Securing of Two and Three Dimensional Information Based on In-line Digital Holography

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1 Securing of Two and Three Dimensional Information Based on In-line Digital Holography Hesham Eldeeb Computer & System Department Electronic Research Institute National Research Center, Dokki, Giza Cairo, Egypt Abstract: - A method that combines the high speed of digital encryption and the high security of optical encryption with the advantages of electronic transmission, storage, and decryption is introduced. The encryption is performed by use of multi-dimensional lock on the hologram plan, providing high security in the encrypted image and a key with many degrees of freedom. We described how our method can be adapted to encrypt either 2D or 3D information. Electronic decryption can be performed and a correlation method, which has several advantages over the FFT, used for reconstruction procedure. The most important of these advantages is that the width and the length of the hologram need not to be equal. This gives the proposed method a unique technique compared to any other security methods. Experimental results for both 2D and 3D information including setup are shown. Key-Words: - Information security, computer generated holography, encryption, decryption, 3D virtual image. 1 Introduction Information hiding is a fast growing research subject that has drawn increasing attention from both academic and business circles as it covers a great number of application areas in the field of information technology to prevent huge economic losses [1]. Recently, a number of optical methods have been proposed for the purpose of information hiding, since the data/image encryption with optical information processing has many inherent advantages such as the capability of parallel processing, the increase in security level. Those optical encryption methods reported either employing an all-optics or a hybrid system to implement the image encryption and decryption. Especially, hybrid systems utilize digital holography to reconstruct and decrypt hidden information, but they all optically recorded the hologram of the object with CCD camera.[2],[3],[4] In this paper, unlike the previously reported optical encryption methods, we present a new method that is based on a concept of a digital optics, which means that we implement the encryption and decryption process totally with digital method. By using this method, one is able to virtually select a wavelength for encoding digital hologram instead of using specific physical light source like a laser with certain wavelength. Also, one is able to freely select spatial position of the signal plane from a much larger and more flexible scope. Thus a dramatic increase of the imperceptibility and security level will be introduced. Secondly, digital recording and encryption technique totally get rid of the physical limitations imposed by optical or electronic hardware, such as the complexity of optical hardware, lack of flexibility, etc [5]. In practical systems, data security is an important issue. Optical encryption techniques provide a high level of security [6-8] because there are many degrees of freedom to encode the information, such as amplitude, phase, wavelength, and polarization. Figure (1) shows the overall block diagram for Secure Communication system using the proposed digital holography multi-dimensional lock method. This article discusses the proposed method for Computer-Generated (CGH) using raytracing technique and encryption procedure in section 2. Section 3 describes the details of decryption part of this method and the reconstruction of virtual image using correlation method. The experimental results for 2D and 3D objects are presented in section 4. The conclusion is discussed in section 5.

2 2D or 3D original data CCD Camera Or CGH Digital Virtual Image of the Original Data Multifunction Encrypt -ion Reconstruction of the Original Data (digital/ optical) Encrypted data Decrypting data Decrypting system using correct functions & parameters Electronic Transmition /Internet Fig.1 Block diagram of the proposed secure communication system 2 Methodology of hologram generation and Encryption procedure 2.1 generation using ray-tracing method Ray-tracing is a convenient way to model optical systems using a computer. This technique simulates the light that reaches the hologram plane by using thousands of geometrical rays, each with a starting point in the reference or object model and a direction of travel. The sum of all input rays comprises a representation of the interference pattern inside the hologram plane [9],[10]. Most of the CGHs described in literature are simply two-dimensional Fourier transforms of a twodimensional image plane. Although the fringe pattern on the plate is calculated by a computer, what is calculated is a Fourier transformation. In these holograms, the principal mode of reconstruction is Fraunhofer diffraction which limits low frequency spatial information to the center of the interference pattern and higher frequency components to the edges. Alternatively, holograms generated via ray tracing distribute information so that different areas of the plate correspond to different perspectives of the object. In the past, however, implementation of this algorithm has not been trivial since the quantity of calculations quickly approached the limits of computing technology [10] Ray-tracing method In computer simulations of holograms by ray tracing, the complex amplitudes that reach the plane of the photographic film from different directions are summed up [9]. The object beam is modeled as if the object were made of point sources. The light wave from a point source can be written as: Eo = Ao exp( ikr) / R (1) where k is the wave number and A o is the light wave amplitude which is real and exp(ikr ) is the phase factor. The time-dependent phase exp( iωt) is ignored in this model since it does not change the final result. R is the distance between the hologram plane and the point ( x, y, z) on the object : / 2 R ( x, y, z, ξ, η) = [( x ξ ) + ( y η) + z ] (2) where ( ξ, η) is the coordinate on the hologram plane. The x y plane is parallel to the ξ η plane, the z -axis is chosen to be perpendicular to the hologram and the hologram is set at z = 0. For simplicity we use a reference beam of normal incidence yielding the approximation N 2 Aj exp( ikrαj ) Iα 1+ (3) AR j= 1 Rαj This equation is used for hologram generation by ray tracing using the computer simulation [9]. 2.2 Encryption procedure The digital holography-based technique is originally not designed for data encryption setup. However as the method enables one to digitally recording and reconstructing the image, it opens possibility to employ this technique for encoding an image as a

3 hologram that irreproducible using the existing intensity based detector [11],[12],[13]. To illustrate our idea, in this paper, we utilize the encryption parameters z, wavelength ( λ ), width (W) and length (L) of hologram. Also, we can divide the hologram into four parts or more and calculating lock functions for example cosine, subtract, logarithm and exponential,.etc., as shown in fig.2. Digital Function(1) Function(3) Fig.2 Data encryption setup with multifunction lock The parameters and the functions can be secretly selected during the process of encrypting and encoding digital hologram, so it is an unknown for unauthorized third parties. If unauthorized parties attempt to decrypt host image without the knowledge of encoding process, it will be extremely difficult to determine the decoding parameters to decode the hologram. As, for example, the virtual wavelength λ can be a value selected from huge numerical range instead of one coming from physical existed light sources. This also implies that a slight mismatch between recording and reconstructing parameters will change the position of reconstructed image a lot, leading to its disappearance in space. Also, the ray-tracing method for recording and the correlation method for reconstruction give the flexibility over FFT to choose the width different than the length of the hologram. This means that if the third party for any reason know the one dimension from the hologram it is so difficult to find the other. The normal distance between the hologram plane and the object plane z at recording phase is also unknown for the third party to reconstruct the correct image. In other words, virtual wavelength λ, z, W, L, and functions of encryption can be utilize to design multi-dimensional lock 3 Methodology of decryption procedure and reconstruction of virtual image 3.1 Decryption procedure Function(2) Function(4) Encrypted After receiving the encrypted hologram in the desired side, the decryption procedure begins. The using of the correct inversed functions will lead to success in decryption of the data, as shown in fig.3. Encrypted Fig(3) Data decryption setup with the correct multifunction key These data, digital hologram, will be an input for the next step in the proposed secure system which is the reconstruction of virtual image. The use of correct reconstruction setup parameters will lead to output the desired image (data). 3.2 The correlation method for virtual 3D image reconstruction In this paper, we use quite a different method which is not related to the ideas that hold for conventional holography by using FFT. This method is based on the correlation between the hologram of a point source and the hologram of real objects [14]. In order to retrieve the 3D pattern A ( x, y, z), we perform the calculation φ i = α Inv(Function(1)) Inv(Function(2)) Inv(Function(3)) Inv(Function(4)) α ( kr ) I cos / R, (4) iα where the summation over α is performed on the surface of the hologram. We approved in pervious paper [15] that this process can retrieve the image, that means, A( x, y, z) ϕi. A quick comparison between conventional holography using FFT method and the correlation method is shown in Table 1 [13],[14],[15]. From Table (1) the correlation method has many facilities in the encryption/decryption process more than FFT method. This gives our security method a unique technique compared to any other methods. 4 Results Although the off-axis setup is widely chosen, there are certain applications that must be performed using the In-line setting. Such applications for example are widely found in particle field analysis. In this paper, the setup is chosen to be In-line setup. As approve for the proposed security method, we tested it with two different objects. iα Decrypted

Table (1) Conventional method using FFT 1 Shape and size of object planes must dependent on those of the hologram plane 2 The sampling points on the hologram and object plane must be equally-spaced

Correlation method Shape and size of object planes can be independent on those of the hologram plane Not necessary Not necessary Separated and segmented hologram can be used.

A 3D object of pyramid model of different planes consisting of 616 points whose size gradually shrinks in the Z-direction is shown in fig(5).

4 Table (1) Conventional method using FFT 1 Shape and size of object planes must dependent on those of the hologram plane 2 The sampling points on the hologram and object plane must be equally-spaced Cartesian grid 3 m The number of grid points must be in the form 2 and must be the same on the hologram and object plane. 4 Separated, segmented and/or curved hologram cannot be used. Correlation method Shape and size of object planes can be independent on those of the hologram plane Not necessary Not necessary Separated and segmented hologram can be used. A 2D object of circle model consists of 284 points, is shown in figure (4). A 3D object of pyramid model of different planes consisting of 616 points whose size gradually shrinks in the Z-direction is shown in fig(5). Inverse of encrypted functions together in the same process. The correlation method is used for reconstruction of the desired image, as shown in fig(7)(a). Reconstruction with wrong parameters will result with incorrect output, as shown in fig(7)(b). Fig.4 2D object of a circle model Fig.5 3D object of pyramid model gradually shrinks in the Z-direction Fig.6.a shows the generated hologram with 400 x 300 pixels of the 2D object using ray tracing method. The wavelength used in this processing is equal to 1GHz. Fig.6.b shows the hologram after encryption by the new method. The four function used in this encryption are cosine, subtract, logarithm and exponential. Sure, the method can be implemented by any other functions as discussed in the pervious sections. Then you can send the encrypted hologram through the internet, or any other communication channel, to the destination side. Therefore even if some unauthorized intruder acquires the encrypted hologram, he would not be able to reproduce the original object image. Only the right decryption can be done, by calculating the a. hologram with 400 x 300 pixels of the 2D object b. the same hologram after encryption by the proposed method Fig.6 CGH of 2D with ray-tracing method a. reconstruction of the desired image b. reconstruction with wrong parameters will result with Incorrect output Fig.7 Reconstruction of 2D object with correlation method

The same processing is done on the 3D object of a pyramid with different z planes, as shown in fig(8) and fig(9). a. hologram with 400 x 300 pixels of the 3D object b.

5 The same processing is done on the 3D object of a pyramid with different z planes, as shown in fig(8) and fig(9). a. hologram with 400 x 300 pixels of the 3D object b. the same hologram after encryption by the proposed method Fig.8 CGH of 3D with ray-tracing method Z0 Z1 Z2 Z3 Fig.9 Reconstruction of the desired image with different cross-sections plans in Z direction with correct multi-dimensional key Reconstruction with wrong parameters will result with Incorrect output, as discussed with the 2D object. The FFT method can not be used for either generation or reconstruction for this processing, as the width of the hologram is different than the length. This gives more security for the proposed method over conventional methods. 5 Conclusion We have developed a method to design a high security system based on computer-generated holography with the advantages of securing electronic transmutation and storage of 2D or 3D information. According to our method, one can design multi-dimensional lock in order to receive a chosen image. The resulting masks can be used for security and encryption, as the desired image will be received in the output plane only when the multidimensional key is known. The correlation method is used for reconstruction of virtual image which give several advantages over the well known FFT method. This gives our security method a unique technique compared to any other methods. Also, dividing the hologram to parts that each of them has different encryption function, this adds more locking difficulty for the third party. Finally, this digital method can be expanded to encrypt/decrypt different kinds of information, such audio signals, video signals, digital images, maps, and other physical signals, while optical methods are limited to optical images. References: [1] Fabien A. P. Petitcolas, Ross J. Anderson, and Markus G. Kuhn, Information Hiding- A Survey, Proceeding of the IEEE, Vol. 87, No. 7, July [2] T. Nomura and S. Mikan, B. Javidi, Optical Image Encryption using an Optimally Designed Encryption Key, Proceedings of the International Workshop on Optical Display and Information Processing, Korea, May 15-16, [3] B. Javidi and T. Nomura, Securing information by use of digital holography, Opt. Letter. 25, 2000, pp [4] S. Lai and M. A. Neifeld, Digital wavefront reconstruction and its application to image encryption, Opt. Comm. 178, (2000), pp [5] Xiang Peng, Lingfeng Yu and Lilong Cai, Double-lock for image encryption with virtual optical wavelength, Optics express Vol. 10, No. 1 January 14, 2002, pp [6] O. Matoba and B. Javidi, Encrypted optical storage with wavelength-key and random phase codes," Appl. Opt. 38, 1999, pp [7] E. Tajahuerce, O. Matoba, S.C. Verrall, and B. Javidi, Optoelectronic information encryption with phaseshifting interferometry, Appl. Opt. 39, (2000), pp [8] L. Yu and L. Cai, Iterative algorithm with a constraint condition for numerical reconstruction of a three dimensional object from its hologram, J. Opt. Soc. Am. A, 18, 2001, pp [9] Stein,A. D., Wang, Z., Leigh, J. S., Jr., Computer generated holograms: A simplified ray-tracing approach, Computer in Physics, vol. 6, 1992, pp [10] Hesham Eldeib, Takashi Yabe, A Fast Computer Holography System and Its Experimental verification, Journal of Computer modeling and simulation in engineering, vol.2,1996, pp

6 [11] Enrique Tajahuerce, Osamu Matoba, Steven C. Verrall, and Bahram Javidi, Optoelectronic information encryption with phase-shifting interferometry, APPLIED OPTICS, Vol. 39, No. 14, May 2000 [12] Thomas J. Naughton, John B. Mc Donald, and Bahram Javidi, Efficient compression of Fresnel fields for Internet transmission of three-dimensional images, Applied Optics, vol. 42, no. 23, August 2003, pp [13] Hesham Eldeeb, Takashi Yabe and Toshio Yoshino, Efficient Design of Computer Algorithm for Improving the Gabor-Type Holography, Proceedings of IEEE international conference on Systems, Man and Cybernetics SMC97, Oct. 1997, Orlando, USA, vol.3, pp [14] Hesham Eldeib, A High Performance Computing Algorithm for Improving the IN-Line Holography, Journal of Computer modeling and simulation in engineering, vol. 4, No.2, May 1999 [15]Tomoyoshi Ito, Hesham Eldeib, Kenji Yoshida, Shinya Takahashi, Takashi Yabe and Tomoaki Kunugi, Special-Purpose Computer for Holography HORN-2, Computer Physics Communications, vol.93, No.1, January 1996, pp

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