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OPTICS LETTERS / Vol. 32, No. 13 / July 1, 2007
Multiple-image hiding in the Fresnel domain Yishi Shi,1,* Guohai Situ,2 and Jingjuan Zhang1 1
College of Physical Sciences, Graduate University of the Chinese Academy of Sciences, Beijing, China 2 School of Electrical, Electronic and Mechanical Engineering, College of Engineering, Mathematics and Physical Sciences, University College Dublin, Belfield, Dublin 4, Ireland *Corresponding author:
[email protected]
Received February 9, 2007; revised April 26, 2007; accepted May 1, 2007; posted May 4, 2007 (Doc. ID 79927); published June 22, 2007 To avoid additive cross-talk among multiple images, they are separated from each other and even from the host image in the hiding and the extraction process by locating them independently in the Fresnel domain with computer-generated double phase-only masks. Gray images and white-ground and black-ground binary images can be hidden in the system at the same time. The hiding capacity of the system as to these three types of multiple image has also been investigated using a cascaded iterative algorithm. The security and the multiplexing parameters of the system are analyzed as well. © 2007 Optical Society of America OCIS codes: 070.2580, 070.4560, 100.0100, 050.1970.
One of most attractive issues in optical security is multiple-image hiding, which is widely applied in multiple-user authentications, content distribution, and improving the efficiency of secret information transmission [1]. Recently proposed hiding techniques include the Hadamard matrix method [2], digital holography [3], and wavelength multiplexing [4]. The multiple-image hiding process of all these methods includes encrypted multiple images’ being added together and then being embedded in the host image. Since the additive cross-talk arising from their mutual disturbances results in the evident decline of the quality of the multiple extractions, these techniques are in fact much more suitable for binary images than for gray images, and the hiding capacity is also limited. Thus, the performance of a multipleimage security system will be improved by depressing cross-talk. Furthermore, in this paper, we thoroughly eliminate additive cross-talk by separating multiple images from each other and even from the host image in the hiding process by using computergenerated double phase-only masks (POMs) in the Fresnel domain. The optical setup of our system is shown in Fig. 1. The original host image and the secret multipleimage set are defined as fo and 兵goi其, respectively, where i represents that goi is the ith image to be hidden. As each plane is vertically located in the axis in the Fresnel domain, only the z coordinate is retained to discriminate the different planes. The distance parameters z0 and 兵zi其 and the operation wavelength all serve as additional keys. They need to fulfill the Fresnel approximation according to the size of double POMs. POM1 is attached to fo in the input plane, while POM2 is in the transform plane. The phase distributions of the double POMs that serve as the phase keys are generated in the computer by the chosen algorithm, which is the hiding process of multiple images. Correspondingly, in the extraction process, when the plane wave of the wavelength perpendicularly illuminates the system, the watermarked image f is modulated sequentially by the double POMs, and the multiple extractions 兵gi其 will be dis0146-9592/07/131914-3/$15.00
played independently in predefined locations 兵zi其. Both digital and optical extraction processes are available. In the latter case, we can realize double POMs with spatial light modulators and obtain the intensities of multiple extractions by CCD. Due to the configuration, the secret multiple images and the host image are all independently located in the Fresnel domain. Therefore, common additive crosstalk is avoided thoroughly in our system, and high quality of multiple extractions and satisfactory hiding capacity are achieved. Next let us briefly describe the cascade phase retrieval algorithm chosen as an example to search the solutions of double POMs. First, their phase distributions and are initialized randomly. If the number of secret multiple images is assumed to be N, each iteration consists of N loops. In the ith loop for the secret image goi in the kth iteration, the host image fo is Fresnel transformed to the output plane of goi and then is inverse Fresnel transformed back to the input plane, with the modulation of double POMs expressed as k 兲兴 = T, uko exp共jk兲 = FrTz 关fo exp共ji+ 0
k 兲兴, gik exp共jk兲 = FrTz 关T exp共ji+ i
共1兲
k 兲 = IFrTz 关goi exp共jk兲兴, uk exp共ji− i
k 兲 = IFrTz 关uk exp共jk兲兴, fk exp共ji− 0
共2兲
where FrTz and IFrTz denote the Fresnel transform and its inversion with the wavelength and the dis-
Fig. 1. Optical setup of multiple-image hiding system. © 2007 Optical Society of America
July 1, 2007 / Vol. 32, No. 13 / OPTICS LETTERS
tance z, respectively, and superscript k means the kth iteration. The subscripts ⫹ and ⫺ are used to distinguish between the phase distributions generated before and within the ith loop, respectively. The input and the output amplitude are constrained to fo and the square root of goi, respectively, for optical detection can acquire only the intensity of goi. In the next loop, goi is changed to go共i+1兲, and the phase distributions of double POMs are replaced by k k , 共i+1兲+ = i−
k k 共i+1兲+ = i− − k .
共3兲
The kth iteration is finished when N loops from go1 to goN are done. We choose the correlation coefficient 共Co兲 between the iterative image and the original secret image to evaluate the convergence of the algorithm defined as Co共t,to兲 = cov共t,to兲共t · t0兲−1 ,
共4兲
where cov共t , to兲 is the cross-covariance between t and to, is the standard deviation, to denotes fo and goi, and t represents fk and gik. The value of Co ranges over [0, 1]. The maximum value of unity means t is perfectly correlated with to. Hence, the undetectability of the watermarked image fk is measured by Co共fk , fo兲 [5], and 兵Co共gk , go兲其 corresponds to the quality of the extracted multiple-image. Either the Co value or the number of iterations can be employed to terminate the iterations. Computer simulations named Group 1 have demonstrated that gray, white-ground, and black-ground binary images can be hidden in the proposed system at the same time. Figure 2(a) is the original host image Lena, and the secret multiple images airplane 共g1兲, Olympic Games 共g2兲, and Beijing 2008 共g3兲 are shown in Figs. 3(a) and 3(b), all with 256 ⫻ 256 pixels of 15.625 m pixel pitch. g1 is a 256 gray-level image. g2 and g3 are the white-ground and the black-ground binary image, respectively. The size of the double POMs is the same as that of the images, and the operation wavelength is 532 nm. The distance z0 between the double POMs is 60 mm, and three secret images are located at 共z1 , z2 , z3兲 of 共20, 60, 100兲 mm away from POM2. We used an Intel
Fig. 2. (a) Host image and (b) initial and (c) result phase key of POM1.
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Pentium 4 CPU 3.2 G PC with 1 G RAM for the simulations. By using the algorithm above, after 14 iterations in 1 min, the Co values of three extracted images are (0.5410, 0.7990, 0.9651) and the average Co is 0.7684. The difference between the minimum and the maximum Co is 0.4241, for each iteration follows the order of g1 → g2 → g3, including the final iteration, which results in an imbalance of the quality of the three extractions. As compensation, another reverse iteration of g2 → g1 is performed additionally. Then, the Co values become (0.9137, 0.8481, 0.8335), the average value of which rises to 0.8651, and their difference declines to 0.0802 consequently. Furthermore, the watermarked image is the same as the host image Lena shown as Fig. 2(a), containing no information of secret multiple images. This result means that the undetectability requirement of the image hiding system is fully satisfied. The extractions after 100 iterations with another reverse iteration in 5 min are presented in Figs. 3(d)–3(f), whose main information is recognizable and details are even quite distinct. The corresponding Co values are (0.9436, 0.9057, 0.8919), and the average Co is 0.9137. After 103 and 104 iterations, both within another reverse one, the average Co values are further improved to 0.9338 and 0.9420, respectively. The randomly initialized and the resulting POM1 are shown in Figs. 2(b) and 2(c), respectively. Actually, the double POMs are both still pseudorandom and statistically independent from each other. As general phase modulation systems [5], noises added to the phase keys bring much greater influence on the quality of multiple extractions than when they are added to the amplitude of the host image, which is verified by simulations not presented here. Providing considerably large key space, the double phase keys are the main security source of the system. The results of Group 1 show that the spatial size and the sampling interval of secret multiple images can be the same as those of the host image. Thus, sufficient spatial domains are utilized for hiding. Even the number of equal multiple images still leads to a much larger hiding capacity compared with other multiple-image hiding techniques. The simulations named Group 2 reveal the variations of the average Co as to three types of image with different N, shown in Fig. 4. The gray, the blackground, and the white-ground multiple-image employed are analogous to the ones in Group 1, except that the intervals among multiple images are designated equally and the values of the other parameters are maintained. As each multiple-image set belongs to just one image type, the previous algorithm is slightly modified. If the white regions are max signal
Fig. 3. (a)–(c) Three types of image to be hidden together; the corresponding extractions are (d)–(f).
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Fig. 4. Hiding capacity of three types of multiple image.
while the blacks are nonsignal in both types of binary image, only the intensities of the max signal regions of the black-ground multiple images are constrained and only the nonsignal regions are constrained for the white-ground ones. The constraints for gray multiple images are still the entire regions. In Fig. 4, the triangles on solid, dashed, and dashed–dotted lines represent the original average Co of the gray, the black-ground, and the white-ground images, respectively. The average Co of gray images decreases fastest with increasing N, for the entire intensity constraints of these images are the strongest and the most complicated of all three types. On the other hand, although nearly the same constraint method is adopted for both types of binary image, the average Co of black-ground type decreases more quickly than that of white-ground type as N rises. This results from the much longer nonsignal regions of the blackground type, whose diffraction limitations are much greater than those of white-ground type. Further, since white-ground images’ signal regions, which occupy most of the spatial domains, are apt to be noisy in iteration process, the original average Co are much smaller than those of the black-ground ones at the same N. Fortunately, the extracted quality of both types of binary multiple-image can be improved by morphological processing methods such as local thresholding [6], which ensures that the max-signal and the nonsignal regions re-emerge. In Fig. 4, the improved average Co of binary images, represented by the circles, are all more than 0.9, with N increasing from 2 to 8. The variation trend of these improved average Co indicates the potential of the system for hiding more binary multiple images. Morphological processing methods are not effective in the case of multiple gray images. However, if the
threshold value of the average Co of recognizable gray images is set to 0.6, as shown in Fig. 4, four gray images still can be hidden. The number of multiple grey images number is less than that of the binary images, for the diffractive evolution of the gray type from one plane to another is most fierce. Nevertheless, as determined by lossless compression algorithm with the Burrows–Wheeler transform, the information content of each gray image is over 30 times that of each binary image employed in simulations. Thus, the actual hiding capacity of multiple gray images is also satisfactory. The multiplexing parameters of the system are the diffractive distances in the Fresnel domain. Computer simulations, not presented here, show that too small or too large intervals of multiple images result in a decline in the extracted quality. Moreover, cooperation of the distance keys with the wavelength keys further expands the key space provided by the double POMs, and higher security of the system is achieved. As an angular spectrum algorithm is employed to compute the Fresnel diffraction in the simulations above, higher accuracy corresponding to exact optical implementation may be obtained either by embedding each employed image in zero padding of suitable size [7] to prevent the Fresnel ripples or by the direct optical implementation of the real-time iterative algorithm [8]. In summary, the proposed multiple-image hiding technique is suitable for gray and binary images. It is hoped that much more powerful algorithms can be used to explore the potential of the multiple-image hiding system and an appropriate scheme of distribution of keys will enhance the security of the system. Additionally, we are preparing an experimental verification of the proposed technique. This study was supported by National Natural Science Foundation of China grant 60577039. References 1. I. J. Cox, M. L. Miller, and J. A. Bloom, Digital Watermarking (Morgan Kaufmann, 2002). 2. J. Kim, J. Choi, and E. Kim, Opt. Eng. 43, 113 (2004). 3. M. He, L. Cai, Q. Liu, X. Wang, and X. Meng, Opt. Commun. 247, 29 (2005). 4. G. Situ and J. Zhang, Opt. Lett. 30, 1306 (2005). 5. Y. Shi, G. Situ, and J. Zhang, J. Opt. A, Pure Appl. Opt. 8, 569 (2006). 6. R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice-Hall, 2002). 7. F. Shen and A. Wang, Appl. Opt. 45, 1102 (2006). 8. J. Hahn, H. Kim, and B. Lee, Opt. Express 14, 11103 (2006).
