## 3 Proposed Framework
### MS Optimization Procedure  
The MS procedure is a non-parametric gradient descent method used to move the particles' state, \(s^{\mathrm{i}}_{\mathrm{k}+1}\) to their new state \(\mathbf{s}^{\mathrm{i}}_{\mathrm{k}+1}\) that is more precise and accurate. MS is initialized in each \(N_{\mathrm{r}}\) features and applied to all the particles that essentially gives us as many new locations for each particle. These \(N_{\mathrm{r}}\) locations are merged, over each particle, using median over them. Evaluating the MS procedure for every particle using multi-features is apparently an intensive computation. However, in the proposed MS optimization technique, the complexity is reduced by an order of magnitude. The weights are estimated for each pixel of the object that is first segmented using normalized cut algorithm based on [PERSON] and [PERSON] [31]. We pick only a fraction of samples from the reference and candidate regions according to these estimates. Essentially, the proposed MS operates only on a fraction of samples. This improvement reduces its complexity by an order of magnitude. This subsection describes our proposed innovation introduced in the MS optimization method. The heart of any MS procedure is the maximization of the Bhattacharryya coefficient that is given by Equation (6):  
\[\rho_{\mathrm{1:}N_{\mathrm{r}}}\equiv\rho[\mathbf{p}(\mathrm{y}),\,\mathbf{q} ]=\sum_{\mathrm{u}=1}^{\mathrm{m}}\sqrt{\mathbf{p}(\mathrm{y}).\mathbf{q_{u}}} \tag{6}\]  
The Bhattacharryya coefficient, \(\rho_{\mathrm{1:}N_{\mathrm{r}}}\), is maximized over all \(N_{\mathrm{r}}\) features. \(\mathbf{q_{u}}\) and \(\mathbf{p_{u}}\) are m-bin discrete color histograms of the target and candidates respectively. These bins are simply a series of small intervals that divides the whole range into smaller ones to reduce computational load. These densities are given from Equation (7) through Equation (8):  
\[\mathbf{q_{u}}=C\sum_{\mathrm{i}=1}^{N_{\mathrm{h}}}\mathrm{k}\Big{(}\| \mathbf{x}^{*}_{\mathrm{i}}\|^{2}\Big{)}\delta[\mathrm{b}(\mathbf{x}^{*}_{ \mathrm{i}})-\mathrm{u}] \tag{7}\]  
\[\mathbf{p_{u}}(\mathrm{y})=C\sum_{\mathrm{i}=1}^{N_{\mathrm{h}}}\mathrm{k} \Big{(}\|\frac{\mathrm{y}-\mathbf{x}_{\mathrm{i}}}{\mathrm{h}}\|^{2}\Big{)} \delta[\mathrm{b}(\mathbf{x}_{\mathrm{i}})-\mathrm{u}] \tag{8}\]  
where, \(\left\{\mathbf{x}^{*}_{\mathrm{i}}\right\}_{\mathrm{i}=1\dots N_{\mathrm{h}}}\) and \(\left\{\mathbf{x}_{\mathrm{i}}\right\}_{\mathrm{i}=1\dots N_{\mathrm{h}}}\) are the pixels of the target and candidates respectively. \(\delta\) is the delta function equal to 1 only at the particular bin \(\mathrm{u}\) and 0 otherwise. \(N_{h}\) represents a fraction of random samples inside the target and candidate. In order to represent \(\mathbf{q_{u}}\), \(\mathbf{p_{u}}\) as densities, we multiply by a coefficient \(C\) to restrict the summations \(\sum_{\mathrm{u}=1}^{\mathrm{m}}\mathbf{p_{u}}\) and \(\sum_{\mathrm{u}=1}^{\mathrm{m}}\mathbf{q_{u}}\) to 1. \(\mathrm{k}\) is a monotonic decreasing convex kernel. Traditionally Bhattacharryya coefficient is maximized using densities that are spatially weighted by an Epanechnikov kernel. However, these multivariate kernels are not good at dealing with non-rigid objects. We instead, employed a monotonic decreasing function to select a fraction of samples picked from the object and its immediate surroundings for spatial weighting. By using only a fraction of samples, the computational load reduces considerably. Equation (9) computes the similarity measure based on the distance between the target and candidates using a fraction of samples from them. \[\mathrm{d}(y)=\sqrt{1-\rho[\mathbf{p}(y),\mathbf{q}]} \tag{9}\]  
The MS procedure maximizes Equation (6), which consequently minimizes Equation (9). The MS optimization process is an iterative process and is initialized in the previous frame with the target position \(\mathbf{y}_{0}\). The new location \(\mathbf{y}_{1}\) is evaluated based on the convergence of Equation (6).