1. Introduction

Due to its inherent advantages of high computing speed and parallelism, information security techniques based on different optical theories and means have received more and more attention in the past decades [1–4]. Among them, the double random phase encoding (DRPE) is the most classical and provides a powerful impetus for the related research in the field of optical information processing. Although a plaintext can be encrypted into a complex-valued ciphertext image with stationary white noise distribution, DRPE is vulnerable to some common attacks such as known-plaintext attack and chosen-plaintext attack. To destroy its intrinsic linearity to enhance the corresponding resistance, the phase-truncated Fourier transforms are applied under the framework of DRPE. In addition, DRPE is further extended into different transform domains, where additional parameters can be considered as the secret keys to increase the difficulty of cracking cryptosystems. For example, gyrator domain [5,6], Fresnel domain [7,8] and fractional Fourier domain [9–11] are the most commonly used domains for information encryption, data hiding and image authentication. Most importantly, additional optical parameters in different domains can be served as secret keys, which greatly expands the key space and increases the difficulty of cracking. A variety of cryptosystems based on multifarious optical principles have been developed [12–22], and it is believed that these technologies will play very critical roles in the field of information security in the future.

Differing from the traditional optical imaging system, where the scene is illuminated by a uniform light field and detected with a pixelated detector array, single-pixel imaging that can be traced back to the Fly-Spot camera [23] has great advantages of high signal-to-noise ratio, wide spectral range and low cost, especially in low-light and non-visible light band imaging environment. So far, single-pixel imaging has been widely applied in more and more fields, such as 3D shape measurement [24], dynamic target imaging [25], motion estimation [26], and optical information security [27–32]. In order to reduce the distortion of reconstructed objects, a set of orthogonal transform base patterns are usually used to modulate the illumination light in the process of imaging. Therefore, the single-pixel imaging can be categorized into Hadamard single-pixel imaging [33], discrete cosine single-pixel imaging [34], wavelet transform single-pixel imaging [35], and Fourier single-pixel imaging [36–41]. Among them, the imaging efficiency and accuracy of Fourier single-pixel imaging is higher than others. Although a large number of illumination patterns are required to obtain Fourier spectrum coefficients in imaging, more and more research works have been developed in the field of optical information security based on Fourier single-pixel imaging. Chen et al. experimentally demonstrated that a non-imaging object recognition can be realized by obtaining the hash value in the Fourier domain, where the object image is illuminated with sinusoid patterns and the corresponding Fourier spectrum is collected using the four-step phase-shift algorithm [42]. Zhang et al. designed a compact mechanism to simultaneously implement the encryption and compression of an object image. In the encryption process, a random index map is used to choose the Fourier basis to form the measurement matrix, with which the object image is sparsely sampled [43]. Zhang et al. proposed a compressive optical steganography similar to a wireless broadcast system. Before broadcasting, modulated patterns to generate the Fourier spectrum of secret image are obtained using a generalized phase retrieval algorithm. In the decryption process, the secret image can be recovered from the sparse spectrum using the convex optimization algorithms [44]. Furthermore, Zhao et al. developed an optical encryption scheme based on ghost imaging, where the object image can be reconstructed using the four-step or the three-step phase shift method. In the imaging process, special speckles with designed orders of the fractional Fourier transform are applied to illuminate the object image [45]. To further improve the measurement efficiency of the Fourier single-pixels imaging, a lot of research works also have been reported in recent years. Deng et al. proposed a two-step phase shift method to reduce the required number of illumination patterns, and the object image can be recovered even at a sampling ratio of 20% [46]. Meng et al. designed a variable density random sampling matrix to obtain the sparse spectrum of the object image. Compared with equal probability random sampling and circular sampling, this scheme can efficiently reduce the number of samples while maintaining image quality [47].

As it can be seen from the above discussion, for most optical information security methods based on Fourier single-pixel imaging, a large number of illumination patterns should be projected to record the reflected light intensity of the object image to be protected, which obviously reduce the measurement efficiency of imaging. To solve this issue, an efficient optical multiple-image authentication method based on sparse sampling and multiple logistic maps is proposed. In this method, the spatial frequencies used for sampling object images are randomly divided into different groups by the aid of the sequence generated with a logistic map, and four binarized Fourier sinusoid patterns are generated for each frequency using the Floyd-Steinberg error diffusion dithering algorithm. Afterwards, illumination patterns in different groups are sequentially projected to the surface of different object images, and the corresponding reflected light intensities are collected using the bucket detector. Finally, based on the sequence generated with another logistic map, a real-valued sequence obtained by the catenation of all reflected light intensities is scrambled as the ciphertext. In the authentication process, the sparse Fourier spectrum of each object image can be obtained using the four-step phase-shift algorithm, which yields to the noisy reconstructed result. Although the content of reconstructed results cannot be visually observed, the existence of original object images can be verified using the modified nonlinear correlation with high discrimination capability. Due to high sensitivity, the initial values and bifurcation parameters of logistic maps are considered as secret keys, which both enhances the security of the cryptosystem and is very convenient for the key management. Most importantly, the authentication of object images can be efficiently implemented with only a significantly small part of illumination patterns, i.e., at a very low sampling ratio.

The rest of this paper is organized as follows. In Section 2, the proposed optical multiple-image authentication method based on Fourier single-pixel imaging and logistic maps is introduced in detail. In Section 3, simulated and optical experiments are carried out along with security and robustness analyses. Finally, a brief conclusion is summarized in Section 4.

2. Method description

In this section, the details of the proposed optical multiple-image authentication method is introduced based on Fourier single-pixel imaging and multiple logistic maps, along with the generation and binarization of Fourier basis patterns are presented.

2.1 Fourier single-pixel imaging

As an important optical technique, the perfect reconstruction of an object image can be implemented using Fourier single-pixel imaging, where the recovered result is almost the same as the ground truth. This is principally because the Fourier spectrum of the object image can be obtained with the help of illumination patterns with the sinusoidal distribution. These patterns applied in the process of Fourier spectrum acquisition construct a complete orthogonal set, which can be mathematically expressed as

Referencing Eq. (1), Eq. (4) and triangle transformation, this Fourier spectrum can be calculated using the four-step phase-shifting algorithm as

From Eq. (5), it can be seen that not only the noise error caused by environmental illumination but also the average intensity of the object image can be effectively removed. Finally, applying the inverse Fourier transform on $F({{f_x},{f_y}} )$, the object image $Q({x,y} )$ can be reconstructed as

2.2 Binarization and randomly selection of Fourier basis patterns

As a major evaluation indicator, the measurement efficiency determines whether a single-pixel imaging method can be applied in practical applications. For the Fourier single-pixel imaging system, there are two main obstacles to be solved for enhancing the measurement efficiency. One is to improve the projection frequency of illumination patterns, and the other is to reduce the number of patterns projected on the object image.

Due to the advantages of higher reflectivity, higher light throughput and higher frame rates, the digital micro-mirror device (DMD) has been widely used in the process of single-pixel imaging [35]. Because DMD can generate a large number of binary patterns per-second, it is the best way to improve the projection frequency of illumination patterns by modulating the wave field with DMD. However, it should be realized that DMD generates grayscale patterns by temporal dithering at the expense of time and will work much slower for grayscale sinusoid patterns. In this case, even if the high-speed DMD is used, a longer data acquisition time is still needed to collect the Fourier spectrum of an object image. To deal with this problem, sinusoid patterns should be converted to binary Fourier mode to facilitate the working of DMD with the high refresh rate. During the binarization process, the positive or negative residual quantization error will inevitably occur for each pixel. In the proposed multiple-image authentication method, the Floyd-Steinberg error diffusion dithering algorithm is used to spread the quantization error of a pixel onto its adjacent pixels according to the distribution shown as

After a sinusoid pattern is scanned from left to right and top to bottom, its average quantization error will be close to zero.

Although the object image can be reconstructed with high visual quality by making the use of the four-step phase-shifting algorithm, the number of sinusoid illumination patterns are too large to efficiently implement real-time imaging. If an object image with $M \times N$ pixels is fully sampled, a total of $4MN$ illumination patterns should be generated using Eq. (1). As shown in Fig. 1, the distribution matrix of spatial frequencies are displayed, where each element is a tuple and represents a spatial frequency. For each frequency, four illumination patterns with different phase such a $\textrm{0,}{{\mathrm{\ \pi }} / \textrm{2}}\mathrm{,\ \pi ,}{{\mathrm{\ and\ 3\pi }} / \textrm{2}}$ are formed to collect the total intensity of reflected light. Additionally, because the Fourier spectrum is conjugate symmetric, half of spatial frequencies are redundant. Thus, it is enough that only the top half of spatial frequencies in Fig. 1 are sampled in the process of imaging, and the bottom half of frequencies can be automatically complemented based on the conjugate symmetry. Namely, the object image can be reconstructed at a sampling ratio of 50%. However, although only $2MN$ patterns are needed to project onto the object image, it is still difficult to improve the measurement efficiency significantly, especially for high-resolution imaging. In the proposed multiple-image authentication method, there is no need to reconstruct the object image perfectly. Even if the reconstruction result is noisy, as long as the existence of the object image can be authenticated. Therefore, the number of illumination patterns can be further reduced. For one of object images, a certain number of spatial frequencies in the top half of Fig. 1 are randomly selected, with which a series of illumination patterns are formed to collect the intensity of reflected light.

 

Fig. 1. The distribution matrix of spatial frequencies.

Download Full Size | PDF

To randomly select spatial frequencies, the sequence generated based on logistic map is applied. Given the initial value and the bifurcation parameter, denoted as ${x_0}$ and $\mu$, respectively, a random sequence can be engendered using the iterative form of logistic map, which is mathematically described as

It should be noted that the initial value and the bifurcation parameter of logistic map as well as the integer $K$ can be considered as the secret key to enhance the security of the authentication system in the proposed multiple-image authentication method.

2.3 Ciphertext generation and multiple-image authentication

Based on the theory of Fourier single-pixel imaging and randomly selection of binary Fourier basis patterns, the related diagram of the ciphertext generation is depicted in Fig. 2, where ${Q_{(i )}}({i = 1,2,\ldots ,L} )$ denotes an object image with $W \times H$ pixels and ${M_{SF}}$ is the distribution matrix of spatial frequencies with $M \times N$ tuple elements as shown in Fig. 1. Notably, the constraint between the size of an object image and that of the distribution matrix should be satisfied as $W = d \times M$ and $H = d \times N$ in the proposed multiple-image authentication method, where the scale factor $d$ is positive integer. The generation process of the ciphertext can be described as

 

Fig. 2. The flowchart of the generation process of the ciphertext for multiple object images. Note:${M_{SF}}$ is the spatial frequency matrix, and FSI denotes the Fourier single-pixel imaging.

Download Full Size | PDF

From the above description, it is obvious that not only the high measurement efficiency but also the enhanced security of the authentication method can be simultaneously realized. On the one hand, the size of the spatial frequency matrix as shown in Fig. 1 is much smaller than that of object images, and an object image is imaging at the lower sampling ratio. When $L$ object images with $W \times H$ pixels are sampled based on the spatial frequency matrix with $M \times N$ elements and spatial frequencies in the top half of the matrix are used, the sampling ratio is only ${1 / {({2ddL} )}}$. For example, if there are 4 object images to be authenticated, where the size of object images is $128 \times 128$ pixels and the spatial frequency matrix is $64 \times 64$ elements, the scale factor $d = 2$ and an object image can be imaging at the sampling ratio of 3.13%, namely only 512 frequency coefficients should be collected for an object image. On the other hand, initial values and bifurcation parameters of two logistic maps can be considered as secret keys due to their high sensitivity. In the same time, the positive integer ${K_2}$ used in the scrambling process of reflected light intensities also can be used as an important secret key. Thus, the security level of the proposed multiple-image authentication system is significantly improved.

Compared with the generation process of ciphertext, the authentication process as shown in Fig. 3 is relatively simple, and the primary aspects should be paid attention as follows

3. Results and analysis

In this section, the feasibility and the validity of the proposed multiple-image authentication method are demonstrated with both simulation and optical experiments. To quantitatively evaluate the performance of the proposed multiple-image authentication method, correlation coefficient (CC) that represents the degree of consistency between two images is calculated, which plays a crucial role in analyzing the quality of the reconstructed image. In the proposed method, the CC value between the original object image and the corresponding reconstruction is mathematically calculated as

 

Fig. 3. The flowchart of the authentication process for multiple object images. Note: iFFT is the inverse Fourier transform.

Download Full Size | PDF

 

Fig. 4. Four object images to be authenticated.

Download Full Size | PDF

3.1 Simulations

In the following simulations, besides the feasibility, the security and robustness of the proposed method are analyzed from different aspects such as key sensitivity, noise attack, and occlusion attack and so on. In the generation process of illumination patterns, the average intensity and the intensity modulation coefficient in Eq. (1) are set to $a = 127$ and $b = 127$ respectively. Considering the influence of ambient light, the coefficient $k$ in Eq. (3) is set to 0.5, and the ambient light response ${D_n}$ is set to 10. As shown in Fig. 1, the size of the frequency matrix is set to $64 \times 64$ elements. In the generation process of the ciphertext, one group of initial value and bifurcation parameter are set to ${x_{01}} = 0.35$ and ${\mu _1} = 3.999997$, respectively, and another group are set to ${x_{02}} = 0.25$ and ${\mu _2} = 3.999998$. In the same time, two integers ${K_1}$ and ${K_2}$ are set to 2000 and 3000, respectively. In the authentication process of multiple object images, the nonlinearity strength $p$ in Eq. (9) is set to 0.4, and the enhancement coefficient $\varpi$ in Eq. (11) is set to 4.

In the generation process of the ciphertext, all 2048 spatial frequencies in the top half of the frequency matrix are used to construct illumination patterns. These frequencies are scrambled and divided into 4 groups on average, where a group of spatial frequencies is allocated to an object image. Thus, only 512 spatial frequencies are applied to an object image, i.e., an object image is sampled at a very low sampling ratio of 3.13%, which is less than 5% of Nyquist limit. Because four sinusoid patterns are generated for a spatial frequency, a total of 2048 patterns are projected to record the reflected light intensities for an object image. Therefore, a real-valued sequence composed of 8192 reflected light intensities is eventually obtained as the ciphertext for multiple object images. As shown in Fig. 5, the ciphertext is plotted, from which no valid information can be concluded. Moreover, it can be seen that the corresponding distribution is basically random and multiple object images are fully encrypted.

 

Fig. 5. The distribution of the ciphertext.

Download Full Size | PDF

To enhance the security level, initial values and bifurcation parameters of two logistic maps as well as the positive integer ${K_2}$ are considered as secret keys. When all secret keys are correct, the decrypted images are shown in Fig. 6(a)-(d). The corresponding CC values are 0.4226, 0.4043, 0.3419, and 0.5755, respectively. It can be seen that the details related to original object images cannot be clearly discerned with the naked eyes. This is mainly because only 512 spatial frequencies are used to sample an object image and only the sparse Fourier spectrum is recovered in the authentication process. However, the nonlinear correlation can be applied to authenticate these original object images [48]. As displayed in Fig. 6(e)-(h), the authentication results are calculated between Fig. 6(a)-(d) and downsampled object images of Fig. 4, respectively, from which it can be observed that there is one relatively significant peak in the center of each nonlinear correlation distribution. That is to say, these distributions can efficiently disclose the existence of original object images, and the feasibility of the proposed method has been demonstrated.

 

Fig. 6. (a)-(d) The decrypted object images with $64 \times 64$ pixels, and (e)-(h) the corresponding correlation distribution maps.

Download Full Size | PDF

The obvious advantage of considering the initial values and the bifurcation parameters of two logistic maps as secret keys is the high sensitivity to the slight variation, which can potentially improve the security of the proposed method. As shown in Fig. 7(a)-(d), the first object image as shown in Fig. 4(a) decrypted only with one of incorrect keys including ${x_{01}}$ and ${\mu _1}$ are displayed, where the absolute deviation of ${x_{01}}$ is ${10^{ - 15}}$ and that of ${\mu _1}$ is ${10^{ - 15}}$, respectively. In these figures, no valid information about the first object image can be visually observed. For the initial value, the CC curve between four object images and their corresponding reconstructed results is plotted in Fig. 7(e). For the bifurcation parameter, the CC curve between them is plotted in Fig. 7(f). Obviously, when the correct initial value or bifurcation parameter is applied, the CC value is the highest, otherwise it will decrease significantly for tiny variation. As shown in Fig. 8(a)-(d), the same results can be obtained for two secret keys including ${x_{02}}$ and ${\mu _2}$ of another logistic map. The related CC values are plotted in Fig. 8(e) and (f). From Fig. 7 and 8, it can be easily concluded that the proposed method is very sensitive to these secret keys. In addition, the integer ${K_2}$ as the secret key is also highly sensitive to tiny variation. As shown in Fig. 9(a) and (b), the first object image is decrypted with the absolute deviation of 1, where no any content is discerned. The related CC curve is plotted in Fig. 9(c). Once this secret key is changed, the CC value will decrease sharply. Only considering secret keys of two logistic maps, the key space of the proposed method is approximately ${10^{61}}$ and is very huge to resist against the brute-force attack. For other original images, similar conclusion can be obtained.

 

Fig. 7. (a)-(b) The first object image decrypted with ${x^{\prime}_{01}} = {x_{01}} - {10^{ - 15}}$,${x^{\prime}_{01}} = {x_{01}} + {10^{ - 15}}$, ${\mu ^{\prime}_1} = {\mu _1} - {10^{ - 15}}$, and ${\mu ^{\prime}_1} = {\mu _1} + {10^{ - 15}}$, respectively, where the size is $64 \times 64$ pixels, (e) the CC curve with variation of the initial value, and (f) the CC curve with the variation of the bifurcation parameter.

Download Full Size | PDF

 

Fig. 8. (a)-(b) The first object image decrypted with ${x^{\prime}_{02}} = {x_{02}} - {10^{ - 16}}$, ${x^{\prime}_{02}} = {x_{02}} + {10^{ - 16}}$, ${\mu ^{\prime}_2} = {\mu _2} - {10^{ - 15}}$, and ${\mu ^{\prime}_2} = {\mu _2} + {10^{ - 15}}$, respectively, where the size is $64 \times 64$ pixels, (e) the CC curve with variation of the initial value, and (f) the CC curve with the variation of the bifurcation parameter.

Download Full Size | PDF

 

Fig. 9. (a)-(b) The first object image decrypted with ${K^{\prime}_2} = {K_2} - 1$ and ${K^{\prime}_2} = {K_2} + 1$, respectively, where the size is $64 \times 64$ pixels, (c) the related CC curve.

Download Full Size | PDF

Also, the nonlinear correlation maps between decrypted results as shown in Fig. 7(a)-(d) and the downsampled object image of Fig. 4(a) are calculated and displayed in Fig. 10(a)-(d), respectively. It can be seen intuitively that there are many lower peaks in different locations, but there is no remarkable peak exists in the center of the distribution map. Obviously, if ${x_{01}}$ or ${\mu _1}$ has a tiny change, the authentication of object image still cannot be implemented by the means of nonlinear correlation. For other nonlinear correlation maps such as between decrypted results of Fig. 8(a)-(d) or Fig. 9(a)-(b) and that of Fig. 4(a), the same conclusion is obtained. Therefore, it can be safe to say that the initial values and the bifurcation parameters of two logistic maps as well as ${K_2}$ considered as secret keys will greatly enhance the security of the proposed method. More importantly, these secret keys are real-valued, which will facilitate the management such as storage and transmission.

 

Fig. 10. (a)-(d) The nonlinear correlation map between decrypted results in Fig. 7(a)-(d) and the downsampled object image of Fig. 4(a).

Download Full Size | PDF

During the storage and transmission, the information of the ciphertext may be disturbed by noise. When the ciphertext is an image, different kinds of noise can be added to analyze the corresponding robustness. Differing from this case, for the ciphertext sequence obtained in the generation process shown in Fig. 2, the resistance against the Gaussian noise attack is analyzed in order to evaluate the related performance. Suppose that the ciphertext is disturbed by the Gaussian noise $G$ with zero-mean and 1 standard deviation, the corresponding modal can be mathematically described as

 

Fig. 11. (a)-(c) First original image decrypted with different noise strength, where the size is $64 \times 64$ pixels, and (d)-(f) corresponding nonlinear correlation maps.

Download Full Size | PDF

Generally, it is inevitable to avoid the partial loss of the ciphertext. To test the performance of resisting against this influence, the analysis is performed from three representative cases. Because the ciphertext is a real-valued sequence consisting of measurements of original images, as shown in Fig. 5, the occlusion attack is performed by discarding the first 1/3, middle 1/3, and last 1/3 consecutive data of this sequence, respectively. As shown in Fig. 12(a)-(c), the decrypted results of the first object image are displayed, where the corresponding CC values are 0.3492, 0.3628, and 0.3408, respectively. The first object image is seriously deteriorated, and no distinct structure can be discerned. The corresponding nonlinear correlation maps are displayed in Fig. 12(d)-(f), where a remarkable peak in the center of each map indicates the existence of the related object image. For other object images, the same conclusions are also obtained. Therefore, it is safe to say that the proposed method has high robustness against occlusion attack.

 

Fig. 12. (a)-(c) First original image decrypted in three cases, where the size is $64 \times 64$ pixels, and (d)-(f) corresponding nonlinear correlation maps.

Download Full Size | PDF

In order to evaluate the feasibleness of the proposed method according to grey images, we have done the simulated experiments. As shown in Fig. 13(a)-(d), four gray images to be authenticated are displayed, where the same parameter settings described for binary images are used. When all secret keys are correct, the decrypted images are shown in Fig. 14(a)-(d), and the corresponding nonlinear maps are displayed in Fig. 14(e)-(h). Obviously, every original image can be verified because there is one relatively significant peak in the center of its nonlinear correlation distribution.

 

Fig. 13. (a)-(d) Four original gray images to be authenticated.

Download Full Size | PDF

 

Fig. 14. (a)-(d) Four original images decrypted, where the size is $64 \times 64$ pixels, and (e)-(h) corresponding nonlinear correlation maps.

Download Full Size | PDF

3.2 Optical experiments

A schematic of the optical experiment is displayed in Fig. 15, with which the Fourier single-pixel imaging of the proposed method is executed. In this experiment, illumination patterns are generated based on the top half of the spatial frequency matrix as shown in Fig. 1 by a computer, where the parameters, such as the average intensity and the intensity modulation coefficient and so on, are set to be the same as those used in simulations. Then, a projector is used to project these patterns onto the surface of object images, and a bucket detector is used to record reflected light intensities. Due to the constraints of resource in our laboratory, instead of using the bucket detector without spatial resolution, an industrial camera is applied to collect the total light reflected from the surface of object plane, on which object images are placed. The model of the project and the camera are XMING A62S and IMAGING SOURVE DFK 72AUC02, respectively.

 

Fig. 15. The experiment setup. DLP: digital light projector; BD: bucket detector; IPs: illumination patterns.

Download Full Size | PDF

As shown in Fig. 16(a)-(d), the decrypted object images are displayed, where the image quality are clearly inferior to those as shown in Fig. 6(a)-(d). The corresponding CC values are 0.2131, 0.2365, 0.1790, and 0.4360, respectively. There are some main reasons to yield that the quality of reconstructed object images are seriously destroyed. One is that the experiments suffer from interference due to ambient light, and another is that the nonlinearity of the projector has the undesirable influence on the experiments. However, the authentication of object images still can be implemented using the nonlinear correlation algorithm. As shown in Fig. 16(e)-(h), the nonlinear correlation maps between Fig. 16(a)-(d) and the downsampled result of Fig. 4(a) are displayed. Obviously, there are remarkable peaks existed in the center of these nonlinear correlation distributions, which indicate the existence of object images. In addition, consistent results as those in simulations can be obtained in different aspects, such as key sensitivity, noise attack, and occlusion attack.

 

Fig. 16. (a)-(d) The decrypted object images with $64 \times 64$ pixels, and (e)-(h) the corresponding correlation distribution maps.

Download Full Size | PDF

Compared the obtained results from the experiments (Fig. 16(a)-(d)) with the simulations (Fig. 6(a)-(d)), significant differences are evident. There are some factors to affect the practical optical experiments. Firstly, the practical experiments are best performed in the low-light environment so that only ambient light affects the measurements collected. Thus, after each acquired image is converted to a bucket signal, all measured values vary according to different fringe patterns projected on the original image, which better simulate the actual values collected with a practical bucket detector. Secondly, the size of the camera aperture should be adjusted to avoid obtaining too bright images. Otherwise, almost all measured values vary around a fixed value, and it is difficult to ensure that correct bucket signals are recorded. Thirdly, the focal length should be adjusted according to the distance between the object plane and the camera so that images with clear edges and corners can be obtained. For the same reason, it can guarantee that the produced bucket signals are consistent with the actual values as soon as possible.

4. Conclusion

In summary, an optical multi-image authentication method is proposed based on Fourier single-pixel imaging and multiple logistic maps. In this method, the reconstructed object images can be authenticated, when a very low sampling ratio (i.e., less than 5% of Nyquist limit) is used in the imaging process. Because relatively few illumination patterns are generated to record the reflected light intensities of object images, the measurement efficiency of the proposed method can be greatly improved. Although reconstructed object images are not observed with clear visualization, the existence of original information can be verified using the nonlinear correlation maps. The feasibility and effectiveness of the proposed method have been demonstrated by simulation and optical experiments. Meanwhile, two logistic maps are used to randomly select spatial frequency in the imaging process and scramble the collected intensities, the security level of the proposed method can be greatly enhanced. Also, the performance of resisting against malicious attacks is demonstrated. It is believed that this research not only enriches functionality of the Fourier single-pixel imaging but also provides a fascinating boost to the real-time application in this field.