Securing color information using Arnold transform in gyrator transform domain

Optics and Lasers in Engineering 50 (2012) 772–779

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Securing color information using Arnold transform in gyrator transform domain Muhammad Rafiq Abuturab Department of Physics, Maulana Azad College of Engineering and Technology, Patna 801 115, India

a r t i c l e i n f o

a b s t r a c t

Article history: Received 9 October 2011 Received in revised form 12 December 2011 Accepted 13 December 2011 Available online 30 December 2011

In this paper, we propose a new method for securing color information based on Arnold transform in gyrator transform domain. A color image is first separated into red, green and blue component images, and each of these component images is then independently encrypted into first random phase mask placed at input image plane, and employed first Arnold transform and gyrator transform. The second random phase mask is placed at gyrator transform plane, and employed second Arnold transform and gyrator transform. The system parameters of Arnold transform and gyrator transform in each channel serve as additional keys in color image encryption and decryption, and hence enhances the security of the system. Numerical simulations are presented to confirm the security, validity and possibility of the proposed idea. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Random phase function Arnold transform Gyrator transform

1. Introduction The optical information processing based on optical signal processing and computational technique has been widely used in information security applications because of its inherent advantages of parallel and high speed processing and providing many degrees of freedom with which optical beam may be coded. Refregier and Javidi first time proposed a new optical image encryption method using double random phase encoding (DRPE) technique in the image- and Fourier domains [1]. The image encryption systems using fractional Fourier transform (FRFT) [2], extended FRFT [3–6], lensless optical security method [7] and, with multiple users and multiple security levels [8] have been proposed. The image encryption methods using pixel scrambling operation [9–11] have also been proposed. But, in all of these methods, input images are illuminated by monochromatic light and recovered images lose their color information, which is useful in image processing and practical applications. Zhang and Karim first time reported double random phase encoding system of single-channel color image encryption in Fourier domain [12]. The color image encryptions using wavelength multiplexing based on lensless Fresnel transform holograms [13], FRFT [14], the rotation of color vector based on discrete Hartley transform [15] and the simultaneous encryption of a color and a gray-scale image using single-channel double random phase encoding in the fractional Fourier domain [16] have been proposed. The color image encryption using Arnold

E-mail address: rafi[email protected] 0143-8166/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2011.12.006

transform (ART) and interference method [17] has been proposed in which RGB component images are encrypted into DRPE based on ART and virtual optics. The color image encryption method using discrete fractional random transform (DFRNT) and ART in the intensity-hue-saturation (IHS) color space [18] has been proposed. In this technique, a color image is transformed into IHS color space and I component is encrypted by DFRNT whereas H and S components are encrypted using ART. The color image encryption using Arnold transform and color-blend operation in discrete cosine transform (DCT) [19] has also been presented. In this method, Arnold transform 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 finally twice in DCTs encrypt the resulting image. The color image hiding and extracting using the phase retrieval algorithm in the FRFT domain and ART [20] has been reported. In this scheme, a secret color image is hidden in a host color image using a phase iterative algorithm in FRFT and ART permutation. Recently, polychromatic pattern recognition using color component 3D Arnold transform [21] has been proposed. Recently Rodrigo et al. have introduced optical gyrator transform (GT), formulated its main properties [22], applied it as a tunable optical mode converter and as an optical image processing and designed its flexible optical experimental setup [23]. The image encryptions using GT based on two-step phaseshifting interferometry [24], chaotic random phase masks [25] and Arnold transform [26] have been proposed. Arnold transform (ART) scrambles the matrix-pixel sequence by encoding a single parameter and forms color noise-like image, and reduces key space for storage and transmission applications [26].

M.R. Abuturab / Optics and Lasers in Engineering 50 (2012) 772–779

These properties motivated for using ART. The FRFT as well as the GT belong to the orthosymplectic class of linear canonical integral transforms and correspond to the rotations in the twisted position-spatial frequency planes of phase space [27]. GT is different transform from FRFT as four cross quadratic phase factors exist in its mathematical representation and as the kernel of the fractional FT is a product of the spherical and plane waves whereas the kernel of the GT is a product of hyperbolic and plane waves [28]. These basic differences motivated for using GT domains for color image encryption. In this paper, first time to my knowledge, a color image encryption/decryption based on ART and GT is introduced. A color image is first decomposed into red, green and blue component images, and each of these component images is then independently encrypted into first random phase function mask (RPM) placed at image domain

773

and, performed first ART and GT. The second RPM is placed at GT domain and, performed second ART and GT. The period of transform and iterative number of ART, and rotation angles of GT in each channel are additional keys to recovering the original color image. Numerical simulations are performed to confirm proposed algorithm. 2. Gyrator transform The optical GT at parameter a (called below as transformation angle) Ga of a two-dimensional function fi(xi,yi) associated in firstorder optics with the complex field amplitude is mathematically defined as [22] Z Z þ1 f o ðxo ,yo Þ ¼ Ga ½f i ðxi ,yi Þðxo ,yo Þ ¼ f i ðxi ,yi ÞK a ðxi ,yi xo ,yo Þdxi dyi ð1Þ 1

f xi , yi

f r xi , yi

r1

xi , yi

f g xi , yi

g1

xi , yi

fb xi , yi

b1

xi , yi

A nr1

A n g1

A nb1

G

G

G

r2

r1

x, y

g2

g1

x, y

b2

b1

x, y

A nr 2

A ng 2

A nb 2

G

G

G

r2

g2

b2

E xo , yo Fig. 1. (a) Flowchart corresponding to proposed color image encryption algorithm. (b) The flowchart corresponding to proposed color image decryption algorithm. (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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M.R. Abuturab / Optics and Lasers in Engineering 50 (2012) 772–779

of image matrix. ART is chaotic in unit square and possesses pseudo-numeration property [17–19] and thus it is applied to the two-dimensional image processing. The image is scrambled using ART for n iterative number and the scrambled image can be retrieved by inverse ART for m( ¼p  n) iterative number. Thereby both periodic scrambling transform p and iterative number are keys to decrypting the original image.

The Kernel of GT is defined as   1 ðxo yo þxi yi Þcos aðxi yo þ xo yi Þ exp i2p K a ðxi ,yi ; xo ,yo Þ ¼ sin a 9sin a9 ð2Þ Thus Z Z þ1 1 f i ðxi ,yi Þ f o ðxo ,yo Þ ¼ 9sin a9 1   ðxo yo þ xi yi Þcos aðxi yo þxo yi Þ dxi dyi exp i2p sin a

4. Proposed algorithm

ð3Þ

An RGB color image is an M  N  3 array of color pixels, where each color pixel is a triplet corresponding to the red, green and blue component images of an RGB image at a particular spatial location. The proposed optical encryption and decryption technique is based on double random phase mask encoding using twice ARTs and GTs. The schematic encryption and decryption processes are shown in Figs. 1(a) and (d), respectively. The input color image is first decomposed into three component images. Let f be color image and fr, fg and fb be, respectively, its red, green and blue component images in input plane:

where (xi,yi) and (xo,yo) indicate the input and output dimensionless 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 ¼ 3p/2, it corresponds to the inverse FT with the rotation of coordinates at p/2. When aA[0,2p], the GT can be implemented in the coherent optical system consists of three generalized lenses in which each lens is an assembled set of two thin convergent cylindrical lenses of the same power. For other angles a the Kernel of GT Ka(xi,yi;xo,yo) has constant amplitude and a hyperbolic phase structure. The GT is additive and periodic with respect to parameter a:GaGb ¼ Ga þ b and Ga þ 2p ¼Ga. Ga and G2p  a are reciprocal transforms and hence GT is applied to the two-dimensional image processing.

f ¼ f r þ f g þf b

4.1. Algorithm for encryption The component images fr, fg and fb are, respectively, multiplied by first random phase functions exp½i2pjr1 ðxi ,yi Þ, exp½i2pjg1 ðxi ,yi Þ and exp½i2pjb1 ðxi ,yi Þ within the range [0,2p] of the phase variation in input plane. The random functions are statistically independent and are distributed uniformly in an interval [0,1], where 0 and 1 are, respectively, RGB values of black and white. The ART at iterative numbers nr1, ng1 and nb1 are, respectively, performed over complex distributions for red, green and blue component images and then the GT at rotation angles ar1, ag1 and ab1 are, respectively, performed over the complex distributions for red, green and blue component images, and subsequently multiplied by second random phase functions exp½i2pjr2 ðx,yÞ, exp½i2pjg2 ðx,yÞ and exp½i2pjb2 ðx,yÞ in GT plane.

3. Arnold transform The Arnold transform (ART) operation AM of a two-dimensional function fi(xi,yi) is mathematically defined as [29] " 0#  " # ! x 1 1 xi AM : ¼ mod ,M ð4Þ y0 1 2 yi where (xi,yi) and (x0 ,y0 ) denote, respectively, the position vectors of matrix pixel before and after executing ART. The operator ‘‘mod’’ represents modulus after division operation. The period of transform depends on the parameter M, which represents the size

RPM 1

L1

L2

L1

ð5Þ

RPM 2

L'1

L'2

L'1

Encrypted image

CCD

Input Redcomponent image

SLM 1

GT 1

SLM 2

GT 2

Computer System Fig. 2. Electro-optical hybrid color image architecture of proposed system in which Arnold transform is performed digitally. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

M.R. Abuturab / Optics and Lasers in Engineering 50 (2012) 772–779

The ART at iterative numbers nr2, ng2 and nb2 are, respectively, performed over resulting complex distributions for red, green and blue component images and then the GT at rotation angles ar2, ag2 and ab2 are, respectively, performed over the resulting complex distributions for red, green and blue component images. The three encrypted component images Er, Eg and Eb are obtained in output plane, which are multiplexed to form encrypted color image E and is expressed as E ¼ Er þEg þ Eb

ð6Þ

For simplicity, only encrypted red-component image Er in output plane is expressed as Er ¼ G

ar2

nr2

fA

f½expfi2pjr2 ðx,yÞg

Gar1 fAnr1 ½f r ½expfi2pjr1 ðxi ,yi Þgggg

ð7Þ

 ag1 and  ab1 are, respectively, performed over complex distributions of red, green and blue component images and then ART at iterative numbers mr1, mg1 and mb1 are, respectively, performed over the resulting complex distributions red, green and blue component images, and subsequently multiplied by conjugate of first random phase functions exp½i2pjnr1 ðxi ,yi Þ, exp½i2pjng1 ðxi ,yi Þ and exp½i2pjnb1 ðxi ,yi Þ in input plane. The three decrypted component images Dr, Dg and Db are obtained in input plane, which are multiplexed to form color image and is expressed as D ¼ f ¼ Dr þ Dg þ Db

The decryption process is the inverse of the encryption process. The encrypted color image is first decomposed into encrypted red, green and blue component images. The GT at rotation angles rotation angles  ar2,  ag2 and  ab2 are, respectively, performed over encrypted red, green and blue component images and then ART at iterative numbers mr2, mg2 and mb2 are, respectively, performed over resulting encrypted red, green and blue component images, and subsequently multiplied by conjugate of second random phase functions exp½i2pjnr2 ðx,yÞ, exp½i2pjng2 ðx,yÞ and exp½i2pjnb2 ðx,yÞ in GT plane. Now the GT at rotation angles  ar1,

ð8Þ

For simplicity, only decrypted red-component image Dr in input plane is expressed as Dr ¼ Amr1 fGar1 fAmr2 fGar2 ½Er gexp½i2pjnr2 ðx,yÞgg exp½i2pjnr1 ðxi ,yi Þ

4.2. Algorithm for decryption

775

ð9Þ

Similarly, the encryption and decryption algorithms for green and blue component images can also be obtained using the same method. The proposed algorithm uses three RPMs, six different iterative numbers, six different periodic scrambling transforms and six different rotation angles as keys for the image encryption and decryption techniques. 4.3. Optical realization The GT can be implemented in coherent optical system, which contains three generalized lenses (denoted as L1, L2 and L1) with fixed equal distance z between them. Every generalized lens corresponds to the combination of two identical plano-convex

Fig. 3. 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 transformation angles for component images changed by 0.004 but correct iterative numbers and random phase functions. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

M.R. Abuturab / Optics and Lasers in Engineering 50 (2012) 772–779

5. Numerical simulation

where Io (i,j) and Ii (i,j) are, respectively, output image and input image at pixel position (i,j). (M  M) denotes the total number of pixels of the image.

8000 7000 6000 MSE (AU)

cylindrical lenses of the same power. The first and third identical generalized lenses are rotated with respect to each other having focal length f1 equal to distance z between two consecutive generalized lenses of the setup (the first and second or the second and third generalized lenses) whereas the second generalized lens is fixed having a focal length f2 ¼z/2. 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 performed by proper rotation of these lenses [23]. The proposed scheme for color image encryption using ART and GT can be electro-optically implemented. The electro-optical hybrid setup of the proposed encryption process for red color image is shown in Fig. 2. The blocks made with dotted lines having lenses L1, L2 and L1 indicate the first optical GT and that having lenses L0 1, L0 2 and L0 1 indicate the second optical GT. For simplicity, only the red component input image fr is considered. The red component image attached to a first RPM and displayed on the first SLM (Spatial Light Modulator) in input plane, is digitally permutated by first ART and then optically transformed by first GT. The resulting image attached to a second RPM and displayed on the second SLM in GT plane is digitally permutated by second ART and then optically transformed by second GT. The image so produced is superimposed on the plane reference beam to produce a holographic interference fringe, which is captured and recorded as an off-axis hologram by charged couple device cameras and digitally processed by computer systems. Each RGB color component image independently recorded and processed by the same method is multiplexed to form encrypted color image. The decryption process is the reverse of the encryption procedure.

5000 4000 3000 2000 Red Green Blue

1000 0 0

0.2

0.4

0.6 0.8 1 Transformation angle

1.2

5000 4000 3000 Red Green Blue

2000

MSE ¼

M X M  X  1 ½Io ði,jÞIi ði,jÞ2 M  Mi¼1j¼1

ð10Þ

1000 0 0

20

40

60 80 Iterative number

100

120

8000 7000 6000 MSE (AU)

Numerical simulations have been performed on a Matlab 7.11.0 (R2010b) platform to verify the validity, performance and robustness of proposed technique. The original color image with 512  512  3 pixels and 24 bits is shown in Fig. 3(a). The red, green, and blue components images are, respectively, encrypted independently with the first RPE and using ART at iterative numbers nr1 ¼30, ng1 ¼60 and nb1 ¼90, and GT at transformation angles ar1 ¼0.14, ag1 ¼ 0.28 and ab1 ¼0.42. The resulting red, green, and blue components are, respectively, encrypted independently with the second RPE and performing ART at iterative numbers nr2 ¼120, ng2 ¼150 and nb2 ¼180, and GT at transformation angles ar2 ¼0.56, ag2 ¼0.70 and ab2 ¼0.84. The three encrypted component images obtained are multiplexed to form a noise-like color image as displayed in Fig. 3(b). When all the correct RPMs along with correct iterative numbers of inverse ART and transformation angles of inverse GT are used, the three decrypted component images obtained are multiplexed to form like an original color image (without any noise and distortion) as illustrated in Fig. 3(c). If all the correct RPMs along with correct iterative numbers of inverse ART but transformation angles of inverse GT are changed by 0.004 from their correct values, the three decrypted component images obtained are multiplexed to also form a noise-like image as displayed in Fig. 3(d). This demonstrates that in addition to the RPMs and iterative numbers of ART, transformation angles of GT also serve as strong keys for the proposed security system. The values of iterative numbers and transformation angles used for encryption have been selected arbitrarily. The performance of the proposed technique has been evaluated quantitatively by calculating mean square error (MSE) and defined as

1.4

6000

MSE (AU)

776

5000 4000 Red Green Blue

3000 2000 1000 0 0

20

40

60 80 100 Iterative number

120

140

160

Fig. 4. (a) MSE values as a function of the transformation angles between original red, green. and blue component images and their corresponding decrypted images. (b) MSE values as a function of the iterative numbers between original red, green and blue component images and their corresponding decrypted images. (c) MSE values as a function of iterative numbers between original red, green and blue component images and their corresponding decrypted images. The iterative number n is fixed at 80 for all component images. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

M.R. Abuturab / Optics and Lasers in Engineering 50 (2012) 772–779

The calculated MSE values between original red, green and blue component images and their corresponding encrypted images are, 7.307  103,4.982  103and 5.088  103, respectively, which is very high. That means the color information is lost. The calculated MSE values between original red, green and blue component images and their corresponding decrypted component images with all right keys are 1. 693  10  26, 1.023  10  26 and 9.316  10  27, respectively, which is very low. That means the original color image is completely recovered. The sensitivity of retrieved image to small change in rotation angles is tested. The red, green and blue component images of an original color image are independently decrypted with the inverse GT at each rotation angle changed by 0.004 but with

777

correct random phase functions and iterative numbers. The calculated MSE between original and decrypted red, green and blue component images are 7.003  103,4.613  103 and 4.671  103, respectively. These MSE values are very close to MSE values calculated for encrypted images. This means no color information can be retrieved. Thus the transformation angles of GT are sensitive and provide strong encryption keys in the proposed algorithm. The quality of recovered image is evaluated more accurately by plotting MSE between decrypted image and original image. The MSE values between original red, green and blue component images and their corresponding recovered images calculated against variation in the transformation angles are plotted in

Fig. 5. 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 (d) corresponding reconstructed image from (b), (e) with 70% occlusion, (f) corresponding retrieved image from (e).

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M.R. Abuturab / Optics and Lasers in Engineering 50 (2012) 772–779

Fig. 4(a), the MSE values between original red, green and blue component images and their corresponding retrieved images calculated against variation in the iterative numbers are plotted in Fig. 4(b) and the MSE values between original red, green and blue component images and their corresponding retrieved images calculated against variation in the iterative numbers from iterative number fixed at 80 for all component images are plotted in Fig. 4(c). The robustness test of the proposed method is verified against occlusion attack on encrypted image with 25%, 50% and 70% occlusion sizes are shown in Figs. 5(a), (c) and (e), respectively, and corresponding recovered image are displayed in Figs. 5(b), (d) and (f), respectively. The calculated MSE values between original red, green and blue component images and their corresponding retrieved component images with all right keys from encrypted image with 25% occlusion are 2.458  103, 1.519  103and 1.400  103, respectively. The MSE values between original red, green and blue component images and their corresponding decrypted images with all right keys from encrypted image with 50% occlusion are 4.934  103, 3.095  103 and 2.914  103, respectively. The MSE values between original red, green and blue component images and their corresponding decrypted images with all right keys from encrypted image with 70% occlusion are 7.297  103, 4.954  103 and 5.118  103, respectively. In spite of the high values of MSEs and hence maximum data loss of 25%, 50% and 70% occluded encrypted images, their respective decrypted images with all correct keys can be recognized in vision. The proposed system demonstrates robustness against occlusion attacks and thus offers high security.

The robustness test is further verified against Gaussian and speckle noise attacks on the encrypted image with standard deviation of 0.1, are shown in Figs. 6(a) and (c), respectively and corresponding recovered image are displayed in Figs. 6(b) and (d), respectively. The calculated MSE values between original red, green and blue component images and their corresponding recovered component images with all right keys from encrypted image with Gaussian noise having standard deviation 0.1 are 1.238  103, 1.286  103and 1.287  103, respectively. The MSE values as a function of Gaussian noise attack on the encrypted red, green and blue component images and their corresponding decrypted images are shown in Fig. 7(a). Similarly the MSE values between original red, green and blue component images and their corresponding decrypted images with speckle noise having standard deviation 0.1 are 6.031  103, 4.342  103 and 3.925  103, respectively. The MSE values as a function of speckle noise attack on the encrypted red, green and blue component images and their corresponding decrypted images are displayed in Fig. 7(b). Albeit the high values of MSEs and hence maximum noisy data of standard deviations of 0.1 for Gaussian and speckle noisedencrypted images, their respective decrypted images with all correct keys can be identified in vision. The proposed system illustrates robustness against noised attacks and thus provides high security.

6. Conclusion We have proposed image encryption and decryption based on ART using GT. A color image is first decomposed into red, green

Fig. 6. Robustness test of the proposed method against (a) gaussian noise attack on the. encrypted image with variance 0.1, (b) corresponding recovered image from (a), (c) speckle noise attack on the encrypted image with variance 0.1, (d) corresponding retrieved image from (c).

M.R. Abuturab / Optics and Lasers in Engineering 50 (2012) 772–779

References

1500 1400 1300

Red Green Blue

MSE (AU)

1200 1100 1000 900 800 700 600 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Gaussian noise

0.1

0.11

0.1

0.11

2000 1800 1600

Red Green Blue

1400 MSE (AU)

779

1200 1000 800 600 400 200 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Speckle noise

Fig. 7. (a) MSE values as a function of gaussian noise attack on the encrypted red, green and. blue component images and their corresponding decrypted images. (b) MSE values as a function of speckle noise attack on the encrypted red, green and blue component images 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.)

and blue component images, and each of these component images is then independently encrypted into first random phase function mask placed at input domain executing first ART and GT. The second random phase function mask is placed at GT domain executing second ART and GT. The system parameters of two transforms in each channel serve as additional keys in color image encryption and decryption. Numerical simulations have demonstrated the feasibility of proposed scheme.

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