# Exploring Frontiers of 4D X-ray Tomography
## 3 High-Speed X-ray CT
### Without Sample Rotation  
The tomogram in Figure 7 was obtained by pulling the sample while rotating it at a high speed of 3000 rpm (50 Hz). To further increase the temporal resolution, the sample needs to be rotated at an even higher speed. However, when the rotation speed was increased by several times, the sample was deformed by centrifugal force, so no CT reconstruction image could be obtained. Rotating the specimen at high speed not only causes the centrifugal force to be large but also imposes various restrictions on such experiments, such as difficulty in controlling the environment of the specimen and inapplicability to fluid specimens. Therefore, the authors aimed to develop a method for realizing synchrotron radiation X-ray CT without the need to rotate samples.  
Since it is not possible to rotate the synchrotron radiation source around a sample, the authors developed a multi-beam optical system for millisecond X-ray CT without sample rotation. There are three ways to change the propagation direction of X-rays: reflection, refraction, and diffraction, but the change in propagation direction that can be achieved by the first two methods is on the order of mrad. The authors selected diffraction (Bragg reflection at a single crystal with high perfection), which can change the propagation direction most significantly, due to the need to cover a wide angular range of projection directions in X-ray CT.  
However, the resonance width of highly perfect Bragg reflection is very narrow (about 10 \(\upmu\)rad in terms of the angle of incidence) due to the dynamical diffraction effect of X-rays, which generally requires a very expensive, precise optical system. Therefore, the authors used white synchrotron radiation and aimed to precisely control multiple reflected beams at the same time by precisely curving a thin single-crystal wafer.  
Figure 10a shows the principle of the multi-beaming technique we developed to realize X-ray CT without sample rotation. A 100 \(\upmu\)m-thick Si single crystal was etched into the shape shown in the upper figure of Figure 10a by microfabrication [50], and when it is bent, the small rectangular plates at the top (hereinafter called \"blades\") point in different directions in accordance with the orientation of the lower part of the crystal. If the Si wafer is curved along a hyperbola, the reflected beams will converge toward the sample. As mentioned above, the resonance width of Bragg reflection is very narrow, so the energy bandwidth of the reflected X-rays is very small. Therefore, the transmitted X-rays  
Figure 9: Schematic illustration of how sparse solutions are more easily selected by using \(L_{1}\) norm for \(N=2\) (**(a)**\(M=1\), **(b)**\(M=2\)). Solid line represents line with constant \(L_{1}\) norm, and dashed line represents contour line of \((\parallel\mathbf{y}-\mathbf{A}\mathbf{x}\parallel_{2})^{2}\). In figure, \(\mathbf{x_{0}}\) is point that minimizes \((\parallel\mathbf{y}-\mathbf{A}\mathbf{x}\parallel_{2})^{2}\) under constraint of constant \(L_{1}\) norm, i.e., solution to optimization problem in Equation (7) (solution with \(x_{1}=0\)). Note that cases of \(M\geq 3\) can be illustrated similarly to (**(b)**), but **y** is generally not in \(x_{1}\)-\(x_{2}\) plane.  
can be reused. Figure 10b shows the hyperbolic-shape multi-beam optical element we developed [31]. The element can generate 32 beams covering a range of \(\pm\)70\({}^{\circ}\). Figure 10c is a 3D CT reconstruction image acquired with an exposure time of 1 ms using a 50-\(\upmu\)m diameter tungsten wire as a sample. The spatial resolution of the tomogram was 65 \(\upmu\)m. We used the state-of-the-art CT reconstruction algorithm based on compressed sensing described in Section 3.2.