# Multi-View Synthesis of Sparse Projection of Absorption Spectra Based on Joint GRU and U-Net
## 2 Mathematical Background
### GMResLnet Structure  
In this section, a sparse projection synthesis network is proposed, which incorporates the GRU network with multilevel residual connectivity on the basis of U-Net, and is named GMResUnet, as shown in Figure 4. The combination of the U-Net, the multilevel residual connectivity, and the GRU module with the learnable sequence feature has not been applied in the synthesis of laser absorption spectral projection data from sparse projections. Therefore, this study paid special attention to the sequence characteristics of the TDLAT projection data with respect to the projection angle, where the data from each detector are sinusoidally regular along the detection angle and where U-Net embodies a classical network architecture with encoding and decoding structures. Downsampling is accomplished through a series of successive convolutional and pooling layers, while upsampling involves successive convolutional and deconvolutional layers, with cascade connections between the corresponding layers. Figure 3: Structure diagram of the residual module. The GMResUnet architecture consists of seven convolutional modules, four GRU modules, and multilevel residual connections. During the encoding phase, the encoder's coding module comprises a GRU module, two \(3\times 3\) convolutional layers, and a ReLU activation function. The GRU module extracts sequential features based on the projected view from the projected data, while the convolutional layers capture spatial features. The nonlinear activation function enhances the network's expressive power. Increasing the number of channels enables the capture of rich semantic information. Additionally, a maximum pooling layer is utilized between each coding module to decrease the size of the feature mapping. This gradual downsampling process aids in broadening the sensory field and extracting global features. During the decoding stage, the feature map's spatial resolution is gradually restored through upsampling while simultaneously reducing the number of channels. The decoder's decoding module comprises an inverse convolutional layer and an activation function that gradually recovers the feature map's spatial dimensions. The inverse convolutional layer's gradual upsampling operation recovers detailed image information and enables fine modeling of the local features. Additionally, each decoding module in the decoder is fused twice with the corresponding feature map of the encoder to introduce low-level fine-grained information. Two methods are used to connect the feature maps of the encoder and decoder levels in image restoration. The first method involves splicing the feature maps element-by-element, while the second method involves the summation of the initial feature maps with the feature maps in the decoder via residual connection. The use of jump connections helps to convey low-level details and boundary information, resulting in better restoration of the original image. Additionally, the introduced multilevel residual connections provide noise immunity. By adding the inputs and outputs directly and using jump connections to introduce low-level features into the high-level computation, the network can focus more on the difference between the inputs and outputs. This approach is relatively insensitive to noise because it fits the difference between the inputs and outputs by learning the residual function. Residual learning can improve the network's robustness to noise, accelerate its convergence, and better synthesize sparse projection data. The network uses zero-padding to maintain input and output image size during training. The last convolutional module in the decoding stage has a kernel size of 1, ensuring that each pixel corresponds to a category label. In the GMResUNet framework, the encoder's convolution operation is preceded by the GRU module. The data from the projected viewpoints are treated as time-series data, with each row representing a sequence. This sequence is then recursively processed in behavioral units by the GRU module.