- Email: [email protected]
Structured-illumination-based lensless diffractive imaging and its application to optical image encryption
Optics Communications 285 (2012) 2044–2047
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Structured-illumination-based lensless diffractive imaging and its application to optical image encryption Wen Chen ⁎, Xudong Chen Department of Electrical and Computer Engineering, National University of Singapore, 4 Engineering Drive 3, Singapore 117576, Singapore
a r t i c l e
i n f o
Article history: Received 23 May 2011 Accepted 28 December 2011 Available online 9 January 2012 Keywords: Lensless diffractive imaging Structured illumination Optical image encryption Security analysis Virtual optics
a b s t r a c t In this paper, we propose a new method in the structured-illumination-based lensless diffractive imaging using variable grating pitches. When a phase grating pitch is sequentially changed, a series of diffraction patterns can be recorded by a charge-coupled device (CCD) camera. Subsequently, a phase retrieval algorithm with a rapid convergence rate is developed to recover a high-quality object from the recorded diffraction patterns. The proposed method is further applied to optical image encryption, and simulation results are presented to demonstrate validity of the proposed method. © 2012 Elsevier B.V. All rights reserved.
1. Introduction In recent years, coherent diffractive imaging [1–5] has attracted much attention in various fields, such as biology science. The diffractive imaging is developed based on a single-path beam-propagation strategy, and object information is usually reconstructed using an iterative algorithm between real and reciprocal spaces [1–5]. Gerchberg–Saxton algorithm [6] and Fienup's algorithms (such as hybrid input–output algorithm) [7] are widely considered as the prototypes of phase retrieval algorithms. A support constraint (such as an aperture) is applied in the real space, and the measured diffraction pattern is used to correct object wavefront in the reciprocal space. In practice, the iterative operation may stagnate without the convergence to a satisfactory point, since strong support constraints (such as certain special shapes or separated supports) are usually required especially for the retrieval of complexvalued objects [8]. Some phase retrieval algorithms [9–11], such as oversampling phasing [9,10] and ptychographic technique [11], have been further developed to overcome the stagnation problem and recover complex-valued objects. Recently, alternative methods for phase recovery [12,13] have been investigated for the extraction of complex-valued objects. Faulkner and Rodenburg [12] proposed to laterally move an aperture during the recordings of diffraction patterns. It required an accurate control of aperture movement to achieve proper overlapping areas. Rodrigo et al. [13] proposed to use multiple transformation angles in the gyrator transform domain
⁎ Corresponding author. Tel.: + 65 65166855; fax: + 65 67791103. E-mail address: [email protected] (W. Chen). 0030-4018/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.12.097
combined by Gerchberg–Saxton algorithm [6]. However, the configuration of lens structures for gyrator transform could be complicated and difficult. In this paper, we propose a new method, for the first time to the best of our knowledge, in the structured-illuminationbased lensless diffractive imaging using variable grating pitches. A phase retrieval algorithm is developed to recover object information from the recorded diffraction patterns. Since optical image encryption is considered as an important topic for information security [14–20], an optical cryptosystem is further developed in the Fresnel domain by using the proposed method. 2. Theoretical analysis and results Fig. 1(a) shows a schematic experimental arrangement for the structured-illumination-based lensless diffractive imaging. A phase grating is placed just before the object, and a diffraction pattern is correspondingly recorded by a charge-coupled device (CCD) camera. Wave propagation between the object plane and the CCD plane can be described by [21,22]
Oðξ; ηÞ ¼
j þ∞ þ∞ 1 2π ∫ ∫ Gðx; yÞOðx; yÞ exp −j ρ dx dy; λ −∞ −∞ ρ λ
ð1Þ
where G(x, y) is a phase grating, O(x, y) is a real or complex-valued pffiffiffiffiffiffiffiffi object, O(ξ, η) denotes object wavefront in the CCD plane, j ¼ −1; λ is the wavelength, and ρ(x, y ; ξ, η) is the distance from a point in the object plane to a point in the CCD plane. Wave propagation can be implemented by the Fresnel approximation method, angular spectrum algorithm or convolution method [20,22].
W. Chen, X. Chen / Optics Communications 285 (2012) 2044–2047
GO
CCD
masks are used in optical cryptosystem. The accuracy of a decrypted image is evaluated by correlation coefficient (CC) [20,24].
Light CC ¼
d
(a) G O M1 M2
M3
CCD
……
Light
d1
d2
d3
(b) Fig. 1. (a) A schematic experimental setup for structured-illumination-based lensless diffractive imaging using variable grating pitches; (b) a schematic experimental setup for optical image encryption based on the proposed method. G, Grating; O, Object; M1–M3, Phase-only masks; CCD, Charge-coupled device.
When grating pitch is sequentially changed, a series of diffraction intensity patterns I h(ξ, η) can be recorded and expressed as h i2 h h I ðξ; ηÞ ¼ WP G ðx; yÞOðx; yÞ ;
ð2Þ
where integer h denotes a sequence of intensity recordings (i.e., 1,2,3, ……), | | denotes a modulus operation, and symbol WP denotes wave propagation described in Eq. (1). When the series of diffraction intensity patterns and phase gratings are known, an iterative phase retrieval algorithm is developed to recover the object as follows: (1) start with a real-valued guess (constant or random) of object function in the real space; (2) multiply current guess by a phase grating function G1(x, y), and obtain an exit wave; (3) wave propagation to the CCD plane, and obtain the estimated object wavefront; (4) apply a modulus constraint [i.e., square root of the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi diffraction pattern
2045
crc O; O′ σ O σ O′
;
ð3Þ
where O′(x, y) is the decrypted image, crc denotes crosscovariance, and σ denotes standard deviation. For the sake of brevity, coordinate (x, y) is omitted in Eq. (3). We conduct a numerical experiment as shown in Fig. 1(b) to illustrate validity of the proposed method. Three diffraction intensity maps (i.e., ciphertexts) are recorded during the encryption, when phase gratings shown in Fig. 2(a)–(c) are respectively used. The grating is binary (i.e., 0 and π), and grating pitches are 64 pixels, 32 pixels and 16 pixels in Fig. 2(a)–(c), respectively. Wavelength of the plane illumination wave is 600 nm, and pixel number and pixel size of the CCD camera are 512 × 512 and 4.65 μm, respectively. Phaseonly masks M1–M3 are 2D independent maps randomly distributed in a range of [0, 2π], and axial distances of d1, d2 and d3 are set as 70 mm, 95 mm and 120 mm, respectively. During image decryption, the threshold (Ω) in the proposed phase retrieval algorithm is set as 0.0001. The plaintext (“Baboon” with 512 × 512 pixels and 8 bits) is shown in Fig. 2(d). Three ciphertexts are obtained as shown in Fig. 3(a)–(c), when phase gratings in Fig. 2(a)–(c) are respectively used. It can be seen in Fig. 3(a)–(c) that the plaintext has been fully hidden by using the proposed method, and no information about the plaintext can be observed. A decrypted image is obtained in Fig. 3(d), when security keys are correct. The CC value for Fig. 3(d) is 1, which means the plaintext being fully and accurately extracted. A relationship between the number of iterations and the CC values is further illustrated in Fig. 4. The 62 iterations are required in order to satisfy the threshold in the retrieval algorithm. It can be seen in Fig. 4 that the proposed retrieval algorithm possesses a rapid convergence rate. The performance of security keys is also analyzed during image decryption. Fig. 5(a) and (b) shows decrypted images, when a wavelength error of 5 nm and a wrong phase-only mask M2 respectively
I1 ðξ; ηÞ] to replace real amplitude of object wave-
front obtained in the step (3); (5) propagate back to the object plane and multiple by the conjugate of the phase grating function G1(x, y), and extract a new and improved object estimate; and (6) use object wavefront obtained in the step (5) as a new guess, and apply other phase grating functions [G2(x, y) and q G 3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (x, y)] and the corresponding qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi diffraction patterns [
I2 ðξ; ηÞ and
I3 ðξ; ηÞ] through sequentially
repeating the steps (2)–(5). In this investigation, three diffraction patterns (h amounts to 3) are used, and an iterative error [23,24] is calculated by using the neighboring object estimates obtained after the step (6). If the iterative error is larger than a preset threshold (Ω), object wavefront obtained in the step (6) is used as a new estimate for the next iteration [start from the step (2)]. The proposed method is further applied to optical image encryption in this study, and Fig. 1(b) shows the schematic experimental setup for an optical cryptosystem based on the proposed method. Multiple random phase-only masks (one placed just behind the object) are further used in the optical path, and three diffraction patterns (i.e., ciphertexts) are recorded during the encryption through a modification of phase grating pitch. The phase retrieval algorithm aforementioned is employed to extract a decrypted image, and random phase-only masks should also be considered during image decryption. For simplicity, only three phase-only
(a)
(b)
(c)
(d)
Fig. 2. Phase gratings with a pitch of (a) 64 pixels, (b) 32 pixels and (c) 16 pixels; (d) a plaintext.
2046
W. Chen, X. Chen / Optics Communications 285 (2012) 2044–2047
(a)
(b)
(a)
(b)
(c)
(d)
(c)
(d)
Fig. 3. (a)–(c) Diffraction intensity maps (i.e., ciphertexts) recorded by using phase gratings respectively in Fig. 2(a)–(c) during the encryption; and (d) a decrypted image using correct security keys.
exist during image decryption. The CC values obtained after 2500 iterations are 0.0004 and 0.0005 for Fig. 5(a) and (b), respectively. It can be seen in Fig. 5(a) and (b) that when security keys are wrong, correct information about the plaintext cannot be extracted. For the sake of brevity, the performance of other security keys, such as axial distances and grating pitches, is not presented here. The tolerance to noise contamination and ciphertext occlusions is further investigated. Fig. 5(c) and (d) shows decrypted images, when ciphertexts are respectively contaminated by additive random noise [signal-to-noise ratio (SNR) of 5] and occlusions (12.5%). Random noise [20] is generated by ({Mean[Ih(ξ, η)]}/SNR) × V, where Mean denotes a mean value of the ciphertext and V is a 2D variable randomly distributed in a range of [–0.5, 0.5]. The CC values after 100 iterations are 0.9625 and 0.2499 for Fig. 5(c) and (d), respectively. It can be seen in Fig. 5(c) and (d) that the quality of decrypted images degrades with ciphertext contaminations, but plaintext information can still be observed.
The proposed method can be considered as a virtual-optics technique, thus various grating pitches and more random phaseonly masks can be used which provide a huge key space and a great flexibility for optical image encryption. Some complementary algorithms, such as random shifting [17], can also be integrated into the proposed optical cryptosystem to further enhance the security. In addition, the proposed method can effectively endure the attacks [25–27], since multiple random phase-only masks are used and diffractive imaging instead of interference strategy is applied. It is worth noting that the proposed optical cryptosystem is different from conventional phase-only-mask retrieval strategies [28–30], and in optical image encryption the lensless case can be extended to other transform domains, such as fractional Fourier transform [16,31,32] and gyrator transform [13,33–35].
3. Conclusions In this paper, we have proposed a new method in the structuredillumination-based lensless diffractive imaging using variable grating pitches. The phase retrieval algorithm with a rapid convergence rate is developed to recover high-quality objects from the recorded diffraction patterns. The proposed method is further applied to optical image encryption. The results demonstrate that the proposed method is feasible and effective. The analyses also show that when the proposed opto-electronic technique is applied to encrypt the images (or data), it has a huge key space and possesses a high security. The proposed method can provide a new alternative or open up a new research perspective for optical image encryption, and can also be useful in many other applications, such as highresolution material and biological imaging.
1.1
1
CC Values
Fig. 5. Decrypted images using (a) a wrong wavelength and (b) a wrong phase-only mask M2. Decrypted images when all ciphertexts are contaminated by (c) random noise and (d) occlusions.
0.9
0.8
0.7 0
10
20
30
40
50
60
Iteration Number Fig. 4. A relationship between the number of iterations and CC values obtained during image decryption.
Acknowledgments This work was supported by the Singapore Ministry of Education (MOE) grant under Project No. MOE2009-T2-2-086.
W. Chen, X. Chen / Optics Communications 285 (2012) 2044–2047
References [1] J. Miao, P. Charalambous, J. Kirz, D. Sayre, Nature 400 (1999) 342. [2] W. McBride, N.L. O'Leary, L.J. Allen, Physical Review Letters 93 (2004) 233902. [3] M.A. Pfeifer, G.J. Williams, I.A. Vartanyants, R. Harder, I.K. Robinson, Nature 442 (2006) 63. [4] P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, F. Pfeiffer, Science 321 (2008) 379. [5] V. Elser, I. Rankenburg, P. Thibault, Proceedings of the National Academy of Sciences of the United States of America 104 (2007) 418. [6] R.W. Gerchberg, W.O. Saxton, Optik 35 (1972) 237. [7] J.R. Fienup, Applied Optics 21 (1982) 2758. [8] H.M. Quiney, K.A. Nugent, A.G. Peele, Optics Letters 30 (2005) 1638. [9] J.M. Zou, I. Vartanyants, M. Gao, R. Zhang, L.A. Nagahara, Science 300 (2003) 1419. [10] H. Jiang, D.R. Johnson, C. Song, B. Amirbekian, Y. Kohmura, Y. Nishino, Y. Takahashi, T. Ishikawa, J. Miao, Physical Review Letters 100 (2008) 038103. [11] J.M. Rodenburg, A.C. Hurst, A.G. Cullis, B.R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, I. Johnson, Physical Review Letters 98 (2007) 034801. [12] H.M.L. Faulkner, J.M. Rodenburg, Physical Review Letters 93 (2004) 023903. [13] J.A. Rodrigo, H. Duadi, T. Alieva, Z. Zalevsky, Optics Express 18 (2010) 1510. [14] P. Refregier, B. Javidi, Optics Letters 20 (1995) 767. [15] E.P. Cabré, M. Cho, B. Javidi, Optics Letters 36 (2011) 22.
[16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]
2047
G. Unnikrishnan, J. Joseph, K. Singh, Optics Letters 25 (2000) 887. B. Hennelly, J.T. Sheridan, Optics Letters 28 (2003) 269. G. Situ, J. Zhang, Optics Letters 29 (2004) 1584. X.F. Meng, L.Z. Cai, X.F. Xu, X.L. Yang, X.X. Shen, G.Y. Dong, Y.R. Wang, Optics Letters 31 (2006) 1414. W. Chen, X. Chen, Optics Express 18 (2010) 27095. W. Chen, C. Quan, C.J. Tay, Applied Physics Letters 95 (2009) 201103. J.W. Goodman, Introduction to Fourier Optics, 2nd ed. McGraw-Hill, New York, 1996. W. Chen, X. Chen, C.J.R. Sheppard, Optics Letters 35 (2010) 3817. W. Chen, X. Chen, Optics Express 19 (2011) 9008. X. Peng, P. Zhang, H. Wei, B. Yu, Optics Letters 31 (2006) 1044. X. Peng, H. Wei, P. Zhang, Optics Letters 31 (2006) 3261. W. Stallings, Cryptography and Network Security: Principles and Practice, 4th ed. Prentice Hall, New Jersey, 2006. H.T. Chang, W.C. Lu, C.J. Kuo, Applied Optics 41 (2002) 4825. H.E. Hwang, H.T. Chang, W.N. Lie, Optics Letters 34 (2009) 3917. G. Situ, J. Zhang, Optics Communications 232 (2004) 115. J. Hua, L. Liu, G. Li, Journal of the Optical Society of America A 14 (1997) 3316. Y. Zhang, C.H. Zheng, N. Tanno, Optics Communications 202 (2002) 277. Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, S. Liu, Journal of Optics 12 (2010) 035407. Z. Liu, Q. Guo, L. Xu, M.A. Ahmad, S. Liu, Optics Express 18 (2010) 12033. H. Li, Optics and Lasers in Engineering 47 (2009) 45.
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Structured-illumination-based lensless diffractive imaging and its application to optical image encryption Wen Chen ⁎, Xudong Chen Department of Electrical and Computer Engineering, National University of Singapore, 4 Engineering Drive 3, Singapore 117576, Singapore
a r t i c l e
i n f o
Article history: Received 23 May 2011 Accepted 28 December 2011 Available online 9 January 2012 Keywords: Lensless diffractive imaging Structured illumination Optical image encryption Security analysis Virtual optics
a b s t r a c t In this paper, we propose a new method in the structured-illumination-based lensless diffractive imaging using variable grating pitches. When a phase grating pitch is sequentially changed, a series of diffraction patterns can be recorded by a charge-coupled device (CCD) camera. Subsequently, a phase retrieval algorithm with a rapid convergence rate is developed to recover a high-quality object from the recorded diffraction patterns. The proposed method is further applied to optical image encryption, and simulation results are presented to demonstrate validity of the proposed method. © 2012 Elsevier B.V. All rights reserved.
1. Introduction In recent years, coherent diffractive imaging [1–5] has attracted much attention in various fields, such as biology science. The diffractive imaging is developed based on a single-path beam-propagation strategy, and object information is usually reconstructed using an iterative algorithm between real and reciprocal spaces [1–5]. Gerchberg–Saxton algorithm [6] and Fienup's algorithms (such as hybrid input–output algorithm) [7] are widely considered as the prototypes of phase retrieval algorithms. A support constraint (such as an aperture) is applied in the real space, and the measured diffraction pattern is used to correct object wavefront in the reciprocal space. In practice, the iterative operation may stagnate without the convergence to a satisfactory point, since strong support constraints (such as certain special shapes or separated supports) are usually required especially for the retrieval of complexvalued objects [8]. Some phase retrieval algorithms [9–11], such as oversampling phasing [9,10] and ptychographic technique [11], have been further developed to overcome the stagnation problem and recover complex-valued objects. Recently, alternative methods for phase recovery [12,13] have been investigated for the extraction of complex-valued objects. Faulkner and Rodenburg [12] proposed to laterally move an aperture during the recordings of diffraction patterns. It required an accurate control of aperture movement to achieve proper overlapping areas. Rodrigo et al. [13] proposed to use multiple transformation angles in the gyrator transform domain
⁎ Corresponding author. Tel.: + 65 65166855; fax: + 65 67791103. E-mail address: [email protected] (W. Chen). 0030-4018/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.12.097
combined by Gerchberg–Saxton algorithm [6]. However, the configuration of lens structures for gyrator transform could be complicated and difficult. In this paper, we propose a new method, for the first time to the best of our knowledge, in the structured-illuminationbased lensless diffractive imaging using variable grating pitches. A phase retrieval algorithm is developed to recover object information from the recorded diffraction patterns. Since optical image encryption is considered as an important topic for information security [14–20], an optical cryptosystem is further developed in the Fresnel domain by using the proposed method. 2. Theoretical analysis and results Fig. 1(a) shows a schematic experimental arrangement for the structured-illumination-based lensless diffractive imaging. A phase grating is placed just before the object, and a diffraction pattern is correspondingly recorded by a charge-coupled device (CCD) camera. Wave propagation between the object plane and the CCD plane can be described by [21,22]
Oðξ; ηÞ ¼
j þ∞ þ∞ 1 2π ∫ ∫ Gðx; yÞOðx; yÞ exp −j ρ dx dy; λ −∞ −∞ ρ λ
ð1Þ
where G(x, y) is a phase grating, O(x, y) is a real or complex-valued pffiffiffiffiffiffiffiffi object, O(ξ, η) denotes object wavefront in the CCD plane, j ¼ −1; λ is the wavelength, and ρ(x, y ; ξ, η) is the distance from a point in the object plane to a point in the CCD plane. Wave propagation can be implemented by the Fresnel approximation method, angular spectrum algorithm or convolution method [20,22].
W. Chen, X. Chen / Optics Communications 285 (2012) 2044–2047
GO
CCD
masks are used in optical cryptosystem. The accuracy of a decrypted image is evaluated by correlation coefficient (CC) [20,24].
Light CC ¼
d
(a) G O M1 M2
M3
CCD
……
Light
d1
d2
d3
(b) Fig. 1. (a) A schematic experimental setup for structured-illumination-based lensless diffractive imaging using variable grating pitches; (b) a schematic experimental setup for optical image encryption based on the proposed method. G, Grating; O, Object; M1–M3, Phase-only masks; CCD, Charge-coupled device.
When grating pitch is sequentially changed, a series of diffraction intensity patterns I h(ξ, η) can be recorded and expressed as h i2 h h I ðξ; ηÞ ¼ WP G ðx; yÞOðx; yÞ ;
ð2Þ
where integer h denotes a sequence of intensity recordings (i.e., 1,2,3, ……), | | denotes a modulus operation, and symbol WP denotes wave propagation described in Eq. (1). When the series of diffraction intensity patterns and phase gratings are known, an iterative phase retrieval algorithm is developed to recover the object as follows: (1) start with a real-valued guess (constant or random) of object function in the real space; (2) multiply current guess by a phase grating function G1(x, y), and obtain an exit wave; (3) wave propagation to the CCD plane, and obtain the estimated object wavefront; (4) apply a modulus constraint [i.e., square root of the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi diffraction pattern
2045
crc O; O′ σ O σ O′
;
ð3Þ
where O′(x, y) is the decrypted image, crc denotes crosscovariance, and σ denotes standard deviation. For the sake of brevity, coordinate (x, y) is omitted in Eq. (3). We conduct a numerical experiment as shown in Fig. 1(b) to illustrate validity of the proposed method. Three diffraction intensity maps (i.e., ciphertexts) are recorded during the encryption, when phase gratings shown in Fig. 2(a)–(c) are respectively used. The grating is binary (i.e., 0 and π), and grating pitches are 64 pixels, 32 pixels and 16 pixels in Fig. 2(a)–(c), respectively. Wavelength of the plane illumination wave is 600 nm, and pixel number and pixel size of the CCD camera are 512 × 512 and 4.65 μm, respectively. Phaseonly masks M1–M3 are 2D independent maps randomly distributed in a range of [0, 2π], and axial distances of d1, d2 and d3 are set as 70 mm, 95 mm and 120 mm, respectively. During image decryption, the threshold (Ω) in the proposed phase retrieval algorithm is set as 0.0001. The plaintext (“Baboon” with 512 × 512 pixels and 8 bits) is shown in Fig. 2(d). Three ciphertexts are obtained as shown in Fig. 3(a)–(c), when phase gratings in Fig. 2(a)–(c) are respectively used. It can be seen in Fig. 3(a)–(c) that the plaintext has been fully hidden by using the proposed method, and no information about the plaintext can be observed. A decrypted image is obtained in Fig. 3(d), when security keys are correct. The CC value for Fig. 3(d) is 1, which means the plaintext being fully and accurately extracted. A relationship between the number of iterations and the CC values is further illustrated in Fig. 4. The 62 iterations are required in order to satisfy the threshold in the retrieval algorithm. It can be seen in Fig. 4 that the proposed retrieval algorithm possesses a rapid convergence rate. The performance of security keys is also analyzed during image decryption. Fig. 5(a) and (b) shows decrypted images, when a wavelength error of 5 nm and a wrong phase-only mask M2 respectively
I1 ðξ; ηÞ] to replace real amplitude of object wave-
front obtained in the step (3); (5) propagate back to the object plane and multiple by the conjugate of the phase grating function G1(x, y), and extract a new and improved object estimate; and (6) use object wavefront obtained in the step (5) as a new guess, and apply other phase grating functions [G2(x, y) and q G 3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (x, y)] and the corresponding qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi diffraction patterns [
I2 ðξ; ηÞ and
I3 ðξ; ηÞ] through sequentially
repeating the steps (2)–(5). In this investigation, three diffraction patterns (h amounts to 3) are used, and an iterative error [23,24] is calculated by using the neighboring object estimates obtained after the step (6). If the iterative error is larger than a preset threshold (Ω), object wavefront obtained in the step (6) is used as a new estimate for the next iteration [start from the step (2)]. The proposed method is further applied to optical image encryption in this study, and Fig. 1(b) shows the schematic experimental setup for an optical cryptosystem based on the proposed method. Multiple random phase-only masks (one placed just behind the object) are further used in the optical path, and three diffraction patterns (i.e., ciphertexts) are recorded during the encryption through a modification of phase grating pitch. The phase retrieval algorithm aforementioned is employed to extract a decrypted image, and random phase-only masks should also be considered during image decryption. For simplicity, only three phase-only
(a)
(b)
(c)
(d)
Fig. 2. Phase gratings with a pitch of (a) 64 pixels, (b) 32 pixels and (c) 16 pixels; (d) a plaintext.
2046
W. Chen, X. Chen / Optics Communications 285 (2012) 2044–2047
(a)
(b)
(a)
(b)
(c)
(d)
(c)
(d)
Fig. 3. (a)–(c) Diffraction intensity maps (i.e., ciphertexts) recorded by using phase gratings respectively in Fig. 2(a)–(c) during the encryption; and (d) a decrypted image using correct security keys.
exist during image decryption. The CC values obtained after 2500 iterations are 0.0004 and 0.0005 for Fig. 5(a) and (b), respectively. It can be seen in Fig. 5(a) and (b) that when security keys are wrong, correct information about the plaintext cannot be extracted. For the sake of brevity, the performance of other security keys, such as axial distances and grating pitches, is not presented here. The tolerance to noise contamination and ciphertext occlusions is further investigated. Fig. 5(c) and (d) shows decrypted images, when ciphertexts are respectively contaminated by additive random noise [signal-to-noise ratio (SNR) of 5] and occlusions (12.5%). Random noise [20] is generated by ({Mean[Ih(ξ, η)]}/SNR) × V, where Mean denotes a mean value of the ciphertext and V is a 2D variable randomly distributed in a range of [–0.5, 0.5]. The CC values after 100 iterations are 0.9625 and 0.2499 for Fig. 5(c) and (d), respectively. It can be seen in Fig. 5(c) and (d) that the quality of decrypted images degrades with ciphertext contaminations, but plaintext information can still be observed.
The proposed method can be considered as a virtual-optics technique, thus various grating pitches and more random phaseonly masks can be used which provide a huge key space and a great flexibility for optical image encryption. Some complementary algorithms, such as random shifting [17], can also be integrated into the proposed optical cryptosystem to further enhance the security. In addition, the proposed method can effectively endure the attacks [25–27], since multiple random phase-only masks are used and diffractive imaging instead of interference strategy is applied. It is worth noting that the proposed optical cryptosystem is different from conventional phase-only-mask retrieval strategies [28–30], and in optical image encryption the lensless case can be extended to other transform domains, such as fractional Fourier transform [16,31,32] and gyrator transform [13,33–35].
3. Conclusions In this paper, we have proposed a new method in the structuredillumination-based lensless diffractive imaging using variable grating pitches. The phase retrieval algorithm with a rapid convergence rate is developed to recover high-quality objects from the recorded diffraction patterns. The proposed method is further applied to optical image encryption. The results demonstrate that the proposed method is feasible and effective. The analyses also show that when the proposed opto-electronic technique is applied to encrypt the images (or data), it has a huge key space and possesses a high security. The proposed method can provide a new alternative or open up a new research perspective for optical image encryption, and can also be useful in many other applications, such as highresolution material and biological imaging.
1.1
1
CC Values
Fig. 5. Decrypted images using (a) a wrong wavelength and (b) a wrong phase-only mask M2. Decrypted images when all ciphertexts are contaminated by (c) random noise and (d) occlusions.
0.9
0.8
0.7 0
10
20
30
40
50
60
Iteration Number Fig. 4. A relationship between the number of iterations and CC values obtained during image decryption.
Acknowledgments This work was supported by the Singapore Ministry of Education (MOE) grant under Project No. MOE2009-T2-2-086.
W. Chen, X. Chen / Optics Communications 285 (2012) 2044–2047
References [1] J. Miao, P. Charalambous, J. Kirz, D. Sayre, Nature 400 (1999) 342. [2] W. McBride, N.L. O'Leary, L.J. Allen, Physical Review Letters 93 (2004) 233902. [3] M.A. Pfeifer, G.J. Williams, I.A. Vartanyants, R. Harder, I.K. Robinson, Nature 442 (2006) 63. [4] P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, F. Pfeiffer, Science 321 (2008) 379. [5] V. Elser, I. Rankenburg, P. Thibault, Proceedings of the National Academy of Sciences of the United States of America 104 (2007) 418. [6] R.W. Gerchberg, W.O. Saxton, Optik 35 (1972) 237. [7] J.R. Fienup, Applied Optics 21 (1982) 2758. [8] H.M. Quiney, K.A. Nugent, A.G. Peele, Optics Letters 30 (2005) 1638. [9] J.M. Zou, I. Vartanyants, M. Gao, R. Zhang, L.A. Nagahara, Science 300 (2003) 1419. [10] H. Jiang, D.R. Johnson, C. Song, B. Amirbekian, Y. Kohmura, Y. Nishino, Y. Takahashi, T. Ishikawa, J. Miao, Physical Review Letters 100 (2008) 038103. [11] J.M. Rodenburg, A.C. Hurst, A.G. Cullis, B.R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, I. Johnson, Physical Review Letters 98 (2007) 034801. [12] H.M.L. Faulkner, J.M. Rodenburg, Physical Review Letters 93 (2004) 023903. [13] J.A. Rodrigo, H. Duadi, T. Alieva, Z. Zalevsky, Optics Express 18 (2010) 1510. [14] P. Refregier, B. Javidi, Optics Letters 20 (1995) 767. [15] E.P. Cabré, M. Cho, B. Javidi, Optics Letters 36 (2011) 22.
[16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]
2047
G. Unnikrishnan, J. Joseph, K. Singh, Optics Letters 25 (2000) 887. B. Hennelly, J.T. Sheridan, Optics Letters 28 (2003) 269. G. Situ, J. Zhang, Optics Letters 29 (2004) 1584. X.F. Meng, L.Z. Cai, X.F. Xu, X.L. Yang, X.X. Shen, G.Y. Dong, Y.R. Wang, Optics Letters 31 (2006) 1414. W. Chen, X. Chen, Optics Express 18 (2010) 27095. W. Chen, C. Quan, C.J. Tay, Applied Physics Letters 95 (2009) 201103. J.W. Goodman, Introduction to Fourier Optics, 2nd ed. McGraw-Hill, New York, 1996. W. Chen, X. Chen, C.J.R. Sheppard, Optics Letters 35 (2010) 3817. W. Chen, X. Chen, Optics Express 19 (2011) 9008. X. Peng, P. Zhang, H. Wei, B. Yu, Optics Letters 31 (2006) 1044. X. Peng, H. Wei, P. Zhang, Optics Letters 31 (2006) 3261. W. Stallings, Cryptography and Network Security: Principles and Practice, 4th ed. Prentice Hall, New Jersey, 2006. H.T. Chang, W.C. Lu, C.J. Kuo, Applied Optics 41 (2002) 4825. H.E. Hwang, H.T. Chang, W.N. Lie, Optics Letters 34 (2009) 3917. G. Situ, J. Zhang, Optics Communications 232 (2004) 115. J. Hua, L. Liu, G. Li, Journal of the Optical Society of America A 14 (1997) 3316. Y. Zhang, C.H. Zheng, N. Tanno, Optics Communications 202 (2002) 277. Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, S. Liu, Journal of Optics 12 (2010) 035407. Z. Liu, Q. Guo, L. Xu, M.A. Ahmad, S. Liu, Optics Express 18 (2010) 12033. H. Li, Optics and Lasers in Engineering 47 (2009) 45.