## 2 Proposed Method
### Traffic Sign Classification
#### 2.3.1 Sparse Classifier Optimization Model  
In order to classify traffic signs, we choose \(t\) traffic sign templates from the color image set. We select \(l\) pre-determined overlapping local patches inside each traffic sign template to ensure some important features of the traffic sign are captured in each patch and all important features of the traffic sign are captured by all patches. We then employ the CNN to extract a \(d\)-dimensional deep feature vector for each patch. These feature vectors are utilized to construct the template set \(\mathbf{D}=[\mathbf{D}_{1},\ldots,\mathbf{D}_{t}]\in\mathbb{R}^{d\times(lt)}\), where \(\mathbf{D}_{i}\in\mathbb{R}^{d\times l}\).  
For each of \(n\) traffic sign candidates in color images, we select the same \(l\) pre-determined overlapping local patches within itself and employ the CNN to extract a \(d\)-dimensional deep feature vector for each patch. Using these \(l\) feature vectors, we build the traffic sign candidate matrix \(\mathbf{X}=[\mathbf{X}_{1},\ldots,\mathbf{X}_{n}]\in\mathbb{R}^{d\times(lt)}\). We denote the sparse coefficient matrix corresponding to the \(j\)th traffic sign candidate as \(\mathbf{C}\triangleq\begin{bmatrix}\mathbf{C}_{1}&\cdots&\mathbf{C}_{t} \end{bmatrix}^{\top}\in\mathbb{R}^{(lt)\times l}\), where \(\{\mathbf{C}_{t}\}_{q=1}^{t}\) is an \(l\times l\) matrix indicating the group representation of \(l\) local features of the \(j\)th traffic sign candidate using \(l\) local features of the \(q\)th traffic sign template (\(q=1\ldots t\)). We formulate the following convex model to represent deep features of the \(j\)th traffic sign candidate using \(t\) traffic sign templates in a set:  
\[\underset{\mathbf{C}\in\mathbb{R}^{(lt)\times l}}{\text{minimize }} \left\|\mathbf{X}_{j}-\mathbf{D}\mathbf{C}\right\|_{\text{F}}^{2}+ \lambda\left\|\begin{bmatrix}\mathbf{C}_{1}(:)\ldots\mathbf{C}_{t}(:)\end{bmatrix} ^{\top}\right\|_{1,\infty} \tag{6a}\] \[\text{subject to } \mathbf{C}\geq 0,\] (6b) \[\mathbf{1}_{lt}^{\top}\mathbf{C}=\mathbf{1}_{l}^{\top}, \tag{6c}\]  
The first term in (6a) shows the similarity between a traffic sign candidate matrix \(\mathbf{X}_{j}\) and the traffic sign template set. The second term is a group-sparsity regularization term, which penalizes the objective function in proportion with the number of selected templates. It establishes the \(\left\|\cdot\right\|_{1,\infty}\) minimization on matrix \(\mathbf{C}\) to impose all local patches inside a candidate to jointly select similar few templates. Particularly, the \(l_{1}\) norm minimization on the columns of \(\mathbf{C}\) makes them to be sparse and therefore selects few traffic sign templates for representation. The \(l_{\infty}\) norm minimization on the rows of \(\mathbf{C}\) motivates the group of local patches to jointly select similar few templates. The parameter \(\lambda>0\) is a trade-off to balance the two terms. The constraint (6b) ensures sparse coefficients to be non-negative since a traffic sign candidate may be represented by traffic sign templates dominated by non-negative coefficients. The constraint (6c) ensures that at least one local patch of the template set is selected to represent a local patch in \(\mathbf{X}_{j}\).