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A lensless optical security system based on computer-generated phase only masks
Optics Communications 232 (2004) 115–122 www.elsevier.com/locate/optcom
A lensless optical security system based on computer-generated phase only masks Guohai Situ *, Jingjuan Zhang Department of Physics, Graduate School of the Chinese Academy of Sciences, 19A Yuquan Road, P.O. Box 3908, Beijing 100039, China Received 9 August 2003; received in revised form 12 December 2003; accepted 5 January 2004
Abstract A lensless optical security system based on computer generated phase only masks is proposed in the present paper. These masks are located at determined positions along the direction of propagation so as to decrypt the target image. These positions coordinates are used as encoding parameters as well as the wavelength in the encryption process. Compared with previous studies, the proposed system has three significant advantages: first, it is lensless and therefore can minimize the hardware requirement and is much easier to implement. Second, the proposed system uses the wavelength and the position parameters besides the phase codes as additional keys and consequently achieves much higher security. Finally, the encrypted data can be directly transmitted via communication lines and then decrypted with the correct wavelength and the position parameters at the receiver. Applications and implementation considerations of the proposed system are also discussed. Ó 2004 Elsevier B.V. All rights reserved. PACS: 42.30.K; 42.79.S; 42.30.R Keywords: Optical encryption; Optical data processing; Phase retrieval
1. Introduction Optical information security techniques have received increasing attention. Varied systems for this purpose were proposed [1–16]. Conventional optical encryption techniques are based on the 4-f correlator architecture that uses two fixed random
*
Corresponding author. Tel.: +86-10-8825-6136; fax: +86-108825-6072. E-mail address: [email protected] (G. Situ).
phase-masks in the input and the Fourier plane, respectively, to convert the original image into a stationary white noise. As indicated in previous studies [1–4,16], however, the encrypted data is complex and have to be stored holographically. Wang et al. [12] proposed an alternative technique that iteratively encodes the original image into a phase-only mask (POM) in the Fourier plane of a 4-f correlator. This method was modified by Li et al. [13] to encrypt the image into the POM in the input plane for more convenient application, and by Chang et al. [14] and Situ and Zhang [15] into
0030-4018/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.01.002
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POMs in both planes for higher recovered quality and security. The computer generated POM(s) is (are) also used as the key(s) of the security system. It is difficult for intruders to directly find out the phase distribution of the key(s) because of the property of the encoding algorithms [13,14]. A much simpler but more secure optical security system based on computer generated POMs is proposed in the present paper. Compared with previous studies [12–15], the proposed system is lensless, and therefore minimizes the hardware requirement (i.e., the expensive Fourier lens) and is much easier to implement. Fig. 1 shows the optical setup of the proposed system. Three planes are defined as the input plane in which POM2 is located, the transform plane where POM1 is located in and which is z2 away from the input plane, and the output plane, which is z1 away from the transform plane, respectively. Distance parameters z1 and z2 are determined according to the size of the POMs to satisfy the Fresnel approximation. The target image is thus obtained at the output plane when the system is directly illuminated with an appropriate light wave whose wavelength is one of the parameter for encryption. However, the encryption process is not so straightforward. It is somewhat like a phase retrieval problem. Given the different initial phase distributions of these masks, all the corresponding solutions of the generated phase distributions are different. The generated phase distributions are also determined by the positions of the output plane and the masks, and the operation wavelength used in the encryption process. Therefore, these parameters, as well as the phase codes, can be used as the keys of the system and results in the significant enlargement of the key space. And higher security is thus achieved.
The encryption process in the proposed system is iterative and digital. While the decryption process can be implemented optically or digitally. In the case of optical implementation, the designed phase distributions can be fabricated in the form of phase only masks (either diffractive or reflective elements) with micro-optics fabrication techniques [17]. For digital implementation, on the other hand, the encryption processes is simply performed with a digital computer. This flexibility introduces another advantage: the encrypted data can be directly transmitted over the digital communication lines and then decrypted with correct keys (the positions and the wavelength parameters) at the receiver. For high secure verification, each of these masks together with its corresponding position parameter is assigned to different important person. As a result, the target pattern can be obtained at the output only with all these personsÕ authorization. The iterative algorithm is presented in Section 2. The wavelength sensitivity and the shift tolerance of the proposed system are discussed in Section 3.
2. Encryption algorithm The encryption problem is to encode the target image into the phase functions. For convenience, denote the target image to be gðx0 ; y0 Þ, the distribution of POM1 and POM2 to be exp½jw1 ðx1 ; y1 Þ and exp½jw2 ðx2 ; y2 Þ, respectively. A unity plane wave is assumed to use in the encryption process. Under the Fresnel approximation, the complex amplitude uðx1 ; y1 Þ in the transform plane with respect to exp½jw2 ðx2 ; y2 Þ is expressed as ZZ uðx1 ; y1 Þ ¼ exp½jw2 ðx2 ; y2 Þ hðx1 ; y1 ; x2 ; y2 ; z2 ; kÞdx2 dy2 ;
ð1Þ
where hðx1 ; y1 ; x2 ; y2 ; z2 ; kÞ Fig. 1. Optical setup of the lensless optical security system.
¼
exp½j2pz2 =k jp 2 2 exp ½ðx1 x2 Þ þ ðy1 y2 Þ jkz2 kz2
G. Situ, J. Zhang / Optics Communications 232 (2004) 115–122
is the point spread function (PSF) of the first stage of the system, and k represents the wavelength of the plane wave. For the sake of simplicity, we rewrite Eq. (1) in the form uðx1 ; y1 Þ ¼ FrTk fexp½jw2 ðx2 ; y2 Þ; z2 g;
ð2Þ
where FrT represents the Fresnel transform. Then, uðx1 ; y1 Þ is multiplied by the phase function of POM1 , and then Fresnel transformed to the output plane to obtain a present estimate of the target image, which is expressed as g^ðx0 ; y0 Þ ¼ FrTk fuðx1 ; y1 Þ exp½jw1 ðx1 ; y1 Þ; z1 g:
ð3Þ
Now the encryption problem can be expressed to solve Eq. (3) to obtain two phase functions exp½jw1 ðx1 ; y1 Þ and exp½jw2 ðx2 ; y2 Þ that make the amplitude of g^ðx0 ; y0 Þ equate gðx0 ; y0 Þ. It is a phase retrieval problem, which has been well demonstrated that no analytic solution is available. Hence phase retrieval algorithms must be employed. However, it is a phase retrieval problem with respect to phase distributions in three planes, i.e., the input plane, the transform plane and the output plane, in free space. Therefore conventional iterative algorithms [17] should be modified for this application as follows. Initially, the phases w1 ðx1 ; y1 Þ and w2 ðx2 ; y2 Þ are generated with a random number generator. Then they are Fresnel transformed to the output according to Eq. (3) to obtain an estimate of target image: g^ðx0 ; y0 Þ. If the amplitude of the iterative image j^ gðx0 ; y0 Þj satisfies the convergent criterion, the iteration process stops, and w1 ðx1 ; y1 Þ and w2 ðx2 ; y2 Þ generated at the present iteration are the optimized distributions. That is to say, the target image gðx0 ; y0 Þ is successfully encrypted into w1 ðx1 ; y1 Þ and w2 ðx2 ; y2 Þ. Generally, the convergent criterion can be the mean square error (MSE) between the iterative and the target image. However, in most of the security systems, a correlation operation must be carried out at the output in order to determine whether the users is legal or not, for example, it is true for an authentication verification system. Therefore, the convergent criterion can be measured by the correlation coefficient q between the iterative and the target image, which is defined by
117
n o E ½g E½g½j^ gj E½j^ gj q¼n n o n oo12 ; 2 2 E ½g E½g E ½j^ gj E½j^ gj
ð4Þ
where E½ denotes the expected value operator. In fact, the numerator of Eq. (4) is the cross-covariance between g and g^, and the denominator is the product of the standard deviations of g and g^. The correlation coefficient q gets the maximum value of 1 if g^ is perfectly correlated with g. Otherwise it is a positive number less than 1 and has a minimum value of 0 if g^ is completely uncorrelated with g. However, if the retrieved image g^ðx0 ; y0 Þ does not satisfy the convergent criterion, it is modified to satisfy the target image constraint as follows g^c ðx0 ; y0 Þ ¼ gðx0 ; y0 Þ exp½argf^ gðx0 ; y0 Þg:
ð5Þ
The subscript ÔcÕ represents the constraint operation. Then the modified function g^c ðx; yÞ is inverse Fresnel transformed backward to the transform plane in which the complex amplitude distribution is expressed as ZZ exp½j2pz1 =k u0 ðx1 ; y1 Þ ¼ g^c ðx0 ; y0 Þ jkz1 jp 2 ½ðx1 x0 Þ exp kz1 2 þ ðy1 y0 Þ dx0 dy0 : ð6Þ Analogously, u0 ðx1 ; y1 Þ can be rewritten in a simpler form u0 ðx1 ; y1 Þ ¼ IFrTk f^ gc ðx0 ; y0 Þ; z1 g;
ð7Þ
where IFrT represents the inverse Fresnel transform. Therefore the phase distributions generated at the present iteration are expressed as 0 u ðx1 ; y1 Þ 0 w1 ðx1 ; y1 Þ ¼ arg ð8Þ uðx1 ; y1 Þ and w02 ðx2 ; y2 Þ
( ¼ arg IFrTk
(
u0 ðx1 ; y1 Þ ; z2 exp½jw01 ðx1 ; y1 Þ
)) ; ð9Þ
respectively. Then w1 ðx1 ; y1 Þ and w2 ðx2 ; y2 Þ are replaced by w01 ðx1 ; y1 Þ and w02 ðx2 ; y2 Þ, respectively and substituted into Eq. (3) for the next iteration.
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The iteration process described above is carried out over and over till the convergent criterion is satisfied. This criterion may be measured by the correlation coefficient q is greater than a threshold c, which is equal to, e.g., 0.999. It is seen from (8) and (9) that both the phase distributions of the POMs are modified in each iteration. This efficient searching strategy results in very fast convergent speed. In decryption, the determined phase-masks w1 ðx1 ; y1 Þ and w2 ðx2 ; y2 Þ are placed at their corresponding positions. Once the system is illuminated by a plane wave with the wavelength k, the recovered image g^ðx0 ; y0 Þ, which resembles closely to the target image is obtained at the output and can be detected by a CCD camera.
3. Simulation and analysis The proposed algorithm retains some properties of the conventional iterative algorithm for phase retrieval [17], e.g., the final phase distributions of the generated POMs are determined by the initializations of them. In addition, it is easily seen from the algorithm description in Section 2 that the final distributions are also determined by the operation wavelength and the positions of the masks. In this section, we numerically analyze the wavelength and the axial shifting sensitivity of the proposed system. In simulation, a window of the size 256 256 with a 128 128, 256 grayscales image embedded in the center is used as the target image (Fig. 2). The sizes of both of the POMs are the same as the target image. We assume that the operation wavelength k of the incident unity plane wave for encryption is 600 nm, the position parameters z1 and z2 are 30 and 20 mm, respectively, and the aperture of the output plane is 1.6 mm. The algorithm starts with the initializations of w1 ðx1 ; y1 Þ and w2 ðx2 ; y2 Þ. Then the phase functions are transformed forward and backward alternatively through the iteration process described in Section 2. The correlation coefficient converges very fast, and is greater than 0.95 within five iterations. Then it keeps increasing slowly till the stopping criterion is satisfied. When the iteration process converges,
Fig. 2. The image used as the target image for computer simulation.
the target image is encoded into the w1 ðx1 ; y1 Þ and w2 ðx2 ; y2 Þ, which are noise-like phase distributions (Figs. 3 and 4). When the computer generated POMs are placed in the correct planes, respectively, the recovered image obtained at the output
Fig. 3. The phase distribution of w1 ðx1 ; y1 Þ.
G. Situ, J. Zhang / Optics Communications 232 (2004) 115–122
Fig. 4. The phase distribution of w2 ðx2 ; y2 Þ.
is shown in Fig. 5. The correlation coefficient between this image and the target image is 1. Assume that the POMs designed with the parameters presented in the preceding paragraph are considered. Once the POMs are determined, their phase retardance is independent of the wavelength
Fig. 5. The decrypted image obtained at the output with correct w1 ðx1 ; y1 Þ, w2 ðx2 ; y2 Þ and the corresponding keys.
119
of the decryption beam. Thus, the wavelength only influences the PSF of the system. To determinate the wavelength sensitivity, we shift the decryption wavelength from the encryption wavelength with Dk. The corresponding correlation coefficient between the target and the recovered image as the function of the wavelength difference Dk, which ranges from )50 to 50 nm, is shown in Fig. 6. It indicates that the quality of the recovered image is quite sensitive to the wavelength of the decryption beam. The correlation coefficient decreases very rapidly as Dk increases, and it drops to 0.5 when the difference is about 4 nm. The corresponding retrieved image is shown in Fig. 7. Obviously, it is hardly to distinguish the information of interesting from the noise. Therefore, the wavelength can be used as the key to recover the target image. To evaluate the axial shifting sensitivity, we use the same wavelength for encryption and decryption. The position parameters z1 and z2 are 30 and 20 mm, respectively as indicated above, and the pixel size in all these planes is therefore 0.00625 mm. For the sake of simplicity, we fix the distance between the input and the output plane to be ðz1 þ z2 Þ ¼ 50 mm, and then shift POM1 from its matched position with a distance Dz along the axis. Therefore, the distance between the input POM2 and POM1 , and POM1 and the output plane becomes ðz2 þ DzÞ and ðz1 DzÞ, respectively. Fig. 8 shows the behavior of the correlation coefficient between the recovered and the target image versus
Fig. 6. The correlation coefficient between the target and the recovered image as the function of the wavelength difference between the encryption and the decryption beam.
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Fig. 7. The retrieved image when the wavelength difference is 4 nm.
Fig. 8. The correlation coefficient between the target and the recovered image as the function of the axial offset of POM1 from its matched position.
Dz, which varies from )100 pixels to 100 pixels size. It indicates that the correlation coefficient decreases as jDzj increases. Fig. 8 also shows that the decreasing speed of the correlation coefficient between the target and the corresponding recovered image is different for different operation wavelength. The shorter the wavelength is used for encryption, the slower the decreasing speed of the correlation coefficient as jDzj increases. That is to say, the axial shifting-
tolerance with respect to short wavelength is much better than those with respect to the long. The decrypted images from the data with the encryption wavelength equates 400, 600, 800 and 1000 nm, respectively, when POM1 shifts 125 lm from its matched position are shown in Fig. 9. It is clearly shown in this figure that the axial shiftingtolerance with respect to short wavelength is much better. It can be seen from Fig. 8 that the correlation coefficients between the target image and the corresponding recovered image are 0.8846, 0.7506, 0.6017 and 0.4306, respectively. Thus short wavelength is suggested to use for optical implementation of the security system because the misalignment criterion in this case may be comparatively improved. Contrarily, when long wavelength is used for encryption, a very slight shifting of the position difference jDzj may result in low recovered quality. Thus the long wavelength is favorable in digital implementation because the recovered image in this case is more sensitive to the wavelength difference, and therefore higher security may be introduced. Now that the recovered image is sensitive to the decryption wavelength and the positions of the POMs, these parameters as well as the phase codes can be used as keys of the security system. These additional keys introduce a very useful property to the system. It is easily seen that the generated phase-functions, or essentially, the encrypted data, w1 ðx1 ; y1 Þ and w2 ðx2 ; y2 Þ are real and therefore can be directly transmitted over digital communication lines. Both optical and digital methods can be used at the receiver to recover the original image. For optical method, uploading the two distributions into the spatial light modulator located at corresponding positions and then illuminating the system with an appropriate plane wave, one can obtain the recovered image at the output. It is simpler and more flexible for digital decryption. All that need to do in this case just to transform the encrypted data w1 ðx1 ; y1 Þ and w2 ðx2 ; y2 Þ according to Eq. (3) to the output with correct keys z1 , z2 and k. It is quite difficult for intruders in the Internet to decipher the encrypted data because decrypting the information without the knowledge of the positions and the wavelength keys requires a random search in an infinite key space.
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Fig. 9. The decrypted image from the data with the encryption wavelength equates (a) 400 nm; (b) 600 nm; (c) 800 nm; and (d) 1000 nm when POM1 shifts 125 lm (20 pixels size) from its matched position. The correlation coefficients between the target image and the corresponding recovered image are: 0.8846, 0.7506, 0.6017 and 0.4306, respectively.
This property is not available in a 4-f based system described in [13–15] because there is no additional key besides the phase codes. Intruders therefore can easily recover the original data just simply by performing the following operations: Fourier transform the first phase code into the frequency plane, then multiply the complex distribution by the second phase code and finally inverse Fourier transform the result into the output plane. The technique proposed in [12] may be available for this application because only one phase code is iteratively generated, and the other is generated with a random number generator with a seed. Therefore the latter phase code may be used as the additional key of the system. However, it was shown and pointed out in [14,15] that the recovered quality is not so high because of the
searching strategy and the limitation of the solution space. The proposed system can be used for local authentication verification as well. For this application, the generated phase distributions are treated as the keys of the system and fabricated in the form of transparent (diffractive or reflective) POMs, which are embedded into the personal identification (ID) cards. Note that in such system, at least two phase-codes are generated. They should be embedded into two different ID cards to achieve high security. Thus these ID cards are assigned to two legal users with the corresponding position z1 and z2 of the phase codes as their respective personal identification number (PIN). When the phase codes together with their corresponding position parameter are inputted into the
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system, they are illuminated with the appropriate plane wave to recover an image at the output. And then this recovered image is compared with the target image stored in the database by performing a correlation operation to judge whether or not the input is legal. Intruders cannot access the system even if he could steal all those POMs but without the knowledge of the PINs. Clearly, the level of the security of the system is increasing with the amount of the generated phase codes because the key space would significantly enlarge. Therefore, if higher security is required, the target image can be encoded into more POMs with a cascaded iterative algorithm. The mathematical analysis of this case is similar to that presented in Section 2. When the iteration process stops, all of these phase codes and their corresponding positions are determined.
used as additional keys besides the phase codes of the security system. Higher security than previous techniques is thus achieved. This property would introduce some very significant advantages: the encrypted data generated with the proposed system can be directly transmitted via communication lines and then decrypted with the correct additional keys at the receiver. The proposed technique can be used as a key-sharing system as well.
Acknowledgements The authors would like to thank the anonymous referee for his constructive comments. This study was funded by the National Natural Science Foundation of China under grant 60277027.
4. Conclusion References A lensless optical security system based on computer generated POMs is proposed. The system is more compact, simpler and easier to implement owing to its minimization of the hardware requirement. We mathematically analyze the encoding process. Simulation results show that the encoding algorithm has very fast convergence and good quality for the recovered image. We also investigate the sensitivity of the recovered image to the decryption wavelength difference and the axial shifting of the POMs. Numerical experiment shows that the shorter the wavelength is used for encryption, the slower the decreasing speed of the correlation coefficient between the target and the corresponding recovered image as jDzj increases, and vice versa. Therefore short wavelength is suggested for optical decryption while long wavelength is suitable for digital implementation. Numerical result also indicates that the encryption wavelength and the positions parameters can be
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A lensless optical security system based on computer-generated phase only masks Guohai Situ *, Jingjuan Zhang Department of Physics, Graduate School of the Chinese Academy of Sciences, 19A Yuquan Road, P.O. Box 3908, Beijing 100039, China Received 9 August 2003; received in revised form 12 December 2003; accepted 5 January 2004
Abstract A lensless optical security system based on computer generated phase only masks is proposed in the present paper. These masks are located at determined positions along the direction of propagation so as to decrypt the target image. These positions coordinates are used as encoding parameters as well as the wavelength in the encryption process. Compared with previous studies, the proposed system has three significant advantages: first, it is lensless and therefore can minimize the hardware requirement and is much easier to implement. Second, the proposed system uses the wavelength and the position parameters besides the phase codes as additional keys and consequently achieves much higher security. Finally, the encrypted data can be directly transmitted via communication lines and then decrypted with the correct wavelength and the position parameters at the receiver. Applications and implementation considerations of the proposed system are also discussed. Ó 2004 Elsevier B.V. All rights reserved. PACS: 42.30.K; 42.79.S; 42.30.R Keywords: Optical encryption; Optical data processing; Phase retrieval
1. Introduction Optical information security techniques have received increasing attention. Varied systems for this purpose were proposed [1–16]. Conventional optical encryption techniques are based on the 4-f correlator architecture that uses two fixed random
*
Corresponding author. Tel.: +86-10-8825-6136; fax: +86-108825-6072. E-mail address: [email protected] (G. Situ).
phase-masks in the input and the Fourier plane, respectively, to convert the original image into a stationary white noise. As indicated in previous studies [1–4,16], however, the encrypted data is complex and have to be stored holographically. Wang et al. [12] proposed an alternative technique that iteratively encodes the original image into a phase-only mask (POM) in the Fourier plane of a 4-f correlator. This method was modified by Li et al. [13] to encrypt the image into the POM in the input plane for more convenient application, and by Chang et al. [14] and Situ and Zhang [15] into
0030-4018/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.01.002
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G. Situ, J. Zhang / Optics Communications 232 (2004) 115–122
POMs in both planes for higher recovered quality and security. The computer generated POM(s) is (are) also used as the key(s) of the security system. It is difficult for intruders to directly find out the phase distribution of the key(s) because of the property of the encoding algorithms [13,14]. A much simpler but more secure optical security system based on computer generated POMs is proposed in the present paper. Compared with previous studies [12–15], the proposed system is lensless, and therefore minimizes the hardware requirement (i.e., the expensive Fourier lens) and is much easier to implement. Fig. 1 shows the optical setup of the proposed system. Three planes are defined as the input plane in which POM2 is located, the transform plane where POM1 is located in and which is z2 away from the input plane, and the output plane, which is z1 away from the transform plane, respectively. Distance parameters z1 and z2 are determined according to the size of the POMs to satisfy the Fresnel approximation. The target image is thus obtained at the output plane when the system is directly illuminated with an appropriate light wave whose wavelength is one of the parameter for encryption. However, the encryption process is not so straightforward. It is somewhat like a phase retrieval problem. Given the different initial phase distributions of these masks, all the corresponding solutions of the generated phase distributions are different. The generated phase distributions are also determined by the positions of the output plane and the masks, and the operation wavelength used in the encryption process. Therefore, these parameters, as well as the phase codes, can be used as the keys of the system and results in the significant enlargement of the key space. And higher security is thus achieved.
The encryption process in the proposed system is iterative and digital. While the decryption process can be implemented optically or digitally. In the case of optical implementation, the designed phase distributions can be fabricated in the form of phase only masks (either diffractive or reflective elements) with micro-optics fabrication techniques [17]. For digital implementation, on the other hand, the encryption processes is simply performed with a digital computer. This flexibility introduces another advantage: the encrypted data can be directly transmitted over the digital communication lines and then decrypted with correct keys (the positions and the wavelength parameters) at the receiver. For high secure verification, each of these masks together with its corresponding position parameter is assigned to different important person. As a result, the target pattern can be obtained at the output only with all these personsÕ authorization. The iterative algorithm is presented in Section 2. The wavelength sensitivity and the shift tolerance of the proposed system are discussed in Section 3.
2. Encryption algorithm The encryption problem is to encode the target image into the phase functions. For convenience, denote the target image to be gðx0 ; y0 Þ, the distribution of POM1 and POM2 to be exp½jw1 ðx1 ; y1 Þ and exp½jw2 ðx2 ; y2 Þ, respectively. A unity plane wave is assumed to use in the encryption process. Under the Fresnel approximation, the complex amplitude uðx1 ; y1 Þ in the transform plane with respect to exp½jw2 ðx2 ; y2 Þ is expressed as ZZ uðx1 ; y1 Þ ¼ exp½jw2 ðx2 ; y2 Þ hðx1 ; y1 ; x2 ; y2 ; z2 ; kÞdx2 dy2 ;
ð1Þ
where hðx1 ; y1 ; x2 ; y2 ; z2 ; kÞ Fig. 1. Optical setup of the lensless optical security system.
¼
exp½j2pz2 =k jp 2 2 exp ½ðx1 x2 Þ þ ðy1 y2 Þ jkz2 kz2
G. Situ, J. Zhang / Optics Communications 232 (2004) 115–122
is the point spread function (PSF) of the first stage of the system, and k represents the wavelength of the plane wave. For the sake of simplicity, we rewrite Eq. (1) in the form uðx1 ; y1 Þ ¼ FrTk fexp½jw2 ðx2 ; y2 Þ; z2 g;
ð2Þ
where FrT represents the Fresnel transform. Then, uðx1 ; y1 Þ is multiplied by the phase function of POM1 , and then Fresnel transformed to the output plane to obtain a present estimate of the target image, which is expressed as g^ðx0 ; y0 Þ ¼ FrTk fuðx1 ; y1 Þ exp½jw1 ðx1 ; y1 Þ; z1 g:
ð3Þ
Now the encryption problem can be expressed to solve Eq. (3) to obtain two phase functions exp½jw1 ðx1 ; y1 Þ and exp½jw2 ðx2 ; y2 Þ that make the amplitude of g^ðx0 ; y0 Þ equate gðx0 ; y0 Þ. It is a phase retrieval problem, which has been well demonstrated that no analytic solution is available. Hence phase retrieval algorithms must be employed. However, it is a phase retrieval problem with respect to phase distributions in three planes, i.e., the input plane, the transform plane and the output plane, in free space. Therefore conventional iterative algorithms [17] should be modified for this application as follows. Initially, the phases w1 ðx1 ; y1 Þ and w2 ðx2 ; y2 Þ are generated with a random number generator. Then they are Fresnel transformed to the output according to Eq. (3) to obtain an estimate of target image: g^ðx0 ; y0 Þ. If the amplitude of the iterative image j^ gðx0 ; y0 Þj satisfies the convergent criterion, the iteration process stops, and w1 ðx1 ; y1 Þ and w2 ðx2 ; y2 Þ generated at the present iteration are the optimized distributions. That is to say, the target image gðx0 ; y0 Þ is successfully encrypted into w1 ðx1 ; y1 Þ and w2 ðx2 ; y2 Þ. Generally, the convergent criterion can be the mean square error (MSE) between the iterative and the target image. However, in most of the security systems, a correlation operation must be carried out at the output in order to determine whether the users is legal or not, for example, it is true for an authentication verification system. Therefore, the convergent criterion can be measured by the correlation coefficient q between the iterative and the target image, which is defined by
117
n o E ½g E½g½j^ gj E½j^ gj q¼n n o n oo12 ; 2 2 E ½g E½g E ½j^ gj E½j^ gj
ð4Þ
where E½ denotes the expected value operator. In fact, the numerator of Eq. (4) is the cross-covariance between g and g^, and the denominator is the product of the standard deviations of g and g^. The correlation coefficient q gets the maximum value of 1 if g^ is perfectly correlated with g. Otherwise it is a positive number less than 1 and has a minimum value of 0 if g^ is completely uncorrelated with g. However, if the retrieved image g^ðx0 ; y0 Þ does not satisfy the convergent criterion, it is modified to satisfy the target image constraint as follows g^c ðx0 ; y0 Þ ¼ gðx0 ; y0 Þ exp½argf^ gðx0 ; y0 Þg:
ð5Þ
The subscript ÔcÕ represents the constraint operation. Then the modified function g^c ðx; yÞ is inverse Fresnel transformed backward to the transform plane in which the complex amplitude distribution is expressed as ZZ exp½j2pz1 =k u0 ðx1 ; y1 Þ ¼ g^c ðx0 ; y0 Þ jkz1 jp 2 ½ðx1 x0 Þ exp kz1 2 þ ðy1 y0 Þ dx0 dy0 : ð6Þ Analogously, u0 ðx1 ; y1 Þ can be rewritten in a simpler form u0 ðx1 ; y1 Þ ¼ IFrTk f^ gc ðx0 ; y0 Þ; z1 g;
ð7Þ
where IFrT represents the inverse Fresnel transform. Therefore the phase distributions generated at the present iteration are expressed as 0 u ðx1 ; y1 Þ 0 w1 ðx1 ; y1 Þ ¼ arg ð8Þ uðx1 ; y1 Þ and w02 ðx2 ; y2 Þ
( ¼ arg IFrTk
(
u0 ðx1 ; y1 Þ ; z2 exp½jw01 ðx1 ; y1 Þ
)) ; ð9Þ
respectively. Then w1 ðx1 ; y1 Þ and w2 ðx2 ; y2 Þ are replaced by w01 ðx1 ; y1 Þ and w02 ðx2 ; y2 Þ, respectively and substituted into Eq. (3) for the next iteration.
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The iteration process described above is carried out over and over till the convergent criterion is satisfied. This criterion may be measured by the correlation coefficient q is greater than a threshold c, which is equal to, e.g., 0.999. It is seen from (8) and (9) that both the phase distributions of the POMs are modified in each iteration. This efficient searching strategy results in very fast convergent speed. In decryption, the determined phase-masks w1 ðx1 ; y1 Þ and w2 ðx2 ; y2 Þ are placed at their corresponding positions. Once the system is illuminated by a plane wave with the wavelength k, the recovered image g^ðx0 ; y0 Þ, which resembles closely to the target image is obtained at the output and can be detected by a CCD camera.
3. Simulation and analysis The proposed algorithm retains some properties of the conventional iterative algorithm for phase retrieval [17], e.g., the final phase distributions of the generated POMs are determined by the initializations of them. In addition, it is easily seen from the algorithm description in Section 2 that the final distributions are also determined by the operation wavelength and the positions of the masks. In this section, we numerically analyze the wavelength and the axial shifting sensitivity of the proposed system. In simulation, a window of the size 256 256 with a 128 128, 256 grayscales image embedded in the center is used as the target image (Fig. 2). The sizes of both of the POMs are the same as the target image. We assume that the operation wavelength k of the incident unity plane wave for encryption is 600 nm, the position parameters z1 and z2 are 30 and 20 mm, respectively, and the aperture of the output plane is 1.6 mm. The algorithm starts with the initializations of w1 ðx1 ; y1 Þ and w2 ðx2 ; y2 Þ. Then the phase functions are transformed forward and backward alternatively through the iteration process described in Section 2. The correlation coefficient converges very fast, and is greater than 0.95 within five iterations. Then it keeps increasing slowly till the stopping criterion is satisfied. When the iteration process converges,
Fig. 2. The image used as the target image for computer simulation.
the target image is encoded into the w1 ðx1 ; y1 Þ and w2 ðx2 ; y2 Þ, which are noise-like phase distributions (Figs. 3 and 4). When the computer generated POMs are placed in the correct planes, respectively, the recovered image obtained at the output
Fig. 3. The phase distribution of w1 ðx1 ; y1 Þ.
G. Situ, J. Zhang / Optics Communications 232 (2004) 115–122
Fig. 4. The phase distribution of w2 ðx2 ; y2 Þ.
is shown in Fig. 5. The correlation coefficient between this image and the target image is 1. Assume that the POMs designed with the parameters presented in the preceding paragraph are considered. Once the POMs are determined, their phase retardance is independent of the wavelength
Fig. 5. The decrypted image obtained at the output with correct w1 ðx1 ; y1 Þ, w2 ðx2 ; y2 Þ and the corresponding keys.
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of the decryption beam. Thus, the wavelength only influences the PSF of the system. To determinate the wavelength sensitivity, we shift the decryption wavelength from the encryption wavelength with Dk. The corresponding correlation coefficient between the target and the recovered image as the function of the wavelength difference Dk, which ranges from )50 to 50 nm, is shown in Fig. 6. It indicates that the quality of the recovered image is quite sensitive to the wavelength of the decryption beam. The correlation coefficient decreases very rapidly as Dk increases, and it drops to 0.5 when the difference is about 4 nm. The corresponding retrieved image is shown in Fig. 7. Obviously, it is hardly to distinguish the information of interesting from the noise. Therefore, the wavelength can be used as the key to recover the target image. To evaluate the axial shifting sensitivity, we use the same wavelength for encryption and decryption. The position parameters z1 and z2 are 30 and 20 mm, respectively as indicated above, and the pixel size in all these planes is therefore 0.00625 mm. For the sake of simplicity, we fix the distance between the input and the output plane to be ðz1 þ z2 Þ ¼ 50 mm, and then shift POM1 from its matched position with a distance Dz along the axis. Therefore, the distance between the input POM2 and POM1 , and POM1 and the output plane becomes ðz2 þ DzÞ and ðz1 DzÞ, respectively. Fig. 8 shows the behavior of the correlation coefficient between the recovered and the target image versus
Fig. 6. The correlation coefficient between the target and the recovered image as the function of the wavelength difference between the encryption and the decryption beam.
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Fig. 7. The retrieved image when the wavelength difference is 4 nm.
Fig. 8. The correlation coefficient between the target and the recovered image as the function of the axial offset of POM1 from its matched position.
Dz, which varies from )100 pixels to 100 pixels size. It indicates that the correlation coefficient decreases as jDzj increases. Fig. 8 also shows that the decreasing speed of the correlation coefficient between the target and the corresponding recovered image is different for different operation wavelength. The shorter the wavelength is used for encryption, the slower the decreasing speed of the correlation coefficient as jDzj increases. That is to say, the axial shifting-
tolerance with respect to short wavelength is much better than those with respect to the long. The decrypted images from the data with the encryption wavelength equates 400, 600, 800 and 1000 nm, respectively, when POM1 shifts 125 lm from its matched position are shown in Fig. 9. It is clearly shown in this figure that the axial shiftingtolerance with respect to short wavelength is much better. It can be seen from Fig. 8 that the correlation coefficients between the target image and the corresponding recovered image are 0.8846, 0.7506, 0.6017 and 0.4306, respectively. Thus short wavelength is suggested to use for optical implementation of the security system because the misalignment criterion in this case may be comparatively improved. Contrarily, when long wavelength is used for encryption, a very slight shifting of the position difference jDzj may result in low recovered quality. Thus the long wavelength is favorable in digital implementation because the recovered image in this case is more sensitive to the wavelength difference, and therefore higher security may be introduced. Now that the recovered image is sensitive to the decryption wavelength and the positions of the POMs, these parameters as well as the phase codes can be used as keys of the security system. These additional keys introduce a very useful property to the system. It is easily seen that the generated phase-functions, or essentially, the encrypted data, w1 ðx1 ; y1 Þ and w2 ðx2 ; y2 Þ are real and therefore can be directly transmitted over digital communication lines. Both optical and digital methods can be used at the receiver to recover the original image. For optical method, uploading the two distributions into the spatial light modulator located at corresponding positions and then illuminating the system with an appropriate plane wave, one can obtain the recovered image at the output. It is simpler and more flexible for digital decryption. All that need to do in this case just to transform the encrypted data w1 ðx1 ; y1 Þ and w2 ðx2 ; y2 Þ according to Eq. (3) to the output with correct keys z1 , z2 and k. It is quite difficult for intruders in the Internet to decipher the encrypted data because decrypting the information without the knowledge of the positions and the wavelength keys requires a random search in an infinite key space.
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Fig. 9. The decrypted image from the data with the encryption wavelength equates (a) 400 nm; (b) 600 nm; (c) 800 nm; and (d) 1000 nm when POM1 shifts 125 lm (20 pixels size) from its matched position. The correlation coefficients between the target image and the corresponding recovered image are: 0.8846, 0.7506, 0.6017 and 0.4306, respectively.
This property is not available in a 4-f based system described in [13–15] because there is no additional key besides the phase codes. Intruders therefore can easily recover the original data just simply by performing the following operations: Fourier transform the first phase code into the frequency plane, then multiply the complex distribution by the second phase code and finally inverse Fourier transform the result into the output plane. The technique proposed in [12] may be available for this application because only one phase code is iteratively generated, and the other is generated with a random number generator with a seed. Therefore the latter phase code may be used as the additional key of the system. However, it was shown and pointed out in [14,15] that the recovered quality is not so high because of the
searching strategy and the limitation of the solution space. The proposed system can be used for local authentication verification as well. For this application, the generated phase distributions are treated as the keys of the system and fabricated in the form of transparent (diffractive or reflective) POMs, which are embedded into the personal identification (ID) cards. Note that in such system, at least two phase-codes are generated. They should be embedded into two different ID cards to achieve high security. Thus these ID cards are assigned to two legal users with the corresponding position z1 and z2 of the phase codes as their respective personal identification number (PIN). When the phase codes together with their corresponding position parameter are inputted into the
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system, they are illuminated with the appropriate plane wave to recover an image at the output. And then this recovered image is compared with the target image stored in the database by performing a correlation operation to judge whether or not the input is legal. Intruders cannot access the system even if he could steal all those POMs but without the knowledge of the PINs. Clearly, the level of the security of the system is increasing with the amount of the generated phase codes because the key space would significantly enlarge. Therefore, if higher security is required, the target image can be encoded into more POMs with a cascaded iterative algorithm. The mathematical analysis of this case is similar to that presented in Section 2. When the iteration process stops, all of these phase codes and their corresponding positions are determined.
used as additional keys besides the phase codes of the security system. Higher security than previous techniques is thus achieved. This property would introduce some very significant advantages: the encrypted data generated with the proposed system can be directly transmitted via communication lines and then decrypted with the correct additional keys at the receiver. The proposed technique can be used as a key-sharing system as well.
Acknowledgements The authors would like to thank the anonymous referee for his constructive comments. This study was funded by the National Natural Science Foundation of China under grant 60277027.
4. Conclusion References A lensless optical security system based on computer generated POMs is proposed. The system is more compact, simpler and easier to implement owing to its minimization of the hardware requirement. We mathematically analyze the encoding process. Simulation results show that the encoding algorithm has very fast convergence and good quality for the recovered image. We also investigate the sensitivity of the recovered image to the decryption wavelength difference and the axial shifting of the POMs. Numerical experiment shows that the shorter the wavelength is used for encryption, the slower the decreasing speed of the correlation coefficient between the target and the corresponding recovered image as jDzj increases, and vice versa. Therefore short wavelength is suggested for optical decryption while long wavelength is suitable for digital implementation. Numerical result also indicates that the encryption wavelength and the positions parameters can be
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