Three-dimensional holographic hiding and display using layer-oriented theorem

Optics Communications 453 (2019) 124340

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Three-dimensional holographic hiding and display using layer-oriented theorem Dan Xiao a , Xiao-Wei Li a , Di Wang b , Ying Wang a , Qiong-Hua Wang b ,∗ a b

School of Electronics and Information Engineering, Sichuan University, Chengdu 610065, China School of Instrumentation and Optoelectronic Engineering, Beihang University, Beijing 100191, China

ARTICLE

INFO

Keywords: Three-dimension holographic hiding Layer-oriented theorem Singular value matrix

ABSTRACT Image hiding is an important technology of secure communications, which has a broad application in military field. In this paper, we present a three-dimensional (3-D) holographic hiding and display technology using layer-oriented theorem. As the secret image, the hologram is generated using layer-oriented theorem. Angular spectra from each layer are synthesized based on the fast Fourier transform without paraxial approximation. And the 3-D image, which is coded as a hologram based on layer-oriented theorem, is transformed into a quick response (QR) code and embedded into the singular value matrix of the carrier image. The simulation and optical experimental results show the effectiveness and display application with our proposed technology.

1. Introduction Information security technology is an important branch of signal processing. It has been caused widely attention of many experts and scholars, and also plays an important role in military and civilian fields. Information hiding technology is used to hide the information which needs to be transmitted. Image information hiding can be used as a method of image encryption, as a preprocessing and post-processing of digital image watermarking technology, and also can be combined with other encryption methods. Therefore, the technology has strong practicality and flexibility. With the development of the information technology, 3-D information has more information than two-dimensional (2-D) information, and will gradually become the main body of information exchange. Therefore, 3-D information encryption, especially 3-D information hiding, will be the focus of information security. Among them, holographic technology has a series of advantages and has received extensive attention, therefore it has extraordinary significance for the hiding of 3-D information [1–4]. A 3-D display can enhance visual effects when depth information of the 3-D image is provided. Holographic display, as it can provide all the depth cues which our eyes needed, is a real 3-D display technology. It is also an ideal technique for 3-D image display because it is capable of reconstructing the wavefronts of a 3-D scene [5–8], implementing a high-resolution display [9] and achieving a 3-D dynamic and video display [10–12]. Since double-random phase encoding (DRPE) to encrypt an original image was proposed in 1995 [13], various extension strategies for encryption and hiding have been proposed [14–26]. In DRPE, the input image can be converted into stationary white noise with the help of two ∗

independent random phase-only masks that were placed in the input plane and Fourier plane, respectively. However, during the past decade, DRPE has been reported that it can be attacked under some conditions. Because of the stronger anti-breaking ability of image hiding technology in transform domain than the spatial domain, the current transform image hiding techniques such as wavelet transform, discrete cosine transform (DCT), singular value transform and so on. Based on the fractional wavelet transform (FWT) [16,17], the fractional orders are the additional keys compared with the traditional Fourier transform. Among them, some techniques secure and store the information by multiplying random phases in the input planes or in some fractional domains. But as we all know, the content of the carrier image is rarely changed in the wavelet transform domain, and the effective protection can be obtained. However, since it does not have rotational translation invariance, there is bad robust to resist geometric attacks such as rotation. In Ref. [18], Arnold transform scrambles the pixel position of the blocked sub images of original image at local area, color-blend operation defined by a 3 × 3 matrix exchanges and mixes randomly scrambled RGB components, and DCT changes the pixel values of color image. Arnold transform, color-blend operation, and DCT are performed two times to encrypt the image. The method is effectively for encrypting the 2-D image but it cannot encrypt a 3-D information. And in Ref. [20], scholars have proposed a new image encryption algorithm based on singular value decomposition (SVD) and Arnold transform. An original image is first transformed in fractional domain by FRFT, and then decomposed into three segments by SVD. The results are successful and people cannot obtain the information easily. However, simple singular value transformation, which has strong robust to resist

Corresponding author. E-mail address: [email protected] (Q.-H. Wang).

https://doi.org/10.1016/j.optcom.2019.124340 Received 23 June 2019; Received in revised form 26 July 2019; Accepted 4 August 2019 Available online 6 August 2019 0030-4018/© 2019 Elsevier B.V. All rights reserved.

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Optics Communications 453 (2019) 124340

Fig. 1. The computer-generated hologram (CGH) generated-diagram of our proposed technology . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

( ) where 𝛴 represents the diagonal matrix, 𝛴 = diag 𝜎1 , 𝜎2 , … , 𝜎𝑟 . We know that values of 𝜎1 , 𝜎2 , … , 𝜎𝑟 are the all non-zero eigenvalue, thus, the Eq. (2) can be rewritten as:

geometric attacks such as image rotation and scaling, is weak against Gaussian, salt and pepper, filtering and other attacks. In this paper, we propose a technology for hiding and displaying 3-D image based on holography. The 3-D information is coded as the computer-generated hologram (CGH) using layer-based theorem. Then, the hologram is transformed into a quick response (QR) code as a secret image hidden into the carrier image. When we extract the hologram from hidden image and uploaded to phase-only spatial light modulator (SLM), we can display all information of the 3-D image. With our proposed method, it can be reconstructed in both digital and optical ways, and becomes more flexible.

𝐴=

𝑖=1

𝐿𝑖 (𝑥0 , 𝑦0 ) exp(𝑗

2𝜋 𝑧) 𝜆 𝑖

(3)

Wavelet was originally derived from the shortcomings of Fourier analysis. In classical signal analysis, the transform cannot obtain the time domain and frequency domain characteristics of the signal at the same time. Thus, in order to make up for this deficiency, the wavelet can use a variable time-frequency window to perform local analysis of the signal. In practical applications, wavelet is defined as follows: ( ) 𝑡 − 𝑛𝑏0 𝑎0 𝑚 1 𝜁𝑚,𝑛 (𝑡) = √ 𝜁 (4) 𝑚 𝑎0 𝑎0 𝑚

The computer-generated hologram (CGH) generated-diagram of layer-oriented theorem is shown in Fig. 1 [5]. A 3-D object O is first sliced into many layers along the optical axis with appropriate layer space. Consequently, the 3-D object can be represented as: 𝑂=

𝜎𝑖 𝑢 𝑖 𝑣 𝑖 𝐻

𝑖=1

2. Principle

𝑡 ∑

𝑟 ∑

And the corresponding wavelet transform is defined as shown in Eq. (5): 2 ⟨ ⟩ 𝑓 , 𝜁𝑚,𝑛 = 𝑎0 − 𝑚

+∞

∫−∞

𝑓 (𝑡)𝜁𝑚,𝑛 (𝑡)𝑑𝑡

(5)

In Eqs. (4) and (5), 𝑎0 represents the scaling factor, and 𝑏0 denotes the location parameters, and 𝜁(𝑡) represents the wavelet. If 𝑎0 > 1, the function 𝜁(𝑡) has a stretching effect. On the contrast, if 𝑎0 < 1, the function 𝜁(𝑡) has a contraction effect. The phenomenon is exactly contrary to the Fourier transform. When the signal frequency increases, the window width becomes narrower and the frequency window width increases, which is conducive to improving the resolution of the time domain, and vice versa. The wavelet transform is used to decompose the image through a set of filters to decompose the image into two parts: low frequency and high frequency. Then, the computed low frequency can be decomposed again and form two parts. In this way, the signal is decomposed into different frequencies. And the decomposition processing is shown in Fig. 2. Moreover, 2-D scale function can be obtained by couple of the one-dimensional function 𝜓 1 (𝑥) and 𝜓 2 (𝑦), as follows:

(1)

where (𝑥0 , 𝑦0 ) is the coordinates at the object plane, i is the sequence number of the layer, t is the total number of the layer, 𝐿𝑖 represents the ith layer amplitude, 𝑧𝑖 represents the distance of the ith layer. In Fig. 1, we pick one layer of the object and mark it with a blue box. Then via propagation, the corresponding amplitude and random phase could generate a sub-hologram, and the complex amplitude distribution of the sub-hologram is calculated by the angular spectrum method. We slice the object into i layers and angular spectra from each layer are synthesized as a layer-corresponded sub-hologram based on the fast Fourier transform (FFT). Finally, the phase-only hologram can be obtained from the complex amplitude hologram using phase extracting method. Singular value decomposition is one of the most efficient tools in linear algebra. It is widely used in the field of signal and image processing, system control theory and statistical analysis. We set the matrix 𝐴 ∈ 𝐶 𝑚×𝑛 , then there would be a m-order matrix U and a n-order matrix V and make the Eq. (2) true. ( ) 𝛴 0 𝐴=𝑈 𝑉𝐻 (2) 0 0

𝜓 1 (𝑥, 𝑦) = 𝜓 1 (𝑥)𝜓 2 (𝑦)

(6)

Then, the 2-D wavelet functions can be obtained when 𝜙(𝑥) represents a wavelet function, the expression of these functions as follows: 𝜙1 (𝑥, 𝑦) = 𝜓 1 (𝑥)𝜙2 (𝑦) 2

1

2

𝜙 (𝑥, 𝑦) = 𝜙 (𝑥)𝜓 (𝑦) 2

(7) (8)

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Optics Communications 453 (2019) 124340

Fig. 2. Image wavelet decomposition.

𝜙3 (𝑥, 𝑦) = 𝜙1 (𝑥)𝜙2 (𝑦)

(9)

where the wavelet function is used to express the variation of image gray in different directions: 𝜙1 (𝑥, 𝑦) is used to measure the changes along vertical edge, 𝜙2 (𝑥, 𝑦) is used to measure the changes along horizontal edge and 𝜙3 (𝑥, 𝑦) is used to measure the changes along diagonal line. And in hiding processing, as shown in Fig. 3, firstly, we choose a 3D object as the hiding object and generate a hologram h according the layer-oriented theorem. Considering the size of hologram, the hologram h is coded as a QR code m. Wavelet transform has the characteristics of multi-resolution analysis and can be easily combined with certain characteristics of human visual system. Secondly, performing a discrete wavelet decomposition operation on the carrier 2-D image. Considering the low frequency coefficient stability and strong anti-attack ability, the hologram is embedded in the low frequency sub-band 𝐿𝐿2 . Then, we operate 𝐿𝐿2 using singular value decomposition, and as shown in Eq. (10): 𝐿𝐿2 = 𝑈1 𝑆1 𝑉1𝑇

(10)

At the same time, we also operate the QR code m using the same method, and shown in Eq. (11): 𝑚 = 𝑈2 𝑆 2 𝑉 2 𝑇

Fig. 3. The flow chart with our proposed technology.

Table 1 The PSNR values about the scaling factor k.

(11)

And then we generate a new S, and the value can be obtained by Eq. (12): 𝑆 = 𝑆1 + 𝑘𝑆2

(12)

where k is the scaling factor. Next, S replace the 𝑆1 in Eq. (10) and obtain the new 𝐿𝐿2 ′ . Finally, the final 2-D image can be obtained after inverse discrete wavelet transforming (DWT).

The value of the scaling factor k

0.01

0.02

0.05

0.10

0.20

0.40

PSNR (dB)

64.4575 58.1462 52.9586 44.9256 40.0125 32.9562

where MSE is the mean square error between original image Q and hidden image Q ′ , which can be obtained by Eq. (14): (( )2 ) 𝑁−1 ∑ 𝑀−1 ∑ 𝑄(𝑖, 𝑗) − 𝑄′ (𝑖, 𝑗) 𝑀𝑆𝐸 = (14) 𝑁𝑀 𝑖=0 𝑗=0

3. Experiments and simulation analysis The experiments are performed to prove the feasibility of the proposed technology. The computer simulations with MATLAB programming are accomplished and we carry out the 3-D object scene ‘‘Dragon’’, as shown in Fig. 4(a). And the carrier 2-D image ‘‘Einstein’’, which have 512 × 512 pixels, is shown in Fig. 4(b). With our proposed technology, we generate a hologram which has all information of the 3-D object scene, as shown in Fig. 4(c). And then, we transform the hologram into a QR code and the QR code is operated by SVD method. For the carrier 2-D image, we perform a discrete wavelet decomposition operation on it firstly and then operate the low frequency sub-band 𝐿𝐿2 using SVD method. According to Eq. (12) and the value of scaling factor k is 0.20, let S replace the 𝑆1 in Eq. (10) and obtain the new 𝐿𝐿2 ′ . Finally, the final hidden 2-D image can be obtained after inverse DWT, as shown in Fig. 4(e). For the imperceptibility measure, we calculate the peak signal tonoise ratio (PSNR) and the structural similarity index (SSIM) of the hidden image. And PSNR is computed by: ( ) 2552 𝑃 𝑆𝑁𝑅 = 10 lg (13) 𝑀𝑆𝐸

where M × N denotes the size of the image and Table 1 shows the PSNR values about the scaling factor k. According to previous studies, the watermark has good transparency when the PSNR is greater than 30 dB. It can be seen from Table 1 that when 𝑘 = 0.40, the PSNR is 32.9562 dB, which is close to the critical requirement of watermark transparency of 30 dB. When 𝑘 = 0.20, the PSNR is 40.0125 dB, which is much larger than the 30 dB requirement. Therefore, in order to balance transparency and robustness, the embedding strength that can be chosen is 𝑘 = 0.20. The robustness of the proposed scheme is verified against the noise attack. A salt and pepper random noise, which usually is used to test the robustness against the noise attack, is added to the hidden image. Fig. 5 shows the results of hidden image and the extracted QR code when adding different salt and pepper noise attacks, in which the noise density is represented by d. The experimental results show that the extracted watermark has a good visual effect, and the extracted QR code can be correctly decoded. 3

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Optics Communications 453 (2019) 124340

Fig. 4. The experiment result of originality image. (a) The 3-D object scene ‘‘Dragon’’; (b) the 2-D carrier image ‘‘Einstein’’; (c) the hologram generated by the 3-D object scene; (d) the QR code of the hologram; (e) the final hidden 2-D image when k equals to 0.20.

Fig. 5. The results of hidden image and the extracted QR code when adding different salt and pepper noise attacks: hidden image when (a) 𝑘 = 0.001, (b) 𝑘 = 0.01, (c) 𝑘 = 0.10, (d) 𝑘 = 0.40; extracted QR code when (e) 𝑘 = 0.001, (f) 𝑘 = 0.01, (g) 𝑘 = 0.10, (h) 𝑘 = 0.40.

Table 2 The SSIM between the watermark extracted from the different salt and salt noise attacks and the original watermark.

The SSIM has been proved to be better objective quality assessment metric which exploits the structural similarity in the viewing field. The SSIM can be calculated by ( )( ) 2𝜇𝑥 𝜇𝑦 + 𝑐1 2𝛿𝑥,𝑦 + 𝑐2 𝑆𝑆𝐼𝑀(𝑥, 𝑦) = ( (15) )( 2 ), 𝜇𝑥 2 + 𝜇𝑦 2 + 𝑐1 𝛿𝑥 + 𝛿𝑦 2 + 𝑐2 ( )2 ( )2 𝑐1 = 𝑡1 𝑆 , 𝑐2 = 𝑡2 𝑆

The noise density d

0.001

0.01

0.10

0.20

0.30

0.40

SSIM Is QR code decoded?

0.9996 Yes

0.9980 Yes

0.9598 Yes

0.9101 Yes

0.8626 Yes

0.8102 Yes

(16) And Fig. 6 is a composite image of different intensity Gaussian noise attacks and extracted QR code. The noise variance is represented by 𝛼. It can be seen from the QR code pattern extracted from Fig. 6 that although the quality of the composite carrier image is significantly reduced as the noise intensity increases, the extracted QR code can still be correctly decoded, and the noise intensity is less than 0.2. The extracted watermark image has a pretty good visual effect. And Table 3 shows results of SSIM between the watermark extracted from the different Gaussian noise attacks and the original watermark.

where 𝜇𝑥 and 𝜇𝑦 is the average of x and y, respectively, 𝛿 is variance and 𝛿𝑥,𝑦 is covariance of x and y, S is the dynamic range of the pixel values of image, 𝑡1 = 0.01 and 𝑡2 = 0:03 by default. Its value is used to indicate the robustness of the watermarking algorithm. The closer its value is to 1, the better the watermarking effect is, and the stronger the robustness of the algorithm. Table 2 shows the SSIM between the watermark extracted from the different salt and salt noise attacks and the original watermark. It can be seen from the table that when the added noise density is less than 0.4, the extracted QR code can be correctly decoded. 4

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Optics Communications 453 (2019) 124340

Fig. 6. The results of hidden image and the extracted QR code when adding different Gaussian noise attacks: the hidden image when (a) 𝛼 = 0.001, (b) 𝛼 = 0.01, (c) 𝛼 = 0.10, (d) 𝛼 = 0.40; the extracted QR code when (e) 𝛼 = 0.001, (f) 𝛼 = 0.01, (g) 𝛼 = 0.10, (h) 𝛼 = 0.40.

Fig. 7. The structure of the reconstruction optical system.

the reconstructed object. And the SLM has 1080 × 1920 pixels and video-rate operation (60 Hz). The pixel size is 8 μm × 8 μm. The reconstruction results are shown in Fig. 8, Fig. 8(a)–(c) respective shows the result when the reconstruction plane focused on the front part of the dragon 𝑠 = 160 mm, s = 170 mm and s = 180 mm.

It can be seen from the table that when the Gaussian noise variance is less than 0.4, the SSIM of the algorithm is greater than 0.90. The method has a strong ability to resist Gaussian noise attacks. When we extract the hologram from hidden image and uploaded to SLM, the all information of the 3-D image can be displayed. For achieving it, we set an optical system. The system is shown in Fig. 7 and formed by a red laser, a filter, a collimation lens, a beam splitter (BS), an SLM, a computer and a CCD. The red laser with the wavelength of 671 nm takes turns to pass through the filter and the collimation lens. Thus, the collimated light can be generated. Then, the collimated light irradiates the SLM panel through a BS. At last, the CCD receives

4. Conclusion We present a technology for hiding and displaying 3-D image based on holography in this paper. We code the 3-D information of the image as a hologram based on layer-based theorem. Then, the hologram is transformed into a QR code and as the secret image is hidden into the

Fig. 8. The reconstructed results when the reconstruction plane focused on the front part of the dragon: (a) 𝑠 = 160 mm, (b) s = 170 mm, (c) s = 180 mm.

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Optics Communications 453 (2019) 124340

Table 3 The SSIM between the watermark extracted from the different Gaussian noise attacks and the original watermark. The noise variance

0.001

0.01

0.10

0.20

0.30

0.40

SSIM Is QR code decoded?

0.9965 Yes

0.9950 Yes

0.9871 Yes

0.9641 Yes

0.9365 Yes

0.9012 Yes

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carrier image. When we the extracted the hologram from hidden image and uploaded it to phase-only SLM, we can display all information of the 3-D image. Experimental verify the feasibility of the proposed technology. Acknowledgments This work is supported by the National Key R&D Program of China under Grant No. 2017YFB1002900 and the National Natural Science Foundation of China under Grant No. 61535007. References [1] S. Kishk, B. Javidi, Watermarking of three-dimensional objects by digital holography, Opt. Lett. 28 (3) (2003) 167–169. [2] S. Jiao, C. Zhou, Y. Shi, W. Zou, X. Li, Review on optical image hiding and watermarking techniques, Opt. Laser Technol. 109 (2019) 370–380. [3] Y. Zhang, M. Brady, S. Smith, Segmentation of brain MR images through a hidden Markov random field model and the expectation–maximization algorithm, IEEE Trans. Med. Imaging 20 (1) (2001) 45–57. [4] K. Matsushima, A. Kondoh, A wave-optical algorithm for hidden-surface removal in digitally synthetic full-parallax holograms for three-dimensional objects, Int. Soc. Opt. Photonics 5290 (2004) 90–98. [5] Y. Zhao, L. Cao, H. Zhang, D. Kong, G. Jin, Accurate calculation of computergenerated holograms using angular-spectrum layer-oriented method, Opt. Express 23 (20) (2015) 25440–25449. [6] E. Buckley, Holographic laser projection, J. Disp. Technol. 7 (3) (2011) 135–140. [7] C. Chang, J. Wu, Y. Qi, C. Yuan, S. Nie, J. Xia, Simple calculation of a computergenerated hologram for lensless holographic 3D projection using a nonuniform sampled wavefront recording plane, Appl. Opt. 55 (28) (2016) 7988–7996. [8] D. Wang, N. Li, C. Liu, Q. Wang, Holographic display method to suppress speckle noise based on effective utilization of two spatial light modulators, Opt. Express 27 (8) (2019) 11617–11625.

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