ICOP 2009-International Conference on Optics and Photonics India,30 Oct.-1 Nov.2009
Chandigarh,
IMAGE ENCRYPTION USING CHAOS AND RADIAL HILBERT TRANSFORM Madhusudan Joshi§, Chandra Shakher± and Kehar Singh* § The International Centre for Automotive Technology, IMT Manesar, Haryana (India) ±Instrument Design and Development Centre, Indian Institute of Technology Delhi, New Delhi (India) * Department of Physics, Indian Institute of Technology Delhi, New Delhi (India) § Corresponding author‟s email:
[email protected] Abstract: A new method for image encryption using integral order radial Hilbert transform (RHT) filter in the Fourier transform domain has been proposed. The technique is based on popular double random phase encoding architecture. The conventional random phase masks have been replaced by random chaotic masks. Simulation results have been presented and the schematic for optical implementation has been proposed. The robustness of the technique has been verified against unauthorized access in number of ways. 1.
INTRODUCTION
Optical image encryption techniques have played an important role in the field of optical information processing. Various algorithms [1-4] have been proposed for image encryption, information hiding, and watermarking etc., due to high computational speed, data storage and massive parallelism in optics. Out of all these techniques, the double random phase encoding (DRPE) [5] is the most widely used for image encryption. Using the DRPE, the image to be encrypted is changed to noise like patterns and exact recovery of the image is possible only when the correct random phase functions and other key parameters are correctly used during decryption. The Hilbert transform [6] has widely been used for image processing applications. The RHT is a radially symmetric version of the Hilbert transform and the fractional Hilbert Transform [7-9]. Davis et. al. [10] have proposed its application, for the edge enhancement of images through spatial filtering operations. Recently, an application of the RHT for image encryption has been proposed [11], in the fractional Fourier transform (FRT) [12] domain. In this technique, the RHT mask has been used as a spatial filter to segregate the spatial frequencies into twochannels, and subsequently the encryption is performed using the DRPE algorithm. The fractional orders of the FRT and the random phase masks have been used as keys for encryption. In this paper, a simple but effective technique for optical image encryption using random chaotic masks (RCMs) [13-15] and integral order RHT has been proposed in the FT domain. The integral orders of the RHT and the RCMs used during the process are the keys for encryption and decryption. The simulation results supporting the idea have been presented. The technique can be realized on a conventional 4-f set-up for implementing DRPE algorithm and does not require any special optical arrangement. The robustness of the technique has also been
analyzed against various attacks and perturbations. For that matter, the effects of noise and occlusion on the performance of the proposed system have been investigated [16]. 2. DESCRIPTION OF TECHNIQUE 2.1 Creating a chaos based random mask The process of making a chaos based random mask is described as follows. First, a chaotic map [ r ( x, y ) ] is generated by introducing some input parameters to the map. These input parameters are called seed values and the size of the map respectively. A typical chaotic map of size m n pixels generated based on the mentioned technique is shown in Fig. 1 (i) and its chaotic behavior is plotted in fig. 1 (ii).
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Fig. 1 (i) A typical chaotic map; (ii) Chaotic behaviour of the map The chaotic map obtained in Fig. 1 is real. The RCM is generated from the chaotic map using;
( x, y) = exp[ir ( x, y)]
(1)
2.2 Hilbert transform and RHT The Hilbert transform [6] has been widely used in image processing and phase observation on account of its edge enhancing properties. The generalization of the
ICOP 2009-International Conference on Optics and Photonics Chandigarh, India,30 Oct.-1 Nov.2009 conventional Hilbert transform to fractional counterpart P (integral in this case), in the FT plane to get: has been explained by Lohmann et. al. [7]. Both of its (3) Fp (u, v) Fr (u, v) H p (r, ) , forms are useful to selectively emphasize the features of the input image during the spatial filtering operation [8, 9]. The Hilbert transform forms an edge enhanced where ( x, y ) and ( u,v) represent the coordinates in version of the input image, whereas the fractional the image- and the FT plane respectively. In the next Hilbert transform changes the nature of the edge step; enhancement. The classic Hilbert filter enhances image (4) e( x, y) = 1[ Fp (u, v) 2 (u, v)] only along one dimension. Although it is possible to create 2-dimensional masks by performing the product 1 of two Hilbert masks, but these masks retain the basic is calculated. Here denotes inverse FT, x, y symmetry only [10]. However it is possible to 2 (u, v)] denotes Fourier plane RCM, and maintain basic concept of the Hilbert mask and avoid e( x, y) denotes the encrypted image. This image can the x,y symmetry by making a radial counterpart [10] be recorded interferometrically using a CCD. The order given as; of the RHT mask used during encryption serves as an additional key to encryption apart from the two RCMs, (2) H P (r, ) exp(iP ), RHT mask where the variables (r , ) represent the polar H p (r , ) coordinates and P represents the order of the radial Hilbert transform. The opposite halves of any radial 1 Encrypted image Input image line of the mask have a relative phase difference of Pπ e( x, y) i( x, y) radian. Therefore for each radial line we have the equivalent of a one-dimensional Hilbert transform of RCM1 RCM2 order P 1 ( x, y) 2 (u, v) The first order RHT corresponds to continuous (i) phase jump from 0 to 2π, whereas its higher orders would lead to such multiple jumps. Figs. 2 (i) & 2(ii) RHT mask show the RHT for P=1 and P=4 respectively. It can be H Q (r , ) observed that the RHT for P=4 leads to four phase jumps from 0 to 2π. Besides this, a negative order of the 1 Decrypted image RHT would produce a reverse phase change e.g. from Encrypted d ( x, y) 2π to 0 for P= -1 as is shown in Fig. (iii). We term this image negative order RHT as the inverse RHT (IRHT). The e( x, y) numbers of phase jumps as mentioned are used as a key RCM2 during the encryption and the recovery of the image 2* (u, v) during decryption is possible only when the numbers of (ii) phase jumps are equal as well as reversed. Fig.3 Schematic for (i) Encryption and (ii) Decryption. i( x, y)
i( x, y)
i( x, y)
i( x, y)
i( x, y)
i( x, y)
i( x, y)
i( x, y)
During decryption, the FT of the encrypted image
e( x, y) to be calculated followed by multiplication with complex conjugate of the RCM 1 (u, v) and the (i) (ii) (iii) Fig. 2 RHT mask of order (i) P=1; (ii) P=4, and (iii) P=-1 respectively 2.2 The proposed technique The encryption is performed on the input image using the DRPE architecture and additionally a RHT mask of some order (say P) is also multiplied in the FT plane during encryption [Fig 3 (i)]. The input image i( x, y) after multiplication with a RCM 1 ( x, y) is Fourier transformed to obtain Fr (u, v) . The image Fr (u, v) is then multiplied with a RHT mask of order
a RHT mask of some order (say Q) and their product is represented as :
FQ (u, v) E (u, v)2 * (u, v) H Q (u, v)
(5)
where E (u, v) represents the FT of e( x, y) and „*‟ denotes the complex conjugate. In the final step;
d ( x, y) = 1 FQ (u, v)
(6)
is calculated. Here d ( x, y) is the complex decrypted image. As the decrypted image is recorded as an intensity pattern, it is redundant to use the second RCM. It is pertinent to mention that the correct image
ICOP 2009-International Conference on Optics and Photonics Chandigarh, India,30 Oct.-1 Nov.2009 information is retrieved only when a RHT mask of image when incorrect RCM and RHT of incorrect order order „–P‟ (i.e. negative of the order used during (Q= -9) is used is represented in Fig. 4 (vi). encryption) and the exact complex conjugates of the RCMs used during encryption are applied at decryption The effect of noise [14,15] on the encrypted stage [Fig. 3 (ii)]. data and the random keys has been investigated. The salt & pepper noise (on/off pixels), speckle noise 3. DIGITAL SIMULATION AND DISCUSSION (multiplicative noise with zero mean) and Gaussian noise (with zero mean and a constant variance) have To investigate the quality of encryption, digital been considered for the study. Figs. 5 (i)-(iii) simulations are performed. The input image chosen for respectively show the decrypted Lena images with the encryption is a fingerprint of size 256 × 256 pixels as encrypted image perturbed with salt & pepper noise shown Fig. 4(i). The order of RHT mask used for (density=0.05), speckle (variance = 0.04) - and encryption is chosen as P=8. The encrypted image is Gaussian noise (variance=0.01). Figs. 6 (i)-(iii) shown in Fig. 4(ii). respectively show the decrypted Lena images with the Fourier plane RCM perturbed with salt & pepper noise (density=0.05), speckle (variance = 0.04) - and Gaussian noise (variance=0.01).
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(iii)
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(i) (ii) (iii) Fig. 5 Decrypted images when- (a) salt & pepper noise, (b) speckle noise, and (c) the Gaussian noise is present in the encrypted image.
(i) (ii) (iii) Fig. 6 Decrypted images when- (a) salt & pepper noise, (b) speckle noise, and (c) the Gaussian noise is present in the Fourier plane RCM.
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Fig. 4 (i) Input image; (ii) encrypted image (for P=8); decrypted images (iii) when the random phase functions as well as order of RHT (i.e. Q=8) are correct are correct; (iv) when random phase functions are correct but the order of RHT (Q= 7) is incorrect (v) when order of RHT is correct (Q= -8) but one of the random functions is incorrect; (vi) when one of the random phase functions as well as order of RHT (Q= -9) are incorrect. The decrypted image obtained when correct RCMs as well as RHT mask of correct order (Q= -8) is used is represented in Fig. 4(iii). The decrypted image when correct RCMs and RHT mask of incorrect order (Q= -7) is used is shown in Fig. 4(iv).The decrypted image when incorrect RCM and RHT mask of correct order (Q= -8) is used is shown in Fig. 4(v). The decrypted
The performance of the proposed technique has been analyzed against the occlusion of the encrypted data [16]. Figs. 7 (i)-(iii) respectively show the encrypted images with 25%, 50%, and 75% occlusion and Figs. 7(iv)-7(vi) show the corresponding decrypted images.
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(iv) (v) (vi) Fig. 7 (i) – (iii) Encrypted images with 25%, ) 50%, and 75% occlusion respectively; (iv)(vi) Corresponding decrypted images
ICOP 2009-International Conference on Optics and Photonics Chandigarh, India,30 Oct.-1 Nov.2009 4. CONCLUSION Hilbert transform: theory and experiments”, Opt. Lett., 25 (2000), 99-101. We have proposed a technique for image 11. M. Joshi, C. Shakher, and K. Singh, “Image encryption using the integral orders of the radial Hilbert encryption and decryption using fractional transform as the key parameter for encryption and Fourier transform and radial Hilbert transform ” , decryption. In addition to the random chaos functions Opt. Lasers Eng. 46 (2008), 522-526. have been used in place of random phase functions of 12. H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The the well-known double random phase encoding Fractional Fourier transform with applications technique. Digital simulations are presented to in optics and signal processing. Wiley, demonstrate the robustness of the technique against the Chichester (2001). variation in the key parameters. The system is shown to 13. L. H. Zhang, X. F. Liao, X. B. Wang, “An perform well against occlusion and noise image encryption approach based on chaotic contamination. maps”, Chaos, Solitons & Fractals, 24 (2005), 759–765. ACKNOWLEDGEMENT 14. L. Kocarev, “Chaos-based cryptography: a brief overview”, IEEE Circ Syst Mag., 1 (2001), 6– One of the authors (Madhusudan Joshi) is thankful 21. to The Chief Executive Officer, The National 15. F. Sun, S. liu, Z. li, Z. lu, “A novel encryption Automotive Testing, Research and Infrastructure scheme based on spatial chaos map”, Chaos, Development Project (NATRiP), and The Director, Solitons & Fractals, 38 (2008),631–640. The International Center for Automotive Technology 16. X.F. Meng, L. Z. Cai, M.Z. He, G.Y. Dong and (iCAT) for their support. X. X. Shen, “Cross-talk-free double- image encryption and watermarking with amplitudeREFERENCES phase separate modulations”, J. Opt. A: Pure Appl. Opt., 7 (2005), 624-631. 1. B. Javidi, Optical and digital methods for information security, Springer (2005). 2. G. Unnikrishnan, Investigations on some algorithms and architectures for optical encryption, Ph.D. Thesis, Indian Institute of Technology Delhi, India (2002). 3. R. Tao, Y. Xin and Y. Wang, “Double image encryption based on random phase encoding in fractional Fourier domain”, Opt. Express,24 (2007), 16067-16079. 4. L.F. Chen and D.M. Zhao, “Color information processing (coding and synthesis) with fractional Fourier transforms and digital holography”, Opt. Express, 15(2007),16080-16089. 5. P. Refregier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding”, Opt. Lett., 20 (1995), 767-769. 6. R.B. Bracewell, The Fourier transform and its application, McGraw-Hill, (1965), Chap. 12. 7. A.W. Lohmann, D. Menlovic, Z. Zalevsky, “Fractional Hilbert transform”, Opt. Lett., 21 (1996), 281-283. 8. J.A. Davis, D. E. McNamara and D.M. Cottrell, “Analysis of the fractional Hilbert transform”, Appl. Opt., 37 (1998), 6911-6913. 9. A. W. Lohmann, E. Tepichin and J.G. Ramarez, “Optical implementation of fractional Hilbert transform for two-dimensional objects”, Appl. Opt., 36 (1997), 6620-6626. 10. J.A. Davis, D. E. McNamara, D.M. Cottrell and J. Campos, “Image Processing with radial
