Open Access
Add to BibSonomy
Abstract
Reflected light microscope is a tool for imaging opaque specimens. However, most of the existing reflected light microscopes can only obtain the two-dimensional image of the specimen. Here we demonstrate that with the help of single-pixel imaging, we can develop a reflection light-field microscopy for volumetric imaging. Importantly, using single-pixel imaging, we can digitally adjust the size of the aperture diaphragm of the proposed reflection light-field microscope for changing the depth of field and for achieving three-dimensional differential phase-contrast imaging in an arbitrary direction, without a hardware change. Our approach may benefit various reflective specimens with wide depth information in the semiconductor industry and material science.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Corrections
29 October 2019: A typographical correction was made to the author affiliations.
1. Introduction
Inspection of the defect of the opaque specimen, such as Micro-Nano devices and integrated circuits, is essential in the manufacturing process and during use. The reflected light microscope has been developed for this purpose [1]. However, the conventional reflected light microscope can only obtain the two-dimensional (2D) image of the specimen. Although optical coherence tomography [2] and confocal microscopy [3] have been proven to be useful for reconstructing the three-dimensional (3D) structure of the specimen, their use in image acquisition relies on point-by-point scanning. As a scan-less technology, light-field microscopy enables reconstructing the 3D structure of a specimen slice by slice from a series of images with different depths of focus [4–7]. However, in most previous demonstrations, light-field microscopy was implemented in a transmission mode [8–9], which imposes a serious limitation on its applications.
Although the camera array [10] and micro-lens array [11] have been used to build the reflection light-field microscope, the camera array system is bulky and expensive [12], and the apertures of these light-field microscopes are difficult to be adjusted. Indeed, aperture adjustment is of critical importance in securing correct illumination, contrast, and depth of field (DOF) in microscopic imaging [13]. It is highly desirable to develop a reflection light-field microscope with a digitally tunable aperture.
Here we propose a reflection light-field microscope with a digitally tunable aperture by single-pixel imaging [14–16]. Using the single-pixel imaging method, we can shift the 2D spatial sampling away from the detector onto the illumination side with a set of basis patterns. Then by placing an array of single-pixel detectors at the rear focal plane of the objective, we can sample the 2D angular information of the light rays. Based on the 4D information, we can digitally refocus the specimen in different depths. More importantly, with the proposed system, the aperture diaphragm can be digitally adjusted for changing the image DOF and for achieving 3D differential phase-contrast imaging (DPC) in an arbitrary direction without a hardware change. The validity and feasibility of the proposed method were validated by imaging a USAF1951 resolution target and an integrated circuit chip. Our approach may benefit various reflective specimens with wide depth information.
2. Experimental setup of reflection light-field microscope based on single-pixel imaging
Figure 1(a) shows the schematic diagram of the experimental setup. The light beam coming from a light emitted diode (LED) source (center wavelength: $\textrm{633 nm}$) is directed onto a digital micromirror device (DMD: 0.95 inches, $\textrm{1},\textrm{920} \times 1,080$ pixels, pixel size: 10.8 µm) by a reflecting mirror. We generate a set of Fourier basis patterns on the DMD. By projecting these basis patterns onto the specimen in sequence via a tube lens (Thorlabs: TTL200, focal length: $\textrm{200 mm}$) and an objective (Nikon: $\textrm{20} \times$, NA 0.4), the 2D spatial information about the specimen is encoded into a group of one-dimensional (1D) time-varying light intensity sequences. After reflected by the specimen, the 1D light intensity sequences are collected by an array of single-pixel detectors. Based on the 1D light intensity sequences recorded by every single-pixel detector, we can retrieve an image of the specimen from every single-pixel detector by using the single-pixel imaging algorithm [17–21]. For the sake of convenience, we use a CCD camera (Point Grey: GS3-U3-60QS6C-C, 1 inch CCD, $2,736 \times 2,192$ pixels, pixel size: 4.54 µm) to construct the single-pixel detector array. Each single-pixel detector is formed by binning together adjacent pixels of the camera.
Fig. 1. (a) Schematic diagram of the proposed system; (b) conventional reflected light microscope equipped with Köhler illumination (BS: beam splitter; ${f_{OL}}$: focal length of the objective lens; $f$: focal length of lens 1 and lens 2).
By placing the single-pixel detector array at the rear focal plane of the objective, we can also enable every single-pixel detector recording the 2D angular information of light rays. For example, the leftmost and the rightmost single-pixel detectors shown in Fig. 1(a) can only record the light signals reflected by the specimen from two different directions. Note in the commercial objective the rear focal plane may not be accessible, a 4f relay system can be necessary. In this way, the rear focal plane of the objective is imaged onto the sensor plane of the camera, and every single-pixel detector can simultaneously record the 2D angular information of the light rays and 2D spatial information of the specimen. Finally, we will capture a bright spot image by using the camera, as shown in the lower-left corner of Fig. 1(a).
Based on the 4D information, we can reconstruct the specimen slice by slice from a series of images with different depths of focus. The reconstruction procedure [22] is summarized as follows. (1) Determine the radius R of the bright spot by using the circle detection and fitting algorithms [23]. (2) Determine the reconstruct depth $\Delta z$ from the focal plane of the objective lens. (3) Determine the incident angle $({\theta_{{x_i}},\theta_{{y_i}}})$ for every single-pixel detector via $\theta_{{x_i}} = \arctan \left( {\frac{{{x_i}}}{R}\tan ({\arcsin ({\textrm{NA}} )} )} \right)$ and $\theta _{{y_i}} = \arctan \left( {\frac{{{y_i}}}{R}\tan ({\arcsin ({\textrm{NA}} )} )} \right)$, where $({{x_i},{y_i}} )$ is the pixel coordinate of the i-th single-pixel detector and NA is the numerical aperture of the objective. (4) Determine the shift for every perspective image via $\Delta {x_i} = \frac{{({M\cdot \Delta z\cdot \tan ({{\theta_{{x_i}}}} )} )}}{{{S_{DMD}}}}$ and $\Delta {y_i} = \frac{{({M\cdot \Delta z\cdot \tan ({{\theta_{{y_i}}}} )} )}}{{{S_{DMD}}}}$, where M denotes the magnification of the objective lens and ${S_{DMD}}$ is the size of the unit pixel of Fourier basis patterns. (5) All perspective images are then shifted based on $({\Delta {x_i},\Delta {y_i}} )$. To avoid interpolation error, we shift all the perspective images in the Fourier domain via the phase-shifting factor ${e^{2\pi ({\Delta {x_i}\cdot {f_x} + \Delta {y_i}\cdot {f_y}} )}}$, where $({{f_x},{f_y}} )$ is the frequency coordinates. (6) All the shifted images are summed to yield the refocusing image of the specimen at $\Delta z$.
3. Method of adjusting the aperture digitally for changing the DOF and realizing 3D DPC imaging
Inspired by the principle of aperture adjustment in the Köhler illumination setup [1,13] and by exploiting the principle of Helmholtz reciprocity [24], we can also enable the proposed system having the capability of changing the image DOF. For a light microscope equipped with the Köhler illumination setup, as shown in Fig. 1(b), changing the image DOF is achieved by adjusting the size of the iris diaphragm mounted at the rear focal plane of the objective, because the size of the iris diaphragm determines the cone angle of light that illuminates the specimen, while the cone angle directly affects the image resolution, contrast, and DOF. For the proposed system shown in Fig. 1(a), it is a reciprocal configuration of the imaging system shown in Fig. 1(b). More specifically, we use a DMD to replace the 2D sensor for sampling the 2D spatial information and use an array of single-pixel detectors to replace the Köhler illumination setup for sampling the 2D angular information. Therefore, by exploiting the Helmholtz reciprocity, if we change the size of the single-pixel detector shown in Fig. 1(a), we can also adjust the aperture of the proposed system for changing the image DOF. Different from the conventional imaging system shown in Fig. 1(b), the aperture of the proposed system can be digitally adjusted because each single-pixel detector shown in Fig. 1(a) is formed by binning adjacent pixels of the 2D sensor together.
Furthermore, we can digitally adjust the aperture of the proposed system to realize the 3D DPC imaging in an arbitrary direction. 3D DPC imaging is commonly used to produce DPC images at varying focal planes, where each DPC image represents the gradient variation of the specimen at a different depth [25]. To realize the $0^\circ$ 3D DPC imaging with the proposed system, we first divide the single-pixel detector array into two groups as shown in Figs. 2(a1) and 2(a2). After shifting all the perspective images retrieved from all the single-pixel detectors based on the refocusing depth, we sum the shifted perspective images related the single-pixel detectors (Fig. 2(a1)) to produce images ${I_1}$ and sum the shifted perspective image related the single-pixel detectors (Fig. 2(a2)) to produce images ${I_2}$. Then the 3D phase-contrast image of the specimen can be recovered by ${{({{I_1} - {I_2}} )} \mathord{\left/ {\vphantom {{({{I_1} - {I_2}} )} {({{I_1} + {I_2}} )}}} \right.} {({{I_1} + {I_2}} )}}$. Particularly, we can also divide the single-pixel detector array into two groups in arbitrary directions, such as $30^\circ$ (Figs. 2(b1) and 2(b2)), $60^\circ$ (Figs. 2(c1) and 2(c2)), and $90^\circ$ (Figs. 2(d1) and 2(d2)). Figure 2(e) shows the flow chart on how to divide the single-pixel detector array into two groups in an arbitrary direction $\beta$. Then, following the method of yielding the $0^\circ$ DPC images, we can achieve 3D DPC imaging in arbitrary directions. Note that this DPC imaging system proposed here is similar to the scanning DPC system reported in [26], where a split-detector is used to record the light signals transmitted or reflected by the specimen from different directions. The DPC image is produced by subtracting light signals recorded by two halves of the detector. However, using the proposed system, we can realize 3D DPC imaging in an arbitrary direction without mechanical scanning.
Fig. 2. Layouts of the single-pixel detector array used to produce $\textrm{0}^\circ$(a1-a2), $\textrm{30}^\circ$(b1-b2), $\textrm{60}^\circ$(c1-c2), and $\textrm{90}^\circ$(d1-d2) DPC images; (e) procedure used to divide the single-pixel detectors into two groups in an arbitrary direction.
4. Experimental Results
We conducted three experiments to validate the proposed method. Figure 3 shows a photograph of the experimental setup, where the focal lengths of Lens 1 and Lens 2 are $\textrm{150 mm}$. The size of the reconstructed images is set as $384 \times 384$ pixels. To encode the spatial information of the specimen, we use the Fourier basis patterns in the experiment [27]. To maximize the measurement speed, we convert the grayscale Fourier basis patterns to the binary Fourier basis patterns via the dither algorithm [28]. Furthermore, to ensure the quality of the reconstructed image, we up-sample the binary Fourier basis patterns to be $768 \times 768$ pixels on the DMD. Lastly, to reduce the data storage, each image captured by the camera is down-sampled by 2 times.
4.1 Results of multi-perspective imaging and light-field refocusing
To realize light-field microscopy, we built an annular array of 73 single-pixel detectors from the camera. The layout of the annular single-pixel detector array is shown in the lower-left corner of Fig. 1(a), which includes 5 rings. From the center to the outside, the number of single-pixel detectors in each ring is 1, 3, 6, 9, 18, and 36, respectively. The radii of all the single-pixel detectors were set as 454 µm.
An integrated circuit chip was used as the measurement target. Figures 4(a1)–4(d1) show four perspective images retrieved from the bottommost (Fig. 4(a2)), rightmost (Fig. 4(b2)), topmost (Fig. 4(c2)), and leftmost (Fig. 4(d2)) single-pixel detectors, respectively. It can be seen that when we used the single-pixel detectors (Figs. 4(b2) and 4(d2)) distributed along the horizontal direction to retrieve images, the images of the objects distributed below or above the focal plane were shifted horizontally from each other, as highlighted by the red arrows in Figs. 4(b1) and 4(d1). Similarly, when we used the single-pixel detectors (Figs. 4(a2) and 4(c2)) distributed along the vertical direction to retrieve images, the images of the objects distributed below or above the focal plane were shifted vertically from each other in the retrieved images, as highlighted by the red arrows in Figs. 4(a1) and 4(c1). We also synthesized a Media (Visualization 1) to show the change of the perspective view by changing the position of the single-pixel detector. These experimental results indicate that using the proposed system, we can realize multiple-perspective imaging without mechanical scanning. Particularly, different from the light-field microscope based on a micro-lens array, the multiple-perspective imaging here was achieved without sacrificing the spatial resolution because sampling of spatial information was implemented in the illumination side, while sampling of angular information was implemented in the detection side.
Fig. 4. (a1)–(d1) Perspective images retrieved from the bottommost, rightmost, topmost, and leftmost single-pixel detectors, respectively; (a2)–(d2) locations of the single-pixel detectors. The radii of all the single-pixel detectors were set as 454 µm.
Based on the perspective images retrieved above, we refocused the specimen at different depths by using the digital refocusing procedure mentioned in Section 2. Figure 5 shows three images that are digitally refocused at different depths. To show the entire digital refocusing process from −50 µm to 50 µm, we also synthesized a Media (see Visualization 2) based on all the refocusing images. These experimental results suggest that, by using the proposed system, different parts of the specimen can be digitally refocused at different depths.
Fig. 5. Digitally refocused images. (a) $z ={-} 15$ µm; (b) $z = 10$ µm; (c) $z = 32$ µm. The radii of all the single-pixel detectors were set as 454 µm.
4.2 Results of adjusting the DOF by digitally changing the aperture diaphragm
We tested the reported approach with a USAF1951 resolution target. The USAF resolution target was first placed in the focal plane of the objective lens and later was displaced axially to 10 µm, 20 µm, 40 µm, 70 µm, 100 µm, and 150 µm, respectively. In every position, we reconstructed six images by adjusting the aperture diaphragm. The aperture adjustment was realized by setting the radius of the single-pixel detector as 90.8 µm, 454 µm, 908 µm, 1362 µm, and 2270 µm, respectively, corresponding NA of 0.009, 0.046, 0.092, 0.139, and 0.231, respectively. Figure 6 shows the reconstructed images by changing the radius of the single-pixel detector. Note that we show only the image for distances moving away in a positive direction, but it is seen to be the same for the negative direction. As shown in Fig. 6, when the radius of the single-pixel detector was set as 2270 µm the features of element 1 in group 6 (resolution limit of 7.81 µm per line) are hardly resolved if the specimen was displaced at 40 µm. In contrast, when the radius of the single-pixel detector was reduced to 90.8 µm, the features of element 4 in group 6 (resolution limit of 5.52 µm per line) are resolved in the reconstructed image even though the specimen was displaced at 40 µm. These experimental results suggest that by changing the radius of the single-pixel detector, we can digitally adjust the aperture of the proposed system for changing the DOF of the reconstructed image.
Fig. 6. Images reconstructed by changing the size of the single-pixel detector and displacing the specimen at different positions.
From Fig. 6, we can also evaluate the resolution and DOF of the proposed system. It can be seen that when the specimen is in focus, the smallest features that can be resolved by the proposed method are 1.74 µm per line in the resolution (in group 8, element 2). Different from the conventional light microscopic image, when the specimen is in focus, the resolution of the image reconstructed by our approach will not be changed with the size of the aperture, because the resolution of the reconstructed image is determined by the highest frequency of the structured patterns projected in the experiment [29] and the highest frequency of the projected Fourier basis patterns is lower than the cutoff frequency determined by the objective. In addition, when displacing the specimen away from the focal plane, the resolution of the reconstructed image degrades gradually. Nevertheless, it can be seen from Fig. 6 that with the 0.4 NA objective used in the proposed system, the features of element 4 in group 6 (resolution of 5.52 µm per line) can still be resolved within a range of ${\pm} \textrm{40}$ µm.
We further validated the proposed approach via light-field refocusing. Figure 7 shows four groups of refocusing results by setting the radius of every single-pixel detector as 2270 µm, 1362 µm, 908 µm, and 90.8 µm, respectively, corresponding NA of 0.231, 0.139, 0.092, and 0.009, respectively. Visualization 3 shows the entire digital refocusing process by changing the aperture. As shown in Figs. 7(a1)–7(a6), when we increase the radius of the single-pixel detector, the DOF of the light-field refocusing result becomes shallow. In contrast, if we decrease the radius of the single-pixel detector, the DOF of the refocusing result becomes large, as shown in Figs. 7(d1)–7(d6). These experimental results indicate that the DOF of the proposed light-field microscope can be digitally adjusted by changing the radius of the single-pixel detector. It is noteworthy that all the digitally refocused results shown in Fig. 7 were retrieved from the same measurement data. Therefore, changing the DOF of the light-field microscope here can be achieved without increasing the number of measurements.
Fig. 7. Digitally adjust the DOF of the light-field imaging by setting the radius of the single-pixel detector as 2270 µm (a1-a6), 1362 µm (b1-b6), 908 µm (c1-c6), and 90.8 µm (d1-d6), respectively.
4.3 Results of 3D DPC imaging in an arbitrary direction by changing the aperture diaphragm
By changing the aperture diaphragm digitally, we can further enable the proposed system having 3D DPC imaging capability in an arbitrary direction without a hardware change. To validate it, we used the perspective images retrieved in Subsection 4.1 to produce the 3D DPC images. Figures 8(a1)–8(a3) show the $\textrm{0}^\circ$ 3D DPC images digitally refocused at different depths. Compared with the images shown in Fig. 7, the images produced by the DPC method have a distinctive shadow-cast appearance, rendering the slope, valleys, and other discontinuities in the surface of the specimen into a high-contrast pseudo-relief. Particularly, the proposed system allows us to produce 3D DPC images in arbitrary direction without a hardware change. Figure 8 shows the images by using the $\textrm{0}^\circ$, $\textrm{30}^\circ$, $\textrm{60}^\circ$, $\textrm{90}^\circ$, $\textrm{120}^\circ$, and $\textrm{150}^\circ$ 3D DPC imaging, respectively. Visualization 4 shows the 3D DPC images in different directions. From these results, we can see that, in each direction, the DPC image achieves maximum contrast for the features of a specimen that are perpendicular to the shear axis, while achieving minimum contrast for the features of a specimen that are parallel to the shear axis. These experimental results suggest that using the 3D DPC imaging, the proposed system has the capability to maximize or minimize the contrast effect for the selected specimen feature. This capability is especially critical to highly ordered semiconductors having numerous extended, linear regions intermixed with closely-spaced periodic structures.
Fig. 8. Digitally refocusing images retrieved by using the $\textrm{0}^\circ$(a1-a3), $\textrm{30}^\circ$(b1-b3), $\textrm{60}^\circ$(c1-c3), $\textrm{90}^\circ$(d1-d3), $\textrm{120}^\circ$(e1-e3), and $\textrm{150}^\circ$(f1-f3) DPC imaging, respectively. The radii of all the single-pixel detectors were set as 454 µm.
5. Discussion
One limitation of the proposed method is that it suffers from a long imaging time due to the use of single-pixel imaging. The greater the image size, the longer the imaging time will be because the single-pixel imaging time depends on the size of the reconstructed image. But this problem can be considerably alleviated using compressive sensing [14,30] because the natural image tends to be compressed in the Fourier space. For example, one can reconstruct the perspective images by selectively sampling the Fourier coefficients [19]. Recently, Satat et al. [31] developed a compressive ultrafast sensing method that can greatly reduce the number of measurements for single-pixel imaging. This compressive sensing approach may open up the door to further reduce the imaging time of the proposed method. Importantly, most of the reflective specimens are static. Thus, the imaging effect will not be affected by the imaging time. Moreover, reconstructing images with different DOFs and realizing 3D DPC imaging in arbitrary directions will not increase the measurement time because all these images are retrieved from the same measurement data by using different post-processing methods. Therefore, we believe a number of applications, such as semiconductor industry and material science, may immediately gain from the advantages of the proposed reflection light-field microscope: digital refocusing the specimen in different depths without mechanical scanning, adjusting the DOF and realizing 3D DPC imaging without a hardware change.
Another unwanted effect introduced by decreasing the radius of the single-pixel detector is the diffraction artifact [32,33], as shown in Fig. 6. This effect is similar to that introduced by reducing the condenser aperture in a conventional light microscope. As the aperture decreases, the coherence of the light source increases and diffraction ringing is visible. In contrast, when we increase the aperture size, the diffraction effect will be washed out. Since each light-field refocusing image is computed by shifting and adding all the perspective images together, the diffraction effect is suppressed by the shifting and adding process, as shown in Fig. 7 and Fig. 8. Thus the proposed approach is tolerant to diffraction.
The proposed method requires using an array of single-pixel detectors. For visible-light imaging, we can use a regular camera to build an array of single-pixel detectors. As a regular camera is relatively cheap, the cost of the array of single-pixel detectors is lower. For non-visible waveband imaging, a non-visible waveband camera or an array of single-pixel detectors is required, making the proposed system more expensive than the imaging system based on a single-pixel detector. Given its digitally tunable aperture and volumetric imaging capability, the proposed imaging system we proposed here might open up interesting perspectives for microscopic imaging for the reflective specimen.
6. Conclusion
In summary, we develop a reflection light-field microscope with a digitally tunable aperture by single-pixel imaging. Experimental results demonstrate that the proposed system has the capability of realizing multi-perspective imaging and reconstructing the specimen slice by slice from a series of images without mechanical scanning. Particularly, it allows digitally adjusting the aperture by changing the size of the single-pixel detector, thus we can digitally adjust the DOF of the light-field microscope and achieve 3D DPC imaging in an arbitrary direction without a hardware change. The proposed system may potentially be applied in the semiconductor industry and material science for various specimens with wide depth information.
Funding
National Natural Science Foundation of China (61875074, 61605126, 61605063); Fundamental Research Funds for the Central Universities (21617403).
Acknowledgment
We thank Prof. Chao Zuo from Nanjing University of Science and Technology, Nanjing, China, for his assistance in preparing the specimen.
Disclosures
The authors declare no conflicts of interest.
References
1. D. Murphy, Fundamentals of light microscopy and electronic imaging (John Wiley & Sons, 2002).
2. D. Huang, E. A. Swanson, C. P. Lin, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, K. Gregory, C. Puliafito, and J. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991). [CrossRef]
3. https://www.microscopyu.com/techniques/confocal/confocal-reflection-microscopy.
4. M. Levoy, R. Ng, A. Adams, M. Footer, and M. Horowitz, “Light field microscopy,” ACM T. Graphic. 25(3), 924–934 (2006). [CrossRef]
5. M. Levoy, Z. Zhang, and I. McDowall, “Recording and controlling the 4D light field in a microscope using microlens arrays,” J. Microsc. 235(2), 144–162 (2009). [CrossRef]
6. M. Broxton, L. Grosenick, S. Yang, N. Cohen, A. Andalman, K. Deisseroth, and M. Levoy, “Wave optics theory and 3-D deconvolution for the light field microscope,” Opt. Express 21(21), 25418–25439 (2013). [CrossRef]
7. Z. Lu, J. Wu, H. Qiao, Y. Zhou, T. Yan, Z. Zhou, X. Zhang, J. Fan, and Q. Dai, “Phase-space deconvolution for light field microscopy,” Opt. Express 27(13), 18131–18145 (2019). [CrossRef]
8. K. Guo, Z. Bian, S. Dong, P. Nanda, Y. Wang, and G. Zheng, “Microscopy illumination engineering using a low-cost liquid crystal display,” Biomed. Opt. Express 6(2), 574–579 (2015). [CrossRef]
9. C. Zuo, J. Sun, S. Feng, M. Zhang, and Q. Chen, “Programmable aperture microscopy : A computational method for multi-modal phase contrast and light field imaging,” Opt. Laser. Eng. 80, 24–31 (2016). [CrossRef]
10. X. Lin, J. Wu, G. Zheng, and Q. Dai, “Camera array based light field microscopy,” Biomed. Opt. Express 6(9), 3179–3189 (2015). [CrossRef]
11. J. Liu, D. Claus, T. Xu, T. Keßner, A. Herkommer, and W. Osten, “Light field endoscopy and its parametric description,” Opt. Lett. 42(9), 1804–1807 (2017). [CrossRef]
12. G. Wu, B. Masia, S. Jarabo, Y. Zhang, L. Wang, Q. Dai, T. Chai, and Y. Liu, “Light Field Image Processing: An Overview,” IEEE J. Sel. Top. Signal Process. 11(7), 926–954 (2017). [CrossRef]
13. S. Inoué, Video microscopy (Springer Science & Business Media, 2013).
14. M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Proc. Mag. 25(2), 83–91 (2008). [CrossRef]
15. W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93(12), 121105 (2008). [CrossRef]
16. B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science 340(6134), 844–847 (2013). [CrossRef]
17. P. Clemente, V. Durán, E. Tajahuerce, P. Andrés, V. Climent, and J. Lancis, “Compressive holography with a single-pixel detector,” Opt. Lett. 38(14), 2524–2527 (2013). [CrossRef]
18. N. Radwell, K. Mitchell, G. Gibson, M. Edgar, R. Bowman, and M. Padgett, “Single-pixel infrared and visible microscope,” Optica 1(5), 285–289 (2014). [CrossRef]
19. Z. Zhang, X. Ma, and J. Zhong, “Single-pixel imaging by means of Fourier spectrum acquisition,” Nat. Commun. 6(1), 6255 (2015). [CrossRef]
20. M.-J. Sun, M. P. Edgar, D. B. Phillips, G. M. Gibson, and M. J. Padgett, “Improving the signal-to-noise ratio of single-pixel imaging using digital microscanning,” Opt. Express 24(10), 10476–10485 (2016). [CrossRef]
21. J. Peng, M. Yao, J. Cheng, Z. Zhang, S. Li, G. Zheng, and J. Zhong, “Micro-tomography by single-pixel imaging,” Opt. Express 26(24), 31094–31105 (2018). [CrossRef]
22. G. Zheng, C. Kolner, and C. Yang, “Microscopy refocusing and dark-field imaging by using a simple LED array,” Opt. Lett. 36(20), 3987–3989 (2011). [CrossRef]
23. R. Gonzalez and R. Woods, Digital imaging processing (Addison-Wesley: Massachusetts, 1992).
24. M. Born and E. Wolf, Principle of Optics (Pergamon, 1995).
25. L. Tian, J. Wang, and L. Waller, “3D differential phase-contrast microscopy with computational illumination using an LED array,” Opt. Lett. 39(5), 1326–1329 (2014). [CrossRef]
26. D. K. Hamilton and C. J. R. Sheppard, “Differential phase contrast in scanning optical microscopy,” J. Microsc. 133(1), 27–39 (1984). [CrossRef]
27. Z. Zhang, X. Wang, G. Zheng, and J. Zhong, “Hadamard single-pixel imaging versus Fourier single-pixel imaging,” Opt. Express 25(16), 19619–19639 (2017). [CrossRef]
28. Z. Zhang, X. Wang, G. Zheng, and J. Zhong, “Fast Fourier single-pixel imaging via binary illumination,” Sci. Rep. 7(1), 12029 (2017). [CrossRef]
29. J. Peng, M. Yao, Z. Cai, X. Qiu, Z. Zhang, S. Li, and J. Zhong, “Optical synthetic sampling imaging: Concept and an example of microscopy,” Appl. Phys. Lett. 115(12), 121101 (2019). [CrossRef]
30. K. Marwah, G. Wetzstein, Y. Bando, and R. Raskar, “Compressive light field photography using overcomplete dictionaries and optimized projections,” ACM. T. Graphic. 32(4), 1 (2013). [CrossRef]
31. G. Satat, M. Tancik, and R. Raskar, “Lensless Imaging with Compressive Ultrafast Sensing,” IEEE T. Comput. Imag. 3(3), 398–407 (2017). [CrossRef]
32. L. Tian and W. Laura, “3D intensity and phase imaging from light field measurements in an LED array microscope,” Optica 2(2), 104–111 (2015). [CrossRef]
33. L. Tian and W. Laura, “Quantitative differential phase contrast imaging in an LED array microscope,” Opt. Express 23(9), 11394–11403 (2015). [CrossRef]
