1. Introduction

Ghost imaging (GI) is an unconventional imaging technique that creates an image of an object by using a single-pixel detector and a reference beam. Unlike traditional imaging methods that require an array of detectors or a pixelated detector, GI relies on the correlation between the light transmitted through an object (termed the ‘object beam’) and the reference beam to produce an image [1]. Theoretical studies show that it is possible, in principle, to reduce dose with GI given prior knowledge of the sample [2,3]. X-ray GI [4,5] is an emerging technique, whose foundations are still under development. Before attempting to achieve dose reduction, we must determine how to optimize current GI techniques, e.g., mask design and fabrication, experimental protocols, and image recovery methods. This paper represents a step towards this goal. In particular, GI has the potential to produce an image of an object with a small number of photons, compared to traditional imaging techniques [2,3,6]. When using ionizing radiation, this implies that ghost imaging might be able to produce an image with much less radiation exposure to the patient or object, when compared with conventional imaging. GI has been explored in a range of application areas including medical imaging [7], remote sensing [8,9], 3D imaging [10], astronomy [11], biometric identifications [12,13] and face recognition [14].

In classical ghost imaging [15], a patterned object beam interacts with an object and transmitted or backscattered photons are collected by a single-pixel detector (also called a ‘bucket detector’). The reference beam, however, does not interact with the object. It interacts with some form of beam modulation to create patterned illumination and the transmitted intensity is recorded by a multi-pixel detector. The image of an object is then reconstructed by correlating the object beam and the reference beam (in other words, by correlating the intensities collected by the bucket detector and the multi-pixel detector).

The beam modulation technique or device that is placed upstream of the object (to produce the illumination pattern as recorded in the reference beam) is crucial to the success of ghost imaging. The pattern must be aperiodic [16] and have a well-defined statistical distribution to ensure that the correlation between the reference and object fields is strong enough to reconstruct the image. Furthermore, the choice of illumination patterns can affect the contrast and the resolution of the reconstructed image [17]. Some particular patterns can improve the visibility of particular features in the object, while others can suppress unwanted noise and background signal. Therefore, careful consideration of the illumination patterns is necessary to achieve high-quality ghost images, since the quality of the images is directly related to the chosen patterns.

Spatial light modulators [18,19] or digital micromirror devices [20] can be used to generate such aperiodic structures in the visible-light light band. However, these techniques are not effective for weakly interacting probes such as X-rays and neutrons. The illumination patterns that are typically used in these short-wavelength regimes are non-configurable transversely-displaced masks, which can be either natural or structured masks [17]. Natural masks are made of materials that are not specifically designed for use in imaging, but can still be used to modulate the beam in a random or semi-random way. Some examples of these natural masks are sandpapers [21], copy paper [22], iodized table salt grains [23], and metal foams [24]. They are commonly used because they are cheap and widely available. However, they are not always efficient compared to structured masks because they have pseudo-random and unpredictable patterns. In addition, natural masks should be examined to ensure that they can produce a sufficient number of unique illumination patterns, which is an important requirement for ghost imaging.

Structured masks have specific patterns or shapes that are designed to encode information regarding the object being imaged. They are based on intensity modulation through attenuation- or phase-contrast and generally contain patterns of transparent and opaque regions that can be specifically designed to match the characteristics of the object being imaged and/or the photon energy being used. For instance, He et al. [25] fabricated an Au X-ray modulation mask with 324 pairs (positive and negative) of binary patterns. They matched the thickness of the patterns with the X-ray photon energy being used, such that the transmission through the Au area was less than 1% at that photon energy. Utilizing this mask to generate illumination patterns, they reconstructed ghost images with fewer measurements compared to the traditional scanning methods. This can correspond to reduced radiation exposure, which is beneficial especially in medical and biological imaging. In another X-ray study [26], a fabricated mask with Hadamard patterns and a sandpaper mask were employed separately in GI for comparison of performance. The quality of the ghost images obtained by the Hadamard mask were much higher than the quality of that from the sandpaper, when using the same number of measurements. This can be due to increased contrast and the orthogonality of the Hadamard patterns making this mask less sensitive to noise and other sources of imperfection, improving the quality of the final image. A main drawback of these orthogonal masks is their sensitivity to alignment. They require precise alignment with the imaging system in order to produce high-quality images. Any misalignment can introduce errors or distortions in the final image. Inaccuracy in the fabrication process can also affect the quality of the ghost image [25].

For the emerging field of X-ray ghost imaging [4,5,21,22,27], we have designed and fabricated a range of illumination patterns—including random, fractal, and orthogonal configurations—to explore the advantages and disadvantages of each type of pattern in terms of robustness, sensitivity, and scalability. Moreover, utilizing a laboratory X-ray source, we experimentally demonstrated and evaluated the quality of ghost images obtained by each class of illumination pattern. In addition to GI using individual masks, a series of GI experiments were also performed utilizing a mixture of patterns, to understand the effect on GI quality of combining different types of mask.

The remainder of the paper proceeds as follows. Section 2 considers mask design (Sec. 2.1), mask fabrication (Sec. 2.2), and means for analyzing the quality of the fabricated mask (Sec. 2.3). The experimental ghost imaging setup is also given in this section (Sec. 2.4). Experimental results of X-ray ghost imaging using the fabricated illumination patterns are shown, discussed, and evaluated in Sec. 3. This includes GI with individual random masks (Sec. 3.1), GI with mixtures of random masks (Sec. 3.2), and GI with orthogonal masks (Sec. 3.3). Section 4 provides discussion on the key parameters of each illumination pattern based on the experimental results. Finally, a brief conclusion along with indicators for future work are presented in Sec. 5.

2. Materials and methods

Here we report the materials and methods used throughout the paper. The set of pattern designs, selected and incorporated into the non-configurable X-ray GI mask, is outlined in Sec. 2.1. The fabrication process, employed to realize these designs, is given in Sec. 2.2. The quality of mask fabrication is analyzed in Sec. 2.3. We close by describing the experimental setup, used to perform X-ray ghost imaging by transversely displacing our non-configurable mask, in Sec. 2.4.

2.1 Mask design

We selected multiple binary patterns with different sizes and resolutions (20 $\mathrm{\mu}$m and 40 $\mathrm{\mu}$m), to be incorporated on a 4-inch circular area. All patterns were chosen to be binary, to make the fabrication process efficient. Moreover, binary patterns increase variance, which is important in ghost imaging [17]. These patterns can be divided into two main classes: random structures and orthogonal patterns.

The random structures include:

Images of random, fractal, Gaussian blurred, and Lorentzian blurred patterns are shown respectively in Fig. 1(a), 1(b), 1(c), and 1(d). For more details on each pattern and precise mathematical definitions, see Refs. [17,28].

 

Fig. 1. Subsets of designed illumination patterns: (a) random mask, (b) fractal mask $\mathcal {C}_{\alpha,\beta }$ ($\alpha =$ −1, $\beta$= 0), (c) Gaussian mask ($\sigma =$ 8.5 pixels), (d) Lorentzian mask ($\gamma =$ 14.14 pixels), (e) URA mask, (f) Legendre mask. URA and Legendre masks are orthogonal under translation by an integer number of pixels in horizontal and vertical directions.

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The illumination patterns in the second class are orthogonal in some way. We selected uniformly redundant arrays (URA) and Legendre structures that are orthogonal under translation. The URA orthogonal basis functions are generated by the quadratic residue method [29,30] and the Legendre patterns [31] are constructed by combining the Legendre sequence [32] with the finite Radon transform (FRT), as described in Ref. [33]. Both of these classes of patterns are based on prime numbers and generate a $p \times p$ array where $p$ is prime. Images of example URA and FRT-Legendre masks for $p = 67$ are illustrated in Fig. 1(e) and 1(f) respectively. We also included a set of Hadamard bases [34] for comparison. These are constructed as $n^2$ unique, orthogonal $n \times n$ array patterns where $n$ is a power of two. We selected $n = 32$ (not shown in Fig. 1).

Each illumination pattern, as shown in Fig. 1, can be treated as a single mask and can also be generated with different sizes and resolutions. In our design, we tried to fit a wide variety of masks onto our 4-inch substrate. An overview of our mask design is depicted in Fig. 2. As seen in Fig. 2(a), random, fractal, and orthogonal masks are generated with two different resolutions, i.e. 20 $\mathrm{\mu}$m and 40 $\mathrm{\mu}$m. The orthogonal masks (URA and FRT-Legendre) are also designed with three different sizes: $p = 31$, 67, and 127. The Gaussian and the Lorentzian masks are generated with only 20 $\mathrm{\mu}$m resolution, because they have relatively large structures. The size of all random masks (the top two rows of patterns) is approximately 10 mm $\times$ 10 mm.

 

Fig. 2. (a) Layout of the mask showing different types of illumination patterns on a 4-inch substrate, (b) fabricated mask on a 4-inch $\textrm {SiO}_2$ substrate. The numbers with $\mathrm{\mu}$m units indicate the resolution/pixel size of the masks.

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2.2 Mask fabrication process

The majority of the fabrication process was conducted at the Melbourne Centre for Nanofabrication (MCN), Melbourne, Australia. The mask described in Sec. 2.1 was fabricated on a 4-inch $\textrm {SiO}_2$ wafer with a thickness of 700 $\mathrm{\mu}$m. A schematic of the fabrication process is shown in Fig. 3. The $\textrm {SiO}_2$ wafer was first cleaned with piranha solution to remove organic residues from the surface. Following this, 20 nm of Cr and 100 nm of Au were respectively deposited on the substrate by an Intlvac Nanochrome sputtering tool, as depicted in Fig. 3(a). The deposition rate for both the Cr and Au layers was 16-20 nm/minute. The Cr layer was used as an adhesion layer between the substrate and the Au layer. The Au layer was used as a seed layer for the subsequent electroplating process shown in Fig. 3(e).

 

Fig. 3. Schematic of the fabrication process: (a) thin film deposition of 20 nm of Cr and 100 nm of Au, (b) spin-coating positive photoresist (AZ 40XT in our case) on the Au layer, (c) UV exposure through a physical mask (photolithography process), (d) photoresist development to remove the exposed area, (e) Au electroplating.

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The next step, after the thin-film deposition, was photolithography. A thick photoresist, AZ 40XT-11D, was chosen based on the desired thickness of the final structures. The aim was to achieve 20 to 30 $\mathrm{\mu}$m electroplated gold structures, to attenuate over 90% of the X-ray radiation over the energy range of 18 keV to 23 keV. For this purpose, the thickness of the photoresist should be greater than 35 $\mathrm{\mu}$m. AZ 40XT-11D positive photoresist was spin-coated on the Au layer at 3000 rpm, and then baked at 125 degrees for 5 minutes, as depicted in Fig. 3(b). The spin-coating and the baking processes were repeated two times, to obtain a photoresist thickness of approximately 40 $\mathrm{\mu}$m, based on the technical datasheet provided. At this stage, the wafer was ready for the photolithography process. This step was performed on an EVG6200 mask aligner instrument, using a custom design Chromium photolithography mask, which we fabricated by a direct write lithography machine (Intelligent Micropatterning SF100 XPRESS) at MCN. KLayout software was used to lay out the patterns for lithography. The dose of the UV exposure was set to 300 mJ, which was enough for $\approx$ 40 $\mathrm{\mu}$m thick photoresist, see Fig. 3(c). After the UV exposure, the wafer was baked at 150 degrees for 90 seconds and then developed into a AZ 726 developer solution for 5 minutes, resulting in the situation depicted in Fig. 3(d). The thickness of the photoresist after the development process was measured as approximately 40 $\mathrm{\mu}$m with an optical profilometer (Bruker Contour GT-I). Some optical images, of subsets of the random, fractal, FRT-Legendre and URA masks after the development process, are shown in Fig. 4. The trenches, which can be seen in the images, were filled with Au in the next step, namely the electroplating process depicted in Fig. 3(e).

 

Fig. 4. Optical images of subsets of the fabricated masks after the development process (i.e. step (d) in Fig. 3): (a) random mask, (b) fractal mask, (c) FRT-Legendre mask, (d) URA mask. The scale bars in all images are 100 $\mathrm{\mu}$m.

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The Au electroplating process was conducted on a custom-design instrument at RMIT University, Melbourne, Australia. The electroplating bath was Pur-A-Gold 402 supplied by Macdermid-Enthone and the time of electroplating was approximately 90 minutes. The electroplating time was calculated based on the area of the structures on the wafer. An image of the final result, after Au electroplating, is given in Fig. 2(b). Some parts of the Au layer, at the bottom left side of the wafer, peeled off in the electroplating process. This does not affect the functionality of the masks since the peel zones are outside the mask areas. However, the Hadamard pattern was affected by the peeling, therefore we could not use it in following experiments.

2.3 Mask fabrication quality analysis

Comparing the fabricated mask to the original design is necessary, to verify fabrication accuracy and also to evaluate mask performance. This is more important for orthogonal masks, as any deviation from the design can significantly affect ghost-image quality. A visual comparison, of the transmission images of the fabricated mask (captured using the experimental setup described in Sect. 2.4), with the corresponding original design (ideal masks), is given in Fig. 5. The top row shows transmission images of the fabricated URA, FRT-Legendre, and random masks with a 20 $\mathrm{\mu}$m mask pixel size. The bottom row shows the corresponding ideal masks. Comparing the top-row images to the corresponding bottom-row images shows that the only obvious deviation from the ideal masks is the limited range of transmission, i.e., reduced contrast. Otherwise, the features are captured accurately (except for the 50%-transmission pixel at the center of the FRT-Legendre mask, which was deliberately omitted to create a binary mask; note that setting that single element to 0 or 1 makes a small to negligible difference to that array’s performance).

 

Fig. 5. Comparison of transmission images (a-c) of $31 \times 31$ pixel regions of fabricated masks with corresponding ideal masks (d-f). Mask pixels are 20 $\mathrm{\mu}$m while the image pixel pitch is 2.86 $\mathrm{\mu}$m. The graylevel window is black-0% transmission to white-100% transmission. The transmission range in (a)-(c) is from 0.4 to 0.85. (a) URA mask, (b) FRT-Legendre mask, (c) Random binary mask.

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The quality of the fabricated masks can also be analyzed in Fourier space. Figure 6(i-ii) shows the magnitude of the frequency-space coefficients for all images in Fig. 5. Note that the DC coefficient (central pixel) has been zeroed for the images, i.e., the mask mean has been subtracted. Comparing the frequency-space of the transmission images of the fabricated masks (column (i) in Fig. 6) to the frequency-space of the ideal masks (column (ii)) shows that experimentally-obtained Fourier magnitudes match the ideal case very well. Note that, by design, the orthogonal masks sample Fourier space much more uniformly than the random binary mask. In addition to the Fourier space, we also looked at the corresponding autocorrelation plots, which are given in Fig. 6(iii-iv). Horizontal (x-axis) and vertical (y-axis) profiles through the autocorrelation are presented as a function of image pixels (pixel pitch is 2.86 $\mathrm{\mu}$m). The autocorrelation plots correspond to the point-spread-function (PSF) for GI using the adjoint method [27,35,36]; a sharper PSF is better. All masks have sharp autocorrelation peaks, however, the peak values are much lower than the ideal case, due to reduced image contrast. The tails of the autocorrelation of the fabricated URA mask in Fig. 6(a-iii) are not as flat as those of fabricated FRT mask in Fig. 6(b-iii). This could degrade the GI image quality, as discussed in more detail in Sec. 3.3. The apparent noise in the random-binary-mask autocorrelation tails is also present in the ideal case, indicating it to be a property of the pattern rather than from fabrication defects or noise.

 

Fig. 6. Frequency-space and autocorrelation comparison of transmission images of $31 \times 31$ pixel regions of fabricated masks (see Fig. 5(a-c)) with corresponding ideal masks (see Fig. 5(d-f)). The magnitudes of the Fourier coefficients are presented for the central $31 \times 31$ frequencies of the discrete Fourier transform (DFT); column (i) shows the experimental DFT and column (ii) shows the ideal DFT. Horizontal (x-axis) and vertical (y-axis) profiles through the autocorrelation (or adjoint-GI PSF [27,35,36]) are presented as a function of image pixels (pixel pitch is 2.86 $\mathrm{\mu}$m); column (iii) shows the experimental PSF and column (iv) shows the ideal PSF. (a) URA mask, (b) FRT-Legendre mask, (c) Random binary mask.

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2.4 Experimental X-ray ghost imaging setup

Ghost imaging experiments were carried out using a Thermo-Fisher Scientific Mk2 Heliscan at the National Laboratory for X-ray Micro Computed Tomography (CTLab) at the Australian National University (ANU) in Canberra, Australia. The tube voltage of the laboratory X-ray source was set to 60 kV, with a current of 80 $\mathrm{\mu}$A. The fabricated mask was placed in front of the beam and as close as mechanically possible (approximately 8.87 mm) to the X-ray source. The source-to-detector distance was approximately 835 mm. The detector was a Varex Imaging 4343CB amorphous-silicon flat-panel detector, which employs a CsI scintillator and has a $3072 \times 3072$ pixel array with a pixel pitch of 139 $\mathrm{\mu}$m. Since a 20 $\mathrm{\mu}$m mask pixel projected to nearly 14 pixels at the detector, the images were cropped to $3000 \times 3000$ pixels and then binned by a factor of six, to $500 \times 500$ pixels. GI experiments were performed in two steps. In the first step, the mask was scanned in front of the beam and a total of 961 images of each mask was collected through 31 $\times$ 31 raster scans. The transverse-displacement step size for the random masks was 200 $\mathrm{\mu}$m, and 20 $\mathrm{\mu}$m for the orthogonal masks. In the second step, the sample (a collection of various steel washers in our case) was placed between the mask and the detector, close to the mask. The near-binary test objects used in this work, whilst not able to gauge subtle variations in object transmission intensity, contain shapes that cover a wide range of 2D spatial frequencies required to assess the spatial resolution of our ghost measurements. Radiographs of the mask and the sample (in other words, the bucket images [37]) were then collected at the same positions used in step one. A schematic of the GI experimental setup is shown in Fig. 7. After the data were collected, the Kaczmarz algorithm [38] was employed to iteratively reconstruct images of the sample. The number of iterations and step size for each ghost image reconstruction is reported in the relevant captions. The relevant mathematics and detail of the Kaczmarz GI algorithm is explained in Ref. [17].

 

Fig. 7. GI experimental setup: (a) first step, corresponding to transversely scanning the patterned mask in front of the beam and collecting images of the mask, (b) second step, placing the sample downstream of the mask and collecting bucket images, each of which is then spatially integrated to give the corresponding GI bucket value.

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3. Results and analysis

In this section, X-ray ghost imaging results from the different types of illumination pattern are illustrated, and the quality of the resulting ghost images is analyzed. This section is structured as follows. In Sec. 3.1, we report the GI results from individual random masks, and also examine their scalability. Section 3.2 provides GI results from mixtures of the random masks. GI results with orthogonal masks are presented in Sec. 3.3.

3.1 X-ray GI results with individual random masks

GI was performed with a full set of masks over a 70 $\times$ 70 pixel area, then the field of view was increased to 140 $\times$ 140 pixels, and subsequently to 280 $\times$ 280 pixels (without increasing the number of masks). This enables us to evaluate the quality of the reconstructed images, as we reduce the number of patterns used relative to the amount of information required. Expected images for these three fields of view are depicted in Fig. 8. GI results from individual random masks over an area of 70 $\times$ 70 pixels are illustrated in Fig. 9(f-j). Examples of the random patterns employed—which include fractal masks with 40 $\mathrm{\mu}$m and 20 $\mathrm{\mu}$m mask-pixel sizes, a Gaussian mask with 20 $\mathrm{\mu}$m mask-pixel size, and random masks with 40 $\mathrm{\mu}$m and 20 $\mathrm{\mu}$m mask-pixel sizes—are shown in Fig. 9(a-e). A full set of masks (i.e. 961, as described in Sec. 2.4) was used for all of these reconstructions. As can be seen from the figure, although the quality of the ghost images is different, the object was resolved in all cases using all of the random masks.

 

Fig. 8. Expected X-ray images of the metal washers for different fields of view. (a) 70 $\times$ 70 pixels, (b) 140 $\times$ 140 pixels, and (c) 280 $\times$ 280 pixels. The demagnified pixel pitch is 8.9 $\mathrm{\mu}$m.

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Fig. 9. Top row: example patterns from (a) random fractal mask with 40 $\mathrm{\mu}$m mask-pixel pitch, (b) random binary mask with 40 $\mathrm{\mu}$m mask-pixel pitch, (c) Gaussian mask with 20 $\mathrm{\mu}$m mask-pixel pitch, (d) random fractal mask with 20 $\mathrm{\mu}$m mask-pixel pitch, and (e) random binary mask with 20 $\mathrm{\mu}$m mask-pixel pitch. Bottom row: the respective reconstructed X-ray ghost images with 250 (f), 250 (g), 250 (h), 100 (i), and 500 (j) Kaczmarz iterations. The step size was 0.1 for images (f)-(i) and it was 0.5 for image (j). 961 masks were used for each GI reconstruction. The demagnified pixel pitch is 8.9 $\mathrm{\mu}$m.

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Four different metrics were employed to evaluate the quality of the reconstructed GI images compared to the expected image shown in Fig. 8(a): root mean square error (RMSE), multiscale structural similarity (SSIM) [39], cross-image entropy [40], and Fourier ring correlation (FRC) [41,42]. These metrics were calculated for all the reconstructed GI images in Fig. 9 and plotted in Fig. 10. Note that 1.0-RMSE was plotted instead of RMSE so that ‘a larger value for the metric is better’ in all cases. The line-pairs-per-millimeter metric, plotted on the right axis, defines the spatial resolution that is calculated using FRC. Ghost images with higher similarity, higher cross-image entropy, and lower RMSE are obtained when using fractal and random masks with 40 $\mathrm{\mu}$m resolution. However, ghost images with higher spatial resolution are achieved when using fractal and random masks with 20 $\mathrm{\mu}$m resolution, as expected. Comparing the fractal 20 $\mathrm{\mu}$m and the random 20 $\mathrm{\mu}$m masks, the resolution obtained from the random mask is higher, however, the other three metrics are lower than those for the fractal 20 $\mathrm{\mu}$m mask. This might be due to the noisy background in the reconstructed image which is visible in Fig. 9(j). Another significant observation from the reconstructed ghost images is that resolving an image of the object using the Gaussian mask, which has large feature sizes, shows that resolution is not determined by pattern feature size alone, but also pattern sharpness.

 

Fig. 10. Four different metrics representing the quality of the reconstructed X-ray ghost images in Fig. 9. ‘Frac40’ and ‘Frac20’ are the random fractal masks with 40 $\mathrm{\mu}$m and 20 $\mathrm{\mu}$m mask-pixel pitch, respectively. Similarly, ‘Random40’ and ‘Random20’ are the random masks with 40 $\mathrm{\mu}$m and 20 $\mathrm{\mu}$m mask-pixel pitch, respectively. ‘Gaussian20’ represents the Gaussian mask with a mask-pixel pitch of 20 $\mathrm{\mu}$m.

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We investigated the scalability of the fractal and the random masks with 20 $\mathrm{\mu}$m mask-pixel size. For this purpose, the simulated bucket pixel dimension was increased by binning together larger arrays of pixel values in the 2D detector. The object was then reconstructed with different fields of view: 70 $\times$ 70 pixels, 140 $\times$ 140 pixels, and 280 $\times$ 280 pixels, as shown in Fig. 11. Note that we utilized the same number of masks (i.e. 961), in all cases. The top-row images in Fig. 11 illustrate the GI results for the random 20 $\mathrm{\mu}$m mask and the bottom-row images show the GI results for the fractal 20 $\mathrm{\mu}$m mask. As can be seen from Fig. 11 and the previous analysis (shown in Fig. 10), the resolution of the reconstructed image obtained with the random 20 $\mathrm{\mu}$m mask is higher than that from the fractal 20 $\mathrm{\mu}$m mask at 70 $\times$ 70 pixels area. However, as the field of view increases to 140 $\times$ 140 pixels (see Fig. 11(c)) or 280 $\times$ 280 pixels (see Fig. 11(e)), the resolution of the reconstructed images degrades compared to the resolution of the images reconstructed by the fractal 20 $\mathrm{\mu}$m mask (see Figs. 11(d) and 11(f)). This demonstrates that the random mask is desirable for the specific case for which it is designed and subsequently employed. The fractal mask, however, is spatially scalable and can be used for a range of cases, especially when there is a limited number of masks or when we want to reduce X-ray dose in GI.

 

Fig. 11. X-ray ghost images obtained from the random binary (top row) and the random fractal (bottom row) masks (with 20 $\mathrm{\mu}$m mask-pixel pitch) at different scales but using the same number of masks. Panels (a) and (b) are 70 $\times$ 70 pixels, (c) and (d) are 140 $\times$ 140 pixels, and (e) and (f) are 280 $\times$ 280 pixels. 961 masks are used for each reconstruction. The Kaczmarz iteration numbers are 100 for (b) and 500 for the other images. The step sizes are 0.1 for image (b) and 0.5 for the other images. The demagnified pixel pitch is 8.9 $\mathrm{\mu}$m.

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3.2 X-ray GI results with mixtures of random masks

In this section we show that utilizing different types of mask with different mask-pixel sizes, in a GI experiment, can enhance the quality of the reconstructed image. The key idea is that masks with larger feature size will resolve large parts of the sample, while masks with smaller feature size will add detail to the ghost image and thereby improve its resolution. To demonstrate this concept, a series of X-ray ghost imaging experiments was performed, with a mixture of the random masks having 40 $\mathrm{\mu}$m and 20 $\mathrm{\mu}$m mask-pixel sizes. First, a ghost image was reconstructed by having 100% of the illumination patterns obtained with the random 40 $\mathrm{\mu}$m mask. The total number of masks was 961. Then, the number of the random 40 $\mathrm{\mu}$m masks was reduced (by 25% each time) and the number of the random 20 $\mathrm{\mu}$m masks was increased (by 25% each time), as displayed in Fig. 12. For instance, Fig. 12(b) is a ghost image that is reconstructed from a mixture of 75% random 40 $\mathrm{\mu}$m masks (i.e. 721 masks) and 25% random 20 $\mathrm{\mu}$m masks (i.e. 240 masks). As previously mentioned, the total number of masks was kept constant, at 961 masks, for all of the ghost images. Visually, it can be seen from Fig. 12 that the degree of pixelation in the ghost images reduces as we move from left to right. In other words, the resolution of the ghost images improves as we add more of the random 20 $\mathrm{\mu}$m masks. Conversely, we also observe that the amount of noise increases as we add more of the random 20 $\mathrm{\mu}$m masks.

 

Fig. 12. X-ray ghost images utilizing a mixtures of random binary masks with 40 $\mathrm{\mu}$m mask-pixel pitch (‘R40’) and random binary masks with 20 $\mathrm{\mu}$m mask-pixel pitch (‘R20’). (a) 100% R40, (b) 75% R40 and 25% R20, (c) 50% R40 and 50% R20, (d) 25% R40 and 75% R20, (e) 100% R20. 250 Kaczmarz iterations are used for images (a)-(d) and 500 iterations are used for image (e). The step size was 0.1 for images (a)-(d) and 0.5 for image (e). The demagnified pixel pitch is 8.9 $\mathrm{\mu}$m.

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Beside the visual assessment, the quality of the reconstructed ghost images is also measured using the four metrics (mentioned earlier) and plotted in Fig. 13(a). According to the plot, the SSIM, and 1.0-RMSE have similar trends and their magnitudes decrease as the number of the random 20 $\mathrm{\mu}$m masks increases. The normalized entropy varies only slightly throughout the series of ghost images, namely between 0.24 and 0.21. The line-pairs/mm, however, improves significantly from approximately 19 (where 100% of the masks are random 40) to approximately 30 (where 100% of the masks are random 20) although the trend is not a linear relationship.

 

Fig. 13. (a) Four different metrics representing the quality of the reconstructed X-ray ghost images in Fig. 12. (b) Spatial resolution versus RMSE results from the same ghost images. Data labels represent the percentage of the random binary mask with 20 $\mathrm{\mu}$m mask-pixel pitch starting from 0% (i.e. 100% random mask with 40 $\mathrm{\mu}$m mask-pixel pitch) to 100%.

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Selecting an optimal combination of the masks depends on the purpose of the experiment and the metrics that are important for that purpose. For instance, if RMSE and spatial resolution are crucial, we can plot them against each other as shown in Fig. 13(b) and choose a mask combination that provides ghost images with sufficiently small spatial resolution and acceptably low RMSE. In Fig. 13(b), there is a mix (at 50%) where both spatial resolution and RMSE are relatively small. This might be a reasonable compromise, for this particular object. This mask combination also agrees with visual assessment of the corresponding ghost image, which is illustrated in Fig. 12(c).

Another set of GI experiments was explored that utilized a mixture of the Gaussian mask and the random 20 $\mathrm{\mu}$m mask. Here a larger field of view, namely 140 $\times$ 140 pixel images which correspond to a 62 $\times$ 62 array of mask pixels, was reconstructed. A selection of the images reconstructed using various mask combinations is depicted in Fig. 14; again, a total of 961 masks was used for all cases. Note that this number of masks is only one quarter of the masks that are required for this area, even if the patterns are orthogonal. As can be seen from Fig. 14(a), the reconstructed image using Gaussian masks only is blurry, due to having large feature sizes as well as the limited number of patterns. The reconstructed image using only random 20 $\mathrm{\mu}$m masks (see Fig. 14(e)) is also not desirable, because the random masks are not scalable and have high noise levels (as discussed in Sec. 3.1). However if we look at Fig. 14(d), which is reconstructed with 25% Gaussian masks (240 masks out of 961 masks) and 75% random masks (721 masks), the object is better resolved compared to Fig. 14(e) and also less blurry compared to Fig. 14(a). Thus, this example shows that mixing two masks can improve the quality of the reconstructed X-ray ghost image, without increasing the number of masks or the radiation dose.

 

Fig. 14. X-ray ghost images utilizing a mixture of Gaussian masks with 20 $\mathrm{\mu}$m mask-pixel pitch (‘G’) and random binary masks with 20 $\mathrm{\mu}$m mask-pixel pitch (‘R20’). (a) 100% G, (b) 75% G and 25% R20, (c) 50% G and 50% R20, (d) 25% G and 75% R20, (e) 100% R20. 500 Kaczmarz iterations are used for all images with step size of 0.5. The demagnified pixel pitch is 8.9 $\mathrm{\mu}$m.

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The four metrics that are plotted in Fig. 15(a) measure the quality of the ghost images obtained from ten different combinations of the Gaussian and the random masks, in addition to the ghost images that are reconstructed with 100% of each mask. According to the plot, SSIM and 1.0-RMSE have similar trends and their magnitudes reduce as the number of the random masks increases. The normalized entropy plot behaves similarly, except for the first two values which are almost equivalent. The line-pairs/mm of the ghost images, however, fluctuate as the number of the random masks increases. Since this metric behaves differently, an appropriate mask combination can be selected by compromise. For this purpose, the spatial resolution can be plotted versus RMSE, as depicted in Fig. 15(b). A fair compromise is where the two metrics have low magnitudes (which means higher resolution and lower error). As can be seen from the plot, there are two points (at 0% and 75%) where the spatial resolution and the RMSE have relatively small values. The mask combinations, corresponding to the two points, can provide relatively higher quality ghost images compared to the other ten mask combinations. Moreover, visual assessment can be employed to select one option out of two. The corresponding ghost images are depicted in Fig. 14(a) and (d), which are reconstructed with 100% and 25% of the Gaussian masks respectively. The former image contains less noise and the latter image is sharper (or less blurry) than the first image. Either of these options can be chosen as a fair mask combination, based on the importance of each metric and particular aim of the GI experiment.

 

Fig. 15. (a) Four different metrics representing the quality of the reconstructed X-ray ghost images obtained by Gaussian and random 20 $\mathrm{\mu}$m masks. Some of the images are shown in Fig. 14. (b) Spatial resolution versus RMSE results from the same ghost images. Data labels represent the percentage of the random mask (for each ghost image) starting from 0% (which means 100% of the masks are Gaussian) to 100%.

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3.3 X-ray GI results with orthogonal masks

As mentioned in Sec. 2 and also shown in Fig. 2, twelve orthogonal masks were fabricated on the substrate: URA and FRT-Legendre masks with three different sizes (i.e., $p=31$, 67, and 127) and two different resolutions (i.e., 20 $\mathrm{\mu}$m and 40 $\mathrm{\mu}$m mask-pixel sizes). X-ray GI experiments were performed with one of the URA masks and one of the FRT-Legendre masks with resolution 20 $\mathrm{\mu}$m and size 31 $\times$ 31 pixels. The reconstructed ghost images utilizing 961 masks (31 $\times$ 31) are given in Fig. 16(a) and Fig. 16(b) respectively. As can be seen from the figures, the quality of the ghost images obtained with these two orthogonal masks is much higher compared to the results from the other mask types shown in Sec. 3.1. This is expected because the full set of orthogonal masks have full rank, i.e., constitute a complete representation of the object, and are well conditioned, i.e., exhibit stable reconstruction with respect to noise. This can lead to a more accurate and reliable reconstruction of the object.

 

Fig. 16. X-ray ghost imaging using orthogonal masks: (a) ghost image obtained with all 961 URA mask patterns, (b) ghost image obtained with all 961 FRT-Legendre mask patterns, (c) ghost image obtained with 481 FRT-Legendre mask patterns, (d) ghost image obtained with 961 FRT-Legendre mask patterns over a larger field of view. The demagnified pixel pitch is 8.9 $\mathrm{\mu}$m.

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Comparing the ghost images obtained by the URA and the FRT-Legendre masks in Fig. 16(a-b), it can be seen that the ghost image quality using the URA is not as good as that using the FRT-Legendre mask. This might be explained by the autocorrelation of the URA mask as shown in Fig. 6(a); in particular, the PSF of the fabricated mask had significant features outside the central peak, compared with that for the FRT-Legendre mask. We also noticed that the URA mask was not exactly perpendicular to the beam during the scan, with its normal vector being misaligned with respect to the optical axis by approximately 0.3$^\circ$. This demonstrates how sensitive the orthogonal masks are to mask fabrication and positioning, with slight errors yielding a significant difference in the reconstruction. It is also worth mentioning the fact that the orthogonal masks do not work when not fabricated properly, which is why the Hadamard masks were completely ignored. Even small orthogonal-mask fabrication defects—such as scratches, debris, or a few missing pixels—can introduce major artifacts in the reconstructed image. Conversely, the artifacts in the reconstructed image will be minor for the case of fabrication defects in the random masks. Another point, regarding the sensitivity of the orthogonal masks, is illustrated in Fig. 16(d). There, the 961 FRT-Legendre masks were employed to reconstruct the same object, but over a larger area (100 $\times$ 100 pixels). As can be seen from the figure, the reconstructed object repeats itself, which shows that the orthogonal masks are not scalable at all; the imaging area must be less than or equal to the size of the orthogonal mask.

Despite the sensitivity to fabrication and alignment for orthogonal masks, a ghost image can be reconstructed by utilizing less mask displacements (or measurements) compared with random masks. Figure 16(c) shows a ghost image that is obtained using only half of all possible FRT-Legendre mask displacements used for the ghost image in Fig. 16(b). The RMSE for this ghost image (which is achieved using 481 FRT-Legendre mask displacements) is equivalent to the RMSE of the ghost image achieved using 961 random mask displacements. Thus an image of an object can be reconstructed by fewer measurements using orthogonal masks. This is due to the fact that the orthogonal masks each provide unique information, which can reduce the overall time, X-ray dose, and resources needed for the imaging process.

4. Discussion

We have designed and fabricated a range of random and orthogonal masks for patterned X-ray illumination. The random masks include (i) random binary structures that are either delta-correlated or have various feature sizes (generated through convolution of delta-correlated binary maps with Gaussian or Lorentzian kernels, followed by rebinarization), and (ii) statistically self-similar random-fractal structures that are designed to have no preferred length scale. The orthogonal masks are uniformly redundant arrays and FRT-Legendre structures. The fabricated masks were examined through various ghost-imaging experiments employing a laboratory X-ray source. The data analysis and ghost image-quality evaluations show that random binary masks work well under the specific conditions they are designed for, namely when the characteristic length scale of the mask is matched to the ghost-imaging spatial resolution, but are not spatially scalable beyond this scope. Random-fractal masks, however, were not as suited to any specific scenario but exhibit scalable properties and can be used to reconstruct ghost images at various spatial resolutions. This property can be clearly observed from the reconstructed ghost images in Fig. 11(c) and Fig. 11(d). The random masks, created by convolution of a delta-correlated binary random structure with Gaussian or Lorentzian kernels followed by rebinarization, have larger feature sizes compared with the other masks investigated (see Fig. 1). The ghost images that result from masks that are Gaussian blurred and then rebinarized, e.g., Fig. 9(c), confirm that resolution of the reconstructed X-ray ghost image is not limited to the mask feature size.

We also explored the effect, of mixing different types of mask structures, on the quality of the resulting ghost images. We focused on combining masks with different feature sizes. A series of GI experiments was performed, using different relative contributions of random masks with 40 $\mathrm{\mu}$m and 20 $\mathrm{\mu}$m mask-pixel sizes, and also with different combinations of the Gaussian-blurred and random masks, both with a 20 $\mathrm{\mu}$m mask-pixel size, while keeping the total number of the masks constant. The ghost images reconstructed are illustrated in Fig. 12 and Fig. 14 respectively. The masks with larger feature sizes (random mask with 40 $\mathrm{\mu}$m and Gaussian blurred with 20 $\mathrm{\mu}$m mask-pixel sizes in our case studies) can provide low-frequency information and the masks with smaller feature sizes (random mask with 20 $\mathrm{\mu}$m resolution) can provide high-frequency information. The results presented indicate that combining masks is beneficial and when GI results of individual masks are not satisfactory, mixing the masks can improve the quality of the ghost image without increasing the radiation dose or number of the masks.

To augment the visual assessment, the quality of the ghost images was evaluated using four metrics: RMSE, structural similarity, normalized entropy, and spatial resolution. See Fig. 13 (for the combinations of two random masks) and Fig. 15 (for the combinations of the Gaussian-blurred and the random masks). The first two metrics have similar trends and their values improve as the number of the random masks with 20 $\mathrm{\mu}$m resolution reduces. The changes in the normalized-entropy plot are insignificant throughout the different mask combinations. The spatial-resolution plot presented as line-pairs/mm, however, fluctuates. Hence, choosing an appropriate combination of the masks is a compromise, with the selection depending on the purpose of the experiment and the importance of each metric. A useful device, in this context, is that two metrics can be selected and plotted against each other—such as the plots in Fig. 13(b) and Fig. 15(b)—thereby enabling a combination to be chosen which balances both metrics.

Beside the random masks, GI experiments were conducted with orthogonal masks (URA and FRT-Legendre types). We demonstrated that any small errors in the fabrication process and/or experimental setup cause significant artifact/blurriness in the reconstructed ghost image. Thus, the orthogonal masks are sensitive. They are also restricted in usage and are not spatially scalable at all, as depicted in Fig. 16(d). However, in the appropriate imaging scenario, orthogonal masks can provide the best results (as shown in Fig. 16(b)) and can significantly enhance the performance and efficiency of GI. For our particular suite of experiments, the number of FRT-Legendre masks could be reduced to half and still achieve acceptable ghost-image quality (defined via the RMSE), similar to that using the random mask with 20 $\mathrm{\mu}$m resolution (see Fig. 16(c)).

It is worth mentioning that we also tried to produce X-ray ghost images using a sandpaper mask (a natural random mask commonly employed in X-ray GI [21]), for comparison with our fabricated masks. An 800 grit sandpaper mask with an average particle diameter of 21.8 $\mathrm{\mu}$m was selected to have similar resolution to the fabricated random mask with 20 $\mathrm{\mu}$m pixel size. However, the visibility of the sandpaper mask [43] was too low under hard-X-ray illumination (less than 1%) compared with the visibility of our fabricated mask (which was more than 41%); an X-ray ghost image could not be recovered above noise, even when utilizing 7696 sandpaper masks. The same object was resolved using only 961 fabricated random masks. We can conclude that masks should have relatively high contrast to obtain ghost images. A benefit of custom designed masks (similar to our fabricated masks) is that the thickness of the structures can be matched to the photon energy of the X-rays being used, to maximize mask contrast.

We have limited our study to an almost-binary object, in all experiments reported in the present paper. While this is consistent with preceding X-ray and neutron ghost-imaging studies [5,21–23,27,44], it would be interesting and timely to seek to extend these experiments to the case of non-binary samples, with their associated range of different attenuation values. We would expect the ghost-imaging reconstructions to be more challenging for such objects, as quantified by (and quantifiable by) the various error metrics explored in the present paper. We will pursue such a study of laboratory-source X-ray ghost imaging for non-binary samples, in a future publication.

It would be interesting to investigate the utility of our patterned masks, beyond the context of X-ray ghost imaging that has been experimentally studied in the present paper. Examples include (i) use of our mask designs, applied to different mask materials, for neutron ghost imaging [23] and electron ghost imaging [45], (ii) use of our masks for the technique of X-ray speckle tracking [46–48], which has hitherto employed spatially random masks such as sandpaper that are not designed and fabricated in a precisely controllable manner, and (iii) use of our masks for the technique of ghost projection [28,49–51], which may be thought of as a reversed form of classical ghost imaging, whereby an arbitrary distribution of radiant exposure may be written up to an additive constant, by transversely displacing a single illuminated non-configurable mask.

The fabrication of masks that provide specific transmission patterns for bucket imaging enables one to consider other mask designs that shape the intensity profile of an incident beam. A diffuse incident beam could then be used as a probe to scan and image objects with reduced local radiation intensity and hence less radiation damage. Fabricating beam-shaping masks based on Huffman arrays [52] provides such an opportunity. Huffman-like arrays have a sharp delta-like autocorrelation with relatively small and sparse off-peak contributions, whose effects can be removed by simple deblurring [53].

We close this discussion with several miscellaneous remarks, regarding possible future extensions of the work presented in our paper. (i) We encountered a trade-off between spatial resolution and high-frequency noise, in Sec. 3.2. Such a noise–resolution trade-off is well known in a broader imaging context (see e.g. Reference [54], and papers cited therein). It would be interesting to further investigate the noise–resolution relation, in applications of our masks to X-ray ghost imaging, in particular, to demonstrate that the degree of noise-resolution trade-off is difficult to predefine a priori for a specific mask mixture, as shown in Fig. 13 and Fig. 15. A suitable starting point for the development of this point is explained in Ref. [6]. (ii) Let $M_1$ and $M_2$ be two particular GI image-quality metrics, that one seeks to jointly minimize in the context of a particular application. One could form a composite metric $M=M_1^2+\tau M_2^2$, where $\tau \ge 0$ is a real parameter that governs the relative importance of the two metrics, and then seek to minimize $M$ for a particular X-ray ghost-imaging application of our masks in which both metrics are considered to be important. Other composite metrics could also be explored, such as $M=M_1M_2$ or $M=M_1^q+\tau M_2^q$, where $q>0$. Interested readers may consider the theoretical results regarding Pareto optimality for multi-objective optimization in Ref. [55, Chapter 3] for more details. (iii) There may be certain contexts in which one does not necessarily want to reconstruct an X-ray ghost image as such, but rather seeks to reconstruct a particular sample parameter (or parameters), for example the porosity and Euler-number density of a rock sample [56], or whether or not a given manufactured object is unacceptably likely to be defective. It would be interesting to extend the work of the present paper into such non-imaging settings. (iv) Our mask-fabrication strategy can be employed to give precisely matched positive–negative mask pairs, that could be employed for the technique of X-ray differential ghost imaging [35,57,58]. This strategy may assist with improving X-ray GI quality metrics, e.g. in the presence of slow variations in the illumination intensity.

5. Conclusion

Various types of structured illumination pattern, including random and orthogonal structures, were designed for X-ray illumination and fabricated with gold on a glass substrate. The random structures included random binary masks, blurred random binary masks that were rebinarized, and random fractal structures; the orthogonal structures corresponded to uniformly redundant arrays and Legendre masks constructed using the finite Radon transform. Multiple X-ray ghost imaging experiments were performed with individual classes of mask pattern, as well as combinations of mask classes, using a laboratory X-ray source. We examined which mask properties are important under experimental conditions, and analyzed the quality of the studied X-ray ghost images through several different metrics. We believe that our fabricated masks can significantly enhance the performance and efficiency of X-ray ghost imaging, relative to natural masks, and suggest that they may also be useful for X-ray ghost projection and X-ray speckle tracking.