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Abstract
In contrast to conventional X-ray imaging systems, the lobster eye lens, serving as a pivotal component for X-ray focusing, presents the potential for downsizing X-ray backscatter imaging systems. This study reports the successful implementation of a pioneering non-contact staring X-ray backscatter imaging experiment, with the target positioned 1.5 meters away from the system and employing a tube voltage of 60 kV for the X-ray light source. The system is built upon a novel high aspect ratio (500) meridian lobster eye lens, employing a laboratory low illuminance desktop light source and a commercial X-ray detector to achieve high-resolution focused imaging of hard X-rays. Point spread function testing and a series of imaging experiments were carried out to assess the resolution and optimal imaging photon energy of the proposed system. Furthermore, according to the characteristics of the point spread function of the cross image of the lobster eye lens, we proposed an image processing algorithm. The experimental results demonstrate that, after processing, the Structural Similarity (SSIM) Index of the backscatter image and the ground truth image can be improved from an average of 0.0526 to 0.5758. Our research significantly contributes to the advancement of a new generation of X-ray backscatter imaging systems.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
Corrections
Yuxiang Yang, Mingzhao Ouyang, Longhui Li, Yingjun Zhang, Shizhang Ma, and Yuegang Fu, "Staring X-ray backscatter imaging based on ultra-high aspect ratio lobster eye lens: erratum," Opt. Express 32, 18744-18745 (2024)https://opg.optica.org/oe/abstract.cfm?uri=oe-32-11-18744
1. Introduction
X-ray imaging plays a crucial role in non-destructive inspection, finding widespread applications in medical diagnosis, security, industrial testing, and various other fields. Traditional transmission X-ray imaging systems excel at detecting metallic or inorganic materials but face challenges in accurately imaging low-atomic-number materials. Moreover, these systems necessitate placing the detector and source on opposing sides of the target, imposing limitations on their practical applications [1]. Compared to X-ray transmission systems, X-ray backscattering imaging is more effective for low-atomic-number materials due to its robust Compton scattering, leading to improved imaging results for such targets [2]. At present, a traditional scanning X-ray backscatter system developed by American Science & Engineering, Inc. (AS&E) utilizes a patented “Flying spot” technology [3]. However, unlike the staring imaging system, this system necessitates relative motion between the test object and scanning beam, which limits its imaging distance and resolution.
Incorporating the lobster-eye lens into the backscatter imaging system is an effective solution to address the limitations of traditional imaging methods. The lobster-eye optics were initially proposed in 1979 [4], which is inspired by the reflective compound eyes of macruran crustaceans, and was first applied to astronomy as an X-ray telescope. The device consist of thousands of mirror boxes arranged on a common center of curvature [5], its reflect-base focusing mode and spherical structure give lobster-eye optics unique focusing ability and a wide field of view (FOV) [6,7]. In 2005, the combination of lobster-eye optics and X-ray backscattering inspection was first proposed by Gertsenshteyn and his associate [8]. In 2007, V. Grubsky et al. firstly utilized two intersecting sets of parallel gold-coats silicon wafers with 1 mm spacing to form a $14 \times 12$ mic-channel plates (MCPs) to carry out X-ray backscattering imaging experiments on a cleaner container placed behind the obstacle [9]. In 2016, Jie Xu, Xin Wang et al proposed a treble-lens Schmidt objective on the basis of the original double-lens Schmidt geometry by dividing one of the two sets of reflectors into two separate sets of reflectors [10]. They stacked treble sets of the surface data of D263 optical glass to form a Schmidt objective and carried out a imaging experiments.
Since the lobster eye lenses are reflective focusing based on the total reflection theorem, for optimum focusing [11], it require channels must be ultra-high aspect ratio and roughness of the sidewalls of the channels must less than 1 nm [12], which aways a challenge for manufacture process and coat metallic material on to the inner wall surface of such a narrow square channel.
In 2022, Li et al. from North Night Vision Technology Co., LTD (NNVT) report the first trial of a Lobster eye micropore optics with a high aspect ratio (500) using microchannel plate technology and atomic layer deposition technology [13]. This method provides a possibility for lobster eye micropore optics used for hard X-rays.
We cooperated with North Night Vision Technology Co., LTD to carry out research on the combination of this ultra-high aspect ratio lobster-eye optics with X-ray backscatter imaging system. In this paper, a non-contact (target is 1.5 m away from the imaging system) X-ray backscatter imaging system based on high aspect ratio (500) lobster eye lens is proposed, which utilized a laboratory low illumination table-top light source and a commercial X-ray detector to achieve focused imaging of hard X-ray (60 kev) in laboratory environment. Furthermore, an image processing algorithm is introduced, tailored to the unique characteristics of the imaging method. This algorithm effectively enhances high-resolution objects in backscatter images. The Structure Similarity (SSIM) Index is introduced as an evaluation index to assess the processing quality of the proposed algorithm.
2. Experimental system procedure
The experiment employed the IXS series wide-spectrum high-frequency integrated X-ray generator provided by VJ Technologies. The tube voltage of the generator was programmable within the range of 30 to 160 kV, corresponding to a photon energy range of 20 keV to 106.675 keV. Radiation intensity could be controlled by adjusting the tube current from 0.1 mA to 3 mA, and the cone beam had a divergence angle of 30 degrees.
As the test object, a polyamide material letter ‘S’ measuring 58 mm in length and 34 mm in width was selected. The letter was positioned 220 mm from the source, ensuring that the cone beam completely covered it. For the detector, a Hamamatsu X-ray image intensifier camera C7336-52 was used, comprising a high-resolution (75µm) x-ray image intensifier V10905P coupled to a CMOS camera with a resolution of $1920(h )\times 1440(v )$ pixels in a $73mm \times 55mm$ image plane. The detector employed Csl as the input phosphor material to convert X-rays into visible light, was mounted 1800 mm from the backscatter target.
Different from the rectangular microchannel array proposed by Angel [4], the meridional lobster-eye lens is a tapered microchannel array, where each extension plane of the reflective surface of the microchannel inner wall points to the center of the sphere, and all channels are arranged on the spherical surface. When the incidence Angle of the grazing incident X-ray entering the lobster eye lens is less than the critical Angle of total reflection, the rays may have several times reflections on the inner wall of the channel [4,14,15]. As shown in Fig. 1(c), the rays with odd times of reflections form both vertical and the horizontal channels walls will eventually reach the real focal point to form the central bright spot like the brown rays, and the rays that experienced no or an even times of reflections in both directions will produce “arms” of the cross resembling the green and black rays. The distribution of all rays on a detector is the composition of all the reflection rays in two perpendicular directions [11,16]. Therefore, the point spread function (PSF) of lobster-eye lens includes a real focal point, two cross-focal arms, and low-radiation background around the focus point on the detector plane. Since the critical Angle is determined by the energy of the X-ray, for optimum focusing, it require the ultra-high aspect ratio and low roughness of the sidewalls of the channels [11,12,17]. As show in Fig. 1(a) and (b), a $42.5mm \times 42.5mm \times 2.5mm$ high aspect ratio meridional lobster-eye lens made by North Night Vision Technology Co., LTD (NNVT) with a radius of curvature of $750mm$ and an x-ray focal length of $375mm$ was utilized in the experiment. NNVT firstly utilized the microchannel plate and atomic layer technology to firstly achieved ultra-high aspect ratio of 500, an inner wall roughness of ∼0.5 nm and an Ir-coated thickness of 19 nm. The diameter of each square channel in the lens is $5\mu m$ and the wall thickness of the channel is $1\mu m$. Install the lobster eye lens in the clamp and place it in the light path, mount 300 mm away from the detector and 1500 mm away from the test object.
Fig. 1. The meridional lobster-eye lens we used and its focusing schematic. (a). Lobster-eye lens in the clamp place in the light path. (b). Ultra-high aspect ratio lobster eye lens we used in experiment. (c). Focusing schematic of the lobster-eye lens.
The schematic of the experimental setup of x-ray backscatter imaging base on lobster eye lens is shown in Fig. 2.
Fig. 2. Experimental setup of x-ray backscatter imaging experiment. The axis of light source - lens - detector and the axis of light source - target are at an acute angle of 45 degrees.
3. Imaging experiment
To validate the proposed backscatter imaging method, a series of imaging experiments were conducted using the aforementioned experimental setup, as illustrated in Fig. 2. The evaluation encompassed testing the performance of imaging system across varying X-ray energy levels and assessing its resolution. The Structural Similarity Index (SSIM) will serve as a metric for quantitatively evaluating the imaging quality.
3.1 Energy analysis
Affected by its distinctive structure and unique focusing mechanism, lobster-eye lenses exhibit different focusing efficiency for incident X-rays of different energies. This characteristic stands as a crucial determinant of image quality.
Previous research [11,13,18] demonstrates that the maximum efficiency remains constant and is determined by the critical angle of total reflection when considering only the structural impact. The energy of the incident ray determines its corresponding critical angle, leading to different focusing efficiencies of the lobster-eye lens for different energy rays. The maximum efficiency, denoted as C, can be expressed by the following formula:
The wide-spectrum tungsten anode harrow X-ray tube used in this experiment has a following correspondence between the tube voltage $V({kv} )$ and the limiting short-wave ${\lambda _0}$
where $hc/e = 1.2399kv \cdot nm$ represent the Voltage-wavelength conversion factor, the relationship between maximum intensity wavelength (peak wavelength) ${\lambda _p}$ and ${\lambda _0}$ isReferring to the formular (2) and (3), it can be obtained that the tube voltage of the light source should be set at $46.5kv$ for the lobster-eye lens to attain maximum focusing efficiency.
To verify the above theoretical calculations, we conducted experiments to assess the PSF of the lobster-eye lens at various incident X-ray energies to measure the diameter of the central focal point which reflects the efficiency of focusing. Since the minimum step of the tube voltage of the laboratory table-top X-ray source we used is 5 kv, the tube voltage is set to 40,45,50,55,60 and 65 kv respectively, the corresponding peek intensities of the spectrum are 26.668, 30.002, 33.336, 36.667, 40 and 43.337 kev respectively. Some exemplary obtained results are shown in Fig. 3.
Fig. 3. The PSF of lobster-eye lens under different photon energies (each pixel/75 $\mu m$). (a) is $26.668kev$, (b) is $33.336kev$, (c) is $43.337kev$.
Table 1 illustrates the diameter of the focal point in the PSF under varying energy conditions. It reveals that the focal point diameter in the PSF reaches its minimum value at a peak spectrum intensity of $33.336kev$, the corresponding tube voltage is 50 kV. This observation suggests that the lens achieves optimal focusing efficiency under these conditions, with a deviation of $2.336kev$ from the theoretical value of $31kev$.
Table 1. The diameter of the focal point in the PSF under different energy
Different from the PSF testing in which X-rays radiate from light source then enter the lens directly, in the backscatter imaging, the wavelength of scattering rays will be larger than the wavelength of incident rays after Compton scattering. For an incident photon of energy ${E_0} = h\nu $, the relationship between the energy of the incident and scattered rays can be expressed by the Compton equation [19], by substituting the theoretical optimum photon energy ${E_1} = 31kev$ for lobster-eye lens and the scattering angle we set at ${135^\circ }$ into the Compton equation, the radiant energy of light source ${E_1}$ should be $36.319kev$ in backscatter imaging, according to formula (2) and (3), the tube voltage of the light source should be $54.47kv$.
Therefore, to further verify the relationship between image quality and incident X-ray energy, we obtain the sequences of backscatter imaging result for the cases where the peek intensities of the spectrum are operates at $33.336kev$, $36.667kev$, $40kev$, $43.337kev$, $46.67kev$ and $53.338kev$ respectively which the corresponding tube voltage is $50kv$, $55kv$, $60kv$, $65kv$, $70kv$ and $80kv$. Since the image is difficult to distinguish due to low contrast when the tube voltage is lower than 60 kv. Figure 4 only shows the image with tube voltage above 60 kv.
As shown in the Fig. 4(b), an image was created with the target position and size precisely matching the backscatter image, serving as a ground truth. The Structural Similarity (SSIM) Index, which characterizes the similarity of two images by analyzing brightness, contrast, and structure, is introduced as an evaluation index for imaging quality under varying photon energy conditions. With the SSIM value ranging from 0 to 1, a closer proximity to 1 indicates heightened similarity between the two images. In this study, we conduct a comparative analysis of the structural similarity between backscatter images and the ground truth, then the SSIM value close to 1 represents better imaging quality. The output SSIM results of the image under different energy are given in Table 2.
Fig. 4. The test object ‘S’ and its backscatter image under different energy rays with an exposure time of 350s. (a) The target ‘S’ for the imaging experiment. (b). Ground truth for the target ‘S’. (c)–(f) Imaging result under different peak intensity of the light source spectrums, where (c) represents $40kev$, (d) represents $43.337kev$, (e) represents $46.67kev$, (f) represents $53.338kev$.
Table 2. The output SSIM result of the image under different energy
Table 2 illustrates that the backscatter image attains optimal quality when the peak intensity of the light source spectrum is 40 keV. According to the Compton equation, the corresponding energy of Compton scattering is calculated to be 33.644 keV, aligning with the consistent results obtained from PSF testing.
The results from both the PSF and backscatter imaging experiments reveal that the system achieves optimum quality when the ray energy is approximately $33kev$, resulting in a system resolution of about $5^{\prime}\; 19^{\prime\prime}$ based on the focal point diameter of 0.6958 mm at this energy level.
It's noteworthy that the measured optimum energy value deviates by approximately $2kev$ from the theoretical value of $31kev$. After analysis, we consider the reason for the deviation may be as follows: First of all, a small amount of lead is doped on the coating to prevent X-rays from directly penetrating the inner wall of the microchannel, these lead-doped coatings therefore affect the Angle of reflection of the incident X-rays. Secondly, the focusing efficiency of the lens is not the only factor that determines the final image quality, the X-ray image intensifier camera used in experiment is a secondary detection type X-ray detector. the phosphor Csl used to convert incident X-rays has different conversion efficiency for different energies of X-rays, which makes the energy at the optimal focusing efficiency not the optimal conversion efficiency of the scintillator.
3.2 Complex object imaging
To assess the resolution capabilities of the system with complex targets, the letters ‘C’, ‘U’, ‘S’, ‘T’, ‘N’ with a length of 43 mm were chosen to construct the targets ‘CUST’ and ‘NS’, which the spacing between each letter in the target was approximately 3 mm for imaging. The backscattering imaging outcomes of the two complex targets under a radiation energy of $40\; kev$ are depicted in Fig. 5.
Fig. 5. Two complex test objects and their backscatter image. (a) The target ‘CUST’ for the imaging. (b) Ground truth for the target ‘CUST’. (c) The backscatter imaging result of ‘CUST’. (d) The target ‘NS’ for the imaging. (e) Ground truth for the target ‘NS’. (f) The backscatter imaging result of ‘NS’.
As shown in Fig. 5(b) and (e), we similarly utilized two images precisely matching the target position and size in the backscatter image as the ground truth for comparative evaluation of imaging quality. The output SSIM results are shown in Table 3.
Table 3. The output SSIM result of the image under the target of ‘CUST’ and ‘NS’.
Table 3 demonstrates that, compare with imaging for single simple object, the proposed system maintains consistent imaging quality for complex targets, which indicates that the X-ray backscatter imaging system based on the novel high aspect ratio lobster-eye lens exhibits high spatial resolution.
4. Processing algorithm for backscatter image
In the imaging experiments presented in the previous section (Section 3), it becomes evident that, when compared with traditional focusing imaging, backscatter imaging necessitates a longer integration time due to the weak Compton scattering signal and the substantial impact of air attenuation over the extended imaging distance. Consequently, the imaging outcomes are significantly influenced by noise, making it challenging to extract useful information directly from the original backscattering image. In this work, an image processing algorithm is proposed for the backscattered image, which effectively reduces the necessary acquisition time of the image and enhances the target information in the backscattered image.
The proposed algorithm mainly consists of 3 parts, which are cruciform background removing, image denoising and image enhancement. The flow block diagram of the algorithm we proposed is shown in Fig. 6.
4.1 Removing the cruciform background
Considering the imaging mechanism of the high aspect ratio of the meridian lobster eye lens outlined in Section 2 and the imaging characteristics of the point spread function as illustrated in Fig. 4 and Fig. 5, the target in the backscatter image based on the lobster eye lens is concealed within a cruciform background. We found that a method of subtracting rows and columns can effectively remove the cruciform background.
The method of removing the cruciform background can be expressed as the following formula
According to the formula, the processing of cruciform background removing can be described as follows: extract each the columns and rows from the original image which do not contain target and expand each these column and row into M${\times} $N matrices, subtract these matrices from the original image respectively, then sum the obtained results to take the mean.
4.2 Noise reduction
In the staring backscatter x-ray imaging system there are a variety of imaging noise source, the dominant cause of noise is due to the quantum fluctuations in the x-ray beam because of the low light level caused by weak Compton scattering [20,21]. The Poisson distribution can be used to model the arrival of photons and their expression by electron counts on detectors [22] as the follow formula
Poisson noise is proportional to the underlying signal intensity, which makes very difficult to separating signal from noise. Therefore, a Variance stabilization (Anscombe’s) transformation (VST) which can approximate the Poisson noise to the Gaussian white noise model is applied into the denoising processing [21–23]. The Anscombe’s transformation and its inverse can be described by the following formular [22]
where $I({x,y} )$ and ${I_A}({x,y} )$ indicate the original and transformed images, respectively.After Anscombe’s transformation, the noise in backscatter image can be treated as Gaussian white noise, which utilized a Gaussian low pass filter can effectively reduce it and avoid ringing artifacts, the filter can be described by
where ${D_L}$ is the Cut-off frequency of Gaussian low-pass filter, and $D({u,v} )$ is the distance between the position of $({u,v} )$ and the center of the frequency domain in a $M \times N$ image, which expression is4.3 Target enhancement
Due to the characteristics of staring X-ray imaging, certain rays will directly pass through the lobster eye lens and reach the detector during imaging. These transmitted rays will appear as uniform bright blocks in the image. Therefore, the influence of these bright blocks ought to be attenuated before enhancing the target in the image. Upon observation, we noted that most of these bright blocks are smooth, having slightly lower gray values than the target.
Therefore, the target enhancement we proposed consist with three steps: Gaussian high-pass filtering, histogram equalization and connected component analysis.
The high-pass filter is capable of filtering the low-frequency part of the image (smooth region) and enhancing the high-frequency part (such as the image boundary region). Therefore, utilizing a Gauss high-pass filter can suppress the smooth bright blocks and enhance the target boundary. The filter can be expressed as the following formula:
where ${D_H}$ is the Cut-off frequency of Gaussian High-pass filter, and $D({u,v} )$ is the distance between the position of $({u,v} )$ and the center of the frequency domain.After High-pass filtering, the Histogram equalization is introduced to stretch the pixel value of the target, which means broaden the gray values with more pixels in the image and merges values with fewer pixels to increase the image contrast, which the gray mapping function can expressed as
After filtering, most of the smooth bright blocks will eliminated, but some with high frequency components are still retained. Due to similar pixel values, this retained noise is stretched to the target level in the histogram equalization process. However, we observed that that the area of this retained blocks after high-pass filtering is generally smaller than that of the target, thus the Connected components analysis (CCA) is introduced to distinguish the target from the noise by the area of each connected domain, and to preserve the connected domain with a larger area [24]. The connected component analysis we utilized can be described as: transform the histogram equalization results into binary image, then classical two-pass sequential region labelling is used to identify and label each connected component to calculating the number of pixels possessed by each connected component. The mean value of the number of pixels contained by each connected component is taken as the threshold, and the connected component who is smaller than the threshold is regarded as noise to remove and retain the area larger than the threshold, so as to prevent the target from being misidentified as noise when if there exist a stray block whose area is larger than the target.
4.4 Image processing results and discussion
We applied the aforementioned processing methods to enhance the 60 kv image, which exhibited the best imaging quality, the processing results are shown in the Fig. 7. Since the low contrast in the images, we performed a logarithmic transformation of Fig. 7(a) to (c) for clearer display, the image before the logarithmic transformation for subfigure (a) is depicted in Fig. 4(c).
Fig. 7. The result of each step of the algorithm. (a) Original image. (b) The result of removing the cruciform background. (c) Anscombe’s transformation. (d) Gaussian low-pass filtering. (e) Inverse Anscombe’s transformation. (f) Gaussian high-pass filtering. (g) Histogram equalization. (h) Connected component analysis. (i) The ground truth of ‘S’.
For comparison, Fig. 8 illustrates the results without the bright block removing step, where Subfigure (a) shows the results with only noise reduction and histogram equalization of the image, while Subfigure (b) shows the enhancement results without high-pass filtering and only Connected Components Analysis for bright block removal. We utilized the SSIM index to evaluate the processing results, the output results are shown in Table 4.
Table 4. Comparison of SSIM value between original image and processing result
Table 4 demonstrates that, in comparison with the original image, the target information in the result processed by the proposed algorithm has been significantly enhanced. The Structural Similarity (SSIM) with the ground truth has increased from 0.0509 to 0.6401. This validates the effective processing of the backscatter image based on the high aspect ratio lobster eye lens by the proposed algorithm. furthermore, without the bright block removing steps, the values of the SSIM index will decrease to 0.2059 and 0.0553, respectively, which means that these bright blocks significantly affect the processing results in this scenario.
The weak Compton scattering signal in backscattering imaging system requires the long exposure time. In the imaging experiment in Section 3, to ensure the imaging quality, the imaging needs 350 seconds of exposure time to achieve the effect of 0.0509 SSIM value. After processing with the proposed algorithm, the exposure time can be greatly reduced. As shown in Fig. 9 and Table 5, the SSIM of 0.6199 is still achieved when the exposure time is 110 seconds after processing, which is not much lower than 0.6401 at 350 seconds, but is down to 0.5812 when exposure time is decrease to 100 seconds.
Fig. 9. Images under different exposure time, where (a) is 100 seconds, (b) is 110 seconds and (c) is 350 seconds.
Table 5. The output SSIM value of the backscatter image under different exposure time.
In order to verify whether the algorithm is also applicable to backscatter images of complex targets, we also processed the imaging results in Section 3.2, the obtained results is shown in Fig. 10 and the output SSIM value is shown in table.6.
Table 6 demonstrates that the structural similarity of the target ‘CUST’ is improved from 0.0482 to 0.4936, and the structural similarity of the target ‘NS’ is improved from 0.0763 to 0.5443 after processing, which indicates that the proposed algorithm can still have a good processing effect for complex target backscatter images.
Table 6. Comparison of SSIM value between original image and the processing result
5. Conclusion
In this paper, we introduced an X-ray backscatter imaging method based on an ultra-high-aspect (500) ratio lobster-eye lens. This method enables staring imaging with hard X-rays at a distance of 1.5 meters in a laboratory environment. A series of imaging experiments were conducted to measure the optimal imaging energy and system resolution. Additionally, we proposed an image processing algorithm tailored to the characteristics of this imaging method. Experimental results demonstrated that the structural similarity index of the backscatter image and the ground truth image improved from an average of 0.0526 to 0.5758 after processing, which indicates that the proposed algorithm can effectively enhance the target. Although the theoretical resolution can be very high with an aspect ratio of 500, the actual resolution, determined by the MPO fabrication process [12,13], was measured at $5^{\prime}{19^{^{\prime\prime}}}$, which presents a deviation from the theoretical resolution, suggesting a focus for improvement in future work. Nonetheless, our experimental research has established the feasibility of this novel X-ray gaze imaging system based on lobster-eye lenses, offering potential for lighter and smaller X-ray fluoroscopy imaging equipment.
Funding
Education Department of Jilin Province (JJKH20210814KJ); 111 Project (D17017); National Natural Science Foundation of China (61705018).
Acknowledgments
Mingzhao Ouyang thanks the National Natural Science Foundation of China for help identifying collaborators for this work.
Disclosures
The authors declare no conflicts of interest.
Data availability
All relevant data supporting the findings of this research are available upon request. We are committed to ensuring the accessibility and integrity of our data for the purpose of verification, replication, and further exploration by the scientific community.
For inquiries regarding data access, interested researchers may contact Mingzhao Ouyang via email. We will promptly respond to requests and provide the necessary information to facilitate a thorough examination of the presented results.
We believe that open access to our data will contribute to the advancement of knowledge in the field of optics and enhance the overall impact of our research.
References
1. R. Lanier, Recent Developments in X-Ray Imaging Technology (2012), p. LLNL-TR-587512, 1053658.
2. D.-C. Dinca, J. R. Schubert, J. Callerame, et al., “X-ray backscatter imaging,” in C. S. Halvorson, D. Lehrfeld, T. T. Saito, eds., (2008), p. 694516.
3. J. Callerame, “X-ray backscatter imaging: Photography through barriers,” Powder Diffr. 21(2), 132–135 (2006). [CrossRef]
4. J. R. P. Angel, “Lobster eyes as X-ray telescopes,” ApJ 233, 364 (1979). [CrossRef]
5. M. F. Land, “Eyes with mirror optics,” J. Opt. A: Pure Appl. Opt. 2(6), R44–R50 (2000). [CrossRef]
6. W. Yuan, L. Amati, J. K. Cannizzo, et al., “Perspectives on Gamma-Ray Burst Physics and Cosmology with Next Generation Facilities,” in Gamma-Ray Bursts, D. Götz, M. Falanga, Z. Dai, E. Le Floc’h, N. Tanvir, B. Zhang, eds., Space Sciences Series of ISSI (Springer Netherlands, 2016), Vol. 61, pp. 237–279.
7. V. Dániel, A. Inneman, L. Pína, et al., “X-ray Lobster Eye all-sky monitor for rocket experiment,” in R. HudecL. Pina, eds. (2017), p. 1023503.
8. M. Gertsenshteyn, T. Jannson, G. Savant, et al., “Staring/focusing lobster-eye hard x-ray imaging for non-astronomical objects,” in R. B. James, L. A. Franks, A. Burger, eds. (2005), p. 59220N.
9. V. Grubsky, M. Gertsenshteyn, T. Jannson, et al., “Non-scanning x-ray backscattering inspection systems based on x-ray focusing,” (n.d.). [CrossRef]
10. J. Xu, X. Wang, Q. Zhan, et al., “A novel lobster-eye imaging system based on Schmidt-type objective for X-ray-backscattering inspection,” Rev. Sci. Instrum. 87(7), 073103 (2016). [CrossRef]
11. M. Ouyang, X. Zhao, W. He, et al., “Structural design method of the meridional lobster-eye lens with optimal efficiency,” Appl. Opt. 58(33), 9033 (2019). [CrossRef]
12. L. Li, C. Zhang, G. Jin, et al., “Study on the optical properties of Angel Lobster eye X-ray flat micro pore optical device,” Opt. Commun. 483, 126656 (2021). [CrossRef]
13. L. Li, Y. Zhang, M. Ouyang, et al., “Fabrication and Performance of Lobster Eye X-Ray Micro Pore Optics with the Ultra-high Aspect Ratio,” Publ. Astron. Soc. Pac. 134(1041), 115002 (2022). [CrossRef]
14. H. N. Chapman, K. A. Nugent, and S. W. Wilkins, “X-ray focusing using square channel-capillary arrays,” Rev. Sci. Instrum. 62(6), 1542–1561 (1991). [CrossRef]
15. T. H. K. Irving, A. G. Peele, and K. A. Nugent, “Optical metrology for analysis of lobster-eye x-ray optics,” Appl. Opt. 42(13), 2422 (2003). [CrossRef]
16. A. G. Peele and K. A. Nugent, “Lobster-eye x-ray optics: a rapid evaluation of the image distribution,” Appl. Opt. 37(4), 632 (1998). [CrossRef]
17. V. Grubsky, M. Gertsenshteyn, T. Jannson, et al., “Lobster-eye infrared focusing optics,” in E. L. DereniakR. E. Sampson, eds., (2006), p. 62950F.
18. S. Ma, M. Ouyang, Y. Fu, et al., “Analysis of Imaging Characteristics of Wide-field Lobster Eye Lens,” J. Phys.: Conf. Ser. 2597(1), 012010 (2023). [CrossRef]
19. C. Tarpau, J. Cebeiro, M. K. Nguyen, et al., “Analytic Inversion of a Radon Transform on Double Circular Arcs With Applications in Compton Scattering Tomography,” IEEE Trans. Comput. Imaging 6, 958–967 (2020). [CrossRef]
20. Z. G. Gui, P. C. Zhang, J. H. Zhang, et al., “A X-ray image sharpening algorithm using nonlinear module,” The Imaging Science Journal 60(1), 3–8 (2012). [CrossRef]
21. P. Kittisuwan, “Medical image denoising using simple form of MMSE estimation in Poisson–Gaussian noise model,” Int. J. Biomath. 09(02), 1650020 (2016). [CrossRef]
22. J. Wilder, A Review of: “ Image Processing and Data Analysis, the Multiscale ApproachJ-L Starck and F. Murtagh, eds., IIE Transactions 31, 581–582 (Cambridge University Press, 1999).
23. R. J. Ferrari and R. Winsor, “Digital Radiographic Image Denoising Via Wavelet-Based Hidden Markov Model Estimation,” J Digit Imaging 18(2), 154–167 (2005). [CrossRef]
24. X. Jiang, “Significance linked connected component analysis plus,” Ph. D., University of Missouri–Columbia (2018).
