## 3 Proposed Framework
### Feature Ranking  
Unlike the domains of image understanding and medical applications where offline feature ranking and selection are successfully adopted, the broad spectrum of applications in the computer vision domain is directly tied to real-time processing. We have used a simple ranking criterion adopted by [PERSON] [9] to accommodate multiple features that are considered by the tracker on the fly. The ranking module ranks a pool of features based on the variance ratio between the background and foreground. The top ranked features are input into the AMF-MSPF framework, as a result it takes care of the changing variations such as background clutter and abrupt illumination changes. In this subsection, we describe the feature ranking module. Our feature space F is formed by a linear combination of the RGB color components having coefficients \(\mathrm{c_{1}},\mathrm{c_{2}},\) and \(\mathrm{c_{3}}\):  
\[\mathrm{F}=\{\mathrm{c_{1}R+c_{2}G+c_{3}B}\} \tag{1}\]  
where, \(\mathrm{c}\in[-2-1\ 0\ 1\ 2]\) and a combination of which produces \(5^{3}\) possible candidates in our feature space. Filtering out the redundant and useless cases, leave us with around 50 features. Let \(\mathrm{fg(i)}\) and \(\mathrm{bg(i)}\) be the normalized discrete densities of background and foreground respectively. These densities are discretized to 16 bits for efficiency. The likelihood ratio of these densities is given by Equation (2).  
\[\ell(\mathrm{i})=\log\frac{\mathrm{fg(j)}}{\mathrm{bg(j)}} \tag{2}\]  
Multiple likelihood images are generated using these likelihood ratios that are to be used by the MS procedure. We have used the traditional definition of variance, \(\mathrm{var(x)=E(x)^{2}-E(x)^{2}}\), to calculate the variance of \(\ell(\mathrm{i})\) w.r.t the object and background densities to maximize the inter-class variance between them. Equations (3) and (4) are used to calculate the variance:  
\[\mathrm{v}(\ell;\mathrm{fg})=\mathrm{E}\big{[}\ell^{2}(\mathrm{j})\big{]}-( \mathrm{E}[\ell(\mathrm{j})])^{2}=\sum_{\mathrm{j}}\mathrm{fg(j)}\ell^{2}( \mathrm{j})-\big{[}\sum_{\mathrm{j}}\mathrm{fg(j)}\ell(\mathrm{j})\big{]}^{2} \tag{3}\]  
\[\mathrm{v}(\ell;\mathrm{bg})=\mathrm{E}\big{[}\ell^{2}(\mathrm{j})\big{]}-( \mathrm{E}[\ell(\mathrm{j})])^{2}=\sum_{\mathrm{j}}\mathrm{bg(j)}\ell^{2}( \mathrm{j})-\big{[}\sum_{\mathrm{j}}\mathrm{bg(j)}\ell(\mathrm{j})\big{]}^{2} \tag{4}\]  
Figure 1: Adaptive multi-feature framework for meanshift based Particle Filter (AMF-MSPF).  
In the final ranking step, Equation (5) finds the Variance Ratio (\(VR\)). Essentially, the inter-class variance of the foreground and the background is maximized for the feature space \(\mathbf{F}\).  
\[VR(\mathrm{L};\mathrm{fg},\mathrm{bg})=\frac{\mathrm{v}(\ell;(\mathrm{fg}+ \mathrm{bg})/2)}{[\mathrm{v}(\ell;\mathrm{fg})+\mathrm{var}(\ell;\mathrm{bg})]} \tag{5}\]  
This variance ratio is sorted for the top \(N_{\mathrm{r}}\) features which indicates the highest discrimination score. The likelihood, corresponding to features with the highest score, are used to form new likelihood images. These images are used to initialize the MS optimization procedure that produces new \(N_{\mathrm{r}}\) locations for each particle. Symbolically Equations (1)-(5) is represented as : \(N_{\mathrm{r}}=\mathrm{featureRnk}(\mathbf{F})\), the ranking module that produces \(N_{\mathrm{r}}\) top features. The MS optimization procedure is explained in the next subsection.