# Algorithms in Tomography and Related Inverse Problems--A Review
## 3 Multi-Slice Fusion in CT Reconstruction
Experimental results demonstrate that multi-slice fusion is highly effective in significantly reducing artifacts and improving resolution compared to alternative reconstruction methods. In the realm of 4D X-ray CT imaging, a dynamic object undergoes rotation, and multiple 2D projections (radiographs) of this object are captured at various angles. The core challenge lies in reconstructing the 4D array of X-ray attenuation coefficients using these measurements, where the four dimensions are allocated as follows: three for spatial dimensions and the fourth for time. It is defined that \(N_{t}\) is the number of time points, the number of measurements at each time point is denoted by \(M_{n}\), and each time point of the 4D volume \(N_{s}\) is the number of voxels. For each time point \(n\) within the range \(\{1,\ldots,N_{t}\}\), defined \(y_{n}\in\mathbb{R}^{M_{n}}\) encapsulates the sinogram measurements taken at time \(n\), and \(x_{n}\in\mathbb{R}^{N_{s}}\) serves as the vector representation of the 3D volume containing X-ray attenuation coefficients for that time point. Combining all measurements yields a comprehensive measurement vector \(y=\left[y_{1}^{T},\ldots,\ y_{N_{t}}^{T}\right]^{T}\in\mathbb{R}^{M}\). Here, \(M\) represents the total number of measurements, which can be expressed as \(M=\sum_{n=1}^{N_{t}}M_{n}\). Similarly, the 3D volumes at each time point can be stacked to form a vectorized 4D volume, denoted as \(x=\left[x_{1}^{T},\ldots,x_{N_{t}}^{T}\right]^{T}\in\mathbb{R}^{N}\), where \(N=N_{t}N_{s}\) represents the total number of voxels within the 4D volume. Recovering the 4D volume of attenuation coefficients x from the series of sinogram measurements \(y\) is actually the 4D reconstruction problem. The provided text describes a 4D reconstruction problem in the context of 4D X-ray CT imaging, where a dynamic object undergoes rotation, and multiple 2D projections are captured at various angles. The goal is to reconstruct the 4D array of X-ray attenuation coefficients using these measurements. The reconstruction is formulated using a Maximum A Posteriori (MAP) approach, incorporating data fidelity and a 4D regularizer. Here is an outline of the algorithm steps mentioned in the text:  
1. Formulate the reconstruction problem using the MAP approach: \[x^{*}=arg\min_{x}\{l(x)+\beta h(x)\}\]
2. Express the data fidelity term \(l(x)\) as the sum of squared differences between sinogram measurements and the forward model: \[l(x)=\frac{1}{2}\sum_{n=1}^{N_{t}}\lVert y_{n}-A_{n}x_{n}\rVert_{\Lambda_{n}}^ {2}\]
3. Define the weight matrix \(\Lambda_{n}=diag\{c\ exp\{-y_{n}\}\}\) to address non-uniform noise variance by approximating it through a Gaussian approximation [95; 115] of the underlying Poisson noise. 4. Formulate each \(H_{k}(x):\mathbb{R}^{N}\ \rightarrow\mathbb{R}^{N}\) as the MAP estimation for a Gaussian denoising problem, where \(h_{k}(x),\ k=1,\ldots,K\) represents a prior model, and \(\sigma\) is the noise standard deviation. \[H_{k}(x)=arg\min_{x\in\mathbb{R}^{N}}\{\frac{1}{2\sigma^{2}}\lVert x-z\rVert_{ 2}^{2}+h_{k}(z)\}\]  
Figure 4: This visual representation showcases the proposed innovative multi-slice fusion approach. Each CNN denoiser is designed to function within the temporal dimension and two spatial dimensions. These CNN denoisers are seamlessly integrated with the measurement model to generate a 4D reconstruction that is inherently regularized. Step 5.: Modify the optimization problem to incorporate \(K\) different regularizers, resulting in a consensus equilibrium formulation. \[x^{*}=arg\min_{x}\{l(x)+\frac{\beta}{K}\sum_{k=1}^{K}h_{k}(x)\}\] Step 6.: Define proximal maps \(L(x)=arg\min_{z\in\mathbb{R}^{N}}\left\{l(z)+\frac{1}{2\sigma^{2}}\|x-z\|_{2}^ {2}\right\}\) : \(\mathbb{R}^{N}\rightarrow\mathbb{R}^{N}\) and \(H_{k}(x)\) : \(\mathbb{R}^{N}\rightarrow\mathbb{R}^{N}\) for each term in the optimization problem.