# Exploring Frontiers of 4D X-ray Tomography
## 3 High-Speed X-ray CT
### With Compressed Sensing  
If we try to further shorten the acquisition time, the number of projections has to be reduced because the frame rate cannot be increased while maintaining the field of view. Figure 8a shows a tomogram of a polypropylene (PP) sphere with a diameter of 3/32 inches acquired at 2.0 ms. Here, the projection images were obtained by the same experimental setup as that of Figure 5, but the Fourier transform method [43] was used instead of the fringe scanning method. In the Fourier transform method, a differential phase image can be obtained from a projection image with fine moire fringes, so that higher temporal resolution is realized than that of the fringe scanning method, although the spatial resolution becomes worse up to the period of the moire fringes. We obtained the projection images at a frame rate of 37,500 frames per second with a field of view of 256 \(\times\) 256 and rotated the sample at a speed of 29,605 rpm. To increase the field of view, the so-called offset CT was used: the two projections from opposite directions were combined to obtain a single projection image. Since only 76 projections for a half turn of the sample can be obtained, using the classical CT reconstruction algorithm (filtered back-projection algorithm) results in radial artifacts as shown in the left pad in the figure. A compressed sensing-based CT reconstruction method has recently attracted attention as a solution to this problem [28; 44; 45; 46; 47; 48; 49]. In this method, images are recovered from a small number of observation data samples by using the inherent sparsity of images. Sparsity is often inherent in images obtained from the natural world. For example, if we carry out a wavelet transform of an image, extract only 10% of its components, and carry out an inverse transform, the resulting image will be almost indistinguishable from the original. In other words, even if the number of data samples is insufficient, the original image can be obtained by applying an appropriate linear transformation (called sparsify transformation) since the number of non-zero components to be recovered can be approximated to a small number. Figure 7: Real-time 4D observation of rubber tensile fracture process (10-ms temporal resolution) [41; 42]. Upper: rotation device for high-speed X-ray CT, lower: 4D tomograms of tensile fracture process of rubber. The problem of compressed sensing can be mathematically generalized as follows. Suppose that an original physical quantity, \(f(\mathbf{r})\), results in the observed data \(y_{i}\) (i = 1, \(\cdots\), \(M\)), as expressed in the following equation. \[y_{i}=\int f(\mathbf{r})\psi_{i}(\mathbf{r})d\mathbf{r}\enspace, \tag{2}\]  
where the inner product with \(\psi_{i}(\mathbf{r})\) expresses the \(i\)th sampling. Additionally, suppose that the original physical quantity \(f(\mathbf{r})\) can be expanded by the basis \(\phi_{j}(\mathbf{r})\) (\(j=1,\cdots\), \(N\)) of the sparsify transformation as follows  
\[f(\mathbf{r})=\sum_{j=1}^{N}x_{j}\phi_{j}(\mathbf{r})\enspace. \tag{3}\]  
Substituting Equation (3) into Equation (2) and defining \(A_{ij}\equiv\int\psi_{i}(\mathbf{r})\phi_{j}(\mathbf{r})d\mathbf{r}\), we obtain  
\[y_{i}=\sum_{j=1}^{N}A_{ij}x_{j}\enspace, \tag{4}\]  
that is, the vector \(\mathbf{y}\) with components \(y_{i}\) can be expressed by the product of the matrix \(\mathbf{A}\) and the vector \(\mathbf{x}\) with components of \(A_{ij}\) and \(x_{j}\), respectively. If \(M=N\), we can find x by calculating the inverse; if \(M>N\) and there is no \(\mathbf{x}\) satisfying \(\mathbf{y}=\mathbf{A}\mathbf{x}\), we can find the least-squares solution (the solution that minimizes (\(\left\|\ \mathbf{y}-\mathbf{A}\mathbf{x}\right\|_{2}\)) by using the so-called Moore-Penrose pseudo-inverse matrix, for example. If \(M<N\), the solution is indefinite and becomes an ill-posed problem. However, if the number \(K\) of non-zero components of \(\mathbf{x}\) is small and \(M\geq K\), then there is a possibility of finding the \(K\) non-zero components.