Abstract

In this paper, a novel secure optimal image watermarking scheme using an encrypted gyrator transform computer generated hologram (CGH) in the contourlet domain is presented. A new encrypted CGH technique, which is based on the gyrator transform, the random phase mask, the three-step phase-shifting interferometry and the Fibonacci transform, has been proposed to produce a hologram of a watermark first. With the huge key space of the encrypted CGH, the security strength of the watermarking system is enhanced. To achieve better imperceptibility, an improved quantization embedding algorithm is proposed to embed the encrypted CGH into the low frequency sub-band of the contourlet-transformed host image. In order to obtain the highest possible robustness without losing the imperceptibility, particle swarm optimization algorithm is employed to search the optimal embedding parameter of the watermarking system. In comparison with other method, the proposed watermarking scheme offers better performances for both imperceptibility and robustness. Experimental results demonstrate that the proposed image watermarking is not only secure and invisible, but also robust against a variety of attacks.

© 2014 Optical Society of America

1. Introduction

With the rapid development of the information technology and the growth of the Internet, acquisition, exchange, and transmission of digital multimedia have become quite convenient. Meanwhile, digital media can be manipulated or reproduced easily without the loss of information, and even illegally distributed through the Internet. Hence, the copyright protection of digital multimedia has become an important issue. In order to prevent any copyright forgery, misuse or violation, digital watermarking, which allows for the imperceptibly embedding information in an original multimedia data, provides a promising way for the copyright protection of digital media [15].

In the past ten decade, lots of image watermarking methods have been proposed [27]. According to the domain in which the watermark is applied, the watermark can be classified into two categories: spatial domain or transform domain (such as discrete Fourier transform (DFT), discrete cosine transform (DCT), and discrete wavelet transform (DWT)) [25]. Compared with transform domain watermarking techniques, spatial domain methods are simple and fast, but are not robust against attacks. Among the transform domain techniques, DWT-based techniques are more popular, since DWT has a number of advantages over other transforms including space-frequency multiresolution representation, localization, superior HVS modeling, linear complexity, and adaptivity [6]. Even though DWT is popular, powerful, and familiar among watermarking methods, it has its own limitations in capturing the directional information such as smooth contours and the directional edges of the image [7]. This problem is addressed by contourlet transform (CT) proposed by Do et al. [8]. The CT possesses multiscale and time frequency localization properties of wavelet in addition to directionality and anisotropy [7]. Hence CT is considered as an improvement over wavelet in terms of efficiency. Therefore, we focus on CT domain watermarking in this work.

In general, the characteristics of digital watermarking are as follows [1]: (1) robustness, (2) imperceptibility, (3) capacity, (4) security, and (5) verifiability. However, in the watermarking system, there exist conflict requirements between the two major properties – robustness and imperceptibility which are essential in preserving the security of multimedia data from unauthorized usage [5]. In other words, if the watermarking is wanted to be more robust, then the quality of the watermarked digital data may be sacrificed. The design of an optimal watermarking always involves a trade-off between these requirements. Therefore, watermarking problem can be considered as an optimization problem. But most of existing watermarking methods usually employ pre-defined embedding rules and determine their embedding parameters, such as embedding strengths or thresholds, either empirically or experimentally. However, it is difficult to determine optimal watermarking parameters empirically or experimentally since watermarking methods have larger parameter space. As a result, these watermarking techniques do not exhibit desirable performance.

Recently, many watermarking methods which employ intelligent optimization techniques, such as genetic algorithm (GA) [5] and particle swarm optimization (PSO) [9], to optimize the robustness and imperceptibility have been proposed. These watermarking methods have attracted much attention because of their promising performance. Compared with other similar optimization techniques, such as GA which suffers from important limitations of high computational costs which eventually results in low convergence speed [10], PSO has some advantages as the following [10]: 1) PSO is easier to implement and there are fewer parameters to adjust; 2) PSO has a more effective memory capability since every particle remembers its own previous best value as well as the neighborhood best; 3) Because all the particles use the information related to the most successful particle, PSO is more efficient in maintaining the diversity of the swarm. In order to achieve a tradeoff between imperceptibility and robustness, PSO is utilized to search the optimal embedding parameter in this study.

In recent years, a new approach which uses digital holograms as the watermark signal has been investigated [1117]. Takai and Mifune proposed that the Fourier transform hologram is added to the spatial domain of the host image [11].The major drawback of this method is that the host image must be low-pass filtered to remove the high-frequency components before the superposition stage. Therefore the quality of the watermarked image is seriously degraded. To achieve the high quality of the watermarked image, Chang and Tsan proposed that the hologram is superposed on the DCT middle frequency coefficients of the host image [12]. But the method is weak in resisting a JPEG compression attack [12]. Wang et al. developed a method that the computer generated hologram (CGH) is also inserted into the DCT domain of the image [13]. This method also cannot resist the compression attack effectively. With quantization index modulation technique, Okman and Akar proposed that the hologram is added to the wavelet domain of the host image [14]. But the watermarking quantizers are determined experimentally in this method. Based on the geometric correction method, we presented a geometric robust watermarking scheme which embeds the CGH into the wavelet-transformed image [15]. However, the image quality may be decreased when the absolute values of the variations of the embedded wavelet coefficients are greater than half of the quantization step. Furthermore, the security of the methods mentioned above is less investigated and seriously affected. To increase the security, Cai et al. proposed that the holograms, which are obtained by three-step phase-shifting interferometry (PSI) [18] and encrypted using the random phase mask (RPM) [19], are added to the host images in the spatial domain after multiplying by a weighting factor [16]. But it is inconvenient for the copyright protection of an image because three interferograms are needed to be inserted into three same or different host images by using this watermarking method. To avoid the above drawback, the effect of embedding only real, imaginary, or phase parts of the hologram is studied in [17]. However, the quality of reconstruction of the extracted watermark is seriously degraded. In addition, the both above methods are less concerned with either the watermarked image quality or the robustness criteria.

In this paper, a secure optimized CGH-based image watermarking scheme is presented to guarantee high security, good imperceptibility, and robustness to withstand attacks. To obtain the high security of the watermarking scheme, based on the gyrator transform (GT) [20], the RPM, the three-step PSI and the Fibonacci transform [21], a novel encrypted CGH method which offers a huge key space is developed to generate a hologram of a watermark. In the embedding procedure, the encrypted CGH is inserted into the host image using the proposed improved quantization embedding algorithm which offers better imperceptibility. Different from the previously suggested holography-based methods, the mark hologram is inserted into the image in the contourlet transform domain. Moreover, to achieve the highest possible robustness without losing the imperceptibility, PSO is utilized to find the optimal embedding parameter which can satisfy the both imperceptibility and robustness requirements of the watermarking system. Compared with the method in [15], the proposed watermarking scheme not only provides better performance for imperceptibility, but also can resist most of common attacks more effectively. The experimental results demonstrate that the proposed watermarking scheme is not only secure and invisible, but also robust against different attacks.

2. Related background

2.1 Gyrator transform

The GT is a linear canonical integral transform which produces the rotation in the twisted position-spatial frequency planes of phase space [20]. Mathematically, the GT of a two-dimensional function f(x,y) can be expressed as

Gα(u,v)=GTα[f(x,y)]=f(x,y)|sinα|exp[i2π(xy+uv)cosαxvyusinα]dxdy,
where the Gα(u,v) and f(x,y) are the output and input of the transform, respectively. The symbol GTα represents gyrator operator. α is the transformation angle. The GT can be achieved either by using an optical system [20] or by using a fast algorithm [22].

2.2 Phase-shifting interferometry

Phase-shifting interferometry is an effective way to record complex wave field digitally. A variety of PSI techniques have been developed, including three-step, four-step, etc [23].

Let A(x,y)exp[iφ(x,y)] and Arexp(k) be the complex amplitude distribution of the object wave in the recording plane and the reference wave in that plane at the kth exposure, respectively. Here, Ar is a constant, δk is the phase shift of the reference wave between two adjacent steps and k = 1,2,…,N. The kth interference pattern Ik(x,y) can be represented as [18],

Ik(x,y)=A2(x,y)+2A(x,y)Arcos[φ(x,y)δk]+Ar2.

For a known set {δk}(k = 1,2,...,N), the expression of U(x,y) which is a digital hologram can be derived as a function {Ik} and {δk} [18]. In the three-frame case, N is 3. When δ1 = 0, δ2 = π/2 and δ3 = π, a digital hologram U(x,y) from the three interferograms I1, I2 and I3 can be expressed as [16,18]

U(x,y)=[(I1I3)+i(2I2I1I3)]/(4Ar).

2.3 Watermarking scheme in [15]

In the previous work [15], the mark hologram is suggested embedding into the DWT low frequency sub-band using the following formula.

Bm,n'=Bm,n/ΔΔ+λΔwm,n,
where Bm,n is the (m,n)th pixel of the low frequency sub-band, 0≤wm,n≤1 is the (m,n)th element of the normalized hologram, is the floor operation, Δ>0 is the quantization step and 0<λ<1 is a factor to avoid the errors in the extraction process when wm,n = 1. Using the following formula, the watermark can be extracted.

wm,n'=(Bm,n'Bm,n'/ΔΔ)/(λΔ).

Let ρ=Bm,nBm,n/ΔΔ, then 0≤ρ<Δ. And Eq. (4) can be rewritten as follows.

Bm,n'=Bm,nρ+λΔwm,n.

However, the quality of the watermarked image may be degraded when |-ρ + λΔwm,n|>Δ/2. The reason is analyzed as follows.

From |-ρ + λΔwm,n|>Δ/2, –ρ + λΔwm,n>Δ/2 and –ρ + λΔwm,n<–Δ/2 can be obtained.

For –ρ + λΔwm,n>Δ/2 case, let X = Bm,n–Δ firstly. Then embed wm,n into X to obtain X'=X/ΔΔ+λΔwm,n using Eq. (4). The watermark signal w’m,n can be extracted using Eq. (5).

wm,n'=(X'X'/ΔΔ)/(λΔ)=((Bm,nΔ)/ΔΔ+λΔwm,n((Bm,nΔ)/ΔΔ+λΔwm,n)/ΔΔ)/(λΔ)=(Bm,n/ΔΔ+λΔwm,n(Bm,n/ΔΔ+λΔwm,n)/ΔΔ)/(λΔ)=(Bm,n'Bm,n'/ΔΔ)/(λΔ).

It can be observed from Eq. (5) and Eq. (7) that the extracted signals are the same from B’m,n and X’m,n which were embedded with wm,n, respectively.

On the other hand, the variation between Bm,n and X’, and that between Bm,n and B’m,n can be calculated as follows.

|Bm,nX'|=|Bm,n(X/ΔΔ+λΔwm,n)|=|Bm,n((Bm,nΔ)/ΔΔ+λΔwm,n)|=|Bm,n(Bm,n/ΔΔΔ+λΔwm,n)|=|Δ(ρ+λΔwm,n)|.
|Bm,nBm,n'|=|Bm,n(Bm,n/ΔΔ+λΔwm,n)|=|(ρ+λΔwm,n)|.

Since 0≤ρ<Δ, 0≤wm,n≤1, Δ>0, 0<λ<1 and –ρ + λΔwm,n>Δ/2, the following inequality can be obtained according to Eq. (8) and Eq. (9).

0<|Bm,nX'|<Δ/2<|Bm,nBm,n'|<Δ.

It can be seen from inequality (10) that the variation between Bm,n and X’ is less than that between Bm,n and B’m,n. Hence, the quality of the watermarked image can be improved if Bm,n is replaced by Bm,n –Δ when –ρ + λΔwm,n>Δ/2.

For –ρ + λΔwm,n<–Δ/2 case, let Y = Bm,n + Δ. Similar to the procedure mentioned above, the formula and the inequality can be obtained as follows.

wm,n'=(Y'Y'/ΔΔ)/(λΔ)=(Bm,n'Bm,n'/ΔΔ)/(λΔ).
0<(|Bm,nY'|=|ρλΔwm,nΔ|)<Δ/2<|Bm,nBm,n'|<Δ.

Therefore, the invisibility of the watermarking method can also be improved if Bm,n is replaced by Bm,n + Δ when –ρ + λΔwm,n<–Δ/2.

2.4 Particle swarm optimization

PSO is an evolutionary optimization technique which is proposed by Kennedy and Eberhart [24]. Birds’ swarm behavior is simulated in the PSO model. The swarm is modeled by particles in d-dimensional search space. Every particle i has its own position xid(it) and velocity vid(it). At the start of the PSO process, the position xid(it) and the velocity vid(it) of each particle are initialized randomly. Every particle in the swarm is a part of the solution set. According to its best historical experience and the best experience of the whole swarm, every particle in the swarm moves iteratively. A fitness function is used to evaluate the effectiveness of the particles in the swarm. During the iteration, let pid and pgd, which are called the local best value and the global best value [9], denote the best position among all positions that the ith particle has reached, and the global best position that all the particles have reached, respectively. After each iteration it, the velocity and the position of every particle are updated by the following formula [9].

vid(it+1)=ωvid(it)+c1rand()(pidxid(it))+c2rand()(pgdxid(it)).
xid(it+1)=xid(it)+vid(it+1).
where c1 and c2 are constants and ω is the inertia weight; rand() is a random number function and its values are between 0 and 1; it shows the iteration number. Inertia weight is used to balance the search ability of the algorithm over global and local exploration and exploitation. The inertia weight proposed in [25] is used in this work.
ω(it)=ωstart(ωstartωend)it/itmax.
where it is the current iteration number and itmax denotes the maximum iteration number; and ωstart and ωend are the initial and final values of inertia weight, respectively. Earlier studies on PSO have shown that the applications have good performances when c1 = 2, c2 = 2, ωstart = 0.9 and ωend = 0.4 [9]. Hence, c1, c2, ωstart and ωend are 2, 2, 0.9 and 0.4 in this study, respectively.

3. The proposed method

3.1 Encrypted gyrator transform computer generated hologram

CGH is a holographic technique that the diffraction pattern to be used as the hologram is generated by computer and then the image is reconstructed numerically. In this work, a secure encrypted CGH method is proposed based on the Fibonacci transform, the GT, the RPM and three-step PSI.

3.1.1 Generation of the encrypted hologram

The encrypted CGH is generated according to the following steps:

  • 1) Divide the input image f0(x0,y0) into nonoverlapping blocks with t × t pixels. Assume that the size of the image is m × n, and m and n are the integral multiples of t respectively, f0(x0,y0) has (m/t) × (n/t) blocks after splitting. Let Bk,l be the (k,l)th block in f0(x0,y0). Here, 1≤km/t and 1≤ln/t.
  • 2) Generate a positive integer matrix D size with (m/t) × (n/t) randomly. Let Dk,l which is a random positive integer be the (k,l)th element of the matrix.
  • 3) To increase the security, the blocks obtained in step 1) are scrambled by the Fibonacci transform [21] which is usually used to make the change of sequence of image pixels. The parameter of Fibonacci transform can be considered as the key for decryption. In the scrambling process, the pixels of block Bk,l are scrambled by using Fibonacci transform with the parameter Dk,l. Repeat the procedure until all the blocks in f0(x0,y0) are permuted. Let f(x0,y0) denote the scrambled image.
  • 4) The scrambled image f(x0,y0) is multiplied by a virtual RPM R1, mathematically represented by function f(x0,y0)exp[i2πp(x0,y0)], where p(x0,y0) is a random function distributed uniformly in the interval [0,1]. Through the use of the fast GT algorithm [22] at angle α1, the resulting complex distribution U1(x1,y1) can be represented by
    U1(x1,y1)=GTα1{f(x0,y0)exp[i2πp(x0,y0)]}.
  • 5) The complex amplitude U1(x1,y1) is multiplied with a virtual RPM R2, exp[i2πq(x1,y1)], where q(x1,y1) is also a random function distributed uniformly in the interval [0,1]. By using the GT algorithm at angle α2, the object wave becomes
    U2(u,v)=GTα2{U1(x1,y1)exp[i2πq(x1,y1)]}=A'(u,v)exp[iφ'(u,v)],

    where A’(u,v) and φ’(u,v) are the amplitude and the phase of U2(u,v) respectively.

  • 6) Similar to Huang’s method [26], let |A’(u,v)|max = 1 and Ar = 1, three interference patterns which are generated by computer can be obtained according to the three-step PSI.
    I1=0.5A'(u,v)Ar{1+cos[φ'(u,v)δ1]}=0.5A'(u,v){1+cos[φ'(u,v)]}.
    I2=0.5A'(u,v)Ar{1+cos[φ'(u,v)δ2]}=0.5A'(u,v){1+sin[φ'(u,v)]}.
    I3=0.5A'(u,v)Ar{1+cos[φ'(u,v)δ3]}=0.5A'(u,v){1cos[φ'(u,v)]}.

    Here, δ1 = 0, δ2 = π/2 and δ3 = π.

  • 7) According to Eq. (3), the encrypted hologram U(u,v) can be achieved as
    U(u,v)=[(I1I3)+i(2I2I1I3)]/(4Ar)=[(I1I3)+i(2I2I1I3)]/4=A(u,v)exp[iφ(u,v)],

    where A(u,v)∈[0, 0.25] and φ(u,v)∈[0, 2π] are the amplitude and the phase of U(u,v) respectively. Noting that (I1-I3) = A’(u,v)cos[φ’(u,v)] and (2I2-I1-I3) = A’(u,v)sin[φ’(u,v)], the amplitude A(u,v) = [(I1-I3)2 + (2I2-I1-I3)2]1/2/4 = |A’(u,v)|/4. Since |A’(u,v)|max = 1, A(u,v)∈[0, 0.25].

    The parameters of the proposed method, including the random integer matrix D, the random phase code q(x1,y1) and the angles of GTs α1 and α2, form a very large key space enhancing the security level of the encryption system. Only when all the right keys are simultaneously used for decryption can CGH be reconstructed correctly.

3.1.2 Reconstruction of the encrypted hologram

The encrypted hologram can be reconstructed according to the following steps:

  • 1) Apply a GT to U(u,v) with angle -α2.
  • 2) Multiply the resulting complex distribution by the conjugate of the RPM R2.
  • 3) Make another GT with angle -α1. The resulting distribution function of reconstruction can be expressed as
    U'(x0,y0)=GTα1{GTα2[U(u,v)]exp[i2πq(x1,y1)]}.
  • 4) Take the amplitude f’(x0,y0) of U’(x0,y0).
  • 5) Divide the f’(x0,y0) into nonoverlapping blocks B’ = {B’k,l} with t × t pixels. Then the decrypted image can be obtained after applying the inverse Fibonacci transform to every block B’k,l with Dk,l.

Figure 1 shows the generation and reconstruction processes of the encrypted hologram. In Fig. 1 (b), conj(R2) is the conjugate of the RPM R2.

 figure: Fig. 1

Fig. 1 The diagram of the generation and reconstruction processes of the encrypted hologram. (a) The generation procedure for the encrypted CGH. (b) The reconstruction procedure for the encrypted CGH.

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3.2 The proposed watermarking method

This section describes the overall scheme of the encrypted CGH-based watermarking. First, it illustrates how the embedding process is performed. Then, the extraction process is explained.

3.2.1 Watermark embedding algorithm

The mark CGH is embedded into the low frequency sub-band of the contourlet-transformed host image. The watermarking process is presented as follows.

  • 1) According to the method presented in sub-section 3.1.1, the encrypted hologram WM = Awexp(w) of the original watermark W is generated by computer. Here, Aw∈[0, 0.25] and φw∈[0, 2π] are the amplitude and the phase of WM respectively.
  • 2) To avoid the watermarked data becoming complex, the to-be-embedded watermark signal is consisted of Aw and φw instead of the whole hologram. In general, greater robustness can be achieved if significant modifications are made to the host image. Hence, since Aw is too small, the robustness will be decreased if Aw is inserted into the host image directly. In addition, the watermark signal should be in the interval [0,1] when the embedding Eq. (4) is used. However, φw belongs to [0, 2π]. Therefore, the to-be-embedded watermark can be obtained using the following formula.
    wm=[TΨ]=[1Awφw/(2π)],

    where T = 1- Aw and Ψ = φw/(2π) belong to [0.75,1] and [0,1] respectively. Suppose the size of W is M × N, the size of wm is 2M × N.

  • 3) To weaken the correlation of watermark pixels, wm is scrambled by Fibonacci transform with the parameter z. Here, z is a random positive integer.
  • 4) Decompose the normalized host image with n-level CT to obtain the low frequency sub-band CL.
  • 5) To overcome the drawback of method in literature [15], which is analyzed in sub-section 2.3, wm is embedded into CL according to the following formula.
    CLξ,η'={(CLξ,ηΔ)/ΔΔ+λΔwmξ,η,ifρ+λΔwmξ,η>Δ/2(CLξ,η+Δ)/ΔΔ+λΔwmξ,η,ifρ+λΔwmξ,η<Δ/2CLξ,η/ΔΔ+λΔwmξ,η,otherwise,

    where ρ=CLξ,ηCLξ,η/ΔΔ, CLξ,η is the (ξ,η)th pixel of the low frequency sub-band, wmξ,η is the (ξ,η)th element of wm, Δ and λ have the same meaning to those in sub-section 2.3.

  • 6) With inverse CT, the watermarked image is generated.

    The transform parameter z, the random matrix D, the random phase code q(x1,y1) and the angles of GTs α1 and α2 are the private keys for watermark extraction.

3.2.2 Watermark extraction

The watermark can be extracted without the host image. Watermark extraction algorithm steps are as follows.

  • 1) Decompose the watermarked image with n-level CT to obtain the low frequency sub-band CL’. Then extract the watermark signal wm’ using the following formula.
    wmξ,η'=(CLξ,η'CLξ,η'/Δ)/(λΔ).
  • 2) Apply inverse Fibonacci transform to wm’ with z.
  • 3) According to Eq. (23), take T’ and Ψ’ from wm’. Let A’w = 1-T’ and φ’w = 2πΨ’. Then the mark hologram WM’ = A’wexp(iφ’w) is obtained.
  • 4) With D, q(x1,y1), α1 and α2, the encrypted hologram WM’ can be reconstructed according to the method described in sub-section 3.1.2,

3.3 Performance improvement using PSO

To obtain the trade-off between imperceptibility and robustness, the PSO is used to search the optimal quantization step Δ in Eq. (24). One quality value of the watermarked image and five robustness values of the extracted watermark from the attacked images are used to model a fitness function. The fitness function are defined as

Fitness=PSNR/100+1.1×(j=15NCCj)/5.
where the peak signal to noise ratio (PSNR), which is usually used to measure the watermarked image quality, is computed as [9]
PSNR=10log10(MaxPV2/MSE)(dB),
MSE=(m=1Mn=1N(Hm,nH'm,n)2)/(M×N),
where MSE is the mean squared error, Hm,n and H’m,n are the original and the watermarked images respectively. The sizes of the both images are M × N and MaxPV is the maximum pixel value of the image.

The normalized cross correlation (NCC) is used to indicate the similarity between two images. In our case, it is used to evaluate the similarity between the extracted watermark and the embedded watermark. The value of NCC defines the strength of robustness of extracted watermark from the attacked image [27]. It can be calculated by the following formula [27].

NCC=k=1Kl=1L(Wk,lW'k,l)/k=1Kl=1L(Wk,l)2,

where Wk,l and W’k,l are the original and the extracted watermarks respectively. The sizes of the both watermarks are K × L. Since PSNR is far greater than NCC, the PSNR is divided by 100 and the average NCC is multiplied by 1.1 as shown in Eq. (26) to balance the effects of PSNR and NCC values for fitness function. Because an ideal digital image watermarking scheme should be able to resist both intentional and unintentional attacks, five attacks of the most common intentional and unintentional attacks [28] are considered, namely Gaussian low-pass filter 3 × 3, median filter 3 × 3, cropping 25%, JPEG compression of 30% quality and horizontal motion of six pixels within this work. Every attack corresponds to a different NCC value by using Eq. (29).

A particle, which represents the quantization step of watermark, is a one-dimension vector and is used to find suitable quantization step for embedding watermark. That is, finding the optimal Δ for Eq. (24) as well as finding the max value for fitness function as shown in Eq. (26). The flowchart of the parameter optimization of watermarking using PSO is illustrated in Fig. 2. If the number of iteration is less than the max iteration number of PSO, for each particle, its velocity and position are updated according to Eq. (13) and Eq. (14), and the watermark is embedded into the host image and the PSNR of the watermarked image is calculated. Then apply the five attacks mentioned above to the watermarked image respectively, and calculate the corresponding NCCs after watermark extraction. With PSNR and NCCs, fitness value is obtained by the fitness function. In each iteration, pid and pgd are updated according to the fitness values. The above procedure is repeated until the iteration number is equal to the max iteration number. In Fig. 2, it_num and Maxnum are the iteration number and the max iteration number, respectively.

 figure: Fig. 2

Fig. 2 PSO based optimization process of watermarking.

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4. Experiment and results

Experiments were carried out to verify the performance of the proposed watermarking scheme. The simulations in the experiment were implemented using MATLAB.

4.1 Generation of the encrypted mark hologram

Based on the method mentioned in section 3.1, the encrypted gyrator transform CGH was generated. Figure 3 illustrates a sample watermark and its reconstruction. The image “STU” of size 32 × 64 shown in Fig. 3(a) was adopted to generate the encrypted mark hologram. In the experiment, t = 16, α1 = 245°, α2 = 275° and the elements of D are the random positive integers between 1 and 1000.

 figure: Fig. 3

Fig. 3 A sample watermark generated by computer. (a) image “STU” of size 32 × 64; (b), (c), (d) interferograms I1, I2 and I3, respectively. (e), (f) the amplitude and the normalized phase of the gyrator transform hologram. (g) reconstruction of the hologram with all correct keys.

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Figures 3(b)-3(d) show the three interferograms I1, I2 and I3, which are generated by computer. The amplitude and the normalized phase of the hologram obtained from Eq. (21) are shown in Figs. 3(e) and 3(f), respectively. Figure 3(g) is the reconstructed image with the correct D, q(x1,y1), α1 and α2.

4.2 Test for the effectiveness of the proposed watermarking scheme

Experiments were carried out to investigate the effectiveness of the presented watermarking scheme. The PSNR and the NCC are used to measure the watermarked image and the extracted watermark respectively. It should be noted that the larger PSNR, the better imperceptibility. If a method has a higher NCC, it is more robust.

Figures 4(a)-4(d) show 4 host images with size 512 × 512, which include Goldhill, Elaine, Butterfly and Scenic. According to Eq. (23), the watermark signal shown in Fig. 4(e) is consisted of the amplitude and the phase of the encrypted hologram.

 figure: Fig. 4

Fig. 4 The host images and watermark. (a) Gold hill. (b) Elaine. (c) Butterfly. (d) Scenic. (e) the watermark signal consisted of the amplitude and the phase of the encrypted hologram.

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In the Experiments, the normalized host images were decomposed with 3-level CT. In order to weaken the correlation of watermark pixels, Fig. 4(e) is permuted by Fibonacci transform with z = 136. The optimal quantization steps for different host images are obtained using PSO mentioned in sub-section 3.3. The parameters of the PSO model c1, c2, ωstart and ωend are set to 2, 2, 0.9 and 0.4 as given in sub-section 2.4. The iteration is 100 and the number of particle is 20. The factor λ is 0.998.

4.2.1 Test results for imperceptibility

According to the parameters given above, the proposed watermarking scheme is implemented on host images. The PSO convergence history for the proposed method is shown in Fig. 5. As seen from the figure, the PSO algorithm will converge in a specific value after several iterations. The corresponding quantization step of this value is used as the optimal quantization step. After the PSO based optimization process of watermarking, the optimal Δs for Goldhill, Elaine, Butterfly and Scenic are 0.2012, 0.1916, 0.2034 and 0.1924, respectively.

 figure: Fig. 5

Fig. 5 The PSO convergence history for the proposed method. The green line, the pink line, the blue line and the red line are the convergence curves for Goldhill, Elaine, Butterfly and Scienic, respectively.

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Figure 6 shows the results of the proposed watermarking scheme without any attacks. Figure 6(a), Fig. 6(c), Fig. 6(e) and Fig. 6(g) are the watermarked images, and Fig. 6(b), Fig. 6(d), Fig. 6(f) and Fig. 6(h) depict the corresponding reconstructions of the extracted mark holograms. The PSNR values of the watermarked images, which are compared with the scheme in [15], are listed in Table 1. In general, a processed image is acceptable to the human eyes if its PSNR is greater than 30 dB [29]. The larger the PSNR, the better is the image quality. It can be observed from Fig. 6 and Table 1 that all the PSNRs of the watermarked images are larger than 40 db. The high PSNR values prove the good imperceptibility of the proposed scheme. From Table 1, the proposed method is better than the scheme in [15] in terms of the imperceptibility.

 figure: Fig. 6

Fig. 6 The watermarked images and the corresponding reconstructions of the extracted mark holograms without any attacks. (a) The watermarked “Gold hill”(PSNR = 41.284). (b) reconstructed watermark of (a). (c) the watermarked “Elaine” (PSNR = 42.225). (d) reconstructed watermark of (c). (e) the watermarked “Butterfly” (PSNR = 40.456). (f) reconstructed watermark of (e). (g) the watermarked “Scenic” (PSNR = 42.481). (h) reconstructed watermark of (g).

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Tables Icon

Table 1. PSNR Values of Watermarks Extracting without Any Attacks Comparing between Proposed Method and Scheme in [15]

4.2.2 Test results for robustness

To evaluate the robustness of the proposed scheme, various attacks, which include filtering, noise addition, compression, cropping, blurring, brighten, darken, rescaling, painting, etc., have been performed.

For all attack cases, the NCC values obtained from both proposed method and scheme in [15] are presented in Table 2. Some attacked watermarked images of the proposed method are shown in Fig. 7. Figure 8 shows the corresponding reconstructions of the watermarks which are extracted from the above attacked images. It can be observed from the Table 2 and Fig. 8 that the proposed method has good robustness against a variety of attacks. As can be seen from Fig. 8, the reconstructed images are clear enough to be recognized. From the experimental results listed in Table 2, it is clear that the proposed scheme outperforms the scheme in [15] under most attacks in terms of NCC. However, the NCC values of cropping and painting attacks of the proposed method are smaller than those of the method in [15].The reason is analyzed as follows.

Tables Icon

Table 2. NCC Values of Watermarks Extracting under Attacks Comparing between Proposed Method and Scheme in [15]

 figure: Fig. 7

Fig. 7 The attacked watermarked images. (a) Gaussian low-pass filtering (PSNR = 26.28). (b) median filtering (PSNR = 28.34). (c) average filtering (PSNR = 30.47). (d) Gaussian noise (PSNR = 29.56). (e) salt & pepper noise (PSNR = 27.64). (f) JPEG (PSNR = 32.24). (g) JPEG2000 (PSNR = 34.88). (h) cropping (PSNR = 11.69). (i) unsharp (PSNR = 26.33). (j) horizontal motion (PSNR = 28.07). (k) circular average (PSNR = 24.63). (l) brighten (PSNR = 19.83). (m) darken (PSNR = 18.06). (n) rescaling (PSNR = 26.26). (o) painting (PSNR = 17.84).

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 figure: Fig. 8

Fig. 8 The corresponding reconstructions of the holograms extracted from the distorted watermarked images in Fig. 7. (a) Gaussian low-pass filtering. (b) median filtering. (c) average filtering. (d) Gaussian noise. (e) salt & pepper noise. (f) JPEG. (g) JPEG2000. (h) cropping. (i) unsharp. (j) horizontal motion. (k) circular average. (l) brighten. (m) darken. (n) rescaling. (o)painting.

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As can be seen from Fig. 7(h) and Fig. 7(o), the pixel values of the cropped or painted regions in the attacked images are set to 0 and 255. Hence, the details of the cropped or painted regions are removed. In image processing, the high frequency coefficients denote the details of the image. So the corresponding high frequency information of the processed regions mentioned above is lost. Furthermore, on the one hand, the change of high frequency coefficients affects the low frequency coefficients in CT domain [30]. On the other hand, the change of high frequency coefficients does not affect the low frequency coefficients in DWT domain [30]. Therefore, for the image processed by cropping or painting, the variations of the low frequency sub-band coefficients in CT domain are greater than those in DWT domain. Consequently, with the extraction algorithm mentioned in sub-section 3.2.2, the distortion of the watermark extracted from the CT low frequency sub-band is more seriously than that of the watermark which is extracted from the DWT low frequency sub-band when the watermarked image suffers from attacks of cropping or painting. As a result, the NCC values of the watermarks, which are extracted from the watermarked images undergone the both attacks mentioned above, of the proposed method are less than those of the method in [15].

4.2.3 Test results for security

The sensitivity of the reconstruction of the extracted hologram to small changes of the keys D, q(x1,y1), α1 and α2 is tested. Figure 9(a), which cannot be recognized, is the reconstructed image with correct q(x1,y1), α1 and α2 but with D = D + 1. Figures 9(b) and 9(c) are the images reconstructed with correct D, α1 and α2 but without RPM R2 and with q(x1,y1) whose transverse position has been shifted by just one pixel, respectively. These two images cannot be distinguished too. The effects of small changes of the angles α1 and α2 on the reconstructed images are shown in Figs. 9(d) and 9(e). It can be observed from Fig. 9(d) and 9(e) that the images become unrecognizable when the relative errors of α1 and α2 reach approximately 1.4% and 0.8% even if the other three parameters are correct. As can be seen from Fig. 9(f), even with right D and q(x1,y1), the reconstructed image cannot be recognized when α1 and α2 both have relative errors of 0.6%. The results show that the proposed watermarking scheme has good security characteristics. Only when all the correct keys are simultaneously used for reconstruction can information be accessed.

 figure: Fig. 9

Fig. 9 The reconstructions of the watermarks which were extracted from watermarked Goldhill. (a) reconstructed image with correct q(x1,y1), α1 and α2 and with D = D + 1; (b) reconstructed image with correct D, α1 and α2 when RPM R2 is not used; (c) reconstructed image with correct D, α1 and α2 when q(x1,y1) is shifted transversely by one pixel; (c) reconstructed image with correct D, q(x1,y1) and α2 when α1 has a relative error of 1.4%; (d) reconstructed image with correct D, q(x1,y1) and α1 when α2 has a relative error of 0.8%; (e) reconstructed image with correct D and q(x1,y1) when α1 and α2 both have relative errors of 0.6%.

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5. Conclusion

This paper presents a novel secure optimal method of robust image watermarking based upon encrypted CGH using PSO algorithm. The proposed method is blind and does not need the host image during the extraction process.

A new encrypted CGH technique, which is based on the gyrator transform, the RPM, the three-step PSI and the Fibonacci transform, has been proposed to produce a hologram of a watermark. Compared with conventional optical holography, the proposed encrypted CGHs have the advantage of being easily and effectively generate by computer. Besides, the method can provide a huge key space, which enhances the security level. One needs to specify all the right keys to reconstruct the CGH correctly.

In the proposed scheme, the encrypted CGH is superposed onto the contourlet domain of the host image using the improved quantization embedding algorithm. To optimize both the watermarked image quality and robustness of watermarking, PSO is applied to search the optimal quantization step of embedding watermark. In the optimization process, a fitness function is defined based on the two conflicting requirements of imperceptibility and robustness, and then the quantization step can be obtained adaptively by invoking PSO algorithm. Through the PSO method, the perceptual transparency and robustness can be optimized. As a result, in the proposed method the values of the PSNR of the watermarked images are always greater than 40 dB. The extensive experimental works have also shown that the watermarking scheme has a satisfactory performance for different common image processing operations, including filtering, noise addition, JPEG/JPEG2000 compression, cropping, blurring, brighten, darken, rescaling, etc. In addition, with the good security feature of the encrypted CGH, the security strength of the watermarking system is enhanced.

Acknowledgments

This work is partly supported by the 2013 higher school discipline and specialty construction project in Guangdong Province (2013KJCX0127), the 2013 quality project of Department of Education of Guangdong Province, the Science and Technology Program of Chaozhou (2013G01) and the scientific project of Hanshan Normal University (QD20111108 and LT201201). The author thanks the reviewers for their helpful suggestions and comments.

References and links

1. C. C. Lai, “An improved SVD-based watermarking scheme using human visual characteristics,” Opt. Commun. 284(4), 938–944 (2011). [CrossRef]  

2. L. D. Li and B. L. Guo, “Localized image watermarking in spatial domain resistant to geometric attacks,” Int. J. Electron. Commun. 63(2), 123–131 (2009). [CrossRef]  

3. A. Poljicak, L. Mandic, and D. Agic, “Discrete Fourier transform–based watermarking method with an optimal implementation radius,” J. Electron. Imaging 20(3), 033008 (2011). [CrossRef]  

4. A. Phadikar, S. P. Maity, and B. Verma, “Region based QIM digital watermarking scheme for image database in DCT domain,” Comput. Electr. Eng. 37(3), 339–355 (2011). [CrossRef]  

5. P. Surekha and S. Sumathi, “Implementation of genetic algorithm for a dwt based image watermarking scheme,” IJSC 2(1), 244–252 (2011).

6. P. Meerwald and A. Uhl, “A survey of wavelet-domain watermarking algorithms,” Proc. SPIE 4314, 505–516 (2001). [CrossRef]  

7. B. C. Mohan and S. S. Kumar, “Robust Digital Watermarking Scheme using Contourlet Transform,” IJCSNS 8(2), 43–51 (2008).

8. M. N. Do and M. Vetterli, “The contourlet transform: An efficient directional multiresolution image representation,” IEEE Trans. Image Process. 14(12), 2091–2106 (2005). [CrossRef]   [PubMed]  

9. O. Fındık, İ. Babaoğlu, and E. Ülker, “A color image watermarking scheme based on hybrid classification method: Particle swarm optimization and k-nearest neighbor algorithm,” Opt. Commun. 283(24), 4916–4922 (2010). [CrossRef]  

10. Y. del Valle, G. K. Venayagamoorthy, S. Mohagheghi, J.-C. Hernandez, and R. G. Harley, “Particle Swarm Optimization: Basic Concepts, Variants and Applications in Power Systems,” IEEE Trans. Evol. Comput. 12(2), 171–195 (2008). [CrossRef]  

11. N. Takai and Y. Mifune, “Digital watermarking by a holographic technique,” Appl. Opt. 41(5), 865–873 (2002). [CrossRef]   [PubMed]  

12. H. T. Chang and C. L. Tsan, “Image watermarking by use of digital holography embedded in the discrete-cosine-transform domain,” Appl. Opt. 44(29), 6211–6219 (2005). [CrossRef]   [PubMed]  

13. S. Z. Wang, S. J. Huang, X. P. Zhang, and W. Wu, “Hologram-based watermarking capable of surviving print-scan process,” Appl. Opt. 49(7), 1170–1178 (2010). [CrossRef]   [PubMed]  

14. O. E. Okman and G. B. Akar, “Quantization index modulation-based image watermarking using digital holography,” J. Opt. Soc. Am. A 24(1), 243–252 (2007). [CrossRef]   [PubMed]  

15. J. Z. Li, “Robust image watermarking scheme against geometric attacks using a computer-generated hologram,” Appl. Opt. 49(32), 6302–6312 (2010). [CrossRef]   [PubMed]  

16. L. Z. Cai, M. Z. He, Q. Liu, and X. L. Yang, “Digital image encryption and watermarking by phase-shifting interferometry,” Appl. Opt. 43(15), 3078–3084 (2004). [CrossRef]   [PubMed]  

17. M. Z. He, L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-only encryption and watermarking based on phase-shifting interferometry,” Appl. Opt. 44(13), 2600–2606 (2005). [CrossRef]   [PubMed]  

18. L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28(19), 1808–1810 (2003). [CrossRef]   [PubMed]  

19. P. Refregier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20(7), 767–769 (1995). [CrossRef]   [PubMed]  

20. N. Singh and A. Sinha, “Gyrator transform-based optical image encryption, using chaos,” Opt. Lasers Eng. 47(5), 539–546 (2009). [CrossRef]  

21. M. M. Sathik and S. S. Sujatha, “A novel DWT based invisible watermarking technique for digital images,” Int. Arab J. e-Technol. 2(3), 167–172 (2012).

22. Z. J. Liu, D. Z. Chen, J. P. Ma, S. Y. Wei, Y. L. Zhang, J. M. Dai, and S. T. Liu, “Fast algorithm of discrete gyrator transform based on convolution operation,” Optik (Stuttg.) 122(10), 864–867 (2011). [CrossRef]  

23. S. Zhang, D. Van Der Weide, and J. Oliver, “Superfast phase-shifting method for 3-D shape measurement,” Opt. Express 18(9), 9684–9689 (2010). [CrossRef]   [PubMed]  

24. J. Kennedy and R. Eberhart, “Particle swarm optimization,” in Proceedings of IEEE International Conference on Neural Networks (Institute of Electrical and Electronics Engineers, New York, 1995), pp. 1942–1948. [CrossRef]  

25. M. Pluhacek, R. Senkerik, I. Zelinka, and D. Davendra, “Multiple choice strategy for PSO algorithm - performance analysis on shifted test functions,” in Proceedings of European Council for Modeling and Simulation,W. Rekdalsbakken, R. T. Bye and H. Zhang, ed. (Academic, Aalesund, Norway, 2013), pp. 393–397.

26. Z. L. Yu and G. F. Jin, Computer-Generated Hologram (Tsinghua University Press, Beijing, 1984), Chap. 4.

27. M. A. Qureshi, A. Aziz, B. Ahmed, A. Khalid, and H. Munir, “Comparative Analysis and Implementation of Efficient Digital Image Watermarking Schemes,” Int. J. Comp. Elec. Eng. 4(4), 558–561 (2012). [CrossRef]  

28. H. Elazhary, “A fast, blind, transparent, and robust image watermarking algorithm with extended torus automorphism permutation,” Int. J. Comp. Appl. 32(4), 34–41 (2011).

29. T. H. Chen, G. Horng, and S. H. Wang, “A robust wavelet-based watermarking scheme using quantization and human visual system Model,” Pakistan Journal of Information and Technology 2(3), 213–230 (2003). [CrossRef]  

30. H. H. Song, S. Y. Yu, X. K. Yang, L. Song, and C. Wang, “Contourlet-based image adaptive watermarking,” Signal Process-Image. 23(3), 162–178 (2008). [CrossRef]  

References

  • View by:

  1. C. C. Lai, “An improved SVD-based watermarking scheme using human visual characteristics,” Opt. Commun. 284(4), 938–944 (2011).
    [Crossref]
  2. L. D. Li and B. L. Guo, “Localized image watermarking in spatial domain resistant to geometric attacks,” Int. J. Electron. Commun. 63(2), 123–131 (2009).
    [Crossref]
  3. A. Poljicak, L. Mandic, and D. Agic, “Discrete Fourier transform–based watermarking method with an optimal implementation radius,” J. Electron. Imaging 20(3), 033008 (2011).
    [Crossref]
  4. A. Phadikar, S. P. Maity, and B. Verma, “Region based QIM digital watermarking scheme for image database in DCT domain,” Comput. Electr. Eng. 37(3), 339–355 (2011).
    [Crossref]
  5. P. Surekha and S. Sumathi, “Implementation of genetic algorithm for a dwt based image watermarking scheme,” IJSC 2(1), 244–252 (2011).
  6. P. Meerwald and A. Uhl, “A survey of wavelet-domain watermarking algorithms,” Proc. SPIE 4314, 505–516 (2001).
    [Crossref]
  7. B. C. Mohan and S. S. Kumar, “Robust Digital Watermarking Scheme using Contourlet Transform,” IJCSNS 8(2), 43–51 (2008).
  8. M. N. Do and M. Vetterli, “The contourlet transform: An efficient directional multiresolution image representation,” IEEE Trans. Image Process. 14(12), 2091–2106 (2005).
    [Crossref] [PubMed]
  9. O. Fındık, İ. Babaoğlu, and E. Ülker, “A color image watermarking scheme based on hybrid classification method: Particle swarm optimization and k-nearest neighbor algorithm,” Opt. Commun. 283(24), 4916–4922 (2010).
    [Crossref]
  10. Y. del Valle, G. K. Venayagamoorthy, S. Mohagheghi, J.-C. Hernandez, and R. G. Harley, “Particle Swarm Optimization: Basic Concepts, Variants and Applications in Power Systems,” IEEE Trans. Evol. Comput. 12(2), 171–195 (2008).
    [Crossref]
  11. N. Takai and Y. Mifune, “Digital watermarking by a holographic technique,” Appl. Opt. 41(5), 865–873 (2002).
    [Crossref] [PubMed]
  12. H. T. Chang and C. L. Tsan, “Image watermarking by use of digital holography embedded in the discrete-cosine-transform domain,” Appl. Opt. 44(29), 6211–6219 (2005).
    [Crossref] [PubMed]
  13. S. Z. Wang, S. J. Huang, X. P. Zhang, and W. Wu, “Hologram-based watermarking capable of surviving print-scan process,” Appl. Opt. 49(7), 1170–1178 (2010).
    [Crossref] [PubMed]
  14. O. E. Okman and G. B. Akar, “Quantization index modulation-based image watermarking using digital holography,” J. Opt. Soc. Am. A 24(1), 243–252 (2007).
    [Crossref] [PubMed]
  15. J. Z. Li, “Robust image watermarking scheme against geometric attacks using a computer-generated hologram,” Appl. Opt. 49(32), 6302–6312 (2010).
    [Crossref] [PubMed]
  16. L. Z. Cai, M. Z. He, Q. Liu, and X. L. Yang, “Digital image encryption and watermarking by phase-shifting interferometry,” Appl. Opt. 43(15), 3078–3084 (2004).
    [Crossref] [PubMed]
  17. M. Z. He, L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-only encryption and watermarking based on phase-shifting interferometry,” Appl. Opt. 44(13), 2600–2606 (2005).
    [Crossref] [PubMed]
  18. L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28(19), 1808–1810 (2003).
    [Crossref] [PubMed]
  19. P. Refregier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20(7), 767–769 (1995).
    [Crossref] [PubMed]
  20. N. Singh and A. Sinha, “Gyrator transform-based optical image encryption, using chaos,” Opt. Lasers Eng. 47(5), 539–546 (2009).
    [Crossref]
  21. M. M. Sathik and S. S. Sujatha, “A novel DWT based invisible watermarking technique for digital images,” Int. Arab J. e-Technol. 2(3), 167–172 (2012).
  22. Z. J. Liu, D. Z. Chen, J. P. Ma, S. Y. Wei, Y. L. Zhang, J. M. Dai, and S. T. Liu, “Fast algorithm of discrete gyrator transform based on convolution operation,” Optik (Stuttg.) 122(10), 864–867 (2011).
    [Crossref]
  23. S. Zhang, D. Van Der Weide, and J. Oliver, “Superfast phase-shifting method for 3-D shape measurement,” Opt. Express 18(9), 9684–9689 (2010).
    [Crossref] [PubMed]
  24. J. Kennedy and R. Eberhart, “Particle swarm optimization,” in Proceedings of IEEE International Conference on Neural Networks (Institute of Electrical and Electronics Engineers, New York, 1995), pp. 1942–1948.
    [Crossref]
  25. M. Pluhacek, R. Senkerik, I. Zelinka, and D. Davendra, “Multiple choice strategy for PSO algorithm - performance analysis on shifted test functions,” in Proceedings of European Council for Modeling and Simulation,W. Rekdalsbakken, R. T. Bye and H. Zhang, ed. (Academic, Aalesund, Norway, 2013), pp. 393–397.
  26. Z. L. Yu and G. F. Jin, Computer-Generated Hologram (Tsinghua University Press, Beijing, 1984), Chap. 4.
  27. M. A. Qureshi, A. Aziz, B. Ahmed, A. Khalid, and H. Munir, “Comparative Analysis and Implementation of Efficient Digital Image Watermarking Schemes,” Int. J. Comp. Elec. Eng. 4(4), 558–561 (2012).
    [Crossref]
  28. H. Elazhary, “A fast, blind, transparent, and robust image watermarking algorithm with extended torus automorphism permutation,” Int. J. Comp. Appl. 32(4), 34–41 (2011).
  29. T. H. Chen, G. Horng, and S. H. Wang, “A robust wavelet-based watermarking scheme using quantization and human visual system Model,” Pakistan Journal of Information and Technology 2(3), 213–230 (2003).
    [Crossref]
  30. H. H. Song, S. Y. Yu, X. K. Yang, L. Song, and C. Wang, “Contourlet-based image adaptive watermarking,” Signal Process-Image. 23(3), 162–178 (2008).
    [Crossref]

2012 (2)

M. M. Sathik and S. S. Sujatha, “A novel DWT based invisible watermarking technique for digital images,” Int. Arab J. e-Technol. 2(3), 167–172 (2012).

M. A. Qureshi, A. Aziz, B. Ahmed, A. Khalid, and H. Munir, “Comparative Analysis and Implementation of Efficient Digital Image Watermarking Schemes,” Int. J. Comp. Elec. Eng. 4(4), 558–561 (2012).
[Crossref]

2011 (6)

H. Elazhary, “A fast, blind, transparent, and robust image watermarking algorithm with extended torus automorphism permutation,” Int. J. Comp. Appl. 32(4), 34–41 (2011).

Z. J. Liu, D. Z. Chen, J. P. Ma, S. Y. Wei, Y. L. Zhang, J. M. Dai, and S. T. Liu, “Fast algorithm of discrete gyrator transform based on convolution operation,” Optik (Stuttg.) 122(10), 864–867 (2011).
[Crossref]

A. Poljicak, L. Mandic, and D. Agic, “Discrete Fourier transform–based watermarking method with an optimal implementation radius,” J. Electron. Imaging 20(3), 033008 (2011).
[Crossref]

A. Phadikar, S. P. Maity, and B. Verma, “Region based QIM digital watermarking scheme for image database in DCT domain,” Comput. Electr. Eng. 37(3), 339–355 (2011).
[Crossref]

P. Surekha and S. Sumathi, “Implementation of genetic algorithm for a dwt based image watermarking scheme,” IJSC 2(1), 244–252 (2011).

C. C. Lai, “An improved SVD-based watermarking scheme using human visual characteristics,” Opt. Commun. 284(4), 938–944 (2011).
[Crossref]

2010 (4)

2009 (2)

N. Singh and A. Sinha, “Gyrator transform-based optical image encryption, using chaos,” Opt. Lasers Eng. 47(5), 539–546 (2009).
[Crossref]

L. D. Li and B. L. Guo, “Localized image watermarking in spatial domain resistant to geometric attacks,” Int. J. Electron. Commun. 63(2), 123–131 (2009).
[Crossref]

2008 (3)

B. C. Mohan and S. S. Kumar, “Robust Digital Watermarking Scheme using Contourlet Transform,” IJCSNS 8(2), 43–51 (2008).

Y. del Valle, G. K. Venayagamoorthy, S. Mohagheghi, J.-C. Hernandez, and R. G. Harley, “Particle Swarm Optimization: Basic Concepts, Variants and Applications in Power Systems,” IEEE Trans. Evol. Comput. 12(2), 171–195 (2008).
[Crossref]

H. H. Song, S. Y. Yu, X. K. Yang, L. Song, and C. Wang, “Contourlet-based image adaptive watermarking,” Signal Process-Image. 23(3), 162–178 (2008).
[Crossref]

2007 (1)

2005 (3)

2004 (1)

2003 (2)

L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28(19), 1808–1810 (2003).
[Crossref] [PubMed]

T. H. Chen, G. Horng, and S. H. Wang, “A robust wavelet-based watermarking scheme using quantization and human visual system Model,” Pakistan Journal of Information and Technology 2(3), 213–230 (2003).
[Crossref]

2002 (1)

2001 (1)

P. Meerwald and A. Uhl, “A survey of wavelet-domain watermarking algorithms,” Proc. SPIE 4314, 505–516 (2001).
[Crossref]

1995 (1)

Agic, D.

A. Poljicak, L. Mandic, and D. Agic, “Discrete Fourier transform–based watermarking method with an optimal implementation radius,” J. Electron. Imaging 20(3), 033008 (2011).
[Crossref]

Ahmed, B.

M. A. Qureshi, A. Aziz, B. Ahmed, A. Khalid, and H. Munir, “Comparative Analysis and Implementation of Efficient Digital Image Watermarking Schemes,” Int. J. Comp. Elec. Eng. 4(4), 558–561 (2012).
[Crossref]

Akar, G. B.

Aziz, A.

M. A. Qureshi, A. Aziz, B. Ahmed, A. Khalid, and H. Munir, “Comparative Analysis and Implementation of Efficient Digital Image Watermarking Schemes,” Int. J. Comp. Elec. Eng. 4(4), 558–561 (2012).
[Crossref]

Babaoglu, I.

O. Fındık, İ. Babaoğlu, and E. Ülker, “A color image watermarking scheme based on hybrid classification method: Particle swarm optimization and k-nearest neighbor algorithm,” Opt. Commun. 283(24), 4916–4922 (2010).
[Crossref]

Cai, L. Z.

Chang, H. T.

Chen, D. Z.

Z. J. Liu, D. Z. Chen, J. P. Ma, S. Y. Wei, Y. L. Zhang, J. M. Dai, and S. T. Liu, “Fast algorithm of discrete gyrator transform based on convolution operation,” Optik (Stuttg.) 122(10), 864–867 (2011).
[Crossref]

Chen, T. H.

T. H. Chen, G. Horng, and S. H. Wang, “A robust wavelet-based watermarking scheme using quantization and human visual system Model,” Pakistan Journal of Information and Technology 2(3), 213–230 (2003).
[Crossref]

Dai, J. M.

Z. J. Liu, D. Z. Chen, J. P. Ma, S. Y. Wei, Y. L. Zhang, J. M. Dai, and S. T. Liu, “Fast algorithm of discrete gyrator transform based on convolution operation,” Optik (Stuttg.) 122(10), 864–867 (2011).
[Crossref]

del Valle, Y.

Y. del Valle, G. K. Venayagamoorthy, S. Mohagheghi, J.-C. Hernandez, and R. G. Harley, “Particle Swarm Optimization: Basic Concepts, Variants and Applications in Power Systems,” IEEE Trans. Evol. Comput. 12(2), 171–195 (2008).
[Crossref]

Do, M. N.

M. N. Do and M. Vetterli, “The contourlet transform: An efficient directional multiresolution image representation,” IEEE Trans. Image Process. 14(12), 2091–2106 (2005).
[Crossref] [PubMed]

Elazhary, H.

H. Elazhary, “A fast, blind, transparent, and robust image watermarking algorithm with extended torus automorphism permutation,” Int. J. Comp. Appl. 32(4), 34–41 (2011).

Findik, O.

O. Fındık, İ. Babaoğlu, and E. Ülker, “A color image watermarking scheme based on hybrid classification method: Particle swarm optimization and k-nearest neighbor algorithm,” Opt. Commun. 283(24), 4916–4922 (2010).
[Crossref]

Guo, B. L.

L. D. Li and B. L. Guo, “Localized image watermarking in spatial domain resistant to geometric attacks,” Int. J. Electron. Commun. 63(2), 123–131 (2009).
[Crossref]

Harley, R. G.

Y. del Valle, G. K. Venayagamoorthy, S. Mohagheghi, J.-C. Hernandez, and R. G. Harley, “Particle Swarm Optimization: Basic Concepts, Variants and Applications in Power Systems,” IEEE Trans. Evol. Comput. 12(2), 171–195 (2008).
[Crossref]

He, M. Z.

Hernandez, J.-C.

Y. del Valle, G. K. Venayagamoorthy, S. Mohagheghi, J.-C. Hernandez, and R. G. Harley, “Particle Swarm Optimization: Basic Concepts, Variants and Applications in Power Systems,” IEEE Trans. Evol. Comput. 12(2), 171–195 (2008).
[Crossref]

Horng, G.

T. H. Chen, G. Horng, and S. H. Wang, “A robust wavelet-based watermarking scheme using quantization and human visual system Model,” Pakistan Journal of Information and Technology 2(3), 213–230 (2003).
[Crossref]

Huang, S. J.

Javidi, B.

Khalid, A.

M. A. Qureshi, A. Aziz, B. Ahmed, A. Khalid, and H. Munir, “Comparative Analysis and Implementation of Efficient Digital Image Watermarking Schemes,” Int. J. Comp. Elec. Eng. 4(4), 558–561 (2012).
[Crossref]

Kumar, S. S.

B. C. Mohan and S. S. Kumar, “Robust Digital Watermarking Scheme using Contourlet Transform,” IJCSNS 8(2), 43–51 (2008).

Lai, C. C.

C. C. Lai, “An improved SVD-based watermarking scheme using human visual characteristics,” Opt. Commun. 284(4), 938–944 (2011).
[Crossref]

Li, J. Z.

Li, L. D.

L. D. Li and B. L. Guo, “Localized image watermarking in spatial domain resistant to geometric attacks,” Int. J. Electron. Commun. 63(2), 123–131 (2009).
[Crossref]

Liu, Q.

Liu, S. T.

Z. J. Liu, D. Z. Chen, J. P. Ma, S. Y. Wei, Y. L. Zhang, J. M. Dai, and S. T. Liu, “Fast algorithm of discrete gyrator transform based on convolution operation,” Optik (Stuttg.) 122(10), 864–867 (2011).
[Crossref]

Liu, Z. J.

Z. J. Liu, D. Z. Chen, J. P. Ma, S. Y. Wei, Y. L. Zhang, J. M. Dai, and S. T. Liu, “Fast algorithm of discrete gyrator transform based on convolution operation,” Optik (Stuttg.) 122(10), 864–867 (2011).
[Crossref]

Ma, J. P.

Z. J. Liu, D. Z. Chen, J. P. Ma, S. Y. Wei, Y. L. Zhang, J. M. Dai, and S. T. Liu, “Fast algorithm of discrete gyrator transform based on convolution operation,” Optik (Stuttg.) 122(10), 864–867 (2011).
[Crossref]

Maity, S. P.

A. Phadikar, S. P. Maity, and B. Verma, “Region based QIM digital watermarking scheme for image database in DCT domain,” Comput. Electr. Eng. 37(3), 339–355 (2011).
[Crossref]

Mandic, L.

A. Poljicak, L. Mandic, and D. Agic, “Discrete Fourier transform–based watermarking method with an optimal implementation radius,” J. Electron. Imaging 20(3), 033008 (2011).
[Crossref]

Meerwald, P.

P. Meerwald and A. Uhl, “A survey of wavelet-domain watermarking algorithms,” Proc. SPIE 4314, 505–516 (2001).
[Crossref]

Mifune, Y.

Mohagheghi, S.

Y. del Valle, G. K. Venayagamoorthy, S. Mohagheghi, J.-C. Hernandez, and R. G. Harley, “Particle Swarm Optimization: Basic Concepts, Variants and Applications in Power Systems,” IEEE Trans. Evol. Comput. 12(2), 171–195 (2008).
[Crossref]

Mohan, B. C.

B. C. Mohan and S. S. Kumar, “Robust Digital Watermarking Scheme using Contourlet Transform,” IJCSNS 8(2), 43–51 (2008).

Munir, H.

M. A. Qureshi, A. Aziz, B. Ahmed, A. Khalid, and H. Munir, “Comparative Analysis and Implementation of Efficient Digital Image Watermarking Schemes,” Int. J. Comp. Elec. Eng. 4(4), 558–561 (2012).
[Crossref]

Okman, O. E.

Oliver, J.

Phadikar, A.

A. Phadikar, S. P. Maity, and B. Verma, “Region based QIM digital watermarking scheme for image database in DCT domain,” Comput. Electr. Eng. 37(3), 339–355 (2011).
[Crossref]

Poljicak, A.

A. Poljicak, L. Mandic, and D. Agic, “Discrete Fourier transform–based watermarking method with an optimal implementation radius,” J. Electron. Imaging 20(3), 033008 (2011).
[Crossref]

Qureshi, M. A.

M. A. Qureshi, A. Aziz, B. Ahmed, A. Khalid, and H. Munir, “Comparative Analysis and Implementation of Efficient Digital Image Watermarking Schemes,” Int. J. Comp. Elec. Eng. 4(4), 558–561 (2012).
[Crossref]

Refregier, P.

Sathik, M. M.

M. M. Sathik and S. S. Sujatha, “A novel DWT based invisible watermarking technique for digital images,” Int. Arab J. e-Technol. 2(3), 167–172 (2012).

Singh, N.

N. Singh and A. Sinha, “Gyrator transform-based optical image encryption, using chaos,” Opt. Lasers Eng. 47(5), 539–546 (2009).
[Crossref]

Sinha, A.

N. Singh and A. Sinha, “Gyrator transform-based optical image encryption, using chaos,” Opt. Lasers Eng. 47(5), 539–546 (2009).
[Crossref]

Song, H. H.

H. H. Song, S. Y. Yu, X. K. Yang, L. Song, and C. Wang, “Contourlet-based image adaptive watermarking,” Signal Process-Image. 23(3), 162–178 (2008).
[Crossref]

Song, L.

H. H. Song, S. Y. Yu, X. K. Yang, L. Song, and C. Wang, “Contourlet-based image adaptive watermarking,” Signal Process-Image. 23(3), 162–178 (2008).
[Crossref]

Sujatha, S. S.

M. M. Sathik and S. S. Sujatha, “A novel DWT based invisible watermarking technique for digital images,” Int. Arab J. e-Technol. 2(3), 167–172 (2012).

Sumathi, S.

P. Surekha and S. Sumathi, “Implementation of genetic algorithm for a dwt based image watermarking scheme,” IJSC 2(1), 244–252 (2011).

Surekha, P.

P. Surekha and S. Sumathi, “Implementation of genetic algorithm for a dwt based image watermarking scheme,” IJSC 2(1), 244–252 (2011).

Takai, N.

Tsan, C. L.

Uhl, A.

P. Meerwald and A. Uhl, “A survey of wavelet-domain watermarking algorithms,” Proc. SPIE 4314, 505–516 (2001).
[Crossref]

Ülker, E.

O. Fındık, İ. Babaoğlu, and E. Ülker, “A color image watermarking scheme based on hybrid classification method: Particle swarm optimization and k-nearest neighbor algorithm,” Opt. Commun. 283(24), 4916–4922 (2010).
[Crossref]

Van Der Weide, D.

Venayagamoorthy, G. K.

Y. del Valle, G. K. Venayagamoorthy, S. Mohagheghi, J.-C. Hernandez, and R. G. Harley, “Particle Swarm Optimization: Basic Concepts, Variants and Applications in Power Systems,” IEEE Trans. Evol. Comput. 12(2), 171–195 (2008).
[Crossref]

Verma, B.

A. Phadikar, S. P. Maity, and B. Verma, “Region based QIM digital watermarking scheme for image database in DCT domain,” Comput. Electr. Eng. 37(3), 339–355 (2011).
[Crossref]

Vetterli, M.

M. N. Do and M. Vetterli, “The contourlet transform: An efficient directional multiresolution image representation,” IEEE Trans. Image Process. 14(12), 2091–2106 (2005).
[Crossref] [PubMed]

Wang, C.

H. H. Song, S. Y. Yu, X. K. Yang, L. Song, and C. Wang, “Contourlet-based image adaptive watermarking,” Signal Process-Image. 23(3), 162–178 (2008).
[Crossref]

Wang, S. H.

T. H. Chen, G. Horng, and S. H. Wang, “A robust wavelet-based watermarking scheme using quantization and human visual system Model,” Pakistan Journal of Information and Technology 2(3), 213–230 (2003).
[Crossref]

Wang, S. Z.

Wei, S. Y.

Z. J. Liu, D. Z. Chen, J. P. Ma, S. Y. Wei, Y. L. Zhang, J. M. Dai, and S. T. Liu, “Fast algorithm of discrete gyrator transform based on convolution operation,” Optik (Stuttg.) 122(10), 864–867 (2011).
[Crossref]

Wu, W.

Yang, X. K.

H. H. Song, S. Y. Yu, X. K. Yang, L. Song, and C. Wang, “Contourlet-based image adaptive watermarking,” Signal Process-Image. 23(3), 162–178 (2008).
[Crossref]

Yang, X. L.

Yu, S. Y.

H. H. Song, S. Y. Yu, X. K. Yang, L. Song, and C. Wang, “Contourlet-based image adaptive watermarking,” Signal Process-Image. 23(3), 162–178 (2008).
[Crossref]

Zhang, S.

Zhang, X. P.

Zhang, Y. L.

Z. J. Liu, D. Z. Chen, J. P. Ma, S. Y. Wei, Y. L. Zhang, J. M. Dai, and S. T. Liu, “Fast algorithm of discrete gyrator transform based on convolution operation,” Optik (Stuttg.) 122(10), 864–867 (2011).
[Crossref]

Appl. Opt. (6)

Comput. Electr. Eng. (1)

A. Phadikar, S. P. Maity, and B. Verma, “Region based QIM digital watermarking scheme for image database in DCT domain,” Comput. Electr. Eng. 37(3), 339–355 (2011).
[Crossref]

IEEE Trans. Evol. Comput. (1)

Y. del Valle, G. K. Venayagamoorthy, S. Mohagheghi, J.-C. Hernandez, and R. G. Harley, “Particle Swarm Optimization: Basic Concepts, Variants and Applications in Power Systems,” IEEE Trans. Evol. Comput. 12(2), 171–195 (2008).
[Crossref]

IEEE Trans. Image Process. (1)

M. N. Do and M. Vetterli, “The contourlet transform: An efficient directional multiresolution image representation,” IEEE Trans. Image Process. 14(12), 2091–2106 (2005).
[Crossref] [PubMed]

IJCSNS (1)

B. C. Mohan and S. S. Kumar, “Robust Digital Watermarking Scheme using Contourlet Transform,” IJCSNS 8(2), 43–51 (2008).

IJSC (1)

P. Surekha and S. Sumathi, “Implementation of genetic algorithm for a dwt based image watermarking scheme,” IJSC 2(1), 244–252 (2011).

Int. Arab J. e-Technol. (1)

M. M. Sathik and S. S. Sujatha, “A novel DWT based invisible watermarking technique for digital images,” Int. Arab J. e-Technol. 2(3), 167–172 (2012).

Int. J. Comp. Appl. (1)

H. Elazhary, “A fast, blind, transparent, and robust image watermarking algorithm with extended torus automorphism permutation,” Int. J. Comp. Appl. 32(4), 34–41 (2011).

Int. J. Comp. Elec. Eng. (1)

M. A. Qureshi, A. Aziz, B. Ahmed, A. Khalid, and H. Munir, “Comparative Analysis and Implementation of Efficient Digital Image Watermarking Schemes,” Int. J. Comp. Elec. Eng. 4(4), 558–561 (2012).
[Crossref]

Int. J. Electron. Commun. (1)

L. D. Li and B. L. Guo, “Localized image watermarking in spatial domain resistant to geometric attacks,” Int. J. Electron. Commun. 63(2), 123–131 (2009).
[Crossref]

J. Electron. Imaging (1)

A. Poljicak, L. Mandic, and D. Agic, “Discrete Fourier transform–based watermarking method with an optimal implementation radius,” J. Electron. Imaging 20(3), 033008 (2011).
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Commun. (2)

C. C. Lai, “An improved SVD-based watermarking scheme using human visual characteristics,” Opt. Commun. 284(4), 938–944 (2011).
[Crossref]

O. Fındık, İ. Babaoğlu, and E. Ülker, “A color image watermarking scheme based on hybrid classification method: Particle swarm optimization and k-nearest neighbor algorithm,” Opt. Commun. 283(24), 4916–4922 (2010).
[Crossref]

Opt. Express (1)

Opt. Lasers Eng. (1)

N. Singh and A. Sinha, “Gyrator transform-based optical image encryption, using chaos,” Opt. Lasers Eng. 47(5), 539–546 (2009).
[Crossref]

Opt. Lett. (2)

Optik (Stuttg.) (1)

Z. J. Liu, D. Z. Chen, J. P. Ma, S. Y. Wei, Y. L. Zhang, J. M. Dai, and S. T. Liu, “Fast algorithm of discrete gyrator transform based on convolution operation,” Optik (Stuttg.) 122(10), 864–867 (2011).
[Crossref]

Pakistan Journal of Information and Technology (1)

T. H. Chen, G. Horng, and S. H. Wang, “A robust wavelet-based watermarking scheme using quantization and human visual system Model,” Pakistan Journal of Information and Technology 2(3), 213–230 (2003).
[Crossref]

Proc. SPIE (1)

P. Meerwald and A. Uhl, “A survey of wavelet-domain watermarking algorithms,” Proc. SPIE 4314, 505–516 (2001).
[Crossref]

Signal Process-Image. (1)

H. H. Song, S. Y. Yu, X. K. Yang, L. Song, and C. Wang, “Contourlet-based image adaptive watermarking,” Signal Process-Image. 23(3), 162–178 (2008).
[Crossref]

Other (3)

J. Kennedy and R. Eberhart, “Particle swarm optimization,” in Proceedings of IEEE International Conference on Neural Networks (Institute of Electrical and Electronics Engineers, New York, 1995), pp. 1942–1948.
[Crossref]

M. Pluhacek, R. Senkerik, I. Zelinka, and D. Davendra, “Multiple choice strategy for PSO algorithm - performance analysis on shifted test functions,” in Proceedings of European Council for Modeling and Simulation,W. Rekdalsbakken, R. T. Bye and H. Zhang, ed. (Academic, Aalesund, Norway, 2013), pp. 393–397.

Z. L. Yu and G. F. Jin, Computer-Generated Hologram (Tsinghua University Press, Beijing, 1984), Chap. 4.

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Figures (9)

Fig. 1
Fig. 1 The diagram of the generation and reconstruction processes of the encrypted hologram. (a) The generation procedure for the encrypted CGH. (b) The reconstruction procedure for the encrypted CGH.
Fig. 2
Fig. 2 PSO based optimization process of watermarking.
Fig. 3
Fig. 3 A sample watermark generated by computer. (a) image “STU” of size 32 × 64; (b), (c), (d) interferograms I1, I2 and I3, respectively. (e), (f) the amplitude and the normalized phase of the gyrator transform hologram. (g) reconstruction of the hologram with all correct keys.
Fig. 4
Fig. 4 The host images and watermark. (a) Gold hill. (b) Elaine. (c) Butterfly. (d) Scenic. (e) the watermark signal consisted of the amplitude and the phase of the encrypted hologram.
Fig. 5
Fig. 5 The PSO convergence history for the proposed method. The green line, the pink line, the blue line and the red line are the convergence curves for Goldhill, Elaine, Butterfly and Scienic, respectively.
Fig. 6
Fig. 6 The watermarked images and the corresponding reconstructions of the extracted mark holograms without any attacks. (a) The watermarked “Gold hill”(PSNR = 41.284). (b) reconstructed watermark of (a). (c) the watermarked “Elaine” (PSNR = 42.225). (d) reconstructed watermark of (c). (e) the watermarked “Butterfly” (PSNR = 40.456). (f) reconstructed watermark of (e). (g) the watermarked “Scenic” (PSNR = 42.481). (h) reconstructed watermark of (g).
Fig. 7
Fig. 7 The attacked watermarked images. (a) Gaussian low-pass filtering (PSNR = 26.28). (b) median filtering (PSNR = 28.34). (c) average filtering (PSNR = 30.47). (d) Gaussian noise (PSNR = 29.56). (e) salt & pepper noise (PSNR = 27.64). (f) JPEG (PSNR = 32.24). (g) JPEG2000 (PSNR = 34.88). (h) cropping (PSNR = 11.69). (i) unsharp (PSNR = 26.33). (j) horizontal motion (PSNR = 28.07). (k) circular average (PSNR = 24.63). (l) brighten (PSNR = 19.83). (m) darken (PSNR = 18.06). (n) rescaling (PSNR = 26.26). (o) painting (PSNR = 17.84).
Fig. 8
Fig. 8 The corresponding reconstructions of the holograms extracted from the distorted watermarked images in Fig. 7. (a) Gaussian low-pass filtering. (b) median filtering. (c) average filtering. (d) Gaussian noise. (e) salt & pepper noise. (f) JPEG. (g) JPEG2000. (h) cropping. (i) unsharp. (j) horizontal motion. (k) circular average. (l) brighten. (m) darken. (n) rescaling. (o)painting.
Fig. 9
Fig. 9 The reconstructions of the watermarks which were extracted from watermarked Goldhill. (a) reconstructed image with correct q(x1,y1), α1 and α2 and with D = D + 1; (b) reconstructed image with correct D, α1 and α2 when RPM R2 is not used; (c) reconstructed image with correct D, α1 and α2 when q(x1,y1) is shifted transversely by one pixel; (c) reconstructed image with correct D, q(x1,y1) and α2 when α1 has a relative error of 1.4%; (d) reconstructed image with correct D, q(x1,y1) and α1 when α2 has a relative error of 0.8%; (e) reconstructed image with correct D and q(x1,y1) when α1 and α2 both have relative errors of 0.6%.

Tables (2)

Tables Icon

Table 1 PSNR Values of Watermarks Extracting without Any Attacks Comparing between Proposed Method and Scheme in [15]

Tables Icon

Table 2 NCC Values of Watermarks Extracting under Attacks Comparing between Proposed Method and Scheme in [15]

Equations (29)

Equations on this page are rendered with MathJax. Learn more.

G α (u,v)=G T α [f(x,y)]= f(x,y) |sinα| exp[i2π (xy+uv)cosαxvyu sinα ]dxdy,
I k (x,y) = A 2 (x,y)+2A(x,y) A r cos[φ(x,y) δ k ]+ A r 2 .
U(x,y) =[( I 1 I 3 )+i(2 I 2 I 1 I 3 )]/(4 A r ).
B m,n ' = B m,n /Δ Δ+λΔ w m,n ,
w m,n ' =( B m,n ' B m,n ' /Δ Δ)/(λΔ).
B m,n ' = B m,n ρ+λΔ w m,n .
w m,n ' =(X' X'/Δ Δ)/(λΔ)=( ( B m,n Δ)/Δ Δ+λΔ w m,n ( ( B m,n Δ)/Δ Δ+λΔ w m,n )/Δ Δ)/(λΔ) =( B m,n /Δ Δ+λΔ w m,n ( B m,n /Δ Δ+λΔ w m,n )/Δ Δ)/(λΔ)=( B m,n ' B m,n ' /Δ Δ)/(λΔ).
| B m,n X ' |=| B m,n ( X/Δ Δ+λΔ w m,n )|=| B m,n ( ( B m,n Δ)/Δ Δ+λΔ w m,n )| =| B m,n ( B m,n /Δ ΔΔ+λΔ w m,n )|=|Δ(ρ+λΔ w m,n )|.
| B m,n B m,n ' |=| B m,n ( B m,n /Δ Δ+λΔ w m,n )|=|(ρ+λΔ w m,n )|.
0<| B m,n X'|<Δ/2<| B m,n B m,n ' |<Δ.
w m,n ' =(Y' Y'/Δ Δ)/(λΔ)=( B m,n ' B m,n ' /Δ Δ)/(λΔ).
0<(| B m,n Y'|=|ρλΔ w m,n Δ|)<Δ/2<| B m,n B m,n ' |<Δ.
v id (it+1)=ω v id (it)+ c 1 rand()( p id x id (it))+ c 2 rand()( p gd x id (it)).
x id (it+1)= x id (it)+ v id (it+1).
ω(it)= ω start ( ω start ω end )it/i t max .
U 1 ( x 1 , y 1 )=G T α 1 {f( x 0 , y 0 )exp[i2πp( x 0 , y 0 )]}.
U 2 (u,v)=G T α 2 { U 1 ( x 1 , y 1 )exp[i2πq( x 1 , y 1 )]}=A'(u,v)exp[iφ'(u,v)],
I 1 =0.5A'(u,v) A r {1+cos[φ'(u,v) δ 1 ]}=0.5A'(u,v){1+cos[φ'(u,v)]}.
I 2 =0.5A'(u,v) A r {1+cos[φ'(u,v) δ 2 ]}=0.5A'(u,v){1+sin[φ'(u,v)]}.
I 3 =0.5A'(u,v) A r {1+cos[φ'(u,v) δ 3 ]}=0.5A'(u,v){1cos[φ'(u,v)]}.
U(u,v)=[( I 1 I 3 )+i(2 I 2 I 1 I 3 )]/(4 A r ) =[( I 1 I 3 )+i(2 I 2 I 1 I 3 )]/4 =A(u,v)exp[iφ(u,v)],
U'( x 0 , y 0 )=G T α 1 {G T α 2 [U(u,v)]exp[i2πq( x 1 , y 1 )]}.
wm=[ T Ψ ]=[ 1 A w φ w /(2π) ],
C L ξ,η ' ={ (C L ξ,η Δ)/Δ Δ+λΔw m ξ,η , if ρ+λΔw m ξ,η >Δ/2 (C L ξ,η +Δ)/Δ Δ+λΔw m ξ,η , if ρ+λΔw m ξ,η <Δ/2 C L ξ,η /Δ Δ+λΔw m ξ,η , otherwise ,
w m ξ,η ' =(C L ξ,η ' C L ξ,η ' /Δ )/(λΔ).
Fitness=PSNR/100+1.1×( j=1 5 NC C j )/5.
PSNR=10 log 10 (MaxP V 2 /MSE) (dB) ,
MSE=( m=1 M n=1 N ( H m,n H ' m,n ) 2 )/(M×N),
NCC= k=1 K l=1 L ( W k,l W ' k,l )/ k=1 K l=1 L ( W k,l ) 2 ,

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