https://doi.org/10.1007/s11042-023-17682-y
Locally optimum watermark decoder based on fast quaternion generic polar complex exponential transform
Si-yu Zhang1·Chun-peng Wang2·Yao-ru Sun1 ·Jun Yang1·Shi-qing Gao1
Received: 15 January 2022 / Revised: 29 September 2023 / Accepted: 21 November 2023
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023
Abstract
Digital image watermarking, as an important image content security protection technology, has higher research value. It is well-known that imperceptibility, robustness, and watermark payload are three paramount factors to evaluate model performance. Recent methods based on statistical modeling strategy effectively trade-off between imperceptibility and robustness.
However, capacity and time complexity are not allowed to ignore. Furthermore, how to select robust modeling objects, appropriate statistical models, and decision rules is one of the major issues in statistical watermark detection. To this end, we propose a color image watermark detector in robust fast quaternion generic polar complex exponential transform (FQGPCET) magnitudes domain, which fully considers the human visual system (HVS) and the statistical properties of image signals. Specifically, we adopt the Cauchy-Rayleigh distribution to model the probability density function of the FQGPCET magnitudes. Then, local high entropy image blocks are used as a new embedding domain for watermark messages due to their strong robustness. Furthermore, we develop a novel optimum multibit watermark decoder based on maximum likelihood theory. Experimental results are evaluated employing a set of standard color images in terms of watermark capacity, imperceptibility, and robustness. The proposed scheme provides better performance against all types of attacks in comparison with other existing methods.
Keywords Watermarking·Fast quaternion generic polar complex exponential transform· Maximum likelihood decoder·Cauchy-Rayleigh distribution
1 Introduction
In the current wireless mobile internet society, digital multimedia data can be more rapidly and conveniently transmitted. However, many issues are ubiquitous and pervasive, such as
B
Yao-ru Sun1 Department of Computer Science and Technology, School of Electronic and Information Engineering, Tongji University, Shanghai 201804, China
2 Shandong Provincial Key Lab of Computer Networks, School of Cyber Security, Qilu University
illegal copying, manipulation, and redistribution. Thus, preventing illegal acts and main- taining multimedia data transmission secure and effective has become more significant for researchers. Digital watermarking is regarded as an effective technology-wise method to protect intellectual property, owner identification, and data authentication [1]. It can secretly hide additional messages (i.e., the watermark) into multimedia signals (e.g., audio, video, images). In recent years, watermarking technology based on different schemes has been developed to resolve image security issues, including zero-watermarking and lossless water- marking schemes. Watermarking techniques of images can be classified based on visible [2]
and invisible methods. Visible watermark methods can present ownership messages directly on the marked multimedia. However, it is difficult to deter malicious attempts of quality distortion for original host images. Compared with visible watermarking methods, invisible methods are more robust in privacy protection and hold greater value for applications.
It is well known that imperceptibility, robustness, as well as payload, are three paramount factors for evaluating the performance of image watermarking. Imperceptibility means the perceptual similarity between the watermarked image and the original vision. Robustness refers to the resistance against different types of common or geometric attacks in signal processing. The payload of a watermark refers to the total amount of messages, which can be embedded into digital multimedia. It is worth noting that how to transmit more mes- sages and effectively resolve the conflict between imperceptibility and robustness remains a serious challenge. Fortunately, the statistical properties of the transform coefficients have recently drawn extensive attention. Many watermarking algorithms based on statistical theory have been proposed in different transform domains, such as DWT (discrete wavelet trans- form) [3], DCT (discrete cosine transform) [4], DFT (discrete Fourier transform) [5], shearlet transform, contourlet transform and DST (discrete shearlet transform) [6], and DNST (dis- crete non-separable shearlet transform) [7]. However, transform coefficients are fragile to resist various attacks. Furthermore, it is difficult to describe the correlation between RGB channels for color images. Moment-based watermarking methods have become more popu- lar in recent years. They have robust geometric invariance and favorable image description capability, such as PHFMs (polar harmonic Fourier moments) [8] and TFrCOMs (trinion fractional-order continuous orthogonal moments) [9]. In this regard, quaternion-type orthog- onal image moments, which are particularly promising tools, have been studied, such as QRHFMs (quaternion radial harmonic Fourier moments) [10], QZMs (quaternion Zernike moments) [11], QPHTs (quaternion polar harmonic transforms) [12], QPCET (quaternion polar complex exponential transform) [13], and QGPCET (quaternion generic polar complex exponential transform) [14].
Embedding and extraction are the two main processes for a watermarking system. Typ- ically, to maintain the trade-off between invisibility and robustness, the properties of the human visual system (HVS) [15] can be considered during the embedding process. Watermark embedding techniques are usually based on additive, multiplicative, or quantization-based methods. Recently, the multiplicative rule has proven to be more robust and invisible than other methods. The extraction process is usually divided into non-blind and blind algorithms.
It is known that non-blind algorithms require original data, whereas a system is called blind if it works under the assumption that the original host data would not be accessible at detection.
Clearly, the blind watermark algorithm plays an important role in the value of the applica- tion. In general, the performance of a statistical model-based watermark detector is highly dependent on the accuracy of the statistical model itself and the applicability of the decision criterion [16]. In view of this, choosing appropriate models and decision rules to analyze
statistical properties becomes more important in watermark detection. Commonly used sta- tistical models include the K-Bessel function [17], GG (generalized Gaussian), Weibull [4], bivariate Gaussian [18], tLS (t location-scale) [19], alpha-stable [20], vector-based Gaussian hidden Markov tree (HMT) [21], and 2D-GARCH (2-D generalized autoregressive condi- tional hetero-scedasticity) models [22]. In terms of decision rules, the LMP (locally most powerful) test [23], LRT (likelihood ratio test) [20], LLRT (Bayesian log-likelihood ratio test) [19], and min-max test are used.
Although significant progress has been made in research on moments-based watermarking methods, a major problem still persists. These moments-based methods are used for calcu- lations according to their definition. However, these methods are inefficient, numerically unstable, and inaccurate, seriously affecting their calculation accuracy and performance. To address this problem, in this paper, the fast quaternion generic polar complex exponential transform (FQGPCET) as a novel calculation strategy is used to transform the original image.
Then, the watermark message is hidden in the magnitude coefficients with the highest image entropy blocks, and a multiplicative color watermarking framework is proposed using the Cauchy-Rayleigh distribution. For the detection process, maximum likelihood decision the- ory is utilized to extract watermark bits without the original host image in the implementation process. Experimental results on different test data, watermark messages, and all types of attacks discuss the performance of the designed methods in comparison with other literature.
More specifically, the major contributions of this work are the following:
• The high-precision strategy of FQGPCET is employed for watermarking applications in the case of color images, which develops superiority in time complexity, calculation accuracy, and robustness.
• The watermark is hidden in robust high-entropy blocks using a multiplicative strategy, which effectively adapts to the HVS properties for improved imperceptibility while enhancing robustness. In particular, we achieve a higher average PSNR of 53.22 and an average BER of 0.0087 at 1024 bits compared to existing works.
• We use the Cauchy-Rayleigh distribution as an excellent statistical modeling tool. Fur- thermore, a blind watermark framework is proposed based on FQGPCET magnitudes, which promises a local optimum decoder using maximum likelihood decision theory.
• Our watermark capacity is larger than other existing methods, and extensive experimental results prove its superior performance.
The remainder of this paper is structured as follows. The related work is introduced in Section2. In Section3, we analyze robust FQGPCET and use Cauchy-Rayleigh as a statistical tool. Section4provides a watermark embedding and detection process. In Section5, we provide extensive experimental results and compare them with state-of-the-art methods.
Finally, conclusions are drawn in Section6.
2 Related work
In general, there are two major types during the watermark detection process: detecting and decoding watermarked messages. In terms of detecting non-Gaussian data, a watermark decoder or detector can be formulated as a statistical decision problem that detects weak signals in noise. In recent years, many excellent algorithms have been proposed to construct an
optimum watermark detector by a proper statistical distribution. In light of their applications, we will discuss the watermark detector and decoder separately.
In some applications of watermarking, it may only be necessary to consider whether a specific watermark message is present in the received signal. For instance, [24] designed an embedding algorithm based on a multiplicative approach that selected contourlet coefficients extracted from a given image to represent the edge features of a sparse image. Furthermore, an optimal detector was derived from the LRT.[4] proposed the use of DCT AC and DC coef- ficients for watermarking. In their scheme, the watermark was hidden in the DC coefficients of an image, which can effectively balance the requirements for transparency and robustness.
A color image watermarking scheme was designed in [25] to verify ownership and image authentication, where the inter-channel dependencies accounted for RGB channels of the sparse coefficients using the hidden Markov model (HMM). [26] designed a polar complex exponential transform-based zero-watermarking algorithm by regarding magnitudes with the same order and magnitudes with the same repetition as the reference values of each mag- nitude, which solved the low discrimination issues for multiple medical images. A robust multiple zero-watermarking scheme for color medical images was presented in [27]. This scheme was based on local feature regions and quaternion polar harmonic Fourier moments, and stable feature points were extracted from the original image using a speeded-up robust feature operator. [14] proposed zero-watermarking techniques to protect intellectual property rights by performing the fast quaternion generic polar complex exponential transform and asymmetric tent map. However, the visual quality was unsatisfactory.
To solve the problem of digital copyright protection, decoding watermark messages has become an effective method in many applications. [17] proposed a blind multibit water- marking approach in the nonsubsampled shearlet transform (NSST) domain. During the embedding stage, the hidden message was embedded into the most significant NSST direc- tional subband with the highest energy. In [28], a statistical method for watermark extraction was presented by exploiting the normal inverse Gaussian as a prior for the contourlet coeffi- cients. The watermark decoder was constructed using the maximum likelihood (ML) criterion in both noisy and clean environments, resulting in closed-form analytical expressions for the decoder. [29] used vector Cauchy to fit the nonsubsampled contourlet transform (NSCT) coefficients and proposed a local optimum decoder using the statistical model. A blind audio watermark scheme using bivariate generalized Gaussian distributions was presented in [30], wherein both the local statistical properties and inter-scale dependencies of the stationary wavelet transform coefficients were taken into account. During the insertion process, an adap- tive nonlinear watermark embedding strength function is designed. [12] presented a statistical model-based watermarking system based on a quaternion polar harmonic transform (QPHT).
At the receiver, the ML decoder was designed by using the K-Bessel function to capture the QPHT magnitude coefficients, but it proved to be computationally expensive. [23] proposed a multiplicative image watermarking scheme based on the NSST, where NSST coefficients were modeled by the ranked set sample method. At the receiver, the locally most powerful test criterion was applied. A blind image watermarking algorithm was designed in [7], where the Gaussian-Cauchy mixture-based vector HMT was used to model the DNST coefficients.
The watermark data were inserted into the DNST high entropy blocks by modifying the robust singular values.
The above statistics-based watermarking scheme still faces some difficulties. In view of this, our proposed scheme mainly focuses on FQGPCET color images while resisting various attacks effectively.
3 Fast quaternion generic polar complex exponential transform 3.1 Definition of QGPCET
In 2011, a class of rotation-invariant orthogonal moments named the generic polar complex exponential transform (GPCET) was designed by Hoang et al. [31]. GPCET has numerous beneficial properties over existing methods, such as stability and completeness. In addition, GPCET has mainly distinctive properties depending on the value of a parameter, making it more suitable for capturing local features for image representation and recognition. Hoang et al. improved and expanded the theory of [31] in [32]. Some efficient calculation strategies for GPCET were proposed in [33] to replace inefficient direct computation. In 2019, Yang et al.
[14] extended the excellent properties of GPCET to color image processing, and QGPCET was proposed by using the algebra of quaternion for color image watermarking.
Quaternions, which were first described by the Irish mathematician Hamilton [34], consist of three imaginary parts and one real part as follows:
q=a+bi+cj+dk (1)
wherei,j, andkare three imaginary units, anda,b,c, anddare real numbers. Quaternions can be regarded as a non-commutative extension of complex numbers, which obey the following relations:
i2= j2=k2=i j k= −1,i j = −j i=k,j k= −k j=i,ki= −i k= j (2) A quaternionq is called a pure quaternion when the real parta =0.The modulus and conjugate of a quaternion are respectively expressed as:
|q|=
a2+b2+c2+d2,q∗=a−bi−cj−dk (3) According to (2) and (3), the conjugate of product forq1andq2are expressed as:
(q1·q2)∗=q2∗·q1∗ (4)
In terms of a continuous color image f(r, θ),the three imaginary units fR(r, θ),fG(r, θ), and fB(r, θ)of the quaternions are used to represent the red, green, and blue components, respectively. Whena=0, f(r, θ)can be encoded as an array of pure quaternions:
f(r, θ)= fR(r, θ)i+ fG(r, θ)j+ fB(r, θ)k (5) Compared with the traditional method of employing luminance channels, the integrity of image information can be better reflected by the quaternion-based color image processing method, and the image color information can be better preserved. Since it is known from (2) that the multiplication of quaternions is non-commutative, each quaternion transform has two different forms [35]. To this end, we can provide two definitions of QGPCET. The left-side QGPCET is first defined by:
HnmsL = 1
0
2π
0
Vnms∗ (r, θ;μ)· f(r, θ)r dr dθ
=
1 2π srs−2
e−μ2nπrse−μmθf(r, θ)r dr dθ
(6)
The right-side QGPCET can be obtained by reversing the orders of image as follows:
HnmsR = 1
0
2π
0
f(r, θ)·Vnms∗ (r, θ;μ)r dr dθ
= 1
0
2π
0
f(r, θ)
srs−2
2π e−μ2nπrse−μmθr dr dθ
(7)
whereμis a unit pure quaternion chosen asμ=(i+j+k)/√
3 in this paper.
According to (4), the relationship betweenHnmsL andHnmsR is derived as:
(HnmsL )∗=
⎧⎨
⎩ 1
0
2π
0
srs−2
2π ·e−μ(2nπrs+mθ)f(r, θ)r dr dθ
⎫⎬
⎭
∗
= 1
0
2π
0
srs−2
2π ·(e−μ(2nπrs+mθ)f(r, θ))∗r dr dθ
= 1
0
2π
0
srs−2
2π ·(f(r, θ))∗(e−μ(2nπrs+mθ))∗r dr dθ
= − 1
0
2π
0
srs−2
2π · f(r, θ)eμ(2nπrs+mθ)r dr dθ
= −H(−n)(−m)sR
(8)
Based on the principle of orthogonal function, as a color image, f(r, θ)can be approxi- mately reconstructed from the QGPCET coefficients. fˆ(r, θ)is denoted as a reconstructed color image as follows:
fˆ(r, θ)= K n=−K
K m=−K
Vnms(r, θ;μ)·HnmsL
≈
+∞
n=−∞
+∞
m=−∞
Vnms(r, θ;μ)·HnmsL
(9)
3.2 Robustness analysis and fast calculation
QGPCET can be used as a powerful tool in many image processing tasks. To this end, QGPCET magnitudes can provide many robust coefficients as an effective carrier to insert watermark bits.
Rotation invariance.To prove the rotation invariant property, the color image generated after the rotation angleαis denoted by f(r, θ)= f(r, θ+α).Thus, f(r, θ)and fr(r, θ) will obey the following rules:
HˆnmsL = 1
0
2π
0
R∗ns(r;μ)e−μmθf(r, θ+α)r dr dθ
= 1
0
2π
0
R∗ns(r;μ)e−μm(θ−α)f(r, θ)r dr dθ
=eμmα 1
0
2π
0
R∗ns(r;μ)e−μmθf(r, θ)r dr dθ
=eμmαHnmsL (10)
It is easy to obtain that | ˆHnmsL |=| eμmαHnmsL |=| HnmsL |. Similarly, | ˆHnmsR |=|
HnmsR eμmα|=| HnmsR |. Therefore, the QGPCET magnitudes are invariant to image rotation.
Scaling invariance.In terms of the image function, it is assumed to be defined on a contin- uous domain, and the QGPCET coefficients are scaling-invariant. However, the normalization of images requires re-sampling and re-quantifying in an actual processing environment, so scaling invariance is approximate.
Next, we will formally consider the calculation issues in this subsection. The direct cal- culation of QGPCET leads to the problem of inefficiency, inaccuracy, and instability. Since we will use a fast calculation strategy, the method will be hereinafter referred to as fast quaternion generic polar complex exponential transform (FQGPCET). This fast calculation strategy provides an intrinsic correlation between the GPCET and QGPCET. QGPCET can be calculated quickly and accurately by using the FGPCET.
Substituting (5) into (6) and combining (2), the formula is as follows:
HnmsL = [Re(Hnms(R))+μI m(Hnms(R))] ·i + [Re(Hnms(G))+μI m(Hnms(G))] ·j + [Re(Hnms(B))+μI m(Hnms(B))] ·k
=AnmsL +BnmsL i+CnmsL j+DnmsL k
(11)
where
AnmsL = − 1
√3[I m(Hnms(R))+I m(Hnms(G))+I m(Hnms(B))]
BnmsL = 1
√3[I m(Hnms(G))−I m(Hnms(B))] +Re(Hnms(R)) CnmsL = 1
√3[I m(Hnms(B))−I m(Hnms(R))] +Re(Hnms(G)) DnmsL = 1
√3[I m(Hnms(R))−I m(Hnms(G))] +Re(Hnms(B))
(12)
Here,Hnms(R),Hnms(G), andHnms(B)denote the GPCET coefficients for the R, G, and B channels, respectively. The original color image can be reconstructed by a finite number of FQGPCETs using the following form:
fˆ(r, θ)= ˆfA(r, θ)+ ˆfB(r, θ)i+ ˆfC(r, θ)j+ ˆfD(r, θ)k (13) where
fˆA(r, θ)= Re(ALnms)− 1
√3[I m(BnmsL )+I m(CnmsL )+I m(DnmsL )]
fˆB(r, θ)= Re(BnmsL )+ 1
√3[I m(AnmsL )+I m(CnmsL )−I m(DnmsL )]
fˆC(r, θ)= Re(CnmsL )+ 1
√3[I m(ALnms)−I m(BnmsL )+I m(DnmsL )]
fˆD(r, θ)= Re(DnmsL )+ 1
√3[I m(ALnms)+I m(BnmsL )−I m(CnmsL )] (14)
Here, the value of fˆA(r, θ)is very close to 0. fˆB(r, θ), fˆC(r, θ), and fˆD(r, θ)represent the R, G, and B components of the reconstructed color image, respectively.ALnms,BnmsL ,CnmsL , andDnmsL represent the reconstruction matrices ofAnmsL ,BnmsL ,CnmsL , andDnmsL , respectively.
ALnms= K n=−K
K m=−K
ALnmsVnms(r, θ;μ)≈
+∞
n=−∞
+∞
m=−∞
AnmsL Vnms(r, θ;μ)
BnmsL = K n=−K
K m=−K
BnmsL Vnms(r, θ;μ)≈
+∞
n=−∞
+∞
m=−∞
BnmsL Vnms(r, θ;μ)
CnmsL = K n=−K
K m=−K
CnmsL Vnms(r, θ;μ)≈ +∞
n=−∞
+∞
m=−∞
CnmsL Vnms(r, θ;μ)
DnmsL = K n=−K
K m=−K
DnmsL Vnms(r, θ;μ)≈
+∞
n=−∞
+∞
m=−∞
DnmsL Vnms(r, θ;μ)
(15)
3.3 Statistical modeling of FQGPCET magnitudes
In recent years, highly non-Gaussian properties of most transform coefficients have been proven using many methods. However, the statistical properties of local moment magni- tudes are neglected. In this subsection, the marginal coefficient distribution characteristics of local FQGPCET magnitudes are analyzed in detail. First, the original color image with 512×512 pixels is divided into 4096 non-overlapping equal-sized blocks with 8×8 pixels.
Next, the value of each FQGPCET block is calculated and any FQGPCET selected from the accurate moment sets. Last, we obtain FQGPCET. The histograms of FQGPCET magnitude coefficients are depicted in Fig.1. A sharp peak close to zero and its own tails are heavier compared to the Gaussian probability density function (PDF). To illustrate further the above conclusion, the non-Gaussian property of local FQGPCET magnitudes is validated using the kurtosis value. For univariate datax(x1,x2, . . . ,xn), the kurtosis value is defined as:
kx= 1 ns4
n i=1
(xi−x)4 (16)
wherex is the mean of x and sis the standard deviation. Kurtosis values are computed for four different color images, which are 29.8912, 35.1399, 29.1795, and 30.6622. These kurtosis values are larger than the standard value of the Gaussian distribution (equal to 3).
Therefore, we can show that the FQGPCET magnitudes have highly non-Gaussian properties.
For FQGPCET magnitudes, it can also be concluded here that the local moment coefficients are suitable to embed the watermark information.
Fig. 1 Histograms of local FQGPCET magnitudes: (a) Barbara, (b) Peppers, (c) Lena, (d) Baboon
3.4 CR modeling of FQGPCET magnitudes
We employ the FQGPCET magnitudes to design a watermark detector algorithm. Next, a proper distribution model for the PDF of the FQGPCET magnitudes is required. In this paper, five different distributions will be used to fit the FQGPCET magnitudes, which contain Gamma, Weibull, Weibull-Rayleigh, Exponential, and Cauchy-Rayleigh distributions.
Modeling lifetime phenomena is a main problem in many scientific fields. However, clas- sical models no longer describe new phenomena with the advancement of science. Recently, many researchers have attempted to extend the classical distribution. As a result, many new flexible models appeared and proved to be more acceptable. The Cauchy-Rayleigh (CR) dis- tribution, also named the generalized Cauchy distribution [36], has received great attention as a descriptor of SAR features. The CR distribution is the particular case of a heavy-tailed.
The heavy-tailed Rayleigh distribution is well known for fitting the amplitude coefficients, which provides more flexibility for the purpose of modeling. The PDF of the CR distribution with scale parameterγ >0 and a random amplitude variable 0≤ x ≤ ∞is obtained as follows:
f(x;γ )= xγ
(x2+γ2)32 (17)
This PDF is associated with the amplitude on which the components are jointly distributed as Cauchy. The cumulative distribution function (CDF) of this model is derived as follows:
g(x;γ )=1− γ
x2+γ2 (18)
As mentioned, in addition to the CR distribution, Weibull, Weibull-Rayleigh, Gamma, and exponential distributions are employed to fit the magnitude components of coefficients.
The Weibull distribution [4] is a special case of the generalized Gamma distribution with a long tail. It can capture the characteristics of the pulse-shape distribution whenβ <1. The PDF and CDF with shape parameterβ >0 and scale parameterα >0 are respectively given as:
f(x;α, β)= β α
x α
β−1
ex p
−x α
β
(19) g(x;α, β)=1−ex p
−x α
β
(20) where the random amplitude variablex ≥0.
The Weibull-Rayleigh distribution [37] is a three-parameter lifetime model, and its PDF and CDF are expressed as follows:
f(x;b,c, γ )= cx γb2
x2 2γb2
c−1
ex p
− x2
2γb2 c
(21)
g(x;b,c, γ )=1−ex p
− x2
2γb2 c
(22) wherex≥0,shape parameterc>0,location parameterb>0, and scale parameterγ >0.
The Gamma distribution [38] is a two-parameter family of curves, and the PDF forx ≥0 is
f(x;α, β)=βαxα−1
(α) ex p(−xβ) (23)
whereα > 0 andβ > 0 are referred to as shape and scale parameters of the Gamma distribution, respectively.(α)is the Gamma function evaluated atα.The CDF for Gamma distribution is denoted as:
g(x;α, β)= 1
(α)γ (α, βx) (24)
The Exponential distribution [39] is a continuous probability distribution used for mod- eling. The PDF and CDF formulas are given by:
f(x;μ)= 1 μex p
−x μ
(25) g(x;μ)=1−ex p
−x μ
(26) where the parameterμis also equal to the standard deviation.
Figure2shows the histogram of magnitude components as well as the best-fitted CR, Weibull, Weibull-Rayleigh, Gamma, and exponential distributions for five different color test images. Figure2shows the high efficiency of the CR distribution for modeling the FQGPCET magnitude coefficients. To quantify the findings, the Kolmogorov-Smirnov (K- S) test is applied as a common measure for continuous distributions, which describes the distance between the CDF of the reference and that of samples. Here, the K-S test is used to measure the difference between the magnitude component histogram and the proposed models. The K-S value is the maximum difference betweenF0(x)andFn(x), i.e.,
K −S= max
−∞<x<∞|F0(x)−Fn(x)| (27) whereF0(x)andFn(x)denote the empirical sample CDF and reference CDF, respectively.
The K-S test results of Gamma, Weibull, Weibull-Rayleigh, exponential, and CR distribu- tions for local FQGPCET magnitudes are presented in Table1. According to Table1, the K-S test result for the CR model is the smallest, and the CR model fits the empirical data better than other distributions. Therefore, the CR distribution is used to fit FQGPCET magnitudes in this paper. From the above results, we presented a statistical modeling watermarking system by utilizing the FQGPCET method in the next section.
Fig. 2 Histograms of the empirical data and PDFs of Weibull, Weibull-Rayleigh, Cauchy-Rayleigh, Gamma as well as Exponential for (a) Barbara, (b) Peppers, (c) Lena, (d) Baboon
Table 1 K-S values of FQGPCET magnitudes for color test image. The best results are shown in bold Color image Gamma Weibull Exponential Weibull-Rayleigh Cauchy-Rayleigh
Lena 0.2292 0.1677 0.8573 0.2007 0.0657
Barbara 0.1854 0.1227 0.6249 0.1706 0.0963
Baboon 0.1976 0.1304 0.5880 0.1648 0.0437
Peppers 0.2286 0.1720 0.7000 0.2241 0.0452
Average 0.2102 0.1482 0.6926 0.1901 0.0627
4 The proposed watermarking algorithm 4.1 Digital watermark embedding
LetI = {f(x,y),0≤x <P,0≤y<Q}be the host color image.W = {wl,0≤l<L}
is used as the pseudo-random noise generator to generate a watermark message sequence with values{−1,1}.FQGPCET magnitudesxi(i =1,2, . . . ,L)are employed to insert the watermark message.yi(i=1,2, . . . ,L)are denoted the watermarked coefficients.
In general, the human visual system (HVS) considers the limitations of human visual sensitivity, making it impossible for human vision to distinguish the difference between signals. In view of this, HVS can provide a favorable embedding space for watermarks to improve image imperceptibility. Based on the visual masking rules, we mainly implement two aspects. On the one hand, high-entropy block regions are calculated by using the FQGPCET magnitudes, which are stable and contain more texture than other image blocks. On the other hand, the edge regions of entropy blocks are chosen for embedding to alleviate the influence of smooth regions in high-entropy blocks. In addition, we adopt a multiplicative rule to embed different watermark bits, which can adapt to HVS properties to the greatest extent than additive ones. Our specific steps of the embedding scheme are realized in Algorithm 1.
The watermark embedding scheme is depicted in Fig.3. Note that in step 3, some robust FQGPCET magnitudes in the shaded area are selected to insert the watermark W. The
Fig. 3 The key components of the proposed watermark embedding
Algorithm 1Embed the watermarkWintoI.
Input:Original carrier imageI, the binary watermark messageW.
Output:Watermarked imageI∗.
Step1:Segment the original carrier imageI intoNblock(Nblock ≥ L)non-overlapping equal-sized local blocks.
Step2:Calculate and select higher ranking blocks in the estimated entropy measure for watermark embedding.
Step3:Lhigh-entropy local blocks are computed based on the FQGPCET magnitudes. Here, the accurate moment order and basis parameter are set toT=15 ands=2. The corresponding matrixBl(l=1,2, . . . ,L) is obtained.
Step4:By modifying the selected FQGPCET magnitudes, the watermark informationWwith “-1” or “1” is hidden in the local image blocks. Here, the multiplicative rule is as follows:
xi=
xi(1+λwl) wl=1i s embedded
xi(1−λwl) wl= −1i s embedded,xi∈Bl (28)
wherexiandxi denote the original and watermarked FQGPCET magnitudes, respectively.λindicates the watermark inserting strength factor.
Step5:The watermarked image blockfwis calculated given by:
fw= fo− fr+fr (29)
where forepresents the original image block. fr represents the image blocks reconstructed based on the initially selected FQGPCET, and fris the watermarked FQGPCET.
Step6:Combine theLwatermarked image blocks with theNblock−Lnon-watermarked image blocks to obtain the watermarked imageI∗.
coefficients in this area do not differ significantly and exhibit good statistical characteristics.
Furthermore, the impact of the embedded watermarks on the image is minimal, thereby ensuring better imperceptibility and stability.
We provide robustness verification of high-entropy blocks. Figure4(a) presents L=1000 bits local image blocks with N×N=8×8 pixels. After undergoing various attacks on the local image blocks in Fig.4(a), Fig.4(b) provides some stable image blocks. In Fig.4(b), stable image blocks for Lena, Sailboat, Baboon, Flowers, and Motocross are 84.3%, 84.1%, 73.8%, 79.1%, and 80.1%, respectively. In Fig.4(c), L=128 bits of stable high-entropy blocks are selected for watermark embedding. From the above data, the robustness of high-entropy blocks is fully verified.
4.2 Digital watermark detection
An important requirement for copyright protection is to assess the copyright authenticity of the media. To meet this requirement, watermark existence or watermark signal detection needs to be verified from the watermarked image. Therefore, a digital blind watermark decoder should be developed, which can be achieved by employing the ML decision rule in this work. With a blind decoder in mind, the host color image acts are assumed to be noisy environments, and the FQGPCET magnitudes are modeled based on the C-R distribution. The watermark decoder without the host image and the corresponding scheme flow chart is depicted in Fig.5. Next, the watermarkW∗can be detected from watermarked I∗, and the corresponding scheme is implemented in Algorithm 2. By utilizing the GA-based MLE algorithm [40], we can estimate the scale parameterγ of the watermarked FQGPCET magnitudes.
Fig. 4 High-entropy blocks robustness test for image Lena, Sailboat, Baboon, Flowers and Motocross. (a) original blocks image, (b) various attacks blocks image, (c) watermark length L=128 bits
5 Experiments
In our work, the simulation experimental platform is MATLAB R2020a, which is tested on the CVG-UGR [41] and Kodak test databases [42]. All experimental results are performed with an Intel Core i7 CPU 2.6GHz, 16GB memory, and a Microsoft Windows 10 Ultimate oper- ating system. Twenty standard test images with 512×512×24 bits are divided into different numbers of non-overlapping local image blocks. In addition, the watermark pseudoran- dom sequences used include 128/256/512/1024/2048 bits and binary images. The proposed
Fig. 5 The key components of the proposed watermark extracting
Algorithm 2Extract the watermarkW∗fromI∗.
Input:Watermarked imageI∗,
Output:The binary watermarkW∗=Wl∗,0≤l<L.
Step1:Segment the watermarked imageI∗into non-overlapping equal-sized localNblock(Nblock≥L)blocks.
Step2:Lhigh-entropy local blocks are calculated and selected based on the FQGPCET magnitudes as the same as the watermark embedding process.
Step3:To derive the optimum behavior of the ML decoder, a binary hypothesis test is constructed by the multiplicative embedding function in (28) as follows:
H1:yi=xi(1+λw∗l), wl∗=1
H0:yi=xi(1−λw∗l), wl∗= −1 (30)
wherexi∈Bl,xiandyiare the original and watermarked coefficients.
Step4:The FQGPCET magnitudes represent a random sample that is assumed to be drawn from some underlying PDF. An optimum decoder using ML decision for theLFQGPCET magnitudes is formulated as:
i∈B l
fY(yi|wl∗=1) H>1 H<0
i∈B l
fY(yi|w∗l = −1) (31)
Step5:The natural logarithm is applied on both sides, the log-likelihood ratioL(y)is derived as:
L(y)= i∈B l
ln fY(yi|w∗l =1) fY(yi|w∗l = −1)
H1
><
H0
0 (32)
where
fY(yi|w∗l = ±1)= 1 1±λfx
yi 1±λ
(33)
Here,fx(x)is denoted the PDF of the FQGPCET magnitudes.
Step6:According to the C-R model, the PDFs of the C-R distribution under each hypothesis is substituted into (33) fC−R(yi|1)=1+λ1 γ
yi 1+λ
yi
1+λ 2
+γ23 2
fC−R(yi|0)=1−λ1 γ yi
1−λ yi
1−λ 2+γ23
2
(34)
whereγis defined by the scale parameter.
Step7:Substituting (34) into (32), the final locally optimum watermark decoderLl(y)can be obtained as:
Ll(y)= i∈B l
ln11−λ+λ+ i∈B l ln
γ yi 1+λ
yi
1−λ 2+γ23
2 yi
1−λ 2+γ2
3 2γ yi
1−λ
(35)
Step8:Thelth watermark bit that exists in the FQGPCET magnitudes is decoded as:
w∗l =
1 Zl(y)≥τl
−1 Zl(y) < τl (36)
where
Zl(y)= i∈B
l ln
γ yi 1+λ
yi
1−λ 2
+γ23 2 yi
1−λ 2+γ23
2γ yi 1−λ
(37)
τl= i∈B l
ln1+λ
1−λ (38)
Here,τlis the threshold.
watermarking scheme is evaluated in comparison with other state-of-the-art schemes [6,12,
5.1 Performance evaluation from QGPCET and FQGPCET
To evaluate the scaling invariance of the QGPCET, we take a standard color Lena image.
Figure6shows the QGPCET modulus coefficients for the color image Lena under various conditions. The QGPCET modulus coefficients remain almost unchanged against various noises and geometric transforms. Thus, the QGPCET modulus coefficients are chosen as an effective carrier to embed the watermark bits.
The quality of the reconstructed image is an important criterion to measure the image reconstruction capability. To this end, we implement some reconstruction experiments of the proposed FQGPCET in this paper. The reconstructed images and reconstruction error images obtained for Lena and Peppers under the condition of 128×128 pixels are shown in Fig.7. (multiplied by 5 for better display). It can be observed that the more orders used, the closeness of the reconstructed image to the original image. In addition, the edges of the reconstructed image are better defined with less jaggedness. The mean-square reconstruction error (MSRE) [14] is used to evaluate the quality of the reconstruction, which is defined as:
M S R E= +∞
−∞
+∞
−∞(f(x,y)− ˆf(x,y))2d xd y +∞
−∞
+∞
−∞ f(x,y)2d xd y (39) where f(x,y)and fˆ(x,y)are the original and reconstructed images, respectively.
The average MSRE values are provided at the basis parameters s=0.5, s=1, and s=2 in Fig.8. Figure9shows the MSRE values for the Lena image at 128×128 pixels using the proposed FQGPCET method and other comparison methods, including QPHT [12], QPHFM [44], QEM [45], QRHFM [46], and QZM [47]. It is clear from the above experimental results that the proposed FQGPCET method is much better than other methods while inheriting the advantages of high accuracy, low complexity, and stability.
Fig. 6 The QGPCET modulus coefficients for Lena undergo various attacks:(a) original host image, (b) Median filtering 3×3, (c) JPEG compression (50), (d) Sharpening, (e) Salt and peppers noise, (f) Gaussian noise, (g) Rotation (15), (h) Scaling (0.7)
Fig. 7 The reconstructed images and reconstructed error images based on the QGPCET with size 128×128 (moment orders K=10,20,30,40,50,60,70). (a) Lena, (b) Peppers
5.2 Invisibility experiment
The peak signal-to-noise ratio (PSNR) is used as an evaluation criterion to evaluate the difference in visual quality between the original image and the watermarked image. It is denoted as:
P S N R=10 log10
⎡
⎢⎢
⎢⎣3×P×Q 2552 3
j=1
p x=1
q y=1
[I(x,y,j)−I∗(x,y,j)]2
⎤
⎥⎥
⎥⎦ (40)
whereI(x,y,j)andI∗(x,y,j)are the pixel values at point(x,y,j)in the original image and watermarked image, respectively. 3×P×Q is the color image size. According to theory [12], PSNR>38dB can be regarded as a criterion to measure imperceptibility.
Figure10shows six test images (Lena, Barbara, Baboon, Sailboat, Flowers, and Hats) of size 512×512 with 1024 bits watermark. The watermarked images and their original carrier version utilizing the presented algorithm are described in Fig.10(a) and (b). The absolute differences between the original and watermarked images are shown in Fig.10(c). Here, the corresponding PSNR values from left to right are 54.1853 dB, 51.8790 dB, 52.3374 dB, 51.3137 dB, 52.9091 dB, and 56.6968 dB, respectively. From the above observations, most of the watermark message is hidden inside the edge details of test images, which suggests that the proposed watermarking scheme has achieved outstanding invisibility. It is worth noting that the value of PSNR is affected by the watermark capacity.
Next, we present some comparative results. Figure11shows the relationship between the watermark capacity and the average PSNR values. The smaller the watermark capacity, the larger the average PSNR values. In addition, we also compare the proposed watermarking
Fig. 8 The average MSRE values at different moment order values of K using the FQGPCET in this paper
sequences. Obviously, the proposed scheme provides higher average PSNR values as the watermark capacity increases, indicating its better imperceptibility. To maintain the balance between algorithm computational complexity and performance, we set N=64×64 as the number of non-overlapping blocks. This means that 8×8 pixels represent the size of each block. Non-overlapping blocks with different sizes of 16×16, 32×32, 64×64, and 128×128 are shown in Fig.12. From Fig. 12, our proposed scheme demonstrates high invisibility compared with the method in [12]. The comparison results of the imperceptibility experiments are obtained in Table2. Here, twenty 128 bits watermark pseudorandom sequences are used
Fig. 9 MSRE values computation with different comparison methods at the moment order K=4, 8, 12, 16, 20,
Fig. 10 Original, watermarked and differences images by utilizing the designed algorithm: (a) original carrier images, (b) watermarked images, (c) absolute difference images. Multiplied by 20 for better display
to embed into the luminance component. Furthermore, the carrier test images are extended to RGB images for a fair comparison with other watermarking algorithms.
5.3 Detector evaluation and robustness experiment
We perform several experiments to test the performance of the proposed ML watermark decoder in the FQGPCET domain. A high entropy FQGPCET magnitude block is randomly
Fig. 11 The average PSNR values under different watermark capacity
Fig. 12 The relationship between the average PSNR values and the number of blocks
selected to embed watermark bits "-1" or "1". Additionally, we calculate their threshold and response value under the two conditions. Figure13gives the performance of the watermark detector and calculates T(ML), which represents the cumulative response value minus the accumulation threshold value in different detectors. From Fig.13, we can see that watermark bit "1" can be effectively decoded to extract when the value of T(ML) is positive; otherwise, extract watermark bit "-1".
To fully evaluate the robustness of our proposed watermarking scheme, as evaluation metrics, the bit error rate (BER) is used to evaluate the fidelity of the extracted watermark messageW∗with respect to its original versionW. The calculation formula of this metric is defined as (41).
B E R= B
m×n (41)
wheremandnare the length and width of the original watermark message, respectively. In our case,m=n=32.Bindicated the number of error bits in (41).
In this section, BER values are employed to measure algorithm robustness against different types of attacks. Figure14shows a part of the attacked image, which includes JPEG, salt and pepper, median filtering, Gaussian filtering, cropping, rotation, and gamma correction. For the
Table 2 The comparison of our algorithm with references [12,17,21,28,29], and [43] by PSNR values. Bold values highlight the best experiment results for the comparison of the algorithms
Test image Ours [43] [28] [21] [29] [17] [12]
Lena 59.39 46.35 50.15 52.15 53.20 45.32 56.13
Barbara 59.02 47.21 47.25 50.08 52.15 42.52 53.17
Peppers 60.07 45.05 51.31 53.15 53.41 47.15 54.53
Girl 60.01 49.76 48.56 54.10 54.52 46.65 55.85
Baboon 59.30 48.43 50.24 50.18 51.09 43.72 52.72
Fig. 13 The detection results under different attacks: (a) without attack, “1” (Left): T(ML)=273.6940; “- 1” (Right): T(ML)= -329.1364; (b) JPEG (QF=10), “1” (Left): T(ML)=253.6977; “-1” (Right): T(ML)=
-224.4999; (c) Gaussian filter (9×9), “1” (Left): T(ML)=280.3305; “-1” (Right): T(ML)= -348.3036; (d) Gamma correction (0.75), “1” (Left): T(ML)=253.8884; “-1” (Right): T(ML)= -337.9943
