## 3 Proposed Method
### Parallel Search Strategy  
Given the target state in current frame, the goal of tracking is to locate it in the subsequent frames. In the particle filter framework, plenty of particles with associated weights are used to estimate the posterior pdf of target state. Generally speaking, the target candidates are compared to the templates in turn. For original L1 tracker, each candidate is represented sparsely in the linear space spanned by target templates and trivial templates. Then, target candidate with minimum reconstruction error in target template subspace is taken as the optimal estimation. In such structure, plenty \(\ell_{1}\) minimization must be solved in each frame which is equal to the number of particles; thus, it has to suffer heavy computational burden. Besides, it is not reliable to estimate the target state only with one state particle. In fact, target state is a continuous variable and it is more reliable to estimate it with more particles with associated weights. The weights can be obtained by the similarity measurement between candidates and templates. In fact, all target candidates can be divided into two categories. Those containing true target belong to the first category. By contraries, background is dominate in the second category. It is more reasonable to estimate the optimal target state only with the true candidates. A parallel search strategy based on sparse representation is proposed to select efficient target candidates which are used to estimate the optimal target state. At the \(t\) frame, plenty of state particles \(X_{t}=\{x_{t}^{1},x_{t}^{2},\cdots,x_{t}^{N}\}\) are obtained by sampling, and the corresponding measurement are target candidates \(Y_{t}=\{y_{t}^{1},y_{t}^{2},\cdots,y_{t}^{N}\}\). In fact, the number of particles is much lager than that of target templates \(T_{t}=\{T_{t}^{1},T_{t}^{2},\cdots,T_{t}^{n}\}\), which means \(n\ll N\). Given that target templates and true candidates lie in the same linear subspace, each template can be represented as a linear combination of the true candidates. Sparse constraints on the coefficient guarantee that only true target candidates can be selected. In principle, the coefficients can be any real numbers without any restrictions; thus, candidates with reversed intensity patterns may be selected. It is necessary to impose nonnegative constraints on the coefficient to eliminate such false candidates; thus, the entries can reflect the associated weights of efficient target candidates. Then the optimal state can be estimated by the linear combination of such weighted states. The above idea can be described as follows. For each target template \(T_{t}^{i}\), all target candidates constitute an overcomplete dictionary \((Y_{t}\in R^{d\times N},d<N)\) then the weights of target candidates can be obtained simultaneously by solving  
\[\hat{\alpha}_{t}^{i}=arg\min_{\alpha_{t}^{i}}\|Y_{t}\alpha_{t}^{i}-T_{t}^{i}\|_{ 2}^{2}+\lambda\|\alpha_{t}^{i}\|_{1}\quad s.t.\quad\alpha_{t}^{i}\succcurlyeq 0 \tag{4}\]  
where \(\hat{\alpha}_{t}^{i}\in R^{N}\) is the coefficient for the \(i\)th target template. When the target template are used to search for efficient target states, their contribution are different. It can be measured with the reconstruction error; thus, the weights of target templates can be defined as  
\[w_{T}{}_{t}^{i}=e^{-\tau\|Y_{t}\alpha_{t}^{i}-T_{t}^{i}\|_{2}^{2}} \tag{5}\]  
where \(\tau\) is used to control the shape of weight distribution. Then the coefficients of all target templates can be fused as  
\[\hat{\alpha}_{t}=\frac{1}{n}\sum_{i=1}^{n}w_{T}{}_{t}^{i}\times\hat{\alpha}_{ t}^{i} \tag{6}\]  
whose nonzero entries denote the associated weights of efficient state particles. The normalized weights of states are obtained by  
\[w_{X}{}_{t}^{i}=\frac{\hat{\alpha}_{t}(i)}{\sum_{i=1}^{N}\hat{\alpha}_{t}(i)} \tag{7}\]  
where \(\sum_{i=1}^{N}w_{X}{}_{t}^{i}=1\). Due to the fact that the target state is a continuous vector, particles with associated weights are used to approximate the posterior pdf.