## 3 Proposed Framework
### MS Optimization Procedure
By expanding the Taylor series around coefficient \(\mathbf{p_{u}}(y_{0})\), the linear approximation of Equation (6) is obtained after some manipulation [6; 7; 8]:  
\[\rho_{1:N_{\mathrm{F}}}[\mathbf{p}(y),\mathbf{q_{u}}]\approx\frac{1}{2}\sum_{ \mathrm{u}=1}^{\mathrm{m}}\sqrt{\mathbf{p_{u}}(y_{0}).\mathbf{q_{u}}}+\frac{1} {2}\sum_{\mathrm{u}=1}^{\mathrm{m}}\mathbf{p_{u}}(y)\sqrt{\frac{\mathbf{q_{u} }}{\mathbf{p_{u}}(y_{0})}} \tag{10}\]  
\(y\) substituting Equation (8) in Equation (10), we obtain  
\[\rho_{1:N_{\mathrm{F}}}[\mathbf{p}(y),\mathbf{q_{u}}]\approx\frac{1}{2}\sum_{ \mathrm{u}=1}^{\mathrm{m}}\sqrt{\mathbf{p_{u}}(y_{0}).\mathbf{q_{u}}}+\frac{ \mathrm{C}}{2}\sum_{\mathrm{i}=1}^{N_{\mathrm{f}}}\omega_{\mathrm{i}}\mathrm{ k}\bigg{(}\big{\|}\frac{\mathrm{y-x_{i}}}{\mathrm{h}}\big{\|}^{2}\bigg{)} \tag{11}\]  
where, \(\omega_{\mathrm{i}}=\sum_{\mathrm{u}=1}^{\mathrm{m}}\sqrt{\frac{\mathbf{q_{u }}}{\mathbf{p_{u}}(y_{0})}}\delta[\mathrm{b}(\mathrm{x_{i}})-\mathrm{u}]\). The first term, of Equation (11), is actually the previous location of the target which is not dependent on the new coordinate \(y\). Therefore, we only maximize the second term of the equation to get the new target location \(\mathbf{y}_{1}\). \[\mathbf{y}_{1}=\frac{\sum_{\mathrm{i}=1}^{N_{\mathrm{f}}}\mathrm{x_{i}}\omega _{\mathrm{i}}\mathrm{g}\bigg{(}\big{\|}\frac{\mathrm{y_{0}-x_{i}}}{\mathrm{h}} \big{\|}^{2}\bigg{)}}{\sum_{\mathrm{i}=1}^{N_{\mathrm{f}}}\omega_{\mathrm{i}} \mathrm{g}\bigg{(}\big{\|}\frac{\mathrm{y_{0}-x_{i}}}{\mathrm{h}}\big{\|}^{2} \bigg{)}} \tag{12}\]  
where \(\mathrm{g}(\mathrm{x})\) is a constant for the kernel profile that reduces Equation (12) to a simple weighted distance average as in Equation (13). \[\mathbf{y}_{1}=\frac{\sum_{\mathrm{i}=1}^{N_{\mathrm{f}}}\mathrm{x_{i}} \omega_{\mathrm{i}}}{\sum_{\mathrm{i}=1}^{N_{\mathrm{f}}}\omega_{\mathrm{i}}} \tag{13}\]  
Update \(\mathbf{p}(y_{1})\) and evaluate \(\rho_{1:N_{\mathrm{F}}}[\mathbf{p}(y),\mathbf{q}]\) until \(\|y_{1}-y_{0}\|<\epsilon\) for every \(N_{\mathrm{F}}\) feature and \(\epsilon\) is usually 1 pixel. The particles along the gradient direction of the MS vector move to the local extreme of a probability density function and that determines the best location i.e., where Bhattacharyya coefficient is the highest. The MS procedure is applied to every particle over all the top \(N_{\mathrm{F}}\) features given as in Equation (14). \[[\mathbf{y}_{1:N_{\mathrm{F}}}^{1:N_{\mathrm{s}}}]=\mathrm{Mn}\mathrm{S} \mathrm{f}\mathrm{t}^{1:N_{\mathrm{F}}}\Big{(}\mathbf{s}_{\mathrm{k}+1}^{1:N_{ \mathrm{s}}}\Big{)} \tag{14}\]  
\(\mathbf{y}_{1:N_{\mathrm{F}}}^{1:N_{\mathrm{s}}}\) are the new \(N_{\mathrm{F}}\) estimate locations, for each \(N_{\mathrm{s}}\) particles, that are combined to form new estimates using Equation (15). \[\Big{[}\mathbf{s}_{\mathrm{k}+1}^{1:N_{\mathrm{s}}}\Big{]}=\mu\mathrm{e} \mathrm{d}\mathrm{i}\Big{(}\mathbf{y}_{1:N_{\mathrm{F}}}^{1:N_{\mathrm{s}}} \Big{)} \tag{15}\]  
\(\mathbf{\hat{s}}_{\mathrm{k}+1}\) are the new estimate of the particles that are now closer to their local maxima than \(\mathbf{s}_{\mathrm{k}+1}\). Figure 1 depicts the working of our framework. The top \(N_{\mathrm{F}}\) features along with the initialized particles serves as the input to the MS modules. This gives us new probable locations of the object for all the selected features. Or in other words, the particles after processing through MS, gives rise to \(N_{\mathrm{F}}\) locations for each particle. The median over all the locations gives us the particles that are more accurate i.e., \(\mathbf{\hat{s}}_{\mathrm{k}+1}\). These new hypotheses are used to estimate the posterior density through Equation (20) in the next subsection.