Optical hiding based on single-input multiple-output and binary amplitude-only holograms via the modified Gerchberg-Saxton algorithm

Lina Zhou; Yin Xiao; Zilan Pan; Yonggui Cao; Wen Chen; Wen Chen

doi:10.1364/OE.428564

1. Introduction

In recent years, we have witnessed a great stride made in optical technologies and their applications in the field of optical security for data transmission and data storage [1–4]. Optical encryption and hiding usher in a new era of information security, which promotes the evolution of cryptography and steganography. The prevalence of optical means widely applied for cryptography and steganography can be ascribed to its major properties, i.e., parallel processing and intrinsic parameters of optical setups (e.g., amplitude, phase, polarization and wavelength) [2–4]. Double random phase encoding (DRPE) [5] was the first optical method implemented by using two random phase-only masks respectively located at the input plane and the Fourier plane. Over the decades, DRPE scheme has been developed and extended to various domains, e.g., Fresnel domain, Gyrator domain and fractional Fourier domain [6–8]. Many studies related to optical cryptography and steganography have also been conducted based on computational ghost imaging, computer-generated hologram, diffractive imaging and other optical techniques [1–4,9–12].

The optical technologies aforementioned are usually applied to optical single-image encoding and hiding, which are inadequate for the coming age of data explosion. To meet the requirement of large-data hiding and transmission, optical multiple-image encoding and hiding have to be studied [13–20]. In a comparison with optical single-image hiding, optical multiple-image hiding conceal some different secret images into a host image, which overcomes the challenges existing in data storage and data transmission. Current optical multiple-image hiding schemes originate from multiplexing techniques. Multiplexing provides the possibility to store big data on a single crystal, and massive hiding strategies have been developed attributing to sensitivity to various optical parameters. Similarly, optical multiplexing was proposed to implement optical multiple-image encoding and hiding by exploiting sensitiveness of optical parameters [13–20]. Over the past decades, optical multiple-image encoding and hiding have drawn great attention owing to some striking properties, e.g., the enlarged hiding capacity, facile transmission and effective storage etc. For the multiplexing, information of secret images is superposed to generate a host image, and then each secret image can be retrieved from the host image by using its corresponding optical keys or parameters. For instance, wavelength multiplexing was proposed for optical multiple-image encoding, in which different images can be retrieved by using the corresponding wavelengths. Similarly, position multiplexing, polarization multiplexing, spectral multiplexing, angular multiplexing, aperture modulation and theta modulation based multiple-image encoding and hiding have been developed by using the intrinsic properties of optical techniques [14–20]. However, these multiplexing strategies are restricted by optical calibration problems and the limited number of secret images to be hidden. In addition, optical multiple-image encoding and hiding have also been reported by using iterative phase retrieval algorithms [6,18]. However, these iterative algorithms are usually complex and time-consuming, and the hiding capacity is still limited. Hence, traditional optical multiple-image hiding methods suffer from a limited number of secret images to be processed, difficult calibration of optical setups and the relatively complicated hiding procedure. It is still impossible to retrieve a large number of different secret images from a single host image. Moreover, it is highly desirable that optical implementation complexity can be reduced.

In this paper, a new optical hiding method based on single-input multiple-output (SIMO) and binary amplitude-only holograms (AOHs) via the modified Gerchberg-Saxton algorithm (MGSA) is proposed. The SIMO means that one and only one host image is transmitted, and multiple outputs (i.e., different secret images) can be retrieved from the single host image by using security keys or parameters. In the proposed method, the host image is a binary AOH which is directly generated by using MGSA. In contrast to traditional optical multiple-image hiding methods, the proposed method provides an approach for generating the host image without a superposition of all secret images. Note that the proposed method is different from image sharing and watermarking concepts. When size of the host image is large enough, a large number of different secret images can be retrieved from the host image as verified by our simulations and optical experiments. Moreover, eavesdropping analyses, noise contamination and occlusion contamination have also been numerically and experimentally conducted to demonstrate feasibility and robustness of the proposed method. It is demonstrated by the simulations and optical experiments that the proposed method using the SIMO and binary AOHs via MGSA can greatly enhance the hiding capacity with the reduced optical implementation complexity.

2. Optical hiding based on SIMO and binary AOHs via MGSA

2.1. MGSA

It was shown in conventional Gerchberg-Saxton algorithm [6,18] that the approximate phase distribution of a target image can be iteratively retrieved through a randomly initialized phase distribution and a predefined propagation function, e.g., Fourier transform and Fresnel transform. The Gerchberg-Saxton algorithm has also been developed to retrieve the propagation function. However, there are few studies to apply or modify the Gerchberg-Saxton algorithm to retrieve amplitude information. Amplitude information is usually omitted in conventional Gerchberg-Saxton algorithms, and it is meaningful to explore the field of amplitude holography [19]. In the Gerchberg-Saxton algorithm [6,18], wave propagation from the hologram plane to the Fourier plane can be described by

(1)$$E\textrm{(}x,y\textrm{)} = \frac{{exp\textrm{(}jkf\textrm{)}}}{{j\lambda f}}exp\left[ {\frac{{jk}}{{2f}}\textrm{(}{x^2} + {y^2}\textrm{)}} \right] \times FT \{{exp}[j\varphi \textrm{(}{x_h},{y_h}\textrm{)}]\},$$

where (x_h, y_h) and (x, y) denote coordinates respectively in the hologram plane and the Fourier plane, φ(x_h, y_h) denotes phase distribution of a hologram, FT denotes Fourier transform, k denotes wavenumber, λ denotes the wavelength, $j = \sqrt { - 1} \textrm{,}$ and f denotes focal length. To remove imaginary part for generating amplitude-only hologram, conjugate of the hologram can be used. Then, wave propagation from the hologram plane to the Fourier plane can be described by

(2)$$FT \textrm{\{ }exp[j\varphi \textrm{(}{x_h},{y_h}\textrm{)}]\textrm{ + }exp[ - j\varphi \textrm{(}{x_h},{y_h}\textrm{)}]\textrm{\} } = FT \textrm{\{ 2cos[}\varphi \textrm{(}{x_h},{y_h}\textrm{)]\} ,}{\kern 1pt}$$

where the image obtained at the Fourier plane contains the retrieved (real) image and a twin image.

Figure 1(a) shows a flow chart of the proposed MGSA for generating AOHs. It starts with an inverse Fourier transform (IFT) of a target image T with a randomly initialized phase φ_n to generate an intermediate phase φ’_n. Subsequently, phase pattern φ’_n is constrained by a cosine function to generate an AOH cos(φ’_n), and then the generated AOH cos(φ’_n) is Fourier transformed to obtain an updated phase pattern φ_n+₁. Finally, the target image T with the updated phase pattern φ_n+₁ is further inverse Fourier transformed in a new iteration. Once the preset number of iterations is completed, the final AOH is used as the approximated AOH of the target image T. In the case that amplitude retrieval would be too challenging to optically implement, binarization operation is further proposed in the iterative process to reduce the complexity as shown in Fig. 1(b). The generated AOH cos(φ’_n) is binarized by using a simple method as follows:

(3)$$\cos (\varphi {^{\prime}_{nm}}) = \left\{ \begin{array}{l} 1{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} if {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \cos (\varphi {^{\prime}_{nm}}) \ge \overline {\sum\limits_m {\textrm{co}s(\varphi {^{\prime}_{nm}})} } \\ 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} if {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \cos (\varphi {^{\prime}_{nm}}) < \overline {\sum\limits_m {\textrm{co}s(\varphi {^{\prime}_{nm}})} } \end{array} \right.,$$

where m denotes each pixel position in the generated AOH cos(φ’_n), ∑ denotes summation of all the pixel values, and $\overline {\sum\limits_m {\textrm{co}s(\varphi {^{\prime}_{nm}})} }$ denotes mean value of all the pixel values. The binarized hologram φ’’_n is further Fourier transformed in the iterative process to update the phase pattern. After the preset number of iterations is completed, an approximated binary AOH of the target image T is generated.

Fig. 1. Flow charts of the proposed MGSA: (a) Flow chart of the proposed MGSA for generating AOHs, and (b) flow chart of the proposed MGSA for generating binary AOHs.

Abstract

1. Introduction

2. Optical hiding based on SIMO and binary AOHs via MGSA

2.1. MGSA

2.2. Redundancy of binary AOHs

2.3. Optical hiding using SIMO and binary AOHs via MGSA

3. Simulations and optical experiments

3.1. Simulation results and discussion

3.2. Security and robustness analyses

3.3. Optical experiments and result discussion

4. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (25)

Equations (6)

Optics Express