1. Introduction
X-ray computed tomography (CT) is a powerful three-dimensional (3D) imaging method for the non-invasive inner exploration of materials [
1] and biological samples [
2]. X-ray CT at storage ring sources is traditionally considered a benchmark for investigations in the soft [
3] and hard [
4] X-ray regimes; it enables access to length scales as low as 1 µm and 50 nm spatial resolutions in the fields of microtomography [
5] and nanotomography [
6], respectively. However, X-ray CT inevitably requires up to thousands of projections to build a single high-resolution 3D volumetric image.
The strong reduction in the number of projections for achieving 3D or quasi-3D images is an important objective today. Quasi-3D encompasses all of the techniques—such as stereoscopy or holography—that partially sample the object in 3D. The advantage of single-shot quasi-3D imaging over traditional CT is twofold: (a) it is suitable for fast dynamic processes, and (b) it intrinsically avoids the need to rotate the sample. (a) The temporal resolution of an acquisition is reduced from the duration of a scan to the duration of a single exposure. This can signify the ability to track flow dynamics in perfusion CT, intravascular tools in interventional imaging, changes induced by fast chemical reactions in operating energy storage devices, or crack nucleation and propagation in material deformation experiments, all in 3D. (b) The required sample rotation in synchrotron CT can interact with fluid samples, and can strongly influence their state and behaviour. The ability to resolve their 3D structure without rotating them can unlock unprecedented studies of their unaltered functioning state (e.g., particle image velocimetry of opaque fluids); moreover, it can enable the reconstruction of specimens with anisotropic or non-cylindrical shape (e.g., biological sample slices).
Several attempts have been made to recover quasi-3D information from a pair of views using X-ray stereoscopy [
7,
8] and a single sample orientation using X-ray ankylography [
9]. Recently, visible plenoptic imaging has attracted attention given its ability to record depth information in one view. From a single exposure, plenoptic imaging enables the generation of a stack of images located at several distances along the optical axis. The technique is based on a combination of an objective lens and an array of microlenses [
10,
11]. To date, this method is still restricted to visible light, due to the complexity of using X-ray optics. In the literature there can be found numerical simulations emulating the operating mode of an X-ray plenoptic system with limited-angle tomography setups [
12,
13]. A preliminary experimental demonstration of plenoptic X-ray imaging was performed by Sowa et al. in so-called multipoint projection geometry [
14,
15]. Using a laboratory source, Sowa et al. developed an X-ray plenoptic microscope version based on polycapillary focusing optics, instead of using a microlens array [
14]. As consequence, the setup shows a limited depth resolution due to the limited angular sampling of the polycapillary devices and a lateral resolution restricted to micro-sized sources [
14]. This article shows an advancement of the optical design required for the implementation of an X-ray plenoptic microscope, thus enabling an increase in the angular and spatial sampling of the incoming X-rays, as well as enhancement of the refocusing ability. Here, the first X-ray plenoptic microscope based on an X-ray optics array is presented and used in a flexible configuration [
16]. The microscope was created by combining a Fresnel zone plate (FZP)—i.e., the objective lens—with a movable array of FZPs—i.e., the microlens array—placed in front of a detector. This implementation is flexible, enabling different plenoptic imaging configurations on the same setup. By slightly modifying the distances between the optical elements, it is possible to switch from the focused plenoptic imaging geometry [
11] to the classical plenoptic imaging configuration [
10], both known in visible light photography. Any change of distances implies a change of angular and spatial light field sampling, and of the related lateral and longitudinal resolutions. Therefore, the different microscope parameters can be adjusted according to the acquisition needs. The X-ray microscope geometry adopted in this work reproduces the focused plenoptic camera configuration [
11]. The X-ray plenoptic microscope implemented here was thoroughly tested by imaging USAF test targets placed at different positions, and by computationally refocusing the recorded raw plenoptic images. To increase the spatial sampling, the image acquisition was performed by scanning the FZP array 16 times. Finally, the acquired micro-images were stitched leading to better resolved image features compared to the initial number of available micro-images. The achieved spatial and longitudinal resolutions were measured on the refocused images considering different approaches, and the estimated values were compared to the theoretical calculations.
3. Results
Figure 2a shows characteristic features in the raw plenoptic images produced by the X-ray microlens array. Due to a mismatch in size between the OSA around the main FZP and the microlenses’ diameter, several grey lines corresponding to 0th order transmission can be observed. Superimposed on these lines are circles with black structures at their centres. A closer look at these circles (
Figure 2b) reveals that the inner black structures are images of a small part of the sample produced by the +1st diffraction order (called micro-images). The use of a beam stop array placed after the microlens array prevents the 0th order (red dotted circle in
Figure 2b) from mixing with the +1st order (yellow dotted circle in
Figure 2b). The plenoptic information is contained only in the yellow circle (1st order). Only the signal from these pixels can be used for data treatment. The objective lens magnifies the front and the back targets by approximately 30.6 and 29.6 times, respectively. The microlenses magnify by a factor 0.2, leading to a total magnification of ~6. The beam divergence at the objective lens exit was estimated at around 0.3 mrad, while for the microlenses, the entrance divergence was 0.11 mrad. This mismatch of divergences generated small micro-images, ~20 µm in diameter (i.e., 22 pixels), separated by a distance corresponding to the microlens pitch of 110 µm (
Figure 2a). In order to achieve a denser sampling, a new image was taken with a shift of the microlens array by a quarter of the inter-lens spacing, i.e., by 27.5 µm; these two images were then merged. This procedure was repeated three times in both the horizontal and vertical directions, generating a stitched plenoptic raw image, composed of 36 × 36 micro-images (
Figure 2c).
The raw data were treated using a dedicated algorithm (see Materials and Methods) that can operate on all plenoptic configurations (focused and unfocused) [
22].
Figure 3 shows the comparison between the images obtained using the objective lens only (
Figure 3a), using the objective lens and the microlens array corresponding to classical plenoptic imaging (
Figure 3b), and the resulting plenoptic image after stitching (
Figure 3c).
The three images were obtained with the same position of the targets adjusted such that the first image is in focus, thus placed at a position of approximately Z
0exp = 128.7 mm from the objective lens. However, for
Figure 3a the detector was situated at the position of the intermediate image plane (
Figure 1), while it was moved further from the objective lens in order to insert the microlens array for the plenoptic configuration. This leads to different magnifications in
Figure 3a–c. The 2 µm spaced lines of the first and second targets are recognizable on the right part of the reconstructed plenoptic image (
Figure 3b); they appear dashed, and not plain as they are in reality (
Figure 3a). In addition, on the left-hand side the number “2” of the first target (TP1) is barely visible, and the lines of the back target (TP2) with spacing lower than 1 µm are not resolved; this is due to the low number of microlenses (9 × 9) used for sampling the intermediate image created by the main lens.
Figure 3c displays the reconstructed image of the targets using the synthetic 36 × 36 micro-images. Compared to
Figure 3b, the improvement is striking. The noise has been strongly reduced thanks to the addition of 16 images. In addition, the true image can now be resolved, showing the plain lines instead of the dashed lines seen in
Figure 3b. Additionally, the number “2” is now clear and readable (
Figure 3c). The image of the back target remains blurred due to being out of focus. This first step of the study shows the quality of the plenoptic reconstruction.
The first dataset was taken with the first test target (TP1) situated close to the best experimental in-focus position. The second target (TP2) was placed 1.3 mm from TP1. An image stack was computationally produced by varying the focus position of TP1 from 128.6 mm to 129.5 mm in 100 µm steps in order to find its most accurate in-focus position. Four consecutive refocused images of TP1 are displayed in
Figure 4a,d,g,j, within a 0.3 mm long range.
Figure 4d,g exhibit images of TP1 with the highest contrast over the full scan, implying that both refocused planes can be considered to be in focus at the same time. This was expected, since the separation of these two reconstruction planes is lower than the theoretical depth of field [
22], found to be 0.3 mm in this demonstration. Image processing allowed Z
0exp to be defined as 128.75 ± 0.05 mm. In all of the computationally refocused positions, TP2 was blurred, since it was placed ~1 mm away from the depth of field of the whole plenoptic system. Experimental results were compared with simulated images. To precisely characterise the plenoptic X-ray microscope, the algorithm used for reconstructing the data was reversed, allowing us to generate synthetic plenoptic raw images [
21,
22] (see Materials and Methods) of the three TP1 bars labelled 1, 2, and 3 in
Figure 4g. All parameters of the simulation were chosen to be the same as in the experiment, considering TP1 placed at Z
0exp = 128.7 mm. The simulated refocused images are displayed in
Figure 4c,f,i,l, and are a zoomed view of TP1’s bars. The simulated images of the bars are compared with the magnified views of the refocused images of the bars shown in
Figure 4b,e,h,k. The visual agreement between modelling and experiment is good for all of the refocused positions. The ability of the plenoptic X-ray microscope to perform digital refocusing and defocusing from a single acquisition was subsequently tested. The assembly of TP1 and TP2 was positioned at different distances from the FZP. For each raw image acquired, a refocused image stack was reconstructed by the refocusing algorithm, and the image showing the highest contrast on either TP1 or TP2 was selected. The highest contrast was determined by visual inspection, as there was a marked difference between the in-focus image corresponding to the highest contrast and the other unfocused images.
Figure 5 shows three refocused images acquired at different positions of the assembly of TP1 and TP2. The first position labelled “0” in
Figure 5 corresponds to
Figure 4d at Z
0exp = 128.7 mm. Moving the target assembly closer to the main lens by 0.5 mm (position 1,
Figure 5), both targets appear out of focus, although the image of TP2 is slightly sharper and more resolved compared to the structures in
Figure 4d,g. Moving the assembly once more towards the main lens by 0.5 mm (position 2,
Figure 5), the image of TP1 becomes totally blurred. Due to a small change in contrast between positions 1 and 2, section profiles were not sufficient to define the best focusing distance for TP2 (see
Appendix A,
Figure A4). Thus, the long bar of TP2, indicated by the yellow arrow in each sub-image of
Figure 5, was identified as a reference feature to determine the focusing trend for TP2. While this line is single (see
Appendix A,
Figure A3) and well defined at position 2, it looks doubled at position 1, and even more doubled at position 0. This is evidence that TP2 is defocused at positions 0 and 1, while its sharpness improves moving towards position 2; however, position 2 is not quite its exact focal plane. Similarly, the orange arrow indicates the focusing evolution of TP1 for the three different cases. The chart border, indicated by the orange arrow, is sharp and single at position 0, and then it becomes doubled at position 2. In this condition, TP2 was positioned at 0.3 mm from the initial position of TP1. This indicates a depth of field smaller than 0.3 mm, and confirms the plane of best focus to be around Z
0 = 128.75 mm. TP1 and TP2 are never completely in focus at the same time.
Spatial resolution is a standard parameter used to find the focal plane for classical cameras, and it is used here to confirm the position of the best digitally refocused plane. The spatial resolution was estimated by two independent techniques giving similar values. The first estimation was done by estimating the point-spread function (PSF) directly from the gradient of the intensity along a line crossing a sharp edge. This measurement was performed on the large bars labelled 5 and 6 in
Figure 4g. The derivative generated a Gaussian shape for the PSF with a resolution of ~420 ± 60 nm. The resolution estimation was obtained averaging over five pixels. However, the signal-to-noise ratio was too low (2–3) for ensuring an accurate measurement. Alternatively, it can be considered that the narrower black bars (labelled 1–4 in
Figure 4g) are the convolution of a 1 µm width square function with a Gaussian PSF [
23]. Thus, the second estimation was performed by convolving the 1 µm bars (labelled 1 and 4) with a Gaussian function of different widths. The results are full width at half-maximum values, and are displayed in
Figure 6a,b. The measured resolution varies from 350 nm ± 50 nm to 660 nm ± 50 nm.
Figure 7 displays the cross-sections of bars 1–3, extracted from both experimental and numerical images (
Figure 4h,i, respectively) obtained at Z
0 = 128.8 mm, corresponding to the plane of best focus. The theoretically computed curve is in good agreement with the experimental measurement. It is worth noting that the shape of the three bars obtained experimentally is accurately reproduced numerically. As shown in
Figure 6, the modification of the bar shape from a top hat (left and centre bars) to a Gaussian shape (right bar) indicates a variation in the spatial resolution across the image.
As a final step of the study of this plenoptic X-ray microscope, the longitudinal resolution of the refocused images was investigated by scanning the spatial resolution over a wide range of refocused positions.
Figure 8 displays both numerical and experimental spatial resolutions versus position along the optical axis. Only the first target TP1 located at Z
0exp = 128.8 mm was considered. Both experimental and numerical resolutions were calculated using the image of bar 2 (in
Figure 4g), and treated using the convolution technique explained above (
Figure 6). The overall variation in the spatial resolution with the refocused position matches very well between experiment and modelling, although numerical images always produce a slightly better resolution. The best spatial resolution equals 0.4 µm for modelling, and compares well to the 0.45 µm found in the experiment. The discrepancy from the focal plan increases for the highest positions due to the stronger influence of noise when the image of the bar becomes more and more blurred. The experimental curve (in green) shows a kind of plateau of best resolution centred at Z
0exp = 128.71 mm and extending over ~80 µm, while the numerical curve (black curve) has a less pronounced plateau, extending over ~100 µm. These two values, 80 and 100 µm, correspond to the experimental and numerical longitudinal resolutions, respectively. Finally, the voxel size achieved with this first demonstration is equal to 0.45 µm × 0.45 µm × 80 µm.