## 2 Materials and Methods
### Other Related Algorithms  
Sparse Representation-based Classification (SRC) is based on linear regression through the punishment of regression coefficients. The SRC [10,11] algorithm introduces the L1 norm to constrain the regression coefficients so that more zero values are included in the regression coefficient, equivalent to using the Lasso regression model. The SRC algorithm first encodes the test samples as a sparse linear combination of all the training samples. It then makes the final decision by comparing which category has the smallest error.  
The sparse coefficient of the SRC model can be equivalent according to the Lasso regression model:  
\[\hat{\beta}=\operatorname*{argmin}_{\beta}\left\|y-X\beta\right\|_{2}^{2}+ \lambda\left\|\beta\right\|_{1}, \tag{8}\]where \(X\) is all the training samples, \(y\) is the test sample, \(\beta\) is the regression coefficient of \(X\), and \(\left\|\cdot\right\|_{1}\) is the L1 norm.  
After determining the sparse coefficient of the SRC algorithm, the SRC algorithm is similar to the LRC algorithm, and the same equations as those of (3), (6), and (7) can use the Euclidean distance as a measure to determine the category of the test sample.  
Since there is no analytical solution to the Lasso problem, its computational complexity is much greater than the classifier. Later, the L2 norm was introduced to restrict the regression coefficient, that is, the classification method (Collaborative Representation-based Classification, CRC).  
The CRC [14] problem is equivalent to the ridge regression problem. It is similar to the SRC algorithm in that it similarly encodes the test samples as a linear combination of all the training samples. The CRC algorithm performs the L2 norm constraint on the regression coefficient and its sparse coefficient:  
\[\hat{\beta}=\underset{\beta}{\operatorname{argmin}}\|y-X\beta\|_{2}^{2}+ \lambda\|\hat{\beta}\|_{2}^{2}, \tag{9}\]  
where \(X\) is all the training samples, \(\beta\) is the regression coefficient of \(X\), \(\lambda\) is regularization coefficient, and \(\left\|\cdot\right\|_{2}\) is the L2 norm.  
The regression coefficients in Equation (9) have analytical solutions with the expression:  
\[\hat{\beta}=\left(X^{T}X+\lambda I\right)^{-1}X^{T}y \tag{10}\]  
By combining Equations (3), (6), and (7), the residuals of the CRC algorithm and the predicted test sample category can be expressed as:  
\[d_{i}(y)=\frac{\left\|y-X_{i}\hat{\beta}_{i}\right\|_{2}}{\left\|\hat{\beta}_{ i}\right\|_{2}},i=1,2,\ldots,n. \tag{11}\]  
\[\operatorname{identity}(y)=\underset{i}{\operatorname{argmin}}\,d_{i}(y),i=1, 2,\ldots,n, \tag{12}\]  
where \(\hat{\beta}_{i}\) is the coefficient of the \(i\)-th category of training samples.  
The SRC and CRC algorithms improve the linear regression classifier by restricting the L1 and L2 norm on the regression coefficient, respectively, which can somewhat suppress the influence of noise on the linear model. Nevertheless, the CRC algorithm is more computational than the SRC algorithm.  
The Euler Sparse Representation-based Classification (ESRC [17]) algorithm is similar to the SRC algorithm, which takes Euler distance as a measure and expands intra-class and inter-class distance. The multiple of the inter-class distance will be greater than the multiple of the in-class distance in some data, thus improving the robustness of the SRC algorithm. The ESRC method involves an implementation process, and details of image mapping to the complex space process can found in [17].