1. Introduction
Object tracking has become a very important research topic for many years since it can be used in many applications [
1,
2]. We can find a variety of approaches for object tracking in related literature [
3,
4,
5]. For object tracking problems with linear or Gaussian system models, the Kalman filter (KF) can be applied to obtain optimal solutions [
6]. For problems with nonlinear system models, many nonlinear filtering methods have been proposed, such as the extended Kalman filter (EKF) [
7,
8] and unscented Kalman filter (UKF) [
9,
10], which are usually implemented to provide Gaussian approximation to the posterior probability density function in the state space. In recent years, the particle filter [
11] has attracted a lot of researchers’ attention. A particle filter is a powerful tool for object tracking based on sequential Monte Carlo methods under a Bayesian estimation framework [
12]. The core of particle filter is to represent the probability density of object state by a set of random samples with their associated weights and how to allocate particles to a high-probability density area. Due to this sample-based method, particle filters are able to represent a variety of complex probability densities distribution with nonlinear and non-Gaussian dynamic systems. Thus, particle filters have been applied with great success to a variety of object tracking problems [
13,
14].
However, a standard particle filter (SPF) is not always satisfactory which may confront the particle degeneracy [
15]. That is to say, all but a few particles have negligible weights after a few iterations. Consequently, a large amount of computational efforts is devoted to updating particles whose contribution to the approximation to posterior probability density function is almost zero. For many years, much attention has been paid to solve this problem of particle degeneration. Some novel strategies were proposed to solve this problem, and many improved variants of SPF were presented. An important strategy to overcome this problem is to design better proposal distributions. One such approach is the auxiliary particle filter (APF) [
16] which improves some deficiencies of the SPF algorithm when dealing with tailed observation densities. However, requiring a large number of particles is its drawback. In addition to the APF, there are other methods, such as auxiliary extended and auxiliary unscented Kalman particle filters [
17], genetic particle filter (GPF) [
18], risk-sensitive particle filters (RSPF) [
19], and so on. Many new methods have been recently proposed in the last few years. Ding et al. proposed a new selective sampling importance resampling (SSIR) particle filter framework [
20]. This framework integrates an auto-regressive filter to improve the process of sample generation. Shabat et al. proposed an accelerating particle filter using randomized multi-scale and fast multi-pole type methods [
21]. The accelerating particle filter samples a small subset from the source particles using matrix decomposition methods.
In recent years, a new approach has been proposed [
22,
23]. This approach incorporates grey prediction [
24] into particles filter (called GP-PF) based on grey system theory, in order to solve the sample degeneracy problem. The basic idea of the GP-PF is that new particles are sampled by both the state transition prior and the grey prediction method. Since the grey prediction algorithm is able to predict the system state based on historical measurements other than establishing an a priori dynamic model, the GP-PF can significantly alleviate the sample degeneracy problem, which is common in SPF, especially when it is used for object target tracking. The main advantages of GP-PF lie in two aspects: one is that it can obtain the better proposal distributions and the other is that it does not require building any a priori model of the object. However, because of the inherent drawbacks of grey prediction, the weakness of GP-PF is that it cannot obtain good tracking accuracy, though GP-PF can reduce the loss rate of object tracking to the maximum extent.
In order to further improve the tracking accuracy, this paper incorporates fuzzy theory into standard grey prediction used in a particle filter for object tracking. The proposed Fuzzy-Grey-Prediction based particle filter (called FuzzyGP-PF) has both the inherent advantages of fuzzy theory and grey prediction and, thus, can improve the object tracking performances.
The rest of this paper is organized as follows: in
Section 2, we introduce the most common grey prediction model GM(1,1) (Grey Model for first-order equation with one variable) firstly, then we proposed the Fuzzy-Grey-Prediction model (FGP). In
Section 3, we introduce the particle filter briefly firstly and then we proposed the object tracking algorithm based on the FGP.
Section 4 shows the experiments and results in detail. Finally, the paper is concluded in
Section 5.
2. Fuzzy Grey Prediction
2.1. Grey Prediction Model
The most common Grey-Prediction model is GM(1,1). The concept of the grey prediction is briefly described as follows [
25]:
Let
be a non-negative reference data sequence in real space:
and take the accumulated generating operation (AGO) on
. Then the first order AGO sequence
can be described by:
where:
Then sequence
can be obtained by applying the Adjacent-Mean operation to
:
where:
The equation:
is called a grey differential equation of GM(1,1), where the parameters
a and
b are called the development coefficient and the grey input, respectively. The equation:
is called the whitening equation corresponding to the grey differential equation of Equation (6). In order to find out the solution of the above differential equation, the parameters
a and
b must he decided. They can be solved by means of the Least-Square method as follows:
where:
Once a and b in whitening Equation (7) are decided, the value of reference data sequence x at time instant k + 1 can be obtained as follows:
First, the AGO grey data sequence can be obtained:
Then, the forecasting value of x can be calculated by an IAGO (inverse accumulated generating operation):
2.2. Fuzzy-Grey-Prediction Model
Grey systems theory is developed to study problems of “small samples and poor information”. These problems studied by grey systems theory cannot be handled successfully by using either probability statistics. Grey systems theory has no special requirements and restrictions on data sequence. Thus, its application is very broad. In the research of system, because of the existence of internal and external disturbances, when the grey prediction model is used directly, the stability and precision of the prediction cannot meet the demand. Buffer operator theory is an important aspect of grey system theory and one of the main features of the theory. In this paper, a new fuzzy buffer operator is constructed based on grey system theory “the new information priority” principle and the theory of fuzzy mathematics. It can weaken some randomness to show regularity successfully by excluding the impact of external interference.
A new fuzzy grey prediction model (called FGP) is established, and the new model is applied to the particle filter applications. The basic idea of fuzzy grey prediction is modeling to the fuzzy buffer sequence using GM(1,1), and the original data sequence is converted into the fuzzy buffer sequence by using the proposed fuzzy buffer operator.
We set an original data series is a time series data
X = {
x1,
x2,…,
xn}, and we can specify a fuzzy coefficient for each moment of the data to distinguish the historical data impact on model results. The fuzzy coefficient can be decided by a fuzzy membership function. For maneuvering target tracking, this paper chooses the typical membership function of real number field R: rising half-Cauchy distribution membership function:
We can compute all of the fuzzy coefficients
S = {
s1,
s2,…,
sn} when the parameter
a,
b,
c are decided, while the choice of parameter
a,
b,
c may be problem-dependent, it needs a concrete analysis of specific situations. Then, the converted fuzzy buffer sequence can be obtained by a proposed fuzzy buffer operator as follows:
Obviously, according to Equation (12), the last value of the sequence is not changed, while the other data in the sequence is weakened or strengthened, which depends on the value of the fuzzy coefficients. Therefore, the proposed fuzzy buffer operator is simple, but very practical.
Then, modeling to the fuzzy buffer sequence using GM(1,1), the two parameters of FGP a and b can be computed, and substituting the obtained parameters a and b into GM(1,1) to predict the required result, which completes a solution process of FGP.
3. Fuzzy-Grey-Prediction Based Particle Filter
3.1. Particle Filter
In this section the proposed Fuzzy-Grey-Prediction-based particle filter algorithm (called FuzzyGP-PF) is presented. For the sake of completeness, we briefly review the particle filter.
Due to intractable integrals, a particle filter is commonly used to approximate it by a set of random particles (or samples) drawn from a probability distribution. The posterior probability density function of the target state given the observations is represented by a set of weighted particles
with their associated weights
, where
t is the time index,
i is the particle index and
N is the number of particles, i.e.:
In the processing of particle filters, the approximation can be achieved by performing the following four recursive steps, namely sampling, calculating particle weighting, state estimation, and resampling:
Step 1: Sampling: generating new particles for , where is the proposal distribution. The most popular choice of the proposal distribution is transition prior due to its simplicity.
Step 2: Calculating weight: the weight
to the particle
is determined by its observation likelihood, i.e.:
Step 3: State estimation: the mean state of the new particle set, which specifies the position of object being tracked, can be calculated using a minimum mean square error (MMSE) estimator as follows:
Step 4: Resampling: resampling is an auxiliary step, which is used to alleviate the particle degeneracy problem inevitably encountered over time [
9]. Drawing new particles
from the above set of particles
based on the particle weights according to a resampling algorithm. It is implemented by multiplying particles with high weights multiple times and diminishing particles with relatively low weights.
3.2. Proposed Tracking Algorithm
The key idea of the proposed tracking algorithm is that the history state sequence is utilized as prior information to predicting and sampling a part of particles for generating proposal distribution in particle filter. That is to say sampling particles can be achieved by following two ways: one is a few of the particles sampled from the proposal distribution produced by predicting the state of the object using FGP, and the other is sampled by the SPF, as in the operation of the state transition prior.
The outline of the tracking algorithm FuzzyGP-PF as follows, where N is the total number of particles, and Ngrey is the number of particles generated by fuzzy grey prediction; L is the length of the data series used for fuzzy grey prediction algorithm.
The proposed tracking algorithm |
for , initialization |
Initialize particles ; |
end for |
for |
if , normal Sampling |
for |
Generate samples according to motion model of target tracking; |
end for |
else |
for |
Generate samples by making a prediction for state of target using fuzzy grey prediction; |
end for |
for |
Normal sampling, generate samples according to motion model of target tracking; |
end for |
end if |
for , calculating the weight of particles, ; end for |
for , normalize the weight: ; end for |
State estimation: |
Resample to obtain a new particle set |
end for |