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A blind image detection method for information hiding with double random-phase encoding
ARTICLE IN PRESS Optics & Laser Technology 41 (2009) 590–595
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Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
A blind image detection method for information hiding with double random-phase encoding Yuan Sheng, Zhou Xin , Chen Jian-guo, Xiao Yong-liang, Liu Qiang Department of Opto-electronics Science and Technology, Sichuan University, 29, South Yi-huan Road, Chengdu 610064, China
a r t i c l e in f o
a b s t r a c t
Article history: Received 3 June 2008 Received in revised form 11 October 2008 Accepted 21 October 2008 Available online 6 December 2008
In this paper, a blind image detection method based on a statistical hypothesis test for information hiding with double random-phase encoding (DRPE) is proposed. This method aims to establish a quantitative criterion which is used to judge whether there is secret information embedded in the detected image. The main process can be described as follows: at the beginning, we decompose the detected gray-scale image into 8 bit planes considering it has 256 gray levels, and suppose that a secret image has been hidden in the detected image after it was encrypted by DRPE, thus the lower bit planes of the detected image exhibit strong randomness. Then, we divide the bit plane to be tested into many windows, and establish a statistical variable to measure the relativity between pixels in every window. Finally, judge whether the secret image exists in the detected image by operating the t test on all statistical variables. Numerical simulation shows that the accuracy is quite satisfactory, when we need to distinguish the images carrying secret information from a large amount of images. & 2008 Elsevier Ltd. All rights reserved.
Keywords: Double random-phase encoding (DRPE) Information hiding t test
1. Introduction Information hiding is to hide secret information in a medium imperceptibly for the purpose of covert communication. In recent years, information hiding technique [1,2] has become an active research area due to its potential applications, such as copyright protection and military communication. On the other hand, people need to extract secret information from its carriers in some conditions, especially in military applications. Because the digital mediums (such as audio, video and pictures) are increasingly furnished on Internet in modern society, there is a tremendous trend to use information hiding techniques on these mediums to transmit message by enemy. So, it becomes more and more important to improve the capability of discriminating suspicious objects from a large number of images. The purpose of steganalysis is to make a decision whether the secret information is embedded in a media. Currently, researchers have proposed many methods to detect the existence of secret information, such as w2 analysis method proposed by Westfeld and others [3], regular singular (RS) steganalysis method using the digital image’s space correlation proposed by Fridrich et al. [4], the raw quick pairs (RQP) method [5] to detect the existence of hidden information in a color image, the steganalysis method of least significant bit (LSB) [6] based on the differential histogram, and so
Corresponding author. Tel.: +86 28 85227168.
E-mail address: [email protected] (Z. Xin). 0030-3992/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2008.10.007
on. All of these methods are used for steganalysis of the LSB information hiding method. Since its proposition by Refregier and Javidi in 1995 and with the succeeding joint efforts of researchers [7–9], the double random-phase encoding (DRPE) has shown great potential in the information hiding area [10–12]. Also it is necessary to find an effective way to detect whether there is any secret information embedded in the carriers by DRPE technique. But at present, the detection method suited for DRPE is rarely reported. And all of those steganalysis methods mentioned above are not applicable to the information hiding with DRPE. In this paper, utilizing the relativity between pixels in different areas of the digital image’s lowest bit plane, we propose a statistical blind detection method for information hiding method with DRPE.
2. A brief description for information hiding method with DRPE The DRPE process can be expressed as [7] gðx; yÞ ¼ FT 1 fFT½f ðx; yÞ yðx; yÞ jðu; vÞg
(1)
where f(x, y) denotes the image to be hidden (the secret image), and FT, FT1 represent the Fourier transform and the inverse Fourier transform, respectively. y(x, y) and j(u, v) are the two random-phase functions in space domain and frequency domain, respectively. g(x, y) denotes the encrypted image, which is a complex image.
ARTICLE IN PRESS Y. Sheng et al. / Optics & Laser Technology 41 (2009) 590–595
Let h(x, y) and c(x, y) denote the host image and the combined image, respectively. c(x, y) can be expressed as [10] cðx; yÞ ¼ hðx; yÞ þ agðx; yÞ
(2)
where a is a constant representing the superposition weight. Obviously, c(x, y) is also a complex image. Because DRPE technique has good diffusion and confusion properties, the information hiding method based on DRPE can resist several type attacks, such as occlusion, JPEG compression, fading, filtering, and so on [10]. The computer simulation results for these distortions and the detail decryption method for the secret image from the combined image is described in Ref. [10].
3. The blind image detection method for information hiding with DRPE technique based on a statistical hypothesis test From the above description, we know that the information hiding method with DRPE can be summarized as that the product, of the encrypted image and a small superposition weight a, is superposed to the host image. The influence of this product is mainly on the lower bit planes of the host image, because the superposition weight a is very small. In contrast, the LSB information hiding method is to replace the lowest bit plane of the host image by some secret data according to a certain rule, so there is no impact on other bit planes. Superposition and replacement are the essential difference between these two information hiding methods. It is this difference that makes some detection methods be applicable to LSB steganography, but not to information hiding with DRPE. In the following steps, we propose a detection method which can be applied to the information hiding method with DRPE. 3.1. The bit planes of a gray-scale image and the combined image Every gray-scale image with 256 gray levels (such as Fig. 1(a)) can be decomposed into 8 bit planes as shown in Fig. 1(b)–(i) from the highest bit plane 7 to the lowest bit plane 0. Here, we can find two features: first, if the higher bit plane exhibits randomness, so does the corresponding area of the lower bit plane; second, the randomness is gradually stronger from the bit plane 7–0, but the
591
relativity still exists between different pixels in some areas of the lowest bit plane. Subsequently, we take Fig. 1(a) as the host image, and another gray-scale image shown in Fig. 2(a) as the image to be hidden (the secret image). According to the information hiding method described in Section 2, the combined image is obtained and shown in Fig. 2(b). Its 8 bit planes are displayed in Fig. 2(c)–(i). Here, the superposition weight a is 0.03. Comparing Fig. 1 with Fig. 2, we can find that the lower bit planes of the combined image exhibit strong randomness. Its reason is that the secret image is superposed to the host image after it has been encrypted. Thus, we can judge whether or not the secret image is hidden in the detected image only by the relativity between the pixels in its lower bit planes. 3.2. Steps of the blind image detection method The main idea of this method is that: we take a statistical hypothesis test on the lowest bit plane of the detected image. If the test result shows that the lowest bit plane exhibits strong randomness, we suspect that a secret image is hidden in the detected image; otherwise, we think there is no secret image in the detected image. The quantitative criterion can be expressed as follows: Firstly, we suppose that the secret image is encrypted and hidden in the detected image, thus its lowest bit plane should exhibit strong randomness. In general, the probability of two gray values, 0 and 1, are 1/2 [13]. So we can make several inferences as below. 1 1 ; pðr i ¼ 1Þ ¼ ) 2 2 1 1 2 2 pðr i ¼ 0Þ ¼ ; pðr i ¼ 1Þ ¼ ) 2 2 1 1 Eðr i Þ ¼ ; Eðr 2i Þ ¼ ) 2 2 1 Dðr i Þ ¼ Eðr 2i Þ ½Eðr i Þ2 ¼ 4 1 1 p0 ¼ pðr i r j ¼ 0Þ ¼ ; p1 ¼ pðr i r j ¼ 1Þ ¼ ) 2 2 1 Eðr i r j Þ ¼ 2
pðr i ¼ 0Þ ¼
Fig. 1. (a) A gray-scale image with 256 gray levels; (b)–(i) its 8 bit planes from the bit plane 7 to 0, orderly.
(3)
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Fig. 2. (a) Secret image; (b) combined image (real part) when a ¼ 0.03; (c)–(j) the 8 bit planes of the combined image from the bit plane 7–0, orderly.
where ri and rj are the gray values of the ith and jth pixels in the lowest bit plane. p(d) denotes the probability, E(d) the mathematical expectation, D(d) the variance, and ‘‘’’ expresses the XOR (Exclusive OR) operation. Secondly, suppose the lowest bit plane with w h pixels, and W is any one window with c c pixels in the bit plane. The window W is defined as W ¼ fðxi ; yj Þjxp pxi pxpþc ; yq pyj pyqþc g p 2 ð1; w c þ 1Þ;
q 2 ð1; h c þ 1Þ
(4)
Every pixel’s gray value r1,r2,y,rn can be gotten by the orderly scanning within a window, where n is the pixel’s number in a window. According to the supposition, variables r1, r2, y, rn are randomly independent. Therefore, we establish a one-dimensional (1D) statistical variable T, which could be used to measure the relativity between pixels in the sequence. It can be written as Pnd Pnd Pnd nd r i r iþd i¼1 r i Þ i¼1 r i r iþd 2 pi¼1 ffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Pnd 1 n d 2 Dð i¼1 r i Þ X 2 nd nd ¼ pffiffiffiffiffiffiffiffiffiffiffiffi r r iþd i¼1 i 2 nd
T¼
Eð
(5)
where d is the displacement of the XOR operation (i.e. the distance between the two pixels ri and ri+d). Then, a statistical variable T can be derived from every window. So a group of statistical variables, which reflect the relativity
between different pixels in different areas of the lowest bit plane, can be obtained through the above described process. These variables compose a statistical sample TA (supposing there are M windows in the bit plane to be tested); that is T A ¼ fT i jT 1 ; T 2 ; ; T M g
(6)
where M ¼ (wc+1)*(hc+1). According to the central limit theorem [14,15], TA trends to exhibit N(0,1) distribution (the standard normal distribution) when nd-N. Finally, we carry out the statistical hypothesis test on the sample TA. Because the variance of the population is unknown, we choose the t test [14,15] as a quantitative criterion to judge whether the relativity exists or not between the different pixels in the lowest bit plane. The t test can be expressed as t¼
m m0
pffiffiffiffiffi tðM 1Þ S= M
(7)
where m is the mean of the sample TA. m0 is the mean of the standard normal distribution, so m0 ¼ 0. S is the unbiased estimation of the population’s standard deviation [14,15]. t(d) denotes the t distribution. We make judgments under the condition that the significance level a is given, and the fraction 1a is called the confidence coefficient or the degree of confidence [14,15] H0 ¼ m ¼ m0 ¼ 0 The secret image is hidden in the detected image
H0 ¼ mam0 ¼ 0 The secret image is not hidden in the detected image
ARTICLE IN PRESS Y. Sheng et al. / Optics & Laser Technology 41 (2009) 590–595
This is a two-tailed test; that is, when jtjXt a=2 ðM 1Þ
(8)
we reject H0, otherwise, accept H0. 3.3. The method discussion From the above description, one can learn that the main idea of this method is to divide the lowest bit plane into a lot of windows, then establish an area statistical variable, which reflects the relativity between pixels in each window, and finally use all the variables’ value to measure the relativity between pixels in the whole bit plane. Of course, we can directly calculate the statistical variable in the whole bit plane without using the window method, but obviously this will reduce the statistical variable’s value, and cannot accurately reflect the relativity between pixels, especially for those images whose some areas of the lowest bit plane, with high relativity between pixels are decentralized. However, if we use the window method, this means that the relativity between pixels in part of the area is separated from the whole bit plane. The window method plays a role of amplification and improves the detection accuracy.
4. The electro-optical system for information hiding detection In the method mentioned above, XOR operation is a most timeconsuming step. The authors of Ref. [16] have done an experiment to implement the optical XOR operation. Therefore, we quote their electro-optical setup shown in Fig. 3 [16] to illustrate that XOR operation can be quickly achieved by the optical method because of its inherent parallel and high-speed processing capability. This setup can also be used in our method. At the beginning, the lowest bit plane of the detected image is divided into M windows with the size of c c. Then, pixels from the first row to the (c1)th row compose the image A as shown in Fig. 3, and from the second row to the last row compose the image B (in this method, we take the displacement of the XOR operation d ¼ c). The XOR operation of the two images A and B can be achieved by utilizing spatial light modulators (SLMs) that modulate the polarization [16]. In Fig. 3, MA and MB are SLMs that are used to implement the parallel input. The read light is an s-polarized light. The reflected light
593
rotates its polarization by p/2 in the write light incident region (i.e. the place where the pixel’s value is 1). On the other hand, the read light does not change its polarization in the nonwrite light incident region (i.e. the place where the pixel’s value is 0). MB works with the same principle as MA. Only when the corresponding pixel’s values of A and B are different, the read light rotates its polarization and changes into p-polarized light. Otherwise, it is still the s-polarized light. Thus the XOR operation is implemented when the light goes through the p-polarizer (Pp). Finally, the XOR operation results are imported into the computer for further processing. In Fig. 3, BS is beam splitter; CL is cylindrical lens.
5. Numerical simulation and result analysis To test the feasibility of the method proposed in this paper, numerical simulations are carried out on JHH image database, which was collected by Hateren [17]. The first 4000 images with 128 192 pixels are selected from the database. We hide different secret images in the former 2000 images, respectively, thus the combined images are composed; here, the superposition weight of a ¼ 0.03. And the latter 2000 images without secret image are taken as the original images. Otherwise, we take W ¼ 16 16 as the size of window, and d ¼ 16 as the displacement value which is equal to the column number of pixels in a window. If the sample size of TA is large enough, say M1X30, the distribution of TA does not differ considerably from the standard normal [14,15]. Here, the sample size of M ¼ 20001, so under the condition that the significance level of a ¼ 0.05 is given t a=2 ðM 1Þ ¼ t 0:025 ðM 1Þ z0:025
(9)
where z0.025 represents the z value above which we find an area equal to 0.025 in the curve of the standard normal distribution [14,15]. Index upon the appendix of the Refs. [14,15], we can know z0.025 ¼ 1.96. Now, we take the t test on every statistical sample TA obtained from every detected image. The results are shown in Fig. 4. From Fig. 4, we can see that the test results of all the images with secret image (the combined images) fall in the acceptance region (i.e. |t|pt0.025(M1)E1.96), and of most images, without secret image (the original images) fall outside the acceptance region. According to statistics, the values of |t| for 248 images without secret image are o1.96, thus the false alarm rate is 12.4%.
Read light (s)
(c-1)*c Reflecting mirror MA
A
BS1 CL The image to be detected
The lowest bit plane
M
2
1
The M windows CL A XOR B
B
CCD
MB
computer (c-1)*c
BS2
Fig. 3. Illustration of the electro-optical system for information hiding detection.
Pp
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Fig. 4. The values of |t| for bit plane 0 of (a) original images and (b) combined images.
Fig. 5. The values of |t| for bit plane 1 of (a) original images and (b) combined images.
To solve the problem of false alarm, we pay attention to such facts: the information hiding method with DRPE limits the value of superposition weight a, generally, it should not be less than 102; otherwise, the receiver will not recover the secret image from the combined image. Under this condition, the bit plane 1 of the combined image still exhibits strong randomness. However, the relativity between pixels in the bit plane 1 is higher than that in the bit plane 0 for the original image, so we can test on the bit plane 1 of an image to reduce the false alarm rate. Fig. 5 shows the test results for the bit plane 1. The superposition weight a is also 0.03. From Fig. 5, we can see that the test results of all the combined images still fall in the acceptance region, so do the test results of some original images, but the number decreases significantly. There are only 76 original images whose |t| values are o1.96, thus the false alarm rate is 3.8%. Because the relativity between pixels in the bit plane 1 of the original image is higher than that in the bit plane 0, we can directly test on the bit plane 1 of an image for the information hiding method with DRPE. Thus, it can improve the detection accuracy and reduce the false alarm rate.
6.. Conclusion In this paper, considering the differences between the information hiding method with DRPE technique and LSB
steganography method, we propose a novel steganalysis method, which can blindly detect the existence of secret image for information hiding method with DRPE technique. In this method, the use of the statistical hypothesis test makes the method more scientific, so the test results are more credible. To improve the processing speed, we quote an electro-optical system from Ref. [16], which utilizes the inherent parallel and high-speed processing capability of optics. In our numerical experiments, a database composed of 4000 images is utilized to evaluate the performance of our proposed method and the numerical simulation results support this method. References [1] Katzenbeisser S, Petitcolas FP. Information hiding techniques for steganography and digital watermarking. Artech House, New York; 2000. [2] Cox IJ, Miller ML, Bloom JA. Digital watermarking. New York: Harcourt Science and Technology; 2001. [3] Westfeld A, Pfitzmann A. Attacks on steganographic systems. In third international workshop on information hiding, Lecture Notes in Computer Science 1999; 1768: p. 61–76. [4] Fridrich J, Goljan M, Du R. Detecting LSB steganography in color and gray-scale images. Mag IEEE Multimedia 2001;22-8 [Special Issue on Security]. [5] Fridrich J, Du R, Long M. Steganalysis of LSB encoding in color images. In: Proc IEEE Int Conf Multimedia Expo. New York; 2000, p. 1279–82. [6] Zhang T, Xi JP. A new approach to reliable detection of LSB steganography in natural images. Signal Process 2003;83:2085–93. [7] Refregier P, Javidi B. Optical image encryption based on input plane and Fourier plane random encoding. Opt Lett 1995;20:767–9.
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[8] Zhou X, Yuan S, Wang SW, Xie J. Affine cryptosystem of double-random-phase encryption based on the fractional Fourier transform. Appl Opt 2006;45: 8434–9. [9] Hennelly BM, Sheridan JT. Optical encryption and the space bandwidth product. Opt Commun 2005;247:291–305. [10] Kishk S, Javidi B. Information hiding technique with double phase encoding. Appl Opt 2002;41:5462–70. [11] Rosen J, Javidi B. Hidden images in halftone pictures. Appl Opt 2001;40: 3346–53. [12] Zhou X, Lai D, Yuan S, Li DH, Hu JP. A method for hiding information utilizing double-random phase-encoding technique. Opt Laser Technol 2007;39:1360–3.
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[13] Wang SZ, Zhang XP, Zhang KW. Digital steganography and steganalysis. Tsing Hua University press; 2005 [in Chinese], p. 22–23. [14] Walpole RE, Meyers HR, Meyers LS, Ye K. Probability & statistics for engineers & scientists. Englewood, Cliffs, NJ: Prentice Hall; 2002. [15] Rohatgi VK. An introduction to probability theory and mathematical statistics. New York: Wiley Inc.; 1976. [16] Fukushima S, Kurokawa T, Suzuki H. Optical implementation of parallel digital adder and subtractor. Appl Opt 1990;29:2099–106. [17] Hateren JH, Schaaf A. Independent component filters of natural images compared with simple cells in primary visual cortex. Proc Soc London B 1998;265:359–66 [http://hLaboratoryphys.rug.nl/imlib/index.html].
Contents lists available at ScienceDirect
Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
A blind image detection method for information hiding with double random-phase encoding Yuan Sheng, Zhou Xin , Chen Jian-guo, Xiao Yong-liang, Liu Qiang Department of Opto-electronics Science and Technology, Sichuan University, 29, South Yi-huan Road, Chengdu 610064, China
a r t i c l e in f o
a b s t r a c t
Article history: Received 3 June 2008 Received in revised form 11 October 2008 Accepted 21 October 2008 Available online 6 December 2008
In this paper, a blind image detection method based on a statistical hypothesis test for information hiding with double random-phase encoding (DRPE) is proposed. This method aims to establish a quantitative criterion which is used to judge whether there is secret information embedded in the detected image. The main process can be described as follows: at the beginning, we decompose the detected gray-scale image into 8 bit planes considering it has 256 gray levels, and suppose that a secret image has been hidden in the detected image after it was encrypted by DRPE, thus the lower bit planes of the detected image exhibit strong randomness. Then, we divide the bit plane to be tested into many windows, and establish a statistical variable to measure the relativity between pixels in every window. Finally, judge whether the secret image exists in the detected image by operating the t test on all statistical variables. Numerical simulation shows that the accuracy is quite satisfactory, when we need to distinguish the images carrying secret information from a large amount of images. & 2008 Elsevier Ltd. All rights reserved.
Keywords: Double random-phase encoding (DRPE) Information hiding t test
1. Introduction Information hiding is to hide secret information in a medium imperceptibly for the purpose of covert communication. In recent years, information hiding technique [1,2] has become an active research area due to its potential applications, such as copyright protection and military communication. On the other hand, people need to extract secret information from its carriers in some conditions, especially in military applications. Because the digital mediums (such as audio, video and pictures) are increasingly furnished on Internet in modern society, there is a tremendous trend to use information hiding techniques on these mediums to transmit message by enemy. So, it becomes more and more important to improve the capability of discriminating suspicious objects from a large number of images. The purpose of steganalysis is to make a decision whether the secret information is embedded in a media. Currently, researchers have proposed many methods to detect the existence of secret information, such as w2 analysis method proposed by Westfeld and others [3], regular singular (RS) steganalysis method using the digital image’s space correlation proposed by Fridrich et al. [4], the raw quick pairs (RQP) method [5] to detect the existence of hidden information in a color image, the steganalysis method of least significant bit (LSB) [6] based on the differential histogram, and so
Corresponding author. Tel.: +86 28 85227168.
E-mail address: [email protected] (Z. Xin). 0030-3992/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2008.10.007
on. All of these methods are used for steganalysis of the LSB information hiding method. Since its proposition by Refregier and Javidi in 1995 and with the succeeding joint efforts of researchers [7–9], the double random-phase encoding (DRPE) has shown great potential in the information hiding area [10–12]. Also it is necessary to find an effective way to detect whether there is any secret information embedded in the carriers by DRPE technique. But at present, the detection method suited for DRPE is rarely reported. And all of those steganalysis methods mentioned above are not applicable to the information hiding with DRPE. In this paper, utilizing the relativity between pixels in different areas of the digital image’s lowest bit plane, we propose a statistical blind detection method for information hiding method with DRPE.
2. A brief description for information hiding method with DRPE The DRPE process can be expressed as [7] gðx; yÞ ¼ FT 1 fFT½f ðx; yÞ yðx; yÞ jðu; vÞg
(1)
where f(x, y) denotes the image to be hidden (the secret image), and FT, FT1 represent the Fourier transform and the inverse Fourier transform, respectively. y(x, y) and j(u, v) are the two random-phase functions in space domain and frequency domain, respectively. g(x, y) denotes the encrypted image, which is a complex image.
ARTICLE IN PRESS Y. Sheng et al. / Optics & Laser Technology 41 (2009) 590–595
Let h(x, y) and c(x, y) denote the host image and the combined image, respectively. c(x, y) can be expressed as [10] cðx; yÞ ¼ hðx; yÞ þ agðx; yÞ
(2)
where a is a constant representing the superposition weight. Obviously, c(x, y) is also a complex image. Because DRPE technique has good diffusion and confusion properties, the information hiding method based on DRPE can resist several type attacks, such as occlusion, JPEG compression, fading, filtering, and so on [10]. The computer simulation results for these distortions and the detail decryption method for the secret image from the combined image is described in Ref. [10].
3. The blind image detection method for information hiding with DRPE technique based on a statistical hypothesis test From the above description, we know that the information hiding method with DRPE can be summarized as that the product, of the encrypted image and a small superposition weight a, is superposed to the host image. The influence of this product is mainly on the lower bit planes of the host image, because the superposition weight a is very small. In contrast, the LSB information hiding method is to replace the lowest bit plane of the host image by some secret data according to a certain rule, so there is no impact on other bit planes. Superposition and replacement are the essential difference between these two information hiding methods. It is this difference that makes some detection methods be applicable to LSB steganography, but not to information hiding with DRPE. In the following steps, we propose a detection method which can be applied to the information hiding method with DRPE. 3.1. The bit planes of a gray-scale image and the combined image Every gray-scale image with 256 gray levels (such as Fig. 1(a)) can be decomposed into 8 bit planes as shown in Fig. 1(b)–(i) from the highest bit plane 7 to the lowest bit plane 0. Here, we can find two features: first, if the higher bit plane exhibits randomness, so does the corresponding area of the lower bit plane; second, the randomness is gradually stronger from the bit plane 7–0, but the
591
relativity still exists between different pixels in some areas of the lowest bit plane. Subsequently, we take Fig. 1(a) as the host image, and another gray-scale image shown in Fig. 2(a) as the image to be hidden (the secret image). According to the information hiding method described in Section 2, the combined image is obtained and shown in Fig. 2(b). Its 8 bit planes are displayed in Fig. 2(c)–(i). Here, the superposition weight a is 0.03. Comparing Fig. 1 with Fig. 2, we can find that the lower bit planes of the combined image exhibit strong randomness. Its reason is that the secret image is superposed to the host image after it has been encrypted. Thus, we can judge whether or not the secret image is hidden in the detected image only by the relativity between the pixels in its lower bit planes. 3.2. Steps of the blind image detection method The main idea of this method is that: we take a statistical hypothesis test on the lowest bit plane of the detected image. If the test result shows that the lowest bit plane exhibits strong randomness, we suspect that a secret image is hidden in the detected image; otherwise, we think there is no secret image in the detected image. The quantitative criterion can be expressed as follows: Firstly, we suppose that the secret image is encrypted and hidden in the detected image, thus its lowest bit plane should exhibit strong randomness. In general, the probability of two gray values, 0 and 1, are 1/2 [13]. So we can make several inferences as below. 1 1 ; pðr i ¼ 1Þ ¼ ) 2 2 1 1 2 2 pðr i ¼ 0Þ ¼ ; pðr i ¼ 1Þ ¼ ) 2 2 1 1 Eðr i Þ ¼ ; Eðr 2i Þ ¼ ) 2 2 1 Dðr i Þ ¼ Eðr 2i Þ ½Eðr i Þ2 ¼ 4 1 1 p0 ¼ pðr i r j ¼ 0Þ ¼ ; p1 ¼ pðr i r j ¼ 1Þ ¼ ) 2 2 1 Eðr i r j Þ ¼ 2
pðr i ¼ 0Þ ¼
Fig. 1. (a) A gray-scale image with 256 gray levels; (b)–(i) its 8 bit planes from the bit plane 7 to 0, orderly.
(3)
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Fig. 2. (a) Secret image; (b) combined image (real part) when a ¼ 0.03; (c)–(j) the 8 bit planes of the combined image from the bit plane 7–0, orderly.
where ri and rj are the gray values of the ith and jth pixels in the lowest bit plane. p(d) denotes the probability, E(d) the mathematical expectation, D(d) the variance, and ‘‘’’ expresses the XOR (Exclusive OR) operation. Secondly, suppose the lowest bit plane with w h pixels, and W is any one window with c c pixels in the bit plane. The window W is defined as W ¼ fðxi ; yj Þjxp pxi pxpþc ; yq pyj pyqþc g p 2 ð1; w c þ 1Þ;
q 2 ð1; h c þ 1Þ
(4)
Every pixel’s gray value r1,r2,y,rn can be gotten by the orderly scanning within a window, where n is the pixel’s number in a window. According to the supposition, variables r1, r2, y, rn are randomly independent. Therefore, we establish a one-dimensional (1D) statistical variable T, which could be used to measure the relativity between pixels in the sequence. It can be written as Pnd Pnd Pnd nd r i r iþd i¼1 r i Þ i¼1 r i r iþd 2 pi¼1 ffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Pnd 1 n d 2 Dð i¼1 r i Þ X 2 nd nd ¼ pffiffiffiffiffiffiffiffiffiffiffiffi r r iþd i¼1 i 2 nd
T¼
Eð
(5)
where d is the displacement of the XOR operation (i.e. the distance between the two pixels ri and ri+d). Then, a statistical variable T can be derived from every window. So a group of statistical variables, which reflect the relativity
between different pixels in different areas of the lowest bit plane, can be obtained through the above described process. These variables compose a statistical sample TA (supposing there are M windows in the bit plane to be tested); that is T A ¼ fT i jT 1 ; T 2 ; ; T M g
(6)
where M ¼ (wc+1)*(hc+1). According to the central limit theorem [14,15], TA trends to exhibit N(0,1) distribution (the standard normal distribution) when nd-N. Finally, we carry out the statistical hypothesis test on the sample TA. Because the variance of the population is unknown, we choose the t test [14,15] as a quantitative criterion to judge whether the relativity exists or not between the different pixels in the lowest bit plane. The t test can be expressed as t¼
m m0
pffiffiffiffiffi tðM 1Þ S= M
(7)
where m is the mean of the sample TA. m0 is the mean of the standard normal distribution, so m0 ¼ 0. S is the unbiased estimation of the population’s standard deviation [14,15]. t(d) denotes the t distribution. We make judgments under the condition that the significance level a is given, and the fraction 1a is called the confidence coefficient or the degree of confidence [14,15] H0 ¼ m ¼ m0 ¼ 0 The secret image is hidden in the detected image
H0 ¼ mam0 ¼ 0 The secret image is not hidden in the detected image
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This is a two-tailed test; that is, when jtjXt a=2 ðM 1Þ
(8)
we reject H0, otherwise, accept H0. 3.3. The method discussion From the above description, one can learn that the main idea of this method is to divide the lowest bit plane into a lot of windows, then establish an area statistical variable, which reflects the relativity between pixels in each window, and finally use all the variables’ value to measure the relativity between pixels in the whole bit plane. Of course, we can directly calculate the statistical variable in the whole bit plane without using the window method, but obviously this will reduce the statistical variable’s value, and cannot accurately reflect the relativity between pixels, especially for those images whose some areas of the lowest bit plane, with high relativity between pixels are decentralized. However, if we use the window method, this means that the relativity between pixels in part of the area is separated from the whole bit plane. The window method plays a role of amplification and improves the detection accuracy.
4. The electro-optical system for information hiding detection In the method mentioned above, XOR operation is a most timeconsuming step. The authors of Ref. [16] have done an experiment to implement the optical XOR operation. Therefore, we quote their electro-optical setup shown in Fig. 3 [16] to illustrate that XOR operation can be quickly achieved by the optical method because of its inherent parallel and high-speed processing capability. This setup can also be used in our method. At the beginning, the lowest bit plane of the detected image is divided into M windows with the size of c c. Then, pixels from the first row to the (c1)th row compose the image A as shown in Fig. 3, and from the second row to the last row compose the image B (in this method, we take the displacement of the XOR operation d ¼ c). The XOR operation of the two images A and B can be achieved by utilizing spatial light modulators (SLMs) that modulate the polarization [16]. In Fig. 3, MA and MB are SLMs that are used to implement the parallel input. The read light is an s-polarized light. The reflected light
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rotates its polarization by p/2 in the write light incident region (i.e. the place where the pixel’s value is 1). On the other hand, the read light does not change its polarization in the nonwrite light incident region (i.e. the place where the pixel’s value is 0). MB works with the same principle as MA. Only when the corresponding pixel’s values of A and B are different, the read light rotates its polarization and changes into p-polarized light. Otherwise, it is still the s-polarized light. Thus the XOR operation is implemented when the light goes through the p-polarizer (Pp). Finally, the XOR operation results are imported into the computer for further processing. In Fig. 3, BS is beam splitter; CL is cylindrical lens.
5. Numerical simulation and result analysis To test the feasibility of the method proposed in this paper, numerical simulations are carried out on JHH image database, which was collected by Hateren [17]. The first 4000 images with 128 192 pixels are selected from the database. We hide different secret images in the former 2000 images, respectively, thus the combined images are composed; here, the superposition weight of a ¼ 0.03. And the latter 2000 images without secret image are taken as the original images. Otherwise, we take W ¼ 16 16 as the size of window, and d ¼ 16 as the displacement value which is equal to the column number of pixels in a window. If the sample size of TA is large enough, say M1X30, the distribution of TA does not differ considerably from the standard normal [14,15]. Here, the sample size of M ¼ 20001, so under the condition that the significance level of a ¼ 0.05 is given t a=2 ðM 1Þ ¼ t 0:025 ðM 1Þ z0:025
(9)
where z0.025 represents the z value above which we find an area equal to 0.025 in the curve of the standard normal distribution [14,15]. Index upon the appendix of the Refs. [14,15], we can know z0.025 ¼ 1.96. Now, we take the t test on every statistical sample TA obtained from every detected image. The results are shown in Fig. 4. From Fig. 4, we can see that the test results of all the images with secret image (the combined images) fall in the acceptance region (i.e. |t|pt0.025(M1)E1.96), and of most images, without secret image (the original images) fall outside the acceptance region. According to statistics, the values of |t| for 248 images without secret image are o1.96, thus the false alarm rate is 12.4%.
Read light (s)
(c-1)*c Reflecting mirror MA
A
BS1 CL The image to be detected
The lowest bit plane
M
2
1
The M windows CL A XOR B
B
CCD
MB
computer (c-1)*c
BS2
Fig. 3. Illustration of the electro-optical system for information hiding detection.
Pp
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Fig. 4. The values of |t| for bit plane 0 of (a) original images and (b) combined images.
Fig. 5. The values of |t| for bit plane 1 of (a) original images and (b) combined images.
To solve the problem of false alarm, we pay attention to such facts: the information hiding method with DRPE limits the value of superposition weight a, generally, it should not be less than 102; otherwise, the receiver will not recover the secret image from the combined image. Under this condition, the bit plane 1 of the combined image still exhibits strong randomness. However, the relativity between pixels in the bit plane 1 is higher than that in the bit plane 0 for the original image, so we can test on the bit plane 1 of an image to reduce the false alarm rate. Fig. 5 shows the test results for the bit plane 1. The superposition weight a is also 0.03. From Fig. 5, we can see that the test results of all the combined images still fall in the acceptance region, so do the test results of some original images, but the number decreases significantly. There are only 76 original images whose |t| values are o1.96, thus the false alarm rate is 3.8%. Because the relativity between pixels in the bit plane 1 of the original image is higher than that in the bit plane 0, we can directly test on the bit plane 1 of an image for the information hiding method with DRPE. Thus, it can improve the detection accuracy and reduce the false alarm rate.
6.. Conclusion In this paper, considering the differences between the information hiding method with DRPE technique and LSB
steganography method, we propose a novel steganalysis method, which can blindly detect the existence of secret image for information hiding method with DRPE technique. In this method, the use of the statistical hypothesis test makes the method more scientific, so the test results are more credible. To improve the processing speed, we quote an electro-optical system from Ref. [16], which utilizes the inherent parallel and high-speed processing capability of optics. In our numerical experiments, a database composed of 4000 images is utilized to evaluate the performance of our proposed method and the numerical simulation results support this method. References [1] Katzenbeisser S, Petitcolas FP. Information hiding techniques for steganography and digital watermarking. Artech House, New York; 2000. [2] Cox IJ, Miller ML, Bloom JA. Digital watermarking. New York: Harcourt Science and Technology; 2001. [3] Westfeld A, Pfitzmann A. Attacks on steganographic systems. In third international workshop on information hiding, Lecture Notes in Computer Science 1999; 1768: p. 61–76. [4] Fridrich J, Goljan M, Du R. Detecting LSB steganography in color and gray-scale images. Mag IEEE Multimedia 2001;22-8 [Special Issue on Security]. [5] Fridrich J, Du R, Long M. Steganalysis of LSB encoding in color images. In: Proc IEEE Int Conf Multimedia Expo. New York; 2000, p. 1279–82. [6] Zhang T, Xi JP. A new approach to reliable detection of LSB steganography in natural images. Signal Process 2003;83:2085–93. [7] Refregier P, Javidi B. Optical image encryption based on input plane and Fourier plane random encoding. Opt Lett 1995;20:767–9.
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