## 2 Materials and Methods
### The Proposed Method  
The workflow of the BPLRC method is shown in Figure 1 under the assumption that the image is divided into 5 pieces. When the image only retains one block, the principle of the BPLRC algorithm is equivalent to Module LRC; when the image retains all blocks, the principle of the BPLRC algorithm is equivalent to LRC. The proposed method contains Module LRC that is reserved with a full block and blocks that only retain one block. Theoretically, its recognition effect will be better than LRC and Module LRC. BPLRC aims to find the best facial features for classification. The proposed method first divides all training images and a test image and divides the number of groups according to the number of reserved blocks. As shown in the figure, the number of arrangements from the first and fifth groups is 5, 10, 10, 5, and 1, respectively. The construction of a linear model according to each scheme then determines the scheme with the minimum residual and obtains the identification result of the scheme. Finally, samples can be continually drawn from the test concentration and the above work can be repeated. Assuming that the training samples contain some noise when the test samples have the same noise, test samples can be linearly represented by the training samples. However, if the training samples contain no noise and the test samples contain a lot of noise, the  
Figure 1: The overall workflow of the BPLRC method (with 5 blocks) [Reprinted with permission from Elsevier [20]. Copyright 2013, Neurocomputing]. linear model will be invalid. Therefore, selecting the sample variables and filtering out the variables with noise in the samples are necessary, which can then help train a more robust linear regression model. The proposed method divides the variables into blocks and performs permutation, combination, and reorganization. The best combination method is then judged by Euclidean distance (residual value) to achieve the purpose of removing continuous noise variables. BPLRC aims to find face characteristics that contain only a small amount of noise or no noise. The following is the basic process of the BPLRC method:  
Suppose the training samples on the \(i\)-th subspace are divided into \(T\) blocks such that each sub-image can be represented as:  
\[U_{i}^{(t)}=[w_{i1}^{(t)},w_{i2}^{(t)},\ldots,w_{ip_{i}}^{(t)}],i=1,2,\ldots,n, \tag{13}\]  
where \(w_{ip_{i}}^{(t)}\) is the grayscale value on the \(p_{i}\) pixel of the \(t\)-th block image from the \(i\)-th class. The number of groups of arrangement schemes is determined by the number of divided blocks, and the block arrangement and combination of the \(i\)-th type of training samples in the first group can be expressed as:  
\[\left\{ \begin{aligned} X_{i}^{(11)}&=[U_{i}^{(1)},O ^{(2)},O^{(3)},O^{(4)},\ldots,O^{(T)}]^{T}\\ &\vdots,m_{1}=1,2,\ldots,C_{T}^{1},\\ X_{i}^{(1m_{1})}&=[O^{(1)},O^{(2)},\ldots,U_{i}^{(m_ {1})},\ldots,O^{(T)}]^{T}\end{aligned} \right.,m_{1}=1,2,\ldots,C_{T}^{1}, \tag{14}\]  
where \(O\) is a zero matrix and \(U_{i}^{(m_{1})}\) is the \(m_{1}\)-th block of class \(i\). Different groups retain a certain number of sub-images. For example, the second group retains 2 sub-images, and the \(T\)-th group retains \(T\) sub-images. Therefore, the 2-\(T\) groups \(i\)-th training sample reorganization can be expressed as:  
\[X_{i}^{(2m_{2})},X_{i}^{(3m_{3})},\ldots,X_{i}^{(tm_{i})},t=1,2,\ldots,T;m_{t} =1,2,\ldots,C_{T}^{t}, \tag{15}\]  
where \(m_{t}\) is the \(m_{t}\)-th arrangement and \(C_{T}^{t}\) is the number of arrangements. Similar to Equations (14) and (15), the recombination test samples of different groups can be expressed as:  
\[y^{(1m_{1})},y^{(2m_{2})},\ldots,y^{(tm_{i})},t=1,2,\ldots,T;m_{t}=1,2,\ldots,C_{T}^{t}. \tag{16}\]  
where \(m_{t}\) is the \(m_{t}\)-th arrangement and \(C_{T}^{t}\) is the number of arrangements. Equations (14)-(16) represent different segmentation arrangements in the image.