Double random phase encoding using phase reservation and compression

Since double random phase encoding (DRPE) was developed by Refregier and Javidi, information security by optical means has attracted more and more attention [1–5]. Its marked advantages, such as multiple-dimensional and parallel processing, have been analyzed and illustrated. In a DRPE system, two statistically independent random phase-only masks are employed and placed in the input image plane and the Fourier domain, respectively. It has been demonstrated that the input image can be converted into stationary white noise [1–3] and a phase-only mask in the Fourier domain acts as the main security key during optical decryption. In recent years, different transforms [6–13], such as the fractional Fourier transform [6] and the Fresnel transform [7–9], have been developed and integrated into DRPE systems, and experimental parameters, such as wavelengths and distances, are also employed as security keys.

Although a number of optical infrastructures have been developed for securing information, a linear characteristic is still maintained in some DRPE systems. It has been proved that attack algorithms [14–17], such as known-plaintext attack, may be employed to attack DRPE systems under some given conditions. Security enhancement approaches [18–20], such as randomized lenses [19], have been further developed and applied for DRPE systems. However, additional or specific optical components may be requested, and the main objective of these security enhancement approaches focuses on generating more security keys, which still cannot fully eliminate the potential weaknesses of DRPE. Recently, a photon-counting strategy has been developed as an effective method for enhancing DRPE security [21]. However, the photon-counting approach may be relatively complicated in practice, and a complex-valued wavefront in the CCD plane is still required during optical decryption [21].

In this paper, we propose a novel and simple method via phase reservation and compression to enhance DRPE security. Only the phase distribution is reserved in the CCD plane, and the amplitude component is not required for optical decryption. A compressed phase distribution in the CCD plane can be generated by randomly selecting some pixels from the original phase distribution or by randomly selecting one small region from the original phase distribution. Since only noise-like distributions can be obtained even using correct security keys during optical decryption, a nonlinear correlation algorithm is further applied for authenticating the decrypted image. It is demonstrated that valid conditions for attack algorithms are broken and high security can be achieved for DRPE systems.

Figure 1 shows a schematic experimental setup for the proposed method. Phase-only masks M1 and M2 are placed in the input image plane and Fourier domain, respectively. Let M₁(x, y) and M₂(μ, ν) respectively denote phase-only masks M1 and M2, which are randomly distributed in the range [0,2π]. Here, symbols (x, y) and (μ, ν) denote the coordinates of the input image plane and phase-only mask (M2) plane, respectively. In this study, the Fourier domain is applied, however it is straightforward to apply other transform domains [6–13] in the proposed optical security system. The optical encoding process can be described by

$$\begin {eqnarray}\label {eq1} H({\xi ,\eta } ) = \mathrm {IFT}({\{ {\mathrm {FT}[{T({x,y} )M_1 ({x,y} )} ]} \}M_2 ({\mu ,\nu } )} ), \end {eqnarray} \tag{ 1 }$$

where T(x, y) denotes an input image, (ξ, η) denotes the coordinate of CCD plane, H(ξ, η) denotes a complex-valued wavefront in the CCD plane, and FT and IFT denote the Fourier transform and inverse Fourier transform, respectively. Digital holography [22, 23], i.e., off-axis or in-line, is usually employed for the recordings during optical encryption.

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It has been found in previous works [14–17] that when the complex-valued wavefront in the CCD plane and the geometrical parameters are assumed as known parameters, phase-only masks M1 and M2 might be extracted by using some attack algorithms, such as a known-plaintext attack [16]. In this study, phase reservation and compression are implemented in the CCD plane, which can effectively break the valid conditions for attack algorithms. For instance, when a known-plaintext attack [16] is applied, the complex-valued wavefront in the CCD plane (i.e., one condition) is usually assumed as a known parameter. However, the proposed method makes only the compressed phase map (in the CCD plane) accessible to authorized receivers or the attackers. The phase distribution P(ξ, η) = H(ξ, η)/|H(ξ, η)| is reserved, and the amplitude part is not required or can be omitted. Subsequently, a simple approach is developed to generate the compressed phase map $\hat {P}({\xi ,\eta } )$: random selection of some pixels from the phase distribution P(ξ, η). Alternatively, a small region can be randomly selected from the phase distribution P(ξ, η) to generate the compressed phase map $\hat {P}({\xi ,\eta })$.

During image decryption, only the compressed phase map $\hat {P}({\xi ,\eta } )$ is available to the receivers, which is Fourier transformed to the phase-only mask (M2) plane with the modulation of the conjugate of phase-only mask M2.

$$\begin {eqnarray}\label {eq2} W({\mu ,\nu } ) = \{ {\mathrm {FT}[{\hat {P}({\xi ,\eta } )} ]} \}[{M_2 ({\mu ,\nu } )} ]^\ast , \end {eqnarray} \tag{ 2 }$$

where the asterisk denotes the complex conjugate. Subsequently, wavefront W(μ, ν) is inverse Fourier transformed to the input image plane.

$$\begin {eqnarray}\label {eq3} \hat {T}({x,y} ) = \vert {\mathrm {IFT}[{W({\mu ,\nu } )} ]} \vert , \end {eqnarray} \tag{ 3 }$$

where $\hat {T}({x,y} )$ denotes the decrypted image and | | denotes the modulus operation. Since an amplitude-only input image is encoded, the phase-only mask M1 can be omitted during optical decryption.

The mean squared error (MSE) is calculated to evaluate the quality of the decrypted images.

$$\begin {eqnarray}\label {eq4} \mathrm {MSE} = \frac {1}{K}\sum \sum [{T({x,y} ) - \hat {T}({x,y} )} ]^2 , \end {eqnarray} \tag{ 4 }$$

where K denotes the total number of image pixels (i.e., 512 × 512). In addition, the peak signal-to-noise ratio (PSNR) can also be used as an effective evaluation parameter, which is calculated by [24]

$$\begin {eqnarray}\label {eq5} \mathrm {PSNR} = 10\,\log _{10} \frac {T_{\max }^2} {({1/ K} )\sum \sum {[{T({x,y} ) - \hat {T}({x,y} )} ]^2} }. \end {eqnarray} \tag{ 5 }$$

Since only the compressed phase map $\hat {P}({\xi ,\eta } )$ is available to the receivers during image decryption, noise-like distributions are generated even when using correct security keys during the decoding. We further apply correlation algorithms, i.e., nonlinear correlation [21, 25–29], to verify the decrypted image.

$$\begin {eqnarray}\label {eq6} C(x,y) &=& \vert \mathrm {IFT} (\vert \{ \mathrm {FT}[\hat {T}({x,y})] \}\{ \mathrm {FT}[T(x,y) ]\}^*\vert ^{b - 1}\nonumber \\ && {\times \,}\{ \mathrm {FT}[\hat {T}(x,y ) ] \}\{ \mathrm {FT}[T(x,y ) ]\}^*) \vert ^2, \end {eqnarray} \tag{ 6 }$$

where C(x, y) denotes the generated nonlinear correlation map, and b denotes the strength of the applied nonlinearity [21, 25–30]. The parameter b is usually evaluated by using peak-to-correlation and discrimination ratio metrics  [21, 25, 27, 28].

Compared with photon-counting DRPE [21], simple compression methods have been developed without the relatively complicated modification in our study. In addition, only compressed phase information is required for optical decryption rather than the complex-valued wavefront in photon-counting DRPE  [21]. Since only the compressed phase distribution is employed during optical decryption, experimental implementation of optical decryption becomes feasible and easy. In practice, the compressed phase distribution in the CCD plane can be embedded into one phase-only spatial light modulator. To clearly illustrate the encoding and decoding processes aforementioned, flow charts are shown in figures 2(a) and (b), respectively.

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A simulated experiment shown in figure 1 is employed for numerically illustrating the validity of the proposed method. A collimated plane wave is first generated for the illumination with light of wavelength 632.8 nm. The Fourier domain is used in the proposed optical security system, however it is straightforward to apply other transform domains [6–13], such as the Fresnel transform. Figure 3(a) shows a gray-scale input image (i.e., plaintext) with 512 × 512 pixels. Figures 3(b) and (c) show phase-only masks M1 and M2 placed in the input image plane and Fourier domain, respectively. The phase values of these masks are randomly distributed in the range [0,2π]. During optical encoding, many technologies, such as holography [22, 23], can be used to record intensity patterns, and complex-valued wavefront H(ξ, η) in the CCD plane can be correspondingly extracted.

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In this study, the phase distribution P(ξ, η) = H(ξ, η)/|H(ξ, η)| is reserved, and the proposed compression method is further applied to generate the compressed phase map $\hat {P}({\xi ,\eta } )$. During optical decryption, only the compressed phase map $\hat {P}({\xi ,\eta } )$ is available in the CCD plane, and the amplitude part is not required. Figure 4(a) shows the phase distribution P(ξ, η), and figure 4(b) shows the compressed phase map $\hat {P}({\xi ,\eta } )$ in the CCD plane. The inset in figure 4(b) shows a small enlarged part obtained from the compressed phase map $\hat {P}({\xi ,\eta } )$. In this case, only 10% of the total pixels are randomly selected from the phase distribution P(ξ, η) to generate the compressed phase map $\hat {P}({\xi ,\eta } )$.

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Figure 5(a) shows a decrypted image when the correct security keys are used during the decoding. The MSE for figure 5(a) is 1.8395 × 10⁴. As seen in figure 5(a), no information about the input image can be visually rendered. Since the decrypted image obtained by using correct keys still contains some invisible but valuable plaintext information, a correlation algorithm, such as a nonlinear one [21, 25–30], is further applied to verify the decrypted image. Figure 5(b) shows the authentication distribution, when the decrypted image in figure 5(a) is nonlinearly correlated with the input image. The strength of applied nonlinearity is set as 0.30. It can be seen in figure 5(b) that one remarkable peak is generated at the center, and the decrypted image has been effectively and correctly verified. We further test the performance of the security keys in the proposed optical encoding system. When the phase-only mask M2 is wrong, a decrypted image is obtained and shown in figure 5(c). The MSE for figure 5(c) is 1.8396 × 10⁴. Figure 5(d) shows the authentication distribution, when the decrypted image in figure 5(c) is nonlinearly correlated with the input image. Only a noisy distribution can be generated, and multiple peaks are simultaneously observed. It is illustrated in figures 5(b) and (d) that the unique characteristics of the DRPE system have been maintained and correct verification results can be generated. In the proposed optical security system, fewer or more pixels can be arbitrarily selected from the phase distribution P(ξ, η) for generating the compressed phase map $\hat {P}({\xi ,\eta } )$. Figure 6(a) shows a decrypted image when the correct keys are used and only 7% of the total pixels are randomly selected to generate the compressed phase map $\hat {P}({\xi ,\eta } )$. The MSE for figure 6(a) is 1.8408 × 10⁴. Figure 6(b) shows the corresponding authentication distribution, when the decrypted image in figure 6(a) is nonlinearly correlated with the input image. Figures 6(c) and (d) show another set of decryption and authentication results, when 15% of the total pixels are randomly selected from P(ξ, η) to generate the compressed phase map $\hat {P}({\xi ,\eta } )$. The MSE for figure 6(c) is 1.8379 × 10⁴. It seen from figures 6(a)–(d) that the proposed optical security system provides great flexibility in verifying the decrypted images without direct observation of the plaintext information.

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Since the intensity patterns recorded in the CCD plane could be contaminated by noise, the performance of the proposed method against noise contamination is further illustrated. Here, it is assumed that white noise (zero mean with 20 000 variance) is added to all the recorded intensity patterns (i.e., phase-shifting holographic system), hence the compressed phase distribution $\hat {P}({\xi ,\eta } )$ in the CCD plane is correspondingly contaminated. Figure 7(a) shows a decrypted image obtained when the correct keys are used and only 15% of the total pixels are randomly selected (under noise contamination), and figure 7(b) shows the authentication distribution corresponding to figure 7(a). The MSE and PSNR for figure 7(a) are 1.8380 × 10⁴ and 4.5913, respectively. It can be seen in figure 7(b) that the decrypted image can still be correctly verified, and the proposed method shows high robustness against noise contamination. To clearly illustrate the proposed method, several parameters due to the change in percentage of the maintained phase are also demonstrated here. Figure 7(c) shows the relationship between the percentage of maintained phase information and the PSNR, and figure 7(d) shows the relationship between the percentage of maintained phase information and the correlation peak values. It can be seen in figures 7(c) and (d) that the proposed method is feasible and effective, and a large percentage range of maintained phase information (in the CCD plane) can be designed to verify the decrypted images without direct plaintext observations.

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In figures 4–7, the compressed phase map $\hat {P}({\xi ,\eta } )$ is generated by randomly selecting some pixels from the phase distribution P(ξ, η). We find that a small phase region can also be arbitrarily cropped from the map P(ξ, η) to generate the compressed phase map $\hat {P}({\xi ,\eta } )$. Figure 8 shows a compressed phase map (180 × 180 pixels) cropped from the map P(ξ, η), which is used for optical decryption. Figure 9(a) shows a decrypted image, when the correct security keys are used during the decoding. The MSE for figure 9(a) is 1.8387 × 10⁴. Figure 9(b) shows the authentication distribution when the decrypted image in figure 9(a) is nonlinearly correlated with the input image. The strength of applied nonlinearity is also set as 0.30. It is illustrated in figure 9(b) that one remarkable peak can be generated at the center, and the decrypted image is effectively verified. When the phase-only mask M2 is wrong, a decrypted image is obtained, as shown in figure 9(c). The MSE for figure 9(c) is 1.8388 × 10⁴. Figure 9(d) shows the authentication distribution when the decrypted image in figure 9(c) is nonlinearly correlated with the input image. It can be seen in figure 9(d) that when the wrong security key is used only a noisy distribution can be generated. Figures 10(a)–(d) show more sets of optical decryption and authentication results, when different phase region sizes (i.e., 160 × 160 pixels and 150 × 150 pixels) are cropped to generate the compressed phase map $\hat {P}({\xi ,\eta } )$. The MSEs for figures 10(a) and (c) are 1.8396 × 10⁴ and 1.8401 × 10⁴, respectively. As illustrated in figures 9 and 10, high flexibility and high security can be guaranteed in the proposed optical security system.

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The advantages of the proposed method and its comparison to previous works are briefly summarized as follows: (1) only the compressed phase map is available or required for optical decryption. However, in photon-counting DRPE [21], complex-valued wavefront in the CCD plane should be used. Hence, the proposed method, i.e., phase reservation and compression, can facilitate optical information transmission or storage. (2) Since only the phase distribution is required in the CCD plane during image decryption, experimental implementation of optical decryption becomes feasible and easy. In practice, the compressed phase distribution can be embedded into a phase-only spatial light modulator during optical decoding. (3) In photon-counting DRPE [21], a photon-counting strategy should be applied to generate the compressed wavefront in the CCD plane, which might be relatively complicated compared with the simple compression approaches developed in our study. (4) In attack algorithms [15, 16], the complex-valued wavefront in the CCD plane is usually assumed as a known parameter. The proposed method can break the valid conditions for attack algorithms, since only the compressed phase map is available in the CCD plane for optical decryption. (5) No visible information about the input image can be extracted even using the correct security keys during optical decryption, which can provide an additional security layer for the DRPE system.

We have proposed a novel and simple method via phase reservation and compression for enhancing DRPE security. During optical encoding, only the compressed phase distribution is reserved in the CCD plane and the amplitude part is not required or can be omitted. During optical decoding and authentication, a nonlinear correlation algorithm is further applied to verify the decrypted image. The results illustrate that the proposed method is feasible and effective, and possesses several advantages over previous works. The developed strategy can effectively facilitate optical information transmission or storage, since only the compressed phase map is required in the CCD plane for optical decryption. Simple approaches are developed to generate the compressed phase map, and the use of a complex-valued wavefront can be avoided during image decryption. In addition, valid conditions for the attack algorithms are broken, and a new security layer is established for DRPE system since only noise-like distributions can be obtained even using the correct keys during optical decryption.

This work was supported by the Singapore Temasek Defence Systems Institute under grant TDSI/11–009/1A.