisfies ( )jk nc r′ ≥

(4) Calculate the particle covariance matrix eigenvector

and eigenvalue as: ( ) ( ) ( )[ , ] = eigen( )j j jk k kV d ∑

(5) Calculate new state, i.e. new mean, ( ) ( ) ( ) ( )n j j jk k k kx x V d randn′ = + , where randn is

Gaussian distribution random number. (6) Calculate the covariance matrix of new particle

( ) ( )covnk kx I′ ′∑ =

(7) Normalize each particle weight of new particle set ( ) 1nkw N′ =

8. Compute the final state output for this system at the

current time ( ) ( ) ( ) ( )( )1

; ,N

i i ik k k k k

ix out w N x x

== Σ∑

In the filtering process of Gaussian kernel particle filter, for each particle, we firstly predict the mean and covariance according to the system dynamic equation. Then we obtain observation which corresponds to the predicted mean. At last, we resample and update the mean and covariance of each Gaussian kernel. The current mean and covariance can be used to construct a new Gaussian proposal distribution at next time, and a new particle set is sampled from this distribution.

IV. TRACKING FAST TARGET USING EVOLUTIONARY STRATEGY BASED GAUSSIAN KERNEL PARTICLE FILTER Gaussian kernel particle filter uses fewer kernel functions

to estimate the posterior probability than the traditional particle filter, thus the computation cost decreases. However, after several iterations, some kernels can converge to the same kernel, which also exist the impoverishment problem. We further propose an improved Gaussian kernel particle filter based on evolutionary strategy. We only use evolutionary strategy to achieve sub-optimal mode. Evolutionary strategy without the crossover operator, which is embedded in the process of particle filtering, makes particles to redistribute to the local mode of the posterior probability effectively. The iterative process has two functions: searching iteratively sub-optimal mode and realizing implicitly iterative resampling. The first function can redistribute particles to the local mode the posterior probability effectively, which can produce more appropriate local expression for the posterior probability. It is similar to the proposed particle filter based mean shift in [11]. Mean shift is applied to each particle to achieve the local mode, but evolutionary strategy can do it parallely. After particles are redistributed to their local mode, we only use fewer particles to maintain multi-mode of particle set. From [20], we can see the sampling process of sampling importance resampling particle filter is similar to the evolutionary process. So we consider the importance sampling and resampling process of sampling importance resampling particle filter correspond to mutation and selection operation of evolutionary strategy respectively. In sampling importance resampling particle filter, the importance sampling process samples a particle

( )ikx according to the importance density (proposal

distribution) ( ) ( )( )1 1:,i ik k kq x x y− . Moreover, in importance

sampling process, offspring are generated from parent ( )1ikx −

using ( )( )1ikf x − and disturbance ( )

1ikv − , which can be viewed

as a mutation operation from the perspective of the evolutionary algorithm. On the other hand, the particle

resampling probability is proportional to the weight ( )ikw in

particle filtering, which can correspond to the selection process according to the fitness value in the evolutionary algorithm.

From the above description, we can see that, to prevent

290

impoverishment problem, sequential importance resampling is introduced into Gaussian kernel particle filter. Gaussian probability density function set generated from a proposal distribution can produce a new Gaussian probability function set through resampling according to

( ) ( )( ) ( ) ( )( )( ) ( ); , ; ,i i j j jr k k k k k k kP N x x N x x wΣ = Σ = (5)

and each particle weight is reset to 1N

. This can be named as

Gaussian kernel resampling particle filter. The iterative resampling algorithm of evolutionary strategy based Gaussian kernel particle filter is a selection process based on evolution operation. The iterative resampling process is as

follows. We get an initial particle set ( ){ }1ikx − , 1, 2, ,i N=

from importance function ( )1 1: 1k kq x y− − . Then we can

obtain particle set ( ){ },i jkx , 1, 2, ,j M= from importance

function ( )( )1 1:,ik k kq x x y− . Thus each particle weight is

computed by

( ) ( )( )( ) ( ) ( )( )( ) ( )( ), ,

1 1,1 ,

1 1:,

i j i j ik k k ki j i

k k i j ik k k

p y x p x xw w

q x x y

− −

−

−

= (6)

Normalize each weight:

( )( )

( )

,,

,

1

i ji j kk M

i jk

j

www

=

=∑

(7)

Then obtain each mean:

( ) ( ) ( ), ,

1

Mi i j i jk k k

j

w xμ=

=∑ (8)

Compute each covariance:

( ) ( ) ( ) ( )( ) ( ) ( )( ), , ,

1

M Ti i j i j i i j ik k k k k k

j

w x xμ μ=

Σ = − −∑ (9)

Update each weight:

( )

( ) ( )

( ) ( )

,1

1

,1

1 1

Mi j ik k

i jk N M

i j ik k

i j

w ww

w w

−=

−= =

=∑

∑∑ (10)

After the evolutionary operation, in the new M N× number particles, sort the Gaussian kernel function set

( ) ( )( ){ }, ,; ,i j i jk k kN x x Σ according to the descending, and

retain first N Gaussian kernels with larger weight to form a

new particle set ( ) ( )( ){ }; ,i ik k kN x x Σ , 1, 2, ,i N= . Then

normalize each weight by

( )( )

( )

1

ii kk N

nk

n

www

=

=∑

(11)

V. EXPERIMENTAL RESULTS AND DISCUSSIONS In the following experiments, the target region is modeled

by kernel density estimation. The tracked target is initialized by hand. The first test sequence was downloaded from the website: groups.inf.ed.ac.uk /vision /CAVIAR/. The image resolution is 384 288× pixels. The target moves slowly in this video. Fig.1 shows the tracking results of a man head using three different particle filter algorithms respectively. From left to right, the images show the first, 49th, 105th frame of the test sequences. The first row shows the tracking results using the traditional particle filter with sampling 220 particles. The second row shows the tracking results using Gaussian kernel particle filter with sampling 65 particles. The third row shows the tracking results using evolutionary strategy based Gaussian kernel particle filter with sampling 25 particles and performing 4 times evolutionary iteration each time.

the first frame the 49th frame the 105th frame Fig 1 Tracking slow target using three kinds of particle filters respectively

Fig. 2, Fig. 3 and Fig. 4 show the tracking results towards

the people on the motorcycle employing the traditional particle filter, Gaussian kernel particle filter and evolutionary strategy based Gaussian kernel particle filter respectively. The moving speed of the tracked target is tow times faster than pedestrians. The traditional particle filter uses 580 particles, Gaussian kernel particle filter employs 180 particles, and evolutionary strategy based Gaussian kernel particle filter uses 80 particles and performs 4 times evolutionary strategy iteration. The resolution of this sequence is 640 480× pixels, the tracked target are initialized by hand. From left to right and top to bottom in the fig. 2 and fig. 3, the figures show the 3rd, 7th, 10th, 16th frame of test sequences respectively. We can see that the first two algorithms lose the target at the 16th frame. In fig. 4, from left to right and top to bottom, figure shows the 3rd, 7th, 16th, 24th frame of test sequences respectively. We can see that evolutionary strategy based Gaussian kernel particle filter can track the target robustly throughout the whole sequence.

291

the 3rd frame the 7th frame

the 10th frame the 16th frame

Fig 2 Fast target tracking based on the traditional particle filter

the 3rd frame the 7th frame

the 10th frame the 16th frame

Fig 3 Fast target tracking based on Gaussian kernel particle filter

292

the 3rd frame the 7th frame

the 16th frame the 24th frame

Fig 4 Fast target tracking using evolutionary strategy based Gaussian kernel particle filter

From the above results, we can see that the tracking algorithm based on the traditional particle filter needs the most particles and the effect of tracking fast target is not good; compared to it, the tracking algorithm based on Gaussian kernel particle filter can use fewer particles to track slow target steadily, but not to track fast target; the tracking algorithm based on evolutionary strategy based Gaussian kernel particle filter can use the fewest particles to track slow and fast target robustly.

VI. CONCLUSION This paper mainly presents how to prevent the

impoverishment problem of particle filter using Gaussian kernel and evolutionary strategy for visual tracking. Initially, we describe the sampling algorithm of the traditional particle filter. Then we presents visual tracking algorithm based on Gaussian kernel particle filter. Finally, we introduce evolutionary strategy to further improve the Gaussian kernel particle filter. In the experiments, we respectively employ three different particle filters to track slow and fast target in the video. From the results, we can demonstrate, compared to the traditional particle filter and Gaussian kernel particle filter, evolutionary strategy based Gaussian kernel particle filter can prevent the impoverishment problem effectively and only use the fewest particles to track fast target robustly.

REFERENCES [1] Maybeck P S, Jensen R L, Harnly D A. An Adaptive Extended

Kalman Filter for Target Image Tracking, IEEE Transactions on Aerospace and Electronic Systems, 1981: 173-180

[2] Arulampalam M, Maskell S, Gordon N, Clapp T. A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking, IEEE Transactions on Signal Processing, 2002: 174-188.

[3] Isard M, Blake A. Contour Tracking by Stochastic Propagation of Conditional Density, Proceedings of 4th European Conference on Computer Vision, Springer-Verlag, 1996: 343-356.

[4] Liu J, Chen R. Sequential Monte Carlo Methods for Dynamic Systems, Journal of the American Statistical Association, 1998: 1032-1044.

[5] Amaud D, Simon G, Chistophe A. On sequential Monte Carlo sampling methods for Bayesian filtering, Statistics and Computing, 2000: 197-208.

[6] MacCormick J, Blake A. A Probabilistic Exclusion Principle for Tracking Multiple Objects, Proceedings of International Conference on Computer Vision, 1999: 572-578.

[7] MacCormick J, Blake A. Partitioned Sampling, Articulated Objects and Interface-quality Hand Tracking, Lecture Notes in Computer Science, 2000: 3-19.

[8] Gilks W R, Berzuini C. Following a moving target monte carlo inference for dynamic Bayesian models, Journal of the Royal Statistical Society B, 200l: 127-146.

[9] Clapp T C. December statistical methods for the processing of communications data, Cambridge, UK: University of Cambridge, 2000.

[10] Pitt M, Shephard N. Filtering via simulation: Auxiliary particle filters, Journal of the American Statistical Association, 1999: 590-599.

[11] Chang C, Ansari R. Kernel Particle Filter: Iterative Sampling for Efficient Visual Tracking, Proceedings of International Conference on Image Processing, 2003: 977-980.

[12] Lehn-Schioler T, Erdogmus D, Principe J C. Parzen Particle Filters, Proceedings of International Conference on Acoustics, Speech, and Signal Processing, 2004: 781-784.

[13] Goldberg D E. Genetic algorithms in search, strategy and machine learning, Addison-Wesley Pub. Co., Massachusetts, 1989.

[14] Kanazawa K, Koller D, Russell S. Stochastic simulation algorithms for dynamic probabilistic networks, Proceedings of 11th Annual Conference of Uncertainty and AI, 1995: 346-351.

[15] Higuchi T. Monte Carlo filter using the genetic algorithm operators, Journal of Statistics Computer Simulation, 1997: 1-23.

[16] Zhang B T. A Bayesian framework for evolutionary computation, Processing of the 1999 Congress on Evolutionary Computation, 1999: 722-728.

[17] Xiaoping Wang, Liming Cao. Genetic Algorithm – theory, application and software realization, 2002.

[18] Harik G, Paz E C, Miller B. The Gamblers Ruin Problem, Genetic Algorithms, and the Sizing of Populations, Proceedings of the 5th International Conference on Evolution Computation, 1997: 7-12.

[19] Baker J E. Reducing bias and inefficiency in the selection algorithm, Genetic Algorithms and Their Applications, MIT, Cambridge, MA, 1987: 14-21.

[20] Schwefel H P. Evolution and Optimum Seeking, 1995.

293