1. Introduction

X-ray phase tomography, which reconstructs the volume distribution of the refractive index inside an object, has been performed based on various techniques for the measurement of phase shift and differential phase shift of X-rays passing through the object [1–4]. Thanks to the high sensitivity of these techniques to materials mainly consisting of light elements, X-ray phase tomography has exhibited innovative visualization performance for biological tissues and polymers.

However, existing reports on X-ray phase tomography have concerned static cases, because a certain scan time is necessary even when bright synchrotron radiation is used. In order to ensure phase sensitivity, a monochromatic and collimated beam is normally used, forcing the tomographic scan time to be a few hours. Although it is obvious that dynamic X-ray phase tomography will lead to various attractive applications, there has been no development toward a time-resolved experiment with a time resolution below one second.

X-ray phase imaging with an X-ray Talbot interferometer [5] consisting of two transmission gratings has attracted attention and has been used for various imaging applications [6–10]. X-ray Talbot interferometry has some advantages that cannot be expected from other phase imaging methods. In this study, we make full use of one of these advantages; a Talbot interferometer functions with X-rays of a broad energy bandwidth because transmission gratings are used in a forward-diffraction geometry.

An X-ray Talbot interferometer functions with energy bandwidth ΔE/E of 10⁻¹, showing an image quality comparable to that by monochromatic X-rays. Even when the bandwidth is broader than 10⁻¹, X-ray phase imaging/tomography is still possible although the image quality is reduced to some extent. A normal double-crystal monochromator, whose typical bandwidth is 10⁻³, is used widely at synchrotron radiation beamlines. The X-ray beam through the double-crystal monochromator is too monochromatized to be used for Talbot interferometry, losing X-ray flux in vain.

Therefore, we attempt at speedup of X-ray phase tomography with white synchrotron radiation [11]. Not only shortening of the CT scan time but also dynamic observation can now be realized. For a polymer material, time-resolved X-ray phase tomography (i.e., four-dimensional X-ray phase tomography) was successfully demonstrated [12]. In this paper, we demonstrate dynamic observation of a living worm with a time resolution of 0.5 s. We investigate the image quality of organs and the effect of the exposure to white synchrotron radiation. Finally, we discuss the problems to be solved to improve image quality.

2. Image reconstruction procedure

In order to perform phase tomography, the phase shift or differential phase shift must be measured quantitatively. In X-ray Talbot interferometry, the fringe-scanning method [13], or phase-stepping technique, is normally adopted by displacing one of the gratings in the direction parallel to its diffraction vector. A moiré image is measured at every position of the grating displacement. A single image of the differential phase shift is calculated using these multiple moiré images. The phase-stepping scan time is therefore long including the time for mechanical movement of the grating and readout, transfer, and storage of images.

Because the differential phase shift is calculated at each pixel independently, spatial resolution is not degraded through this process. However, the fringe-scanning method is obviously not convenient for high-speed imaging. In this study, we therefore adopted the Fourier-transform method [14], which generates a differential phase image from a single moiré image, although the spatial resolution is not comparable with the case by the fringe-scanning method.

The Fourier-transform method requires carrier fringes in the images to be processed. We produced carrier fringes by inclining gratings to each other around the optical axis; that is, rotation moiré fringes were produced. Here, the grating diffraction vector is assumed to be parallel to the x axis. Then, the spacing of rotation moiré fringes is given by d/θ in the y direction and carrier frequency f ₀ by θ/d, where θ (≪ 1) is the inclination angle and d is the pitch of the gratings. The moiré fringe pattern with the carrier fringes is expressed with a Fourier expansion form [15] as

By calculating the Fourier-transform of Eq. (1), extracting the first-order term, obtaining α ₁(x, y, z ₀ ,t) by inverse Fourier-transform, and then calculating its argument, $\tilde{\phi }$_x(x, y,t) is estimated [11]. Strictly speaking, because of imperfection of gratings, a built-in pattern is generated even without a sample. Therefore, $\tilde{\phi }$_x(x, y,t) is actually obtained by calculating the argument of α ₁(x, y, z ₀,t)/α _1,0(x, y, z ₀), where the denominator is obtained without the sample in the same manner.

Thus, $\tilde{\phi }$_x(x, y,t) is calculated from a single moiré pattern, allowing dynamic observation. However, we need to accept the fact that spatial resolution in the y direction is limited by the carrier fringe spacing in principle because higher-frequency information is lost in extracting only the first order. Therefore, finer carrier fringes are preferable for better spatial resolution, and can be generated by increasing θ, although the generation of too-fine rotation moiré fringes may have the drawback of visibility loss.

For a CT scan, the sample rotation axis is set along the y axis. The sample is rotated continuously, and a movie of the differential phase image is obtained over several turns through the Fourier transform method.

3. Experimental method

Experiments were performed at beamline 14C1 of the Photon Factory, the High Energy Accelerator Research Organization, Japan with the setup shown in Fig. 1. We used white synchrotron radiation from a 5-Tesla vertical wiggler installed in the 2.5-GeV storage ring, which was operated with the single-bunch mode (50 mA). In addition to beamline windows, such as Be and Kapton, a 0.5-mm thick Al filter was used. The flux density in front of a sample placed about 37 m downstream from the source was calculated by SPECTRA [17], as shown in the inset of Fig. 1. By the Al filter, soft X-rays below 10 keV were considerably removed. The power density was estimated to be 0.15 W/mm².

 

Fig. 1 Experimental setup (side view) of four-dimensional X-ray phase tomography with a Talbot interferometer consisting of a phase grating (G1) and an amplitude grating (G2) and white synchrotron radiation. The inset show the calculated flux density at the sample position.

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We used a set of a phase grating and an amplitude grating, whose pitch was 5.3 μm [18]. When the moiré-fringe visibility was measured as a function of the distance between the gratings, a distance of 326 mm exhibited the best visibility. In this study, the grating distance was therefore set to 326 mm, which is optimal for 28.8-keV X-rays. The gratings were aligned so that their diffraction vectors were almost vertical since the vertical size of the electron bunches in the storage ring was much smaller than their horizontal size, and the spatial coherence was therefore better in the vertical direction than in the horizontal direction. The vertical bunch size was 45 μm (standard deviation). The spatial coherence length in vertical direction at 28.8 keV was therefore 5.6 μm. This value meets the requirement that the spatial coherence length be larger than the pitch of the phase grating for the operation of the Talbot interferometer.

We observed a living larva of nokona regalis, which is a popular fishing worm, for demonstration. The worm sample was glued softly onto the inner wall of an tube of low-density polyethylene whose inner diameter was about 7 mm. The tube was placed in front of the phase grating, and its rotation axis for the CT scan was set horizontally. Thus, in-plane differentiation of the phase shift was sensed in tomography. The beam size, however, was long in the vertical direction because we used the vertical wiggler, and the field of view along the sample rotation axis was 13 mm. Therefore, as described later, two measurements were performed with different worms for the head half and the tail half. The sample was rotated continuously at a speed of 1.0 rps.

An X-ray image detector, which was composed of a 20 μm-thick Y₃Al₅O₁₂:Ce phosphor screen (P46, Hamamatsu), a coupling lens system (AA60, Hamamatsu), and a CMOS (Complementary Metal Oxide Semiconductor) camera (pco.1200hs, PCO AG), was aligned behind the amplitude grating. The CMOS camera had a 1280 × 1024 pixel array and the effective pixel size was 12.7 μm including the slight demagnification of the coupling-lens system. The camera was operated at a frame rate of 499.3 f/s, and the exposure time per image was about 2 ms.

Figure 2(a) shows a differential phase image of a worm in a tube generated through the Fourier transform method (the corresponding real-time movie is shown in Media 1). A sinogram on the position indicated by the dashed line in Fig. 2(a), the length of which corresponds to about 5.5 turns (that is, about 5.5 s), is shown in Fig. 2(b). A phase tomogram was reconstructed by extracting a 180° portion from the sinogram, as shown in Fig. 2(c), because synchrotron radiation can be assumed to be a parallel beam. The reconstruction of a phase tomogram from the sinogram was performed by the filtered backprojection method with the special filter [16] used for the direct reconstruction of $\tilde{\delta }$ from $\tilde{\phi }$_x. The frames of the phase tomogram movie were created by shifting the origin of the extracted sinogram portion used to reconstruct tomograms. The step of the shift was set not to 180° but to 36° to generate a movie showing smoother movement. Thus, the movie of the phase tomogram shown later is formed with 10 f/s. This procedure was performed for every sectional position, and a four-dimensional phase tomogram was thus finally reconstructed.

 

Fig. 2 Procedure for the reconstruction of a four-dimensional phase tomogram movie: (a) differential phase image ( Media 1); (b) sinogram corresponding to the position indicated by the dashed line in (a); and (c) phase tomogram reconstructed from the portion indicated by the box in (b). A movie of the phase tomogram is generated by shifting the box along the sinogram.

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4. Results

Figure 3 shows phase tomograms obtained for a living larva of nokona regalis at 10-slice (127-μm) intervals from the head to the middle of its body, which were reconstructed from the data for 0.5 s just after the start of exposure. Corresponding Media 2 shows a section scan movie. Many void structures were found inside the body. Because the grayscale value in the void is the same as that of the outside (air), the structure is considered to be the respiratory tract. A contrast depicting organs is also detected, although it is partially unclear due to motion artifacts.

 

Fig. 3 Phase tomograms of the head half of a living larva of nokona regalis at 10-slice (127-μm) intervals just after the start of exposure. All phase tomograms are given sequentially in Media 2.

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Volume-rendering movies of the four-dimensional phase tomography data are given in Media 3 and Media 4, and representative views at 0, 1, 2, 3, and 4 s after the start of the exposure are shown in Fig. 4. Because of the limited beam size of synchrotron radiation, two measurements were performed for the head half and tail half using different worms. Here, the tube holder is cropped and only the body of the worm is shown.

 

Fig. 4 Rendering views of the phase tomogram of the living larva of nokona regalis at 0, 1, 2, 3, and 4 s after the start of exposure to white synchrotron radiation. Because the beam size was smaller than the whole body of the worm, its head part (lower row) and tail part (upper row) are shown. Real-time movies of the four-dimensional phase tomograms are presented in Media 3 and Media 4, respectively.

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In Fig. 4 (lower row) and Media 4, tubular and pod-like features corresponding to the void regions seen in Fig. 3 are clearly seen. Two pod-like regions are paired and connected to the respiratory tracts on the sides. Note that the pod-like features deflated soon after exposure. Figure 5 and Media 5 and Media 6 are given to show the change more clearly. Movement of the tubular feature was also detected at the tail region soon after exposure, as shown in Fig. 4 (upper row) and Media 3, where the tubule near the tail end moved quickly. These movements are speculated to be a response to the heating caused by white synchrotron radiation (0.15 W/mm²); after the measurement, the worms died.

 

Fig. 5 Surface-rendering views of the tubular and pod-like features just after exposure (left) and after 4.9 s (right). The view at 0 s from different directions is seen in Media 5. The time evolution of the features is shown in Media 6.

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5. Discussion

As mentioned, the Fourier transform method was adopted to realize four-dimensional X-ray phase tomography. A drawback to this approach is that the spatial resolution in the direction of the carrier fringe frequency is degraded. In the measurements presented, the direction parallel to the sample rotation axis corresponds to the problem. In order to improve the spatial resolution, we need to introduce fine carrier fringes. As a result, we carried out the presented experiments with carrier fringes of 77.5-μm intervals. When finer carrier fringes were attempted by rotation moiré, we suffered from greater fringe visibility reduction, which was a crucial problem in ensuring image quality.

Two causes of the visibility reduction are pointed out. One is an insufficient modulation transfer function (MTF) value at the higher spatial frequency of our image detector, whose spatial resolution was mainly limited by the screen thickness (20 μm). The other is the effect of the spatial coherence of X-rays. When the grating is greatly inclined around the optical axis, the shift of the ±1st diffraction beams from the grating has a horizontal component to some extent. Then, the spatial coherence of the synchrotron radiation in the horizontal direction affects the performance of Talbot interferometry. Because the spatial coherence length in the horizontal direction is roughly ten times smaller than that in the vertical direction, this effect should be considered when fine carrier fringes are introduced. To avoid this drawback and generate finer carrier fringes with sufficient visibility, the use of gratings of a smaller pitch is effective, overcoming the difficulty in the fabrication of the second grating, which requires a high-aspect-ratio structure.

If the spatial resolution of an image detector is sufficiently smaller than the grating pitch, the second grating is no longer necessary, and the self-image can be resolved directly [19]. The self-image has a high-frequency intensity pattern, and is processed by the Fourier transform method. Therefore, four-dimensional X-ray phase tomography is also attainable with such a configuration.

The spatial resolution in the plane perpendicular to the sample rotation axis is related to the time resolution. The sample was rotated continuously in this study, while, for static observation, a sample is rotated step by step and images of the stationary sample are acquired. Such motion may cause annoying vibration when applied in high-speed phase tomography. Therefore, the sample was rotated continuously at a speed of 1.0 rps in a tube about 7 mm in diameter, accepting motion blur to some extent. The CMOS camera was operated at a frame rate of about 500 f/s. Maximal motion blur is therefore calculated to be 44 μm. Note that the effect due to sample motion, which causes motion artifacts, should be discussed separately.

In this study, white synchrotron radiation was used only with a 0.5-mm Al filter, and the spectral bandwidth was not optimal for the Talbot interferometer. It is clear that better image quality would be attained if the spectral bandwidth were tailored to the Talbot interferometer. Bandpass control must be effective for reducing unnecessary dose to the sample as well. We applied approximately several seconds exposure to the worm sample till the onboard memory (4 GB) of the CMOS camera was full. The exposure was actually fatal to the living worm.

Simply, the use of a filter, which has an appropriate absorption edge, is effective in suppressing higher-energy X-ray photons to some extent. In order to tailor the spectrum to be suitable for an X-ray Talbot interferometer, bandpass filtering with a multilayer mirror may be more effective. However, the size of the field of view (i.e., beam size) is of crucial importance in imaging applications. A longer multilayer mirror must be fabricated to make a thicker beam because multilayer mirrors are used under a grazing incidence condition in the hard X-ray region.

As another possible approach, we have the idea shown in Fig. 6, where three gratings (G1, G2, G3) are used. A periodic intensity pattern corresponding to G1 is formed by the fractional Talbot effect at the position of G2. Note that the periodic intensity pattern is clear at a specific energy depending on the G1–G2 distance while the periodic intensity patterns at different X-ray energies are more blurry. Therefore, G2, which must be an amplitude type and has a pitch equal to the period of the intensity pattern, can filter out the X-rays of a specific energy range. The X-rays passed through G2 form the self-image of G2 at the position of G3. Thus, the set of G1 and G2 functions as a bandpass filter based on the fractional Talbot effect, and the set of G2 and G3 is the Talbot interferometer. In this case, a sample should be placed behind G2 because the filtering performance is negatively influenced when the sample is located between G1 and G2. The grating opening ratio, which is 0.5 in the case of normal X-ray Talbot interferometers, is the key parameter in filtering performance. The opening size should be appropriately narrowed to attain sufficient band-pass filtering performance.

 

Fig. 6 X-ray Talbot interferometer with the function of a bandpass filter. The system consists of three gratings (G1, G2, G3), and G2 and G3 must be amplitude gratings. The set G1–G2 functions as a energy filter and the set G2–G3 is an Talbot interferometer.

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In this study, a Talbot interferometer was used with white synchrotron radiation to realize four-dimensional X-ray phase tomography. The Fourier transform method used in this scheme is also powerful with other opportunities when high-speed imaging is crucial. For example, systems consisting of three gratings [20] (different from Fig. 6) and/or coarse amplitude gratings [21, 22] are operated with a conventional X-ray generator, opening up practical applications for X-ray phase imaging. The Fourier transform method has been adopted in some works [22–24], considering that shortening the exposure time is important for practical applications even for two-dimensional imaging. The presented approach therefore functions with various forms of X-ray phase imaging, opening up wide applications.

Finally, a comment is made on the novel contrast generated by grating interferometry [25]. The contrast is obtained from the same data set measured for phase imaging by calculating the reduction in visibility of fringes caused by a sample. The contrast origin is considered to be ultra-small angle scattering by the microstructures distributed in a sample; scattered X-rays no longer contribute to the formation of fringes, consequently reducing visibility [26]. Although the microstructures cannot be resolved by system spatial resolution with phase contrast, their distribution is visualized in the visibility image. Note that this visibility contrast can be obtained in the process of the Fourier transform method with 2α ₁(x, y, z ₀ ,t)/α ₀(x, y, z ₀ ,t). High-speed imaging and tomography based on this contrast mechanism is therefore possible as well. A preliminary result has been reported elsewhere [27].

The obtained volume movie has never been acquired by any other techniques to our knowledge, stimulating entomological interest. Extensive study will be performed by observing many other kinds of samples in collaboration with biologists. Furthermore, the application to the field of material science and engineering is also attractive. For example, phase separation dynamics in polymer blend, deformation and destruction of composite materials would be observed dynamically. We expect that the presented technique opens up a new frontier of X-ray imaging.

6. Conclusion

Because of the advantage of Talbot interferometry, which functions with X-rays of a broad energy bandwidth, high-speed X-ray phase imaging is possible in combination with white synchrotron radiation. For phase measurement, the Fourier transform method was adopted instead of the fringe-scanning method. Four-dimensional X-ray phase tomography for a living worm was successfully demonstrated with a time resolution of 0.5 s. Clear movement in the body was revealed particularly in the respiratory tract. An idea for the improvement of image quality by band-pass filtering was also mentioned. The presented approach will open up opportunities for four-dimensional observation of biological objects and others, whose static structures have only been detectable with conventional X-ray phase tomography.