Secured single-pixel broadcast imaging

Zibang Zhang; Shuming Jiao; Manhong Yao; Xiang Li; Jingang Zhong

doi:10.1364/OE.26.014578

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.

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 $P_{t} (x, y)$ $(t = 1, 2, 3, \dots, N)$ 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 $I (x, y)$ , $P_{t} (x, y)$ and $i (t)$ can be described by Eq. (1),

i (t) = \iint I (x, y) P_{t} (x, y) d x d y .

The illumination patterns

P_{t} (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 $P_{t} (x, y)$ with the single-pixel measurements $i (t)$ , described by Eq. (2).

I (x, y) \approx 〈 i (t) P_{t} (x, y) 〉 - 〈 i (t) 〉 〈 P_{t} (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 $M \times N$ pixels, referring to the plaintext. The image can be represented by an $M \times N$ -element matrix. First, the image is reshaped to an $(M \times N) \times 1$ -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 $M \times N$ -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.

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 $P_{t} (x, y)$ multiplied with Hadamard coefficients $i_{t} (u, v)$ ( $1 \leq t \leq N_{m}$ ). 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).

i_{t} (u, v) = H {I (x, y)} = \iint I (x, y) P_{t} (x, y) d x d y,

I (x, y) = H^{- 1} {i_{t} (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

H^{- 1} {}

denotes an inverse 2-D Hadamard transform. To perform encryption, we can apply a permutation operation

E_{K}

to the basis pattern as Eq. (5),

P_{E} (x, y) = E_{K} {P_{t} (x, y)},

where

P_{E} (x, y)

is the encrypted basis pattern after applying permutation to

P_{t} (x, y)

with the key

K

. As Eq. (6) shows, the corresponding single-pixel measurement is

i_{K} (u, v)

when the encrypted pattern

P_{E} (x, y)

is used for illumination.

i_{K} (u, v) = \iint I (x, y) E_{K} {P_{t} (x, y)} d x d y = \iint E_{K}^{- 1} {I (x, y)} P_{t} (x, y) d x d y,

where

E_{K}^{- 1}

refers to another permutation operation that is in an inverse order as

E_{K}

. 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) = E_{K} {H^{- 1} {i_{K} (u, v)}} .

For unauthorized users without the key $K$ , the original object image cannot be recovered from $i_{K} (u, v)$ 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 $M_{b}$ × $M_{b}$ pixels. As Fig. 3 shows, when $M_{b}$ 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.

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) $M_{b} = 8$ .

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The energy is evenly distributed in the entire transformation domain. By increasing the block size $M_{b}$ , 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 $M_{b}$ 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 $M_{b}$ and sampling ratios. The Hadamard basis patterns for illumination are permutated with three permutation block sizes $M_{b} = 1$ , $M_{b} = 4$ , and $M_{b} = 8$ . 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.

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.

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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.

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

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The results show that the auto-correlation coefficients for ciphertext images are lower than those for the plaintext especially when $M_{b} = 1$ 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.

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

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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 $N_{m}$ ). 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.

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.

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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 $M_{b}$ increases, the quality in the former scheme becomes closer to the latter. When the sampling ratio $N = N_{m}$ , 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 $M_{b} = 8$ the quality of decryptions is close to that of reconstructions without encryption, which demonstrates that a larger $M_{b}$ 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.

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

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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 $M_{b} = 1$ , 2, 4, and 8, and encrypted Hadamard patterns with Key 2 and $M_{b} =$ 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)].

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 $M_{b}$ , 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 $M_{b}$ values and unencrypted systems are identical. The experimental results are consistent with the simulation results.

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.

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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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	Shift (pixel)
	0	1	4	8
Plaintext image (‘Cameraman’)	1	0.9592	0.8465	0.7442
Ciphertext image with $M_{b} = 1$ [Fig. 4(d)]	1	0.0071	0.0030	0.0007
Ciphertext image with $M_{b} = 4$ [Fig. 4(e)]	1	0.7233	0.0123	−0.0465
Ciphertext image with $M_{b} = 8$ [Fig. 4(f)]	1	0.8394	0.4108	−0.0380

	Plaintext image (‘Cameraman’)	Reference image 1 [Fig. 4(g)]	Reference image 2 [Fig. 4(h)]
Ciphertext image with $M_{b} = 1$ [Fig. 4(d)]	−0.0062	0.0002	0.0014
Ciphertext image with $M_{b} = 4$ [Fig. 4(e)]	0.0062	0.0012	0.0101
Ciphertext image with $M_{b} = 8$ [Fig. 4(f)]	−0.0010	0.0439	0.0113

Object image size (pixel)	Computational time for inverse 2D Hadamard transform	Computational time for permutation decryption
Object image size (pixel)	Computational time for inverse 2D Hadamard transform	$M_{b} = 1$	$M_{b} = 4$	$M_{b} = 8$
128 × 128	0.860 ms	0.073 ms	0.073 ms	0.074 ms
256 × 256	2.596 ms	0.291 ms	0.283 ms	0.288 ms
512 × 512	15.059 ms	1.873 ms	1.775 ms	1.729 ms
1024 × 1024	84.098 ms	10.098 ms	9.301 ms	8.862 ms

	Shift (pixel)
	0	1	4	8
Plaintext image (‘Cameraman’)	1	0.9592	0.8465	0.7442
Ciphertext image with $M_{b} = 1$ [Fig. 4(d)]	1	0.0071	0.0030	0.0007
Ciphertext image with $M_{b} = 4$ [Fig. 4(e)]	1	0.7233	0.0123	−0.0465
Ciphertext image with $M_{b} = 8$ [Fig. 4(f)]	1	0.8394	0.4108	−0.0380

	Plaintext image (‘Cameraman’)	Reference image 1 [Fig. 4(g)]	Reference image 2 [Fig. 4(h)]
Ciphertext image with $M_{b} = 1$ [Fig. 4(d)]	−0.0062	0.0002	0.0014
Ciphertext image with $M_{b} = 4$ [Fig. 4(e)]	0.0062	0.0012	0.0101
Ciphertext image with $M_{b} = 8$ [Fig. 4(f)]	−0.0010	0.0439	0.0113

Secured single-pixel broadcast imaging

Abstract

1. Introduction

2. Principle

2.1 Single-pixel imaging with random patterns and basis patterns

2.2 Proposed single-pixel imaging with encrypted basis patterns

3. Results and discussion

3.1 Simulation results

3.2 Experimental results

4. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (7)

Tables (3)

Equations (7)

Optics Express