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Secured single-pixel broadcast imaging

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Abstract

Active single-pixel imaging (also known as illumination-modulated single-pixel imaging) employs a spatial light modulator to illuminate a scene with structured patterns. The scheme of active single-pixel imaging is similar to a wireless broadcast system, allowing that multiple receivers use a single-pixel detector to capture an image simultaneously from a different place. The use of basis patterns allows for high-quality reconstructions and an efficient sampling process, but the public knowledge of the basis patterns is not a favorable feature for security applications. In order to develop a secured broadcast single-pixel imaging system, we propose to employ block-permutated Hadamard basis patterns for illumination. The randomness in permutation operations introduces strong security characteristics for the system. Both simulation and experimental results demonstrate our proposed scheme has satisfactory imaging quality and efficiency. This work generates a new insight for the application of single-pixel imaging and provides a solution for developing a secured imaging system for non-visible wavebands.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Single-pixel imaging is a novel imaging scheme that allows one to capture a scene without using a conventional two-dimensional (2-D) imaging sensor. Single-pixel imaging has some unique features [1]. It allows for the use of a low-cost single-pixel detector instead of an expensive pixelated detector, which has demonstrated essential for imaging over invisible wavebands [2]. It is easier and less expensive to manufacture single-pixel detectors with a large active area, which makes single-pixel imaging techniques more suitable for imaging under weak-light conditions than conventional imaging techniques. With these advantages, single-pixel imaging has been recently extensively applied in different fields such as terahertz imaging [3], remote sensing [4], three-dimensional (3-D) imaging [5–7], microscopy [8], scattering imaging [9], hyperspectral imaging [10], X-ray imaging [11], shadowless imaging [12], and multi-modality imaging [13]. In addition, the research of encrypted single-pixel imaging has attracted much attention in recent years [14–21].

To acquire spatial information using a single-pixel detector, single-pixel imaging modulates either illumination or detection light signals to encode spatial information into a one-dimensional (1-D) light signals. The resulting 1-D signals can be collected by a detector without spatial resolution (i.e. single-pixel detector or bucket detector). Approaches based on illumination modulation are termed active single-pixel imaging. We find some similarities between an active single-pixel imaging system and a wireless broadcast system. The resulting light signals emitted by the objects under modulated illumination in an active single-pixel imaging system are similar to the amplitude modulated signals emitted from a base station in a broadcast system. The utilized single-pixel detector in an active single-pixel imaging system is like a receiving antenna in a broadcast system. Thus, active single-pixel imaging has a property that as long as the detector receives the light signals one is able to recover the object image. Consequently, single-pixel detectors need not direct line-of-sight to the object. This property enables single-pixel broadcast imaging [22]. Objects under modulated illumination emit light signals. As such, multiple receivers located at different places can capture the same image simultaneously. However, this property would lead to risks of information leakage, because there is no way to ban any unauthorized receivers to collect the broadcasted light signals. Thus, developing a secured single-pixel broadcast imaging system is meaningful.

When the object to be imaged is confidential, the object image needs protection. It will be favorable that the detected light signals and the reconstructed object image are in an encrypted format. It is also favorable that optical imaging and information encryption are performed simultaneously with the same system. In previous works, optical encryption approaches [23] have been extensively investigated, such as 4-f lens system [24], holography [25], optical interference [26], photo counting technology [27], joint transform correlator [28], and ptychography [29].

An encrypted single-pixel imaging system can be formulated as a symmetrical optical encryption model–the object image is considered as the plaintext, the illumination patterns as the encryption (decryption) key, and the single-pixel measurements as the ciphertext. The object image (plaintext) can be recovered from the single-pixel measurements (ciphertext) only when the illumination (key) is known.

In previous works [14–21], random illumination patterns have been used in encrypted single-pixel imaging systems, because the randomness of the illumination is a suitable key for image information protection. However, from the perspective of imaging quality and sampling efficiency, random patterns are not an optimal choice in single-pixel imaging. Although one can use methods such as compressive sensing [30], Gerchberg-Saxton iteration [31] and error correction coding [18,32,33] to improve the imaging quality, the results are not satisfactory enough, which limits practicability of the technique. Recently, single-pixel imaging techniques using basis patterns [34,35] have been demonstrated able to reproduce high-quality reconstructions, but the public knowledge of the basis patterns is not a favorable feature for security applications. Thus, encrypted basis-scan single-pixel imaging has not been reported yet.

In this paper, we propose a novel encrypted basis-scan single-pixel imaging technique which is achieved by pixel-level permutation for every single basis pattern. We experimentally demonstrate that the proposed technique not only provides satisfactory security level but also preserves the merit of high reconstruction quality and sampling efficiency. The proposed technique enables a secured single-pixel broadcast system. This work generates a new insight for the application of single-pixel imaging. It also provides a solution for developing a secured imaging system for non-visible wavebands.

2. Principle

2.1 Single-pixel imaging with random patterns and basis patterns

As Fig. 1 shows, the proposed secured broadcast imaging scheme is based on an active single-pixel imaging system. At the illumination end (a sender), a spatial light modulator (the digital projector in the figure) is employed to generate structured illuminations onto an object scene. The 2-D spatial information of the object scene is encoded into a 1-D intensity sequence. As such, the detection end (a receiver) is able to capture a scene by collecting scattering light with a single-pixel detector. By using encrypted patterns for illumination, the 2-D spatial information stored in the 1-D intensity sequence is protected by a key. The object image (plaintext) can be recovered from the single-pixel measurements (ciphertext) only when the key is known. Otherwise, the reconstruction is still in a ciphertext form.

 figure: Fig. 1

Fig. 1 Proposed secured single-pixel broadcast imaging system.

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In a general single-pixel imaging model, a sequence of N 2-D intensity patterns Pt(x,y) (t=1,2,3,,N) is employed to illuminate the object image I(x,y). A 1-D single-pixel measurements i(t) is recorded by a single-pixel detector. The mathematical relationship between I(x,y), Pt(x,y) and i(t) can be described by Eq. (1),

i(t)=I(x,y)Pt(x,y)dxdy.
The illumination patterns Pt(x,y) can be in various forms, such as random intensity patterns for ghost imaging [14] or basis patterns for basis-scan single-pixel imaging [34,35].

For random patterns, the image can be approximately reconstructed by correlating the illumination patterns Pt(x,y) with the single-pixel measurements i(t), described by Eq. (2).

I(x,y)i(t)Pt(x,y)i(t)Pt(x,y),
where 〈∙〉 denotes the ensemble average of N illuminations. The reconstructed image quality gradually improves as the number of illumination N increases. To our best knowledge, random patterns are employed in all previous works [14–21] about encrypted single-pixel imaging. The use of random patterns leads to a disadvantage of noise contamination and low sampling efficiency.

2.2 Proposed single-pixel imaging with encrypted basis patterns

In this paper, basis-scan single-pixel imaging scheme is adopted. Basis-scan single-pixel imaging uses corresponding basis patterns for illumination. For example, Hadamard single-pixel imaging [34] uses Hadamard basis patterns while Fourier single-pixel imaging [35] uses Fourier basis patterns (also known as sinusoidal patterns). Due to the orthogonality of basis patterns, basis-scan single-pixel imaging allows for perfect reconstruction in fully-sampled condition. By utilizing the sparsity of natural scene images, basis-scan single-pixel imaging is able to reconstruct an image with sharp features even under under-sampling condition (that is, measurements less than total number of image pixels). Thus, basis-scan single-pixel imaging exhibits advantages of high imaging sampling speed efficiency and high imaging quality. However, the basis patterns are public knowledge and constant for different object images.

To encrypt object images, we can exploit the relativity between the illumination patterns and the object image so as to achieve object image encryption equivalently. Hadamard basis patterns are chosen to be encrypted, as they are binary and, therefore, suitable for high-speed generation by using a digital micro-mirror device (DMD). It should be noted that DMD-based fast Fourier single-pixel imaging techniques have also been reported recently. Z. Zhang et al. proposed to binary originally grayscale Fourier basis patterns by using a spatial dithering strategy [36]. Y. Zhang et al. proposed to binary Fourier basis patterns by using an adaptive spatial filter [37]. J. Yang et al. proposed to generate grayscale Fourier basis patterns by using computer-generated binary holograms [38]. Thus, the proposed scheme is applicable for Fourier single-pixel imaging. For basis patterns encryption, we propose to use permutation. Permutation (or scrambling) refers to the rearrangement of elements in a different order for a given sequence. An example of permutation operation performed on a 2-D intensity pattern is illustrated in Fig. 2. The image to be encrypted is in a size of M×N pixels, referring to the plaintext. The image can be represented by an M×N-element matrix. First, the image is reshaped to an (M×N)×1-element array. Given a key which indicates the exact pixel position mapping relationship, the plaintext array is then permutated accordingly which results in a ciphertext array at the same length. Finally, the ciphertext array is reshaped back to an M×N-element matrix and the encrypted image is derived. With the correct key, the original image can be recovered (decrypted) from the permutated result. Otherwise, the decryption is still in a ciphertext form revealing no plaintext information. In addition, the permutated Hadamard patterns are still binary so that it permits high-speed single-pixel imaging. Although permutation has been previously adopted in many optical encryption and optical information hiding works [39–46], it is the first time that permutation is used in basis-scan single-pixel imaging.

 figure: Fig. 2

Fig. 2 An example of permutation operation on a 2-D intensity pattern.

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Hadamard single-pixel imaging is based on Hadamard transform. A Hadamard transform can decompose an arbitrary grayscale image into a superposition of Hadamard binary basis pattern Pt(x,y) multiplied with Hadamard coefficients it(u,v) (1tNm). In single-pixel imaging, if the Hadamard basis patterns are sequentially employed as illumination patterns, the single-pixel measurements are equivalent Hadamard coefficients of object image I(x,y), illustrated in Eq. (3). Thus, the object image can be computationally reconstructed from the single-pixel measurements through an inverse 2-D Hadamard transform, illustrated in Eq. (4).

it(u,v)=H{I(x,y)}=I(x,y)Pt(x,y)dxdy,
I(x,y)=H1{it(u,v)},
where (x,y) is the coordinate in the spatial domain, (u,v) is the coordinate in the Hadamard domain, H{} denotes a 2-D forward Hadamard transform, and H1{} denotes an inverse 2-D Hadamard transform. To perform encryption, we can apply a permutation operation EK to the basis pattern as Eq. (5),
PE(x,y)=EK{Pt(x,y)},
where PE(x,y) is the encrypted basis pattern after applying permutation to Pt(x,y) with the key K. As Eq. (6) shows, the corresponding single-pixel measurement is iK(u,v) when the encrypted pattern PE(x,y) is used for illumination.
iK(u,v)=I(x,y)EK{Pt(x,y)}dxdy=EK1{I(x,y)}Pt(x,y)dxdy,
where EK1 refers to another permutation operation that is in an inverse order as EK. In mathematical terms, when two matrices are multiplied, the permutation of one matrix is equivalent to the permutation of another matrix with inverse order, yielding the same multiplication product. Consequently, we exploit the relativity between the illumination patterns and the object image and encrypt the object image by applying encryption on the illumination patterns. This implies that the Hadamard spectrum of a permutated object image (rather than original object image) is acquired if permutated Hadamard basis patterns are employed in encrypted single-pixel imaging. The final decryption and reconstruction from recorded spectrum can be implemented by an inverse Hadamard transform plus another permutation operation, given by Eq. (7).

I(x,y)=EK{H1{iK(u,v)}}.

For unauthorized users without the key K, the original object image cannot be recovered from iK(u,v) and the security protection of imaging result can be achieved.

In practice, the permutation operation is not necessarily performed on pixel level but also on a ‘super pixel’ level. ‘Super pixel’ refers to pixel binning, that is, an image block consists of Mb × Mb pixels. As Fig. 3 shows, when Mb increases, the security strength of permutation encryption is weakened since the key space is reduced but the imaging quality and speed can be enhanced. This can be explained in the following. The basis-scan single-pixel imaging requires much smaller number of illuminations than conventional single-pixel imaging with random illumination patterns for two reasons. First, the basis patterns are orthogonal for a transform while random intensity patterns are not. Second, smooth images of a natural scene are characterized by that the energy concentrates at the low frequency band in the transformation domain. Thus, the acquisition of only partial spectrum components is sufficient to reconstruct object image with acceptable quality. When the object image is permutated on a pixel basis and becomes a random intensity image, the orthogonal basis criterion is still satisfied but the low frequency concentration criterion is no longer satisfied.

 figure: Fig. 3

Fig. 3 (a) One original Hadamard basis pattern and corresponding permutated Hadamard basis patterns for different Mb: (b) Mb=1; (c) Mb=2; (d) Mb=4; (e) Mb=8.

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The energy is evenly distributed in the entire transformation domain. By increasing the block size Mb, the permutated object image tends to be smoother and the low frequency concentration criterion can be satisfied again to some extent, which is favorable for efficient data acquisition. In terms of imaging quality and sampling efficiency, the proposed technique has an advantage over random pattern single-pixel imaging, but has certain degradation compared with conventional basis-scan single-pixel imaging without encryption. The value of Mb can be appropriately designed to balance the security strength and imaging quality.

3. Results and discussion

The proposed technique is verified by both computer simulation and optical experiments in this section.

3.1 Simulation results

A test image ‘cameraman’ with 256 × 256 pixels is encrypted and decrypted for different values of Mb and sampling ratios. The Hadamard basis patterns for illumination are permutated with three permutation block sizes Mb=1, Mb=4, and Mb=8. In Fig. 4, we show the reconstructed images decrypted by using the correct key (upper row) and an incorrect key (bottom row). As the results show, the object is not recognizable from the images decrypted with an incorrect key, which demonstrates the effectiveness of the proposed technique in terms of security.

 figure: Fig. 4

Fig. 4 Decrypted image of our proposed encrypted Hadamard single-pixel imaging scheme in numerical simulation (sampling ratio = 100%) with correct key when (a) Mb=1; (b) Mb=4; (c) Mb=8; with incorrect key when (d) Mb=1; (e) Mb=4; (f) Mb=8; (g) Reference image 1 for security evaluation; (h) Reference image 2 for security evaluation.

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For further demonstration, we quantitatively evaluate the security of the proposed technique by employing two commonly used methods, auto-correlation coefficients of neighboring pixels and plaintext-ciphertext cross-correlation coefficients [47].

For a plaintext image of a natural scene, the intensity values of neighboring pixels are highly correlated. Instead, the correlation is low for an ideal encrypted ciphertext. Thus, calculating the auto-correlation coefficients of neighboring pixels is an effective approach for quantitative evaluation of the degree of security. Table 1 shows the auto- correlation coefficients for the simulation reconstructions (including the plaintext and the reconstructions without decryption). To calculate the auto-correlation coefficients, we shift the images horizontally with 0, 1, 4, and 8 pixels, respectively.

Tables Icon

Table 1. Auto-correlation of neighboring pixels in plaintext image and ciphertext image.

The results show that the auto-correlation coefficients for ciphertext images are lower than those for the plaintext especially when Mb=1 or when the pixel shift is equal or larger than block size, which is favorable for information protection.

Additionally, the results of security degree evaluated by the cross correlations between plaintext image and ciphertext are shown in Table 2. It can be seen that the cross correlation values are rather low, which indicates a good performance in terms of security. For comparison, we also calculate the cross correlations between the ciphertexts and two other reference images [Figs. 4(g) and 4(h)]. As the results show, the cross correlations tend to zero. There is no significant difference between plaintext-ciphertext cross correlations and reference-ciphertext cross correlations. Some of the plaintext-ciphertext cross correlations [for example, Figs. 4(d) and 4(f)] are even lower than the reference-ciphertext counterparts. It indicates that any attacker has difficulties in identifying which plaintext image the ciphertext image corresponds to.

Tables Icon

Table 2. Cross correlation coefficients between plaintext image and ciphertext image.

According to the results given by both security level evaluation methods, we conclude that the proposed technique is able to provide a considerable security level.

In the second simulation, we verify the proposed technique in terms of sampling efficiency. As are sparse in Hadamard domain, images of a natural scene can be recovered for under-sampling (the number of illuminations N is smaller than the total number of patterns Nm). In practice, one can acquire the Hadamard transform of an image with a ‘zig-zag’ order from the origin [48]. In Fig. 5, we show the initial reconstruction and decryption pairs for different permutation block sizes and sampling ratios. Additionally, we take ghost imaging in comparison. To quantitatively evaluate the quality of reconstructions, structural similarity index (SSIM) [49] is employed.

 figure: Fig. 5

Fig. 5 Simulation results. The first panel shows the recovered encrypted image and final decrypted image by the proposed technique with Key 1 for Mb=1 and different sampling ratios. The second panel shows the recovered encrypted image and final decrypted image by the proposed technique with Key 2 for Mb=2 and different sampling ratios. The third panel shows the recovered encrypted image and final decrypted image by the proposed technique with Key 3 for Mb=4 and different sampling ratios. The fourth panel shows the reconstructions by original Hadamard single-pixel imaging without encryption for different sampling ratios. The fifth panel shows the reconstructions by conventional encrypted single-pixel imaging (or ghost imaging) with random illumination patterns for different sampling ratios. The red number at the right-bottom corner of each image is the corresponding SSIM value.

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As the figure shows, the original object image is not recognizable from the initial reconstructions. Only after permutation decryption with the correct key, the plaintext image appears. The figure also shows that the reconstructed image quality improves as the number of illumination patterns (sampling ratio) increases in all schemes (including encrypted Hadamard single-pixel imaging, conventional Hadamard single-pixel imaging without encryption, and encrypted single-pixel imaging with random patterns). However, the reconstructed image quality improves very slowly by increasing illumination numbers for single-pixel imaging with random patterns. It is evident that the proposed technique can achieve significantly better reconstructed image quality than encrypted single-pixel imaging with random patterns under the same number of illuminations. In other words, much less illuminations are required in the proposed technique to achieve the same imaging quality, which indicates a significantly faster imaging speed. As the results shown in Fig. 5, the imaging quality of the proposed technique is degraded compared with conventional Hadamard single-pixel imaging. However, as Mb increases, the quality in the former scheme becomes closer to the latter. When the sampling ratio N=Nm, in both encrypted and non-encrypted Hadamard single-pixel imaging, the reconstructed image quality can be identical to the original image.

Relatively, the security strength with a smaller block size is higher than the one with larger block size. For Mb=8 the quality of decryptions is close to that of reconstructions without encryption, which demonstrates that a larger Mb substantially results in better efficiency.

The computational time of the proposed technique in image reconstruction and decryption is evaluated. The results are shown in Table 3. Please note that the results are the averaged computational time for 10,000 loops. We implement image reconstruction and decryption using MATLAB R2014a on the computer equipped with an Intel Xeon E3-1231 v3 @3.4GHz CPU, 8GB memory, and Windows 7 Ultimate x64 operating system. For the proposed technique, image reconstruction refers to an inverse 2-D Hadamard transform and decryption refers to a permutation process [Eq. (7)]. As the results show, the computational time for both inverse Hadamard transform and permutation decryption is at a millisecond (ms) level, which can be negligible. Thus, we conclude that the proposed technique is computationally efficient.

Tables Icon

Table 3. Calculation time of object image reconstruction and decryption in proposed scheme.

3.2 Experimental results

We perform an experiment to demonstrate the proposed secured single-pixel broadcast imaging system. The schematic diagram of our experimental set-up is shown in Fig. 6. We use two photodiodes to simulate an unauthorized receiver (Photodiode 1, PD1) who has an incorrect key and an authorized receiver (Photodiode 2, PD2) who has the correct key for decryption. Both photodiodes are hidden from the object scene behind a piece of ground glass. A pair of toys is used as a complex 3-D scene. We use a DLP development kit (Texas Instruments DLP 4100) for spatial light illumination. The DLP development kit is equipped with a 0.7-inch DMD which has 1024 × 768 micro mirrors and the mirror size is 13.6 × 13.6 μm2. We generate three sets of patterns–original Hadamard patterns, encrypted Hadamard patterns with Key 1 and Mb=1, 2, 4, and 8, and encrypted Hadamard patterns with Key 2 and Mb=1, 2, 4, and 8. A 3-watt white LED is used for the illumination source. In order to eliminate noise caused by background illumination, we employ differential Hadamard single-pixel imaging, that is, a pattern followed by its inversed pattern is projected. The resolution of illumination patterns is 256 × 256 pixels. Thus, the total number of measurements of fully-sample reconstruction is 131,072 ( = 256 × 256 × 2). The generated patterns are displayed on the DMD and projected onto the scene through a lens system. Please note that the lens system is from a commercial digital projector (Toshiba T-95). A photodiode (HAMAMATSU S1227-1010BR) is used to collect the intensity of resulting light fields from the scene. The electronic signals are transferred to the computer via a data acquisition board [National Instruments USB-6343 (BNC)].

 figure: Fig. 6

Fig. 6 Experimental set-up of our proposed secured single-pixel broadcast imaging system.

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As the results shown in Fig. 7, the object scene is not recognizable from the recovered encrypted images for different sampling ratios and values of Mb, which demonstrates our proposed technique can hide information well under different situations. The results for these images further decrypted with the correct key and an incorrect key are shown in Fig. 7. The results verify the effectiveness of the proposed technique. Figure 7 also presents a comparison among different sampling ratios. As the sampling ratio increases, the decrypted images become clearer and less noisy. When the sampling ratio reaches 100%, the results for encrypted systems with different Mb values and unencrypted systems are identical. The experimental results are consistent with the simulation results.

 figure: Fig. 7

Fig. 7 Experiments results: the top four panels shows the final decrypted images by the unauthorized receiver (PD1) and the authorized receiver (PD2) for different sampling ratios and different values of Mb; the bottom-most panel shows reconstructed results from original Hadamard single-pixel imaging without encryption.

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4. Conclusion

We propose an encrypted Hadamard single-pixel imaging system to achieve secured broadcast imaging in this paper. We employ permutated Hadamard basis patterns, instead of random intensity illumination patterns, for securing object image information. The randomness in permutation operations introduces strong security characteristics for the system. The trade-off between security strength and reconstructed image quality for varying permutation block sizes is discussed. Simulation and experimental results demonstrate our proposed technique has significantly better imaging quality and higher sampling efficiency than conventional encrypted single-pixel imaging systems using random intensity illumination patterns. This work generates a new insight for the application of single-pixel imaging and provides a solution for developing a secured imaging system for non-visible wavebands.

Funding

National Natural Science Foundation of China (NSFC) (61475064); Chinese Postdoctoral Science Foundation (2017M622763); Pearl River Talent Plan (Postdoctoral Scheme, 2016), Guangdong Province.

Acknowledgments

The authors thank Dr. Bowen Jiang for her precious time and invaluable comments on this work, Xiao Ma for drawing Fig. 1, Dr. Shiping Li for her help with the experimental equipment preparation, and Qinqiu Fang for linguistic assistance.

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31. W. Wang, X. Hu, J. Liu, S. Zhang, J. Suo, and G. Situ, “Gerchberg-Saxton-like ghost imaging,” Opt. Express 23(22), 28416–28422 (2015). [CrossRef]   [PubMed]  

32. S. Jiao, W. Zou, and X. Li, “Noise Removal for Optical Holographic Encryption from Telecommunication Engineering Perspective,” in Digital Holography and Three-Dimensional Imaging, 2017 OSA Technical Digest Series (Optical Society of America, 2017), paper Th2A–5.

33. S. Jiao, Z. Jin, C. Zhou, W. Zou, and X. Li, “Is QR code an optimal data container in optical encryption systems from an error-correction coding perspective?” J. Opt. Soc. Am. A 35(1), A23–A29 (2018). [CrossRef]   [PubMed]  

34. N. Radwell, K. J. Mitchell, G. M. Gibson, M. P. Edgar, R. Bowman, and M. J. Padgett, “Single-pixel infrared and visible microscope,” Optica 1(5), 285–289 (2014). [CrossRef]  

35. Z. Zhang, X. Ma, and J. Zhong, “Single-pixel imaging by means of Fourier spectrum acquisition,” Nat. Commun. 6(1), 6225 (2015). [CrossRef]   [PubMed]  

36. Z. Zhang, X. Wang, G. Zheng, and J. Zhong, “Fast Fourier single-pixel imaging via binary illumination,” Sci. Rep. 7(1), 12029 (2017). [CrossRef]   [PubMed]  

37. Y. Zhang, J. Suo, Y. Wang, and Q. Dai, “Doubling the pixel count limitation of single-pixel imaging via sinusoidal amplitude modulation,” Opt. Express 26(6), 6929–6942 (2018). [CrossRef]   [PubMed]  

38. J. Yang, L. Gong, X. Xu, P. Hai, Y. Shen, Y. Suzuki, and L. V. Wang, “Motionless volumetric photoacoustic microscopy with spatially invariant resolution,” Nat. Commun. 8(1), 780 (2017). [CrossRef]   [PubMed]  

39. B. Hennelly and J. T. Sheridan, “Optical image encryption by random shifting in fractional Fourier domains,” Opt. Lett. 28(4), 269–271 (2003). [CrossRef]   [PubMed]  

40. M. He, Q. Tan, L. Cao, Q. He, and G. Jin, “Security enhanced optical encryption system by random phase key and permutation key,” Opt. Express 17(25), 22462–22473 (2009). [CrossRef]   [PubMed]  

41. S. Liu and J. T. Sheridan, “Optical encryption by combining image scrambling techniques in fractional Fourier domains,” Opt. Commun. 287, 73–80 (2013). [CrossRef]  

42. X. F. Meng, L. Z. Cai, X. L. Yang, X. X. Shen, and G. Y. Dong, “Information security system by iterative multiple-phase retrieval and pixel random permutation,” Appl. Opt. 45(14), 3289–3297 (2006). [CrossRef]   [PubMed]  

43. P. W. M. Tsang, T.-C. Poon, and W. K. Cheung, “Intensity image-embedded binary holograms,” Appl. Opt. 52(1), A26–A32 (2013). [CrossRef]   [PubMed]  

44. P. W. M. Tsang, T.-C. Poon, and A. S. M. Jiao, “Embedding intensity image in grid-cross down-sampling (GCD) binary holograms based on block truncation coding,” Opt. Commun. 304, 62–70 (2013). [CrossRef]  

45. S. Jiao and P. Tsang, “A hologram watermarking scheme based on scrambling embedding and image inpainting,” in Digital Holography and Three-Dimensional Imaging, 2015 OSA Technical Digest Series (Optical Society of America, 2015), paper DT2A–4.

46. Z. Zhuang, S. Jiao, W. Zou, and X. Li, “Embedding intensity image into a binary hologram with strong noise resistant capability,” Opt. Commun. 403, 245–251 (2017). [CrossRef]  

47. N. K. Pareek, V. Patidar, and K. K. Sud, “Image encryption using chaotic logistic map,” Image Vis. Comput. 24(9), 926–934 (2006). [CrossRef]  

48. 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]   [PubMed]  

49. Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: From error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004). [CrossRef]   [PubMed]  

References

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  6. Z. Zhang and J. Zhong, “Three-dimensional single-pixel imaging with far fewer measurements than effective image pixels,” Opt. Lett. 41(11), 2497–2500 (2016).
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  27. E. Pérez-Cabré, H. C. Abril, M. S. Millán, and B. Javidi, “Photon-counting double-random-phase encoding for secure image verification and retrieval,” J. Opt. 14(9), 094001 (2012).
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  28. T. Nomura and B. Javidi, “Optical encryption using a joint transform correlator architecture,” Opt. Eng. 39(8), 2031–2036 (2000).
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  29. Y. Shi, T. Li, Y. Wang, Q. Gao, S. Zhang, and H. Li, “Optical image encryption via ptychography,” Opt. Lett. 38(9), 1425–1427 (2013).
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  30. O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95(13), 131110 (2009).
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  31. W. Wang, X. Hu, J. Liu, S. Zhang, J. Suo, and G. Situ, “Gerchberg-Saxton-like ghost imaging,” Opt. Express 23(22), 28416–28422 (2015).
    [Crossref] [PubMed]
  32. S. Jiao, W. Zou, and X. Li, “Noise Removal for Optical Holographic Encryption from Telecommunication Engineering Perspective,” in Digital Holography and Three-Dimensional Imaging, 2017 OSA Technical Digest Series (Optical Society of America, 2017), paper Th2A–5.
  33. S. Jiao, Z. Jin, C. Zhou, W. Zou, and X. Li, “Is QR code an optimal data container in optical encryption systems from an error-correction coding perspective?” J. Opt. Soc. Am. A 35(1), A23–A29 (2018).
    [Crossref] [PubMed]
  34. N. Radwell, K. J. Mitchell, G. M. Gibson, M. P. Edgar, R. Bowman, and M. J. Padgett, “Single-pixel infrared and visible microscope,” Optica 1(5), 285–289 (2014).
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  35. Z. Zhang, X. Ma, and J. Zhong, “Single-pixel imaging by means of Fourier spectrum acquisition,” Nat. Commun. 6(1), 6225 (2015).
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  36. Z. Zhang, X. Wang, G. Zheng, and J. Zhong, “Fast Fourier single-pixel imaging via binary illumination,” Sci. Rep. 7(1), 12029 (2017).
    [Crossref] [PubMed]
  37. Y. Zhang, J. Suo, Y. Wang, and Q. Dai, “Doubling the pixel count limitation of single-pixel imaging via sinusoidal amplitude modulation,” Opt. Express 26(6), 6929–6942 (2018).
    [Crossref] [PubMed]
  38. J. Yang, L. Gong, X. Xu, P. Hai, Y. Shen, Y. Suzuki, and L. V. Wang, “Motionless volumetric photoacoustic microscopy with spatially invariant resolution,” Nat. Commun. 8(1), 780 (2017).
    [Crossref] [PubMed]
  39. B. Hennelly and J. T. Sheridan, “Optical image encryption by random shifting in fractional Fourier domains,” Opt. Lett. 28(4), 269–271 (2003).
    [Crossref] [PubMed]
  40. M. He, Q. Tan, L. Cao, Q. He, and G. Jin, “Security enhanced optical encryption system by random phase key and permutation key,” Opt. Express 17(25), 22462–22473 (2009).
    [Crossref] [PubMed]
  41. S. Liu and J. T. Sheridan, “Optical encryption by combining image scrambling techniques in fractional Fourier domains,” Opt. Commun. 287, 73–80 (2013).
    [Crossref]
  42. X. F. Meng, L. Z. Cai, X. L. Yang, X. X. Shen, and G. Y. Dong, “Information security system by iterative multiple-phase retrieval and pixel random permutation,” Appl. Opt. 45(14), 3289–3297 (2006).
    [Crossref] [PubMed]
  43. P. W. M. Tsang, T.-C. Poon, and W. K. Cheung, “Intensity image-embedded binary holograms,” Appl. Opt. 52(1), A26–A32 (2013).
    [Crossref] [PubMed]
  44. P. W. M. Tsang, T.-C. Poon, and A. S. M. Jiao, “Embedding intensity image in grid-cross down-sampling (GCD) binary holograms based on block truncation coding,” Opt. Commun. 304, 62–70 (2013).
    [Crossref]
  45. S. Jiao and P. Tsang, “A hologram watermarking scheme based on scrambling embedding and image inpainting,” in Digital Holography and Three-Dimensional Imaging, 2015 OSA Technical Digest Series (Optical Society of America, 2015), paper DT2A–4.
  46. Z. Zhuang, S. Jiao, W. Zou, and X. Li, “Embedding intensity image into a binary hologram with strong noise resistant capability,” Opt. Commun. 403, 245–251 (2017).
    [Crossref]
  47. N. K. Pareek, V. Patidar, and K. K. Sud, “Image encryption using chaotic logistic map,” Image Vis. Comput. 24(9), 926–934 (2006).
    [Crossref]
  48. 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] [PubMed]
  49. Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: From error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
    [Crossref] [PubMed]

2018 (3)

2017 (6)

J. Yang, L. Gong, X. Xu, P. Hai, Y. Shen, Y. Suzuki, and L. V. Wang, “Motionless volumetric photoacoustic microscopy with spatially invariant resolution,” Nat. Commun. 8(1), 780 (2017).
[Crossref] [PubMed]

Z. Zhang, X. Wang, G. Zheng, and J. Zhong, “Fast Fourier single-pixel imaging via binary illumination,” Sci. Rep. 7(1), 12029 (2017).
[Crossref] [PubMed]

Z. Zhuang, S. Jiao, W. Zou, and X. Li, “Embedding intensity image into a binary hologram with strong noise resistant capability,” Opt. Commun. 403, 245–251 (2017).
[Crossref]

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] [PubMed]

Y. Qin and Y. Zhang, “Information encryption in ghost imaging with customized data container and XOR operation,” IEEE Photonics J. 9(2), 1–8 (2017).
[Crossref]

S. Li, Z. Zhang, X. Ma, and J. Zhong, “Shadow-free single-pixel imaging,” Opt. Commun. 403, 257–261 (2017).
[Crossref]

2016 (7)

L. Bian, J. Suo, G. Situ, Z. Li, J. Fan, F. Chen, and Q. Dai, “Multispectral imaging using a single bucket detector,” Sci. Rep. 6(1), 24752 (2016).
[Crossref] [PubMed]

D. Pelliccia, A. Rack, M. Scheel, V. Cantelli, and D. M. Paganin, “Experimental x-ray ghost imaging,” Phys. Rev. Lett. 117(11), 113902 (2016).
[Crossref] [PubMed]

Z. Zhang and J. Zhong, “Three-dimensional single-pixel imaging with far fewer measurements than effective image pixels,” Opt. Lett. 41(11), 2497–2500 (2016).
[Crossref] [PubMed]

M. J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7, 12010 (2016).
[Crossref] [PubMed]

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Pérez-Cabré, M. S. Millán, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap on optical security,” J. Opt. 18(8), 083001 (2016).
[Crossref]

J. Wu, Z. Xie, Z. Liu, W. Liu, Y. Zhang, and S. Liu, “Multiple-image encryption based on computational ghost imaging,” Opt. Commun. 359, 38–43 (2016).
[Crossref]

X. Li, X. Meng, X. Yang, Y. Yin, Y. Wang, X. Peng, W. He, G. Dong, and H. Chen, “Multiple-image encryption based on compressive ghost imaging and coordinate sampling,” IEEE Photonics J. 8(4), 1–11 (2016).
[Crossref]

2015 (5)

W. Chen and X. Chen, “Ghost imaging using labyrinth-like phase modulation patterns for high-efficiency and high-security optical encryption,” Europhys. Lett. 109(1), 14001 (2015).
[Crossref]

S. Zhao, L. Wang, W. Liang, W. Cheng, and L. Gong, “High performance optical encryption based on computational ghost imaging with QR code and compressive sensing technique,” Opt. Commun. 353, 90–95 (2015).
[Crossref]

W. Wang, X. Hu, J. Liu, S. Zhang, J. Suo, and G. Situ, “Gerchberg-Saxton-like ghost imaging,” Opt. Express 23(22), 28416–28422 (2015).
[Crossref] [PubMed]

Z. Zhang, X. Ma, and J. Zhong, “Single-pixel imaging by means of Fourier spectrum acquisition,” Nat. Commun. 6(1), 6225 (2015).
[Crossref] [PubMed]

R. S. Aspden, N. R. Gemmell, P. A. Morris, D. S. Tasca, L. Mertens, M. G. Tanner, R. A. Kirkwood, A. Ruggeri, A. Tosi, R. W. Boyd, G. S. Buller, R. H. Hadfield, and M. J. Padgett, “Photon-sparse microscopy: visible light imaging using infrared illumination,” Optica 2(12), 1049 (2015).
[Crossref]

2014 (2)

2013 (6)

Y. Shi, T. Li, Y. Wang, Q. Gao, S. Zhang, and H. Li, “Optical image encryption via ptychography,” Opt. Lett. 38(9), 1425–1427 (2013).
[Crossref] [PubMed]

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] [PubMed]

W. Chen and X. Chen, “Ghost imaging for three-dimensional optical security,” Appl. Phys. Lett. 103(22), 221106 (2013).
[Crossref]

S. Liu and J. T. Sheridan, “Optical encryption by combining image scrambling techniques in fractional Fourier domains,” Opt. Commun. 287, 73–80 (2013).
[Crossref]

P. W. M. Tsang, T.-C. Poon, and W. K. Cheung, “Intensity image-embedded binary holograms,” Appl. Opt. 52(1), A26–A32 (2013).
[Crossref] [PubMed]

P. W. M. Tsang, T.-C. Poon, and A. S. M. Jiao, “Embedding intensity image in grid-cross down-sampling (GCD) binary holograms based on block truncation coding,” Opt. Commun. 304, 62–70 (2013).
[Crossref]

2012 (3)

M. Tanha, R. Kheradmand, and S. Ahmadi-Kandjani, “Gray-scale and color optical encryption based on computational ghost imaging,” Appl. Phys. Lett. 101(10), 101108 (2012).
[Crossref]

B. I. Erkmen, “Computational ghost imaging for remote sensing,” J. Opt. Soc. Am. A 29(5), 782–789 (2012).
[Crossref] [PubMed]

E. Pérez-Cabré, H. C. Abril, M. S. Millán, and B. Javidi, “Photon-counting double-random-phase encoding for secure image verification and retrieval,” J. Opt. 14(9), 094001 (2012).
[Crossref]

2010 (2)

B. I. Erkmen and J. H. Shapiro, “Ghost imaging: from quantum to classical to computational,” Adv. Opt. Photonics 2(4), 405–450 (2010).
[Crossref]

P. Clemente, V. Durán, V. Torres-Company, E. Tajahuerce, and J. Lancis, “Optical encryption based on computational ghost imaging,” Opt. Lett. 35(14), 2391–2393 (2010).
[Crossref] [PubMed]

2009 (2)

2008 (3)

Y. Zhang and B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33(21), 2443–2445 (2008).
[Crossref] [PubMed]

W. Chan, K. Charan, D. Takhar, K. Kelly, R. Baraniuk, and D. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93(12), 121105 (2008).
[Crossref]

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78(6), 061802 (2008).
[Crossref]

2006 (2)

2004 (1)

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: From error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

2003 (1)

2000 (2)

E. Tajahuerce and B. Javidi, “Encrypting three-dimensional information with digital holography,” Appl. Opt. 39(35), 6595–6601 (2000).
[Crossref] [PubMed]

T. Nomura and B. Javidi, “Optical encryption using a joint transform correlator architecture,” Opt. Eng. 39(8), 2031–2036 (2000).
[Crossref]

1995 (1)

Abril, H. C.

E. Pérez-Cabré, H. C. Abril, M. S. Millán, and B. Javidi, “Photon-counting double-random-phase encoding for secure image verification and retrieval,” J. Opt. 14(9), 094001 (2012).
[Crossref]

Ahmadi-Kandjani, S.

M. Tanha, R. Kheradmand, and S. Ahmadi-Kandjani, “Gray-scale and color optical encryption based on computational ghost imaging,” Appl. Phys. Lett. 101(10), 101108 (2012).
[Crossref]

Alfalou, A.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Pérez-Cabré, M. S. Millán, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap on optical security,” J. Opt. 18(8), 083001 (2016).
[Crossref]

Andrés, P.

Aspden, R. S.

Baraniuk, R.

W. Chan, K. Charan, D. Takhar, K. Kelly, R. Baraniuk, and D. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93(12), 121105 (2008).
[Crossref]

Barrera, J. F.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Pérez-Cabré, M. S. Millán, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap on optical security,” J. Opt. 18(8), 083001 (2016).
[Crossref]

Bian, L.

L. Bian, J. Suo, G. Situ, Z. Li, J. Fan, F. Chen, and Q. Dai, “Multispectral imaging using a single bucket detector,” Sci. Rep. 6(1), 24752 (2016).
[Crossref] [PubMed]

Bovik, A. C.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: From error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Bowman, A.

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] [PubMed]

Bowman, R.

N. Radwell, K. J. Mitchell, G. M. Gibson, M. P. Edgar, R. Bowman, and M. J. Padgett, “Single-pixel infrared and visible microscope,” Optica 1(5), 285–289 (2014).
[Crossref]

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] [PubMed]

Boyd, R. W.

Bromberg, Y.

O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95(13), 131110 (2009).
[Crossref]

Brosseau, C.

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[Crossref]

Pérez-Cabré, E.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Pérez-Cabré, M. S. Millán, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap on optical security,” J. Opt. 18(8), 083001 (2016).
[Crossref]

E. Pérez-Cabré, H. C. Abril, M. S. Millán, and B. Javidi, “Photon-counting double-random-phase encoding for secure image verification and retrieval,” J. Opt. 14(9), 094001 (2012).
[Crossref]

Pinkse, P. W. H.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Pérez-Cabré, M. S. Millán, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap on optical security,” J. Opt. 18(8), 083001 (2016).
[Crossref]

Poon, T.-C.

P. W. M. Tsang, T.-C. Poon, and W. K. Cheung, “Intensity image-embedded binary holograms,” Appl. Opt. 52(1), A26–A32 (2013).
[Crossref] [PubMed]

P. W. M. Tsang, T.-C. Poon, and A. S. M. Jiao, “Embedding intensity image in grid-cross down-sampling (GCD) binary holograms based on block truncation coding,” Opt. Commun. 304, 62–70 (2013).
[Crossref]

Qin, Y.

Y. Qin and Y. Zhang, “Information encryption in ghost imaging with customized data container and XOR operation,” IEEE Photonics J. 9(2), 1–8 (2017).
[Crossref]

Rack, A.

D. Pelliccia, A. Rack, M. Scheel, V. Cantelli, and D. M. Paganin, “Experimental x-ray ghost imaging,” Phys. Rev. Lett. 117(11), 113902 (2016).
[Crossref] [PubMed]

Radwell, N.

M. J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7, 12010 (2016).
[Crossref] [PubMed]

N. Radwell, K. J. Mitchell, G. M. Gibson, M. P. Edgar, R. Bowman, and M. J. Padgett, “Single-pixel infrared and visible microscope,” Optica 1(5), 285–289 (2014).
[Crossref]

Refregier, P.

Rivenson, Y.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Pérez-Cabré, M. S. Millán, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap on optical security,” J. Opt. 18(8), 083001 (2016).
[Crossref]

Ruggeri, A.

Scheel, M.

D. Pelliccia, A. Rack, M. Scheel, V. Cantelli, and D. M. Paganin, “Experimental x-ray ghost imaging,” Phys. Rev. Lett. 117(11), 113902 (2016).
[Crossref] [PubMed]

Shapiro, J. H.

B. I. Erkmen and J. H. Shapiro, “Ghost imaging: from quantum to classical to computational,” Adv. Opt. Photonics 2(4), 405–450 (2010).
[Crossref]

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78(6), 061802 (2008).
[Crossref]

Sheikh, H. R.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: From error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Shen, X. X.

Shen, Y.

J. Yang, L. Gong, X. Xu, P. Hai, Y. Shen, Y. Suzuki, and L. V. Wang, “Motionless volumetric photoacoustic microscopy with spatially invariant resolution,” Nat. Commun. 8(1), 780 (2017).
[Crossref] [PubMed]

Sheridan, J. T.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Pérez-Cabré, M. S. Millán, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap on optical security,” J. Opt. 18(8), 083001 (2016).
[Crossref]

S. Liu and J. T. Sheridan, “Optical encryption by combining image scrambling techniques in fractional Fourier domains,” Opt. Commun. 287, 73–80 (2013).
[Crossref]

B. Hennelly and J. T. Sheridan, “Optical image encryption by random shifting in fractional Fourier domains,” Opt. Lett. 28(4), 269–271 (2003).
[Crossref] [PubMed]

Shi, Y.

Silberberg, Y.

O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95(13), 131110 (2009).
[Crossref]

Simoncelli, E. P.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: From error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Situ, G.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Pérez-Cabré, M. S. Millán, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap on optical security,” J. Opt. 18(8), 083001 (2016).
[Crossref]

L. Bian, J. Suo, G. Situ, Z. Li, J. Fan, F. Chen, and Q. Dai, “Multispectral imaging using a single bucket detector,” Sci. Rep. 6(1), 24752 (2016).
[Crossref] [PubMed]

W. Wang, X. Hu, J. Liu, S. Zhang, J. Suo, and G. Situ, “Gerchberg-Saxton-like ghost imaging,” Opt. Express 23(22), 28416–28422 (2015).
[Crossref] [PubMed]

Soldevila, F.

Stern, A.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Pérez-Cabré, M. S. Millán, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap on optical security,” J. Opt. 18(8), 083001 (2016).
[Crossref]

Sud, K. K.

N. K. Pareek, V. Patidar, and K. K. Sud, “Image encryption using chaotic logistic map,” Image Vis. Comput. 24(9), 926–934 (2006).
[Crossref]

Sun, B.

M. J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7, 12010 (2016).
[Crossref] [PubMed]

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] [PubMed]

Sun, M. J.

M. J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7, 12010 (2016).
[Crossref] [PubMed]

Suo, J.

Suzuki, Y.

J. Yang, L. Gong, X. Xu, P. Hai, Y. Shen, Y. Suzuki, and L. V. Wang, “Motionless volumetric photoacoustic microscopy with spatially invariant resolution,” Nat. Commun. 8(1), 780 (2017).
[Crossref] [PubMed]

Tajahuerce, E.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Pérez-Cabré, M. S. Millán, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap on optical security,” J. Opt. 18(8), 083001 (2016).
[Crossref]

E. Tajahuerce, V. Durán, P. Clemente, E. Irles, F. Soldevila, P. Andrés, and J. Lancis, “Image transmission through dynamic scattering media by single-pixel photodetection,” Opt. Express 22(14), 16945–16955 (2014).
[Crossref] [PubMed]

P. Clemente, V. Durán, V. Torres-Company, E. Tajahuerce, and J. Lancis, “Optical encryption based on computational ghost imaging,” Opt. Lett. 35(14), 2391–2393 (2010).
[Crossref] [PubMed]

E. Tajahuerce and B. Javidi, “Encrypting three-dimensional information with digital holography,” Appl. Opt. 39(35), 6595–6601 (2000).
[Crossref] [PubMed]

Takhar, D.

W. Chan, K. Charan, D. Takhar, K. Kelly, R. Baraniuk, and D. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93(12), 121105 (2008).
[Crossref]

Tan, Q.

Tanha, M.

M. Tanha, R. Kheradmand, and S. Ahmadi-Kandjani, “Gray-scale and color optical encryption based on computational ghost imaging,” Appl. Phys. Lett. 101(10), 101108 (2012).
[Crossref]

Tanner, M. G.

Tasca, D. S.

Torres-Company, V.

Torroba, R.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Pérez-Cabré, M. S. Millán, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap on optical security,” J. Opt. 18(8), 083001 (2016).
[Crossref]

Tosi, A.

Tsang, P. W. M.

P. W. M. Tsang, T.-C. Poon, and A. S. M. Jiao, “Embedding intensity image in grid-cross down-sampling (GCD) binary holograms based on block truncation coding,” Opt. Commun. 304, 62–70 (2013).
[Crossref]

P. W. M. Tsang, T.-C. Poon, and W. K. Cheung, “Intensity image-embedded binary holograms,” Appl. Opt. 52(1), A26–A32 (2013).
[Crossref] [PubMed]

Vittert, L. E.

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] [PubMed]

Wang, B.

Wang, L.

S. Zhao, L. Wang, W. Liang, W. Cheng, and L. Gong, “High performance optical encryption based on computational ghost imaging with QR code and compressive sensing technique,” Opt. Commun. 353, 90–95 (2015).
[Crossref]

Wang, L. V.

J. Yang, L. Gong, X. Xu, P. Hai, Y. Shen, Y. Suzuki, and L. V. Wang, “Motionless volumetric photoacoustic microscopy with spatially invariant resolution,” Nat. Commun. 8(1), 780 (2017).
[Crossref] [PubMed]

Wang, W.

Wang, X.

Z. Zhang, X. Wang, G. Zheng, and J. Zhong, “Fast Fourier single-pixel imaging via binary illumination,” Sci. Rep. 7(1), 12029 (2017).
[Crossref] [PubMed]

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] [PubMed]

Wang, Y.

Wang, Z.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: From error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Welsh, S.

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] [PubMed]

Wu, J.

J. Wu, Z. Xie, Z. Liu, W. Liu, Y. Zhang, and S. Liu, “Multiple-image encryption based on computational ghost imaging,” Opt. Commun. 359, 38–43 (2016).
[Crossref]

Xie, Z.

J. Wu, Z. Xie, Z. Liu, W. Liu, Y. Zhang, and S. Liu, “Multiple-image encryption based on computational ghost imaging,” Opt. Commun. 359, 38–43 (2016).
[Crossref]

Xu, X.

J. Yang, L. Gong, X. Xu, P. Hai, Y. Shen, Y. Suzuki, and L. V. Wang, “Motionless volumetric photoacoustic microscopy with spatially invariant resolution,” Nat. Commun. 8(1), 780 (2017).
[Crossref] [PubMed]

Yamaguchi, M.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Pérez-Cabré, M. S. Millán, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap on optical security,” J. Opt. 18(8), 083001 (2016).
[Crossref]

Yang, J.

J. Yang, L. Gong, X. Xu, P. Hai, Y. Shen, Y. Suzuki, and L. V. Wang, “Motionless volumetric photoacoustic microscopy with spatially invariant resolution,” Nat. Commun. 8(1), 780 (2017).
[Crossref] [PubMed]

Yang, X.

X. Li, X. Meng, X. Yang, Y. Yin, Y. Wang, X. Peng, W. He, G. Dong, and H. Chen, “Multiple-image encryption based on compressive ghost imaging and coordinate sampling,” IEEE Photonics J. 8(4), 1–11 (2016).
[Crossref]

Yang, X. L.

Yao, M.

Yin, Y.

X. Li, X. Meng, X. Yang, Y. Yin, Y. Wang, X. Peng, W. He, G. Dong, and H. Chen, “Multiple-image encryption based on compressive ghost imaging and coordinate sampling,” IEEE Photonics J. 8(4), 1–11 (2016).
[Crossref]

Zhang, S.

Zhang, Y.

Y. Zhang, J. Suo, Y. Wang, and Q. Dai, “Doubling the pixel count limitation of single-pixel imaging via sinusoidal amplitude modulation,” Opt. Express 26(6), 6929–6942 (2018).
[Crossref] [PubMed]

Y. Qin and Y. Zhang, “Information encryption in ghost imaging with customized data container and XOR operation,” IEEE Photonics J. 9(2), 1–8 (2017).
[Crossref]

J. Wu, Z. Xie, Z. Liu, W. Liu, Y. Zhang, and S. Liu, “Multiple-image encryption based on computational ghost imaging,” Opt. Commun. 359, 38–43 (2016).
[Crossref]

Y. Zhang and B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33(21), 2443–2445 (2008).
[Crossref] [PubMed]

Zhang, Z.

Zhao, S.

S. Zhao, L. Wang, W. Liang, W. Cheng, and L. Gong, “High performance optical encryption based on computational ghost imaging with QR code and compressive sensing technique,” Opt. Commun. 353, 90–95 (2015).
[Crossref]

Zheng, G.

Zhong, J.

Zhou, C.

Zhuang, Z.

Z. Zhuang, S. Jiao, W. Zou, and X. Li, “Embedding intensity image into a binary hologram with strong noise resistant capability,” Opt. Commun. 403, 245–251 (2017).
[Crossref]

Zou, W.

S. Jiao, Z. Jin, C. Zhou, W. Zou, and X. Li, “Is QR code an optimal data container in optical encryption systems from an error-correction coding perspective?” J. Opt. Soc. Am. A 35(1), A23–A29 (2018).
[Crossref] [PubMed]

Z. Zhuang, S. Jiao, W. Zou, and X. Li, “Embedding intensity image into a binary hologram with strong noise resistant capability,” Opt. Commun. 403, 245–251 (2017).
[Crossref]

Adv. Opt. Photonics (1)

B. I. Erkmen and J. H. Shapiro, “Ghost imaging: from quantum to classical to computational,” Adv. Opt. Photonics 2(4), 405–450 (2010).
[Crossref]

Appl. Opt. (3)

Appl. Phys. Lett. (4)

O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95(13), 131110 (2009).
[Crossref]

W. Chan, K. Charan, D. Takhar, K. Kelly, R. Baraniuk, and D. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93(12), 121105 (2008).
[Crossref]

M. Tanha, R. Kheradmand, and S. Ahmadi-Kandjani, “Gray-scale and color optical encryption based on computational ghost imaging,” Appl. Phys. Lett. 101(10), 101108 (2012).
[Crossref]

W. Chen and X. Chen, “Ghost imaging for three-dimensional optical security,” Appl. Phys. Lett. 103(22), 221106 (2013).
[Crossref]

Europhys. Lett. (1)

W. Chen and X. Chen, “Ghost imaging using labyrinth-like phase modulation patterns for high-efficiency and high-security optical encryption,” Europhys. Lett. 109(1), 14001 (2015).
[Crossref]

IEEE Photonics J. (2)

X. Li, X. Meng, X. Yang, Y. Yin, Y. Wang, X. Peng, W. He, G. Dong, and H. Chen, “Multiple-image encryption based on compressive ghost imaging and coordinate sampling,” IEEE Photonics J. 8(4), 1–11 (2016).
[Crossref]

Y. Qin and Y. Zhang, “Information encryption in ghost imaging with customized data container and XOR operation,” IEEE Photonics J. 9(2), 1–8 (2017).
[Crossref]

IEEE Trans. Image Process. (1)

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: From error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Image Vis. Comput. (1)

N. K. Pareek, V. Patidar, and K. K. Sud, “Image encryption using chaotic logistic map,” Image Vis. Comput. 24(9), 926–934 (2006).
[Crossref]

J. Opt. (2)

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Pérez-Cabré, M. S. Millán, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap on optical security,” J. Opt. 18(8), 083001 (2016).
[Crossref]

E. Pérez-Cabré, H. C. Abril, M. S. Millán, and B. Javidi, “Photon-counting double-random-phase encoding for secure image verification and retrieval,” J. Opt. 14(9), 094001 (2012).
[Crossref]

J. Opt. Soc. Am. A (2)

Nat. Commun. (3)

M. J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7, 12010 (2016).
[Crossref] [PubMed]

Z. Zhang, X. Ma, and J. Zhong, “Single-pixel imaging by means of Fourier spectrum acquisition,” Nat. Commun. 6(1), 6225 (2015).
[Crossref] [PubMed]

J. Yang, L. Gong, X. Xu, P. Hai, Y. Shen, Y. Suzuki, and L. V. Wang, “Motionless volumetric photoacoustic microscopy with spatially invariant resolution,” Nat. Commun. 8(1), 780 (2017).
[Crossref] [PubMed]

Opt. Commun. (6)

P. W. M. Tsang, T.-C. Poon, and A. S. M. Jiao, “Embedding intensity image in grid-cross down-sampling (GCD) binary holograms based on block truncation coding,” Opt. Commun. 304, 62–70 (2013).
[Crossref]

S. Liu and J. T. Sheridan, “Optical encryption by combining image scrambling techniques in fractional Fourier domains,” Opt. Commun. 287, 73–80 (2013).
[Crossref]

Z. Zhuang, S. Jiao, W. Zou, and X. Li, “Embedding intensity image into a binary hologram with strong noise resistant capability,” Opt. Commun. 403, 245–251 (2017).
[Crossref]

S. Zhao, L. Wang, W. Liang, W. Cheng, and L. Gong, “High performance optical encryption based on computational ghost imaging with QR code and compressive sensing technique,” Opt. Commun. 353, 90–95 (2015).
[Crossref]

J. Wu, Z. Xie, Z. Liu, W. Liu, Y. Zhang, and S. Liu, “Multiple-image encryption based on computational ghost imaging,” Opt. Commun. 359, 38–43 (2016).
[Crossref]

S. Li, Z. Zhang, X. Ma, and J. Zhong, “Shadow-free single-pixel imaging,” Opt. Commun. 403, 257–261 (2017).
[Crossref]

Opt. Eng. (1)

T. Nomura and B. Javidi, “Optical encryption using a joint transform correlator architecture,” Opt. Eng. 39(8), 2031–2036 (2000).
[Crossref]

Opt. Express (5)

Opt. Lett. (6)

Optica (3)

Phys. Rev. A (1)

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78(6), 061802 (2008).
[Crossref]

Phys. Rev. Lett. (1)

D. Pelliccia, A. Rack, M. Scheel, V. Cantelli, and D. M. Paganin, “Experimental x-ray ghost imaging,” Phys. Rev. Lett. 117(11), 113902 (2016).
[Crossref] [PubMed]

Sci. Rep. (2)

Z. Zhang, X. Wang, G. Zheng, and J. Zhong, “Fast Fourier single-pixel imaging via binary illumination,” Sci. Rep. 7(1), 12029 (2017).
[Crossref] [PubMed]

L. Bian, J. Suo, G. Situ, Z. Li, J. Fan, F. Chen, and Q. Dai, “Multispectral imaging using a single bucket detector,” Sci. Rep. 6(1), 24752 (2016).
[Crossref] [PubMed]

Science (1)

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] [PubMed]

Other (3)

S. Jiao, W. Zou, and X. Li, “Noise Removal for Optical Holographic Encryption from Telecommunication Engineering Perspective,” in Digital Holography and Three-Dimensional Imaging, 2017 OSA Technical Digest Series (Optical Society of America, 2017), paper Th2A–5.

Z. Zhang and J. Zhong, “Single-pixel Broadcast Imaging,” in Computational Optical Sensing and Imaging, 2015 OSA Technical Digest Series (Optical Society of America, 2015), paper CT4F–3.

S. Jiao and P. Tsang, “A hologram watermarking scheme based on scrambling embedding and image inpainting,” in Digital Holography and Three-Dimensional Imaging, 2015 OSA Technical Digest Series (Optical Society of America, 2015), paper DT2A–4.

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Figures (7)

Fig. 1
Fig. 1 Proposed secured single-pixel broadcast imaging system.
Fig. 2
Fig. 2 An example of permutation operation on a 2-D intensity pattern.
Fig. 3
Fig. 3 (a) One original Hadamard basis pattern and corresponding permutated Hadamard basis patterns for different M b : (b) M b = 1 ; (c) M b = 2 ; (d) M b = 4 ; (e) M b = 8 .
Fig. 4
Fig. 4 Decrypted image of our proposed encrypted Hadamard single-pixel imaging scheme in numerical simulation (sampling ratio = 100%) with correct key when (a) M b = 1 ; (b) M b = 4 ; (c) M b = 8 ; with incorrect key when (d) M b = 1 ; (e) M b = 4 ; (f) M b = 8 ; (g) Reference image 1 for security evaluation; (h) Reference image 2 for security evaluation.
Fig. 5
Fig. 5 Simulation results. The first panel shows the recovered encrypted image and final decrypted image by the proposed technique with Key 1 for M b = 1 and different sampling ratios. The second panel shows the recovered encrypted image and final decrypted image by the proposed technique with Key 2 for M b = 2 and different sampling ratios. The third panel shows the recovered encrypted image and final decrypted image by the proposed technique with Key 3 for M b = 4 and different sampling ratios. The fourth panel shows the reconstructions by original Hadamard single-pixel imaging without encryption for different sampling ratios. The fifth panel shows the reconstructions by conventional encrypted single-pixel imaging (or ghost imaging) with random illumination patterns for different sampling ratios. The red number at the right-bottom corner of each image is the corresponding SSIM value.
Fig. 6
Fig. 6 Experimental set-up of our proposed secured single-pixel broadcast imaging system.
Fig. 7
Fig. 7 Experiments results: the top four panels shows the final decrypted images by the unauthorized receiver (PD1) and the authorized receiver (PD2) for different sampling ratios and different values of M b ; the bottom-most panel shows reconstructed results from original Hadamard single-pixel imaging without encryption.

Tables (3)

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Table 1 Auto-correlation of neighboring pixels in plaintext image and ciphertext image.

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Table 2 Cross correlation coefficients between plaintext image and ciphertext image.

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Table 3 Calculation time of object image reconstruction and decryption in proposed scheme.

Equations (7)

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i ( t ) = I ( x , y ) P t ( x , y ) d x d y .
I ( x , y ) i ( t ) P t ( x , y ) i ( t ) P t ( x , y ) ,
i t ( u , v ) = H { I ( x , y ) } = I ( x , y ) P t ( x , y ) d x d y ,
I ( x , y ) = H 1 { i t ( u , v ) } ,
P E ( x , y ) = E K { P t ( x , y ) } ,
i K ( u , v ) = I ( x , y ) E K { P t ( x , y ) } d x d y = E K 1 { I ( x , y ) } P t ( x , y ) d x d y ,
I ( x , y ) = E K { H 1 { i K ( u , v ) } } .

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