# Exploring Frontiers of 4D X-ray Tomography
## 3 High-Speed X-ray CT
### With Compressed Sensing
The goal of compressed sensing is to solve the problem of recovering the sparse vector \(\mathbf{x}\) from \(\mathbf{y}\) under the constraint of \(\mathbf{y}=\mathbf{A}\mathbf{x}\). The problem can be formulated as  
\[\text{minimize}\parallel\mathbf{x}\parallel_{0}\text{ subject to }\mathbf{y}- \mathbf{A}\mathbf{x}=\mathbf{0}\enspace, \tag{5}\]  
where the \(L_{p}\)-norm \(\parallel\mathbf{x}\parallel_{p}\) of \(\mathbf{x}\) is defined by  
\[\parallel\mathbf{x}\parallel_{p}= \begin{cases}\left(\sum_{j=1}^{N}\left|x_{j }\right|^{p}\right)^{\frac{1}{p}}&(p>0),\\ \lim_{\varepsilon\rightarrow+0}\left(\sum_{j=1}^{N}\left|x_{j}\right|^{ \varepsilon}\right)^{\frac{1}{p}}&(p=0).\end{cases}  \tag{6}\]  
That is, \(\parallel\mathbf{x}\parallel_{0}\) in Equation (5) is called the \(L_{0}\)-norm, which is defined as the number of non-zero components of vector \(\mathbf{x}\). If we know where the non-zero components are, we can find x by extracting only the non-zero components and calculating the inverse matrix,  
Figure 8: Phase tomogram of polypropylene (PP) sphere taken with measurement time of 2.0 ms [28].Copyright (2017) The Japan Society of Applied Physics (**a**) Reconstruction result using conventional filtered back-projection algorithm. (**b**) Reconstruction result using TV norm. but the difficulty of the problem lies in the fact that we do not know where the non-zero components are. The problem setup in Equation (5) can be applied to X-ray CT. In this case, the vector of all the pixel values of the projection images acquired from various projection directions corresponds to the observed data \(\mathbf{y}\), \(f(\mathbf{r})\) corresponds to a tomogram, and the transformation coefficients with the sparsify transformation correspond to \(\mathbf{x}\). In the case where the observed data \(\mathbf{y}\) contains random errors, the constraint in Equation (5) can be included as a penalty term, resulting in the following optimization problem. \[\text{minimize}\;\lambda\parallel\mathbf{x}\parallel_{0}+\parallel\mathbf{y }-\mathbf{A}\mathbf{x}\parallel_{2}^{2}\;, \tag{7}\]  
where \(\parallel\mathbf{y}-\mathbf{A}\mathbf{x}\parallel_{2}\) is the square root of the sum of the squared components of the vector \(\mathbf{y}-\mathbf{A}\mathbf{x}\) (i.e., the distance in Euclidean space), called the \"\(L_{2}\) norm.\" The hyperparameter \(\lambda\) (\(>\)0) regulates the degree of sparsity; the larger \(\lambda\), the stronger the degree of sparsity, and the closer the solution to that of Equation (5). The \(L_{0}\)-norm optimization problem in Equation (7) is known to be an NP-hard problem when \(M\), \(N\), and \(K\) are large, and it can be solved numerically by replacing it with the \(L_{1}\)-norm optimization problem as follows. \[\text{minimize}\;\lambda\parallel\mathbf{x}\parallel_{1}+\parallel\mathbf{y }-\mathbf{A}\mathbf{x}\parallel_{2}^{2}\;, \tag{8}\]  
where the \(L_{1}\)-norm \(\parallel\mathbf{x}\parallel_{1}\) is the sum of the absolute values of each component of the vector \(\mathbf{x}\). Equation (8) can be interpreted as a formulation of the problem of finding the \(\mathbf{x}\) that minimizes \(\parallel\mathbf{y}-\mathbf{A}\mathbf{x}\parallel_{2}^{2}\) under the constraint that \(\parallel\mathbf{x}\parallel_{1}\) is constant, using Lagrange's undecided multiplier \(\lambda\). Using the \(L_{1}\) norm instead of the \(L_{0}\) norm makes it easier to choose a sparse solution. The reason can be intuitively explained as shown in Figure 9. We consider a two-dimensional vector \(\mathbf{x}\) (i.e., \(N=2\)) for simplicity. In the figure, a line with constant \(\parallel\mathbf{x}\parallel_{1}\) and a contour line of \(\parallel\mathbf{y}-\mathbf{A}\mathbf{x}\parallel_{2}^{2}\) are drawn. It can be seen that the solution that minimizes \(\parallel\mathbf{y}-\mathbf{A}\mathbf{x}\parallel_{2}^{2}\) under the constraint of constant \(\parallel\mathbf{x}\parallel_{1}\) is \(\mathbf{x}_{0}\) on a coordinate axis in the figure, which is a sparse solution (the solution with \(x_{1}=0\)).