## 3 Path Feature Recognition
### Feature Extraction
Then, the covariance matrix **C** of the samples in the feature space **H** is:  
\[\textbf{C}=\frac{1}{n}\sum_{j=1}^{n}\varphi\Big{(}x_{j}\Big{)}\varphi\Big{(}x _{j}\Big{)}^{\text{T}} \tag{3}\]  
(4) Assume that eigenvalues \(\lambda\geq s\) and eigenvectors \(\textbf{V}\in\textbf{H}\backslash\{0\}\) both exist, which satisfy:  
\[\lambda\textbf{V}=\textbf{C}\textbf{V} \tag{4}\]  
Then, all solutions \(V\) with \(\lambda\
eq\text{U}\) lie in the span of \(\varphi(x_{1}),\ldots,\varphi(x_{n})\). Based on this finding, we can construct a group of equations, i.e.,:  
\[\lambda\Big{(}\varphi(x_{k})^{\text{T}}\cdot\textbf{V}\Big{)}=\varphi(x_{k})^ {\text{T}}\cdot\textbf{C}\textbf{V}\quad\text{for all }k=1,\ldots,n \tag{5}\]  
There is certainly the possibility that a set of coefficients \(\alpha_{i}\) (\(i=1,\ldots,n\)) satisfy:  
\[\textbf{V}_{k}=\sum_{i=1}^{n}\alpha_{i}^{k}\varphi(x_{i}) \tag{6}\]  
(5) Substitution of Equations (3) and (6) into Equation (5) yields:  
\[\lambda\sum_{i=1}^{n}\alpha_{i}\Big{(}\varphi(x_{k})^{\text{T}}\varphi(x_{i}) \Big{)}=\frac{1}{n}\sum_{i=1}^{n}\alpha_{i}\Bigg{(}\varphi(x_{k})^{\text{T}} \cdot\sum_{j=1}^{n}\varphi\Big{(}x_{j}\Big{)}\Bigg{)}\Bigg{(}\varphi\Big{(}x_{ j}\Big{)}^{\text{T}}\cdot\varphi(x_{i})\Bigg{)}\text{for all }k=1,\ldots,n \tag{7}\]  
Normally, the specific form of \(\phi\) is unknown. Hence, the Gaussian radial basis kernel function \(\kappa\) is employed as a replacement, where **K** is the kernel function matrix related to \(\kappa\). \[(\textbf{K})_{ij}=\kappa\Big{(}x_{i},x_{j}\Big{)}=\varphi(x_{i})^{\text{T}} \cdot\varphi\Big{(}x_{j}\Big{)} \tag{8}\]  
(6) Upon substituting Equation (8) into Equation (7), one obtains:  
\[n\lambda\alpha=K\alpha \tag{9}\]  
where \(\alpha\) denotes the column vector containing entries. \(\alpha_{1},\ldots,\ \alpha_{n}\). Once the solutions of Equation (9) are found, the nonzero eigenvalue problem can be settled. Let \(\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{n}\) denote the eigenvalues of **K**, i.e., the solutions \(n\lambda\) of Equation (9), and \(\alpha^{1},\ldots,\alpha^{n}\) are the complete set of eigenvectors corresponding to the eigenvalues, with \(\lambda_{p}\) being the first nonzero eigenvalue. The eigenvalues \(\lambda_{i}\) and the eigenvectors \(\alpha^{i}=\Big{(}\alpha_{1}^{i},\alpha_{2}^{i},\ldots,\alpha_{n}^{i}\Big{)}\) of the matrix **K** can be solved using the Jacobi iteration method [33]. (7) In order to extract the principal components, the projections of the samples onto eigenvectors \(\mathbf{V}_{k}\) (\(k=p,\ldots\)\(n\)) in the feature space \(\mathbf{H}\) need to be calculated. Let \(\mathbf{x}\) be a new sample, with an image \(\varphi(\mathbf{x})\) in the space \(\mathbf{H}\). Then, the projections are the nonlinear principal components of the sample, shown as:  
\[\mathbf{h}_{k}(x)=\ \mathbf{V}_{k}^{\mathrm{T}}\cdot\mathbf{\varphi}(\mathbf{x})=\sum_{i=1} ^{n}\alpha_{i}^{k}\Big{(}\mathbf{\varphi}(\mathbf{x}_{i})^{\mathrm{T}}\cdot\mathbf{\varphi} (\mathbf{x})\Big{)}=\sum_{i=1}^{n}\alpha_{i}^{k}\kappa(\mathbf{x}_{i},\mathbf{x}) \tag{10}\]  
where \(\mathbf{h}_{k}(\mathbf{x})\) is the \(k^{\mathrm{th}}\) nonlinear principal component of the sample \(\mathbf{x}\). (8) The contribution rate of the principal components is evaluated. Select \(s\) principal components whose rate of contribution is the most important, and whose sum of contribution is greater than 85%. Then, a new sample matrix with a reduced size of \(n\times\)s is generated as an input when the BPNN-based classifier is trained.