⇦Chinese Physics Letters, 2020, Vol. 37, No. 4, Article code 044208 Energy-Selective X-Ray Ghost Imaging * Yu-Hang He (何雨航)1,2, Ai-Xin Zhang (张艾昕)1,2, Wen-Kai Yu (俞文凯)3, Li-Ming Chen (陈黎明)4**, Ling-An Wu (吴令安)1,2** Affiliations 1Institute of Physics, Chinese Academy of Sciences, Beijing 100191 2University of Chinese Academy of Sciences, Beijing 100049 3School of Physics, Beijing Institute of Technology, Beijing 100081 4Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240 Received 13 February 2020, online 24 March 2020 *Supported by the Science Challenge Project (No. TZ2018005), the Civil Space Project (No. D040301-1), the National Natural Science Foundation of China (Nos. 11991073, 11721404, and 61975229), the National Key R&D Program of China (Nos. 2017YFA0403301 and 2018YFB0504302), and the Key Program of CAS (No. XDB17030500).
**Corresponding authors. Email: wula@iphy.ac.cn; lmchen@iphy.ac.cn Citation Text: He Y H, Zhang A X, Yu W K, Chen L M and Wu L A et al 2020 Chin. Phys. Lett. 37 044208    Abstract X-ray ghost imaging (XGI) has opened up a new avenue for damage-free medical imaging. Here energy-selective spectroscopic XGI under poor illumination is demonstrated with a single-pixel detector for the first time. The key device was a specially fabricated Au mask incorporating a new modulation pattern design, by which means images of a real object were obtained with a spatial resolution of 10 μm and a spectral energy resolution of about 1.5 keV. Compressed sensing was also introduced to improve the image quality. Our proof-of-principle experiment extends the methodology of XGI to make possible the retrieval of spectral images with only a single-pixel detector, and paves the way for potential applications in many fields such as biology, material science and environmental sensing. DOI:10.1088/0256-307X/37/4/044208 PACS:42.30.Va, 42.30.Wb, 87.57.-s, 87.59.bd © 2020 Chinese Physics Society Article Text Ghost imaging (GI) is an attractive technique which does not require a conventional camera to image an object directly. In a classic GI system,[1–4] the incident light is separated by a beam splitter into a reference arm and object arm. In the former, the spatial distribution of each illumination is characterized by a high resolution array detector such as a charge-coupled device (CCD) to provide a series of reference patterns. In the latter, the beam is modulated by the object and collected by a single-pixel detector, which can only measure intensity, to produce so-called 'bucket' signals. The two arms both acquire partial information, but neither alone can form a complete image. The image is retrieved by calculating the second-order correlation of the reference patterns and corresponding bucket signals; that is where the 'ghost' in GI comes from.[5] Another more efficient option to realize GI is known as computational ghost imaging[6,7] or single-pixel imaging,[8–11] where the beam splitter and CCD are replaced by a spatial light modulator or digital mirror device so that a beamline with a pre-programmed speckle distribution can be generated. Based on this, an inexpensive high resolution single-pixel camera can be fabricated at silicon-insensitive optical wavelengths, such as infrared and terahertz.[12,13] Unfortunately however, for the x-ray region, there are no suitable optical devices for realizing GI. In theory, lensless x-ray GI (XGI) is perfectly feasible, and two possible schemes have been demonstrated successfully. One relies on a crystal beamsplitter to perform XGI in a classical way, but this demands high flux intensities so the x-ray source is limited to those of huge synchrotron facilities.[14–16] Another is based on intensity modulation, in which case speckles are generated by absorption in a set of modulation masks that transmit a series of x-ray patterns.[17–20] In this scheme, ultralow intensity XGI has even been demonstrated.[20] For real applications of GI, less exposure and data processing times with better imaging quality is always desirable, and many schemes have been devised to achieve this,[21–23] as well as three-dimensional (3-D) imaging.[15,24,25] However, these methods cannot fully characterize the target sample, as there is a lack of spectral information. Spectral GI with visible light[26–30] has already been demonstrated, and promises to be of great value in ghost lidar, remote sensing and microscopy. For analysis of biological samples and in material science, x-ray radiography is invaluable, therefore it would be of great significance if the methodology of XGI could be extended to imaging in the spectral domain. Generally, it is not possible with only a position-sensitive detector or spectrometer to obtain spatial and spectral information at the same time.[31] Instead, multi-acquisition is necessary in which case we have to perform a measurement scan of one parameter for a given setting of the other, change the setting, repeat the scan, change the setting again, and so on. A typical example is dual-energy x-ray radiography or tomography in medical imaging,[32] which requires fast switching of a high-voltage system or a set of multiple monochromatic sources to perform spectral scanning; this is both expensive and time consuming. For GI, the situation can be greatly improved. The image of a spectral object can be defined as $T$($x, y, z, \lambda$), which is a 4-D data set that includes the wavelength $\lambda$. The outstanding feature of GI is the adoption of a single-pixel detector, which can greatly lower the demands on the detector while further dimensional information can be obtained by later processing. Usually, a common photodiode can act as the bucket detector to measure the intensity fluctuation $B_{m}$, where $m$ is the exposure serial number in a GI measurement. In this case, a 2-D image $T(x, y)$ can be retrieved. Based on the correlation within a limited longitudinal coherence length, if reference patterns at different depths are pre-recorded, one can even obtain a 3-D image $T(x, y, z)$ of a transmissive object via optical coherence tomography.[33] Furthermore, similar to direct imaging, multi-measurements with monochromatic sources in every spectral region can be taken to extract spatial information of the specimen. However the imaging quality depends completely on the source, which suffers limitations from light flux, bandwidth, and acquisition time. For XGI, there is a much better alternative way, that is, we can use a bucket spectrometer to gather the intensity fluctuations ${B}_{m}(\lambda_{k})$, where $k$ denotes the measurement number and $\lambda_{k}$ is the wavelength of interest. In computational GI, the bucket signals acquired in a certain spectral region can be written as $$\begin{align} B_{m}(\lambda_{k})=\,&\int {\it\Psi}_{0}\left(x,y \right) I_{m}\left(x,y \right)T_{\rm mask}\left(x,y,\lambda_{k} \right)\\ &\cdot T_{\rm obj}\left(x,y,\lambda_{k} \right)dxdy,~~ \tag {1} \end{align} $$ where ${\it\Psi}_{0}\left(x,y \right)$ is the intensity distribution of the source, $x$ and $y$ are spatial co-ordinates in the sample plane, $I_{m}\left(x,y \right)$ is the pre-designed mask pattern distribution that performs pure intensity modulation of the beam, and $T_{\rm mask}\left(x,y,\lambda_{k} \right)$ and $T_{\rm obj}\left(x,y,\lambda_{k} \right)$ are the transmittance functions of the modulation mask and object at the incident wavelength $\lambda_{k}$, respectively. The image $T_{\rm obj}\left(x,y,\lambda_{k} \right)$ can then be retrieved from the second-order correlation function[1] $$ G(x, y, \lambda_{k})=\langle \Delta B_m(\lambda_k)\cdot \Delta I_m(x,y)\rangle ,~~ \tag {2} $$ where $\langle\cdot\rangle $ denotes an ensemble average over $m$ measurements, $\Delta B_{m}(\lambda_{k})=B_{m}(\lambda_{k}) - \langle B_m(\lambda_k)\rangle $, and $\Delta I_{m}(x,y)=I_{m}(x,y) - \langle I_{m}(x,y)\rangle $. In this Letter we report the first proof-of-concept demonstration of energy-selective XGI, by which means images of real objects with high resolution and reasonable energy resolution are obtained with just a single-pixel detector. This so-called bucket spectrometer is quite simple and does not have to be conducted at a synchrotron facility; it can be performed in the lab with just a table-top x-ray source, and even with very low illumination. The experimental setup is shown in Fig. 1. An x-ray tube (Incoatec Source Iµs) operating at 45 kV and 10 µA emits a polychromatic beam which mainly consists of spectrally continuous bremsstrahlung with a quasi-monoenergetic characteristic of ${K}_{\alpha} =8.04$ keV and ${K}_{\beta} =8.9$ keV, as shown in Fig. 2(a). This is then modulated in sequence by the patterns in a 2 inch square mask which is set 44.5 cm from the source. The sample object is placed immediately behind as close as possible, about 1 cm away, so that the projected x-ray distribution can be considered the same as the pattern in the mask, a necessary condition for intensity modulated GI. To block unwanted radiation, a 3 mm thick square Cu aperture was inserted between the object and the bucket detector. Finally, a bucket spectrometer[34] composed of a hybrid sensor, preamplifier, digital pulse processor, and specialized software was placed 25 cm behind the aperture. The sensor (Amptek XR-100-CdTe) has an effective area of $5 \times 5$ mm$^{2}$, and an energy resolution of 1.5 keV with very high detection efficiency over the range from 3 to 45 keV. Its key element is a 1 mm thick cadmium telluride diode that converts the x-ray photons to digital pulses; the output signals are reshaped and amplified to an acceptable intensity and then processed to produce an energy spectrum. A digitally controlled metal shutter was placed right after the x-ray tube, synchronized with the detector. The acquisition time of each exposure frame was 2 s, which is limited by the data transfer speed. All the devices were mounted on precise motor or manual translation stages and aligned by a red guide laser which was prealigned with the x-ray beamline. After all the spectral information had been recorded, the XGI images were retrieved by traditional second-order correlation or compressed sensing (CS) algorithms.[35,36] In our case, we adopted the popular TVAL3 algorithm,[37] which is based on the classic augmented Lagrangian multiplier plus an alternating direction and a non-monotone line search technique. By this means the quality of the spectral images could be enhanced even more.
cpl-37-4-044208-fig1.png
Fig. 1. Experimental setup for energy-selective XGI. Bottom right inset: components of the bucket detector system and corresponding functions.
The modulation mask was composed of 324 pairs of patterns designed in a positive-negative style, that is, each pair consisted of a binary matrix and its reverse. This new arrangement[38] is much superior to the commonly used matrices for the same sampling rate. Each pattern consists of $128 \times 128$ pixels of size 10$\times 10$ µm, which gives an illumination area of $1.28 \times 1.28$ mm and an image resolution of around 10 µm. Actually, to fabricate a modulation board with a linewidth of 10 µm, the depth-to-width ratio of the etching process must be taken into consideration. In our case, a 12 µm thick layer of gold foil was electroplated onto a 2 inch square 500 µm thick SiO$_{2}$ substrate, with a transmissivity of 0.8% at 8 keV. The depth-to-width ratio could reach nearly $1\!:\!1$, but due to inhomogeneity of the electrical field distribution during electroplating the edges of the board were slightly thinner than the central parts. Restricted by the size of the board, only 324 pairs of patterns could be etched; ion milling was chosen in preference to electrochemical etching as it is more precise. To check the modulation effects in different wavelength regions, two patterns were randomly selected for a test measurement. As shown in Fig. 2, the bucket spectra of Figs. 2(b) and 2(c) differ greatly compared with the original source output spectrum of Fig. 2(a). Excluding some particular situations, the penetrating capacity of x-rays increases as the photon energy increases. Therefore, the bucket spectrum directly reflects the spectral response of the object superimposed upon the original source spectrum, normalized with respect to the mask transmissivity in a given bandwidth. On the other hand, when different patterns are used, the bucket spectrum would be somewhat different, as can be seen from Figs. 2(b) and 2(c), indicating efficient modulation in our energy-selective XGI experiment.
cpl-37-4-044208-fig2.png
Fig. 2. (a) A typical spectrum of the x-rays emitted from the micro-focus tube. [(b), (c)] Typical bucket spectra of the sample for two different modulation patterns.
The sample was a stencil of the letters CAS, etched similar to the modulation mask, in a 12 µm thick Au foil on a Si substrate. Its image taken by a scanning electron microscope (SEM) is shown in Fig. 3, which includes a top view and an oblique view to show its depth. The width of each line in the letters CAS is 100 µm and the gap between two letters about 50 µm. Of course, to demonstrate energy-selective XGI more conspicuously it would be best to image a complex sample composed of several materials, each having a different spectral response to the x-ray wavelengths. However, due to the limited illumination area it was difficult to find such a sample. Since this was our first attempt to realize energy-selective XGI, we thought it more important to check its feasibility with a pure amplitude absorption object in a proof-of-principle experiment.
cpl-37-4-044208-fig3.png
Fig. 3. SEM image of the sample viewed from the top (left), and obliquely (right).
cpl-37-4-044208-fig4.png
Fig. 4. Experimental results for the ranges of 1–45, 15–20 and 20–25 keV. (a)–(c) Spectral images retrieved by traditional XGI. (d)–(f) Spectral XGI images retrieved by CS. (g)–(i) Line profiles of the areas enclosed by the dotted lines in the GI and CS images above: red squares, GI; black dots, CS.
The whole spectrum of the x-ray photons ranged from 1 to 45 keV, which was determined by the voltage of the x-ray tube generator. In this case reasonable images can be retrieved with traditional GI or the TVAL3 algorithm when the entire bucket data is integrated, as shown in Figs. 4(a) and 4(d). For spectral imaging, bucket data in the ranges 15–20 keV and 20–25 keV were selected, in which case the average photon counts were about 500 and 1000 per frame, respectively. As can be seen, energy-selective images in these regions can also be extracted successfully. The 15–20 keV and 20–25 keV images retrieved by GI are shown in Figs. 4(b) and 4(c), and those by TVAL3 in Figs. 4(e) and 4(f), respectively. It is interesting that although the count rate in the low energy region is much lower, the image quality seems better. This is probably due to the slight differences in transmission of the modulation mask for photons of different energies. For images in the higher spectral region, a mask with higher modulation efficiency should be adopted. For a more quantitative estimate of the image contrast and to identify the edges of the object more precisely, we extracted the line profiles of the cross-sections enclosed in the red dotted-line areas of Figs. 4(a)–4(f), as shown in Figs. 4(g)–4(i). The red squares indicate values obtained from traditional GI and the black dots from the TVAL3 CS algorithm; each point was calculated from the average grey value of three adjacent vertical pixels. It should be mentioned that the images retrieved by CS and GI do not have the same contrast scale, so they were all adjusted and normalized into 8-bit grey tone plots for better visualization. It can be seen that when the TVAL3 algorithm is used, the image edges appear much sharper than in the GI case. This is because at very low sampling rates there is much more noise so the distribution of the grey values in GI is more random, while CS can reduce the noise through multiple iterations. Overall, the XGI images still appear somewhat blurred. The main reason for this is the very few number of exposures taken, much less than that of traditional raster scanning which, according to Nyquist's theorem, would require a full $128 \times 128 = 16384$ measurements. Here, by introducing our suitably arranged deterministic modulation matrix, the sampling number can be reduced by a factor of 25, which also means less radiation damage. Another reason for the blurring is inaccuracies in the fabricated mask. Although great care was taken in the etching process, it is inevitable that the pixels are not ideally sharp; when material is sputtered out from a pixel pit, the remaining edges have a gradient of nearly 30$^{\circ}$. This distortion of the patterns will introduce much noise into the measurement matrix of the image recovery algorithm and cause serious blurring. If this problem could be solved, the image quality would be much better. In conclusion, based on intensity modulation, we have successfully demonstrated energy-selective XGI with a bucket spectrometer at a sampling rate of about 4%, even under poor illumination. Using a specially fabricated Au modulation mask incorporating a new pattern design, images with 10 µm spatial resolution and 1.5 keV spectral resolution have been obtained. The experimental setup is simple and convenient to operate, the key devices are inexpensive and easily available, so it is possible to obtain spectral images directly and more efficiently compared with traditional scanning methods. Furthermore, if a pulsed x-ray source were used, the time needed for a complete exposure could be reduced to a few minutes or even seconds. Through the development of more sophisticated fabrication methods and better algorithms such as deep learning, the image quality and field of view could also be greatly improved. Although our results are preliminary, they demonstrate the immense potential of energy-selective XGI, which we hope will soon see practical applications in biology, material analysis, environment sensing, and so forth. 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