A New Method of Anti-interference Matching Under Foreground Constraint for Target Tracking
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摘要: 传统模型匹配跟踪方法没有充分考虑目标与所处图像的关系，尤其在复杂背景下，发生遮挡时易丢失目标.针对上述问题，提出一种前景约束下的抗干扰匹配（Anti-interference matching under foreground constraint，AMFC）目标跟踪方法.该方法首先选取图像帧序列前m帧进行跟踪训练，将每帧图像基于颜色特征分割成若干超像素块，利用均值聚类组建簇集合，并通过该集合建立判别外观模型；然后，采用EM（Expectation maximization）模型建立约束性前景区域，通过基于LK（Lucas-Kanade）光流法框架下的模型匹配寻找最佳匹配块.为了避免前景区域中相似物体的干扰，提出一种抗干扰匹配的决策判定算法提高匹配的准确率；最后，为了对目标的描述更加准确，提出一种新的在线模型更新算法，当目标发生严重遮挡时，在特征集中加入适当特征补偿，使得更新的外观模型更为准确.实验结果表明，该算法克服了目标形变、目标旋转移动、光照变化、部分遮挡、复杂环境的影响，具有跟踪准确和适应性强的特点.Abstract: The relation between a moving target and its image has not been fully considered in traditional model-matching tracking methods. The tracking drift problem may frequently occur when the target is occluded under a complex background. In this paper, a novel target tracking method, anti-interference matching under foreground constraint (AMFC), is proposed to solve this kind of problem. First, the method selects several initial frames from a vedio sequence for tracking training. Each of these frames is divided into several super-pixel blocks based on its color feature. These super-pixel blocks are combined into cluster sets by a mean shift algorithm to construct a discrimination appearance model. Then, a constrained foreground region is established using the expectation maximization (EM) model and a matching process is conducted based on the Lucas-Kanade (LK) optical flow method in order to select the optimum matching block. A decision-making algorithm is introduced to avoid the interference caused by similar targets in the foreground region, so as to increase the accuracy of the matching process. Moreover, in order to provide a more accurate target representation, an algorithm for appearance model online-updating is proposed. When a severe occlusion occurs, this algorithm can append appropriate feature compensations to the feature sets to improve the accuracy of the appearance model. Experimental results indicate that the proposed approach can provide superior tracking accuracy and adaptability, especially in the context of target deformation, target rotational movements, illumination changes, partial occlusion, and complex background.
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1. Introduction
Fractional calculus and fractional differential equations have received considerable interest in the recent forty years. Fractional derivative means that the order of differentiation can be an arbitrary real number and even it can be a complex number. Fractional derivative modelling has been applied to many scientific and engineering fields, such as quantum mechanics [1], viscoelasticity and rheology [2], electrical engineering [3], electrochemistry [4], biology [5], biophysics and bioengineering [6], signal and image processing [7], mechatronics [8], and control theory [9]-[11]. Although few mathematical issues of fractional derivative remain unsolved, most of the difficulties have been overcome, and the applications of fractional calculus in above fields indicate that the fractional models can depict the property and behavior of a real-world problem more accurately. For a comprehensive review of fractional calculus, we refer readers to some monographs [12]-[14] and references therein. In contrast to integer order derivative, the way of identifying fractional derivative is not unique. There are several types of definitions, such as Riemann-Liouville derivative, Caputo derivative, Grünwald-Letnikov derivative, and so on. More details can be found in [13, Chapter 2]. In the recent years, the study of dynamical system with fractional order derivative becomes more and more popular [15]-[19]. Moreover, the dynamics in fractional dynamical system is more interesting.
Returning back to the fractional derivative, since it has several different definitions, how to develop a generalized form which can unify all the existing fractional derivatives becomes one important topic in fractional calculus [20]-[22]. Recently, a class of new generalized fractional integral and generalized fractional derivative is introduced in [22]. The new generalized fractional integral and generalized fractional derivative depend on a scale function and a weight function, which makes them more general. When the scale function and the weight function reduce to some specific cases, the generalized fractional operators will reduce to Riemann-Liouville fractional integral, Riemann-Liouville fractional derivative and Caputo fractional derivative and so on. However, the study of this new generalized fractional integral and generalized fractional derivative are in the very beginning stage now [23]-[26]. In [24], we show that in generalized fractional diffusion equation, the scale function allows the response domain to be scaled differently. It is required that the scale function should be strictly monotonically increasing or decreasing. A convex increasing scale function will compress the response domain towards to the initial time. A concave increasing scale function will stretch the response domain away from the initial time. The weight function allows the response to be assessed differently at different time, since in many applications, we may require an event to be weighed differently at different time point. For example, modeling of memory of a child may require a heavy weight at current time point, whereas the same for an older person may require more weight on the past. To be an initial attempt of application to chaotic dynamical systems, in this paper, we define a class of new generalized fractional chaotic systems by replacing the original derivatives with the new generalized fractional derivative, then apply a finite difference scheme to study the numerical solutions of two different generalized fractional chaotic systems, namely generalized fractional Lotka-Volterra system (GFLVS) and generalized fractional Lorenz system (GFLS). Their complex dynamics will be discussed, and the dynamic behavior depending on the weight and scale function will be shown graphically.
The rest of this paper is organized as follows: In Section 2, the preliminaries of fractional calculus are given. The new generalized fractional integral and generalized fractional derivative are shown. A finite difference approach for solving equations with generalized fractional derivative is carried out. In Section 3, we define the chaotic systems using the generalized fractional derivative of Caputo type, i.e., the GFLVS and GFLS. Some interesting dynamics of those two systems are shown graphically. Finally, the conclusions are drawn in Section 4.
2. Mathematical Preliminaries
In this section, we introduce the preliminaries of generalized fractional derivatives, and show a proper numerical method for differential equations with such derivatives.
2.1 Generalized Fractional Calculus
Let us begin with the common fractional operators. In calculus, the $n$ -fold integral of an integrable function $u(t)$ is defined as
$ I^{n}u(t)=\overbrace{\int^t_0\cdots\int^t_0}^{n\ {\rm times}}u(s)ds\cdots{ds}= \int^t_0\frac{(t-s)^{n-1}}{(n-1)!}u(s)ds $
where $t\geq{0}$ , and $u(0)$ is well-defined. Replacing the positive integer $n$ by a real number $\alpha>0$ , we have the following definition.
Definition 1[13]: The left Riemann-Liouville fractional integral of order $\alpha>0$ of a function $u(t)$ is defined as
$ \begin{align} \left(I^{\alpha}_{0+}u\right)(t) = \frac{1}{\Gamma(\alpha)}\int^t_0(t-s)^{\alpha-1}u(s)ds \end{align} $
(1) provided the integral is finite, where $\Gamma(\alpha)$ is the Gamma function.
The Riemann-Liouville fractional integral plays an important role in defining fractional derivatives. There are two basic approaches to define the fractional derivative, i.e., "first integration then differentiation" and "first differentiation then integration". The corresponding fractional derivatives are called Riemann-Liouville fractional derivative and Caputo fractional derivative, and the definitions are given as follows.
Definition 2[13]:The left Riemann-Liouville fractional derivative of order $n-1 < \alpha < n$ of a function $u(t)$ is defined as
$ \begin{align} \left(D^{\alpha}_{0+}u\right)(t) = \frac{1}{\Gamma(n-\alpha)}\left(\frac{d^n}{dt^n}\right) \int^t_0(t-s)^{n-\alpha-1}u(s)ds \end{align} $
(2) provided the right side of the identity is finite.
Definition 3[13]: The left Caputo fractional derivative of order $n-1 < \alpha < n$ of a function $u(t)$ is defined as
$ \begin{align} \left({^cD}^{\alpha}_{0+}u\right)(t) = \frac{1}{\Gamma(n-\alpha)}\int^t_0(t-s)^{n-\alpha-1}u^{(n)}(s)ds \end{align} $
(3) provided the right side of the identity is finite.
Besides above, there also exist right Riemann-Liouville integral and derivative, and right Caputo fractional derivative [13]. Mathematically, the Riemann-Liouville and Caputo fractional operators are used in applications frequently. In most real-world models, we always employ the left Caputo fractional derivative. One reason is that we will study generalized fractional dynamical system later, and the derivative is taken with respect to time variable. In physical models, time is always running forward. The other reason is that in the differential equations with Caputo fractional derivative, the initial conditions are taken in the same form as for integer-order differential equations which have clear physical meanings in the practical application and can be easily measured [14]. In what follows, we will introduce the generalized fractional integral and derivative proposed in [22]. They extend nearly all the existing fractional operators. Now we list the generalized fractional integral and derivative defined on positive half axis. They will be used to define the generalized fractional chaotic systems in next section.
Definition 4[22]: The left generalized fractional integral of order $\alpha>0$ of a function $u(t)$ with respect to a scale function $\sigma(t)$ and a weight function $w(t)$ is defined as
$ \begin{align} \left(I^{\alpha}_{0+;[\sigma, w]}u\right)(t) = \frac{[w(t)]^{-1}}{\Gamma(\alpha)}\int^{t}_{0} \frac{w(s)\sigma'(s)u(s)}{[\sigma(t)-\sigma(s)]^{1-\alpha}}ds \end{align} $
(4) provided the integral exists, where $\sigma'(s)$ indicates the first derivative of the scale function $\sigma$ .
Definition 5[22]: The left generalized derivative of order $m$ of a function $u(t)$ with respect to a scale function $\sigma(t)$ and a weight function $w(t)$ is defined as
$ \begin{align} \left(D^m_{[\sigma, w;L]}u\right)(t) = [w(t)]^{-1}\left[\left(\frac{1}{\sigma'(t)}D_t\right)^m(w(t)u(t))\right] \end{align} $
(5) provided the right-side of equation is finite, where $m$ is a positive integer.
Definition 6[22]: The Caputo type left generalized fractional derivative of order $\alpha>0$ of a function $u(t)$ with respect to a scale function $\sigma(t)$ and a weight function $w(t)$ is defined as
$ \begin{align} \left(D^{\alpha}_{0+;[\sigma, w]}u\right)(t) = \left(I^{m-\alpha}_{0+;[\sigma, w]}D^m_{[\sigma, w;L]}u\right)(t) \end{align} $
(6) provided the right-side of equation is finite, where $m-1\leq$ $\alpha$ $ < $ $m$ , and $m$ is a positive integer. Particularly, when $0 < $ $\alpha$ $ < $ $1$ , we have
$ \begin{align} \left(D^{\alpha}_{0+;[\sigma, w]}u\right)(t) = \frac{[w(t)]^{-1}}{\Gamma(1-\alpha)}\int^{t}_{0} \frac{[w(s)u(s)]'}{[\sigma(t)-\sigma(s)]^{\alpha}}ds. \end{align} $
(7) 2.2 Finite Difference Method
Now we introduce a finite difference method for solving differential equations with generalized fractional derivative. Consider the following generalized fractional differential equation:
$ \begin{align} \begin{cases} \left(D^{\alpha}_{0+;[\sigma, w]}u\right)(t)=f(t, u(t)), \quad 0 < t\leq T\\ u(0)=u_0 \end{cases} \end{align} $
(8) where $0 < \alpha < 1$ and $T$ is the final time. Without loss of generality, on a uniform mesh $0=t_0 < t_1 < \cdots < $ $t_j < $ $t_{j+1} < \cdots < t_N=T$ , the Caputo type generalized fractional derivative of $u(t)$ can be approximated as
$ \begin{align} (D^{\alpha}_{0+;[\sigma, w]}& u)(t_{j+1}) \nonumber\\ &= \frac{[w(t_{j+1})]^{-1}}{\Gamma(1-\alpha)}\int^{t_{j+1}}_{0}\frac{[w(s)u(s)]'} {[\sigma(t_{{j+1}})-\sigma(s)]^{\alpha}}ds \nonumber\\ &= \frac{w_{j+1}^{-1}}{\Gamma(1-\alpha)}\sum^{j}_{k=0}\int^{t_{k+1}}_{t_k} \frac{[w(s)u(s)]'}{\left[\sigma(t_{{j+1}})-\sigma(s) \right]^{\alpha}}ds \nonumber\\ & \approx \frac{w_{j+1}^{-1}}{\Gamma(1-\alpha)}\sum^{j}_{k=0}\int^{t_{k+1}}_{t_k} \frac{\frac{w_{k+1}u_{k+1}-w_ku_k}{t_{k+1}-t_k}} {\left[\sigma_{j+1}-\sigma(s)\right]^{\alpha}}ds \nonumber\\ & \approx\sum^{j}_{k=0}\left(A^j_ku_{k+1}-B^j_ku_k\right) \end{align} $
(9) where
$ \begin{align*} A^j_k =&\ \frac{w^{-1}_{j+1}w_{k+1}}{\Gamma(2-\alpha)(\sigma_{k+1}-\sigma_k)} \\ & \times \left[(\sigma_{j+1}-\sigma_k)^{1-\alpha}- (\sigma_{j+1}-\sigma_{k+1})^{1-\alpha}\right]\\ B^j_k =&\ \frac{w^{-1}_{j+1}w_{k}}{\Gamma(2-\alpha)(\sigma_{k+1}-\sigma_k)} \\ & \times \left[(\sigma_{j+1}-\sigma_k)^{1-\alpha}- (\sigma_{j+1}-\sigma_{k+1})^{1-\alpha}\right] \end{align*} $
$k=0, 1, 2, \ldots, j$ , $u_j=u(t_j)$ , $w_j=w(t_j)$ , and $\sigma_j=\sigma(t_j)$ .
Therefore, we obtain the finite difference scheme:
$ \begin{align} \sum^{j}_{k=0}\left(A^j_ku_{k+1}-B^j_ku_k\right)=f(t_{j+1}, u_{j+1}) \end{align} $
(10) and the corresponding iteration scheme as
$ \begin{align} u_{j+1}=\begin{cases} \frac{1}{A^j_j}\left[f_j-\sum\limits^{j-1}_{k=0} \left(A^j_ku_{k+1}-B^j_ku_k\right)+B^j_ju_j \right], \\ \qquad \qquad \qquad \qquad \qquad\qquad j=1, 2, \ldots, N-1\\ \frac{1}{A^0_0}\left(f_0+B^0_0u_0\right), \qquad\qquad \ \, j=0 \end{cases} \end{align} $
(11) where $f_j=f(t_j, u_j)$ .
In what follows, we will apply this method to solve the generalized fractional chaotic systems. The numerical analysis of the above scheme can be found in [26].
3. Dynamic Behavior of Generalized Fractional Chaotic Systems
In this section, we introduce two nonlinear dynamical systems but redefine them with Caputo type generalized fractional derivative. The classical and fractional senses are special cases of the new generalized fractional system below.
3.1 Generalized Fractional Lotka-Volterra and Generalized Fractional Lorenz System
Replacing the derivative with the generalized fractional derivative defined by (7), we define the generalized fractional Lotka-Volterra system (GFLVS) as
$ \begin{align} \begin{cases} D^{\alpha_1}_{0+;[\sigma, w]}x = ax - bxy + mx^2 - sx^2z\\ D^{\alpha_2}_{0+;[\sigma, w]}y = -cy +dxy\\ D^{\alpha_3}_{0+;[\sigma, w]}z = -pz + sx^2z \end{cases} \end{align} $
(12) where $0 < \alpha_1, \alpha_2, \alpha_3 < 1$ ( $\alpha_1$ , $\alpha_2$ , $\alpha_3$ can be the equal or different) are the orders of the derivative and parameters $a$ , $b$ , $c$ , $d$ are positive. $a$ represents the natural growth rate of the prey in the absence of predators, $b$ represents the effect of predator on the prey, $c$ represents the natural death rate of the predator in the absence of prey, $d$ represents the efficiency and propagation rate of the predator in the presence of prey, and $m$ , $p$ , $s$ are positive constants.
By selecting the parameters $a=1$ , $b=1$ , $c=1$ , $d=1$ , $m$ $=$ $2$ , $s=2.7$ , $p=3$ and the initial condition $[x_0, y_0, z_0]$ $=$ $[1.5, 1.5, 1.5]$ , when $\alpha_1=\alpha_2=\alpha_3=0.95$ , (12) represents the generalized fractional Lotka-Volterra chaotic system and the phase portraits of the system (12) are described through Figs. 1(a) and 1(b). In Fig. 1(a), the chaotic phenomenon is shown. Moreover, the GFLVS reduces to the fractional Lotka-Volterra system as $\sigma(t)=t$ and $w(t)=1$ . In Fig. 1(b), we see that when the scale function is specified as a power function, and the weight function is taken as an exponential function, the chaotic attractor vanishes and then a stable equilibrium point appears.
图 1 Phase portraits of GFLVS (top row, (a) and (b)) and GFLS (bottom row, (c) and (d)).Fig. 1 Phase portraits of GFLVS (top row, (a) and (b)) and GFLS (bottom row, (c) and (d)).Similarly, we define the generalized fractional Lorenz system (GFLS) as
$ \begin{align} \begin{cases} D^{\alpha_1}_{0+;[\sigma, w]}x = r(y-x)\\ D^{\alpha_2}_{0+;[\sigma, w]}y = x(\rho-z)-y\\ D^{\alpha_3}_{0+;[\sigma, w]}z = xy-\beta{z} \end{cases} \end{align} $
(13) where $r$ is the Prandtl number, $\rho$ is the Rayleigh number and $\beta$ is the size of the region approximated by the system. The fractional order $0 < \alpha_1, \alpha_2, \alpha_3 < 1$ may take different values.
By taking the parameters $r=10$ , $\rho=28$ , $\beta= {8}/{3}$ , and the initial condition $[x_0, y_0, z_0]=[0.5, 0.5, 0.5]$ , when $\alpha_1$ $=$ $\alpha_2=\alpha_3=0.99$ , (13) represents the generalized fractional Lorenz chaotic system and the phase portraits of the system (13) are described through Figs. 1(c) and 1(d). In Fig. 1(c), the chaotic attractor of fractional Lorenz system is presented. When we take scale function as a power function, and weight function as exponential function, the GFLS remains chaotic. However, the shape of the attractor changes, which is shown in Fig. 1(d).
3.2 Analysis of the Influence of Scale and Weight Functions
Now we analyze the influence of the scale and weight functions on the responses of generalized fractional differential equation. For simplicity, we consider
$ \begin{align} D^{\alpha}_{0+;[\sigma, w]}u(t) = Au(t) + f(t) \end{align} $
(14) where $A\neq{0}$ is a constant.
Equation (14) is equivalent to
$ \begin{align} \frac{[w(t)]^{-1}}{\Gamma(1-\alpha)}\int^{t}_{0} \frac{[w(s)u(s)]'}{[\sigma(t)-\sigma(s)]^{\alpha}}ds = Au(t) + f(t). \end{align} $
(15) Let $v(t) = w(t)u(t)$ , we have
$ \begin{align} \frac{1}{\Gamma(1-\alpha)}\int^{t}_{0} \frac{v(s)'}{[\sigma(t)-\sigma(s)]^{\alpha}}ds = Av(t) + w(t)f(t). \end{align} $
(16) According to [13], we deduce the solution of (16) as:
$ \begin{align} v(t) =&\ E_{\alpha}\left(A[\sigma(t)-\sigma(0)]^{\alpha}\right)v_0 \nonumber\\ & +\int^t_0(\sigma(t)-\sigma(s))^{\alpha-1} \nonumber\\ &\times E_{\alpha, \alpha}[A(\sigma(t)-\sigma(s))^{\alpha}]w(s)f(s)ds \end{align} $
(17) which implies that
$ \begin{align} u(t)=&\ \frac{w(0)}{w(t)}E_{\alpha}\left(A[\sigma(t)-\sigma(0)]^{\alpha}\right)u_0 \nonumber\\ & +\frac{1}{w(t)}\int^t_0(\sigma(t)-\sigma(s))^{\alpha-1} \nonumber\\ & \times E_{\alpha, \alpha}[A(\sigma(t)-\sigma(s))^{\alpha}]w(s)f(s)ds \end{align} $
(18) where $u_0$ is the initial condition, and $E$ is the Mittag-Leffler function.
In (18), we observe that how the weight and scale functions influence the behavior of (14). First of all, the weight function cannot be zero in the domain, otherwise solution $u(t)$ will go to infinity. Second, the scale function cannot be periodic, and if it is, the generalized fractional derivative will be infinity at $t=s$ . For an intuitive comprehension, we present some numerical simulations in the following.
3.3 Dynamics of GFLVS and GFLS Depend on Scale and Weight Functions
The fractional chaotic systems are sufficiently generalized by using the generalized fractional derivative, since many existing fractional derivatives, as well as integer order derivatives, are special cases of the generalized fractional derivative. In our numerical experiments, we find many interesting dynamical behaviors of generalized fractional chaotic systems which are never found in common fractional or integer order chaotic systems. Here we present some particular simulation results. However, our discussion depends on Figs. 2 and 3, and others figures are not shown here.
图 2 Influence of scale function $\sigma(t)$ on GFLVS (top row, (a) and (b)) and GFLS (bottom row, (c) and (d)).Fig. 2 Influence of scale function $\sigma(t)$ on GFLVS (top row, (a) and (b)) and GFLS (bottom row, (c) and (d)).
图 3 Influence of weight function $w(t)$ on GFLVS (top row, (a) and (b)) and GFLS (bottom row, (c) and (d)).Fig. 3 Influence of weight function $w(t)$ on GFLVS (top row, (a) and (b)) and GFLS (bottom row, (c) and (d)).First, we simulate the influence of scale function on dynamics of chaotic systems. In GFLVS, we take fractional order $\alpha_1=\alpha_2=\alpha_3=0.95$ , weight function $w(t)=\exp(1.2t)$ , and other parameters are the same as before. In GFLS, we select fractional order $\alpha_1=\alpha_2=\alpha_3=0.99$ , weight function $w(t)=\exp(0.1t)$ , and other parameters are the same as before. The dynamic behaviors of GFLVS and GFLS with scale function $\sigma(t)=t$ and $t^{1.14}$ are individually presented in Fig. 2.
Second, we simulate the influence of weight function on dynamics of chaotic systems. In GFLVS, we take fractional order $\alpha_1=\alpha_2=\alpha_3=0.95$ , scale function $\sigma(t)=t$ , and other parameters are the same as before. In GFLS, we select fractional order $\alpha_1=\alpha_2=\alpha_3=0.99$ , scale function $\sigma(t)=t$ , and other parameters are the same as before. The dynamic behaviors of GFLVS with weight function $w(t)$ $=$ $\exp(0.8t)$ , $\exp(1.3t)$ , and GFLS with weight function $w(t)=\exp(2+0.5t)$ and $\exp(2+0.2t)$ are presented in Fig. 3.
Finally, to end this section, we make some remarks based on the numerical experiments above. Some other figures are not listed here for shortening the length of paper.
1) The GFLVS is chaotic with scale function $\sigma(t)=t$ , weight function $w(t)$ is a nonzero constant, and fractional order $\alpha_i=0.95$ , $i=1, 2, 3$ [27]. However, From Fig. 1(a), Fig. 2(a) and Fig. 3(a), we may see that as the weight function varies, the chaotic attractor vanishes and then a limit cycle emerges or the system converges to a stable equilibrium point. Furthermore, from Fig. 2(a) and Fig. 2(b), we observe that as the scale function varies, the limit cycle tends to be a stable equilibrium point. From Fig. 3(a) and Fig. 3(b), it is shown that as the weight function varies, the limit cycle can be generated from a stable equilibrium point.
2) The GFLS is chaotic with scale function $\sigma(t)=t$ , weight function $w(t)$ is a nonzero constant, and fractional order $\sum^{3}_{i=1}\alpha_i>2.91$ [28]. In simulation, on one hand, Figs. 1(c) and 1(d), indicate that with suitable scale and weight functions, the GFLS also has a chaotic attractor. On the other hand, Fig. 1(c), Fig. 1(d), Fig. 2(c), Fig. 2(d), and Fig. 3(d) imply that the scale and weight functions can influence the shape and position of chaotic attractor. From Figs. 3(c) and 3(d), we observe that with some suitable weight function, the chaotic attractor tends to be an asymptotically stable equilibrium point.
3) Our previous work [23]-[26] verified that in generalized fractional integral and generalized fractional derivative, the basic property of scale function $\sigma(t)$ is that it changes the time axis, which means that if the time domain is specified as $[0, T]$ , then the response of the dynamical system is obtained over $[\sigma(0), \sigma(T)]$ , provided the scale function is monotone increasing. Since the chaotic dynamical systems are sensitive to the initial conditions, when we take different scale functions in generalized fractional chaotic system, many different dynamical behaviors will be drawn.
4) A similar observation to weight function can be found in [23]-[26], which shows that in generalized fractional integral and generalized fractional derivative, the basic property of weight function $w(t)$ is that it puts different weights for function in different positions of domain. The classical fractional operators have memory property which makes them excellent tools to model the diffusion process with heredity. Generally, in left Caputo type generalized fractional derivative, the monotonic increasing weight function is coincident with the inner memory property of fractional operator, while the monotonic decreasing weight function can destroy this inner property. One can also follow our numerical method and try other scale and weight functions in numerical experiments.
5) In Figs. 2 and 3, one can observe that both changing the scale and weight functions make the systems change between different dynamical behavior (e.g., limit cycle and stable equilibrium point). These phenomena can be regarded as general cases for generalized fractional chaotic systems. We shall guess that either scale function or weight function would influence the dynamics of generalized fractional chaotic systems. In Fig. 2, the weight function is fixed so that the influence of scale function on GFLVS and GFLS is presented. Similarly, in Fig. 3, the scale function is fixed so that the influence of weight function on GFLVS and GFLS is shown. From (18), we clearly see that the scale function plays an important role in scaling the long time behavior of dynamics since it is located in the generalized exponential function, and the weight function provides a different average since it lies inside the integral, and it is a variable coefficient simultaneously. Apparently, the behavior of function $u$ depends on the changing of scale and weight functions.
4. Conclusions
In this paper, we presented a class of new generalized fractional chaotic system, using the new generalized fractional derivative proposed recently. Many dynamical systems with integer or fractional order derivatives can be extended by replacing the derivative with the generalized fractional derivative. Therefore, the new generalized fractional dynamical systems considered in this paper can exhibit more complex dynamic behaviors. In simulations, we show that the dynamical behaviors of such systems not only depend on fractional order, but also depend on the scale and weight functions.
Acknowledgement: The author is grateful to Professor O. P. Agrawal (SIUC, USA) for introducing him theory of generalized fractional calculus, suggesting the basic idea of this paper, as well as his kind help and continuous encouragement in the recent years.
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图 3 $\sigma$值与平均中心误差之间的关系
Fig. 3 The relationship between the $\sigma$ value and average center error
图 8 11种跟踪算法在12组图像序列中的跟踪结果
Fig. 8 Tracking results of the 11 algorithms in the 12 image sequences
表 1 实验图像序列信息
Table 1 The information of the test image sequences
图像序列 光照变化 遮挡 形变 复杂背景 旋转 Girl $\surd$ $\surd$ Deer $\surd$ $\surd$ Bird2 $\surd$ $\surd$ Football $\surd$ $\surd$ Lemming $\surd$ $\surd$ $\surd$ $\surd$ Woman $\surd$ $\surd$ Bolt $\surd$ $\surd$ $\surd$ CarDark $\surd$ David1 $\surd$ $\surd$ $\surd$ David2 $\surd$ Singer1 $\surd$ $\surd$ $\surd$ Basketball $\surd$ $\surd$ $\surd$ $\surd$ 
下载: 导出CSV
表 2 不同跟踪算法的平均中心误差
Table 2 Average center errors of different tracking algorithms
图像序列 ASLA FRAG SCM VTD L1APG CT OAB TLD LOT SPT AMFC Girl 36.76 24.27 $\mathbf{3.47}$ 8.64 25.51 19.45 4.68 7.66 20.28 4.73 4.76 Deer $\mathbf{6.74}$ 87.64 37.62 7.93 38.76 52.13 16.77 25.34 29.44 28.76 11.92 Bird2 20.12 14.96 11.53 46.24 25.44 37.87 26.16 10.23 47.61 $\mathbf{6.32}$ 6.97 Football 16.62 15.38 9.64 13.57 11.31 17.44 9.28 13.82 7.15 11.27 $\mathbf{6.28}$ Lemming 47.52 112.63 75.43 60.73 142.28 37.82 73.74 32.68 14.48 $\mathbf{9.41}$ 11.39 Woman 76.57 118.28 22.24 107.69 115.57 104.58 30.94 137.47 118.41 23.36 $\mathbf{21.71}$ Bolt 62.42 73.62 9.37 49.17 132.48 38.74 129.56 141.24 17.62 15.23 $\mathbf{8.44}$ CarDark 5.30 6.23 6.32 28.72 $\mathbf{3.44}$ 18.70 39.86 21.36 24.18 21.58 8.35 David1 $\mathbf{3.57}$ 84.41 4.38 48.95 5.76 9.69 31.26 8.92 37.84 23.29 9.36 David2 8.94 67.51 6.72 3.51 $\mathbf{3.23}$ 69.83 36.34 6.73 3.97 8.48 9.21 Singer1 45.62 57.83 17.85 12.53 49.84 31.07 36.27 22.17 16.64 $\mathbf{10.14}$ 10.25 Basketball 106.62 18.42 116.24 9.48 84.47 79.41 37.13 95.46 127.83 12.38 $\mathbf{7.73}$ 平均 36.40 56.77 26.73 33.10 53.17 43.06 39.33 43.59 38.79 14.58 9.70 注：粗体为最优结果.
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表 3 不同跟踪算法的跟踪重叠率
Table 3 Tracking overlap ratio of different tracking algorithms
图像序列 ASLA FRAG SCM VTD L1APG CT OAB TLD LOT SPT AMFC Girl 0.31 0.41 $\mathbf{0.74}$ 0.69 0.39 0.29 0.73 0.56 0.43 0.71 0.73 Deer $\mathbf{0.69}$ 0.11 0.47 0.63 0.45 0.34 0.56 0.41 0.50 0.53 0.61 Bird2 0.66 0.70 0.74 0.41 0.63 0.44 0.58 0.80 0.43 $\mathbf{0.84}$ 0.83 Football 0.61 0.63 0.71 0.65 0.68 0.62 0.68 0.65 0.73 0.69 $\mathbf{0.75}$ Lemming 0.71 0.43 0.53 0.57 0.39 0.74 0.56 0.77 0.83 $\mathbf{0.86}$ 0.86 Woman 0.27 0.14 0.59 0.15 0.15 0.17 0.48 0.12 0.14 0.59 $\mathbf{0.61}$ Bolt 0.53 0.47 0.76 0.55 0.23 0.58 0.20 0.16 0.72 0.73 $\mathbf{0.77}$ CarDark $\mathbf{0.82}$ 0.82 0.81 0.43 $\mathbf{0.82}$ 0.72 0.38 0.46 0.44 0.43 0.79 David1 $\mathbf{0.83}$ 0.23 0.82 0.53 0.80 0.77 0.57 0.79 0.55 0.62 0.76 David2 0.68 0.21 0.69 0.73 $\mathbf{0.74}$ 0.02 0.33 0.69 0.73 0.68 0.64 Singer1 0.58 0.55 0.73 0.74 0.57 0.64 0.63 0.69 0.72 $\mathbf{0.76}$ 0.76 Basketball 0.19 0.63 0.17 0.67 0.25 0.27 0.59 0.21 0.14 0.68 $\mathbf{0.69}$ 平均 0.57 0.45 0.65 0.56 0.51 0.47 0.52 0.53 0.53 0.68 0.73
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表 4 不同跟踪算法的平均运行速度
Table 4 Average running speeds of different tracking algorithms
图像序列 ASLA FRAG SCM VTD L1APG CT OAB TLD LOT SPT AMFC Girl 5.31 6.32 0.65 2.74 1.76 38.21 17.51 26.84 0.79 0.47 3.68 Deer 6.24 4.78 0.97 2.67 1.64 31.63 14.72 27.17 0.83 0.41 3.56 Bird2 5.74 5.43 0.67 2.58 1.49 27.06 9.94 26.58 0.65 0.56 3.09 Football 6.15 5.68 0.61 3.14 1.61 36.73 19.67 26.63 0.93 0.76 3.41 Lemming 6.78 6.27 0.69 2.77 1.75 28.15 10.46 26.70 0.71 0.37 3.39 Woman 8.45 6.41 0.57 2.16 1.53 32.42 11.31 26.32 0.66 0.43 4.37 Bolt 7.04 3.97 0.46 2.21 1.63 27.18 8.66 24.74 0.61 0.29 3.06 CarDark 7.23 4.23 0.48 2.49 1.74 26.79 10.35 25.13 0.59 0.36 2.95 David1 7.84 4.48 0.53 3.47 2.03 34.22 14.75 26.47 0.67 0.54 3.86 David2 5.87 5.25 0.48 2.68 1.44 36.36 16.68 25.89 0.73 0.66 2.97 Singer1 5.31 4.96 0.52 2.91 1.73 28.19 10.43 26.31 0.71 0.43 3.42 Basketball 7.91 6.23 0.89 2.34 2.04 25.81 9.09 24.53 0.62 0.34 2.89
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