- Email: [email protected]
Color image encoding in dual fractional Fourier-wavelet domain with random phases
Optics Communications 282 (2009) 3433–3438
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Color image encoding in dual fractional Fourier-wavelet domain with random phases Linfei Chen a,b, Daomu Zhao b,* a b
The School of Science, Hangzhou Dianzi University, Hangzhou 310018, China Department of Physics, Zhejiang University, Hangzhou 310027, China
a r t i c l e
i n f o
Article history: Received 5 February 2009 Received in revised form 8 May 2009 Accepted 21 May 2009
PACS: 42.30.d 42.30.Va 42.30.Kq
a b s t r a c t A new cryptology in dual fractional Fourier-wavelet domain is proposed in this paper, which is calculated by discrete fractional Fourier transform and wavelet decomposition. Different random phases are used in different wavelet subbands in encryption. A new color image encoding method is also presented with basic color decomposition and encryption respectively. All the keys, including random phases and fractional orders in R, G and B three channels, should be correctly used in decryption, otherwise people cannot obtain the totally correct information. Some numerical simulations are presented to demonstrate the possibility of the method. It would have widely potential applications in digital color image processing and protection. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Fractional Fourier transform Wavelet transform Encryption Decryption
1. Introduction During the last decade, network security and data encryption have become an important and high profile issues. Innovative encryption techniques need to be developed for effective data encryption [1–9]. The image processing techniques including denoising, pattern recognition, image compression, encryption, fusion, etc., are mostly realized in computer based on wavelet or wavelet packet transforms and carried out digitally [2–6]. The wavelet transform (WT) is successfully used in transient signal processing, data compression, optical correlators, sound analysis, etc. In the last few years, several optical security systems using double random phases are proposed for image encryption [10–24]. These methods are realized based on some new transforms, such as Fourier transform, Fresnel transform, fractional Fourier transform (FRFT), extended anamorphic FRFT, fractional wavelet transform, etc. Recently, some color image encryption methods are proposed with the development of cryptology and technology [3,7–9,19–22]. In many cases, we need the color information, not only for its beautiful in vision but also for its useful in practical applications. In this paper, we propose a color image encryption method with random phases in fractional Fourier-wavelet domain. The information is encrypted * Corresponding author. Tel.: +86 57185610016; fax: +86 57187951328. E-mail address: [email protected] (D. Zhao). 0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.05.044
with more keys compared to the existed methods such as Ref. [18], and that would strengthen the encryption security in a certain extent. The fractional orders, random phases, even the wavelet function and wavelength can be selected as the keys in encryption. And also, the color information is protected by dividing an authentic color image into three segments, each of which is encrypted independently with different fractional orders and different wavelength. Then in each segment, the information is also decomposed into several parts by WT, with each part coded with different random phases. Therefore these keys can be assigned to a group of authorized persons or saved in several places. Only in the case that all these persons present their corresponding portions and synthesize the whole authentic image correctly, can they access the correct information. This method, combined FRFT with WT, would have widely potential applications in digital color image processing and protection. The drawback of the previous method is that the information is decomposed into one-level with WT in this paper, but it can be improved by using fractional wavelet packet transform method in our further work. This paper is organized as follows. Section 2 gives the basic theory of FRFT and WT, and a new transformation named fractional wavelet transform is also presented. In Section 3, we introduce the method used in this paper that is information encryption in fractional Fourier-wavelet domain with random phases. Then, some computer simulations are used to demonstrate the possibility. While in Section
3434
L. Chen, D. Zhao / Optics Communications 282 (2009) 3433–3438
4, the technique of color image encoding in fractional Fourier-wavelet domain is proposed and verified by some simulation results. Finally, some conclusions are outlined in Section 5. 2. Fractional Fourier transform and wavelet transform WT is widely used in digital signal processing. It has two forms: continuous and discrete, and the later form is desired in practical applications. Using discrete WT, an image can be decomposed into a low resolution smooth image and a number of detailed information. Fig. 1 is a WT with two-level decomposition. In the first level analysis, the image is decomposed into a smooth subband (LL1), a horizontal (HL1) part, a vertical (LH1) subband, and a diagonal (HH1) part. With a second level analysis, LL1 subband can be further decomposed into LL2, HL2, LH2 and HH2. The information would be further analyzed by the same way in the higher level decomposition. The definition of FRFT in optics was proposed by Lohmann [25], Ozaktas et al. [26,27] in 1993, and its basic theory is the propagation of light within a medium with continuously varying refractive index. But it has been known in mathematics for more than 70 years. Since it is the generalization of Fourier transform, FRFT is widely applied in optical and digital information processing. According to the definition, its one-dimensional mathematical expression is
F½f ðxÞðx0 Þ ¼
expfi½psgnðsin /1 Þ=4 /1 =2g
Z
jkfs1 sin /1 j1=2 x2 þ x02 xx0 exp ip 2pi f ðxÞdx: kfs1 tan /1 kfs1 sin /1
ð1Þ
where /1 = p1p/2, which corresponds to the rotating angle in Wigner domain and p1 is the fractional order. k is wavelength of the input light and fs1 is standard focal length. FRFT has some important properties such as linearity and continuity, and satisfies Parseval theorem and shift theorem. So an inverse FRFT can be realized with negative fractional orders. Several digital algorithms for FRFT have been proposed, including algorithms based on discrete Fourier transforms and convolution, direct discrete FRFT algorithms, etc. Based on the FRFT and WT, a new transform named fractional wavelet transform was proposed in optics by Mendlovic and Zalevky in 1997 [28]. And the 2-D hybrid fractional wavelet transform of the signal f(x, y) is defined as
¼ expfi½psgnðsin /1 Þ=4 /1 =2g Wðamn ; bÞ jkfs1 sin /1 j1=2
expfi½psgnðsin /2 Þ=4 /2 =2g
jkfs2 sin /2 j1=2 x2 þ x02 xx0 exp ip 2p i kfs1 tan /1 kfs1 sin /1 2 02 0 y þy yy exp ip 2p i f ðx; yÞ kfs2 tan /2 kfs2 sin /2 Z Z Z Z
0
0
hamn b ðx0 ; y0 Þdxdydx dy :
ð2Þ
LL2 HL2 HL1 LH2 HH2
LH1
HH1
Fig. 1. A two-level wavelet decomposition.
Digitally, it is first realized by an FRFT and then a WT. Its backreconstructing formula can be written as exp i psgn sin/01 =4 /01 =2 f ðx;yÞ ¼ jkfs1 sin/01 j1=2 Z Z ( XXZ Z 1 exp i psgn sin/02 =4 /02 =2 1 F C am an jkfs2 sin/02 j1=2 m n ) u; a mÞ expðj2pub 0 ;j2pmb 0 Þdb 0 db 0 ðx0 ; y0 Þ Wða ; bÞHða mn
m
n
x
x2 þ x02 xx0 exp ip 0 2pi 0 kfs1 tan/1 kfs1 sin/1 y2 þ y02 yy0 0 0 dx dy ; exp ip 0 2pi 0 kfs2 tan/2 kfs2 sin/2
y
x
y
ð3Þ
where /01 ¼ /1 ; /02 ¼ /2 . A new image encryption method with fractional wavelet transform is presented, and the information is disturbed by the FRFT before WT, therefore it is protected by the first time in encryption. 3. Information encryption in dual fractional Fourier-wavelet domain with random phases The proposed method in this paper is also based on FRFT and WT with different random phases multiplied in each subband in fractional Fourier-wavelet domain. The encoding and decoding processes are illustrated in Fig. 2. Fig. 2a is encoding process. We use a one-level WT as an example. First, an original image is fractional Fourier transformed with fractional orders p1 and p2 before a WT. Then, after a one-level WT, the information is decomposed into four parts (LL1, HL1, LH1, LL1), which are multiplied by four different random phases to be an encrypted image. The random phases are independently selected, therefore people can keep them in several places, and of course, strengthen the encryption security. The process of FRFT is quite important in encryption, because we cannot only add the encoding keys with fractional orders to protect the information, but also disturb the information of the original image before WT that makes the energy not be mostly included in the LL1 subband and the four segments are equally important in encryption or decryption. The two fractional orders, four random phases, even the wavelet function and wavelength can be selected as the keys in encryption. Fig. 2b shows the decoding process. If the encrypted image is multiplied by the right phase keys which are the conjugated random phases in each four segments, and after a p1, p2 orders FRFT and an inverse WT, the image can be finally recovered in decryption process. Fig. 3 gives our simulation results. Fig. 3a is an original image with 256 256 pixels. Fig. 3b is its encryption result, which are normalized absolute values of the encrypted pixels. Fig. 3c shows a decrypted image with incorrect random phase keys. Fig. 3d is the recovered image with incorrect fractional orders in the inverse FRFT in decryption. Fig. 3e is the result when wavelength is incorrectly illuminated at the input plane in decryption. Fig. 3f is the simulation result with incorrect random phase keys, incorrect fractional orders and also incorrect wavelength in recovery. Fig. 3g is one partial recovery result, while the random phase keys in LL1 and HH1 are correct, but they are incorrect in LH1 and HL1 in decryption. It is shown that parts of the image are recovered but the other parts are obscured to be recognized. Finally, Fig. 3h is the correctly recovered image. 4. Color image encoding in fractional Fourier-wavelet domain As we know, an RGB color image is the composition of R, G and B three components. Recently, we have proposed an optical color
3435
L. Chen, D. Zhao / Optics Communications 282 (2009) 3433–3438
P1 , P 2
Original image
FRFT
R1
R2
R3
R4
Encrypted image
WT
(a) Encrypted image
R 1*
R 2*
R 3*
R 4*
iWT
Recovered inage
iFRFT
-P 1 , -P 2
(b) Fig. 2. The encryption and decryption processes. (a) The encoding process; (b) the decoding process.
Fig. 3. Computer simulation results. (a) An original image with 256 256 pixels; (b) the encryption result; (c)–(f) the incorrect decryption results; (g) a partial recovery result; (h) the correctly recovered image.
image encryption method with Fresnel transforms and wavelength multiplexing technique. An original color image is decomposed into R, G and B three components, and each component is encrypted or decrypted independently. All the keys in the three channels are important in decryption. Only when all of them are correct, can the image be recovered finally. In this paper, we study the color image encryption and decryption in dual fractional Fourier-wavelet domain with random phases. The basic technique is the same as that in Part 3. And Fig. 4 gives the whole encryption and decryption flows. Fig. 4a is the encryption flow. An RGB color image is first decomposed into R, G and B three components, and each component is encrypted independently with an FRFT, a WT and four different random phases. Then the three encoded information are recomposed to be an RGB coded image. Fig. 4b is the decryption process. The three segments of the RGB encoded information are decoded respectively with three series of keys, and then recomposed together to be the reconstructed image after decoding. As shown in Fig. 5, it can be found the encryption process and its effect. Fig. 5a is a JPEG color image with 512 512 pixels. Fig. 5b is its color encryption result, with k1 = 700 nm, k2 = 546.1 nm, p3 = p4 = 0.50; k3 = 435.8 nm, p1 = p2 = 0.40; p5 = p6 = 2/3. f = 300 mm and haar wavelet function is mother wavelet function, which is optionally chosen as one of the exam-
R
FRFT p1 p 2
R11 R21
WT
λ1 RGB color image
G
B
R31 R41
FRFT p3 p 4
R12 R22
WT
λ2
R32 R42
FRFT p5 p 6
RGB coded image
R13 R23
WT
λ3
R33 R43
(a) R
RGB coded image
G
B
* R11* R 21
iWT
* * R31 R 41
λ1
* R12* R22 * * R32 R42
* R13* R23 * 33
R
iFRFT -p1 -p2
* 43
R
iWT
iFRFT -p3 -p4
iWT
iFRFT -p5- p6
λ2
RGB recovered image
λ3
(b) Fig. 4. Color image encryption and decryption flows. (a) The encryption flow; (b) the decryption flow.
3436
L. Chen, D. Zhao / Optics Communications 282 (2009) 3433–3438
ples and people can select other ones in different situations. Fig. 5c is one of the incorrect decryption result with wrong random phases. Fig. 5d is recovered image with incorrect fractional orders p1 = p2 = 0.38; p3 = p4 = 0.35; p5 = p6 = 8/15. Fig. 5e shows the result with p1 = p2 = 0.24; p3 = p4 = 0.20; p5 = p6 = 0.4. It is easy to find that when the fractional orders are quite different from the correct keys, the image could not be seen or only some obscure feature could be found as shown in Fig. 5e. But when the fractional orders are quite close to the correct keys, then some significant characteristics could be seen as shown in the Fig. 5d. Fig. 5f is the correctly reconstructed image. The normalized mean square error (MSE) between the reconstructed image and the original image is often used to verify the quality of the recovered image, and the total MSE is described as
MSE ¼
M X N h i 1 X jf ðmDx; nDyÞ f0 ðmDx; nDyÞj2 ; M N m¼1 n¼1
ð4Þ
where M and N are the pixels of the image, Dx and Dy are the pixel sizes. In Fig. 6, we give the total MSE vs. different fractional orders in RGB three channels, while the red, green and blue lines show the results respectively. We can notice that only when p1 = p2 = 0.4; p3 = p4 = 0.5; p5 = p6 = 2/3, the MSE are minimal and the image can be correctly decrypted, otherwise people would obtain the incorrectly decrypted images with some noise in them. The results well coincide with those of Fig. 5d and e. Fig. 7 is the simulation results with part of the random phases correct and part of them wrong in decryption. Fig. 7a is an RGB picture with 800 600 pixels. Fig. 7b is the result when the random phases in LL1 part are correct but the others are incorrect. We notice that only a quarter of the random keys cannot reconstruct image clearly. Fig. 7c is the result with two correct random phases that are used in LL1 and HH1. While in Fig. 7d, random phases in LH1 and HL1 are correct. In the above two images, some information of the original image is recovered. Fig. 7e shows the recovered image when three quarters of the random phases, in LL1, HL1 and HH1, are correct. In this time, most of the original information is reconstructed. Fig. 7f is the result with the whole correct random phase keys.
Fig. 6. MSE vs. fractional orders in RGB three channels.
Fig. 8 is the simulation results when the keys in some of the three channels (R, G and B channels) are incorrect in decryption. Fig. 8a is an original color image with 2048 1536 pixels. The p1 = p2 = 1.0; encryption parameters are k1 = 700nm, k2 = 546.1 nm, p3 = p4 = 0.50; k3 = 435.8 nm, p5 = p6 = 0.25. Fig. 8b is one of the incorrect decryption images when only the keys in R segment are correct. Fig. 8c is G and B segments correctly decrypted. The color information is changed and incorrectly transferred to people. Fig. 8d is the result when the random keys in LH1 and HH1 of R segment, LL1 and LH1 of G segment and HL1 and HH1 of B segment are correct, but the other keys are wrong in decryption. Fig. 8e is the result with random keys in LL1, LH1 and HH1 of RGB segments correct. Fig. 8f shows the result when keys in all three RGB channels are correctly used in decoding. The keys are increased, so people without keys cannot recover the totally correct information easily.
Fig. 5. Computer simulation results of color image encoding and decoding. (a) An original RGB picture with 512 512 pixels; (b) its color encryption result; (c)–(e) the incorrect decryption results; (f) the correctly reconstructed image. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
L. Chen, D. Zhao / Optics Communications 282 (2009) 3433–3438
3437
Fig. 7. Partial color encoding simulation results. (a) A picture with 800 600 pixels; (b)–(e) the results with the part of the random phase keys correct; (f) the result with whole correct random phase keys. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 8. Another partial color encryption simulation results. (a) An original color image with 2048 1536 pixels; (b)–(e) the results with the keys in some of the three channels incorrectly decrypted; (f) the correct result. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
We also give the total MSE of the partial decryption results. The total MSE between the decryption image in Fig. 7b and the original image in Fig. 7a is about 0.7708, while in Fig. 7c, d and e the results are 0.6803, 0.6658 and 0.5577, respectively, and the corresponding MSE value between the correct decryption image in Fig. 7f and the original image is about 1.277 1027. The values of total MSE are also calculated for the results in Fig. 8. The MSE between Fig. 8b and a is about 0.1613, while the results are 0.0747, 0.1927 and 0.1736 for Fig. 8c, d and e respectively, and the value is 9.6902 1027 between Fig. 8f and the original image. We note that the MSE are different for the different encryption systems, but in each encryption system, the values of MSE well reflect the quality of the decrypted images. 5. Conclusions In summary, we have proposed a color image encryption method in dual fractional Fourier-wavelet domain. FRFT is important, be-
cause it makes the four parts of WT be equally useful in encryption. Fractional orders and four random phases in each channel are the significant keys in encoding. The parameters in all three channels become the keys in encryption and decryption, and the information can be well protected with increasing of keys. When people only have part of the keys, they also cannot get the correct information or even get the wrong information. This method would have largely potential usage in digital color processing and encoding. Acknowledgments This work was supported by the National Natural Science Foundation of China (10874150) and the Program for New Century Excellent Talents in University (NCET-07-0760). References [1] S.H. Shyu, Pattern Recogn. 40 (2007) 1014. [2] H. Cheng, X.B. Li, IEEE Trans. Sign. Proc. 48 (2000) 2439.
3438 [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
L. Chen, D. Zhao / Optics Communications 282 (2009) 3433–3438
K. Martin, R. Lukac, K.N. Plataniotis, Pattern Recogn. 38 (2005) 1111. F. Ahmed, Opt. Exp. 15 (2007) 4804. P.Y. Huang, L.W. Chan, Proc. Lect. Notes Comput. Sci. 2195 (2001) 301. T. Posch, Proc. IEEE-SP Intl. Sym. Time–Frequency Time-Scale Anal. (1992) 143. S. Sudharsanan, IEEE Trans. Cons. Elec. 51 (2005) 1204. R. Lukac, K.N. Plataniotis, Real-Time Imaging 11 (2005) 454. H.H. Nien, C.K. Huang, S.K. Changchien, H.W. Shieh, C.T. Chen, Y.Y. Tuan, Chaos Soliton Fract. 32 (2007) 1070. P. Refregier, B. Javidi, Opt. Lett. 20 (1995) 767. S. Liu, Q. Mi, B. Zhu, Opt. Lett. 26 (2001) 1242. G. Situ, J. Zhang, Opt. Lett. 30 (2005) 1306. G. Unnikrishnan, J. Joseph, K. Singh, Opt. Lett. 25 (2000) 887. E. Tajahuerce, O. Matoba, S.C. Verrall, B. Javidi, Appl. Opt. 39 (2000) 2313. B. Hennelly, J.T. Sheridan, Opt. Lett. 28 (2003) 269. X. Peng, L.F. Yu, L.L. Cai, Opt. Exp. 10 (2002) 41.
[17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
H. Kim, D.H. Kim, Y.H. Lee, Opt. Exp. 12 (2004) 4912. L. Chen, D. Zhao, Opt. Commun. 254 (2005) 361. S.Q. Zhang, M.A. Karim, Microwave Opt. Tech. Lett. 21 (1999) 318. L. Chen, D. Zhao, Opt. Exp. 14 (2006) 8552. L. Chen, D. Zhao, Opt. Exp. 15 (2007) 16080. M. Joshi, C. Shakher, K. Singh, Opt. Commun. 279 (2007) 35. Y. Zhang, C.H. Zheng, N. Tanno, Opt. Commun. 202 (2002) 277. A.M. Youssef, IEEE Signal Proc. Lett. 15 (2008) 77. A.W. Lohmann, J. Opt. Soc. Am. A 10 (1993) 2181. H.M. Ozaktas, Z. Zalevsky, M.A. Kutay, The Fractional Fourier Transforming with Applications in Optics and Signal Processing, Wiley, New York, 2000. [27] H.M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, J. Opt. Soc. Am. A 11 (1994) 547. [28] D. Mendlovic, Z. Zalevsky, D. Mas, J. Garcia, C. Ferreira, Appl. Opt. 36 (1997) 4801.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Color image encoding in dual fractional Fourier-wavelet domain with random phases Linfei Chen a,b, Daomu Zhao b,* a b
The School of Science, Hangzhou Dianzi University, Hangzhou 310018, China Department of Physics, Zhejiang University, Hangzhou 310027, China
a r t i c l e
i n f o
Article history: Received 5 February 2009 Received in revised form 8 May 2009 Accepted 21 May 2009
PACS: 42.30.d 42.30.Va 42.30.Kq
a b s t r a c t A new cryptology in dual fractional Fourier-wavelet domain is proposed in this paper, which is calculated by discrete fractional Fourier transform and wavelet decomposition. Different random phases are used in different wavelet subbands in encryption. A new color image encoding method is also presented with basic color decomposition and encryption respectively. All the keys, including random phases and fractional orders in R, G and B three channels, should be correctly used in decryption, otherwise people cannot obtain the totally correct information. Some numerical simulations are presented to demonstrate the possibility of the method. It would have widely potential applications in digital color image processing and protection. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Fractional Fourier transform Wavelet transform Encryption Decryption
1. Introduction During the last decade, network security and data encryption have become an important and high profile issues. Innovative encryption techniques need to be developed for effective data encryption [1–9]. The image processing techniques including denoising, pattern recognition, image compression, encryption, fusion, etc., are mostly realized in computer based on wavelet or wavelet packet transforms and carried out digitally [2–6]. The wavelet transform (WT) is successfully used in transient signal processing, data compression, optical correlators, sound analysis, etc. In the last few years, several optical security systems using double random phases are proposed for image encryption [10–24]. These methods are realized based on some new transforms, such as Fourier transform, Fresnel transform, fractional Fourier transform (FRFT), extended anamorphic FRFT, fractional wavelet transform, etc. Recently, some color image encryption methods are proposed with the development of cryptology and technology [3,7–9,19–22]. In many cases, we need the color information, not only for its beautiful in vision but also for its useful in practical applications. In this paper, we propose a color image encryption method with random phases in fractional Fourier-wavelet domain. The information is encrypted * Corresponding author. Tel.: +86 57185610016; fax: +86 57187951328. E-mail address: [email protected] (D. Zhao). 0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.05.044
with more keys compared to the existed methods such as Ref. [18], and that would strengthen the encryption security in a certain extent. The fractional orders, random phases, even the wavelet function and wavelength can be selected as the keys in encryption. And also, the color information is protected by dividing an authentic color image into three segments, each of which is encrypted independently with different fractional orders and different wavelength. Then in each segment, the information is also decomposed into several parts by WT, with each part coded with different random phases. Therefore these keys can be assigned to a group of authorized persons or saved in several places. Only in the case that all these persons present their corresponding portions and synthesize the whole authentic image correctly, can they access the correct information. This method, combined FRFT with WT, would have widely potential applications in digital color image processing and protection. The drawback of the previous method is that the information is decomposed into one-level with WT in this paper, but it can be improved by using fractional wavelet packet transform method in our further work. This paper is organized as follows. Section 2 gives the basic theory of FRFT and WT, and a new transformation named fractional wavelet transform is also presented. In Section 3, we introduce the method used in this paper that is information encryption in fractional Fourier-wavelet domain with random phases. Then, some computer simulations are used to demonstrate the possibility. While in Section
3434
L. Chen, D. Zhao / Optics Communications 282 (2009) 3433–3438
4, the technique of color image encoding in fractional Fourier-wavelet domain is proposed and verified by some simulation results. Finally, some conclusions are outlined in Section 5. 2. Fractional Fourier transform and wavelet transform WT is widely used in digital signal processing. It has two forms: continuous and discrete, and the later form is desired in practical applications. Using discrete WT, an image can be decomposed into a low resolution smooth image and a number of detailed information. Fig. 1 is a WT with two-level decomposition. In the first level analysis, the image is decomposed into a smooth subband (LL1), a horizontal (HL1) part, a vertical (LH1) subband, and a diagonal (HH1) part. With a second level analysis, LL1 subband can be further decomposed into LL2, HL2, LH2 and HH2. The information would be further analyzed by the same way in the higher level decomposition. The definition of FRFT in optics was proposed by Lohmann [25], Ozaktas et al. [26,27] in 1993, and its basic theory is the propagation of light within a medium with continuously varying refractive index. But it has been known in mathematics for more than 70 years. Since it is the generalization of Fourier transform, FRFT is widely applied in optical and digital information processing. According to the definition, its one-dimensional mathematical expression is
F½f ðxÞðx0 Þ ¼
expfi½psgnðsin /1 Þ=4 /1 =2g
Z
jkfs1 sin /1 j1=2 x2 þ x02 xx0 exp ip 2pi f ðxÞdx: kfs1 tan /1 kfs1 sin /1
ð1Þ
where /1 = p1p/2, which corresponds to the rotating angle in Wigner domain and p1 is the fractional order. k is wavelength of the input light and fs1 is standard focal length. FRFT has some important properties such as linearity and continuity, and satisfies Parseval theorem and shift theorem. So an inverse FRFT can be realized with negative fractional orders. Several digital algorithms for FRFT have been proposed, including algorithms based on discrete Fourier transforms and convolution, direct discrete FRFT algorithms, etc. Based on the FRFT and WT, a new transform named fractional wavelet transform was proposed in optics by Mendlovic and Zalevky in 1997 [28]. And the 2-D hybrid fractional wavelet transform of the signal f(x, y) is defined as
¼ expfi½psgnðsin /1 Þ=4 /1 =2g Wðamn ; bÞ jkfs1 sin /1 j1=2
expfi½psgnðsin /2 Þ=4 /2 =2g
jkfs2 sin /2 j1=2 x2 þ x02 xx0 exp ip 2p i kfs1 tan /1 kfs1 sin /1 2 02 0 y þy yy exp ip 2p i f ðx; yÞ kfs2 tan /2 kfs2 sin /2 Z Z Z Z
0
0
hamn b ðx0 ; y0 Þdxdydx dy :
ð2Þ
LL2 HL2 HL1 LH2 HH2
LH1
HH1
Fig. 1. A two-level wavelet decomposition.
Digitally, it is first realized by an FRFT and then a WT. Its backreconstructing formula can be written as exp i psgn sin/01 =4 /01 =2 f ðx;yÞ ¼ jkfs1 sin/01 j1=2 Z Z ( XXZ Z 1 exp i psgn sin/02 =4 /02 =2 1 F C am an jkfs2 sin/02 j1=2 m n ) u; a mÞ expðj2pub 0 ;j2pmb 0 Þdb 0 db 0 ðx0 ; y0 Þ Wða ; bÞHða mn
m
n
x
x2 þ x02 xx0 exp ip 0 2pi 0 kfs1 tan/1 kfs1 sin/1 y2 þ y02 yy0 0 0 dx dy ; exp ip 0 2pi 0 kfs2 tan/2 kfs2 sin/2
y
x
y
ð3Þ
where /01 ¼ /1 ; /02 ¼ /2 . A new image encryption method with fractional wavelet transform is presented, and the information is disturbed by the FRFT before WT, therefore it is protected by the first time in encryption. 3. Information encryption in dual fractional Fourier-wavelet domain with random phases The proposed method in this paper is also based on FRFT and WT with different random phases multiplied in each subband in fractional Fourier-wavelet domain. The encoding and decoding processes are illustrated in Fig. 2. Fig. 2a is encoding process. We use a one-level WT as an example. First, an original image is fractional Fourier transformed with fractional orders p1 and p2 before a WT. Then, after a one-level WT, the information is decomposed into four parts (LL1, HL1, LH1, LL1), which are multiplied by four different random phases to be an encrypted image. The random phases are independently selected, therefore people can keep them in several places, and of course, strengthen the encryption security. The process of FRFT is quite important in encryption, because we cannot only add the encoding keys with fractional orders to protect the information, but also disturb the information of the original image before WT that makes the energy not be mostly included in the LL1 subband and the four segments are equally important in encryption or decryption. The two fractional orders, four random phases, even the wavelet function and wavelength can be selected as the keys in encryption. Fig. 2b shows the decoding process. If the encrypted image is multiplied by the right phase keys which are the conjugated random phases in each four segments, and after a p1, p2 orders FRFT and an inverse WT, the image can be finally recovered in decryption process. Fig. 3 gives our simulation results. Fig. 3a is an original image with 256 256 pixels. Fig. 3b is its encryption result, which are normalized absolute values of the encrypted pixels. Fig. 3c shows a decrypted image with incorrect random phase keys. Fig. 3d is the recovered image with incorrect fractional orders in the inverse FRFT in decryption. Fig. 3e is the result when wavelength is incorrectly illuminated at the input plane in decryption. Fig. 3f is the simulation result with incorrect random phase keys, incorrect fractional orders and also incorrect wavelength in recovery. Fig. 3g is one partial recovery result, while the random phase keys in LL1 and HH1 are correct, but they are incorrect in LH1 and HL1 in decryption. It is shown that parts of the image are recovered but the other parts are obscured to be recognized. Finally, Fig. 3h is the correctly recovered image. 4. Color image encoding in fractional Fourier-wavelet domain As we know, an RGB color image is the composition of R, G and B three components. Recently, we have proposed an optical color
3435
L. Chen, D. Zhao / Optics Communications 282 (2009) 3433–3438
P1 , P 2
Original image
FRFT
R1
R2
R3
R4
Encrypted image
WT
(a) Encrypted image
R 1*
R 2*
R 3*
R 4*
iWT
Recovered inage
iFRFT
-P 1 , -P 2
(b) Fig. 2. The encryption and decryption processes. (a) The encoding process; (b) the decoding process.
Fig. 3. Computer simulation results. (a) An original image with 256 256 pixels; (b) the encryption result; (c)–(f) the incorrect decryption results; (g) a partial recovery result; (h) the correctly recovered image.
image encryption method with Fresnel transforms and wavelength multiplexing technique. An original color image is decomposed into R, G and B three components, and each component is encrypted or decrypted independently. All the keys in the three channels are important in decryption. Only when all of them are correct, can the image be recovered finally. In this paper, we study the color image encryption and decryption in dual fractional Fourier-wavelet domain with random phases. The basic technique is the same as that in Part 3. And Fig. 4 gives the whole encryption and decryption flows. Fig. 4a is the encryption flow. An RGB color image is first decomposed into R, G and B three components, and each component is encrypted independently with an FRFT, a WT and four different random phases. Then the three encoded information are recomposed to be an RGB coded image. Fig. 4b is the decryption process. The three segments of the RGB encoded information are decoded respectively with three series of keys, and then recomposed together to be the reconstructed image after decoding. As shown in Fig. 5, it can be found the encryption process and its effect. Fig. 5a is a JPEG color image with 512 512 pixels. Fig. 5b is its color encryption result, with k1 = 700 nm, k2 = 546.1 nm, p3 = p4 = 0.50; k3 = 435.8 nm, p1 = p2 = 0.40; p5 = p6 = 2/3. f = 300 mm and haar wavelet function is mother wavelet function, which is optionally chosen as one of the exam-
R
FRFT p1 p 2
R11 R21
WT
λ1 RGB color image
G
B
R31 R41
FRFT p3 p 4
R12 R22
WT
λ2
R32 R42
FRFT p5 p 6
RGB coded image
R13 R23
WT
λ3
R33 R43
(a) R
RGB coded image
G
B
* R11* R 21
iWT
* * R31 R 41
λ1
* R12* R22 * * R32 R42
* R13* R23 * 33
R
iFRFT -p1 -p2
* 43
R
iWT
iFRFT -p3 -p4
iWT
iFRFT -p5- p6
λ2
RGB recovered image
λ3
(b) Fig. 4. Color image encryption and decryption flows. (a) The encryption flow; (b) the decryption flow.
3436
L. Chen, D. Zhao / Optics Communications 282 (2009) 3433–3438
ples and people can select other ones in different situations. Fig. 5c is one of the incorrect decryption result with wrong random phases. Fig. 5d is recovered image with incorrect fractional orders p1 = p2 = 0.38; p3 = p4 = 0.35; p5 = p6 = 8/15. Fig. 5e shows the result with p1 = p2 = 0.24; p3 = p4 = 0.20; p5 = p6 = 0.4. It is easy to find that when the fractional orders are quite different from the correct keys, the image could not be seen or only some obscure feature could be found as shown in Fig. 5e. But when the fractional orders are quite close to the correct keys, then some significant characteristics could be seen as shown in the Fig. 5d. Fig. 5f is the correctly reconstructed image. The normalized mean square error (MSE) between the reconstructed image and the original image is often used to verify the quality of the recovered image, and the total MSE is described as
MSE ¼
M X N h i 1 X jf ðmDx; nDyÞ f0 ðmDx; nDyÞj2 ; M N m¼1 n¼1
ð4Þ
where M and N are the pixels of the image, Dx and Dy are the pixel sizes. In Fig. 6, we give the total MSE vs. different fractional orders in RGB three channels, while the red, green and blue lines show the results respectively. We can notice that only when p1 = p2 = 0.4; p3 = p4 = 0.5; p5 = p6 = 2/3, the MSE are minimal and the image can be correctly decrypted, otherwise people would obtain the incorrectly decrypted images with some noise in them. The results well coincide with those of Fig. 5d and e. Fig. 7 is the simulation results with part of the random phases correct and part of them wrong in decryption. Fig. 7a is an RGB picture with 800 600 pixels. Fig. 7b is the result when the random phases in LL1 part are correct but the others are incorrect. We notice that only a quarter of the random keys cannot reconstruct image clearly. Fig. 7c is the result with two correct random phases that are used in LL1 and HH1. While in Fig. 7d, random phases in LH1 and HL1 are correct. In the above two images, some information of the original image is recovered. Fig. 7e shows the recovered image when three quarters of the random phases, in LL1, HL1 and HH1, are correct. In this time, most of the original information is reconstructed. Fig. 7f is the result with the whole correct random phase keys.
Fig. 6. MSE vs. fractional orders in RGB three channels.
Fig. 8 is the simulation results when the keys in some of the three channels (R, G and B channels) are incorrect in decryption. Fig. 8a is an original color image with 2048 1536 pixels. The p1 = p2 = 1.0; encryption parameters are k1 = 700nm, k2 = 546.1 nm, p3 = p4 = 0.50; k3 = 435.8 nm, p5 = p6 = 0.25. Fig. 8b is one of the incorrect decryption images when only the keys in R segment are correct. Fig. 8c is G and B segments correctly decrypted. The color information is changed and incorrectly transferred to people. Fig. 8d is the result when the random keys in LH1 and HH1 of R segment, LL1 and LH1 of G segment and HL1 and HH1 of B segment are correct, but the other keys are wrong in decryption. Fig. 8e is the result with random keys in LL1, LH1 and HH1 of RGB segments correct. Fig. 8f shows the result when keys in all three RGB channels are correctly used in decoding. The keys are increased, so people without keys cannot recover the totally correct information easily.
Fig. 5. Computer simulation results of color image encoding and decoding. (a) An original RGB picture with 512 512 pixels; (b) its color encryption result; (c)–(e) the incorrect decryption results; (f) the correctly reconstructed image. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
L. Chen, D. Zhao / Optics Communications 282 (2009) 3433–3438
3437
Fig. 7. Partial color encoding simulation results. (a) A picture with 800 600 pixels; (b)–(e) the results with the part of the random phase keys correct; (f) the result with whole correct random phase keys. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 8. Another partial color encryption simulation results. (a) An original color image with 2048 1536 pixels; (b)–(e) the results with the keys in some of the three channels incorrectly decrypted; (f) the correct result. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
We also give the total MSE of the partial decryption results. The total MSE between the decryption image in Fig. 7b and the original image in Fig. 7a is about 0.7708, while in Fig. 7c, d and e the results are 0.6803, 0.6658 and 0.5577, respectively, and the corresponding MSE value between the correct decryption image in Fig. 7f and the original image is about 1.277 1027. The values of total MSE are also calculated for the results in Fig. 8. The MSE between Fig. 8b and a is about 0.1613, while the results are 0.0747, 0.1927 and 0.1736 for Fig. 8c, d and e respectively, and the value is 9.6902 1027 between Fig. 8f and the original image. We note that the MSE are different for the different encryption systems, but in each encryption system, the values of MSE well reflect the quality of the decrypted images. 5. Conclusions In summary, we have proposed a color image encryption method in dual fractional Fourier-wavelet domain. FRFT is important, be-
cause it makes the four parts of WT be equally useful in encryption. Fractional orders and four random phases in each channel are the significant keys in encoding. The parameters in all three channels become the keys in encryption and decryption, and the information can be well protected with increasing of keys. When people only have part of the keys, they also cannot get the correct information or even get the wrong information. This method would have largely potential usage in digital color processing and encoding. Acknowledgments This work was supported by the National Natural Science Foundation of China (10874150) and the Program for New Century Excellent Talents in University (NCET-07-0760). References [1] S.H. Shyu, Pattern Recogn. 40 (2007) 1014. [2] H. Cheng, X.B. Li, IEEE Trans. Sign. Proc. 48 (2000) 2439.
3438 [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
L. Chen, D. Zhao / Optics Communications 282 (2009) 3433–3438
K. Martin, R. Lukac, K.N. Plataniotis, Pattern Recogn. 38 (2005) 1111. F. Ahmed, Opt. Exp. 15 (2007) 4804. P.Y. Huang, L.W. Chan, Proc. Lect. Notes Comput. Sci. 2195 (2001) 301. T. Posch, Proc. IEEE-SP Intl. Sym. Time–Frequency Time-Scale Anal. (1992) 143. S. Sudharsanan, IEEE Trans. Cons. Elec. 51 (2005) 1204. R. Lukac, K.N. Plataniotis, Real-Time Imaging 11 (2005) 454. H.H. Nien, C.K. Huang, S.K. Changchien, H.W. Shieh, C.T. Chen, Y.Y. Tuan, Chaos Soliton Fract. 32 (2007) 1070. P. Refregier, B. Javidi, Opt. Lett. 20 (1995) 767. S. Liu, Q. Mi, B. Zhu, Opt. Lett. 26 (2001) 1242. G. Situ, J. Zhang, Opt. Lett. 30 (2005) 1306. G. Unnikrishnan, J. Joseph, K. Singh, Opt. Lett. 25 (2000) 887. E. Tajahuerce, O. Matoba, S.C. Verrall, B. Javidi, Appl. Opt. 39 (2000) 2313. B. Hennelly, J.T. Sheridan, Opt. Lett. 28 (2003) 269. X. Peng, L.F. Yu, L.L. Cai, Opt. Exp. 10 (2002) 41.
[17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
H. Kim, D.H. Kim, Y.H. Lee, Opt. Exp. 12 (2004) 4912. L. Chen, D. Zhao, Opt. Commun. 254 (2005) 361. S.Q. Zhang, M.A. Karim, Microwave Opt. Tech. Lett. 21 (1999) 318. L. Chen, D. Zhao, Opt. Exp. 14 (2006) 8552. L. Chen, D. Zhao, Opt. Exp. 15 (2007) 16080. M. Joshi, C. Shakher, K. Singh, Opt. Commun. 279 (2007) 35. Y. Zhang, C.H. Zheng, N. Tanno, Opt. Commun. 202 (2002) 277. A.M. Youssef, IEEE Signal Proc. Lett. 15 (2008) 77. A.W. Lohmann, J. Opt. Soc. Am. A 10 (1993) 2181. H.M. Ozaktas, Z. Zalevsky, M.A. Kutay, The Fractional Fourier Transforming with Applications in Optics and Signal Processing, Wiley, New York, 2000. [27] H.M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, J. Opt. Soc. Am. A 11 (1994) 547. [28] D. Mendlovic, Z. Zalevsky, D. Mas, J. Garcia, C. Ferreira, Appl. Opt. 36 (1997) 4801.