# Mmpal _sciences_
## 2 Proposed Method  
### Multi-Layer Feature Fusion PCANet  
Inspired by DenseNet [28], PCANet (MPCANet) is proposed to fuse different levels of feature information. MPCANet first concatenates the original image features and the output features of the first layer of the network at the channel level (channel concatenation), and then it takes the concatenated results as the input of the second layer of the network; therefore, MPCANet uses more feature information than PCANet. MPCANet includes the input layer, convolution layer, output layer, and classification layer. Its specific design process is as follows:  
(1) Input layer of MPCANet: First, block sampling is performed on each input training image. Assuming that there are \(N\) training images of \(m\times n\) size, the filter size is \(k_{1}\times k_{2}\). Taking each pixel of the image \(l_{i}^{1}\) as the center, \(l\,k_{1}\times k_{2}\) blocks are obtained and vectorized. This allows the input image to build a matrix shaped like \(X_{i}\). \[X_{i}=\big{[}x_{i,1},x_{i,2},\cdots,x_{i,\widetilde{m}\widetilde{m}}\big{]}eR ^{k_{1}k_{2}\times\widetilde{m}\widetilde{m}} \tag{1}\]  
where \(\widetilde{m}=m-k_{1}+1\), \(\widetilde{m}=n-k_{2}+1\). Then, the average value of the column is subtracted from each column to obtain the demeaned matrix \(\overline{X}_{i}=\big{[}\overline{x}_{i,1},\overline{x}_{i,2},\cdots, \overline{x}_{i,\widetilde{m}\widetilde{m}}\big{]}\). This is repeated for all \(N\) training images, and all \(\overline{X}_{i}\) are put together to form a large matrix \(X\). \[X=\big{[}\overline{X}_{1},\overline{X}_{2},\cdots,\overline{X}_{N}\big{]}eR ^{k_{1}k_{2}\times N\widetilde{m}\widetilde{m}} \tag{2}\]  
\(X\) is the input of MPCANet. (2) The first layer convolution: Let the number of filters be \(L_{1}\). Firstly, \(XX^{\text{T}}\) is decomposed by SVD, and the feature vectors \(V_{l}^{1}=\left\{V_{1}^{1},V_{2}^{1},\cdots,V_{L_{1}}^{1}\right\}\) corresponding to the first \(L_{1}\) main eigenvalues of \(XX^{\text{T}}\) are obtained. Subsequently, each feature vector is represented \begin{table}
\begin{tabular}{c c c} \hline \hline
**Methods** & **Advantages** & **Disadvantages** \\ \hline ESRC [9] & Intraclass variations of the same person can be shared by others & Weak discrimination and large scale of occlusion dictionary \\ \hline NNAODL [10] & Integrates the error image with training samples to construct the dictionary & In the case of extreme dark illuminations or unconstrained settings, the accuracy is low \\ \hline PCANet [14] & Can be extremely easily and efficiently designed and learned & Easy to cause the loss of image information \\ \hline L1-2D2 PCANet [19] & Uses the L1-norm-based 2 DPCA for filter learning & The training time is long \\ \hline Methods based on deep learning [5; 24; 25; 26] & Good performance and strong learning ability & Takes a lot of training samples and time \\ \hline MMPCANet & Makes full use of the locality and continuity of occlusion space to obtain better spatial information & High storage overhead \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Key differences between a number of previous methods and proposed methods. as a matrix with the size of \(k_{1}\times k_{2}\). Each matrix is regarded as the first layer of MPCANet. In other words, the filter of first layer convolution can be expressed as follows:  
\[W_{l}^{1}=\text{mat}_{k_{1}k_{2}}\Big{(}V_{l}^{1}\Big{)},\ \ W_{l}^{1}eR^{k_{1} \times k_{2}} \tag{3}\]  
In the expression, mat is a function that maps the \(l\)-th feature vector \(V_{l}^{1}\) to matrix \(W_{l}^{1}\). Therefore, for image \(I_{i}^{1}\), the output after the first layer convolution can be obtained by the following expression:  
\[I_{i,l}^{1}=I_{i}^{1}*W_{l}^{1},\ \ i=1,2,\cdots,N;\ l=1,2,\cdots,L_{1} \tag{4}\]  
Mark \(*\) denotes two-dimensional convolution [14]. For each input image, \(L_{1}\) output maps can be obtained after the first layer convolution.