Open Access
Add to BibSonomy
Abstract
Single-pixel imaging (SPI) can perceive the world using only a single-pixel detector, but long sampling times with a series of patterns are inevitable for SPI, which is the bottleneck for its practical application. Developing new patterns to reduce the sampling times might provide opportunities to address this challenge. Based on the Kronecker product of Hadamard matrix, we here design a complete set of new patterns, called Gao-Boole patterns, for SPI. Compared to orthogonal Hadamard basis patterns with elements valued as +1 or −1, our Gao-Boole patterns are non-orthogonal ones and the element values are designed as +1 or 0. Using our Gao-Boole patterns, the reconstructed quality of a target image (N × N pixels) is as high as the Hadamard one but only with half pattern numbers of the Hadamard ones, for both full sampling (N2 for Gao-Boole patterns, 2N2 for Hadamard basis patterns) and undersampling cases in experiment. Effectively reducing the patterns numbers and sampling times without sacrificing imaging quality, our designed Gao-Boole patterns provide a superior option for structural patterns in SPI and help to steer SPI toward practical imaging application.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Single-pixel imaging (SPI) is a novel indirect imaging technique emerging from dual photography presented by Sen et al. in $2005$ [1]. Different from the conventional pixelated imaging techniques that capture images directly, SPI uses a single-pixel detector to obtain the object information [2–10]. Based on the correlation or inverse calculation between the measured light intensities and a set of structural patterns, the objects could be reconstructed through different computational algorithms [11–14]. Generally speaking, there are two kinds of configurations for SPI, one is structural detection, and the other is structural illumination. The major difference between these two configurations is the position of the light modulator. For structural detection, the light modulator is placed between the object and the detector; while for structural illumination, the light modulator is placed between the light source and the object [2,7], which is also named as computational ghost imaging [5]. Essentially, SPI and computational ghost imaging have the same imaging principle and can use the same computational algorithms.
SPI is especially useful in the situations that direct imaging is not applicable, especially in turbulent environments with fog [15,16] or turbid water [17–19]. When applied in biological tissue imaging [16,20,21], SPI technique can lower the X-ray radiation to biological tissue [20]. Also, compared with the conventional silicon-based cameras, SPI is cheaper and more feasible in the invisible wavelengths such as Terahertz [22–24] and infrared range [25,26]. However, each coin has two sides, so does SPI. Due to its unique imaging principle, the intrinsic drawback of SPI is that the imaging time is much longer than that of pixelated imaging. To overcome this shortcoming, an effective method was proposed by Duarte et al. in $2008$ [4], which combined SPI with compressive sensing (CS) [27]. It made recovering images while the number of measurements was fewer than the total pixel in the images possible and avoided a sharp decline in imaging quality while shortening the imaging time [28–36]. Other reconstruction algorithms like conjugate gradient descent based method [7], alternating projection method [37], and gradient descent method [38] were also developed to work in such undersampling situations. Not only algorithms can affect the efficiency of SPI, but the structural patterns applied are also crucial for improving the efficiency of SPI.
Different structural patterns can be applied in SPI, such as Hadamard basis patterns [39–41], Fourier basis patterns [42] and random patterns [43,44]. Hadamard basis patterns are binary of which the elements value is $+1$ or $-1$, while Fourier basis patterns are in grey-scale and random patterns could be either binary or in grey-scale. Since Hadamard basis patterns and Fourier basis patterns have orthogonality and allow $100\%$ reconstruction in theory, they are the most used structural patterns in SPI. It usually takes $2N^2$ measurements for Hadamard basis SPI [40] to fully sample an image with $N\times N$ pixels through differential measurements, since $-1$ value cannot be displayed by light modulators directly and one Hadamard basis needs to be divided into two patterns. Four-step phase-shifting Fourier basis SPI [42] usually requires more than $2N^2$ measurements to fully sample an image with $N\times N$ pixels and the time per measurement taking is longer because the patterns are in gray-scale if digital micromirror devices (DMD) are employed as a pattern generator. It is generally believed that orthogonal patterns (e.g. Hadamard basis patterns) are more efficient than non-orthogonal ones (e.g. random patterns). Can there be a set of non-orthogonal patterns that perform better than orthogonal ones, in particular than the widely used Hadamard basis patterns?
In this work, we design a complete set of non-orthogonal patterns, called Gao-Boole (GB) patterns, and show their good performance in both full sampling and undersampling cases. In section $2$, we introduce the Kronecker product generation approach of GB patterns, then explore their symmetry, non-orthogonality and completeness. We further explain their imaging principle by designing an efficient patterns’ order, called waffle order, for the analysis and comparison of the undersampling measurements. In section $3$, we evaluate their imaging performance and quality in numerical simulations and experiments. The last section finally gives a conclusion.
2. Gao-Boole patterns
2.1 Generation approach
The generation process of GB patterns is based on the construction method of the Hadamard matrix [45–48] of order $N=2^n$ where $n$ is a positive integer, using the Kronecker product. First, list the complete set of Hadamard matrices of order $N=2^1$ (the lowest order), which are named $H_{1}, H_{2}, H_{3}, H_{4}$. They are the bases to build higher-order Hadamard matrices set. $H_{ij}$ stands for a Hadamard matrix of order $N=2^2$, which is generated through $H_{ij}=H_{j}\otimes H_{i}$, where $H_{i}$ and $H_{j}$ are chosen from the above-mentioned $H_{1}, H_{2}, H_{3}, H_{4}$. Below gives the details when $n = 1, 2$, and $3$.
For $n=1$ (i.e. $N=2^1$),
Second, we replace the elements valued $-1$ with $0$ in $H_{ij\cdots }$, which becomes into $P_{ij\cdots }$, representing our GB patterns. We may memorize their names this way, $H$ represents "Hadamard matrix" and $P$ represents "pattern" and "positive". For $n=1$ (i.e. $N=2^1$),
Then, we take $n=4$ and introduce how to generate a $P_{ijkl}$, where the subscript $ijkl$ represent its serial number. Let us take pattern $P_{3412}$ as an example. $P_{3412}$ is formed from $H_{3412}=H_{2}\otimes H_{1}\otimes H_{4}\otimes H_{3}$, after changing the elements $-1$ into $0$.
For a complete set of GB patterns of order $N=2^n$, there are $N^2$ patterns, and each pattern contains $(\frac {N^{2}}{2}+\frac {N}{2})$ elements valued as $1$ and $(\frac {N^{2}}{2}-\frac {N}{2})$ elements valued as $0$. The proportion of the number of $1$ to the number of $0$ is the same for patterns of the same order. Since the elements in GB patterns are either $1$ or $0$, they belong to the Boolean matrices in mathematics and computer science. Meanwhile, they are just a specific category of Boolean matrices that are developed for SPI, for the convenience of distinguishing, we add the last name of the first author in the naming of Gao-Boole patterns. Figure 1 shows the GB patterns set of order $N=2^1$ and order $N=2^2$.
Fig. 1. Generation of the complete Gao-Boole patterns set of order $N=2^{1}$ and order $N=2^{2}$. The white blocks refer to $+1$ and the black blocks refer to $0$.
2.2 Characteristics of Gao-Boole patterns
One interesting characteristic GB patterns possess is their symmetry. The symmetry refers to the symmetry between two GB patterns, but not the symmetry within a pattern itself. For GB patterns of $N=2^{1}$, it is easy to tell that every two of them are symmetrical to each other, whether they are up-down symmetry ($P_{1}$ and $P_{3}$, $P_{2}$ and $P_{4}$), left-right symmetry ($P_{1}$ and $P_{2}$, $P_{3}$ and $P_{4}$), or central symmetry ($P_{1}$ and $P_{4}$, $P_{2}$ and $P_{3}$), as shown in Fig. 2. This is not a coincidence. For an arbitrary GB pattern, we can always find three other patterns symmetrical to it. Figure 2 also shows different symmetrical $P_{ij}$ for order $N=2^2$: each row involves $4$ patterns symmetrical to each other. For a complete set of GB patterns of order $N=2^n$, there are $N^2$ patterns in total and $\frac {N^{2}}{4}$ groups of symmetrical patterns. The symmetry of patterns hints us a clue to re-order them for sampling scenes efficiently, which will be discussed in section $2.4$.
Non-orthogonality is another characteristic of GB patterns, though arising from the orthogonal matrices. Review the orthogonality of Hadamard basis patterns for ease of understanding. The Hadamard basis patterns’ orthogonality refers to the inner product of any two Hadamard basis patterns is $0$, rather than any Hadamard basis pattern is orthogonal. The fundamental reason is that the big Hadamard matrix ($N^{2}\times N^{2}$) which Hadamard basis patterns ($N\times N$) resized from is orthogonal [40]. Our GB patterns are not capable of these properties, thus they are non-orthogonal.
The set of GB patterns is also complete, so the perfect reconstruction for full sampling is guaranteed theoretically. This is because, when we resize the complete set of GB patterns of order $N$ into a larger matrix of order $N^{2}$, the larger matrix has full rank, which is equivalent to saying that our GB patterns are linearly independent. Therefore, any images with $N^{2}$ pixels could be expressed as a linear combination of them, that is, they are complete.
2.3 Reconstruction principle
Our GB patterns are suitable for imaging square scenes with $N^{2}=2^{2n}$ pixels. For the full sampling of an object $X_{0}$ with $N^{2}$ pixels, $I_{m}$ represents the measured light intensities determined by the original image $X_{0}$ and the structural pattern $P_{m}$, $m$ is the measurement number. $I_{m}$ can be expressed as
where $\langle P_{m},X_{0}\rangle _{F}$ is the Frobenius inner product. The recovered image $X_{r}$ is where $a$ and $b$ are constants, $a=\frac {N^{2}}{4}$, $b=(\frac {N^{2}}{4}+\frac {N}{2})X_{total}$ and $X_{total}=\sum _{i=1}^{N}\sum _{j=1}^{N}X_{0}(i,j)$. While for Hadamard basis patterns, $X_{r}=N^{2}X_{0}$, there is no second constant term.The key to the precise reconstruction of an image is to recover the differences among pixels’ values accurately, which hints us that the recovered images may not be a multiple of the original images, and the structural patterns may not be orthogonal.
2.4 Waffle order
For Hadamard basis patterns, various orders can give different imaging efficiencies in undersampling cases. Using the natural ordered [11] Hadamard basis patterns, the recovered images will contain the ghosting of the scenes. The cake-cutting order [39] and the Russian doll order [40] of Hadamard basis patterns are two kinds of efficient orders for Hadamard basis SPI. In the cake-cutting order, each Hadamard basis pattern is taken as a cake, and the number of pieces are counted inside each cake [39]. For the Russian doll order, the Hadamard basis patterns are catalogued into four quarters and are re-ordered within each quarter based on their block number [40]. They are more efficient in undersampling when compared to "sequency" or "dyadic" Hadamard order. Based on the symmetrical characteristics of GB patterns, we here design a new order, called the waffle order, aiming at achieving high efficiency in the undersampling cases for our GB SPI.
To effectively order the GB patterns, we need to divide them into sixteen areas according to their serial numbers first, like the waffles, as shown in Fig. 3(a). In each grid, we have $\frac {N^{2}}{16}$ patterns of which the last two digits of the serial numbers are the same and the first other digits circulate in the order of $[1,2,3,4],[2,1,4,3],[3,4,1,2],[4,3,2,1]$, respectively. Second, we pick one out of every $4$ patterns within each grid according to the selection rules (Fig. 3(b)) forming the new $4$ parts. The selection rules are the same for GB patterns of different orders ($N>4$). Third, we reorder the selected patterns inside each part on basis of their 2D Fourier frequencies, which refers to the sum of the absolute frequencies derived from 2D Fourier transform of each pattern. Figure 4 gives a detailed example to explain the first two steps of waffle order procedures. The natural order of GB patterns refers to the patterns only taking the Step 1 in Fig. 3. The key for the waffle order is to classify the patterns into four parts according to their symmetry and re-order them based on the 2D Fourier frequencies on behalf of their basic patterns’ properties.
Fig. 4. The explaining example of the first two steps of waffle order of the set of Gao-Boole patterns of order $N=2^3$.
3. Results
3.1 Numerical simulations
To test the imaging effectiveness and the efficiency of our GB patterns in principle, numerical simulations with full sampling and different undersampling ratio are performed, as shown in Fig. 5 and Fig. 6. Using the test image "cameraman" with $64\times 64$ pixels and total variation (TV) [7] reconstruction algorithm (compressive algorithm), we compare the imaging performance of the waffle ordered GB patterns with cake-cutting ordered and Russian doll ordered Hadamard basis patterns. To demonstrate the imaging performance of GB patterns in SPI entirely, the SPI comparison of both natural ordered and the random ordered GB patterns and Hadamard basis patterns are also displayed.
Fig. 5. Comparison of "cameraman" SPI reconstruction with $64\times 64$ pixels by using Gao-Boole patterns and Hadamard basis patterns at different sampling ratio in numerical simulations.
Fig. 6. RE, SSIM and PSNR as a function of the sampling ratio of "cameraman" with $64\times 64$ pixels through Gao-Boole SPI and Hadamard basis SPI in numerical simulations.
To quantitatively assess the quality of the reconstructed image, relative error (RE), structural similarity (SSIM) and the peak signal-to-noise ratio (PSNR) are used [13,39,49].
The RE computes the root squared intensity differences of reconstructed and original image pixels relative to the root squared intensity of the original image pixels, which is defined as:
where $||X_{0}||_{F}$ is the Frobenius norm [37]. For RE, the smaller the value, the better the image quality.SSIM compares the structural information of objects in reconstructed and original images, and it is independent of the average luminance and contrast. The detailed definition of SSIM can be found in Ref. [49]. The bigger value of SSIM, the more similar the structure is between them.
The PSNR is the ratio between the maximum possible power of the signal and the noise power. Assume that $X_{0}$ and $X_{r}$ are the original and reconstructed images, respectively. The noise power is described by the mean squared error (MSE):
where $N$ is the width of the images in terms of pixels, $X_{0}(i,j)$ and $X_{r}(i,j)$ are their gray values of the $i$-th and $j$-th column. Then the PSNR is defined as: where the maximum possible intensity has been normalized to 255.Since the total measurements taken (i.e. patterns required) in the full sampling SPI experiments is different for different kinds of patterns, for ease of comparison, we here define the $100\%$ sampling ratio corresponds to $N^{2}$ measurements taken in the SPI experiments of scenes with $N\times N$ pixels. Thus, $100\%$ sampling ratio is the full sampling for GB SPI but only half sampling for Hadamard basis SPI, because one Hadamard basis should be separated into two patterns in experiment to eliminate the elements valued as $-1$. In other words, for a full sampling case, Hadamard patterns required in SPI experiment is double as our GB patterns. It should be mentioned that we use $2N^2$ Hadamard patterns consisting of $1$ or $0$ elements in simulations, rather than $N^2$ Hadamard basis consisting of $1$ or $-1$ elements directly. Each Hadamard pattern has its complementary pattern (i.e., all $0$ and $1$ element are swapped in two patterns) in our simulation for different undersampling cases. It is noted recently that the single-step Hadamard SPI [50] and two-step Fourier SPI [51] can take $N^2$ measurements for full sampling. Since the grey scale properties of Fourier patterns limit their fast imaging with DMD, which is a critical drawback compared to our binary GB patterns and Hadamard basis patterns, here we will not talk more about them and make comparisons mainly between GB patterns and Hadamard basis patterns.
Figure 5 clearly shows that the waffle ordered GB patterns promises a better reconstructed image than the ones for random and natural orders in all undersampling cases. In undersampling cases, the GB patterns trend to maintain the details where the neighboring pixels’ values varied big and fast, while for the Hadamard basis patterns, they are more likely to recover the original details where the adjacent pixels’ value varied slowly. Compared with Hadamard basis cases (cake-cutting, Russian doll, random and natural orders), waffle ordered GB patterns demonstrate the superiority in imaging quality when they take the same measurement numbers. Figure 6 gives clear comparisons of RE, SSIM and PSNR between GB patterns and Hadamard basis patterns under different sampling ratios. It is unavoidable that for different images, the efficiency of patterns’ order is slightly different and even for the same image, the imaging quality is not always rising as the sampling ratio increases, but we can still estimate their overall performance. As we can see, the RE, SSIM and PSNR curves given by using the waffle ordered GB patterns act better than those of cake-cutting and Russian doll ordered Hadamard basis patterns mostly, indicating that the waffle ordered GB patterns is the most efficient among them.
We further investigate the noise-robustness abilities of GB SPI and Hadamard basis SPI by applying additive white Gaussian noise to the measured light intensities in simulation. The noise-robustness of single-step Hadamard SPI and the single-step Hadamard SPI after noise suppression have also been evaluated here, to fully demonstrate the features of GB patterns. The signal-to-noise ratios (SNRs) with different added noise levels are set as $50$ dB, $40$ dB, $30$ dB and $20$ dB and the images are recovered by TV [7] algorithm, as shown in Fig. 7. We find that the GB SPI is less noise-robust than Hadamard basis SPI in the full sampling of scenes, showing that there is a tradeoff between noise-robustness and imaging efficiency as well, though being more noise-robust than both single-step Hadamard SPI and single-step Hadamard SPI after noise suppression.
Fig. 7. In numerical simulations, the full sampling SPI performance of "cameraman" under additive white Gaussian noise using Gao-Boole patterns and Hadamard basis patterns. The reconstruction algorithm used for Gao-Boole and Hadamard basis SPI is TV algorithm. The reconstruction method and the noise suppression algorithm used for single-step Hadamard SPI are the same as that presented in [50]. The images are of $64\times 64$ pixels.
3.2 Experiments
We further test the performance of our proposed GB patterns in real SPI experiments. As shown in Fig. 8(a), in the experiments, we choose the approach of structural detection (i.e. standard SPI configuration) for the measurements, with DMD (F$4100$, Xintong, $1024\times 768$) as the pattern generator, $632$ nm laser (SuperK EHTREME, NKT Photonics) as the light source, $1951$ USAF resolution test target (R$3$L$3$S$1$N, Thorlabs) as the object, and differential ghost imaging (DGI) [52] and TV [7] as the reconstruction algorithms. $768\times 768$ pixels are used on the DMD totally and one pixel in the images corresponds to $12\times 12$ pixels on the DMD. The frame rate of DMD is set as $2000$ frames per second to realize the fast acquisition and simultaneously lower the switching noise and the possibility for frame loss. And the light intensities are collected by a single pixel detector (PDA$100$A$2$, Thorlabs).
Fig. 8. (a) Schematic of SPI experiment setup. (b) The full sampling results of $1951$ USAF resolution test target by using Gao-Boole patterns and Hadamard patterns with DGI and TV reconstruction algorithms. The reconstructed images are of $64\times 64$ pixels.
As shown in Fig. 8(b), the experimental results are consistent with the theoretical simulations. In full sampling cases, the number of measurements taken by using GB patterns is $64\times 64=4096$, half as many as the number of measurements ($2\times 64\times 64=8192$) taken by using Hadamard basis patterns. More importantly, the reconstruction results of GB patterns behaved as well as the Hadamard ones by using both DGI and TV algorithms. Further, we compare the reconstructed images under different sampling ratios, as shown in Fig. 9. After quantitatively analyzing the RE, SSIM and PSNR values of experimental results with different patterns, one can find that the waffle ordered GB SPI shows a better performance compared to the cake-cutting ordered Hadamard basis SPI, single-step Hadamard SPI as well as the single-step Hadamard SPI after noise suppression, which verifies that our GB patterns have superior imaging efficiency when applied in practical SPI systems. They are skilled at restoring the contour of the original objects in the scenes, like the shape of the numbers, the width of the bars and the spacing between two bars.
Fig. 9. Different sampling ratio of $1951$ USAF resolution test target by using Gao-Boole SPI and Hadamard basis SPI and single-step Hadamard SPI. The reconstruction algorithm used for Gao-Boole and Hadamard basis SPI is TV algorithm. The reconstruction method and the noise suppression algorithm used for single-step Hadamard SPI are the same as that presented in [50]. The reconstructed images are of $64\times 64$ pixels.
4. Conclusion
In conclusion, we propose and test our GB patterns together with their unique waffle order in both simulations and experiments. Their non-orthogonality breaks the usual recognition of recovering images with orthogonal patterns and brings a new insight into improving the performance for SPI. Their comparison to the Hadamard basis patterns demonstrates that the complete set of non-orthogonal GB patterns are effective and practical in both full sampling and undersampling cases of SPI. Being an advanced choice for structural patterns, they are valuable for dealing with the situations in which the imaging time is limited and make the applications of SPI techniques more feasible on account of their high imaging efficiency. Besides that, though the waffle order is efficient, there remains possibilities for occurring other better orders, similar to different developed orders for Hadamard basis patterns. The design of the GB patterns broadens our investigation area on finding other possible complete, non-orthogonal (or orthogonal) patterns to achieve a more efficient SPI.
Funding
Science and Technology Development Fund from Macau SAR (FDCT) (0062/2020/AMJ); Multi-Year Research Grant of University of Macau (MYRG2020-00082-IAPME).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. P. Sen, B. Chen, G. Garg, S. R. Marschner, M. Horowitz, M. Levoy, and H. P. A. Lensch, “Dual photography,” ACM Trans. Graph. 24(3), 745–755 (2005). [CrossRef]
2. M. P. Edgar, G. M. Gibson, and M. J. Padgett, “Principles and prospects for single-pixel imaging,” Nat. Photon. 13(1), 13–20 (2019). [CrossRef]
3. G. M. Gibson, S. D. Johnson, and M. J. Padgett, “Single-pixel imaging 12 years on: a review,” Opt. Express 28(19), 28190–28208 (2020). [CrossRef]
4. 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 Process. Mag. 25(2), 83–91 (2008). [CrossRef]
5. H. C. Liu, B. Yang, Q. Guo, J. Shi, C. Guan, G. Zheng, H. Mühlenbernd, G. Li, T. Zentgraf, and S. Zhang, “Single-pixel computational ghost imaging with helicity-dependent metasurface hologram,” Sci. Adv. 3(9), e1701477 (2017). [CrossRef]
6. 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]
7. L. Bian, J. Suo, Q. Dai, and F. Chen, “Experimental comparison of single-pixel imaging algorithms,” J. Opt. Soc. Am. A 35(1), 78–87 (2018). [CrossRef]
8. Z. Zhang, X. Wang, G. Zheng, and J. Zhong, “Fast Fourier single-pixel imaging via binary illumination,” Sci. Rep. 7(1), 12029 (2017). [CrossRef]
9. T. A. Lu, Z. Qiu, Z. Zhang, and J. Zhong, “Comprehensive comparison of single-pixel imaging methods,” Opt. Lasers Eng. 134, 106301 (2020). [CrossRef]
10. E. Hahamovich, S. Monin, Y. Hazan, and A. Rosenthal, “Single pixel imaging at megahertz switching rates via cyclic Hadamard masks,” Nat. Commun. 12(1), 4516 (2021). [CrossRef]
11. P. G. Vaz, D. Amaral, L. F. R. Ferreira, M. Morgado, and J. Cardoso, “Image quality of compressive single-pixel imaging using different Hadamard orderings,” Opt. Express 28(8), 11666–11681 (2020). [CrossRef]
12. L. Wang and S. Zhao, “Fast reconstructed and high-quality ghost imaging with fast Walsh-Hadamard transform,” Photonics Res. 4(6), 240–244 (2016). [CrossRef]
13. X. Yu, R. I. Stantchev, F. Yang, and E. Pickwell-MacPherson, “Super Sub-Nyquist Single-Pixel Imaging by Total Variation Ascending Ordering of the Hadamard Basis,” Sci. Rep. 10(1), 9338 (2020). [CrossRef]
14. X. Yu, F. Yang, B. Gao, J. Ran, and X. Huang, “Deep Compressive Single Pixel Imaging by Reordering Hadamard Basis: A Comparative Study,” IEEE Access 8, 55773–55784 (2020). [CrossRef]
15. M. Bashkansky, S. D. Park, and J. Reintjes, “Single pixel structured imaging through fog,” Appl. Opt. 60(16), 4793–4797 (2021). [CrossRef]
16. K. Soltanlou and H. Latifi, “Three-dimensional imaging through scattering media using a single pixel detector,” Appl. Opt. 58(28), 7716–7726 (2019). [CrossRef]
17. H. Wu, M. Zhao, F. Li, Z. Tian, and M. Zhao, “Underwater polarization-based single pixel imaging,” J. Soc. Inf. Disp. 28(2), 157–163 (2020). [CrossRef]
18. Q. Chen, S. K. Chamoli, P. Yin, X. Wang, and X. Xu, “Active Mode Single Pixel Imaging in the Highly Turbid Water Environment Using Compressive Sensing,” IEEE Access 7, 159390–159401 (2019). [CrossRef]
19. X. Yang, Y. Liu, X. Mou, T. Hu, F. Yuan, and E. Cheng, “Imaging in turbid water based on a Hadamard single-pixel imaging system,” Opt. Express 29(8), 12010–12023 (2021). [CrossRef]
20. A. X. Zhang, Y. H. He, L. A. Wu, L. M. Chen, and B. B. Wang, “Tabletop x-ray ghost imaging with ultra-low radiation,” Optica 5(4), 374–377 (2018). [CrossRef]
21. D. Wu, J. Luo, G. Huang, Y. Feng, X. Feng, R. Zhang, Y. Shen, and Z. Li, “Imaging biological tissue with high-throughput single-pixel compressive holography,” Nat. Commun. 12(1), 4712 (2021). [CrossRef]
22. 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]
23. D. Shrekenhamer, C. M. Watts, and W. J. Padilla, “Terahertz single pixel imaging with an optically controlled dynamic spatial light modulator,” Opt. Express 21(10), 12507–12518 (2013). [CrossRef]
24. R. I. Stantchev, X. Yu, T. Blu, and E. Pickwell-MacPherson, “Real-time terahertz imaging with a single-pixel detector,” Nat. Commun. 11(1), 2535 (2020). [CrossRef]
25. 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]
26. P. Gattinger, J. Kilgus, I. Zorin, G. Langer, R. Nikzad-Langerodi, C. Rankl, M. Gröschl, and M. Brandstetter, “Broadband near-infrared hyperspectral single pixel imaging for chemical characterization,” Opt. Express 27(9), 12666–12672 (2019). [CrossRef]
27. E. Candés and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Prob. 23(3), 969–985 (2007). [CrossRef]
28. M. Filipe, M. A. Francisco, C. Miguel, A. Mehrdad, and F. Faramarz, “High-resolution hyperspectral single-pixel imaging system based on compressive sensing,” Opt. Eng. 51(7), 071406 (2012). [CrossRef]
29. T. Mohammad, P. Il-Yong, A. B. Randy, J. D. Anthony, and J. T. Bruce, “Compressed single pixel imaging in the spatial frequency domain,” J. Biomed. Opt. 22(3), 030501 (2017). [CrossRef]
30. C. B. Li, “An efficient algorithm for total variation regularization with applications to the single pixel camera and compressive sensing,” Rice University (2010).
31. J. Greenberg, K. Krishnamurthy, and D. Brady, “Compressive single-pixel snapshot x-ray diffraction imaging,” Opt. Lett. 39(1), 111–114 (2014). [CrossRef]
32. A. Thompson, “Compressive single-pixel imaging,” Proc. 2nd IMA Conf. Math. Def., Defence Academy, Shrivenham, UK (2011).
33. Q. Chen, S. K. Chamoli, P. Yin, X. Wang, and X. Xu, “Imaging of hidden object using passive mode single pixel imaging with compressive sensing,” Laser Phys. Lett. 15(12), 126201 (2018). [CrossRef]
34. D. Shi, K. Yin, J. Huang, K. Yuan, W. Zhu, C. Xie, D. Liu, and Y. Wang, “Fast tracking of moving objects using single-pixel imaging,” Opt. Commun. 440, 155–162 (2019). [CrossRef]
35. C. Tao, H. Zhu, X. Wang, S. Zheng, Q. Xie, C. Wang, R. Wu, and Z. Zheng, “Compressive single-pixel hyperspectral imaging using RGB sensors,” Opt. Express 29(7), 11207–11220 (2021). [CrossRef]
36. Z. Zhu, H. Chi, T. Jin, S. Zheng, X. Jin, and X. Zhang, “Single-pixel imaging based on compressive sensing with spectral-domain optical mixing,” Opt. Commun. 402, 119–122 (2017). [CrossRef]
37. K. K. Guo, S. W. Jiang, and G. A. Zheng, “Multilayer fluorescence imaging on a single-pixel detector,” Biomed. Opt. Express 7(7), 2425–2431 (2016). [CrossRef]
38. M. R. Hestenes and S. Eduard, “Methods of Conjugate Gradients for Solving,” J. Res. Natl. Bur. Stand. 49(6), 409–436 (1952). [CrossRef]
39. W. K. Yu, “Super Sub-Nyquist Single-Pixel Imaging by Means of Cake-Cutting Hadamard Basis Sort,” Sensors 19(19), 4122 (2019). [CrossRef]
40. M. J. Sun, L. T. Meng, M. P. Edgar, M. J. Padgett, and N. Radwell, “A Russian Dolls ordering of the Hadamard basis for compressive single-pixel imaging,” Sci. Rep. 7(1), 3464 (2017). [CrossRef]
41. Z. Ye, J. Xiong, and H. C. Liu, “Ghost Difference Imaging Using One Single-Pixel Detector,” Phys. Rev. Appl. 15(3), 034035 (2021). [CrossRef]
42. Z. Zhang, X. Ma, and J. Zhong, “Single-pixel imaging by means of Fourier spectrum acquisition,” Nat. Commun. 6(1), 6225 (2015). [CrossRef]
43. J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78(6), 061802 (2008). [CrossRef]
44. 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]
45. J. J. Sylvester, “LX. Thoughts on inverse orthogonal matrices, simultaneous signsuccessions, and tessellated pavements in two or more colours, with applications to Newton’s rule, ornamental tile-work, and the theory of numbers,” Philosoph. Mag. 34(232), 461–475 (1867). [CrossRef]
46. J. L. Walsh, “A Closed Set of Normal Orthogonal Functions,” Am. J. Math. 45(1), 5–24 (1923). [CrossRef]
47. R. E. A. C. Paley, “On Orthogonal Matrices,” J. Math. Phys. 12(1-4), 311–320 (1933). [CrossRef]
48. W. K. Pratt, J. Kane, and H. C. Andrews, “Hadamard transform image coding,” Proc. IEEE 57(1), 58–68 (1969). [CrossRef]
49. S. Wang, A. Rehman, Z. Wang, S. Ma, and W. Gao, “Perceptual Video Coding Based on SSIM-Inspired Divisive Normalization,” IEEE Trans. Image Process. 22(4), 1418–1429 (2013). [CrossRef]
50. Y. Xiao, L. Zhou, and W. Chen, “Direct Single-Step Measurement of Hadamard Spectrum Using Single-Pixel Optical Detection,” IEEE Photonics Technol. Lett. 31(11), 845–848 (2019). [CrossRef]
51. H. Deng, X. Gao, M. Ma, P. Yao, Q. Guan, X. Zhong, and J. Zhang, “Fourier single-pixel imaging using fewer illumination patterns,” Appl. Phys. Lett. 114(22), 221906 (2019). [CrossRef]
52. F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential Ghost Imaging,” Phys. Rev. Lett. 104(25), 253603 (2010). [CrossRef]
