May 1, 2013 / Vol. 38, No. 9 / OPTICS LETTERS
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Optical image encryption via ptychography Yishi Shi,1,2,* Tuo Li,1 Yali Wang,1 Qiankun Gao,1 Sanguo Zhang,3 and Haifei Li4 1 2
College of Material Science and Opto-Electronic Technology, University of Chinese Academy of Sciences, Beijing 100049, China
State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing 100093, China 3
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China 4 Ordnance Repair Factory of Air Force, Guangzhou 510500, China *Corresponding author:
[email protected] Received February 5, 2013; revised March 10, 2013; accepted March 26, 2013; posted March 27, 2013 (Doc. ID 184976); published April 24, 2013
Ptychography is combined with optical image encryption for the first time. Due to the nature of ptychography, not only is the interferometric optical setup that is usually adopted not required any more, but also the encryption for a complex-valued image is achievable. Considering that the probes overlapping with each other is the crucial factor in ptychography, their complex-amplitude functions can serve as a kind of secret keys that lead to the enlarged key space and the enhanced system security. Further, since only introducing the probes into the input of common system is required, it is convenient to combine ptychography with many existing optical image encryption systems for varied security applications. © 2013 Optical Society of America OCIS codes: (100.2000) Digital image processing; (100.4998) Pattern recognition, optical security and encryption; (110.1650) Coherence imaging; (060.4785) Optical security and encryption. http://dx.doi.org/10.1364/OL.38.001425
Ptychography is a powerful technique for coherent imaging [1–3] that has already been applied in the x-ray [4,5], optical [6,7], and electronic domains [8]. Its merits include good quality of both the recovered amplitude and phase distribution and unlimited size of the specimen to be imaged [9]. In ptychography, the specimen is enlightened by multiple probes, generating multiple diffraction patterns in the far field. Since the neighboring probes cover up some common area of the specimen, their diffraction patterns partially carry the common information about the specimen. Using all the diffraction patterns, all the information of the complex-valued specimen can be recovered with a phase retrieval algorithm. Unlike holography, ptychography does not require a reference beam. On the other hand, like holography, it is capable of reconstructing the complex amplitude of the object [3]. Optical security has become an attractive research branch in optics in the past two decades [10]. Recently, some new techniques of optical encryption based on imaging have emerged, such as those based on ghost imaging [11] and diffractive imaging [12], since the procedure for image decryption in an optical security system is quite similar to the process for specimen reconstruction in an imaging system. Some physical factors of imaging are introduced into the optical security system, which bring several new advantages for the purposes of security. In this Letter, we apply ptychography in optically encrypting the complex-amplitude image. It demonstrated that introducing ptychography can not only simplify the architecture of an optical encryption system but also greatly enhance its security by enlarging the key space. Figure 1 shows the schematic of optical image encryption via ptychography. Since it is convenient to apply ptychography to nearly all current optical image encryption systems, we just take the classical optical encryption system with random-phase encoding in both the input and Fourier planes as an example. It is actually suitable at least for other double random-phase encoding systems, such as the systems in the fractional Fourier domain or in the Fresnel domain [10]. The first probe P1 is 0146-9592/13/091425-03$15.00/0
incident to the secret image f o (Lena) located in the input plane. P1 only covers the upper right of f o , as shown with light red color (Probe1 ) in Fig. 1. Then it is modulated by double random-phase-only masks M 1 and M 2 that are fixed in the input plane (x, y) and in the Fourier plane (p, q), respectively. Subsequently, the intensity ID1 of the first diffractive pattern is recorded by a sensor in the output plane. In the same way, the probe is moved to illuminate the upper left, lower left, and lower right of the input image, respectively. Note that the core concept of performing ptychography is that the multiple probes used to scan the specimen require being overlapped with each other. Thus the applied probes P1 to P4 , are partially overlapped, which is shown in Fig. 1 in different colors. The above process of encryption can be expressed as IDi jFTfFTfPi · f o · expjM 1 g · expjM 2 gj2 ;
(1)
where Pi i 1; 2; …; n stands for the probe that enlightens the secret image and IDi is the corresponding intensity of the diffraction pattern. In this case n equals 4. In Eq. (1), FTf•g stands for Fourier transformation. Now
Fig. 1. Schematic of optical image encryption via ptychography: an example with random-phase encoding in both the input and the Fourier planes. © 2013 Optical Society of America
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the procedure of encryption is accomplished. These intensities fIDi g are the cipher texts of the secret image. With the cipher texts and the correct phase keys, the plain text can be extracted by employing the decryption algorithm. From the perspective of mathematics, it can be thought of as a special case of ptychographical iterative algorithm [3] that is described as below: 1. Guess the complex value of the input image f g x; y, and then begin the following iterative process. 2. For the kth iteration, f g x; y is illuminated by the ith probe, which is modulated with M 1 and M 2 in the input and Fourier planes; the intensity is acquired as IDkig jFTfFTfPi · f kig · expjM 1 g · expjM 2 gj2 :
(2)
3. Substitute the amplitude term of IDki x; y with the detected intensity IDio x; y, while preserving the phase: IDki IDio 1∕2 · IDkig ∕jIDkig j:
(3)
4. Take the inverse Fourier transform of IDki x; y to the Fourier plane and divide by M 2 , then take the other inverse Fourier transform and divide by M 1 : f kigN FT−1 fFT−1 IDki · exp−jM 2 g · exp−jM 1 :
(4)
5. Renew f kigN x; y with f ki1gN x; y according to f ki1g f kig βPi f kigN f kig Pi ∕Pi α;
(5)
where α and β are the adjustment factors; the former is used to ensure the denominator is nonzero, and the latter is employed for the iterative adjustment [3]. (6) Repeat the above steps 2 through 5 for all probes and now an entire iteration is completed. (7) Calculate the correlation coefficients (Co) between the decrypted and secret images: Cof ; f o covf ; f o σ f · σ f o −1 ;
(6)
where f and f o denote the recovered and the original amplitudes or phase functions of the object, respectively, covf ; f o is the cross-covariance between f and f o , and σ f is the standard deviation. The value of Co ranges in [0,1]. If it equals 1 the best decryption is achieved. (8) If the Co reaches the threshold value, finish the iterations and obtain the decrypted image f . We have performed a series of simulations to test the proposed technique. Since one important property of ptychography is that not only the amplitude but also the phase distribution of a specimen is retrievable [3], a complex-amplitude image was tested in the simulations. The image to be encrypted is shown as Fig. 2(a) in gray and Fig. 2(b) in color, which present the amplitude and phase terms, respectively. The amplitude term “Lena” and the phase term “baboon” are both 256 × 256 pixels in 0–255 scale, which are distributed in [0,1] and [0,2π], respectively. Two phase keys M 1 and M 2 are statistically independent and randomly distributed in [0,2π]; one of them
Fig. 2. (a) Amplitude, (b) phase term of the secret image, (c) one of phase keys, and (d) a typical diffraction pattern.
is shown as Fig. 2(c). Figure 2(d) gives one typical diffraction pattern corresponding to a certain probe. A set of probes employed in simulations are shown as Fig. 3, in which the upper four and lower four images represent the amplitude and phase distributions of the four probes, respectively. The gray levels are used to denote the different amplitude distributions as shown in Figs. 3(a)–3(d). In Figs. 3(e)–3(h), the colors stand for the differences of phase functions just as in Fig. 2(b), without the color bars. Figure 3(a) and 3(e) can be deemed as the complex amplitude of the original probe, and another three probes are generated by rotating at 90°, 180°, and 270°, respectively. In the experiments, complex-amplitude distributed probes are achievable by use of a spatial light modulator (SLM). The SLM is suitably encoded and located in the front focal plane of a Fourier-transformed lens, conveniently generating the required distributions of probes in the back focal plane of the lens, which is the same plane that the secret image is fixed on. Note that the black regions of the amplitudes in Figs. 3(a)–3(d) indicate where the incident light is totally blocked. Thus the area of any two mutually overlapping probes occupies two quadrants, which simply satisfies the core concept of ptychography. First, let us examine its feasibility. Three kinds of probes have been tested, including the pure-amplitude, the phase-only, and the complex-amplitude probes, the amplitudes and phase distributions of which are already shown in Fig. 3. The corresponding cipher texts are not presented here for brevity. Employing the above decryption algorithm, Fig. 4 shows the amplitude and phase functions of decrypted images all obtained after 200 iterations. Some noises can be found in the decryptions for the pure-amplitude probes. However, for all three kinds of probes, it is clear that we decrypted nearly all the information, consisting of both the amplitude and the phase, and the values of Co are quite closed to 1. Second, the diversity of the probes has been tested for optical encryption. For the sake of brevity, we only show
Fig. 3. Four employed probes. (a)–(d) Amplitudes and (e)–(h) phase distributions of the probes.
May 1, 2013 / Vol. 38, No. 9 / OPTICS LETTERS
Fig. 4. Amplitude and phase function, respectively, of the decrypted complex-valued images corresponding to three different types of probes: (a), (b) pure amplitude; (c), (d) phase only; and (e), (h) complex amplitude.
three types of probes, named as K1, K2, and K3 in Fig. 5, which all belong to the pure-amplitude type without any phase distributions. Note that the K1 probes are just an example, as was the given schematic in Fig. 1. All black regions still indicate the light fully shielded, and it is also implied that the probes in each type satisfy the core requirement of overlapping for ptychography. In several iterations less than 200, we can obtain decryptions of good quality similar to those presented in Fig. 4, which are not shown for brevity. Third, inspired by the above results of the probes’ diversity, we have examined whether the probes can serve as a new kind of secret keys for an optical encryption system combined with ptychography. With the correct phase keys but the wrong probes, the simulation results are presented in Fig. 6 after 2000 iterations of the decryption algorithm. Figures 6(a) and 6(b) show the decryptions when we employed the pure-amplitude probes of type K1 for encryption but decrypted with the disordered probes in K1. Also for the pure-amplitude probes, the probes of type K2 were used to decrypt the cipher texts that were encrypted by the probes in K1, obtaining the decryptions as shown in Figs. 6(c) and 6(d). On the other hand, we tested the phase-only probes that are already shown in Figs. 3(e)–3(h). Figures 6(e) and 6(f) present the amplitudes and phase terms of the decryptions acquired with disordered phase-only probes, while Figs. 6(g) and 6(h) are the results when using the probes
Fig. 5. Diversity of the probes was tested with other series of probes: (a)–(d) K1, (e)–(h) K2, and (i)–(l) K3.
Fig. 6.
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Security enhancement due to ptychography.
with totally different phase distributions. As shown in Figs. 6(a)–6(h), all decryptions are random noises other than only a little shadow-like signal of the probes or the secret image. Further, we have taken other tests regarding the security of the probes and obtained quite similar results to Fig. 6. These simulations indicate that highquality decryptions are not acquirable when using the wrong probes. In other words, the amplitudes and phase functions of the probes can serve as the secret keys. In conclusion, we apply ptychography in optical image encryption. Successfully securing a complex-valued image is achievable for the 4f encryption system via ptychography. In our method, the optical setup is simplified by avoiding the interferometric optical path, and the system security is also enhanced by employing complexamplitude probes that can effectively enlarge the key space. As it is easy to combine ptychography with many current optical image encryption systems, it has potential for varied security purposes, such as double-image or three-dimensional information encryption and hiding. The authors quite appreciate the anonymous reviewers for their constructive comments. This research was supported by NNSF of China (60907004), K. C. Wong Foundation, the President Fund of UCAS, the Fusion Fund of Research and Education, and CAS. References 1. W. Hoppe, Acta Crystallogr. A 25, 495 (1969). 2. W. Hoppe, Ultramicroscopy 10, 187 (1982). 3. J. M. Rodenburg, in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, 2008), Vol. 150, pp. 87–184. 4. J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007). 5. P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, Science 321, 379 (2008). 6. J. M. Rodenburg, A. C. Hurst, and A. G. Cullis, Ultramicroscopy 107, 227 (2007). 7. A. M. Maiden, J. M. Rodenburg, and M. J. Humphry, Opt. Lett. 35, 2585 (2010). 8. M. J. Humphry, B. Kraus, A. C. Hurst, A. M. Maiden, and J. M. Rodenburg, Nat. Commun. 3, 730 (2012). 9. H. M. L. Faulkner and J. M. Rodenburg, Phys. Rev. Lett. 93, 023903 (2004). 10. A. Alfalou and C. Brosseau, Adv. Opt. Photon. 1, 589 (2009). 11. P. Clemente, V. Durán, V. Torres-Company, E. Tajahuerce, and J. Lancis, Opt. Lett. 35, 2391 (2010). 12. W. Chen, X. Chen, and C. J. R. Sheppard, Opt. Lett. 35, 3817 (2010).
