# Moving Object Tracking and Its Application to an Indoor Dual-Robot Path
## 2 Image Processing
### Mean Shift Tracking  
Mean shift algorithm is a simple interactive procedure that shifts each data point to the average of data points in its neighborhood. For Gaussian kernels, mean shift is a gradient mapping. Convergence is guaranteed for mean shift iterations. Cluster analysis is treated as a deterministic problem of finding a fixed point of mean shift that characterizes the data [28]. The pixel from HSL color space is used in clustering. Similar colors of the pixel will be clustered into the same group. Each pixel in the image corresponds to a point in fuzzy color space. Mean shift can obtain locally similar colors of the pixel to achieve the maximum density clustering effect [29]. The fuzzy color histogram of a target is constructed to characterize the tracked object. The target is represented by a square region in the image. By the use of the trained fuzzy cluster and a kernel function, weighting factors with respect to all pixels within the square region are calculated. Mean shift can find the highest color density of sample points from the fuzzy color space of the target objects, and the objects can be tracked continuously. Given the position of the detected moving object, we want to extract its color distribution information. Let pattern image color density function be \(q(u)\), and the candidate image in the color density be a \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Color** & **Hue** & **Saturation Value** & **Lightness Value** \\ \hline Red (object) & 254 & 184 & 92 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Object HSL color values. Figure 5: Classifications of the background environment and object: (**a**) background environment; (**b**) training background color distribution; (**c**) add object in background environment; (**d**) classifications of the background environment and object. function of position \(y\); \(p\) (\(u\), \(y\)) is used to find the position of \(y\), such that \(p\) (\(u\), \(y\)) and \(q(u)\) have the highest similarity [21; 30]. Pattern image color density function is:  
\[q(u)=C\sum_{u=1}^{m}k(\parallel x_{i}^{*}\parallel^{2})\delta[b(x_{i}^{*})-u], \tag{4}\]  
where \(u\) is the HSL full-color image, \(x_{i}^{*}\) is the object position of pattern image (from fuzzy color classification), \(b(x_{i}^{*})\) is the transform of the corresponding color index of \(x_{i}^{*}\), \(k(\parallel x_{i}^{*}\parallel^{2})\) is the kernel function, \(C\) is the normalization factor, \(\delta(x)\) is the Kronecker delta function, and \(\delta\left(x\right)\)=\(\left\{ \begin{array}{cc}0,&if\ x\
eq 0\\ 1,&if\ x=0\end{array} \right..\)  
Candidate image color density function is:  
\[p(u,y)=C_{h}\sum_{u=1}^{n_{h}}k(\parallel\frac{x_{i}-y}{h}\parallel^{2}) \delta[b(x_{i})-u], \tag{5}\]  
where \(x_{i}\) is the real object position of candidate image, \(C_{h}\) is the normalization factor, \(h\) is the kernel radius length, \(y\) is the initial position of the candidate image, and \(k(\frac{\parallel y-x_{i}\parallel^{2}}{h})\) is the kernel function. The highest estimated Bhattacharyya coefficient is:  
\[\rho(y)\equiv\rho[p(u,y),q(u)]=\sum_{u=1}^{m}\sqrt{p(u,y)q(u)}. \tag{6}\]  
[PERSON] expansion of Equation (6) is applied to deduce the color image density (patterns and candidate) and to make the Bhattacharyya coefficient have the maximum similarity value. \[\rho[p(u,y_{1})]\approx\rho[p(u,y_{0})]+\rho[p(u,y_{0})][p(u,y)]-p(u,y_{0})] \tag{7}\]  
\[\rho[p(u,y_{1})]\approx\frac{1}{2}\sum_{u=1}^{m}\sqrt{p(u,y_{0})q(u)}+\frac{1 }{2}\sum_{u=1}  
\[f(x)=\frac{\sum\limits_{i=1}^{n}k\left(\frac{x_{i}-x}{h}\right)w(x_{i})}{h^{d} \sum\limits_{i=1}^{n}w(x_{i})}. \tag{12}\]  
Let \(k(x)=k(\parallel x\parallel^{2})\). The gradient of kernel density is \(\
abla f(x)\). If \(g(x)\) is defined as \(k(x)\) and \(g(x)=-k(x)\), then  
\[