Color image security system based on discrete Hartley transform in gyrator transform domain

Optics and Lasers in Engineering 51 (2013) 317–324

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

Color image security system based on discrete Hartley transform in gyrator transform domain Muhammad Rafiq Abuturab Department of Physics, Maulana Azad College of Engineering and Technology, Patna, Bihar 801113, India

a r t i c l e i n f o

abstract

Article history: Received 12 April 2012 Received in revised form 21 September 2012 Accepted 21 September 2012 Available online 16 October 2012

A novel color image encryption algorithm based on Arnold- and discrete Hartley transform in gyrator transform domain is proposed. In this method, a color image is segregated into red, green, and blue channels. Each channel is permutated by first Arnold transform operation and the spatial distribution of pixel value is changed by first discrete cosine transform at image plane. The resulting image is encoded by discrete Hartley transform. The encoded information is again permutated by second Arnold transform and the spatial distribution of pixel value is changed by second discrete cosine transform at frequency plane, and finally gyrator transform is executed. The period and iterative number of Arnold transform, and transformation angles of gyrator transform in each channel are main keys, which increase the security of the proposed system. The proposed method can be well protected under chosen- and known plaintext attacks. Numerical simulations are conducted to illustrate the security, validity, and feasibility of the proposed method. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Arnold transform Discrete cosine transform Hartley transform Gyrator transform

1. Introduction The optical security systems are of growing interests for image security system because of their distinct advantages of processing two-dimensional complex data in parallel and at high computation speed. Refregier and Javidi proposed a double random-phase encryption technique that encodes a primary image into white stationary noise. The two random-phase masks placed at the image- and Fourier plane of a 4-f correlator serve as encryption keys [1]. A number of optical encryption techniques have been reported [2–11]. In previous encryption approaches, most of images are encrypted and decrypted by a monochromatic light illuminating on, and hence the decrypted images would not preserve their color information any longer. Zhang and Karim introduced a new method for single-channel color image encryption using Fourier transform [12]. Since then significant works on color image encryption, have been reported [13–15]. The color image encryption based on Arnold transform (AT) and color-blend operation in discrete cosine transform (DCT) [16] has been presented. Recently color image hiding and extracting using the phase retrieval algorithm in the fractional Fourier transform (FRFT) domain and AT [17] has been reported. In this method, the secret RGB component images permutated by AT are encrypted by the cascaded phase iterative FRFT algorithm.

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.2012.09.008

The encoded phase information is extracted and embedded in the blue component images of the enlarged host image. The two-dimensional Hartley transform (HT) is mathematically equivalent to the two Fourier transforms (FTs), but it is real valued and, its inverse and direct operations are identical. As HT is real, it contains only amplitude information, which can be illuminated spatially incoherent or coherent light and recorded directly on an intensity-only medium that is much simpler than the FT method. It offers a significant increase in speed of computing the spectra of images and in computing filtering operations on images, and hence digital processing of images by electronic computers is directly benefited by the HT where speed of computation is important [18–20]. The color image encryption by using the rotation of color vector in HT domains has been reported [21]. In this image encryption process, two-random independent angle shifts are introduced to rotate the color vectors composed by the RGB component images in discrete HT domains. Recently an optical image cryptosystem based on HT in the Fresnel transform domain has been proposed [22]. In this scheme, a simpler optical-image encryption architecture of HT based on only one Fresnel transform has been presented. The main theorems and selected functions of GT have been formulated [23]. The experimental implementation of flexible optical scheme has been reported [24]. The gray-scale image encryption schemes based on GT have been presented [25–30]. Recently color image encryption methods using GT have also been proposed [31–35].

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AT scrambles the pixel-matrix sequence by encoding a single parameter and forms a color noise-like image, which can reduce key space for storage and transmission in a practical application [27]. DCT changes the spatial distribution of pixel value of an image and defines real number field. HT is a real FT and can be realized by incoherent optical systems. In contrast to FRFT, GT does not require axial movements. These properties of above transforms motivated for using in color-information security system. In this paper, for the first time to my knowledge, a novel color image security system based on AT and HT in GT domain is presented. A color image is converted into red, green, and blue channels. Each channel is permutated by first AT, changed spatial distribution of pixel value by first DCT at input plane, and executed HT. The transform spectrum is then permutated by second AT, changed spatial distribution of pixel value by second DCT at transform plane, and performed GT. The system parameters of AT and GT in each channel are encryption keys. Numerical simulations are presented to confirm the security, validity, and possibility of the proposed idea.

2. Theoretical description 2.1. Arnold transform The AT operation AM of a two-dimensional function fi(xi,yi) is defined as [36] " 0# " " # ! x 1 1 xi ¼ mod ,M , ð1Þ AM :  y0 1 2 yi 0

0

where coordinates (xi,yi) and (x ,y ) denote the pixel positions before and after AT operation, and ‘‘mod’’ represents modulus after division operation. The period of transform depends on the parameter M, which represents the size of image matrix. AT is chaotic in unit square and possesses pseudo-numeration property [15–17] and thus it is applied to the two-dimensional image processing. The image is scrambled using AT for n iterative number and the scrambled image can be retrieved by inverse AT for m(¼p n) iterative number. Thus, both periodic scrambling transform p and iterative number are keys to retrieve the original image. 2.2. Hartley transform The HT operation of a two-dimensional real function fi(xi,yi) is defined as [18,19] Z þ1 Z þ1       H xo ,yo ¼ f xi ,yi cas½2p xo xi þ yo yi dxi dyi ð2Þ 1

1

and its inverse transform is defined as Z þ1 Z þ1       H xi ,yi ¼ H xo ,yo cas½2p xo xi þ yo yi dxo dyo 1

ð3Þ

1

where cas ¼ cos þsin. According to Hartley transform, it is deduced as Z þ1 Z þ1        H xo ,yo ¼ f xi ,yi cas 2p xo xi þyo yi dxi dyi 1 1        exp ip=4   pffiffiffi ¼ ð4Þ F xo ,yo þ exp ip=2 F xo ,yo 2 the FT of the real function fi(xi,yi). The constant where F(xo,yo) is p   ffiffiffi term exp ip=4 = 2 can be ignored in HT operation, as intensity information is obtained in the output plane. It is obvious that the   HT can be calculated by two FTs with a phase of exp ip=2 between them.

2.3. Gyrator transform The optical GT at parameter a (known as rotation angle) Ga of a two-dimensional function fi(xi,yi) is defined as [23] Z Z þ1          f o xo ,yo ¼ Ga ½f i xi ,yi  xo ,yo ¼ f i xi ,yi K a xi ,yi ; xo ,yo dxi dyi 1

ð5Þ The Kernel of GT is expressed as        xo yo þxi yi cos a xi yo þxo yi 1 exp i2p K a xi ,yi ; xo ,yo ¼ sin a 9sin a9 ð6Þ Then   f o xo ,yo ¼

Z Z þ1   1 f i xi ,yi exp 9sin a9 1      xo yo þxi yi cos a xi yo þxo yi dxi dyi  i2p sin a

ð7Þ

where (xi,yi) and (xo,yo) indicate 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 composed of six thin convergent 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 employed in the two-dimensional image processing.

3. Proposed technique 3.1. Encryption and decryption schemes In encryption process, the input color image f(xi,yi) is disassembled into fr(xi,yi), fg(xi,yi) and fb(xi,yi) as red, green, and blue channels, which are, respectively, permutated by the first AT at iterative numbers nr1 , ng 1 and nb1 , encoded by the first DCT at input plane, and performed HT. The complex distributions for red, green and blue channels are, respectively, permutated by the second AT at iterative numbers nr2 , ng2 and nb2 , encoded by the second DCT at GT plane, and executed GT at rotation angles ar, ag and ab. The three encrypted channels are multiplexed to form encrypted color image at output plane. In decryption process, the encrypted color image is separated into encrypted red, green, and blue channels, performed GT at rotation angles  ar,  ag and  ab, decoded by the second inverse DCT and permutated by the second inverse AT at iterative numbers mr2 , mg2 and mb2 at GT plane. Now the HT are, respectively, executed over resulting complex distributions of red, green and blue channels, decoded by the first inverse DCT and permutated by the first inverse AT at iterative numbers mr1 , mg1 and mb1 at input plane. The three decrypted channels are multiplexed to form color image. The proposed algorithm uses six different iterative numbers, six different periodic scrambling transforms, and three different rotation angles as additional keys for the image encryption and decryption techniques. 3.2. Optical realization The GT experimental setup contains three generalized lenses (denoted as L1,L2 and L1) with fixed equal distances z between

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them. Every generalized lens corresponds to the combination of two identical plano-convex cylindrical lenses of the same power. The first and third identical generalized lenses having a focal length f1 ¼ z are rotated with respect to each other whereas the second generalized lens having a focal length f2 ¼z/2 is fixed. The third generalized lens compensates the undesirable phase modulation introduced by rotation of first and third generalized lenses. The GT operation for different transformation angles a is executed by proper rotation of these lenses [24]. The HT can be implemented in an incoherent/coherent optical system contains Fourier lens L of focal length f and a Michelsontype interferometer [18,20]. A cube corner prism CCP rotates the field F(xo,yo) of one arm through an angle p to obtain the field F(  xo,  yo), a beam splitter BS splits the beam, and a reflective mirror M1 produces a phase difference p/2 between two optical paths. The optical path length in both arms is equal to f. The proposed scheme for color image encryption can be optoelectronically implemented. The opto-electronic setup for red color image is shown in Fig. 1. The Fourier lens with a Michelson-type interferometer represent the optical HT and lenses L1, L2 and L1 indicate the optical GT. For simplicity, only the red component input-image fr(xi,yi) is considered. The collimated beam is split by a beam splitter BS1 into two beams. One-beam serves as object beam, which illuminates red channel on the first SLM (Spatial Light Modulator) is digitally permutated by first AT, changed spatial distribution of

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pixel value by first DCT, and then optically transformed by HT. The sum of real part and imaginary part of Fourier transform implemented by the CCP is displayed on the second SLM, which is digitally permutated by second AT and spatial distribution of pixel value is changed by second DCT. The inclination of another beam can be adjusted to fit the requirements. The beam splitter BS2 allows the addition of these two waves, which in turn is optically transformed by GT. The holographic interference fringes are then captured and recorded as an off-axis hologram by charged coupled device (CCD) camera and digitally processed on a computer system. Each RGB color channel independently recorded and processed by the same method are multiplexed to form encrypted color image. The reverse of the encryption procedure gives decrypted image.

4. Numerical simulation Numerical simulations have been performed on a Matlab 7.11.0 (R2010b) platform to verify 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 shown in Fig. 2(a). The iterative numbers of the first AT are

CCP

L

SLM1 BS1

Collimated beam

BS M1

f

Red channel

SLM2

L1

L2

L1

Reference beam

CCD M2 BS2

Computer System Fig. 1. Opto-electronic color image architecture of the proposed system. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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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) encrypted image, (c) decrypted image with all the correct keys, (d) decrypted image with one of the, transformation angle for each channel changed by 0.006, but all the other parameters, are correct. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

nr1 ¼ 30, ng1 ¼ 60 and nb1 ¼ 90, and that of the second AT are nr2 ¼ 120, ng 2 ¼ 150 and nb2 ¼ 180. The transformation angles of GT are ar ¼0.24, ag ¼0.48 and ab ¼0.72, respectively. The encrypted image obtained by the proposed scheme is shown in Fig. 2(b), which is a color noise-like image. The decrypted image with all correct keys is displayed in Fig. 2(c), which is like an original color image. If only one of the transformation angles of each channel of inverse GT is changed by 0.006 from its correct value, the decrypted images so obtained is also a noiselike image as illustrated in Fig. 2(d). This demonstrates that the transformation angles of GT are strong keys for the proposed security system. To compare the difference between the original image and recovered image, correlation coefficient (CC) is used, and defined as Ef½Ii E½Ii gf½Io E½Io g qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CC ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E f½Ii E½Ii 2 g f½Io E½Io 2 g

ð8Þ

values of CC show that color image cannot be recovered. The calculated CC values between original red, green, and blue channels and their corresponding decrypted component images with all right keys are 1.0000, which indicate that the original color image is completely retrieved. The sensitivity of decrypted image to a small change in rotation angles is checked. The red, green, and blue channels of an original color image are independently decrypted with the inverse GT at only one of the rotation angles of each component image changed by 0.006. The calculated CC values between original and decrypted red, green and blue channels are 2.15  10  2,7.69  10  2 and 1.439  10  1, respectively. Thus, the transformation angles of GT offer sensitive keys in the proposed algorithm. The CC values between original red, green, and blue channels and their corresponding recovered images calculated against variation in, the transformation angles and the iterative numbers are plotted as shown in Fig. 3(a) and (b), respectively. 4.2. Robustness test

where Io and Ii are, respectively, output and input images, and E[  ] denotes the expected value operator. A CC value of unity indicates that the original image is perfectly recovered. The calculated CC values between original red, green and blue channels and their corresponding encrypted images are,1.05  10  2, 2.4  10  3 and 4.4  10  3, respectively. These low

The robustness of the proposed algorithm has been verified against the chosen- and known plaintext attacks [37]. In chosen plaintext attack, Dirac delta function is employed at image plane and the random phase key is obtained at transform plane, which is used as retrieved key in decryption process. Fig. 4(a) and (b) show

M.R. Abuturab / Optics and Lasers in Engineering 51 (2013) 317–324

1.2

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1.2 Red

Red

Green

1 0.8

Blue

0.8 CC value

CC value

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0.6 0.4

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-0.2 0

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0.4 0.6 0.8 1 Transformation angle error

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60 80 Iterative number

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Fig. 3. (a) The CC values as a function of the transformation angles between original red, green, and blue channels and their corresponding decrypted images. (b) The CC values as a function of the iterative numbers between original red, green and, blue channels and their corresponding decrypted images. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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 image using chosen plaintext, attack, (c) Butter fly, first known plain image, (d) Ali, second known plain image, (e) retrieved, image using known plaintext attack.

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, random phase masks at transform plane are eliminated, and then the phase key

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is deduced by solving a linear system of equations. This system has trivial solution, the first phase key equals to zero. Therefore, the phase key at transform plane is obtained using Gauss elimination method, which is used as decryption key to retrieve the image. The algorithms for known plaintext attack are explained as follows: Let f1(xi,yi) and f2(x  i,yi) be two different known-input color images, andexp½i2pj1 xi ,yi , exp½i2pj2 ðx,yÞ be first and second random phase masks at input- and transform plane, respectively. Step I: Obtain the ciphered images corresponding to two known color images.     E1 ðx,yÞ ¼ Iff 1 xi ,yi ½expi2pj1 xi ,yi gexp½i2pj2 ðx,yÞ ð9Þ

      E2 ðx,yÞ ¼ I f 2 xi ,yi expi2pj1 xi ,yi exp i2pj2 ðx,yÞ

ð10Þ

where (x,y) denotes the transform plane coordinate. I{} represents the Fourier transform. Step II: Eliminate random phase masks at transform plane by solving a linear system of equations.     E1 ðx,yÞfIff 2 xi ,yi ½expi2pj1 xi ,yi gg     ¼ E2 ðx,yÞfIff 1 xi ,yi ½expi2pj1 xi ,yi gg ð11Þ     E1 ðx,yÞfIff 2 xi ,yi ½expi2pj1 xi ,yi gg

    E2 ðx,yÞfIff 1 xi ,yi ½expi2pj1 xi ,yi gg ¼ 0

ð12Þ   This system has a trivial solution exp½i2pj1 xi ,yi  ¼ 0, which is not valid as the first random phase mask is not zero.

Step III: Deduce random phase key exp½i2pj2 ðx,yÞ using Gauss elimination method and employ it as retrieve key for decryption of image.   exp i2pj2 ðx,yÞ ¼

E ðx,yÞ   1   Iff 1 xi ,yi ½expi2pj1 xi ,yi g

ð13Þ

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 illustrate that the proposed system has resistant against chosen- and known plaintext attacks. The robustness test of the proposed method checked against occlusion attacks on encrypted images with 50% and 75% occlusion sizes, are shown in Figs. 5(a) and 5(c), respectively, and corresponding recovered images are displayed in Figs. 5(b) and 5(d), respectively. The calculated CC values between original red, green and blue channels and their corresponding retrieved channels with all right keys from encrypted image with 50% occlusion are 2.799  10  1, 4.100  10  1 and 4.659  10  1, respectively, and that with 75% occlusion are 1.172  10  1,1.757  10  1 and 2.274  10  1, respectively. The robustness test is also confirmed against Gaussian and speckle noise attacks on the encrypted images with standard deviations of 0.1 and 0.4, are illustrated in Figs. 6(a) and 6(c), respectively, and corresponding recovered images are displayed in Figs. 6(b) and 6(d), respectively. The calculated CC values between original red, green and blue channels and their corresponding recovered channels with all right keys from encrypted

Fig. 5. The robustness test of the proposed method against occlusion attack on the encrypted, image: (a) with 50% occlusion, (b) corresponding recovered image from (a), (c) with 75% occlusion (d) corresponding retrieved image from (c).

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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 0.4, (d) corresponding, retrieved image from (c).

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0.4 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 Speckle noise

Fig. 7. (a) The CC values as a function of Gaussian noise attacks on the encrypted red, green, and blue channels and their corresponding decrypted images. (b) The CC values as a function of speckle noise attacks on the encrypted red, green, and blue channels and their corresponding recovered images. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

image with Gaussian noise having standard deviation 0.1 are 9.025  10  1, 8.780  10  1 and 8.960  10  1, respectively. Similarly the CC values between original red, green and blue channels and

their corresponding decrypted images with speckle noise having standard deviation 0.4 are 4.770  10  1, 5.343  10  1 and 5.949  10  1, respectively. The CC values between original red, green, and blue channels and their corresponding recovered

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images calculated against variation in standard deviations of, Gaussian noise and speckle noise are drawn in Fig. 7(a) and (b), respectively. The outline of the decrypted images from occludedand noised encrypted images can be identified.

5. Conclusion A novel image encryption and decryption algorithms based on AT and HT in GT domain is presented. In this method, a color image is separated into red, green, and blue channels. Each channel is encoded by first AT operation and first DCT at image plane, and then executed discrete HT. The transformed image is encoded by second AT and second DCT at GT plane, and performed GT. The system parameters of AT and GT in each channel serve as main keys in color image encryption and decryption, which enhance the security of the proposed algorithm. The proposed method is shown to be secured under chosen- and known plaintext attacks. Numerical simulations have demonstrated the feasibility of proposed scheme.

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