## 2 Proposed Method
### Road Extraction  
Since traffic signs are located above road areas, we aim to remove non-road pixels from color images to reduce the search space to exclusively contain road pixels. We employ an image segmentation technique [11] to extract and integrate complementary local features of a group of image pixels in a factorization-based framework. This framework builds multiple representations of image pixels, which are represented in an optimization model to obtain more informative representation of image pixels for segmentation. We then manually select pixels from road and non-road regions and respectively construct two dictionaries from these pixels to store the feature information of road and non-road regions. The feature of a candidate pixel is finally represented by the road and non-road dictionaries in quadratic optimization problems with equality constraints to classify it as either road or non-road. We explain the type of extracted features in Section 3.1. We denote the two dictionaries, which store the features of road and non-road regions, as road dictionary and non-road dictionary, respectively. We classify the candidate pixels that are well presented by the road dictionary in the quadratic optimization problem with equality constraints (i.e., the reconstruction error with respect to road dictionary is less than the reconstruction error with respect to non-road dictionary) as road regions. To do so, we over-segment each image to extract \(n_{os}\) segments (super-pixels) [12]. We then construct \(n_{f}\) feature matrices with each containing one kind of feature with a dimension of \(d_{w}\) for all \(n_{os}\) segments. For the \(w\)th feature matrix \(\mathbf{Y}^{w}\in R^{d_{w}\times n_{os}}\), we model it as follows:  
\[\mathbf{Y}^{w}=\mathbf{T}^{w}\mathbf{Z}^{w}+\mathbf{E}^{w} \tag{1}\]  
where \(\mathbf{T}^{w}\in R^{d_{w}\times n_{b}}\) is a dictionary of \(n_{k}\) words with a dimension of \(d_{w}\), \(\mathbf{Z}^{w}\in R^{n_{b}\times n_{os}}\) is a new representation of the features of all segments using \(n_{k}\) dictionary words, and \(\mathbf{E}^{w}\) is the model error. In the segmentation task, we aim to learn \(\mathbf{Z}^{w}\) by adopting non-negative matrix factorization in the following optimization model:  
\[\underset{\mathbf{T}^{w},\mathbf{Z}^{w}}{\text{minimize}} \|\mathbf{Y}^{w}-\mathbf{T}^{w}\mathbf{Z}^{w}\|_{\text{F}}^{2}\] (2a) subject to \[\mathbf{Z}^{w}\geq 0,\mathbf{T}^{w}\geq 0 \tag{2b}\]  
We use the alternating least square (ALS) method [13] to solve this optimization problem for all feature matrices \(\mathbf{Y}^{w}(w=1,\cdots,n_{f})\). Calculated representation matrices \(\mathbf{Z}^{w}(w=1,\cdots,n_{f})\) contain \(n_{f}\) kinds of complementary feature information of \(n_{os}\) segments. As a result, they are more informative than one matrix containing individual feature [11]. We use a linear combination of matrices \(\left\{Z^{w}\right\}_{w=1}^{n_{f}}\) to model \(\mathbf{Q}\) to represent the final segmentation results. The coefficients of such a linear combination are gathered into a vector \(\mathbf{p}\in R^{n_{f}\times 1}\) to measure the reliability of representations. We formulate the following aggregation model to compute \(\mathbf{Q}\):  
\[\underset{\mathbf{Q},\mathbf{p}}{\text{minimize}} \left\|\sum_{i=1}^{n_{f}}\mathbf{p}_{i}\mathbf{Z}^{i}-\mathbf{Q} \right\|_{\text{F}}^{2}+\gamma Tr(\mathbf{QLQ}^{\top})\] (3a) subject to \[\mathbf{p}\geq 0,\ \mathbf{1}^{T}\mathbf{p}=1 \tag{3b}\]  
where \(\mathbf{1}\) is a ones column vector, \(\gamma\) is a regularization parameter, \(\mathbf{L}\) is a Laplacian matrix, and \(Tr(\cdot)\) is the trace operator. The Laplacian matrix \(L\) is constructed over a graph \(\mathcal{G}\), where its nodes are segments and its edges connect the neighboring segments [11]. The first term is the penalization of representing the violation of the linear model.