# Single-Pixel Imaging Based on Enhanced Multi-Network Prior
## 3 Multi-Network Prior Reconstruction Method
### Sparse Autoencoder Network  
The decision to choose sparse autoencoder as a prior constraint for image reconstruction comes from previous works [43; 44; 45], where [PERSON] et al. proposed sparse autoencoder. A connection has been established between the output Dx(u) of the sparse autoencoder network and the true data density p(u), as described by the following Equation (2).  
\[D_{x}(u)=\frac{\int{(u-\tau)g_{\sigma\tau}(\tau)p(u-\tau)d\tau}}{\int{g_{ \sigma\tau}(\tau)p(u-\tau)d\tau}}, \tag{2}\]  
In the equation above, \(g_{\sigma\tau}(\tau)\) is a Gaussian kernel with a standard deviation of \(\sigma\tau\). This kernel is a smooth function characterized by rotational symmetry and translational invariance.  
It can be seen from Equation (2) that the network output Dx(u) is the weighted average of the images in the input neighborhood. In other words, the neural network can generate a reconstructed image that is very similar to the original image by considering the probability of pixel values in the input image region and the noise in the image. This means that there is a smooth and weighted relationship between the output image of the neural network and the regional pixels of the original image, rather than a simple one-to-one correspondence between the pixels. This equation explains the basic principle of neural network image reconstruction mathematically and clarifies the precondition of neural network as prior information. As prior neural network, it can adjust the relationship between regional pixels.  
In addition, as the neural network loses D(x) - x iterations, D(x) gradually aligns with the input image x. The variation of D(x) can be obtained by taking the differential on both sides of Equation (2), thus obtaining Equation (3). This equation indicates that the error D(x) - x of the autoencoder is proportional to the logarithm of the smooth gradient.  
\[D(x)-x=\sigma_{\tau}^{2}\
abla\text{log}[g_{\sigma\tau}\,*\,p](x), \tag{3}\]  
When the activation rate of the intermediate hidden layer in the sparse autoencoder is limited, it increases the compression degree of the image, improves the modeling of local rules, and eliminates some noise in the image. The parameters of sparse autoencoders are reduced, and the resources consumed are also reduced, making it easier to train. In addition, the feature limit learned through sparse autoencoders is interpretable and more suitable for processing image noise.  
The structure of the SAE network is shown in Figure 2. The SAE network uses a four-layer network with two encoding and two decoding layers. In order to make full use of the information of the measurement dimension, the first layer of the encoding network encodes the input dimension into the initial dimension minus the dimension of the measurement value, and the second layer of the encoding network encodes the output of the first encoding layer dimension into the dimension of the measurement value. In order to enhance the generalization of the SAE network, the two-layer coding matrix is multiplied as the network coding matrix A, and the network decoding matrix B is obtained by solving the group inverse of the network encoding matrix.