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J Opt 38 (1) : 22–28

J Opt 38 (1) : 22–28

RESEARCH ARTICLE

Optical image watermarking using fractional Fourier transform Naveen Kumar Nishchal Received: 13 January 2009 / Accepted: 24 February 2009 © Optical Society of India 2009

Abstract An optical image watermarking scheme using fractional Fourier transform is proposed. The watermark is encrypted using double random fractional order Fourier domain encoding scheme. Encrypted image is water marked into a host image. Embedding watermark sequences into fractional Fourier domain has an important advantage over embedding in spatial domain or in frequency domain. The watermark is recovered by applying corresponding correct fractional orders and random phase masks. The use of fractional Fourier transform offers additional degrees of freedom to enlarge the key size, thus enhancing the level of security. The effect of occlusion of watermarked image on the recovered watermark is also studied. The proposed idea is supported with simulation results.

Key words Watermarking • Fractional Fourier transform • Double random phase encryption.

Naveen Kumar Nischal (*) Department of Physics, Indian Institute of Technology Patna, Patliputra Colony, Patna 800 013, India. E-mail: [email protected]

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Introduction A digital or optical cryptographic system permits only valid key holders access to the encrypted data. But once such data is decrypted, it becomes almost impossible to track its reproduction. With the availability of modern copiers and scanners, the digital media can be easily duplicated without any loss of quality; therefore, the digital products attract the attention of hackers. However, this allows unauthorized illegal use of information, called the data piracy. Piracy of information without appropriate permission from rightful owners not only deprives rights of original creators but also harms innovations. A digital watermark is intended to complement the cryptographic processes. A watermark is a visible or invisible identification code that is permanently embedded in the data and remains present within the data after any decryption process. A good watermarking scheme should meet a number of conditions [1]. For example, the host data quality should not be affected in a significant way by the hidden data. Another important issue with watermarking is the level of security. In other words how hard it is to decode the hidden information by an unauthorized user even if the watermarking technique is known. A transformation domain is needed for embedding the watermark. The domain can be the spatial domain as well as the frequency domain. Several researchers have shown that it would be more robust to embed a watermark in frequency domain [1]. Frequency domain techniques mostly depend on the spread spectrum approach. Therefore, the signal energy present in any signal frequency becomes undetectable. Double random phase encoding technique has been widely used in image encryption, information hiding, and watermarking [1–5]. The technique offers high level of security and is robust to interference from noise and distortion. Optical information processing for encryption and watermarking have generated considerable interest in the optics community in the last one decade [1, 3, 5–12]. Optical implementation has some very promising scalability advantages over their purely electronic counterparts as, in principle, the size of the key can be increased without increasing the processing time. The vast majority of digital watermarking has been reported for one-dimensional or two-dimensional images. In some of the studies watermarking of three-dimensional images have also been reported [13, 14]. Image watermarking using digital holography have been further investigated by several researchers [15–17]. The fractional Fourier transform (FRT) is a generalization of the ordinary Fourier transform with an order parameter α [18]. A Fourier transform is a first order FRT with α = 1. Properties and applications of the ordinary Fourier transform are special cases of those of the FRT. In every area in which Fourier transforms and frequency domain concepts are used, the potential exists for generalization and improvement by using the fractional transform. The generalization of ordinary Fourier transform to the FRT comes at no additional cost in digital computation or optical implementation. Embedding watermark sequences into fractional Fourier domain has an important advantage over embedding in spatial domain or in frequency domain. Watermark in fractional order domain provides extra security against attackers since fractional orders of the transform provides extra degree of freedom [19, 20]. A fast algorithm has been developed for FRT calculation which is also an attractive feature for watermark embedding purposes [18]. In a double random phase encoding scheme, a primary image is converted to a stationary white noise using two statistically independent random phase codes with their phases uniformly distributed in the interval [0, 2π]. In this paper, an image watermarking scheme using double random fractional order Fourier domain encoding is proposed. The two phase codes are respectively placed in the input and in the fractional Fourier domain. Encrypted image is then watermarked in a host image. To successfully recover the watermark one has to use the corresponding correct fractional orders and the random phase codes. Section 2 explains the principle of the watermarking using FRT. As a proof-of-concept simulation study has been carried out. The effect of occlusion of watermarked image on the

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24 recovered watermark has been studied. Section 3 discusses the simulation results.

Principle Let function f (x, y) represent the watermark to be encrypted by double random fractional Fourier domain encoding scheme. The watermark is multiplied with an RPM1, defined as exp 2 Qjr 1Y, I6 , and its FRT of order α is obtained.

>1 6

1 6 C of order 1B  a Q 26 is given by g1Y, I6 as [18]  x y Y I  2 jQ xyYI  dxdy f 1x, y6 s exp 2 Qjr1x, y6 s exp jQ (1) tan B sin B  

A two-dimensional FRT of function f x, y s exp 2 Qjr Y, I

1 6

g Y, I  K

II

1

2

2

2

1

2

1

1

Here (x,y) and ( Y, I ) represent the space and fractional domain coordinates, respectively. The parameter K is defined by

  1 Q sgn1sin B 6  1 B "#  2 $ ! 4

exp -j K

1

sin B1

1

(2)

12

1 6

The function g Y, I is multiplied by an RPM2, defined as exp 2 Qjr1S, T 6 , and an FRT of order obtained, which is given as

II

1B

2

 a2Q

6

2 is

YIST  dYdI 1 6 >g1Y, I6 s exp 2Qjr1Y, I6 C s exp jQ Y Itan BS T  2 jQ sin (3) B  The function e1S, T6 is the encrypted image of watermark f x, y . The RPMs, r x, y and r 1Y, I6 are two independent e S, T  K

2

2

2

2

2

1

6

random functions uniformly distributed in the interval [0, 2π].

1 6

2

1

6

1 6

The encrypted version of watermark, e S, T , is combined with the host image, h S, T . Thus the watermarked

1 6

image, w S, T , is given by

1 6 1 6

1 6

w S, T  h S, T a e S, T

(4)

where 'a' is an arbitrary constant that ensures the invisibility of watermarked image and the robustness of the watermarked image against distortions. The value for a is selected by trial and error.

Computer simulation The results of computer simulation carried out on MATLAB platform (Fig.1(a–e)). Figure 1(a) shows the image of Lena of size 512 × 512 pixels, to be used as a host image. Fig.1(b) shows the watermark of size 512 × 512 pixels. The watermark is encrypted using double random phase fractional domain encoding, as shown in Fig.1(c). The fractional orders used for encryption were a1  0.25 and a 2  0.55 . The orders were selected arbitrarily. The value used for arbitrary constant a was 0.25. The watermarked host image has been shown in Fig.1(d) and the recovered watermark

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25

after using correct fractional orders and correct RPM has been shown in Fig.1(e). The watermark cannot be recovered without using the correct RPM and correct values of fractional orders.

1(a)

1(b)

1(c)

1(d)

Fig. 1 Simulation results: (a) Host image, (b) watermark, (c) encrypted image of watermark, (d) watermarked host image, and (e) recovered watermark after using correct fractional order and correct phase mask

1(e)

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26

The effect of occlusion of watermarked image on recovered watermark was also studied. The results of occlusion have been shown in Figs.2(a–f). Figs.2 (a,c,e) show the watermarked images with 25%, 50%, and 75% occlusion respectively. Figs.2(b,d,f) correspond to the recovered watermark. It was observed that with 50% occlusion the watermark is fully recovered. With 75% occlusion, most of the information content of the watermark is recovered but some information is embedded in noise. One may make intelligent guess to access the full information. It is also possible to apply some image processing tools to recover the full watermark. Thus it can be inferred that the proposed scheme is resistant up to 75% of data loss.

2(a)

2(b)

2(c)

2(d)

Fig. 2 Effect of occlusion of watermarked image on recovered data: (a) 25% occluded watermarked image, (b) recovered watermark due to 25% occlusion, (c) 50% occluded watermarked image, (d) recovered watermark due to 50% occlusion

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2(e)

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2(f)

Fig. 2 (e) 75% occluded watermarked image, and (f) recovered watermark due to 75% occlusion

The presence of watermark can be proved using correlation methods. While detecting the presence of watermark, the correlation between the extracted watermark and the original watermark is compared with the threshold value. If it exceeds the threshold then we can say that the watermark is present. In this study, the obtained value for correlation was 0.92. With occluded data the deterioration in autocorrelation values were observed. The values for autocorrelation between original watermark, the hide image and the recovered watermark were 0.73, 0.54, and 0.32 corresponding to 25%, 50%, and 75% occlusion of watermarked image.

Conclusion A domain optical image watermark embedding scheme has been proposed. The scheme has the advantage that it can be optically implemented employing the conventional double random phase encoding technique. The encrypted image is watermarked into a host image. Embedding watermark in fraction Fourier domain enhances the level of security. To recover the watermark the corresponding correct fractional orders and correct random phase codes are used. As a quality check of the recovered watermark, autocorrelation between original and recovered watermark have been calculated. The effect of occlusion of watermarked image on the recovered watermark has also been studied. Simulation results have been presented in support of the proposed idea.

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