Single-Pixel Compressive Digital Holographic Encryption System Based on Circular Harmonic Key and Parallel Phase Shifting Digital Holography

Abstract

An encryption system that combines compressive sensing (CS) and two-step parallel phase shifting digital holography (PPSDH) using double random phase encoding (DRPE) is presented in this paper. The two-step PPSDH is a linear inline holographic scheme and is much suitable for encrypting the 2D/3D information in a single exposure. The distribution of random phase mask (RPM) in the DRPE is implemented using circular harmonic key which increases the security of the encryption process. In this system, the keys used to encrypt are spatial positions of the planes, wavelength, and rotation of the circular harmonics in RPMs, and CS acts as an additional key that makes the system more secure than the conventional optical encryption methods. At the transmission end, two-step PPSDH is applied to encrypt the object information in single hologram. The digital mirror device (DMD) is placed between the object and a single-pixel detector for acquiring fewer hologram measurements. At the receiver end, the single digital hologram is numerically recovered by using a CS optimization problem. The original complex object field is decrypted from the CS recovered holograms by the inversion of two-step PPSDH process with the help of the correct keys. The numerical simulations are presented for complex 2D and 3D objects to test the feasibility of the proposed encryption and decryption system. The proposed method carried out intensity and phase reconstruction of the original object field using single-pixel compressive imaging. The computer simulation results demonstrated that the encrypted information is highly secured with the rotation of the circular harmonic key. The sensitivity of the decrypted intensity and phase images is also studied with variations of the encrypted keys. The obtained results show that the proposed encryption scheme is feasible and has better security performance and robustness.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Digital holography [1–7] is a promising technique to sense and retrieve 3D object information such as amplitude and phase of the object. The optically generated and digitally sensed holograms are numerically reconstructed to obtain 3D object features. The sensed digital hologram is a real-valued digital image. The retrieved 3D information from the digital hologram is a 2D complex image that contains 3D information of the object in the form of intensity and phase. Thus, the existing 2D image processing algorithms can adapt to process 3D information from 2D digital complex images. The optically generated and digitally sensed holograms are numerically reconstructed to obtain the 3D object features. The sensed digital hologram is a real-valued digital image. Digital holography has been widely used in many applications in the areas of phase contrast imaging [8, 9], 3D microscopy [10, 11], 3D object recognition [6, 12], information security [13, 14], surface shape measurement [15–17], interferometry [18, 19], etc.

The optoelectronic and computational imagining concepts involved in digital holography enable us to sense the compressed holographic data by digital sensors. Later on, it can be reconstructed with good accuracy using computational techniques. The compressive sensing (CS) techniques [20–25] can adapt well to digital holography as the process involves the detection of holographic data in the transform domain. This opens up new avenues for combining compressive data sensing with information security or encryption. Thus, the sensing and computing modalities of digital holography have complemented the CS framework. It has emerged as a secured CS 3D information system with a high data transmission rate and less storage space. The CS [20–25] is a paradigm shift in signal sampling theory that deals with reconstruction of the original signal from fewer samples than those of the Nyquist sampling rate. CS is an iterative procedure to reconstruct the original signal from the incomplete linear measurements by exploiting the sparsity of the signal. In the CS theory, the most commonly used sparse representations of the signal are FFT, DCT, Wavelet, etc. and help in accurate reconstruction of the original signal. CS framework is applied in holographic encryption for sampling and compression in order to reduce the hologram acquisition data and electronic data-processing load at sensors.

In conventional digital holographic encryption/decryption methods, the encrypted information is transmitted at the Nyquist sampling rate. The sender and receiver use a secret key to decrypt the encrypted information. The transmission of hologram requires a large amount of data storage, but the transmission channel bandwidth is limited. Traditionally, holographic encryption methods are implemented based on double random phase encoding (DRPE) [26–31], joint transform correlator (JTC) [32, 33], and ghost imaging (GI) [34], along with the encryptions based on the Fresnel domain [27, 29, 35, 36], Fourier domain, or fractional Fourier domain [37–40]. The CS based encryption methods have the additional advantage that the encryption depends on the measurement matrix (sampling mask) used for the sensing process which will enhance the security level and act as an additional key, which is not used in the conventional optical encryption systems. CS can be implemented in the optical domain using a digital micromirror device (DMD) and a single-pixel detector [41]. A DMD device is a reflective spatial light modulator introduced by Texas instruments that contain many micromirrors. The light incident on the DMD is modulated with the sampling matrix, and then a single-pixel detector records the measurements depending on the orientation of the mirrors [41, 42]. A single-pixel detector sequentially measures the light field that is reflected from the DMD device. The sensed information of a single-pixel detector ($K\ll M$ measurements) is converted from analog to digital information using A/D convertor, and it transmits K random linear measurements of the object under recording. The encrypted information is numerically reconstructed using the CS optimization algorithm [20–25] from fewer measurements. A single-pixel detector can reduce the size and complexity of the sensor at the transmitting end.

In recent years, CS-based optical encryption methods have been increasing the demand for many optical information processing applications [41–53]. The idea of the single-pixel imaging was first proposed by Duarte et al. [41]. Bromberg et al. [43] have demonstrated a real-time pseudo-thermal GI using a combination of the DMD and single-pixel detector. Di et al. [44] have proposed multiple image encryption based on single-pixel compressive holography to encrypt the holograms. The decryption process of each image was carried out using total variation minimization problem. Clemente et al. [45] have adapted a phase shifting digital holography (PSDH) technique with a single-pixel detector to demonstrate the compressive holographic encryption. This method has experimentally reconstructed the amplitude of the ophthalmic lens by retaining only 20% of the hologram pixels. Li et al. [46, 47] have numerically presented a compressive optical image encryption method for inline PSDH by adopting the single-pixel imaging. In this work, the phase shifted digital holograms were acquired using single-pixel detector. In the reconstruction process, the encrypted holograms were reconstructed by retaining various hologram measurements (such as 60%, 80%, and 90%). Finally, the intensity of the original object wavefield was decrypted from the reconstructed holograms by the inversion of PSDH process. Leihong et al. [48] have experimentally presented compressive ghost imaging encryption based on PSDH. The simulation results have shown that the CS scheme improved the security and reconstruction quality of the intensity image. Wang et al. [49] have presented single-pixel compressive holographic encryption by modifying the Mach–Zehnder interferometer setup for multiple-3D-object imaging based on multiple interferences. The numerical simulation results have shown the accurate reconstruction of original 3D object intensity information from encrypted hologram measurements. It has been demonstrated that the cryptosystem provides high encryption capacity and robustness with multiple interferences. Recently, Du et al. [50] have proposed an efficient CS based optical image encryption system using single-pixel imaging. In this method, the input image is encrypted using one random phase mask (RPM) with various recording distances as one-dimensional vector key. It has been numerically presented that the one RPM key at different recording distances minimized the key storage requirement of data transmission and also enhanced the robustness of the encryption system. The encryption systems [44–50] were proposed based on inline schemes such as four-step, three-step, or two-step PSDH. Moreover, the phase reconstruction of the encrypted object field is not presented in the single-pixel compressive holography. The strength of digital holography is the ability to reconstruct the phase information and has great applications in 3D imaging of moving objects. Although the phase reconstruction is accurate in classical PSDH, it is not suitable for moving objects because it involves multiple exposures. On the other hand, to acquire instantaneous 3D image of object wavefield, parallel phase shifting digital holography (PPSDH) [54–56] was proposed. The PPSDH is a single exposure scheme that can be used for imaging moving objects or static objects. In the present work, single-pixel compressive imaging is demonstrated based on PPSDH and circular harmonic keys in DRPE scheme for improving the security level and storage efficiency of holographic data. The encrypted hologram measurements can be realized with much less criteria than those of the Nyquist sampling.

In this paper, a CS based encryption method to encrypt the complex object information by using linear two-step PPSDH and a circular harmonic key in DRPE is demonstrated. The holographic encryption system is implemented using a phase-only key with the random spatial distribution of circular harmonics in the Fresnel domain to improve the system security performance. The single-pixel compressive imaging is adapted with two-step PPSDH for compression of the encrypted holograms in the optical domain. During the encryption process, the two RPMs, the measurement matrix, the distance, and the wavelength of Fresnel transform are encryption keys. Compared with the traditional encryption scheme, the transmitted information can be significantly reduced with single-shot exposure method, and the circular harmonic key is associated with RPM, which improves the security of the system. In the proposed system, the quality of reconstructed hologram is guaranteed with much less rate than that of the Nyquist sampling. The proposed system is validated using computer simulations to show the proof of the concept. The robustness and key space security of the proposed encryption system have been verified with minimal pixel detection. The CS reconstruction framework was implemented using the Total Variation Minimization by Augmented Lagrangian and Alternating Direction Algorithm (TVAL3) [24]. The efficacy of the proposed cryptosystem has been analyzed using various performance metrics, and also the key space security and robustness of our method have been examined.

2. Proposed Encryption and Decryption Scheme Using a Circular Harmonic Key in DRPE

The schematic representation of the optical encryption arrangement of DRPE using the circular harmonic key in the Fresnel domain is shown in Figure 1(a). In this method, the RPM is expressed in terms of polar coordinates using a circular harmonic key [57, 58] to enhance the security of the optical system. To obtain the circular harmonic key, the Cartesian coordinates $x,y$ are converted to the polar coordinates $\rho ,\phi$ using $x=\rho \cos \phi$ and $y=\rho \sin \phi$. Thus, the RPM key can be expressed in the Fresnel domain as$$\begin{array}{}\text{(1)}& P\rho ,\phi =\exp i\Phi \rho ,\phi ,\end{array}$$where $\mathrm{\Phi }$ contains the distribution of random values. The key can be decomposed into circular harmonics [56] with the position $\rho ,\phi$ as the center, which is expressed as follows:$$\begin{array}{}\text{(2)}& \Phi \rho \cos \phi ,\rho \sin \phi =\Phi \rho ,\phi ={\displaystyle \sum _{m=-\infty }^{\infty }{\Phi }_m\rho \exp im\phi },\end{array}$$where m is the order of circular harmonics and ${\mathrm{\Phi }}_m\rho$ can be calculated as follows:$$\begin{array}{}\text{(3)}& {\Phi }_m\rho =\frac{1}{2\pi }{\displaystyle {\int }_0^{2\pi }\Phi \rho ,\phi \exp im\phi \text{d}\phi }\text{.}\end{array}$$

[figure(s) omitted; refer to PDF]

Now, the RPM using a circular harmonic key in the Fresnel domain is defined as follows:$$\begin{array}{}\text{(4)}& P_m\rho ,\phi =\exp i\Re {\Phi }_m\rho ,\phi ,\end{array}$$where $\Re$ is the real part of ${\mathrm{\Phi }}_m\rho ,\phi \text{;\hspace{0.17em}}{\mathrm{\Phi }}_m$ contains the distribution of random values and polar coordinates $\rho =\sqrt{x^2+y^2}$. The circular harmonic given in (4) is rotation-symmetric, and ${\mathrm{\Phi }}_m\rho$ highly depends on the position $\rho ,\phi$. Now, the first ${\text{RPM}}_1$ is expressed as follows:$$\begin{array}{}\text{(5)}& P_m{\rho }_1,{\phi }_1=\exp i\Re {\Phi }_m{\rho }_1,{\phi }_1={\text{RPM}}_1,\end{array}$$and similarly, the second ${\text{RPM}}_2$ can be expressed as follows:$$\begin{array}{}\text{(6)}& Q_m{\rho }_2,{\phi }_2=\exp i\Re {\Phi }_m{\rho }_2,{\phi }_2={\text{RPM}}_2\text{.}\end{array}$$

The inclusion of circular harmonic function [57, 58] in the random phase masks imparts additional degrees of freedom of encoding such as order of harmonics $m$, radius ${\rho }_1,{\rho }_2$, and rotation angle ${\phi }_1,{\phi }_2$ to strengthen the security level of the encryption system. Therefore, the proposed encryption scheme with keys in polar coordinates is a solution to the problem of variance in the decryption process. The degree of rotation angle and the order of the circular harmonic key in the RPMs increase the complexity of DRPE process, thus breaking the keys.

The encryption process involves three different planes, i.e., the input plane, the intermediate Fresnel transform plane, and the output plane. The optical system is illuminated with a plane wave of wavelength λ, and the encrypted complex object information is obtained at the output plane in the Fresnel domain. The object $O{x}_i,y_i$ which is located in the input plane $x_i,y_i$ is bonded with RPM₁. When the plane wave propagates through the system, the random phase mask distribution $P_m{\rho }_1,{\phi }_1$ gets multiplied by $O{x}_i,y_i$ which results in primary random phase encoding. The propagation of this field to the plane at a distance $d_1$ from the input plane and the resulting complex Fresnel field is denoted by $U{x}^{\prime },y^{\prime }$ as given in (7). At distance $d_1$, the field $U{x}^{\prime },y^{\prime }$ gets multiplied by the second random phase mask $Q_m{\rho }_2,{\phi }_2$ resulting in second random phase encoding in the Fresnel plane. This field is further propagated to the output plane by a distance $d_2$ to obtain the Fresnel encrypted field $G{x}_0,y_0$ as given in (8). The free space propagation corresponding to each plane is modeled by a Fresnel transform with the corresponding distance.$$\begin{array}{}\text{(7)}& U{x}^{\prime },y^{\prime };d_1=\frac{e^{ik{d}_1}}{i\lambda d_1}{\displaystyle \iint O{x}_i,y_i{P}_m{\rho }_1,{\phi }_1}\exp \frac{i\pi }{\lambda d_1}{x^{\prime }-x_i}^2+{y^{\prime }-y_i}^2\text{d}x\text{d}y,\text{(8)}& G{x}_0,y_0;d_2=\frac{e^{ik{d}_2}}{i\lambda d_2}{\displaystyle \iint U{x}^{\prime },y^{\prime }}{Q}_m{\rho }_2,{\phi }_2\exp \frac{i\pi }{\lambda d_2}{x_0-x^{\prime }}^2+{y_0-y^{\prime }}^2\text{d}x\text{d}y.\end{array}$$

For simplicity, (7) and (8) can be written as follows:$$\begin{array}{}\text{(9)}& G{x}_0,y_0=\mathrm{ℱ}r{T}_{\lambda ,d_2}{Q}_m{\rho }_2,{\phi }_2\cdot \mathrm{ℱ}r{T}_{\lambda ,d_1}O{x}_i,y_i\cdot P_m{\rho }_1,{\phi }_1.\end{array}$$

Now, decryption process can be expressed as follows:$$\begin{array}{}\text{(10)}& O{x}_i,y_i=\mathrm{ℱ}r{T}_{\lambda ,d_1}^{-1}\mathrm{ℱ}r{T}_{\lambda ,d_2}^{-1}G{x}_0,y_0\cdot Q_m^{\ast }{\rho }_2,{\phi }_2{P}_m^{\ast }{\rho }_1,{\phi }_1,\end{array}$$where $\mathrm{ℱ}rT$ is forward Fresnel transform and $\mathrm{ℱ}r{T}^{-1}$ is inverse Fresnel transform. The decryption process is the same as the encryption shown in (10), but in the reverse direction with complex conjugate of the random phase masks and inverse Fresnel transforms. The Fresnel transform is inverted using the correct keys to obtain the original complex object information.

The 3D complex object used in the encryption system is shown in Figure 1(b). The 3D object construction can be considered as the integration of different sections of complex planes with certain features separated by a small distance $d_0$ which is analogous to the depth information of the object. Computationally, this is realized by combining the front and back plane with features as shown in Figure 1(b) using the Fresnel transform between the planes corresponding to a distance $d_0$. Then, 3D object optical encryption process can be expressed as follows:$$\begin{array}{}\text{(11)}& G{x}_0,y_0=\mathrm{ℱ}r{T}_{\lambda ,d_2}{Q}_m{\rho }_2,{\phi }_2\cdot \mathrm{ℱ}r{T}_{\lambda ,d_1}{P}_m{\rho }_1,{\phi }_1\cdot O_2{x}_i^{\prime },y_i^{\prime }\cdot \mathrm{ℱ}r{T}_{\lambda ,d_0}{O}_1{x}_i,y_i.\end{array}$$

Similarly, the 3D complex object decryption process is given in the following equations:$$\begin{array}{}\text{(12)}& O_2{x}_i^{\prime },y_i^{\prime }=\mathrm{ℱ}r{T}_{\lambda ,d_1}^{-1}\mathrm{ℱ}r{T}_{\lambda ,d_2}^{-1}G{x}_0,y_0.Q_m^{\ast }{\rho }_2,{\phi }_2\cdot P_m^{\ast }{\rho }_1,{\phi }_1,\text{(13)}& O_1{x}_i,y_i=\mathrm{ℱ}r{T}_{\lambda ,d_0}^{-1}{O}_2{x}_i^{\prime },y_i^{\prime },\end{array}$$where $d_0$ is the distance between two planes, and $O_1{x}_i,y_i$ and $O_2{x}_i^{\prime },y_i^{\prime }$ are the decrypted 3D object fields at the front and back plane, respectively. Two important features have to be considered for optical encryption techniques. Firstly, the encrypted field must be resistant to attacks, and secondly, the end user has to decrypt the information without any difficulty at the receiver end. An authentic user with the knowledge of RPMs and their circular harmonic key distributions can only decrypt the encrypted information. Thus, circular harmonics improve the security of the optical encryption and decryption process. In this method, the decryption procedure significantly increases the key space security with the operating wavelength, and the spatial positions of the phase mask between the adjacent planes are additional keys for the robust decryption.

The proposed optical encryption system consists of a single-pixel compressive holographic imaging system based on two-step PPSDH that encrypts the complex object information using DRPE and circular harmonic keys. Figure 2 shows the schematic of the proposed optical setup using Mach–Zehnder interferometric recording geometry. The light beam from the laser is divided into two parts using a polarizing beam splitter (PBS). The half wave plates are used to control the intensity and polarization of the interfering light beams. In the object arm, the complex object information is encrypted using DRPE in the Fresnel domain using two RPMs and circular harmonic keys. A phase array device PSLM is used in the reference wave arm to give the required phase shifts to the reference wave in an interleaved manner as mentioned in Section 3. The interference field formed by the encrypted and reference beams is sampled using a DMD by modulating it with a chosen measurement matrix (i.e., sampling mask) for a specific sample ratio. The output of the DMD is coupled to a single-pixel photodetector using a convex lens. Thus, the incomplete measurements of the encrypted single digital Fresnel hologram on the photodetector are obtained. The mathematical model of the proposed encryption and decryption using PPSDH and single-pixel compressive imaging is presented in Section 3.

[figure(s) omitted; refer to PDF]

3. Two-Step PPSDH Approach for Proposed Encryption and Decryption Using Single-Pixel Compressive Imaging

In this section, we present mathematical framework for numerical demonstration of single-pixel digital holographic scheme based on two-step PPSDH and DRPE, in which the random phase masks use circular harmonic key in the Fresnel domain. The flow diagram of the proposed encryption and decryption procedure is shown in Figure 3. The optical encryption technique uses DRPE in the Fresnel domain based on circular harmonic key as explained in Section 2. Let us consider that the input object field $O{x}_i,y_i$ of size $M\times N$ to be encrypted is multiplied by RPM₁. Then, Fresnel transform is performed through propagation distance $d_1$ and a complex Fresnel field is obtained as described in the following equation:$$\begin{array}{}\text{(14)}& U{x}^{\prime },y^{\prime }=\mathrm{ℱ}r{T}_{\lambda ,d_1}O{x}_i,y_i\cdot {\text{RPM}}_1,\end{array}$$where $\mathrm{ℱ}rT$ represents Fresnel transform. $U{x}^{\prime },y^{\prime }$ field is obtained after RPM1 is multiplied by RPM2. Then, another Fresnel transform is performed, and the obtained encrypted Fresnel field in the DMD plane can be expressed as in the following equation:$$\begin{array}{}\text{(15)}& G{x}_0,y_0=\mathrm{ℱ}r{T}_{\lambda ,d_2}U{x}^{\prime },y^{\prime }.{\text{RPM}}_2.\end{array}$$

[figure(s) omitted; refer to PDF]

Here, RPM₁ and RPM₂ are two independent random phase masks. Now, linear two-step PPSDH [54, 55, 58, 59] is applied for obtaining single-shot inline hologram on the DMD plane, in which a phase array device such as spatial light modulator is used to give phase shifts to the reference wave in an interleaved manner with a periodic phase of $\theta =0,-\pi /2$. The complex encrypted field $G{x}_0,y_0$ is interfered by the reference wave $R_{\theta }{x}_0,y_0$. The single-shot inline hologram $H_{\theta }{x}_0,y_0$ generated on the DMD plane can be expressed as follows:$$\begin{array}{}\text{(16)}& H_{\theta }{x}_0,y_0={G{x}_0,y_0+R_{\theta }{x}_0,y_0}^2,\end{array}$$where$$\begin{array}{}\text{(17)}& R_{\theta }{x}_0,y_0=\frac{R_0{x}_0,y_0\text{\hspace{0.17em}when\hspace{0.17em}}x,y=\text{even,even}}{R_{-\pi /2}{x}_0,y_0\text{\hspace{0.17em}when\hspace{0.17em}}x,y=\text{odd,even}},\hspace{1em}\text{For\hspace{0.17em}},x_0,y_0=0\text{\hspace{0.17em}to\hspace{0.17em}}N-1.\end{array}$$

Therefore, the digital hologram on the DMD plane is governed by the following equation:$$\begin{array}{}\text{(18)}& H_{\theta }{x}_0,y_0=\frac{H_0{x}_0,y_0;0\text{\hspace{0.17em}when\hspace{0.17em}}x,y=\text{even},\text{\hspace{0.17em}even}}{H_{-\pi /2}{x}_0,y_0;-\pi /2\text{\hspace{0.17em}when\hspace{0.17em}}x,y=\text{odd},\text{\hspace{0.17em}even}},\hspace{1em}\text{For},x_0,y_0=0\text{\hspace{0.17em}to}N-1.\end{array}$$

Now, the measurement matrix $\mathrm{\Psi }$ in the DMD is modulated with single-shot hologram $H_{\theta }{x}_0,y_0$. The random measurement matrix is generated numerically using pseudorandom generator sequences of 0 and 1. Then, the modulated light field reflected from a DMD is measured by a single-pixel detector through $M$ computations of the measurement matrix, and $M\times 1$ linear measurements $Y$ are obtained. The output of the single-pixel detector [41, 59] can be numerically obtained by computing the inner product of the interference filed $H_{\theta }$ and the $M$ computations of the random measurement matrix $\mathrm{\Psi }$ with various sample ratios in the DMD plane:$$\begin{array}{}\text{(19)}& Y=\Psi \cdot H_{\theta },\end{array}$$where $Y\in R^{M\times 1}$ is incomplete measurement data, $\mathrm{\Psi }\in R^{M\times N}$ is random vector generated in the DMD, and $H_{\theta }\in R^{N\times 1}$ is interference filed on the DMD plane.The encrypted incomplete linear measurements are transmitted through the channel and are decrypted at the receiver end. Both sender and receiver share secret keys to decrypt the encrypted information. In the encryption process, the linear measurements $K\ll M$ samples with $K$ nonzero entries are considered less than the Nyquist sampling rate. In the decryption process, first the CS framework is used to recover the original single-shot hologram $H_{\theta }$ from incomplete measurements $Y$ by solving optimization problem [24].$$\begin{array}{}\text{(20)}& {\tilde{H}}_{\theta }=\underset{H_{\theta }}{\underbrace{\min }}{\displaystyle \sum _{i=1}^{N^2}{D_i{H}_{\theta }}_1\text{\hspace{0.17em}subject\hspace{0.17em}to\hspace{0.17em}}Y=\Psi \cdot H_{\theta }},\end{array}$$where $D_i$ denotes discrete gradient of the vector $H_{\theta }$ at position $i$. Here, ${\tilde{H}}_{\theta }$ denotes the recovered single-shot hologram using CS method. Now, the individual phase shift holograms ${\tilde{H}}_0{x}_0,y_0$ and ${\tilde{H}}_{-\pi /2}{x}_0,y_0$ are separated from single hologram with periodic loss of pixels, and then the separated holograms’ values of the vacant pixels are interpolated to obtain the new guess values. The complex Fresnel field $G{x}_0,y_0$ on the DMD plane is obtained using Meng’s two-step PSDH formula [43, 60] as given in (21)–(24) and then inverts the encryption process to reconstruct the original object field.$$\begin{array}{}\text{(21)}& G{x}_0,y_0=\frac{{\tilde{H}}_0{x}_0,y_0-a{x}_0,y_0-i{\tilde{H}}_{-\pi /2}{x}_0,y_0-a{x}_0,y_0}{2A_r},\end{array}$$where$$\begin{array}{}\text{(22)}& a{x}_0,y_0=0.5\ast v-{v^2-2w}^{0.5},\text{(23)}& v={\tilde{H}}_0{x}_0,y_0+{\tilde{H}}_{-\pi /2}{x}_0,y_0+2A_r^2,\text{(24)}& w={\tilde{H}}_0{x_0,y_0}^2+{\tilde{H}}_{-\pi /2}{x_0,y_0}^2+4A_r^2.\end{array}$$

The retrieved complex field $G{x}_0,y_0$ is inverse Fresnel propagated by a distance $d_2$ and multiplied by conjugate of RPM₂ to obtain original field $U{x}^{\prime },y^{\prime }$:$$\begin{array}{}\text{(25)}& U{x}^{\prime },y^{\prime }=\mathrm{ℱ}r{T}_{\lambda ,d_2}^{-1}G{x}_0,y_0\cdot {\text{RPM}}_2^{\ast }.\end{array}$$

To reconstruct the original encrypted object filed, $U{x}^{\prime },y^{\prime }$ is inverse-Fresnel-transformed by the distance $d_1$ and then multiplied by the conjugate phase of the RPM₁.$$\begin{array}{}\text{(26)}& O{x}_i,y_i=\mathrm{ℱ}r{T}_{\lambda ,d_1}^{-1}U{x}^{\prime },y^{\prime }\cdot {\text{RPM}}_1^{\ast }.\end{array}$$

The sample ratio of the measurement matrix is calculated using the formula ${C/L}^{\ast }100$, where C is the sample size of compression and L is the size of the original object $M^{\ast }N$. We reduced the sample ratio parameter “C” to study the performance of the hologram compression. The higher the sample size “C,” the higher the image quality. To measure the quality of the decrypted complex images, we calculated mean square error (MSE) and relative error (RE) between the original object image and decrypted image, expressed as follows:$$\begin{array}{}\text{(27)}& \text{MSE}=\frac{1}{M\times N}{\displaystyle \sum _{x_i=1}^M{\displaystyle \sum _{y_i=1}^N\tilde{O}{x}_i,y_i-O{x_i,y_i}^2}},\text{(28)}& \text{RE}=\frac{{\displaystyle {\sum }_{x_i=1}^M{\displaystyle {\sum }_{y_i=1}^N{\tilde{O}{x}_i,y_i-O{x}_i,y_i}^2}}}{{\displaystyle {\sum }_{x_i=1}^M{\displaystyle {\sum }_{y_i=1}^N{O{x}_i,y_i}^2}}}\text{,}\end{array}$$where M and N are the size of the original object, $\tilde{O}{x}_i,y_i$ is the decrypted complex object wave with compression, and $O{x}_i,y_i$ is the original complex object wave. It can be seen from encryption and decryption system that five keys are used: RPM keys with angles ${\phi }_2$ and ${\phi }_2$, distance between two adjacent planes $d_1$ and $d_2$, measurement matrix $\mathrm{\Psi }$, and wavelength $\lambda$, and the radii of circular harmonic ${\rho }_1$ and ${\rho }_2$ values stored in the encryption process act as the additional keys in the decryption process. Thus, the proposed system improves the security of the encryption effectively. In the proposed technique, the two-step PPSDH and circular harmonic key in DRPE are combined to improve the reconstruction accuracy and security performance of the decrypted information. The main idea of the digital holographic technique integrated with the single-pixel detector and CS theory is to obtain both the intensity and phase information of a complex 2D or 3D scene under recording.

4. Simulation Experiments and Discussion

To verify the proposed scheme, two numerical simulations were conducted. The parameters used in the simulations were the wavelength $\lambda =632.8\text{\hspace{0.17em}nm}$; the order of circular harmonic key $m=2$; the rotation angles and radii of the two circular harmonic keys: ${\phi }_1=3^{°}$, ${\phi }_2=6^{°}$ and ${\rho }_1=6.38$, ${\rho }_2=8.29$; and the distance between two adjacent planes is $d_1=d_2=40\text{\hspace{0.17em}mm}$ from the DMD plane.

4.1. 2D Complex Object Encryption and Decryption Results

A binary 2D complex object field “E” of size 64 × 64 was used in the simulation. The intensity of the 2D object was taken as 0.5, and the corresponding phase was considered as $e^{1j}$. In order to visualize the quality of the phase in the reconstruction process, we have chosen constant phase value in the object function. The original complex object field is encrypted using (9) in the proposed system. The original intensity and the phase of the input complex object field are shown in Figures 4(a) and 4(b), respectively. The circular harmonic key of order $m=2$ was used in the random phase mask as shown in Figure 4(c). The encrypted Fresnel field at the DMD plane was produced by using Fresnel transform equations (7)-(8). The measurement matrix $\mathrm{\Psi }$ was used in the simulation with random sequences of 0 and 1. The encrypted complex Fresnel field from the object path was interfered with the retarded reference wave to obtain the interference filed and then modulated with the measurement matrix. The simulated single-shot inline digital Fresnel hologram on DMD plane is shown in Figure 4(d).

[figure(s) omitted; refer to PDF]

At the receiver end, the compressed single-shot digital hologram is recovered with correct $\mathrm{\Psi }$ matrix by solving the unconstrained CS optimization equation as given in (20). The recovered digital holograms from 50% measurements after interpolation are shown in Figures 4(e) and 4(f). The complex Fresnel field was retrieved from the recovered digital holograms using (21)–(24) and inverted the DRPE process as specified in (10). When one of RPM₂ is incorrect, the decrypted intensity and phase distribution are shown in Figures 5(a) and 5(b), respectively. A small change in the decryption distance $d_1=41\text{\hspace{0.17em}mm}$ used in the reconstruction process results in noisy intensity and phase distribution as shown in Figures 5(c) and 5(d). Figures 5(e) and 5(f) show the intensity and phase of the incorrect measurement matrix which is replaced with the original matrix $\mathrm{\Psi }$. The decrypted simulation results with correct keys are shown in Figures 6 and 7.

[figure(s) omitted; refer to PDF]

4.2. 3D Complex Object Encryption and Decryption Simulation Results

The encryption and decryption simulation procedure of the 3D complex object is explained in Section 2. To simulate the 3D complex object, we have considered two transparent objects, “D” and “H,” made of binary images. The light propagates through the first object plane and obtains the features of the first complex object “D.” Then, the obtained features of first object plane are propagated through a distance of $d_0=5\text{\hspace{0.17em}mm}$ using Fresnel propagation (Fresnel transform) and obtain another complex object features “H” in the second plane. Therefore, finally the 3D complex object information is obtained immediately after the second plane which contains depth information of the objects. The intensity and complex phase of the two objects are taken as 1.0 and $e^{1j}$, respectively. The simulation process of DRPE and compression of the interferograms are carried out as explained in Section 4.1. Finally, the 3D complex object scene at two different sections can be reconstructed back with single digital hologram using CS algorithm and then invert the two-step PPSDH process with original keys.

The decrypted results with incorrect keys are shown in Figure 8. The intensity and phase distribution of 3D complex object scene are decrypted at first plane with one of the incorrect RPM₂ keys. The decrypted intensity and phase distribution are shown in Figures 8(a) and 8(b). Figures 8(c) and 8(d) shows the intensity and phase distribution of incorrect distance key, $d_1=41\text{\hspace{0.17em}mm}$, was used for decrypting the complex object at first plane. When an incorrect $\mathrm{\Psi }$ is replaced with measurement matrix, the intensity and phase distribution of 3D object scene decrypted at first plane are shown in Figures 8(e) and8(f).

[figure(s) omitted; refer to PDF]

4.3. Compressive Ratio Performance

Figure 6 shows the relationship between MSE and sample ratio of the decrypted intensity and phase images for different sample ratios. From Figure 6, it can be seen that when the compression ratio is less than 30% measurements than its intensity, phase reconstruction accuracy is feasible for both the simulations, as shown in Figures 7(a)–7(c) and Figures 9(a) and 9(b). Its corresponding phase images are shown in Figures 10(a)–10(c) and Figures 11(a) and 11(b). When the sample ratio is between 30% and 40% measurements, then the decrypted intensity and phase images’ accuracy is acceptable, as shown in Figures 7(c), 10(c), 9(c) and 9(d), and 11(c) and 11(d), respectively.

[figure(s) omitted; refer to PDF]

When the sample ratio is greater than 40% measurements, then the intensity and phase reconstruction quality is similar to the original image, as shown in Figures 7(e) and 7(f), 10(e) and 10(f), 9(e) and 9(f), and 11(e) and 11(f), respectively. Figures 7–11 show the decrypted intensity and phase distribution of both 2D and 3D object simulation cases from various sample ratios. If the correct keys such as wavelength, propagation distance, RPMs, and measurement matrix are used, then the decrypted intensity and phase images for both 2D and 3D simulation cases are shown in Figures 7–11. If the different wrong keys are used in the decryption process, the quality of the decrypted intensity and phase images are affected and fail to reconstruct the originalobject wavefield as shown in Figures 12–18.

[figure(s) omitted; refer to PDF]

4.4. Key Space Security and Sensitivity Analysis

Figures 12–18 show the sensitivity of the keys with a small deviation $\delta$ when the correct keys fail to decrypt the original complex object and completely looks like a noisy image. It can be seen that the RE plot value reaches minimum only when the decrypted keys are the same as the encrypted keys; otherwise, the RE value is maximum. The quality of the decrypted complex object is high when RE value is minimum; otherwise, it appears like a noisy image. Figures 12–18 show the RE plot of the decrypted complex object with deviation in the decrypted keys for both 2D and 3D simulation cases.

From Figures 12 and 13, it can be observed that the RE plot decreases to zero sharply only when the decrypted distance is the same as the simulation recording distance. Figures 12(b) and 13(b) show that the decrypted 2D complex object looks like a noisy image when a wrong distance key, $d_1=\hspace{0.17em}{d}_2=41\text{\hspace{0.17em}mm}$, is used. On the other hand, Figures 12(c), 12(d), 13(c), and 13(d) show the decrypted complex object of the 3D object with wrong distance key, $d_1=\hspace{0.17em}{d}_2=41\text{\hspace{0.17em}mm}$, at first plane and second plane, respectively.

It can be seen from Figures 14 and 15 that the RE plot value decreases to zero sharply with a small variation in the angle of $\pm {0.1}^{°}$ for both simulation cases. When the wrong angle of the RPM₁ key ${\phi }_1={3.1}^{°}$ is used instead of the correct key value $3^{°}$, the decrypted results are shown in Figure 14. Similarly, when the wrong angle of another RPM₂ key ${\phi }_2={6.1}^{°}$ is used to decrypt the complex object for both the simulation cases, it fails to reconstruct original object wave as shown in Figure 15.

It can be seen from Figures 16 and 17 that the RE plot value decreases to zero sharply with a small variation in the radius of $\pm 1.0$ for both 2D and 3D simulation cases. The decrypted results for the incorrect radius of the RPM₁ key when ${\rho }_1=7.38$ are shown in Figure 16. Similarly, the wrong radius of the other mask RPM₂ key, when ${\rho }_2=9.29$, was used for decryption, and it failed to reconstruct the original object wave, as shown in Figure 17.

The sensitivity of the recording wavelength was also studied. The wrong wavelength of $\lambda =632.81\text{\hspace{0.17em}nm}$ was used, and its corresponding decrypted complex object result is shown in Figure 18. Here, it can be seen that the RE plot of the wavelength decreases to zero sharply with a small deviation of $\pm 0.$01 nm as shown in Figure 18(a). From the above results, it can be concluded that the proposed compressive digital Fresnel holographic encryption system is sensitive to the small variations in the decrypted keys such as measurement matrix, distance, wavelength, and rotation of the RPM using circular harmonic key. Therefore, the increase in the sensitivity of the keys results in the difficulty of duplicating the keys in the decryption process.

5. Conclusion

In this paper, an efficient 2D and 3D information security system is proposed using single-pixel compressive digital holography. The proposed system combines CS and digital holography for complex data encryption using minimal pixel detection. The hologram encryption process uses a DRPE scheme in the Fresnel domain using circular harmonic key, and this scheme enhances the security performance of the decryption process with the variation of the rotation key as presented in the simulation results. In addition, the measurement matrix and other digital holographic secret keys make the system more secure than the conventional method. The two-step PPSDH technique and CS based system are combined to improve the resolution of the reconstructed images. Due to the single-pixel imaging and PPSDH, the 3D information transmission and data storage are effectively reduced. The numerical simulation results show that the proposed cryptosystem has capability to decrypt the intensity and phase information of the object wave accurately from highly compressed digital Fresnel hologram. To quantify the reconstructed complex images, MSE and RE are analyzed for different compression ratios and different key attacks. A small deviation in the decrypted key leads to a considerable increase in the RE value and thus failure to reconstruct the original object information. It is verified from the results that the encryption system has high security and robustness and can be used in many information security applications.

Acknowledgments

This work was supported by Science and Engineering Research Board (SERB), Department of Science and Technology, Government of India, under the sanction order no. CRG/2018/003906.

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