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该加密方法可以对多幅不同尺寸和类型的图像信息实现并行加密。为了验证该方法对多幅图像信息的加密性能,文中选取的八幅待加密图像信息如图1所示,分别为512×512个像素的“蛙”和“熊猫”,512×256个像素的“树”和“棋盘”,256×512个像素的“optics”和“光”,256×256个像素的“像”字和“花”,将八幅图像信息分为左右两组进行加密实验。
首先,对原始图像信息进行采样,得到对应的采样图像信息。其中,采样阵列的采样常数d由不同拓扑荷数的螺旋相位板的空间频率来确定。在圆柱坐标下,螺旋相位板的分布可以表示为:
$$ \psi (r,\theta ) = {\text{circ}}\left(\frac{r}{R}\right)\exp ({{i}}m\theta ) $$ (1) 式中:$R$为螺旋相位板的半径;$r$和$\theta $分别为极坐标系中的半径和方位角。螺旋相位板的空间频谱的振幅模量为:
$$ \left| {E(\rho ,\varphi )} \right| = \left| {{\text{FT\{ }}\psi (r,\theta )\} } \right| = \left| {\frac{{{{( - 1)}^{m + 1}}k}}{f}\exp ({{i}}m\varphi ) \int\limits_0^R {r{J_m}\left(\frac{k}{f}r\rho \right){\text{d}}r} } \right| $$ (2) 式中:$k = {{2{\pi}} \mathord{\left/ {\vphantom {{2{\pi}} \lambda}} \right. } {\lambda}}$和$f$分别为入射光的波数和傅里叶透镜的焦距;FT表示傅里叶变换;$\;\rho $和$\varphi $分别为全息图平面上的半径和方位角;${J_m}$表示第一类贝塞尔函数。由于振幅模量具有径向对称性,当方位角为零时,其分布为:
$$ \left| {E(\rho ,0)} \right| = \left| {\frac{{{{( - 1)}^{m + 1}}k}}{f}\int\limits_0^R {r{J_m}\left(\frac{k}{f}r\rho \right){\text{d}}r} } \right| $$ (3) 故在近轴极限下,不同拓扑荷数的螺旋相位板的空间频率可以用基于傅里叶变换的圆环状强度分布来表示,如图2所示。其中,采样常数d与不同拓扑荷下螺旋相位板的空间频率相对应[16],其被定义为最大振幅模量30%时所对应的像素宽度。在计算过程中,激光的波长为$ \lambda $,SLM的单个像素尺寸为$ h $。因此通过以下公式计算得到全息图的最大数值孔径:
图 2 螺旋相位板空间频率及采样阵列的示意图
Figure 2. Schematic diagram of spatial frequency and sampling array of spiral phase plates
$$ {\text{sin}}\theta {\text{ = }}\frac{\lambda }{{2h}} $$ (4) 为了将全息图所携带的全部信息进行光学传输,可计算重建全息图像的傅里叶透镜的最小的数值孔径取值,然后通过计算最大模振幅的30%所对应的像素宽度就可以得出采样常数d的取值。拓扑荷$m = 1,2, 3,4$时采样常数d大小如图3所示。
然后,利用傅里叶变换的频移特性将8个采样图像信息合成为两幅保留全息图,其获取过程示意图如图4所示。采样图像信息分别经过随机相位调制、逆傅里叶变换和频移相位调制后相干叠加构成两个OAM保留全息图$ {F_1}(u,v) $和$ {F_2}(u,v) $。
$$ {g_i}(x,y){\text{ = }}{f_i}(x,y)\exp [{{i}}2{\text{π }} \times {\text{rand}}({m_i},{n_i})] $$ (5) $$ {G_i}(u,v) = {\text{IFT[}}{g_i}(x,y)] $$ (6) $$ {F_1}(u,v) = \sum\limits_{i = 1}^4 {{G_i}(u,v)\exp [{{i}}2{\text{π }}({a_i}u + {b_i}v)} ] $$ (7) $$ {F_2}(u,v) = \sum\limits_{i = 5}^8 {{G_i}(u,v)\exp [{{i}}2{\text{π }}({a_i}u + {b_i}v)} ] $$ (8) 式中:$ {f_i}(x,y) $表示采样图像信息;$\exp [{{i}}2{\text{π }} \times {\text{rand}} ({m_i},{n_i})]$为随机相位因子;$ \text{IFT} [\cdot ] $表示逆傅里叶变换;$\exp [{i}2{\text{π }} \cdot ({a_i}u + {b_i}v)]$为频移相位因子,其中$ ({a_i},{b_i}) $表征各个图像的位移系数,具体数值与坐标轴选取以及成像平面大小有关。文中,分别为“蛙”$ ({a_{\text{1}}},{b_{\text{1}}}){\text{ = }}(1{\text{/}}3,1{\text{/}}6) $,“树”$ ({a_{\text{2}}},{b_{\text{2}}}){\text{ = }}(1{\text{/}}12,{\text{1/6}}) $,“optics”$ ({a_{\text{3}}},{b_{\text{3}}}){\text{ = }}({\text{1/3}}, - {\text{1/3}}) $,“像”$ ({a_{\text{4}}},{b_{\text{4}}}){\text{ = }}(1{\text{/}}12, - {\text{1/3}}) $,“花”$ ({a_5},{b_5}){\text{ = }}( - {\text{1/12}},{\text{1/3}}) $,“光”$ ({a_6},{b_6}){\text{ = }}( - 1{\text{/}}3,{\text{1/3}}) $,“熊猫”$ ({a_7},{b_7}){\text{ = }}( - 1{\text{/}}6, - {\text{1/6}}) $,“棋盘”$ ({a_8},{b_8}){\text{ = }}( - {\text{5/12}}, - {\text{1/6}}) $。
最后,为了在保留全息图中实现OAM选择特性,将拓扑荷$m = - 1$和$m = - 2$的螺旋相位板分别编码到两个OAM保留全息图上,这样就得到了两个OAM选择全息图,将其相干叠加构成最终的OAM复合选择全息图,即密文图像,该过程如图5所示。
图 5 OAM复合选择全息图获取过程的示意图
Figure 5. Schematic diagram of OAM composite selective hologram acquisition process
由图4和图5可知,相比于其他的多图像加密算法,该方法通过两次加密过程可以将多个不同尺寸和类型的待加密图像信息加密为单个全息图,并且密文图像中未显示任何原始信息。该加密系统具有极高的加密灵活性和极大的加密容量,不仅可以在同一拓扑荷下,设计不同的频移因子来加密一组多个图像信息,还可以利用不同拓扑荷对多组图像信息进行同时加密。该方法将OAM光束的特定拓扑荷设定为解密密钥,使得加密系统的安全性得到极大的提高。
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为了验证加密结果的有效性和鲁棒性,本节对加密结果的相关性进行了分析。图像的相关性可以表征图像相邻位置像素值的相关程度,加密图像相关性越低,加密方法的安全性越高。本节分别对原始图像信息和密文图像信息在水平、垂直和对角3个方向上的相关性系数进行了计算,计算结果如表1所示。计算公式如下:
$$ {{\rm{cov}}} (x,y) = \frac{1}{N} \times \sum\limits_{i = 1}^N {[{x_i} - E(x)} ][{y_i} - E(y)] $$ (9) $$ CC = \frac{{{{\rm{cov}}} (x,y)}}{{\sqrt {D(x)D(y)} }} $$ (10) 由表1的计算结果可见,密文图像信息各个方向上的相关性都很低,文中的加密方法有效地降低了高度相关的原始图像信息的统计特性,能够有效抵抗基于像素相关性的统计攻击,具有较高的安全性。
表 1 原始图像信息和密文图像信息的相关性系数
Table 1. Correlation coefficient of original image and ciphertext image
Image
Horizontal correlation
Vertical correlation
Diagonal correlationFrog 0.9977 0.9979 0.9963 Tree 0.9715 0.9711 0.9504 “optics” 0.8661 0.9383 0.8415 “像” 0.9073 0.9106 0.8139 Flower 0.9881 0.9991 0.9977 “光” 0.9884 0.9791 0.9716 Panda 0.9976 0.9909 0.9841 Chessboard 0.9881 0.9861 0.9743 Encrypted text −0.0013 0.0164 0.0072 -
明文敏感性是一种评价加密系统质量的常用指标,明文敏感性越好,加密系统的安全性越好。可以通过计算某一幅明文图像发生微小变化(某一像素值增大0.1和某两个像素值交换)时密文图像的像素值改变率(NPCR)和归一化平均变化强度(UACI),以此来验证文中加密方法的明文敏感性,计算结果如表2所示。计算公式如下:
$$ NPCR = \frac{{\displaystyle\sum\limits_i {\displaystyle\sum\limits_j {z(i,j)} } }}{{M \times N}} $$ (11) $$ UACI = \frac{1}{{M \times N}}\left[\sum\limits_i {\sum\limits_j {\left| {I(i,j) - {I'}(i,j)} \right|} } \right] $$ (12) 式中:当$I(i,j){\text{ = }}{I'}(i,j)$时,$z(i,j){\text{ = 0}}$;当$I(i,j) \ne {I'}(i,j)$时,$z(i,j){\text{ = 1}}$。
表 2 明文敏感性分析
Table 2. Plaintext sensitivity analysis
Index change when pixel
increases by 0.1Index changes when
two pixels are exchangedNPCR UACI NPCR UACI 99.85% 34.75% 99.84% 34.58% 由表2可知,在该加密系统中当某一明文图像信息发生上述两种的微小变化时密文像素值改变率均在99%以上,其归一化平均改变强度均在30%以上,这说明该系统具有较好的明文敏感性。
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多个图像信息的解密可以通过使用特定拓扑荷数的OAM光束和解密光路来实现。文中解密光路设置如图6所示,不同拓扑荷数的OAM光束由SLM1产生,SLM2上加载复合全息图,傅里叶透镜将频域中叠加的各个图像按照对应的频移因子进行分离,最后各个解密图像由CCD在焦平面上接收。
由于密文全息图存在OAM选择性,因此只有在包含$m = 1$和$m = 2$模式的OAM光束照射时才能解密出所有图像信息。通过使用拓扑荷为$m = 1$和$m = 2$的OAM光束以及解密光路进行多个图像信息的解密,其解密结果如图7所示。
图 7 当$m{\text{ = 1,\;2}}$的OAM光束照射时的解密结果
Figure 7. Decryption result when $m{\text{ = 1,\;2}}$ OAM beam is irradiated
将解密结果和原始图像进行对比,采用结构相似性指数SSIM和相关性系数CC来描述两图像间的相似度,SSIM和CC的计算结果如表3所示。结构相似性 (Structural SIMilarity, SSIM),衡量两张图像的相似度,通过SSIM指标可以衡量样本和预测图像之间的亮度、对比度和结构,SSIM指标更加关注图像生成的感知面的图像质量。
表 3 原始图像和解密图像的相似性
Table 3. Similarity of original image and decrypted image
Similarity index Frog Tree “optics” “像” Flower “光” Panda Chessboard SSIM 0.9703 0.9605 0.9645 0.9509 0.9829 0.9757 0.9699 0.9742 CC 0.9943 0.9932 0.9910 0.9924 0.9940 0.9922 0.9954 0.9936 $$ I(x,y) = \frac{{2{\mu _x}{\mu _y} + {c_1}}}{{{\mu _x}^2 + {\mu ^2}_y + {c_1}}} $$ (13) $$ c(x,y) = \frac{{2{\sigma _x}{\sigma _y} + {c_2}}}{{{\sigma _x}^2 + {\sigma ^2}_y + {c_2}}} $$ (14) $$ s(x,y) = \frac{{{\sigma _{xy}} + {c_3}}}{{{\sigma _x}{\sigma _y} + {c_3}}} $$ (15) 公式(13)表示的是图像亮度的相似程度,其中$\; {\mu _x} $以及$ \;{\mu _y} $分别表示图像的$ x $和$ y $的均值;公式(14)表示的所示图像的对比度相似性,其中$ {\sigma _x} $和$ {\sigma _y} $分别表示的是图像$ x $和图像$ y $的方差;而公式(15)表示图像的结构相似度,其中$ {\sigma _{xy}} $表示图像$ x $和图像$ y $的协方差,$ {c_1} $、$ {c_2} $、$ {c_3} $均为常数。将上述的公式联立,可以得到公式(16),SSIM的取值在0~1之间,SSIM的值越大,说明两种图像更相似。
$$ {\rm{SSIM}}(x,y) = \frac{{2({\mu _x}{\mu _y} + {c_1})(2{\sigma _{xy}} + {c_2})}}{{({\mu ^2}_x{\mu ^2}_y + {c_1})({\sigma ^2}_x + {\sigma ^2}_y + {c_2})}} $$ (16)
High capacity optical information encryption technology based on OAM holography and frequency shift
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摘要: 提出了一种基于轨道角动量全息(Orbital Angular Momentum, OAM)和频移的大容量光学信息加密方法。该方法实现了对多个图像信息的并行加密。首先,对多幅原始图像进行采样,采样阵列的采样间隔取决于具有不同拓扑荷数的螺旋相位的空间频率。然后,多个采样图像信息经过随机相位调制、傅里叶变换和频移相位调制后相干叠加构成轨道角动量保留全息图。最后,将不同拓扑荷的螺旋相位分别编码到轨道角动量保留全息图中,得到轨道角动量选择全息图,进行相干叠加后构成最终的单个加密全息图。解密时,轨道角动量复合选择全息图被加载到空间光调制器上,用包含特定拓扑荷数的涡旋光束照射,并经过傅里叶变换获得多个解密信息。该加密系统具有极高的加密灵活性和极大的加密容量,不仅可以在同一拓扑荷下,设计不同的频移因子来并行加密一组多个图像信息,还可以利用不同拓扑荷对多组图像信息进行加密。该方法将涡旋光束的模式设定为一个新的光学密钥,极大地提高了光学加密系统的安全性。此外,该光学加密方法中,待加密图像信息的尺寸不受空间光调制器的像元数量限制,极大地提高了光学实现信息加密的可行性和有效性。仿真实验结果表明该方法具有较高的安全性、抗噪性和抗剪切能力。Abstract:
Objective In order to improve the capacity and security of the optical encryption system, a new method based on OAM holography and frequency shift is proposed. Methods Since the orbital angular momentum eigenstates mathematically constitute a complete set of orthogonal basis vectors, the orbital angular momentum can be used to realize the encoding of multiple image information. This approach achieves parallel encryption of multiple images information using Fourier transform frequency-shift and OAM holography techniques, which eliminates the crosstalk between multiple image information. Firstly, several original images information are sampled, and the sampling constants is determined by the spatial frequency of the helical phase with different helical mode indexes, which is defined as the pixel width corresponding to 30% of the maximum amplitude modulus (Fig.3). Through this process, we obtained multiple different sampled images. The orbital angular momentum-preserving holograms are then generated by the coherent superposition of multiple sampled image information modulated by random phase, Fourier transform and frequency shift phase (Fig.4). Finally, the helical phases of different helical mode indexes are encoded into two orbital angular momentum preserving holograms, and two OAM selective holograms are obtained, they are superposed coherently to form the final OAM composite selected hologram (Fig.5). In decrypted process, the orbital angular momentum compound choice hologram is loaded onto the spatial light modulator, which is illuminated by a vortex beam containing a specific helical mode index and passes through a Fourier lens, a receiving device on the rear focal plane of the lens can receive a plurality of decrypted images (Fig.6). Results and Discussions Compared with other multi-image encryption algorithms, the proposed method can encrypt multiple image information of different sizes and types into a single hologram through two encryption processes, and no original information is displayed in the ciphertext image. The correlation of ciphertext image information in all directions is very low, which effectively reduces the statistical characteristics of highly correlated original image information, and can effectively resist statistical attacks based on pixel correlation, with high security (Tab.1, Tab.2). This encryption system has high encryption flexibility and great capacity. It can not only design different frequency shift factors to encrypt a group of multiple images information in parallel under the same helical mode index, but also has the advantages of high encryption flexibility and high encryption capacity, several groups of image information can also be encrypted by using different helical mode index. Conclusions In this method, the infinite OAM mode of the vortex beam are set as a new optical key, which greatly improves the security of the encryption system. In addition, due to the frequency-shift phase modulation, the size of the image to be encrypted is not limited by the number of pixels in the Spatial light modulator, which greatly improves the feasibility and effectiveness of optical realization of information encryption. The simulation results show that the proposed method has high safety, anti-noise and anti-shear capability (Fig.11, Fig.12). -
Key words:
- optical information encryption /
- OAM /
- holography /
- frequency shift
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表 1 原始图像信息和密文图像信息的相关性系数
Table 1. Correlation coefficient of original image and ciphertext image
Image
Horizontal correlation
Vertical correlation
Diagonal correlationFrog 0.9977 0.9979 0.9963 Tree 0.9715 0.9711 0.9504 “optics” 0.8661 0.9383 0.8415 “像” 0.9073 0.9106 0.8139 Flower 0.9881 0.9991 0.9977 “光” 0.9884 0.9791 0.9716 Panda 0.9976 0.9909 0.9841 Chessboard 0.9881 0.9861 0.9743 Encrypted text −0.0013 0.0164 0.0072 表 2 明文敏感性分析
Table 2. Plaintext sensitivity analysis
Index change when pixel
increases by 0.1Index changes when
two pixels are exchangedNPCR UACI NPCR UACI 99.85% 34.75% 99.84% 34.58% 表 3 原始图像和解密图像的相似性
Table 3. Similarity of original image and decrypted image
Similarity index Frog Tree “optics” “像” Flower “光” Panda Chessboard SSIM 0.9703 0.9605 0.9645 0.9509 0.9829 0.9757 0.9699 0.9742 CC 0.9943 0.9932 0.9910 0.9924 0.9940 0.9922 0.9954 0.9936 -
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