## 2 Combining Colour Characteristics  
### Problem Formulation  
In this paper, the detection principle is utilized for tracking. The main objective is to obtain a classifier that can discriminate the object of interest from its ambient environment in real time when a new frame is received. In frame \(t\), the rectangle \(p_{t}\) represents the object position in picture \(x_{t}\) which is selected from a collection \(C_{t}\) to maximize a fraction:  
\[p_{t}=\arg\max_{p\in C_{t}}f\left(T\left(x_{t},p\right);\theta_{t-1}\right) \tag{1}\]  
where \(f(T(x,p);\theta)\) denotes a fraction of the rectangular window \(p\) in picture \(x\) with the parameters \(\theta\) of the model, and the function \(T\) denotes an image transformation. Moreover, the parameters of the  
Figure 1: Procedure for Our Approach. In Figure 1, the blue line is template-related. In frame \(t\), the HOG features are extracted from the estimated location and used to update the numerator \(A_{t}^{1}\) and denominator \(B_{t}\) in (14). In frame \(t+1\), features are extracted from the predicted target location \(p_{t}\) and convolved with \(H^{t}\) to obtain the template response via (15). The green line is histogram-related. In frame \(t\), features of the target object and background are used to update \(\rho_{t}(O)\) and \(\rho_{t}(B)\) in (20); thus, the coefficient \(\beta_{t}\) in (19) can be obtained. The histogram response is computed via (5). Then, the template response and the histogram response are combined via (3) to obtain the integrated response map without adjusting the scale. The orange line is scale-related. In frame \(t\), a scale-space filter is trained by using the feature from previous location \(p_{t}\). In frame \(t+1\), the scale filter is combined with the response from (3) to obtain the final response \(p_{t+1}\). The red line is high-confidence judgement-related. The high-confidence mechanisms in (21) and (22) are used to judge whether updating model in the current frame is necessary for the prevention of model corruption. model should be selected to minimize the loss function \(L\left(\theta;\chi_{t}\right)\) based on the foregoing pictures and the positions of the target object in these pictures \(\chi_{t}=\left\{\left(x_{i},p_{i}\right)\right\}_{i=1}^{t}\):  
\[\theta_{t}=\arg\min_{\theta\in O}\left\{L\left(\theta;\chi_{t}\right)+\gamma R (\theta)\right\} \tag{2}\]  
The space of model parameters \(\theta\) is represented by \(O\). The regularization term \(R(\theta)\) with weight coefficients \(\gamma\) used to restrict the model complexity and to avoid over-fitting. To realize real-time performance, the problems in \((1)\) and \((2)\) should be covered well, and the functions \(f\) and \(L\) should be selected to render the location of the target object reliable and accurate. A fraction function is proposed which is a linear composition of histogram and template fractions in which the template fraction is obtained from the HOG feature and the histogram fraction is obtained from the CN feature:  
\[f(x)=\gamma_{tmpl}f_{tmpl}(x)+\gamma_{hist}f_{hist}(x) \tag{3}\]  
The template fraction is a linear function of a N-channel feature picture \(\varphi_{x}:\Gamma\rightarrow\mathbb{R}^{N}\), that is acquired from \(x\) and acts on a finite area \(\Gamma\subset\mathbb{Z}^{2}\). The weight vector \(\alpha\) is another N-channel image:  
\[f_{tmpl}(x;\alpha)=\sum\
olimits_{u\in\Gamma}\alpha[u]^{T}\varphi_{x}[u] \tag{4}\]  
The histogram fraction is obtained from an M-channel feature picture \(\psi_{x}:H\rightarrow\mathbb{R}^{M}\) that originates from \(x\) and acts on a finite area \(H\subset\mathbb{Z}^{2}\):  
\[f_{hist}(x,\beta)=g\left(\psi_{x};\beta\right) \tag{5}\]  
In contrast to the template fraction, the histogram fraction is invariant to the spatial arrangement because the proportion of an object colour distribution is relatively constant.