# Algorithms in Tomography and Related Inverse Problems--A Review
## 3 Multi-Slice Fusion in CT Reconstruction  
The complexity of reconstruction problems has evolved beyond the conventional 2D and 3D spatial representations to tackle more intricate 4D and even 5D challenges involving the dimensions of space and time, as well as specific aspects like heart or respiratory phases [94; 95; 96; 97; 98; 99; 100]. This heightened dimensionality in reconstruction offers valuable opportunities to enhance the quality of reconstruction by capitalizing on the inherent patternswithin this multidimensional space. In particular, for time-sensitive imaging applications, we can harness the regularity within the images to reconstruct each frame using fewer data points, thus elevating temporal resolution. Take, for example, the domain of 4D CT imaging, where notable contributions from references [101; 102; 95] have significantly enhanced temporal resolution by leveraging the inherent spatiotemporal regularities of the objects being imaged. These approaches rely on model-based iterative reconstruction (MBIR) techniques [103; 104], which enforce regularity in the 4D data by incorporating straightforward spatiotemporal prior models. Furthermore, there has been a proposition to employ deep learning-based post-processing methods for 4D reconstruction, aiming to further enhance the quality of the reconstructed images [105]. Recent developments have demonstrated the significant potential of previous plug-and-play (PnP) programs, as documented in references [106; 107; 108; 109], to significantly improve the quality of reconstructions. This is achieved by allowing state-of-the-art denoising techniques to be incorporated as model priors within model-based iterative reconstruction (MBIR). Consequently, PnP methods promise a remarkable improvement in reconstruction quality in the context of 4D CT imaging challenges. However, there is a notable limitation when it comes to applying state-of-the-art denoisers such as deep convolutional neural networks (CNN) and BM4D, as they are primarily designed for 2D and occasionally 3D images. Extending these techniques to higher dimensions, as discussed in references [100; 111; 110], presents significant computational and memory challenges. Specifically, adapting CNNs to 4D requires computationally intensive 4D convolutions applied to 5D feature tensor structures. Additionally, training PnP launchers with 4D CNNs requires access to 4D ground truth data, which can be difficult or even impossible to obtain. The novel 4D X-ray CT reconstruction algorithm introduced in [29] leverages multiple low-dimensional CNN denoisers to generate a highly efficient 4D prior model. The methodology, known as \"multipartite fusion\", seamlessly integrates these different low-dimensional priors using a multi-agent equilibrium consensus (MACE) technique [112]. A visual representation of the basic idea behind this approach is provided in Figure 4. A technique called \"multi-slice fusion\" is used, which combines three discrete CNN denominators, each specifically trained to remove additive white Gaussian noise from lower-dimensional slices (hyperplanes) of the 4D object. When the multi-agent equilibrium consensus (MACE) process merges these jammers, it does so while simultaneously enforcing the constraints imposed by each jammer. As a result, the reconstructed images are forced to exhibit smoothness in all four dimensions. This approach, known as multipart fusion, produces excellent quality reconstructions and remains practical for training and computation even when dealing with high-dimensional reconstruction tasks. The solution for MACE can be computed using a variety of algorithms, as documented in references [106; 114; 107; 113]. To implement multipart fusion, distributed heterogeneous clusters are used, where different agent updates are distributed to various cluster nodes. In particular, the computationally intensive cone beam inversion processes are distributed across multiple CPU nodes, while the CNN denoising calculations are simultaneously distributed across multiple GPU nodes.