# Algorithms in Tomography and Related Inverse Problems--A Review
## 5 A Deep Learning-Based 3D Ground-Penetrating Radar Data Inversion
#### DENOISING, Inverter, and Training  
The above key contributions for GPR are provided in stages below. The denoiser has three main stages:  
1. Initial Feature Extraction: * An initial feature extraction module is employed, consisting of a \(3\times 3\times 3\) convolutional layer with \(C_{1}\) channels and \(1\times 1\times 1\) strides. This module captures the initial features (\(F_{0}=\delta(K(\mathbf{y}))\)) from the noisy input C-scans (\(\mathbf{y}\in\mathbb{R}^{D\times H\times W}\)). * The process involves a 3D convolutional layer \(K(\cdot)\)) and a Rectified Linear Unit (ReLU) activation function \((\cdot)\). 2. Feature Learning Modules: * After initial feature extraction, \(m\) feature learning modules are applied, each consisting of two residual blocks and one feature attention block. * Residual blocks utilize identity mapping to address gradient explosion concerns. * Residual learning is formulated for each block (\(F_{1}=\delta(K(\delta(K(F_{0})))+F_{0})\) and \(F_{2}=\delta(K(\delta(K(F_{1})))+F_{1})\)) and then a feature attention block is introduced to emphasize the significance of features. * The attention mechanism involves global average pooling to compute channel-wise statistics and a gating mechanism using fully connected layers and a Sigmoid function. * The attended feature map is generated through channel-wise multiplication and added to the original feature map via a residual connection. 3. Reconstruction Module: * A reconstruction module featuring a one-channel convolutional layer with residual learning is employed. * This module reconstructs the denoised C-scan (\(\mathbf{y}_{\text{D}}=\delta(\mathbf{y}+\mathbf{K}(F_{0}+F_{M_{m}}))\)) using the learned feature representations. In a similar way, the key points about the inverter architecture are as follows: 1. 3D U-Net Structure: * The inverter follows the structure of the 3D U-Net architecture, comprising both an encoder and a decoder with skip connections. 2. Multi-Scale Feature Aggregation (MSFA) Mechanism: * MSFA is introduced within each encoding and decoding block to capture features at various scales effectively. * Each MSFA module includes three \(3\times 3\times 3\) convolutional layers with \(1\times 1\times 1\) strides. * The increased number of convolutional layers deepens the network, enhancing its nonlinear mapping capabilities and facilitating the extraction of larger-scale features from object reflections. 3. Receptive Field (RF) Calculation: * The RF size of the output feature map \(F_{rf}\) generated by the \(f_{th}\) convolutional layer in the MSFA module is calculated using the formula \(r_{f}=r_{f-1}+\left(k_{f}-1\right)\times\prod_{i=1}^{f-1}s_{i}\). * The choice of fixed kernel size (\(k=3\) and \(s=\)1) leads to different RF sizes, allowing for the capture of multiple scales. * Multi-Scale Feature Map Combination:
* Feature maps \(\mathbf{F_{r_{1}}}\in\mathbb{R}^{C_{2}\times D\times H\times W}\), \(\mathbf{F_{r_{2}}}\in\mathbb{R}^{C_{2}\times D\times H\times W}\), and \(\mathbf{F_{r_{3}}}\in\mathbb{R}^{C_{2}\times D\times H\times W}\) with different RF sizes are combined in the channel dimension within each encoding and decoding block. * The consolidated multi-scale feature map \(\mathbf{F_{r_{1}\sim 3}}=Concat(\mathbf{F_{r_{1}}},\mathbf{F_{r_{2}}},\mathbf{F_{r_{3}}}))\) is obtained by concatenating these feature maps. * Efficient Multi-Scale Feature Capture:
* Unlike approaches that introduce additional parallel convolutional layers, the MSFA module directly integrates feature maps from successive convolutional layers with different receptive fields. * This design choice aims to efficiently capture multi-scale features from reflection patterns in GPR C-scans influenced by diverse subsurface object properties. * Overall, the MSFA mechanism is introduced to enhance the network's ability to represent the nonlinear mapping from C-scans to 3D permittivity maps, taking into account multi-scale features in subsurface imaging.