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Wavelet-based image fusion for securing multiple images through asymmetric keys
Optics Communications 335 (2015) 153–160
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Wavelet-based image fusion for securing multiple images through asymmetric keys Isha Mehra, Naveen K. Nishchal n Department of Physics, Indian Institute of Technology Patna, Patliputra Colony, Patna 800 013, India
art ic l e i nf o
a b s t r a c t
Article history: Received 12 July 2014 Accepted 14 September 2014 Available online 26 September 2014
Image fusion is one of the popular methods which provides better quality fused image for interpreting an image data. Discrete wavelet transform based fusion technique is one such method, in which low and high frequency components are merged together to improve the image content. In this paper, we propose this fusion technique for generating asymmetric keys for securing multiple images. An input image to be encrypted is digitally encoded into two phase-only masks employing the principle of optical interference. This process has been repeated for three different input images; however, it can be extended to n images. Now, one of the phase-only masks corresponding to each input image is preserved as a phase key while another set of phase masks are fused together. This fused image is called the encrypted image. Unlike optical asymmetric encryption technique based on amplitude- and phasetruncation approach, here, four asymmetric keys are generated corresponding to each image. Asymmetric keys corresponding to each image, fractional orders, phase-only masks, level of decomposition and type of wavelet, enlarge the key space and hence offer enhanced security. The proposed method is demonstrated through the simulation results. & 2014 Elsevier B.V. All rights reserved.
Keywords: Image fusion Optical asymmetric cryptosystem Wavelet transforms Interference Fractional Fourier transform
1. Introduction In the past few decades, optical information security techniques have been widely studied because of multi-dimensional and parallel processing nature of large storage memories at great speeds [1,2]. The double random phase encoding (DRPE) technique is the basic optical image encryption architecture [3]. Most of the existing encryption schemes deal with binary and gray-scale images. The encryption and decryption are performed with the help of monochromatic light. Thus, the decrypted images do not preserve their color information. Color image encryption has become an important field of research for data security [4], because of presence of all three (red, green, and blue) components. Now-a-days, the multispectral data, which consists of several spectral bands, plays an importance role over the quality and content of the data. The multispectral data received from satellites and airborne sensors are being increasingly available for further processing and analysis for various applications, including remote sensing [5]. The fused/combined information of various sensors is the source of comprehensive data. The fusion of low and high resolution
n
Corresponding author. Tel.: þ 91 612 2552027; fax: þ 91 612 2277383. E-mail address: [email protected] (N.K. Nishchal).
http://dx.doi.org/10.1016/j.optcom.2014.09.040 0030-4018/& 2014 Elsevier B.V. All rights reserved.
multispectral images is a widely used technique because the fused images possess complementary information from different sources. Discrete wavelet transform (DWT) based fusion technique is one such method in which low and high frequency components are merged together. Since many of such data are significant from security point of view, therefore, it becomes necessary to develop an optical encryption technique for securing fused data/images. Any optical cryptosystem is not highly secure until it has enhanced key space with the use of various degrees of freedom. For example, the security of the basic DRPE scheme was improved by employing different optical domains [6,7]. However, due to inherent linearity, such cryptosystems are not highly secure [8–10]. Frauel et al. [11] proved that DRPE resists brute force attack but is susceptible to chosen plain-text and known plain-text attacks. To overcome the problems of linearity in DRPE, amplitudeand phase-truncation based asymmetric cryptosystem has been proposed [12–16]. Recently, an encryption scheme has been proposed in which phase-retrieval algorithm is applied twice for generating two asymmetric keys from intermediate phases [17,18]. However, there are few problems associated with such cryptosystems; one of them is system's complexity for repetitive saving of phase keys in the form of holograms. Secondly, for single step amplitude- and phase-truncation, only one asymmetric key can be generated. For these reasons, the phase-truncation based asymmetric cryptosystem has been proved to be vulnerable against
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I. Mehra, N.K. Nishchal / Optics Communications 335 (2015) 153–160
specific attack [19]. Thereafter, many modifications have been reported in order to improve its security [20–22]. Wavelet transform (WT) is one of the signal processing tools, which is used for analyzing optical signals [23–27]. Fusion techniques play an important role in the entire multi-wavelength phenomenon [28]. Alfalou and Brosseau [29–31] reported a new spectral multiple image fusion analysis based on the discrete cosine transform. Recently, a wavelet based color image fusion scheme for securing data through phase truncation approach and image hiding has been proposed [32]. In this paper, DWT based fusion technique has been utilized for securing multiple data. Each image to be encrypted is digitally encoded into two phase-only masks (POM) in fractional Fourier transform (FRT) domain using the interference principle [33]. One of the POMs is preserved as a phase key while other one is discrete wavelet transformed. The POMs corresponding to different images are fused together using single level DWT for obtaining the encrypted image. Thus, the interference phenomenon and the WT based fusion plays an important role for securing multiple images. Use of the optical interference helps enhance the security of the scheme. In this case, all four asymmetric keys (approximation, horizontal, vertical, and diagonal coefficients) are necessary for successful decryption. This is because, for decryption through interference principle, exact POMs are required. Also, in the interference based phenomenon, the inherent silhouette problem can be overcome through various existing known schemes [14,34]. Without using the concept of interference only the approximation coefficient (one asymmetric key) is enough to decrypt the input image with acceptable quality. Hence, the scheme becomes less secure.
Using Eq. (6), we can easily find POMs corresponding to ith image, jDj M 1ðiÞ ðξ; ηÞ ¼ argðDÞ þ cos 1 ð7Þ 2
2. Principle 2.1. Image encoding into two phase-only masks The proposed encryption process is shown in Fig. 1(a). Let f0 i(x,y) represents one of the input functions to be encrypted. This input image is encoded into two POMs, M1 and M2 analytically [33]. The complex function can be obtained using a random phase mask (RPM), 0
f i ðx; yÞ ¼ f i ðx; yÞexp½i2πψ ðx; yÞ
ð1Þ
where ψ(x,y) is random distribution having between 0 and 1. The FRT of the two POMs consisting of phases, M1 and M2, optically interfere at the decryption plane to provide the complex input image fi(x,y) [33]. This complex distribution can be expressed as, f i ðx; yÞ ¼ ℑα fexp½iM 1ðiÞ ðξ; ηÞg þ ℑα fexp½iM 2ðiÞ ðξ; ηÞg where ℑα {.} represents the FRT operation with order Eq. (2) we can write, exp½iM 1ðiÞ ðξ; ηÞ þ exp½iM 2ðiÞ ðξ; ηÞ ¼ ℑ α f i ðx; yÞ
α. From ð3Þ
ð4Þ
ð5Þ
The two phase masks, exp½iM 1ðiÞ ðξ; ηÞ and exp½iM 2ðiÞ ðξ; ηÞ are the phase-only functions. Therefore, we have D exp½iM 1ðiÞ ðξ; ηÞ2 ¼ fD exp½iM 1ðiÞ ðξ; ηÞg fD exp½iM 1ðiÞ ðξ; ηÞgn ¼ 1
ð8Þ
where arg{.} represents the phase angle. 2.2. DWT based image fusion In this section, POMs generated corresponding to different images are decomposed into wavelet coefficients using DWT [28]. In a one-dimensional (1-D) DWT, an input function f(x) is decomposed into coefficients with the starting scale mo as: pffiffiffiffi W ϕ ðmo ; kÞ ¼ ð1= L0 Þ∑ f ðxÞψ mo ;k ð9Þ k
pffiffiffiffi W ψ ðm; kÞ ¼ ð1= L0 Þ∑ f ðxÞψ m;k
ð10Þ
k
Then Eq. (4) can be written as exp½iM 2ðiÞ ðξ; ηÞ ¼ D exp½iM 1ðiÞ ðξ; ηÞ
M 2ðiÞ ðξ; ηÞ ¼ argfD expðiM 1ðiÞ ðξ; ηÞÞg
ð2Þ
Let us assume that exp½iM 1ðiÞ ðξ; ηÞ þ exp½iM 2ðiÞ ðξ; ηÞ ¼ D
Fig. 1. Block diagram for multiple image encryption scheme using discrete wavelet transform based fusion technique.
ð6Þ
where, L0 represents the scaling parameter of wavelet. Wφ(mo,k) and Wψ(m,k) are termed as approximation and detailed coefficients respectively. In a two-dimensional (2D) DWT, a 1-D DWT is first performed on the rows and on the columns of data. This results in one set of approximation coefficients, Wφ(m,r,s), and η three sets of detailed coefficients, W ψ(m,r,s), where η ¼ {H,V,D} represents horizontal, vertical, and diagonal components. The wavelet coefficients obtained corresponding to M1(1)(ξ.η) are; {WLL1, WHL1, WLH1, WHH1}. The DWT operation of M1(2)(ξ.η) will yield {WLL2, WHL2, WLH2, WHH2}, and POM obtained from ith image will yield {WLLi, WHLi, WLHi, WHHi}, where L and H represent low and high frequency components, respectively. Now, different wavelet coefficients of n number of images are fused together through DWT, in which the high and low frequency components
I. Mehra, N.K. Nishchal / Optics Communications 335 (2015) 153–160
are merged together [28]. W FLL ¼ AVGðW FLL1 ; W FLL2; …W FLLi…; W FLLn Þ
ð11Þ
W FHL ¼ AV GðW FHL1 ; W FHL2;… W FHLi…; W FHLn Þ
ð12Þ
W FLH ¼ AV GðW FLH1 ; W FLH2;… W FLHi…; W FLHn Þ
ð13Þ
W FHH ¼ AV GðW FHH1 ; W FHH2;… W FHHi…; W FHHn Þ
ð14Þ
2.3. Multiple image encryption The position multiplexing of all four coefficients obtained from the previous operation, (Eqs. (11)–(14)), gives the encrypted image. Eðψ ; ϕÞ ¼ fW FLL ; W FHL ; W FLH ; W FHH g
ð15Þ
The decryption process is shown in Fig. 1(b). For decryption, different wavelet coefficients are separated through position demultiplexing, and the coefficients corresponding to ith image, behaving as asymmetric keys are generated as, kiðappÞ ¼ kiðHÞ ¼
W LLi AV GðW FLL1 ; W FLL2;… W FLLi…; W FLLn Þ
W HLi AVGðW FHL1 ; W FHL2;… W FHLi…; W FHLn Þ
ð16Þ
ð17Þ
155
kiðV Þ ¼
W LHi AV GðW FLH1 ; W FLH2;… W FLHi…; W FLHn Þ
ð18Þ
kiðDÞ ¼
W HHi AV GðW FHH1 ; W FHH2;… W FHHi…; W FHHn Þ
ð19Þ
where, ki(app), ki(H), ki(V), and ki(D) represent the approximation, horizontal, vertical, and diagonal keys, respectively, corresponding to ith image. When these keys are applied on the encrypted image, this will lead to individual image coefficients. fW LLi ; W HLi ; W LHn i; W HHi g ¼ fW FLL kiðappÞ ; W FHL kiðHÞ ; W FLH kiðV Þ ; W FHH kiðDÞ g
ð20Þ
Now, the obtained coefficients {WLLi, WHLi, WLHi, WHHi} is inverse discrete WT resulting into the new function, Gi(ξ.η). Then on the basis of interference principle [33], f i ðx; yÞ ¼ ℑα Gi ðξ; ηÞ þ ℑα exp½iM 1 ðξ; ηÞ
ð21Þ
Here i ¼1,2,…n. Thus, use of asymmetric keys generated through Eqs. (16)–(19), along with the correct type and level of decomposition of wavelet, POMs and FRT orders would retrieve all the original images, respectively.
Fig. 2. (a)–(c) Three input images of size 256 256 pixels to be encrypted.
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Fig. 3. (a) First POM, M1(1)(ξ,η) generated corresponding to first input image, f1(x,y), (b) second POM, M2(1)(ξ,η) generated corresponding to first input image, f1(x,y), (c) first POM, M1(2)(ξ,η) generated corresponding to second input image, f2(x,y), (d) second POM, M2(2)(ξ,η) generated corresponding to second input image, f2(x,y), (e) first POM, M1(3)(ξ,η) generated corresponding to third input image, f3(x,y), and (f) second POM, M2(3)(ξ,η) generated corresponding to third input image, f3(x,y).
3. Simulation results and discussion The numerical study has been carried out using MATLAB 7.10. Fig. 2(a), (b), and (c) show the three input images; f1(x,y), f2(x,y), and f3(x,y), each of size 256 256 pixels, respectively. Each of these images are analytically encoded into two pairs of POMs, M1(i)(ξ,η)
and M2(i)(ξ,η) in the FRT domain, as shown from Fig. 3(a)–(f). The arbitrarily chosen fractional orders used in the simulation are 0.45, 0.50, and 0.55 for three images [Fig. 2(a–c)], respectively. The POMs corresponding to the input images are fused together using a “Haar” wavelet. Fig. 4(a)–(d) show all four fused wavelet coefficients, which are called as approximation, horizontal, vertical, and diagonal
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157
Fig. 4. All four fused wavelet coefficients through wavelet based fusion. (a) Approximation fused component, (b) horizontal fused component, (c) vertical fused component, (d) diagonal fused component, and (e) encrypted image.
coefficients, respectively. Fig. 4(e) shows the encrypted image obtained through position multiplexing of all four obtained fused coefficients. For decryption, different coefficients are separated through de-multiplexing. The coefficients corresponding to first image, f1(x,y), which generated the asymmetric keys have been shown in Fig. 5(a)–(d), respectively. When these keys are applied on the encrypted image, this will lead to individual image coefficients.
When these coefficients are multiplexed together, POMs are recovered [Fig. 6(a)–(c)]. For decryption, the recovered POMs optically interfere and the input images are retrieved successfully. The operation is repeated in order to retrieve all images through different asymmetric keys. In this scheme, the encryption scheme comprises of wavelet based fusion technique. But, rather going in a reverse order of this technique, the four decryption keys are
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Fig. 5. Asymmetric keys (a) approximation coefficient key, (b) horizontal coefficient key, (c) vertical coefficient key, and (d) diagonal coefficient key.
generated for an individual image, which are not used during encryption process. Unlike conventional asymmetric encryption which comprises of amplitude- and phase-truncation approach, here, four asymmetric keys are generated corresponding to each image. The decrypted images are obtained using correct keys. The decrypted images using correct asymmetric keys, fractional orders, POMs, level and type of decomposition of wavelet will be identical to original images as shown in Fig. 6(d)–(f). Fig. 7(a) shows the wrong asymmetric wavelet coefficient key and Fig. 7(b) shows the corresponding decrypted image. With these results, we infer that 4n asymmetric keys corresponding to n images, fractional orders, POMs, level of decomposition and type of wavelet, enlarges the key space. Hence multiple images with multiple levels of security can be encrypted. To evaluate the performance of the proposed encryption–decryption scheme, mean square error (MSE) has been calculated between the original and the decrypted image [22]. The MSE values have been calculated for all the three decrypted images which are; 4.56 10 8, 3.05 10 9, and 9.16 10 10, respectively. If wrong asymmetric approximation coefficient key, k10 (app)(ξ,η) is used, the MSE between input image, f1(x,y), and decrypted image, is 19.24, which is very high as compared to the image decrypted with correct keys. Such results verify the effectiveness of the proposed multiple image encryption scheme. To check the security of the proposed scheme, we applied a hybrid attack, which is a combination of brute force attack, specific
attack, and known plain-text attack [11]. In case of brute force attack, every possible asymmetric key are searched. As, all the four asymmetric keys for each image are of 128 128 pixels and if each pixel is assumed to have 16 possible phase values, then number of attempts to acquire all four keys for single image is 164 128 128, and, for n number of images, possible attempts will be 164 128 128n, which is practically inconceivable. We carried out this study for 10,000 iterations, considering encryption keys as public keys, which is a property of specific attack. On the other hand, in case of known plain-text attack, attacker has to find the keys on the basis of input image and encrypted image. As our main aim is to find 4n asymmetric keys corresponding to n images, so, known plain-text attack property is also incorporated in the hybrid attack. The MSE values calculated for 10,000 iterations have been shown in Fig. 7(c). It can be observed that the MSE value never approaches close to zero. Hence, it can be claimed that the proposed scheme is resistant to the hybrid attack.
4. Conclusion DWT based fusion technique aims to merge high and low frequency components in order to improve the quality of the image. This technique has been utilized for the first time for generating asymmetric keys with enhanced level of security suitable for encrypting multiple images. The digitally encoded
I. Mehra, N.K. Nishchal / Optics Communications 335 (2015) 153–160
159
Fig. 6. (a) Recovered POM corresponding to first image, (b) recovered POM corresponding to second image, (c) recovered POM corresponding to third image, (d) decrypted image corresponding to first image, (e) decrypted image corresponding to second image, and (f) decrypted image corresponding to third image.
POMs corresponding to different images are fused together using single level DWT. The number of asymmetric keys will increase with the increase in the level of decomposition of WT. Four asymmetric keys are generated corresponding to each input image. Thus, 4n asymmetric keys corresponding to n input images, fractional orders, POMs, level of decomposition and type of
wavelet enhances the security of such optical asymmetric cryptosystem. It would be straightforward to extend the proposed scheme for securing three-dimensional multispectral images/data. The proposed method can also be applied for encrypted image compression [29–31]. The method can be implemented optically by employing the optical set-up used by Mendlovic [26].
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Fig. 7. (a) Wrong asymmetric key for approximation coefficient used, (b) corresponding decrypted image, and (c) hybrid attack result which shows a plot between number of iterations and MSE in order to decrypt the image.
Acknowledgments The authors acknowledge the funding from the Council of Scientific and Industrial Research (CSIR), Government of India, under Grant no. 03/(1183)/10/EMR-II. References [1] B. Javidi, Optical and Digital Techniques for Information Security, Springer, New York, 2005. [2] O. Matoba, T. Nomura, E. Perez-Cabre, M.S. Millan, B. Javidi, Proc. IEEE 97 (2009) 1128. [3] P. Refregier, B. Javidi, Opt. Lett. 20 (1995) 767. [4] S. Zhang, M.A. Karim, Microw. Opt. Technol. Lett. 21 (1999) 318. [5] C. Pohl, J.L. Van Genderen, Int. J. Remote Sens. 19 (1998) 823. [6] O. Matoba, B. Javidi, Opt. Lett. 24 (1999) 762. [7] N.K. Nishchal, T.J. Naughton, Opt. Commun. 284 (2011) 735. [8] A. Carnicer, M.M. Usategui, S. Arcos, I. Juvells, Opt. Lett. 30 (2005) 1644. [9] U. Gopinathan, D.S. Monaghan, T.J. Naughton, J.T. Sheridan, Opt. Express 14 (2006) 3181. [10] X. Peng, P. Zhang, H. Wei, B. Yu, Opt. Lett. 31 (2006) 1044. [11] Y. Frauel, A. Castro, T.J. Naughton, B. Javidi, Opt. Express 15 (2007) 10253.
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W. Qin, X. Peng, Opt. Lett. 35 (2010) 118. X. Wang, D. Zhao, Opt. Commun. 284 (2011) 148. S.K. Rajput, N.K. Nishchal, Appl. Opt. 51 (2012) 1446. S.K. Rajput, N.K. Nishchal, Appl. Opt. 52 (2013) 4343. I. Mehra, N.K. Nishchal, Opt. Laser Technol. 60 (2014) 1. W. Liu, Z. Liu, S. Liu, Opt. Lett. 38 (2013) 1651. S.K. Rajput, N.K. Nishchal, Appl. Opt. 53 (2014) 418. X. Wang, D. Zhao, Opt. Commun. 285 (2012) 1078. X. Wang, D. Zhao, Appl. Opt. 50 (2011) 6645. X. Ding, X. Deng, K. Song, G. Chen, Appl. Opt. 52 (2013) 467. S.K. Rajput, N.K. Nishchal, Appl. Opt. 51 (2012) 5377. R.K. Young, Wavelet Theory and Its Applications, Kluwer Academic, 1993. K. Martin, R. Lukac, K.N. Plataniotis, Pattern. Recognit. 38 (2005) 1111. L. Chen, D.M. Zhao, Opt. Commun. 254 (2009) 361. D. Mendlovic, N. Konforti, Appl. Opt. 32 (1993) 6542. B. Javidi, P. Ferraro, S. Hong, S. De Nicola, A. Finizio, D. Alfieri, G. Pierattini, Opt. Lett. 30 (2005) 144. B. Javidi, C.M. Do, S.H. Hong, T. Nomura, J. Disp. Technol. 2 (2006) 411. A. Alfalou, C. Brosseau, N. Abdallah, M. Jridi, Opt. Express 19 (2011) 24023. A. Alfalou, C. Brosseau, Adv. Opt. Photon. 1 (2009) 589. A. Alfalou, C. Brosseau, Opt. Lett. 35 (2010) 1914. I. Mehra, N.K. Nishchal, Opt. Express 22 (2014) 5474. Y. Zhang, B. Wang, Opt. Lett. 33 (2008) 2443. P. Kumar, J. Joseph, K. Singh, Appl. Opt. 50 (2011) 1805.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Wavelet-based image fusion for securing multiple images through asymmetric keys Isha Mehra, Naveen K. Nishchal n Department of Physics, Indian Institute of Technology Patna, Patliputra Colony, Patna 800 013, India
art ic l e i nf o
a b s t r a c t
Article history: Received 12 July 2014 Accepted 14 September 2014 Available online 26 September 2014
Image fusion is one of the popular methods which provides better quality fused image for interpreting an image data. Discrete wavelet transform based fusion technique is one such method, in which low and high frequency components are merged together to improve the image content. In this paper, we propose this fusion technique for generating asymmetric keys for securing multiple images. An input image to be encrypted is digitally encoded into two phase-only masks employing the principle of optical interference. This process has been repeated for three different input images; however, it can be extended to n images. Now, one of the phase-only masks corresponding to each input image is preserved as a phase key while another set of phase masks are fused together. This fused image is called the encrypted image. Unlike optical asymmetric encryption technique based on amplitude- and phasetruncation approach, here, four asymmetric keys are generated corresponding to each image. Asymmetric keys corresponding to each image, fractional orders, phase-only masks, level of decomposition and type of wavelet, enlarge the key space and hence offer enhanced security. The proposed method is demonstrated through the simulation results. & 2014 Elsevier B.V. All rights reserved.
Keywords: Image fusion Optical asymmetric cryptosystem Wavelet transforms Interference Fractional Fourier transform
1. Introduction In the past few decades, optical information security techniques have been widely studied because of multi-dimensional and parallel processing nature of large storage memories at great speeds [1,2]. The double random phase encoding (DRPE) technique is the basic optical image encryption architecture [3]. Most of the existing encryption schemes deal with binary and gray-scale images. The encryption and decryption are performed with the help of monochromatic light. Thus, the decrypted images do not preserve their color information. Color image encryption has become an important field of research for data security [4], because of presence of all three (red, green, and blue) components. Now-a-days, the multispectral data, which consists of several spectral bands, plays an importance role over the quality and content of the data. The multispectral data received from satellites and airborne sensors are being increasingly available for further processing and analysis for various applications, including remote sensing [5]. The fused/combined information of various sensors is the source of comprehensive data. The fusion of low and high resolution
n
Corresponding author. Tel.: þ 91 612 2552027; fax: þ 91 612 2277383. E-mail address: [email protected] (N.K. Nishchal).
http://dx.doi.org/10.1016/j.optcom.2014.09.040 0030-4018/& 2014 Elsevier B.V. All rights reserved.
multispectral images is a widely used technique because the fused images possess complementary information from different sources. Discrete wavelet transform (DWT) based fusion technique is one such method in which low and high frequency components are merged together. Since many of such data are significant from security point of view, therefore, it becomes necessary to develop an optical encryption technique for securing fused data/images. Any optical cryptosystem is not highly secure until it has enhanced key space with the use of various degrees of freedom. For example, the security of the basic DRPE scheme was improved by employing different optical domains [6,7]. However, due to inherent linearity, such cryptosystems are not highly secure [8–10]. Frauel et al. [11] proved that DRPE resists brute force attack but is susceptible to chosen plain-text and known plain-text attacks. To overcome the problems of linearity in DRPE, amplitudeand phase-truncation based asymmetric cryptosystem has been proposed [12–16]. Recently, an encryption scheme has been proposed in which phase-retrieval algorithm is applied twice for generating two asymmetric keys from intermediate phases [17,18]. However, there are few problems associated with such cryptosystems; one of them is system's complexity for repetitive saving of phase keys in the form of holograms. Secondly, for single step amplitude- and phase-truncation, only one asymmetric key can be generated. For these reasons, the phase-truncation based asymmetric cryptosystem has been proved to be vulnerable against
154
I. Mehra, N.K. Nishchal / Optics Communications 335 (2015) 153–160
specific attack [19]. Thereafter, many modifications have been reported in order to improve its security [20–22]. Wavelet transform (WT) is one of the signal processing tools, which is used for analyzing optical signals [23–27]. Fusion techniques play an important role in the entire multi-wavelength phenomenon [28]. Alfalou and Brosseau [29–31] reported a new spectral multiple image fusion analysis based on the discrete cosine transform. Recently, a wavelet based color image fusion scheme for securing data through phase truncation approach and image hiding has been proposed [32]. In this paper, DWT based fusion technique has been utilized for securing multiple data. Each image to be encrypted is digitally encoded into two phase-only masks (POM) in fractional Fourier transform (FRT) domain using the interference principle [33]. One of the POMs is preserved as a phase key while other one is discrete wavelet transformed. The POMs corresponding to different images are fused together using single level DWT for obtaining the encrypted image. Thus, the interference phenomenon and the WT based fusion plays an important role for securing multiple images. Use of the optical interference helps enhance the security of the scheme. In this case, all four asymmetric keys (approximation, horizontal, vertical, and diagonal coefficients) are necessary for successful decryption. This is because, for decryption through interference principle, exact POMs are required. Also, in the interference based phenomenon, the inherent silhouette problem can be overcome through various existing known schemes [14,34]. Without using the concept of interference only the approximation coefficient (one asymmetric key) is enough to decrypt the input image with acceptable quality. Hence, the scheme becomes less secure.
Using Eq. (6), we can easily find POMs corresponding to ith image, jDj M 1ðiÞ ðξ; ηÞ ¼ argðDÞ þ cos 1 ð7Þ 2
2. Principle 2.1. Image encoding into two phase-only masks The proposed encryption process is shown in Fig. 1(a). Let f0 i(x,y) represents one of the input functions to be encrypted. This input image is encoded into two POMs, M1 and M2 analytically [33]. The complex function can be obtained using a random phase mask (RPM), 0
f i ðx; yÞ ¼ f i ðx; yÞexp½i2πψ ðx; yÞ
ð1Þ
where ψ(x,y) is random distribution having between 0 and 1. The FRT of the two POMs consisting of phases, M1 and M2, optically interfere at the decryption plane to provide the complex input image fi(x,y) [33]. This complex distribution can be expressed as, f i ðx; yÞ ¼ ℑα fexp½iM 1ðiÞ ðξ; ηÞg þ ℑα fexp½iM 2ðiÞ ðξ; ηÞg where ℑα {.} represents the FRT operation with order Eq. (2) we can write, exp½iM 1ðiÞ ðξ; ηÞ þ exp½iM 2ðiÞ ðξ; ηÞ ¼ ℑ α f i ðx; yÞ
α. From ð3Þ
ð4Þ
ð5Þ
The two phase masks, exp½iM 1ðiÞ ðξ; ηÞ and exp½iM 2ðiÞ ðξ; ηÞ are the phase-only functions. Therefore, we have D exp½iM 1ðiÞ ðξ; ηÞ2 ¼ fD exp½iM 1ðiÞ ðξ; ηÞg fD exp½iM 1ðiÞ ðξ; ηÞgn ¼ 1
ð8Þ
where arg{.} represents the phase angle. 2.2. DWT based image fusion In this section, POMs generated corresponding to different images are decomposed into wavelet coefficients using DWT [28]. In a one-dimensional (1-D) DWT, an input function f(x) is decomposed into coefficients with the starting scale mo as: pffiffiffiffi W ϕ ðmo ; kÞ ¼ ð1= L0 Þ∑ f ðxÞψ mo ;k ð9Þ k
pffiffiffiffi W ψ ðm; kÞ ¼ ð1= L0 Þ∑ f ðxÞψ m;k
ð10Þ
k
Then Eq. (4) can be written as exp½iM 2ðiÞ ðξ; ηÞ ¼ D exp½iM 1ðiÞ ðξ; ηÞ
M 2ðiÞ ðξ; ηÞ ¼ argfD expðiM 1ðiÞ ðξ; ηÞÞg
ð2Þ
Let us assume that exp½iM 1ðiÞ ðξ; ηÞ þ exp½iM 2ðiÞ ðξ; ηÞ ¼ D
Fig. 1. Block diagram for multiple image encryption scheme using discrete wavelet transform based fusion technique.
ð6Þ
where, L0 represents the scaling parameter of wavelet. Wφ(mo,k) and Wψ(m,k) are termed as approximation and detailed coefficients respectively. In a two-dimensional (2D) DWT, a 1-D DWT is first performed on the rows and on the columns of data. This results in one set of approximation coefficients, Wφ(m,r,s), and η three sets of detailed coefficients, W ψ(m,r,s), where η ¼ {H,V,D} represents horizontal, vertical, and diagonal components. The wavelet coefficients obtained corresponding to M1(1)(ξ.η) are; {WLL1, WHL1, WLH1, WHH1}. The DWT operation of M1(2)(ξ.η) will yield {WLL2, WHL2, WLH2, WHH2}, and POM obtained from ith image will yield {WLLi, WHLi, WLHi, WHHi}, where L and H represent low and high frequency components, respectively. Now, different wavelet coefficients of n number of images are fused together through DWT, in which the high and low frequency components
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are merged together [28]. W FLL ¼ AVGðW FLL1 ; W FLL2; …W FLLi…; W FLLn Þ
ð11Þ
W FHL ¼ AV GðW FHL1 ; W FHL2;… W FHLi…; W FHLn Þ
ð12Þ
W FLH ¼ AV GðW FLH1 ; W FLH2;… W FLHi…; W FLHn Þ
ð13Þ
W FHH ¼ AV GðW FHH1 ; W FHH2;… W FHHi…; W FHHn Þ
ð14Þ
2.3. Multiple image encryption The position multiplexing of all four coefficients obtained from the previous operation, (Eqs. (11)–(14)), gives the encrypted image. Eðψ ; ϕÞ ¼ fW FLL ; W FHL ; W FLH ; W FHH g
ð15Þ
The decryption process is shown in Fig. 1(b). For decryption, different wavelet coefficients are separated through position demultiplexing, and the coefficients corresponding to ith image, behaving as asymmetric keys are generated as, kiðappÞ ¼ kiðHÞ ¼
W LLi AV GðW FLL1 ; W FLL2;… W FLLi…; W FLLn Þ
W HLi AVGðW FHL1 ; W FHL2;… W FHLi…; W FHLn Þ
ð16Þ
ð17Þ
155
kiðV Þ ¼
W LHi AV GðW FLH1 ; W FLH2;… W FLHi…; W FLHn Þ
ð18Þ
kiðDÞ ¼
W HHi AV GðW FHH1 ; W FHH2;… W FHHi…; W FHHn Þ
ð19Þ
where, ki(app), ki(H), ki(V), and ki(D) represent the approximation, horizontal, vertical, and diagonal keys, respectively, corresponding to ith image. When these keys are applied on the encrypted image, this will lead to individual image coefficients. fW LLi ; W HLi ; W LHn i; W HHi g ¼ fW FLL kiðappÞ ; W FHL kiðHÞ ; W FLH kiðV Þ ; W FHH kiðDÞ g
ð20Þ
Now, the obtained coefficients {WLLi, WHLi, WLHi, WHHi} is inverse discrete WT resulting into the new function, Gi(ξ.η). Then on the basis of interference principle [33], f i ðx; yÞ ¼ ℑα Gi ðξ; ηÞ þ ℑα exp½iM 1 ðξ; ηÞ
ð21Þ
Here i ¼1,2,…n. Thus, use of asymmetric keys generated through Eqs. (16)–(19), along with the correct type and level of decomposition of wavelet, POMs and FRT orders would retrieve all the original images, respectively.
Fig. 2. (a)–(c) Three input images of size 256 256 pixels to be encrypted.
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Fig. 3. (a) First POM, M1(1)(ξ,η) generated corresponding to first input image, f1(x,y), (b) second POM, M2(1)(ξ,η) generated corresponding to first input image, f1(x,y), (c) first POM, M1(2)(ξ,η) generated corresponding to second input image, f2(x,y), (d) second POM, M2(2)(ξ,η) generated corresponding to second input image, f2(x,y), (e) first POM, M1(3)(ξ,η) generated corresponding to third input image, f3(x,y), and (f) second POM, M2(3)(ξ,η) generated corresponding to third input image, f3(x,y).
3. Simulation results and discussion The numerical study has been carried out using MATLAB 7.10. Fig. 2(a), (b), and (c) show the three input images; f1(x,y), f2(x,y), and f3(x,y), each of size 256 256 pixels, respectively. Each of these images are analytically encoded into two pairs of POMs, M1(i)(ξ,η)
and M2(i)(ξ,η) in the FRT domain, as shown from Fig. 3(a)–(f). The arbitrarily chosen fractional orders used in the simulation are 0.45, 0.50, and 0.55 for three images [Fig. 2(a–c)], respectively. The POMs corresponding to the input images are fused together using a “Haar” wavelet. Fig. 4(a)–(d) show all four fused wavelet coefficients, which are called as approximation, horizontal, vertical, and diagonal
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157
Fig. 4. All four fused wavelet coefficients through wavelet based fusion. (a) Approximation fused component, (b) horizontal fused component, (c) vertical fused component, (d) diagonal fused component, and (e) encrypted image.
coefficients, respectively. Fig. 4(e) shows the encrypted image obtained through position multiplexing of all four obtained fused coefficients. For decryption, different coefficients are separated through de-multiplexing. The coefficients corresponding to first image, f1(x,y), which generated the asymmetric keys have been shown in Fig. 5(a)–(d), respectively. When these keys are applied on the encrypted image, this will lead to individual image coefficients.
When these coefficients are multiplexed together, POMs are recovered [Fig. 6(a)–(c)]. For decryption, the recovered POMs optically interfere and the input images are retrieved successfully. The operation is repeated in order to retrieve all images through different asymmetric keys. In this scheme, the encryption scheme comprises of wavelet based fusion technique. But, rather going in a reverse order of this technique, the four decryption keys are
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Fig. 5. Asymmetric keys (a) approximation coefficient key, (b) horizontal coefficient key, (c) vertical coefficient key, and (d) diagonal coefficient key.
generated for an individual image, which are not used during encryption process. Unlike conventional asymmetric encryption which comprises of amplitude- and phase-truncation approach, here, four asymmetric keys are generated corresponding to each image. The decrypted images are obtained using correct keys. The decrypted images using correct asymmetric keys, fractional orders, POMs, level and type of decomposition of wavelet will be identical to original images as shown in Fig. 6(d)–(f). Fig. 7(a) shows the wrong asymmetric wavelet coefficient key and Fig. 7(b) shows the corresponding decrypted image. With these results, we infer that 4n asymmetric keys corresponding to n images, fractional orders, POMs, level of decomposition and type of wavelet, enlarges the key space. Hence multiple images with multiple levels of security can be encrypted. To evaluate the performance of the proposed encryption–decryption scheme, mean square error (MSE) has been calculated between the original and the decrypted image [22]. The MSE values have been calculated for all the three decrypted images which are; 4.56 10 8, 3.05 10 9, and 9.16 10 10, respectively. If wrong asymmetric approximation coefficient key, k10 (app)(ξ,η) is used, the MSE between input image, f1(x,y), and decrypted image, is 19.24, which is very high as compared to the image decrypted with correct keys. Such results verify the effectiveness of the proposed multiple image encryption scheme. To check the security of the proposed scheme, we applied a hybrid attack, which is a combination of brute force attack, specific
attack, and known plain-text attack [11]. In case of brute force attack, every possible asymmetric key are searched. As, all the four asymmetric keys for each image are of 128 128 pixels and if each pixel is assumed to have 16 possible phase values, then number of attempts to acquire all four keys for single image is 164 128 128, and, for n number of images, possible attempts will be 164 128 128n, which is practically inconceivable. We carried out this study for 10,000 iterations, considering encryption keys as public keys, which is a property of specific attack. On the other hand, in case of known plain-text attack, attacker has to find the keys on the basis of input image and encrypted image. As our main aim is to find 4n asymmetric keys corresponding to n images, so, known plain-text attack property is also incorporated in the hybrid attack. The MSE values calculated for 10,000 iterations have been shown in Fig. 7(c). It can be observed that the MSE value never approaches close to zero. Hence, it can be claimed that the proposed scheme is resistant to the hybrid attack.
4. Conclusion DWT based fusion technique aims to merge high and low frequency components in order to improve the quality of the image. This technique has been utilized for the first time for generating asymmetric keys with enhanced level of security suitable for encrypting multiple images. The digitally encoded
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Fig. 6. (a) Recovered POM corresponding to first image, (b) recovered POM corresponding to second image, (c) recovered POM corresponding to third image, (d) decrypted image corresponding to first image, (e) decrypted image corresponding to second image, and (f) decrypted image corresponding to third image.
POMs corresponding to different images are fused together using single level DWT. The number of asymmetric keys will increase with the increase in the level of decomposition of WT. Four asymmetric keys are generated corresponding to each input image. Thus, 4n asymmetric keys corresponding to n input images, fractional orders, POMs, level of decomposition and type of
wavelet enhances the security of such optical asymmetric cryptosystem. It would be straightforward to extend the proposed scheme for securing three-dimensional multispectral images/data. The proposed method can also be applied for encrypted image compression [29–31]. The method can be implemented optically by employing the optical set-up used by Mendlovic [26].
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Fig. 7. (a) Wrong asymmetric key for approximation coefficient used, (b) corresponding decrypted image, and (c) hybrid attack result which shows a plot between number of iterations and MSE in order to decrypt the image.
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