Robust zero-watermarking algorithm based on polar ...

Multimed Tools Appl DOI 10.1007/s11042-016-4130-7

Robust zero-watermarking algorithm based on polar complex exponential transform and logistic mapping Chun-peng Wang 1 & Xing-yuan Wang 1 & Xing-jun Chen 2 & Chuan Zhang 1

Received: 20 June 2016 / Revised: 20 October 2016 / Accepted: 3 November 2016 # Springer Science+Business Media New York 2016

Abstract This paper introduces a new zero-watermarking algorithm based on polar complex exponential transform (PCET) and logistic mapping. This algorithm takes advantage of the geometric invariance of PCET to improve the robustness of the algorithm against geometric attacks, and the logistic mapping’s sensitivity to initial values to improve the security of the algorithm. First, the algorithm computes the PCET of the original grayscale image. Then it randomly selects PCET coefficients based on logistic mapping, and computes their magnitudes to obtain a binary feature image. Finally, it performs an exclusive-or operation between the binary feature image and the scrambled logo image to obtain the zero-watermark image. At the stage of copyright verification, the image copyright can be determined by performing the exclusive-or operation between the feature image and the verification image and comparing the resulting image to the original logo image. Experimental results show that this algorithm has excellent robustness against geometric attacks and common image processing attacks and better performance compared to other zero-watermarking algorithms. Keywords Accurate coefficient selection . Geometric attack . Logistic mapping . Polar complex exponential transform . Zero-watermarking

* Xing-yuan Wang [email protected] Chun-peng Wang [email protected]

1

Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian 116023, China

2

Operation Software and Simulation Institute, Dalian Navy Academy, Dalian 116018, China

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1 Introduction With the rapid development of Internet technology, illegally copying and distributing copyright-protected digital images has become easier. More and more images are pirated every day, the image security issues have become an important and emergent problem to be solved. Some techniques are proposed to resolve the image security issues in recent years, including image encryption [11, 23], image hash [16, 25], image authentication [9, 40], image steganography [12, 19] and image watermarking [4, 39], etc. As a technique for the copyright protection of digital images, robust digital watermarking [13, 35] was proposed a few years ago. The main principle is to find the image’s invariable features and modify them by adding a watermark. However, for some sensitive images, such as medical and remote sensing images, due to the importance of the details contained in the pixels, any changes may cause distortions that affect the decisions of doctors or remote sensing experts. In order to overcome this limitation, a new technique called the zero-watermarking was developed recently. Different from the traditional robust digital watermarking technique, a zero-watermarking algorithm does not embed any information into host images; instead, it binds the invariant image features with the logo image and saves them in a copyright verification database. Hence, it was named Bzero-watermarking^, as it embeds no information into host images. The zero-watermarking algorithm provides a good balance among robustness of digital watermarking, the information amount carried by the watermark, and imperceptibility. In addition, in the zero-watermarking algorithm there is an independent third-party entity, i.e. the copyright verification database. The database maintains the zero-watermark of the copyright protected images, copyrighted watermarks, user encryption keys, and other assisting information. When there is a copyright conflict, the copyright verification database can provide evidence and suggestions from both technical and legal aspects, which is also an advantage of the zero-watermarking technique over traditional digital watermarking techniques. Due to its unique advantages, many researchers are now devoting their time and resources to the zero-watermarking algorithm. However, since it has not been developed for long, publications about zero-watermarking algorithm are still much fewer than those about traditional digital watermarking. The original concept of zero-watermarking was proposed by Wen et al. [36]. The algorithm they proposed uses high-order cumulants to extract the image features and construct the zero-watermark. Experimental results showed that the zero-watermark generated with this algorithm has excellent performance and robustness against attacks including noise, filtering, JPEG compression, and slight angle rotation. Chen et al. [3] proposed a zero-watermarking system for public copyright verification. Their algorithm extracts the feature-matrix of the original image from lowfrequency band in the wavelet domain and uses it as the watermark matrix. Then, it sends the encrypted watermark matrix and parameters to a reliable third-party certification authority TA, and adds a timestamp to it. Chang and Lin [2] further improved the feature extraction algorithm in their work by performing Sobel edge detection on the low-scale approximation of the original image before feature-matrix extraction, which yielded excellent results. Sang et al. [20] proposed a zero-watermarking algorithm based on neural networks. It uses the neural network model of the image to construct a binary feature-matrix and perform an exclusive-or operation between the resulting matrix and the logo image to generate the zero-watermark images. Boyer et al. [1] proposed a distortion compensation quantitative indexing modulation (DC-QIM) zero-watermarking algorithm, which adopts large deviation theory to evaluate the algorithm’s performance according to

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the subject’s working feature curve and the probability of total classification error. Moreover, compared to the quantitative projection and spread spectrum method, DCQIM has showed some advantages. Tsai et al. [28] developed a zero-watermarking algorithm based on α-trimmed means and support vector machine (SVM) methods, which uses trained SVMs to memorize and retrieve watermarks and uses α-trimmed means method to improve the robustness. Later, Tsai et al. [27] extended this algorithm by using SVMs and particle swarm optimization algorithm (PSOA), which effectively defended against geometric attacks including image rotation, scaling, and translation attacks. Rawatt et al. [17] proposed a zero-watermarking algorithm based on fractional Fourier Transform and vision cryptography techniques. This algorithm uses vision cryptography share to construct a master share and an ownership share. The master share is generated by image features extracted using SVD. The ownership share can be obtained by performing VC technique on the main share together with a meaningful cryptographic image. The ownership share is then sent to TA for storage. Gao and Jiang [5] suggested a robust zero-watermarking algorithm based on Bessel-Fourier moments. This algorithm first performs translation and scaling normalization and then computes the magnitudes of the Bessel-Fourier moments of the normalized image. The magnitudes are then converted to a binary feature image. A verification image is then generated by performing an exclusive-or operation between the binary feature image and the logo image. The verification image is stored at the reliable authority, TA. Lin et al. [22] proposed a spatial domain zero-watermarking algorithm based on the generalized Arnold transform. It first uses the generalized Arnold transform to scramble the original image, then computes the binary feature matrix from the original image based on quantization embedding rule, and finally uses spread spectrum technique and perform an exclusive-or operation between the scrambled logo image and the feature matrix to generate zero-watermark image. All the zero-watermarking algorithms above have implemented the basic functionality of digital media copyright verification. However, there are still a few problems, mainly presented on two aspects: (1) these algorithms have limitations in withstanding geometric attacks. Most zero-watermarking algorithms can only resist common image processing attacks and cannot resist geometric attacks effectively, such as rotation, scaling, and translation. Thus, they are not robust under geometric attacks. (2) The safety of the algorithms is not satisfying. Only regular encryption algorithms were used in those algorithms to ensure security, but those encryption algorithms are relatively easy to crack. In this paper, we propose a robust geometric-attack-resistant zero-watermarking algorithm based on the geometric invariance and stability of the polar complex exponential transform (PCET) and the security of logistic chaos mapping. Experimental results demonstrate that this algorithm can effectively resist geometric attacks and common image processing attacks and has noticeable advantages compared to existing zero-watermarking algorithms. The contributions of the paper are that (1) the properties of the radial basis function of PCET are discussed in detail, and (2) accurate coefficients selection of PCET is discussed, and (3) the geometric invariance of PCET is employed to improve the robustness of the algorithm against geometric attacks, and (4) the logistic mapping’s sensitivity to initial values is used to improve the security of the algorithm. The rest of the paper is structured as following: Section 2 introduces the definition of PCET and discusses its stability. Section 3 introduces the logistic mapping. Our zero-watermarking algorithm is presented in Section 4. Section 5 discusses the experimental results in detail. Finally, Section 6 summarizes this paper.

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2 Polar complex exponential transform 2.1 Definition of polar complex exponential transform Polar complex exponential transform (PCET) is a type of polar harmonic transforms [14, 31, 38]. It is a technique that represents a signal as a superposition of harmonics. The PCET has proven to be a very useful tool for describing an image and has been widely used in many fields, such as pattern recognition and digital watermarking. In polar coordinate system (r, θ), we define the function set Hnm(r, θ), which includes the radial basis function Rn(r) and the angular function exp(jmθ), as: H nm ðr; θÞ ¼ Rn ðrÞexpð jmθÞ;

ð1Þ

where Rn(r) = exp(j2nπr2), n, m = − ∞, ⋯, 0, ⋯, + ∞, 0 ≤ r ≤ 1, 0 ≤ θ ≤ 2π. Rn(r) is orthogonal in a unit circle, i.e. Z 1 1 Rn ðrÞR*n0 ðrÞrdr ¼ δnn0 ; ð2Þ 2 0 where δnn0 is the Kronecker delta and R*n0 ðrÞ is the conjugate of Rn0 ðrÞ. The function set Hnm(r, θ) is orthogonal in the unit circle:

Z

2π Z

1

0

0

H nm ðr; θÞH *n0 m0 ðr; θÞrdrdθ ¼ πδnm δn0 m0 ;

ð3Þ

where π is the normalization factor; δnm, δn0 m0 are Kronecker deltas; and H *n0 m0 ðr; θÞ denotes the conjugate of H n0 m0 ðr; θÞ. The definition of PCET of order n with repetition m is as follows: Z 2π Z 1 1 Pnm ¼ f ðr; θÞH *nm ðr; θÞrdrdθ π 0 0 ð4Þ Z 2π Z 1   1 ¼ f ðr; θÞexp −j2nπr2 expð− jmθÞrdrdθ: π 0 0 According to the theory of complete orthogonal function sets, an image can be reconstructed by infinite orders of PCET coefficients (|n| ≤ nmax, |m| ≤ mmax). The reconstructed image can be written as: 0

f ðr; θÞ ¼

þ∞ X

þ∞ X

n¼−∞

m¼−∞

nmax X

mmax X

n¼−nmax

m¼−mmax

Pnm H nm ðr; θÞ ≈

Pnm H nm ðr; θÞ:

ð5Þ

Figure 1 gives two original grayscale images Lena and Mandrill with 256 × 256 pixels and Fig. 2 shows reconstructed images using PCET coefficients with max order nmax = 5, 10, 15, 20, 25. It can be seen that the reconstructed images using PCET show much visual resemblance to the original image. As more moments are used in the reconstruction process, the reconstructed images get closer to the original image.

Multimed Tools Appl Fig. 1 Original grayscale images: a Lena with 256 × 256 pixels, b Mandrill with 256 × 256 pixels

2.2 Properties of the radial basis function of PCET It is known that the properties of orthogonal moments are determined by the properties of the radial basis function [6, 7]. Figure 3 shows the radial basis functions of PCET, Zernike moments (ZM) [21] and pseudo Zernike moments (PZM) [33]. The number of zeros of radial basis functions of PCET, ZM and PZM of order n with repetition m can be seen in Table 1. It can be seen that the radial basis function of PCET has 2n zeros in the interval 0 ≤ r ≤ 1, while the radial basis functions of ZM and PZM have n−jmj 2 and n − |m| zeros, respectively. That is to say, for the same number of zeros nz, the degree of the radial basis function of PCET is n2z , while the degree of the radial basis functions of ZM and PZM is 2nz + |m| and nz + |m|, respectively. Hence, to describe a same image, the degree of PCET is much lower than that of ZM and PZM. For image moments, the lower degree, the less sensitive to image transform. Therefore, the performance of PCET is superior to that of ZM and PZM on resisting image attacks. Moreover, the distribution and position of the zeros correspond to the capability of the radial basis functions to describe the image information [7]. The first zeros of radial basis functions of PCET, ZM and PZM are shown in Table 2. As can be seen in Table 2, for

Fig. 2 Reconstructed images using PCET coefficients with max order nmax = 5, 10, 15, 20, 25: a Lena, b Mandrill

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a

b 1

4 n=0 n=1 n=2 n=3 n=4 n=5

3

0.6 0.4 Rn|10|(r)

Rn(r)

2

n = 10 n = 12 n = 14 n = 16 n = 18 n = 20

0.8

1 0

0.2 0 −0.2 −0.4 −0.6

−1

−0.8 −2

0

0.2

0.4

0.6

0.8

1

r

c

0

0.2

0.4

0.6

0.8

1

r

1 n = 10 n = 11 n = 12 n = 13 n = 14 n = 15

0.8 0.6 0.4 Rn|10|(r)

−1

0.2 0 −0.2 −0.4 −0.6 −0.8 −1

0

0.2

0.4

0.6

0.8

1

r

Fig. 3 Radial basis functions of PCET, ZM and PZM: a Radial basis function of PCET with n = 0, 1, …, 5, b Radial basis function of ZM with n = 10, 12, …, 20 and m = 10, c Radial basis function of PZM with n = 10, 11, …, 15 and m = 10

one to have five zeros, the first zero of the radial basis functions of PCET, ZM and PZM is located at r = 0.22, r = 0.66 and r = 0.62, respectively. Hence the distribution of zeros of the radial basis function of PCET is more uniform distributed than that of ZM and PZM. Accordingly, PCET is more suitable than ZM and PZM for image zero-watermarking.

2.3 PCET invariants 2.3.1 Rotation invariant Let fr(r, θ) = f(r, θ + α) denotes the rotation version of f(r, θ) by angle α, then the relationship between the PCET of fr(r, θ) and f(r, θ) is as follows:

Table 1 Number of zeros of radial basis functions of PCET, ZM and PZM

Moment

Number of zeros

PCET

2n

ZM

n−jmj 2

PZM

n − |m|

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Table 2 First zeros of radial basis functions of PCET, ZM and PZM

PCET n

Pnm ð f r Þ ¼

1 π

1 ¼ π ¼ ¼

1 π 1 π

Z

2π Z

Z

0 0 2π Z 1

Z

0 2π Z

0 1

Z

0 2π Z

0 1

0

0

1

ZM r

PZM

n

r

n

r NaN

0

NaN

10

NaN

10

1

0.50

12

0.96

11

0.96

2

0.35

14

0.87

12

0.86

3

0.29

16

0.79

13

0.77

4 5

0.25 0.22

18 20

0.72 0.66

14 15

0.69 0.62

  f r ðr; θÞexp − j2nπr2 expð−jmθÞrdrdθ   f ðr; θ þ αÞexp −j2nπr2 expð−jmθÞrdrdθ   f ðr; θÞexp −j2nπr2 expð− jmðθ−αÞÞrdrdθ

ð6Þ

  f ðr; θÞexp −j2nπr2 expð−jmθÞexpð jmαÞrdrdθ

¼ Pnm ð f Þexpð jmαÞ:

Taking the norm on the both sides of (6), we can obtain jPnm ð f r Þj ¼ jPnm ð f Þexpð jmαÞj ¼ jPnm ð f Þjjexpð jmαÞj ¼ jPnm ð f Þj;

ð7Þ

where Pnm(fr) and Pnm(f) are the PCET of fr(r, θ) and f(r, θ) respectively. Therefore, the rotation invariant can be achieved by taking the norm of PCET. In other words, the magnitudes of PCET are invariant to image rotation.

2.3.2 Scale invariant Theoretically, PCET are not invariant to image scaling, but the scale invariance can be obtained by normalizing the image into a unit circle. If a image f(r, θ) with N × N pixels is scaled by a factor k = N/2, the normalized image will be  0  ð8Þ g r ; θ ¼ f ðr=k; θÞ; where 0 ≤ r′ ≤ 1, 0 ≤ r ≤ N/2. In this way, the unit circle is made to cover the same contents of the image, hence the PCET of the image are invariant to image scaling.

2.4 Accurate coefficients selection It is known that the PCET invariants hold only approximately for digital images. For image zero-watermarking, only the accurate coefficients should be used [8, 37]. To select the accurate

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coefficients, the PCET of 256 × 256 pixels grayscale image with constant gray scales 128 is computed. We restrict the max order of PCET up to 9, and the magnitudes of some PCET coefficients are listed in Table 3. From Table 3, we can see that the magnitudes of PCET with repetition m = 4i, i ∈ ℤ are not zero. Hence the PCET coefficients with repetitions m = 4i, i ∈ ℤ are not accurate, so they are not suitable for image zero-watermarking. In other word, the accurate coefficients set is S = {Pnm, m ≠ 4i, i ∈ ℤ}.

2.5 Stability of PCET To verify the stability of PCET, we conducted several experiments using the image Lena with 256 × 256 pixels. Table 4 shows some PCET magnitudes in the accurate coefficients set under various attacks. From Table 4, we can see that the mean relative errors (MRE) between attacked images and the original image of the PCET coefficients are less than 0.03. That means the PCET of images has excellent stability and is suitable for zero-watermarking algorithms.

3 Logistic mapping Logistic mapping [30] is a very simple and widely studied chaotic dynamical system, which can be represented as follows: xiþ1 ¼ λxi ð1−xi Þð0 < xi ≤1Þ;

ð9Þ

where 0 ≤ λ ≤ 4 is the control parameter of logistic chaos system; and xn is the chaos sequence of the mapping. Logistic mapping has the characteristic of period-doubling bifurcation, as shown in Fig. 4. As can be seen in Fig. 4, the iterative sequence generated by the system is a stable value when λ < 3. When λ = 3, the system begins to appear bifurcation. When 3.569945672 ≤ λ ≤ 4, the system is in the chaotic state. With different initial values x0 and y0, non-periodic and nonconverging random sequences X{x0, x1, ⋯, xn} and Y{y0, y1, ⋯, yn} can be generated Table 3 Magnitudes of some PCET coefficients for a grayscale image with constant gray scale 128 m=0

m=1

m=2

m=3

m=4

m=5

m=6

m=7

m=8

m=9

n=0

127.9235

0.0000

0.0000

0.0000

0.0940

0.0000

0.0000

0.0000

0.0541

0.0000

n=1

0.0765

0.0000

0.0000

0.0000

0.0940

0.0000

0.0000

0.0000

0.0541

0.0000

n=2

0.0765

0.0000

0.0000

0.0000

0.0940

0.0000

0.0000

0.0000

0.0542

0.0000

n=3

0.0765

0.0000

0.0000

0.0000

0.0941

0.0000

0.0000

0.0000

0.0542

0.0000

n=4

0.0766

0.0000

0.0000

0.0000

0.0942

0.0000

0.0000

0.0000

0.0543

0.0000

n=5

0.0767

0.0000

0.0000

0.0000

0.0943

0.0000

0.0000

0.0000

0.0543

0.0000

n=6 n=7

0.0767 0.0768

0.0000 0.0000

0.0000 0.0000

0.0000 0.0000

0.0944 0.0945

0.0000 0.0000

0.0000 0.0000

0.0000 0.0000

0.0544 0.0545

0.0000 0.0000

n=8

0.0769

0.0000

0.0000

0.0000

0.0947

0.0000

0.0000

0.0000

0.0547

0.0000

n=9

0.0771

0.0000

0.0000

0.0000

0.0949

0.0000

0.0000

0.0000

0.0548

0.0000

Multimed Tools Appl Table 4 Stability of PCET magnitudes under various attacks Attacks

|P0,1|

|P1,1|

|P2,1|

|P0,2|

|P1,2|

|P2,2|

|P0,3|

|P1,3|

|P2,3|

MRE(%)

Original image

11.6947 5.2909 4.2777 2.9068 5.4469 0.8412 6.4561 0.9790 0.9360 –

Scaling 0.5

11.6573 5.3469 4.1904 2.9279 5.3708 0.8262 6.4570 0.9155 0.8643 2.3869

Scaling 1.3

11.7116 5.2932 4.2808 2.9176 5.4647 0.8775 6.4421 1.0009 0.9429 0.9410

Rotation 5∘ Rotation 45∘

11.6943 5.2836 4.2746 2.9127 5.4335 0.8465 6.4399 0.9759 0.9447 0.3106 11.6880 5.2984 4.2557 2.9083 5.4039 0.8426 6.4213 0.9590 0.9233 0.6297

Salt & pepper noise (0.001)

11.6779 5.2775 4.2539 2.9229 5.4522 0.8354 6.4416 1.0014 0.9198 0.7269

Average 11.6953 5.2897 4.2714 2.9070 5.4449 0.8402 6.4535 0.9780 0.9328 0.0908 filtering (3 × 3) Sharpening

11.6920 5.2995 4.3363 2.9051 5.4669 0.8519 6.4816 0.9878 0.9656 0.8572

JPEG30

11.7039 5.3101 4.3092 2.8693 5.4393 0.8539 6.5001 0.9374 0.9490 1.1593

Light 11.6947 5.2909 4.2777 2.9068 5.4469 0.8412 6.4561 0.9790 0.9360 0.0000 increasing (50) Contrast lowering (50)

11.6947 5.2909 4.2777 2.9068 5.4469 0.8412 6.4561 0.9790 0.9360 0.0000

iteratively, and the cross-correlation of the two sequences is zero. It means that the system is very sensitive to the initial value. Because of the features of initial value sensitivity, logistic mapping is used to select P × Q PCET coefficients from (2nmax + 1)2 PCET coefficients in this paper. The specific process is as follows: (1) The secret key K is used to generate a chaotic sequence X = {xi, 1 ≤ i ≤ P × Q}. 1 0.9 0.8 0.7

xn

0.6 0.5 0.4 0.3 0.2 0.1 0

1

1.5

2

2.5

λ Fig. 4 Bifurcation diagram of logistic mapping

3

3.5

4

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(2) Let Oi = int(Z × xi), 1 ≤ i ≤ P × Q be the indices of the selected PCET coefficients, where Z = (2nmax + 1)2. Then, we can obtain the coordinates Oi of the P × Q selected PCET coefficients.

4 Zero-watermarking algorithm The key of zero-watermarking algorithm is how to construct a zero-watermark image based on the image features instead of embedding a digital watermark to the image. It does not need an inverse transform from the transform domain to the spatial domain, which is necessary for traditional watermarking algorithms. Thus, the computation time is significantly reduced. Zero-watermarking algorithm mainly contains two procedures: zero-watermark generation and zero-watermark verification. Zero-watermark generation uses the PCET coefficients of an image to construct zero-watermark information, while zero-watermark verification is mainly used for copyright verification of the image to be verified.

4.1 Zero-watermark generation Let us denote the original image as I = {f(x, y), 0 ≤ x, y < N} and binary logo image as L = {l(i, j), 0 ≤ i < P, 0 ≤ j < Q}. The procedure of zero-watermark generation is illustrated in Fig. 5 and described below. Step 1: Pre-processing the logo image. In order to eliminate the spatial relationships between pixels in the binary logo image and improve the robustness of the zero-watermarking algorithm, we first adopt the Arnold transform [34] to scramble the logo image L with seed s and obtain L1 = {l1(i, j), 0 ≤ i < P, 0 ≤ j < Q}. Step 2: Construction of feature vector. By computing the nmax-order PCET of the original image I, we can obtain (2nmax + 1)2 PCET coefficients. To improve the algorithm’s security, this paper uses logistic mapping [30] with secret key K to randomly select P × Q coefficients from the accurate coefficients set S and computes the magnitudes to construct the feature  !  vector A ¼ A1 ; A2 ; ⋯; APQ .

Fig. 5 Procedure of zero-watermark generation

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Step 3: Binarization of feature vector.

! We binarize the feature vector A according to the following formula, thus  !  obtaining the binary feature vector B ¼ B1 ; B2 ; ⋯; BPQ .  Bi ¼

1; if Ai ≥T ; 0; if Ai < T

ð10Þ

! where T is the threshold. Here, T is set as the mean of A . Step 4: Generation of zero-watermark image. ! We convert the 1-D binary feature vector B to a 2-D feature image Lf with P × Q pixels. Then, we perform an exclusive-or operation between the feature image Lf, and the scrambled watermark image L1 to generate the verification image Wv = XOR(Lf, L1). Step 5: Digital timestamping. Compute the hash values HS of the seed s obtained in Step 1, the secret key K obtained in Step 2, and the zero-watermark image Wv using the following formula and send the results to the copyright verification authority CA. HS ¼ H ðskKkW v Þ;

ð11Þ

where H(⋅) is the one-way hash function. A hash function [15, 24, 26] compresses messages of any length into a message digest of a fixed length, that is, a hash value. We use message-digest algorithm 5 (MD5) [18] in this paper. The principle of MD5 is as follows: the input information is initialized and is grouped by 512 bits, and then each group is further grouped into 16 sub-groups with 32 bits, and further 4 sub-groups with 32 bits are output after a series of transformations, and finally a hash value with 128 bits is generated by cascading the above four sub-groups. When the copyright verification authority CA receives a HS, it assigns a digital timestamp to the HS. The resulting HSTS is a unique identification for watermark and copyright verification. A digital timestamp, i.e. a digitized stamp containing time information, is a specific type of digital signature. The verification time of a certain medium is also important information in digital copyright protection. Digital timestamping provides protection for such information. The formula of generating timestamps is HS TS ¼ T S TSK ðHS Þ;

ð12Þ

where TSTSK(⋅) is a timestamp function using the TSK key.

4.2 Zero-watermark verification Zero-watermark verification does not require the original images. In order to verify the copyright of image I* we first need to verify HSTS and HS and check the validity of the seed s, the secret key K, and the watermark image Wv. If successful, the process continues, otherwise we stop. The detailed procedure of zero-watermark verification is given in Fig. 6 and described below.

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Fig. 6 Procedure of zero-watermark verification

Step 1: Construction of feature vector. We compute the nmax-order PCET of the image to be verified select P × Q coefficients based on the verified key, and compute the magnitudes to generate the o ! n feature vector A* ¼ A*1 ; A*2 ; ⋯; A*PQ . Step 2: Binarization of feature vector.

! We binarize the feature vector A* to obtain the binary feature vector n o ! B* ¼ B*1 ; B*2 ; ⋯; B*PQ .

Step 3: Generation of verified logo image. We convert the 1-D feature vector to a 2-D feature image L*f with size of P × Q. Then, we perform an exclusive-or operation between the feature image L*f and the zero-watermark image Wv to generate the scrambled logo image L1* = XOR(L*f , Wv). Subsequently, we perform an inverse transformation on L1* to obtain the verified logo image L* using the seed s.

5 Experimental results 5.1 Robustness to various attacks To evaluate the effectiveness and efficiency of the proposed algorithm, we chose 20 test images as well as a logo test image. The original test images are grayscale images with 256 × 256 pixels and the logo test image is a binary image with 32 × 32 pixels. Eight of the original test images and the logo test image are shown in Fig. 7. The max order of PCET was set to 25. We use the peak signal-to-noise ratio (PSNR) [10, 34] to evaluate the quality of attacked images. The definition of PSNR is as follows: 1

0

C B C B  * 2552  N  N C B PSNR I; I ¼ 10logB N N C; C BX X   2A @ * f ðx; yÞ− f ðx; yÞ x¼1 y¼1

ð13Þ

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Fig. 7 Original images and logo image: a Lena, b Peppers, c Airplane, d Splash, e Mandrill, f remote sensing image, g medical image, h military image, i logo image

where I and I* are the original image and the attacked image respectively, N × N is the original image dimension. The robustness of the proposed algorithm is evaluated using the bit error rate (BER) [32] of the retrieved logo image. It is defined as: BER ¼

B  100%; PQ

ð14Þ

where B is the number of erroneously detected bits, P × Q is the logo image dimension. The robustness of the algorithm was evaluated through geometric attacks and common image processing attacks. The experimental results of image Lena are listed in Figs. 8 and 9. Table 5 shows the experimental results to various attacks of Lena, remote sensing image, medical image and military image. Based on the results, we can conclude that our algorithm has good robustness against geometric attacks and common image processing attacks. Note: Due to the size of the scaled image is not the same as that of the original image, the PSNR of the scaled image and the original image doesn’t exist, as shown in Fig. 9(e)–(h).

5.2 Comparison with similar zero-watermarking algorithms In this subsection, we investigate the proposed zero-watermarking algorithm’s performance compared to other zero-watermarking algorithms [2, 3, 5]. In addition to PSNR, the structural similarity index metric (SSIM) [10, 29] value is used to evaluate the image quality, which is defined as follows:      2σxy þ C2 2μ μ þ C 1  * x y   ; SSIM I; I ¼  ð15Þ μ2x þ μ2y þ C1  σ2x þ σ2y þ C2 where μx and σx are the mean value and the standard deviation of I respectively, μy and σy are the mean value and the standard deviation of I* respectively, σxy is the covariance of I and I*,

Multimed Tools Appl

Fig. 8 The attacked image and the retrieved logo image under common image processing attacks: a No attack, PSNR = ∞, BER = 0, b Median filtering (3 × 3), PSNR = 33.0295, BER = 0.0098, c Average filtering (3 × 3), PSNR = 30.3178, BER = 0.0039, d Gaussian filtering (3 × 3), PSNR = 39.0239, BER = 0.0029, e Gaussian noising (0.001), PSNR = 29.9975, BER = 0.0020, f)Salt and peppers noise (0.001), PSNR = 35.3712, BER = 0.0020, g Random noise (10), PSNR = 32.8824, BER = 0.0020, h JPEG compression (30), PSNR = 32.3249, BER = 0.0059, i JPEG compression (40), PSNR = 33.2472, BER = 0.0049, j JPEG compression (50), PSNR = 34.0087, BER = 0.0049, k JPEG compression (70), PSNR = 35.7026, BER = 0.0020, l JPEG compression (90), PSNR = 40.1767, BER = 0, m Light increasing (50), PSNR = ∞, BER = 0, n Light lowering (50), PSNR = ∞, BER = 0, o Contrast increasing (50), PSNR = ∞, BER = 0, p Contrast lowering (50), PSNR = ∞, BER = 0

C1 and C2 are constants. We set C1 = 0.01 and C2 = 0.03 in this paper. We subjected the ‘Lena’ image to the following attacks: Attack 1: Rotation. The image was rotated 2∘ with no cropping and then resized to original size. The PSNR value was reduced to 16.8487 dB and the SSIM value was 0.4748. Attack 2: Scaling. We scaled the image by 0.25 and then rescale back to original size. The PSNR value was 26.0312 dB and the SSIM value was 0.7538. Attack 3: Surround cropping. We applied surround cropping (1/4) on the image and the PSNR value and the SSIM value were 12.8502 dB and 0.5777 respectively.

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Fig. 9 The attacked image and the retrieved logo image under geometric attacks: a Rotation (5∘), PSNR = 16.0579, BER = 0.0049, b Rotation (45∘), PSNR = 12.2550, BER = 0.0049, c Rotation (60∘), PSNR = 11.6443, BER = 0.0039, d Rotation (90∘), PSNR = 11.5737, BER = 0, e Scaling (0.5), BER = 0.0078, f Scaling (0.8), BER = 0.0049, g Scaling (1.3), BER = 0.0049, h Scaling (1.5), BER = 0.0049

Attack 4: Center cropping. The center cropping (1/4) is applied on the image. The PSNR value was reduced to 15.7711 dB and the SSIM value was 0.7416. Attack 5: Gaussian noise. We added zero-mean Gaussian noise with variance 0.01 to the image. The PSNR value and the SSIM value were 20.3344 dB and 0.3347 respectively. Attack 6: JPEG compression. We compressed the image by JPEG with quality factor 30. The PSNR value was 32.3249 dB and the SSIM value was 0.8907. Attack 7: Blurring. We blurred the image until the PSNR value was decreased to 31.1034 dB. The SSIM value was 0.9039. Attack 8: Sharpening. Sharpening was applied on the image and the PSNR value and the SSIM value were 22.4861 dB and 0.7676 respectively. Attack 9: UnZign. The image was suffered the UnZign (6, 6) attack and the PSNR value and the SSIM value were 25.6308 dB and 0.8203 respectively. Attack 10: StirMark and unZign. We applied the StirMark RBA (1.1) and UnZign (6, 6) attacks on the image. The PSNR value was 20.1542 dB and the SSIM value was 0.7416. Table 6 shows comparison results of retrieved BERs with algorithms [2, 3, 5]. It can be seen clearly that the proposed algorithm has a better robustness than algorithms [2, 3, 5] in general, only slightly worse than them on resisting a few attacks.

5.3 False positive ratio and false negative ratio Due to various factors such as illumination, dust particles, moisture and impurities of glass slide, etc., in the photo shooting process, the photos of the same objects taken by different persons are different. In addition, images will also produce distortion in the process of image transmission. The presented algorithm is an image-content-based zerowatermarking algorithm, and hence the tiny gap between the images will have a serious impact on the algorithm. Therefore, theoretically speaking, the presented algorithm has

Multimed Tools Appl Table 5 Experimental results to various attacks of the eight original images in Fig. 7 Attacks

Lena

Peppers Airplane Splash

Mandrill remote sensing image

medical image

military image

No attack

0

0

0

0

0

0

0

0

Median filtering (3 × 3)

0.0098 0.0137

0.0205

0.0137 0.0205

0.0137

0.0127

0.0176

Average filtering (3 × 3)

0.0039 0.0156

0.0234

0.0234 0.0303

0.0176

0.0342

0.0137

Gaussian filtering (3 × 3) Gaussian noising (0.001)

0.0029 0.0049 0.0020 0.0195

0.0078 0.0146

0.0068 0.0078 0.0176 0.0273

0.0029 0.0166

0.0078 0.0039

0.0020 0.0225

Salt and peppers noise (0.001)

0.0020 0.0088

0.0098

0.0137 0.0078

0.0039

0.0049

0.0156

Random noise (10)

0.0020 0

0.0049

0.0020 0.0078

0.0029

0

0.0098

JPEG compression (30)

0.0049 0.0068

0.0059

0.0117

0.0156

0.0078

0.0107

0.0225

JPEG compression (40) JPEG compression (50)

0.0059 0.0068 0.0059 0.0059

0.0068 0.0068

0.0098 0.0117 0.0088 0.0088

0.0078 0.0029

0.0049 0.0029

0.0186 0.0078 0.0068

JPEG compression (70)

0.0020 0.0029

0.0029

0.0039 0.0059

0.0029

0.0020

JPEG compression (90)

0

0.0020

0.0020

0.0020 0.0009

0.0020

0

0.0020

Light increasing (50)

0

0

0

0

0

0

0

0

Light lowering (50)

0

0

0

0

0

0

0

0

Contrast increasing (50)

0

0

0

0

0

0

0

0

Contrast lowering (50)

0

0

0

0

0

0

0

0

Rotation (5°) Rotation (45°)

0.0049 0.0049 0.0049 0.0049

0.0078 0.0059

0.0078 0.0059 0.0068 0.0068

0.0039 0.0049

0.0078 0.0059

0.0039 0.0068

Rotation (60°)

0.0039 0.0039

0

0.0020 0.0049

0.0039

0.0068

0.0020

Rotation (90°)

0

0

0

0

0

0

Scaling (0.5)

0.0078 0.0293

0.0479

0.0273 0.0449

0.0381

0.0391

0.0146

Scaling (0.8)

0.0049 0.0049

0.0137

0.0127 0.0166

0.0059

0.0205

0.0137

Scaling (1.3)

0.0049 0.0088

0.0049

0.0068 0.0156

0.0068

0.0146

0.0137

Scaling (1.5)

0.0049 0.0049

0.0059

0.0059 0.0137

0.0068

0.0117

0.0088

0

0

low false positive ratio (FPR) and false negative ratio (FNR). To validate the FPR and the FNR of the presented algorithm, this subsection designs an experiment as follows: Let the set of images are used to construct the zero-watermark image be Sp and the set of images are not used to construct the zero-watermark image be Sn. The number of images of Sp and Sn is set to P and N respectively. Assume the number of images correctly determined of Sp and Sn is TP and FP respectively, and assume the number of images incorrect determined Sp and Sn is FN and TN respectively. Then P = TP + FN and N = FP + TN hold. The definitions of the FPR FA and the FNR MA are as follows: F A ¼ F P= ðT P þ FPÞ ; M A ¼ FN =ðT P þ FN Þ

ð16Þ

To analyze the FPR and the FNR of the presented algorithm, we use 500 images to construct the zero-watermark image, and the other 500 images are not used to construct the

Multimed Tools Appl Table 6 Comparison of the results between the proposed algorithm and other zero-watermarking algorithms Attacks

Proposed algorithm

Chen [3]

Chang [2]

Gao [5]

Rotation (2°)

0.0732

0.1768

0.2967

0.0897

Scaling (0.25)

0.0127

0.0092

0.0117

0

Surround cropping (1/4)

0.0693

0.1978

0.1601

0.2112

Center cropping (1/4) Gaussian noise (0.01)

0.1025 0.0117

0.2079 0.0285

0.1659 0.0285

0.2254 0.0142

JPEG compression (30)

0.0049

0.0109

0.0084

0

Blurring

0.0059

0.0075

0.0092

0

Sharpening

0.0137

0.0251

0.0285

0.0142

UnZign (6,6)

0.0117

0.0344

0.0621

0.0142

StirMark RBA (1.1)+UnZign (6,6)

0.0645

0.1098

0.1743

0.1047

zero-watermark image, i.e., P = 500, N = 500. Some of the test images are shown in Fig. 10. The threshold is set to 0.1, which indicates the BER of the verified logo image. Through the experiments, we can obtain that TP = 500, FN = 0, TN = 485, FP = 15, and hence the FPR and FNR of the presented algorithm are FA = 2.91 % and MA = 0 % respectively.

5.4 Computation time This subsection gives the computation time of the proposed algorithm. Due to the core of the proposed algorithm is PCET, we firstly analysis the time complexity of PCET. Let I be the original grayscale image with N × N pixels. When computing the PCET, the number of multiplications in the computation of ω order PCET for I is O(N2(2ω + 1)2). Hence the time complexity of PCET is O(N2ω2). For a grayscale image with 256 × 256 pixels, the relationship between the computation time and the max order of PCET is shown in Fig. 11. The size of the original images is 256 × 256 pixels and the max order of PCET is set to 25 in the experiments. The average computation times of 1000 grayscale images in subsection 5.3 are shown in Table 7. The PC used for the experiment has a 2.27 GHz Processor with 2 GB RAM and the operation system is Microsoft Windows 7 Ultimate. The experiment is conducted using MATLAB version 8.6.

Fig. 10 Some of the test images in the experiment of false positive ratio and false negative ratio

Multimed Tools Appl 45 40

Computation time(s)

35 30 25 20 15 10 5 0

0

5

10

15

20

25

30

35

40

45

50

Max order of PCET

Fig. 11 The relationship between the computation time and the max order of PCET

6 Conclusion Most of the existing image zero-watermarking algorithms are very sensitive to geometric attacks and have low security. In this paper, a robust zero-watermarking algorithm based on polar complex exponential transformation (PCET) and logistic mapping is proposed. We first compute the PCET coefficients of the original grayscale image, generate a binary feature image based on the PCET magnitudes selected by logistic mapping, and perform an exclusive-or operation between the binary feature image and the scrambled logo image to generate the zero-watermark image. At the stage of image copyright verification, this algorithm does not require the original image. Instead, it performs an exclusive-or operation between the extracted feature image and the verification image and compares the result with the original logo image to determine the copyright ownership. Experimental results demonstrated that this algorithm can effectively resist geometric attacks and common image processing attacks and has significant advantages over existing zero-watermarking algorithms. Two main advantages are: (1) this algorithm is based on logistic mapping and is thus very sensitive to the initial state, which leads to excellent security; (2) it uses the PCET coefficients of the image as features, which improve its robustness and capability of resisting both common image processing attacks and geometric attacks.

Table 7 Computation time of the proposed algorithm Computation time (s) Zero-watermark generation

10.9474

Zero-watermark verification

10.9046

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Despite the usefulness of our algorithm, there are still a few limitations. The robustness of our method on resisting large scale cropping attacks is poor. In the future, we should improve our algorithm to overcome these shortcomings. In addition, we can apply PCET to color image to obtain the quaternion PCET, which can be used in color image zero-watermark algorithm. Acknowledgments This work was supported by the National Natural Science Foundation of China (Nos: 61672124, 61370145 and 61173183), Program for Liaoning Excellent Talents in University (No: LR2012003).

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Chun-peng Wang received the M.S. degree from the School of Computer and Information Technology, Liaoning Normal University, China, 2013. He is currently pursuing the PhD degree in Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, China. His research mainly includes image watermarking and signal processing.

Xing-yuan Wang received the PhD degree in computer software and theory from Northeast University, China, 1999. From 1999 to 2001, he was a postdoctoral researcher at Northeast University. He is currently a professor with the Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, China. He has published three books and more than 400 scientific papers in refereed journals and proceedings. His research interests include nonlinear dynamics and control, image processing, chaos cryptography, systems biology, and complex networks.

Multimed Tools Appl

Xing-jun Chen received the PhD degree in science from System Engineering Research Institute, Huazhong University of Science and Technogym, China, 2012. He is currently the teacher of Dalian Navy Academy, His research mainly includes artificial intelligence and algorithm design.

Chuan Zhang received the M.S. degree in applied mathematics from Qufu Normal University, China, in 2015. He is currently working toward the PhD degree with the Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, China. His research interests include systems biology and complex networks.