Securing color image using discrete cosine transform in gyrator transform domain structured-phase encoding

Optics and Lasers in Engineering 50 (2012) 1383–1390

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Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng

Securing color image using discrete cosine transform in gyrator transform domain structured-phase encoding Muhammad Rafiq Abuturab Department of Physics, Maulana Azad College of Engineering and Technology, Patna 801115, India

a r t i c l e i n f o

abstract

Article history: Received 14 November 2011 Received in revised form 22 March 2012 Accepted 20 April 2012 Available online 18 June 2012

A new method for securing color image using discrete cosine transform in gyrator transform domain structured-phase encoding is proposed. In this proposal, the structured phase mask is a zone plate phase function. The input color image to be encrypted is decomposed into three channels: red, green, and blue. Each of these channels is encrypted independently by changing its spatial distribution of pixel value by discrete cosine transform, and encoded with structured phase mask. The gyrator transform is performed on resultant spectrum. Structured phase mask, discrete cosine transform, and gyrator transform are employed twice in this proposed method. The construction parameters of structured phase mask and angle parameters of gyrator transform in each channel are principal encryption keys. The schematic electro-optical implementation has been presented. The proposed architecture does not require axial movements. The effectiveness of the proposed algorithm is demonstrated against the chosen and known plaintext attacks. Numerical simulations are made to verify the security, validity, and capability of the proposed method. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Structured phase mask encoding Discrete cosine transform Gyrator transform

1. Introduction With the rapid development of modern communication techniques, information security has become a serious problem. The optical information processing has been widely used as a promising tool in information security applications because of its inherent advantages of parallel and fast processing. In addition, it provides many degrees of freedom with which optical beam may be encoded. Refregier and Javidi proposed a novel doublerandom phase encoding (DRPE) technique in which two statistically independent random phase masks in the input and the Fourier planes are used to encrypt the input image into a stationary white noise [1]. An extension of this technique to the fractional Fourier transform (FRFT) domain [2–6] and Fresnel domain [7] has also been developed. Recently, flexible optical encryption with multiple users and multiple security levels has been proposed [8]. In all these techniques, as a monochromatic light is used to illuminate an input color image, color information of a decrypted image is lost. However, a color image provides more information than a gray-scale image. Zhang and Karim proposed a single-channel color image encryption using DRPE [9]. In this method, an RGB color image is converted into an indexed image before encoding. During the decryption process, the color image is recovered by converting

E-mail address: rafi[email protected] 0143-8166/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.optlaseng.2012.04.011

the decrypted indexed image back into its RGB color image. In recent years, many color image encryption techniques have been proposed [10–13]. The color image encryption using Arnold transform and color-blend operation in discrete cosine transform (DCT) [14] has been presented. In this algorithm, ART scrambles the pixel position of the blocked sub images of original image at local area, color-blend operation defined by a 3  3 matrix (random angle) exchanges and mixes randomly scrambled RGB components, and DCT changes the pixel values of color image. ART, color-blend operation, and DCT are performed two times to encrypt the image. The simultaneous encryption of an RGB and a gray-scale image using byte-level encoding based on singlechannel double random-phase in the FRFT domain [15] has been proposed. Rodrigo et al. introduced an optical gyrator transform (GT), derived its main properties [16], applied it as a new tool in an image processing, and designed the flexible optical experimental setup to perform this transformation [17]. A number of significant works about the image encryption methods using GT [18–21] have been produced. Barrera et al. reported an optical encryption method using a toroidal zone plate (TZP) as a structured phase encoding mask [22] and analyzed its sensitivity to misalignment [23]. TZP and RPM are two different types of phase masks. Unlike RPM, the TZP is difficult to replicate—as it is a multiple-key device and it is easier to position—as its focusing ring can be aligned with the setup axis.

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DCT changes the spatial distribution of pixel value of an image and defines real number field that means output data can be encoded with only real numbers. Lohmann proposed that FRFT could be implemented by using conventional lens system [24]. Sahin et al. introduced a general treatment of two-dimensional FRFT optical systems [25]. They derived several optical designs for the FRFT by using cylindrical lens system in which input–output scale parameters and the residual spherical phase factors can be controlled. However, in all these systems for any transformation angle value the distances among lens and input–output planes as well as lens power have to be changed. GT operation for any transformation angle value can be performed by only proper rotation of cylindrical lenses, where the distance between them and input–output planes are fixed. In contrast to FRFT, GT does not require axial movements. This advantage of GT over FRFT avoids problems related to misalignment, which motivated for using GT in image processing. In this paper, for the first time to my knowledge, a new color image security system based using discrete cosine transform in gyrator transform domain structured-phase encoding is introduced. A color image is separated into red, green, and blue channels. Each of these channels is encrypted independently by DCT in image plane, and encoded with SPM. The GT is executed on resultant image. DCT, SPM, and GT are performed twice in this encryption scheme. The focal length, radius, and illuminating wavelength of SPM, and rotation angles of GT in each channel are essential encryption keys. Numerical simulations are given out to demonstrate the validity of the proposed idea.

2. Fundamental principle 2.1. Structured phase encoding The Fresnel zone plate (FZP) is a diffractive optical element and its transmittance function is defined as   k tðrÞ ¼ exp i ðr 2 Þ ð1Þ 2f where r is the radius, k is the wave number, and f is the focal length of the FZP. The optical axis is assumed to be coincident with the z-axis. As k¼2p/l and Eq. (1) can be written as   p ð2Þ tðrÞ ¼ exp i ðr 2 Þ lf where l denotes the illuminating wavelength. 2.2. Gyrator transform The optical GT operation Ga of a two-dimensional function fi(xi,yi) with parameter a, known as transformation angle, is defined as [16] Z Z þ1 f i ðxi ,yi ÞK a ðxi ,yi ; xo ,yo Þdxi dyi f o ðxo ,yo Þ ¼ Ga ½f i ðxi ,yi Þðxo ,yo Þ ¼ 1

ð3Þ The Kernel of GT is given as   1 ðx y þ xi yi Þcosaðxi yo þ xo yi Þ  exp i2p o o K a ðxi ,yi ; xo ,yo Þ ¼  sina sina ð4Þ Therefore 1  f o ðxo ,yo Þ ¼  sina

Z Z

þ1

1

  ðxo yo þ xi yi Þcosaðxi yo þ xo yi Þ f i ðxi ,yi Þexp i2p dxi dyi sina

ð5Þ

where (xi,yi) and (xo,yo) are the input and output plane coordinates, respectively. For a ¼0, it corresponds to the identity transform, for a ¼ p/2 it reduces to the Fourier transform (FT) with the rotation of coordinates at p/2, for a ¼ p the reverse transform described by the kernel d[(xo,yo)þ(xi,yi)] is obtained, and for a ¼3p/2, it corresponds to the inverse FT with the rotation of coordinates at p/2. When aA[0,2p], the GT can be realized in the coherent optical system using cylindrical lenses. For other angles a, the Kernel of GT Ka(xi,yi;xo,yo) has a constant amplitude and a hyperbolic phase structure. The GT is additive and periodic with respect to parameter a:GaGb ¼Ga þ b and Ga þ 2p ¼Ga. As Ga and G2p  a are reciprocal transforms, it is used in the two-dimensional image processing.

3. Proposed algorithm A color image can be considered as a stack of RGB channels having an M  N  3 array of color pixels, where each color pixel is represented by three RGB values at a specific spatial location. Let f(xi,yi) be color image in input plane and fr(xi,yi), fg(xi,yi), and fb(xi,yi) be its red, green, and blue channels respectively: j k ð6Þ f ðxi ,yi Þ ¼ f r ðxi ,yi Þ, f g ðxi ,yi Þ, f b ðxi ,yi Þ

3.1. Algorithm for encryption The channels fr(xi,yi), fg(xi,yi) and fb(xi,yi) are, respectively, encrypted by the first DCT, subsequently multiplied by first SPMs tr1(xi,yi), tg1(xi,yi) and tb1(xi,yi) in input plane, and then performed the first GT at rotation angles ar1, ag1 and ab1. The transformed distributions for red, green and blue channels are, respectively, encoded by the second DCT, subsequently multiplied by second SPMs tr2(x,y), tg2(x,y) and tb2(x,y) in GT plane, and then executed the second GT at rotation angles ar2, ag2 and ab2. The three encrypted channels Er(xo,yo), Eg(xo,yo) and Eb(xo,yo) are multiplexed to form encrypted color image E(xo,yo), which is obtained as    ð7Þ Eðxo , yo Þ ¼ Er xo ,yo Þ, Eg ðxo ,yo Þ, Eb ðxo ,yo Þ For the sake of clarity, only encrypted red channel Er(xo,yo) is described as Er ðxo ,yo Þ ¼ Gar2 fC r2 fGar1 f½C r1 ðf r ðxi ,yi ÞÞt r1 ðxi ,yi Þggt r2 ðx,yÞg

ð8Þ

The focal length, radius and three different wavelengths for three channels in each SPM, and three different transformation angles for three channels in each GT are principal encryption keys. As SPM and GT are performed twice in this encryption method. In all, there are 16 independent parameters, which are used as keys of the encryption scheme. 3.2. Algorithm for decryption The encrypted color image is first segregated into encrypted red, green, and blue channels. The inverse of second GT at rotation angles rotation angles  ar2,  ag2 and  ab2 are, respectively, executed over encrypted red, green and blue channels, subsequently multiplied by conjugate of second SPMs t nr2 ðx,yÞ, t ng2 ðx,yÞ and t nb2 ðx,yÞ, and then employed inverse of the second DCT in GT plane. The inverse of first GT at rotation angles  ar1,  ag1 and  ab1 are, respectively, executed over transformed distributions of red, green and blue channels, subsequently multiplied by conjugate of first SPMs t nr1 ðxi ,yi Þ, t ng1 ðxi ,yi Þ and t nb1 ðxi ,yi Þ, and then performed inverse of the first DCT in input plane. The three decrypted channels, Dr(xi,yi), Dg(xi,yi) and Db(xi,yi) are multiplexed

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to form color image, which is obtained as   Dðxi ,yi Þ ¼ f ðxi ,yi Þ ¼ Dr ðxi ,yi Þ, Dg ðxi ,yi Þ, Db ðxi ,yi Þ

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interference fringe, which is captured and recorded as an off-axis hologram by charged couple device (CCD) cameras, and digitally processed by computer systems. Each RGB color channel images independently recorded and processed by the same technique are multiplexed to form encrypted color image. The reverse of the encryption procedure gives decrypted image. The space-bandwidth product (SBP) is an optical system metric. The SBP of an optical system is defined either by the number of degree of freedom of the system or a specific area in spatial domain and spatial frequency domain of the system. The SBP for the two-dimensional optical signal is expressed as [26] ZZ ZZ S ¼ dxdy df x df y ¼ Ss  Sf ð11Þ

ð9Þ

For the sake of clarity, only decrypted red-component image Dr(xi,yi) is described as   Dr ðxi ,yi Þ ¼ C r1 fGar1 fC r2 Gar2 ½Er ðxo ,yo Þt nr2 ðx,yÞ gt nr1 ðxi ,yi Þg ð10Þ The asterisks on SPMs indicate their complex conjugates. The subscripts r, g, and b denote the red, green, and blue channels respectively. The same algorithms are used for green and blue channels. 3.3. Optical realization

where Ss is the area of the spatial signal in the spatial domain and Sf is the area of the frequency band of the signal in the spatial frequency domain. The pixel size and the area mainly determine the SBP of the SLM. A smaller pixel size corresponds to a higher SBP. The position stages for the SLMs in the proposed system are not required because the alignment between amplitude and phase components are digitally achieved using a PC [17]. This alignment accuracy is limited by pixel size of 20 mm and thus a higher SBP is obtained.

The GT experimental setup consists of three generalized lenses (denoted as L1, L2, and L1) and two fixed equal intervals z. Each generalized lens is an assembled set of two identical planoconvex cylindrical lenses of the same power. The first and third identical generalized lenses are rotated with respect to each other. Their focal distance f1 equals the distance z between two consecutive generalized lenses of the setup. The second generalized lens of focal length f2(¼ z/2) is fixed. The third generalized lens compensates the undesirable phase modulation introduced by rotation of first and third generalized lenses. The variation of transformation angles a is achieved by proper rotation of these lenses [17]. There are no relations among focal lengths of the generalized lenses and FZP. The electro-optical setup of the proposed encryption process for red color image is shown in Fig. 1. The dotted block with lenses L1, L2, and L1 indicates the first optical GT and that with lenses L0 1, L0 2, and L0 1 represents the second optical GT. For the sake of clarity, only red channel is described. The spatial distribution of pixel value of the red channel is changed digitally by the first DCT, and displayed on the first Spatial Light Modulator (SLM) in input plane. The obtained distribution is transmitted through first FZP, and then transformed optically by first GT. Now the spatial distribution of pixel value of the resulting image is changed digitally by the second DCT, and displayed on the second SLM in GT plane. The obtained complex distribution is transmitted through second FZP, and then transformed optically by second GT. The image so received is superimposed on the plane reference beam to produce a holographic

FZP 1

L1

L2

L1

4. Numerical simulation Numerical simulations have been performed on a Matlab 7.11.0 (R2010b) platform to prove the performance, security, and robustness of proposed technique. 4.1. Performance and security The original color image with 512  512  3 pixels and 24 bits is illustrated in Fig. 2(a). The first FZP as first SPM of focal length f1 ¼4 cm, radius r1 ¼0.2 mm, red wavelength lr1 ¼ 650:0 nm, green wavelength lg1 ¼ 545:0 nm, and blue wavelength lb1 ¼ 450:0 nm is shown in Fig. 2(b). The second FZP as second SPM of focal length f2 ¼5 cm, radius r2 ¼0.3 mm, red wavelength lr2 ¼ 632:8 nm, green wavelength lg 2 ¼ 532:0 nm, and blue wavelength lb2 ¼ 488:0 nm is demonstrated in Fig. 2(c). The angle parameters of the first GT are ar1 ¼ 0:151, ag1 ¼ 0:301 and

FZP 2

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Reference beam Encrypted image

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Input Redcomponent image

SLM 1

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Computer System Fig. 1. Electro-optical color image architecture of proposed system.

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Fig. 2. The results of proposed color image encryption and decryption.(a) Original image with 512  512 pixels and 24 bits used in numerical simulation, (b) first Fresnel zone plate as first structured phase encoding of focal length (f1 ¼4 cm), radius (r1 ¼0.2 mm), red wavelength (lr1 ¼ 650:0 nm), green wavelength (lg1 ¼ 545:0 nm) and blue wavelength (lb1 ¼ 450:0 nm), (c) second Fresnel zone plate as second structured phase encoding of focal length (f2 ¼ 5 cm), radius (r2 ¼ 0.3 mm), red wavelength (lr2 ¼ 632:8 nm), green wavelength (lg 2 ¼ 532:0 nm) and blue wavelength (lb2 ¼ 488:0 nm), (d) encrypted image, (e) decrypted image with all the correct keys, (f) decrypted image with one of the transformation angle for each component image is changed by 0:0081but all other parameters are correct and (g) decrypted image with one of the wavelength for each component image changed by 8 nm but all other parameters are correct.

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Fig. 3. (a) The MSE values as a function of the transformation angles between original red, green and blue channels, and their corresponding decrypted images. (b) The MSE values as a function of the focal lengths of the FZP between original red, green and blue channels, and their corresponding recovered images. (c) The MSE values as a function of the radii of the FZP between original red, green and blue channels, and their corresponding retrieved images. (d) The MSE values as a function of the wavelengths of the FZP between original red, green and blue channels, and their corresponding retrieved images.

ab1 ¼ 0:451, and that of the second GT are ar2 ¼ 0:603 , ag2 ¼ 0:751 and ab2 ¼ 0:901. The image is encrypted in three channels, which are multiplexed to form a noise-like color image as shown in Fig. 2(d). When all the correct keys are used, the three decrypted channels obtained are multiplexed to form like an original color image as displayed in Fig. 2(e). If only one of the transformation angles of each channel of inverse GT is changed by 0:0081 from their correct values, the three decrypted channels obtained are multiplexed to form a noisy image as shown in Fig. 2(f). If only one of the wavelengths of each channel of SPM is changed by 8 nm from their correct values, the three decrypted channels obtained are multiplexed to also form a noisy image as demonstrated in Fig. 2(g). In order to evaluate the performance of the proposed method numerically, the mean square error (MSE) is introduced, and expressed as MSE ¼

M X N   1 X I ði,jÞIo ði,jÞ 2 M  Ni¼1j¼1 i

ð12Þ

where Ii(i,j) and Io(i,j) are input and output images at pixel position (i,j), respectively, and (M  N) denotes the image size. The MSE values between original red, green and blue channels and their corresponding encrypted images are,7.076  103, 4.984  103and 4.937  103 respectively. These high MSE values indicate that color information cannot be recovered. The MSE values between original red, green, and blue channels and their corresponding decrypted channels with all right keys are 1.658  10  26, 1.129  10  26and 9.938  10  27 respectively. These low MSE values mean that color information is retrieved. A small change in rotation angles tests the sensitivity of recovered image. The red, green, and blue channels of an original color image are independently decrypted with the inverse GT at one of the rotation angles of each channel is changed by 0:0081

but remaining parameters are correct. The MSE values between original and decrypted red, green and blue channels are 6.444  103, 4.325  103 and 4.224  103 respectively. This implies that besides the DCT and the construction parameters of SPMs, the transformation angles of GT provide sensitive encryption keys for the proposed system. A small change in wavelength further tests the sensitivity of reconstructed image. The red, green, and blue channels of an original color image are independently decrypted with the SPM for one of the wavelengths of each channel is changed by 8 nm but other parameters are correct. The MSE values between original and decrypted red green, and blue channels are 9.341  103,4.900  103 and 5.615  103 respectively. These MSE values are very high. Thus, wavelengths of SPMs also serve as sensitive encryption keys in the proposed algorithm. The MSE values between original red, green and blue channels and their corresponding recovered images calculated against variation in the transformation angles, the focal lengths of FZP, the radii of FZP and the wavelengths of the FZP are plotted in Fig. 3(a)–(d) respectively. 4.2. Robustness test Robustness of the proposed algorithm has been checked against the chosen and known plaintext attacks [27]. In chosen plaintext attack, Dirac delta function is employed at image plane and the structured phase key is obtained at transform plane, which is used as retrieved key in decryption process. Fig. 4(a) and (b) show corresponding Dirac delta function as plain image and reconstructed image, when chosen plaintext attack is applied on the proposed algorithm. In known plaintext attack, two different (known) input images are used to obtain corresponding ciphered images, and then the keys are deduced by solving a linear system of equations. This system has trivial solution, the first structured phase key equals to zero. Therefore, the structured phase key at transform

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Fig. 4. The robustness test of the proposed method against chosen plaintext and known plaintext attacks: (a) Dirac delta function as plain image, (b) recovered encoded image using chosen plaintext attack, (c) first input plain image, (d) second input plain image and (e) retrieved encoded image using known plaintext attack.

plane is obtained using Gauss elimination method, which is used as decryption key to retrieve the image. Fig. 4(c) and (d) show the two input images used in known plaintext attacks, and Fig. 4(e) shows recovered image, when known plaintext attack is applied on the proposed algorithm. The results imply that the proposed system has resistant against chosen and known plaintext attacks. Robustness of the proposed method is tested against occlusion attack on encrypted image with 25% and 50% occlusion sizes as shown in Fig. 5(a) and (c) respectively, and corresponding recovered images are displayed in Fig. 5(b) and (d). The MSE values between original red, green and blue channels and their corresponding retrieved channels with all right keys from encrypted image with

25% occlusion are 2.357  103, 1.510  103and 1.570  103 respectively, and that with 50% occlusions are 5.677  103, 3.154  103 and 2.997  103 respectively. Despite 25% and 50% loss of occluded encrypted images, their respective decrypted images with all correct keys can be identified. The proposed system demonstrates robustness against occlusion attacks. Robustness of the proposed scheme is further confirmed against Gaussian and speckle noise attacks on the encrypted image with their standard deviations of 0.1 as shown in Fig. 6(a) and (c) respectively, and corresponding recovered images are displayed in Fig. 6(b) and (d). The MSE values between original red, green and blue channels and their corresponding

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Fig. 5. The robustness test of the proposed method against occlusion attack on the encrypted image: (a) with 25% occlusion, (b) corresponding recovered image from (a), (c) with 50% occlusion and (d) corresponding retrieved image from (c).

Fig. 6. The robustness test of the proposed method against: (a) Gaussian noise attack on the encrypted image with variance 0.1, (b) corresponding reconstructed image from (a), (c) speckle noise attack on the encrypted image with variance 0.1 and (d) corresponding retrieved image from (c).

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scheme, a color image is separated into red, green, and blue channels. Each of these channels is then encrypted independently by DCT, and encoded with SPM. The GT is executed on the obtained distribution. During encryption process SPM, DCT, and GT are employed twice continuously. The construction parameters of SPM and rotation angles of GT in each channel are essential keys to reconstruct the original color image. The proposed system is free from axial movements, has benefits of higher space-bandwidth product, and resistant against the chosen and known plaintext attacks. Numerical simulations are conducted to confirm the viability and effectiveness of the proposed algorithm.

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The author is indebted to Late Muhammad Waizul Haque and Late Mehr-un-nisa for their inspiring supports. References

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Speckle noise Fig. 7. (a) The MSE values as a function of Gaussian noise attacks on the encrypted red, green and blue channels and their corresponding decrypted images. (b) The MSE values as a function of speckle noise attacks on the encrypted red, green and blue channels and their corresponding recovered images.

recovered channels with all right keys from encrypted image with Gaussian noise having standard deviation 0.1 are 1.239  103, 1.287  103 and 1.288  103 respectively. The MSE values as a function of Gaussian noise attacks on the encrypted red, green, and blue channels and their corresponding decrypted images are shown in Fig. 7(a). Similarly the MSE values between original red, green and blue channels and their corresponding decrypted images with speckle noise having standard deviation 0.1 are 1.829  103, 1.230  103and 1.116  103 respectively. The MSE values as a function of speckle noise attacks on the encrypted red, green, and blue channels and their corresponding decrypted images are displayed in Fig. 7(b). Even though standard deviations of 0.1 for Gaussian and speckle noised-encrypted images, their decrypted images with all correct keys can be recognized. The proposed system shows robustness against noised attacks.

5. Conclusion A new image encryption and decryption algorithms based on DCT and DSPE in GT domain is introduced. In the proposed

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