Digital watermark detection method based on vector binary condition Weber distribution

Abstract

The invention discloses a vector binary condition Weibull distribution-based digital watermark detection method, which comprises the following steps of: combining non-sampling dual-tree complex wavelet transform with a fast and accurate Chebyshev Fourier moment to obtain a stable UDTCWT-FACHFMs amplitude domain, selecting an accurate moment in the amplitude domain as an embedding position of a watermark, and then, carrying out binary condition Weibull distribution on the basis of the vector binary condition Weibull distribution. Embedding the watermark into a UDTCWT-FACHFMs amplitude domain by using a multiplicative embedding method, then carrying out inverse transformation to obtain a watermark-containing image, and completing the embedding work of the watermark; according to the statistical property of the amplitude coefficient, selecting relatively appropriate binary condition Weber distribution to perform statistical modeling, further deducing the binary condition Weber distribution into a vector form, and performing parameter estimation by using a probability weighted moment method based on power density to obtain an accurate parameter estimator; and finally, in combination with a local optimal decision criterion, a new statistical watermark detector based on vector binary condition Weber distribution is developed, and the embedded watermark information can be accurately detected according to a threshold.

Description

Technical Field

[0001] The present invention belongs to the technical field of copyright security protection of digital images, and relates to a robust image watermark embedding and detection method, and in particular to a digital watermark detection method based on vector binary conditional Weibull distribution. Background Art

[0002] With the rapid development of the times, all kinds of information are highly digitized. Digital watermarking technology, as a popular research technology in the current scientific field, is an important branch of information hiding technology. It can effectively protect the copyright of digital information in complex environments.

[0003] Robustness, imperceptibility and watermark capacity are important indicators for evaluating image watermarking systems, but there is a mutually restrictive relationship between the three. Image watermarking methods based on statistical models can solve this difficult trade-off problem to a certain extent. Although the existing watermarking algorithms based on statistical models have solved some potential problems and achieved some results, there are still some shortcomings. First, although the robustness has been improved to a certain extent by embedding watermark information into the frequency domain coefficients, the robustness effect is still not very ideal when subjected to geometric attacks. Secondly, a single edge statistical model cannot well describe the statistical characteristics of the coefficients. In the existing algorithms, the coefficients are assumed to be independent of each other during modeling, but the correlation between different sub-bands is ignored. Summary of the invention

[0004] The present invention aims to solve the above technical problems existing in the prior art and provides a digital watermark detection method based on vector binary conditional Weibull distribution.

[0005] The technical solution of the present invention is: a digital watermark detection method based on vector binary conditional Weibull distribution, which is carried out according to the following steps:

[0006] Convention: I represents the original host image; I′ represents the watermarked image; L represents the length of the embedded watermark information; x i is the original UDTCWT-FACHFMs amplitude coefficient; y i is the amplitude coefficient of UDTCWT-FACHFMs containing watermark; ω l is the L-bit watermark information to be embedded; λ is the strength of the embedded watermark; m and n represent the rows and columns of the matrix respectively; f(x) represents the probability density function of the vector binary conditional Weibull distribution; C1 and C2 are the covariance matrices of scale one and scale two respectively, det(C1) is the determinant of C1; γ and β are the shape parameters of scale one and scale two respectively; F(x) represents the cumulative distribution function of the vector binary conditional Weibull distribution; PWM1 1,0,s Represents the overall probability weighted moment; PWM2 1,0,sRepresents the probability weighted moment of the Weibull distribution; PWM3 1,0,s represents the probability weighted moment formula corresponding to the vector binary conditional Weibull distribution; s is a positive integer used to specify the weight of the probability distribution in the upper tail; Γ(.) is the gamma function; k represents the shape parameter; H0 refers to the assumption when no watermark is embedded; H1 is the assumption when the watermark is embedded; Λ(y) represents the likelihood ratio; η is the decision threshold; τ is ln(η), which also represents the decision threshold; is the coefficient sample of binding subbands in different directions into one subband at the first scale, is the coefficient sample of binding subbands in different directions into one subband under the second scale, y 1i T and 2i T Represents y 1i and 2i The transposed matrix of ; g(y) is the inverse function of the embedding function;

[0007] a. Initial Setup

[0008] Get the original host image I and initialize the variables;

[0009] b. Watermark Embedding

[0010] b.1 Perform a two-level undecimated dual complex wavelet transform (UDTCWT) on the original host image I, and obtain two scales after the transformation. Each scale contains 6 real high-frequency sub-bands and 6 imaginary high-frequency sub-bands, and each sub-band has the same size as the original host image I;

[0011] b.2 Select the subband with the largest energy as the target subband, divide the target subband into equal-sized non-overlapping blocks, and calculate the entropy value of each block, and sort them from large to small. Select the first L high-entropy blocks with the same length as the watermark according to the order of entropy values;

[0012] b.3 Perform 3rd-order Fast Accurate Chebyshev Fourier Moments (FACHFMs) decomposition on the selected L blocks of high entropy. Each block is transformed from the original 8×8 UDTCWT amplitude coefficient matrix into a 4×7 UDTCWT-FACHFMs amplitude coefficient matrix. In each matrix, the corresponding 1≤m≤4,1≤n≤3 is selected as the watermark embedding position;

[0013] b.4 Use the multiplication rule to modify the UDTCWT-FACHFMs amplitude coefficient, embed the watermark information into each high entropy block, and obtain the UDTCWT-FACHFMs amplitude block containing the watermark information. The multiplication rule is as follows:

[0014] y i =(1+λω l )x i;

[0015] b.5 Perform 3rd-order FACHFMs on the unmodified and modified amplitude coefficients, map the high entropy block containing the watermark information back to the original position, and obtain the UDTCWT subband containing the watermark;

[0016] b.6 Perform a 2nd-level UDTCWT inverse transform on the UDTCWT subband containing the watermark and the unchanged subband to obtain the watermarked image I′;

[0017] c. Vector Bivariate Conditional Weibull Distribution (BCWD) Modeling

[0018] c.1 Perform a two-level UDTCWT on the image to obtain 6 real high-frequency sub-bands and 6 imaginary high-frequency sub-bands at each scale;

[0019] c.2 Divide the obtained UDTCWT high-frequency amplitude subband into uniform 8×8 amplitude coefficient blocks, and select the first L high entropy blocks in each subband;

[0020] c.3 Perform 3rd order FACHFMs on each high entropy block, and obtain L UDTCWT-FACHFMs amplitude coefficient blocks of size 4×7 in each subband, and select accurate moment coefficients from each block to form the training samples of the subband;

[0021] c.4 Bind the amplitude coefficients at the same position on different subbands at the same scale by vectors to obtain training samples at two scales. Input the two sets of training samples as two vector variables into the probability density function of vector BCWD for statistical modeling. The probability density function of vector BCWD is expressed as follows:

[0022]

[0023] d. Probability weighted moment parameter estimation method based on power density

[0024] d.1 Based on the probability weighted moment method (PWMM), the sample PWM is equal to the overall PWM to obtain a scale shape parameter. The overall PWM1 and sample PWM2 are:

[0025] PWM1 1,0,s =E[X(1-F(X)) s ];

[0026]

[0027] d.2 Substitute the obtained parameter value into the following formula to obtain the shape parameter on the second scale:

[0028]

[0029] PWM31,0,s =E[XY(1-F(X,Y)) s ];

[0030] d.3 Parameters C1 and C2 are obtained from the UDTCWT-FACHFMs amplitude coefficient matrix at the first and second scales of the image, respectively;

[0031] e. Constructing a local optimal watermark detector based on vector binary conditional Weibull distribution

[0032] e.1 regards the image watermark detection problem as a weak signal detection problem. If there is a watermark in the image, it is expressed as H1, otherwise it is considered that there is no watermark in the image, expressed as H0. The expressions in the two cases are as follows:

[0033] H0:y i =x i ;

[0034] H1:y i =x i (1+λω l );

[0035] e.2 The likelihood ratio Λ(y) of the watermark detector designed according to the Niemann-Pearson criterion is expressed as follows:

[0036]

[0037] In actual hypothesis testing, the log-likelihood ratio is used instead of the likelihood ratio, and the expression of the log-likelihood ratio is:

[0038]

[0039] Where g(y) is y i =(1+λω l )x i The inverse function of

[0040] e.3 Using the LMP detection criterion, the expression is expanded by Taylor series at λ = 0, ignoring the second order and higher orders, and the LO detector is obtained:

[0041]

[0042] Among them, h LO (y 1i ,y 2i ) represents “local optimal nonlinearity”, and according to the vector binary conditional Weibull distribution model, its expression is:

[0043]

[0044] e.4 Substitute the above formula into l(y i), the final statistical decision formula of the local optimal detector of the vector binary conditional Weibull distribution is obtained as:

[0045]

[0046] in,

[0047] e.5 When the statistic l LOD When (y) is greater than the threshold τ, the detector at the receiving end considers that there is a watermark signal. On the contrary, if the statistic l LOD When (y) is less than the threshold τ, the detector at the receiving end considers that there is no watermark signal.

[0048] The present invention first combines the undecimated dual-tree complex wavelet transform with good multi-resolution characteristics with the fast and accurate Chebyshev Fourier moment that can provide good resistance to local geometric attacks, obtains a stable UDTCWT-FACHFMs amplitude domain, selects the accurate moment therein as the embedding position of the watermark, embeds the watermark into the UDTCWT-FACHFMs amplitude domain using a multiplicative embedding method, and then performs an inverse transform to obtain a watermarked image to complete the watermark embedding work; according to the statistical characteristics of the amplitude coefficient, a more appropriate binary conditional Weibull distribution is selected for statistical modeling, and considering the dependency between sub-bands at different scales and different directions, the binary conditional Weibull distribution is further derived into a vector form, and a probability weighted moment method based on power density is used to estimate parameters to obtain accurate parameter estimates; finally, combined with a local optimal decision criterion, a new statistical watermark detector based on vector binary conditional Weibull distribution is developed, which can accurately detect the embedded watermark information according to the threshold. The experimental results verify the effectiveness of the statistical watermarking method based on binary conditional Weibull distribution proposed in the present invention, as well as the good performance of the constructed image watermark detector.

[0049] Compared with the prior art, the present invention has the following gain effects:

[0050] First, by combining the transform domain with the moment decomposition, the undecimated dual-tree complex wavelet transform with good multi-resolution characteristics is combined with the fast and accurate Chebyshev Fourier moment that can provide good resistance to local geometric attacks, and a stable UDTCWT-FACHFMs amplitude domain is obtained, which improves the robustness of the algorithm.

[0051] Second, the statistical characteristics of the UDTCWT-FACHFMs amplitude are fully taken into account, the dependencies between subbands between scales and directions are considered, the subband coefficients in different directions in the two scales are vector-bound, and the binary conditional Weibull distribution is expanded into a vector form to model the amplitude coefficients; the probability weighted moment method based on power density is used for parameter estimation, which greatly reduces the time complexity of the algorithm.

[0052] Thirdly, combined with the local optimal decision criterion, a new statistical watermark detector based on vector binary conditional Weibull distribution is developed, which enhances the performance of the detector. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 The original grayscale image, the watermarked UDTCWT high frequency subband and the final UDTCWT-FACHFMs amplitude domain according to the embodiment of the present invention.

[0054] Figure 2 This is a diagram of the robustness test results of the UDTCWT-FACHFMs amplitude coefficient according to an embodiment of the present invention.

[0055] Figure 3 This is a diagram showing the verification result of the UDTCWT-FACHFMs amplitude coefficient correlation according to an embodiment of the present invention.

[0056] Figure 4 This is a verification of the statistical characteristics of the UDTCWT-FACHFMs amplitude coefficient in an embodiment of the present invention.

[0057] Figure 5 The figure is a comparison chart of the fitting effects of several different models in the embodiments of the present invention.

[0058] Figure 6 This is a 20-fold difference result diagram of the original grayscale image, the image containing the 1024-bit watermark, and the original image according to an embodiment of the present invention.

[0059] Figure 7 Detection response diagram of the embodiment of the present invention under different attacks.

[0060] Figure 8 This is a test diagram for comparing detection probabilities at different watermark strengths according to an embodiment of the present invention.

[0061] Fig. 9 This is a test chart comparing the AUROC area histograms under different attacks according to an embodiment of the present invention.

[0062] Fig.10 The following is a flow chart of watermark embedding according to an embodiment of the present invention.

[0063] Fig.11 This is a flow chart of watermark detection according to an embodiment of the present invention. DETAILED DESCRIPTION

[0064] The present invention includes four stages: multiplicative watermark embedding, vector binary conditional Weibull distribution modeling, probability weighted moment parameter estimation method based on power density, and construction of a new statistical image detector based on vector binary conditional Weibull distribution to extract watermarks. Fig.10 , Fig.11As shown, follow the steps below:

[0065] Convention: I represents the original host image; I′ represents the watermarked image; L represents the length of the embedded watermark information; x i is the original UDTCWT-FACHFMs amplitude coefficient; y i is the amplitude coefficient of UDTCWT-FACHFMs containing watermark; ω l is the L-bit watermark information to be embedded; λ is the strength of the embedded watermark; m and n represent the rows and columns of the matrix respectively; f(x) represents the probability density function of the vector binary conditional Weibull distribution; C1 and C2 are the covariance matrices of scale one and scale two respectively, det(C1) is the determinant of C1; γ and β are the shape parameters of scale one and scale two respectively; F(x) represents the cumulative distribution function of the vector binary conditional Weibull distribution; PWM1 1,0,s Represents the overall probability weighted moment; PWM2 1,0,s Represents the probability weighted moment of the Weibull distribution; PWM3 1,0,s represents the probability weighted moment formula corresponding to the vector binary conditional Weibull distribution; s is a positive integer used to specify the weight of the probability distribution in the upper tail; Γ(.) is the gamma function; k represents the shape parameter; H0 refers to the assumption when no watermark is embedded; H1 is the assumption when the watermark is embedded; Λ(y) represents the likelihood ratio; η is the decision threshold; τ is ln(η), which also represents the decision threshold; is the coefficient sample of binding subbands in different directions into one subband at the first scale, is the coefficient sample of binding subbands in different directions into one subband under the second scale, y 1i T and 2i T Represents y 1i and 2i The transposed matrix of ; g(y) is the inverse function of the embedding function;

[0066] a. Initial Setup

[0067] Get the original host image I and initialize the variables;

[0068] b. Watermark Embedding

[0069] b.1 Perform a two-level undecimated dual complex wavelet transform (UDTCWT) on the original host image I, and obtain two scales after the transformation. Each scale contains 6 real high-frequency sub-bands and 6 imaginary high-frequency sub-bands, and each sub-band has the same size as the original host image I;

[0070] b.2 Select the subband with the largest energy as the target subband, divide the target subband into equal-sized non-overlapping blocks, and calculate the entropy value of each block, and sort them from large to small. Select the first L high-entropy blocks with the same length as the watermark according to the order of entropy values;

[0071] b.3 Perform 3rd-order Fast Accurate Chebyshev Fourier Moments (FACHFMs) decomposition on the selected L blocks of high entropy. Each block is transformed from the original 8×8 UDTCWT amplitude coefficient matrix into a 4×7 UDTCWT-FACHFMs amplitude coefficient matrix. In each matrix, the corresponding 1≤m≤4,1≤n≤3 is selected as the watermark embedding position;

[0072] b.4 Use the multiplication rule to modify the UDTCWT-FACHFMs amplitude coefficient, embed the watermark information into each high entropy block, and obtain the UDTCWT-FACHFMs amplitude block containing the watermark information. The multiplication rule is as follows:

[0073] y i =(1+λω l )x i ;

[0074] b.5 Perform 3rd-order FACHFMs on the unmodified and modified amplitude coefficients, map the high entropy block containing the watermark information back to the original position, and obtain the UDTCWT subband containing the watermark;

[0075] b.6 Perform a 2nd-level UDTCWT inverse transform on the UDTCWT subband containing the watermark and the unchanged subband to obtain the watermarked image I′;

[0076] c. Vector Bivariate Conditional Weibull Distribution (BCWD) Modeling

[0077] c.1 Perform a two-level UDTCWT on the image to obtain 6 real high-frequency sub-bands and 6 imaginary high-frequency sub-bands at each scale;

[0078] c.2 Divide the obtained UDTCWT high-frequency amplitude subband into uniform 8×8 amplitude coefficient blocks, and select the first L high entropy blocks in each subband;

[0079] c.3 Perform 3rd order FACHFMs on each high entropy block, and obtain L UDTCWT-FACHFMs amplitude coefficient blocks of size 4×7 in each subband, and select accurate moment coefficients from each block to form the training samples of the subband;

[0080] c.4 Bind the amplitude coefficients at the same position on different subbands at the same scale by vectors to obtain training samples at two scales. Input the two sets of training samples as two vector variables into the probability density function of vector BCWD for statistical modeling. The probability density function of vector BCWD is expressed as follows:

[0081]

[0082] d. Probability weighted moment parameter estimation method based on power density

[0083] d.1 Based on the probability weighted moment method (PWMM), the sample PWM is equal to the overall PWM to obtain a scale shape parameter. The overall PWM1 and sample PWM2 are:

[0084] PWM1 1,0,s =E[X(1-F(X)) s ];

[0085]

[0086] d.2 Substitute the obtained parameter value into the following formula to obtain the shape parameter on the second scale:

[0087]

[0088] PWM3 1,0,s =E[XY(1-F(X,Y)) s ];

[0089] d.3 Parameters C1 and C2 are obtained from the UDTCWT-FACHFMs amplitude coefficient matrix at the first and second scales of the image, respectively;

[0090] e. Constructing a local optimal watermark detector based on vector binary conditional Weibull distribution

[0091] e.1 regards the image watermark detection problem as a weak signal detection problem. If there is a watermark in the image, it is expressed as H1, otherwise it is considered that there is no watermark in the image, expressed as H0. The expressions in the two cases are as follows:

[0092] H0:y i =x i ;

[0093] H1:y i =x i (1+λω l );

[0094] e.2 The likelihood ratio Λ(y) of the watermark detector designed according to the Niemann-Pearson criterion is expressed as follows:

[0095]

[0096] In actual hypothesis testing, the log-likelihood ratio is used instead of the likelihood ratio, and the expression of the log-likelihood ratio is:

[0097]

[0098] Where g(y) is y i =(1+λω l )x i The inverse function of

[0099] e.3 Using the LMP detection criterion, the expression is expanded by Taylor series at λ = 0, ignoring the second order and higher orders, and the LO detector is obtained:

[0100]

[0101] Among them, h LO (y 1i ,y 2i ) represents “local optimal nonlinearity”, and according to the vector binary conditional Weibull distribution model, its expression is:

[0102]

[0103] e.4 Substitute the above formula into l(y i ), the final statistical decision formula of the local optimal detector of the vector binary conditional Weibull distribution is obtained as:

[0104]

[0105] in,

[0106] e.5 When the statistic l LOD When (y) is greater than the threshold τ, the detector at the receiving end considers that there is a watermark signal. On the contrary, if the statistic l LOD When (y) is less than the threshold τ, the detector at the receiving end considers that there is no watermark signal.

[0107] Experimental test and parameter setting:

[0108] The environment of this experiment is MATLAB R2018a, and the grayscale images are all 512×512. Download address:

[0109] http: / / decsai.ugr.es / cvg / dbimagenes / index.php .

[0110] Figure 1 The original grayscale image, the watermarked UDTCWT high frequency subband and the final UDTCWT-FACHFMs amplitude domain according to the embodiment of the present invention.

[0111] Figure 1 The three images from left to right are Lena, Barbara and Peppers; (a) is the original image of the three images; (b) represents the UDTCWT high-frequency sub-band with the largest energy in the corresponding image; (c) represents the UDTCWT-FACHFMs amplitude domain obtained from the image.

[0112] Figure 2 This is a diagram of the robustness test results of the UDTCWT-FACHFMs amplitude coefficient according to an embodiment of the present invention.

[0113] Figure 2 The types of attacks on the images are: (a) no attack; (b) JPEG compression (30); (c) JPEG compression (70); (d) additive Gaussian white noise (10); (e) additive Gaussian white noise (30); (f) median filter (3×3); (g) salt and pepper noise (0.03); (h) salt and pepper noise (0.01); (i) gamma noise (1.5); (j) gamma noise (0.75); (k) Gaussian noise (3×3); (l) Gaussian noise (7×7); (m) scaling (0.9); (n) scaling (1.2); (o) flip (H2, V15); (p) flip (H15, V2); (q) cropping (5%); (r) cropping (20%).

[0114] Figure 3 This is a diagram showing the verification result of the UDTCWT-FACHFMs amplitude coefficient correlation according to an embodiment of the present invention.

[0115] Figure 3 (a) Scatter plot of correlation between scales of UDTCWT-FACHFMs amplitude coefficients; (b) Scatter plot of correlation between directions of UDTCWT-FACHFMs amplitude coefficients; (c) Scatter plot of correlation within sub-bands of UDTCWT-FACHFMs amplitude coefficients.

[0116] Figure 4 This is a verification of the statistical characteristics of the UDTCWT-FACHFMs amplitude coefficient in an embodiment of the present invention.

[0117] Figure 5 The figure is a comparison chart of the fitting effects of several different models in the embodiments of the present invention.

[0118] Figure 6 This is a 20-fold difference result diagram of the original grayscale image, the image containing the 1024-bit watermark, and the original image according to an embodiment of the present invention.

[0119] Figure 6(a), (b), and (c) are the original grayscale images of Lena, Barbara, and Peppers respectively; (d), (e), and (f) are the watermarked images of the three images respectively; (g), (h), and (i) are the difference images between the original images and the watermarked images of the three images respectively (enlarged 20 times); (j), (k), and (l) are the relief effects of the difference images of the three images respectively (enlarged 20 times).

[0120] Figure 7 Detection response diagram of the embodiment of the present invention under different attacks.

[0121] Figure 7 The attack types tested are: (a) additive white Gaussian noise; (b) cropping; (c) Gaussian noise; (d) JPEG compression.

[0122] Figure 8 This is a test diagram for comparing detection probabilities at different watermark strengths according to an embodiment of the present invention.

[0123] Figure 8 The test results of three images are shown in Figure 1: (a) Lena; (b) Peppers; (c) Couple.

[0124] Figure 8 Comparative literature used in:

[0125] [1] Xiang-yang Wang, Xin Shen, Jia-lin Tian, ​​Pan-pan Niu, and Hong-yingYang. Locally optimum image watermark detector based on statistical modeling of SWT-EFMs magnitudes, Journal of Information Security and Applications 65(2022):103105.

[0126] [2]Sadegh Etemad,and Maryam Amirmazlaghani.A new multiplicativewatermark detector in the contourlet domain using t Location-Scaledistribution,Pattern Recognition 77(2018):99-112.

[0127] [3] Xiang-yang Wang, Pan-pan Niu, Jing Tian, ​​and Jia-lin Tian. A new statistical image watermark detector in RHFMs domain using beta-exponentialdistribution, Soft Computing 26(2022):9707-9727.

[0128] Fig. 9 This is a test chart comparing the AUROC area histograms under different attacks according to an embodiment of the present invention.

[0129] Fig. 9 The attack types are: (a) cropping; (b) gamma noise; (c) Gaussian noise; (d) rotation; (e) scaling; (f) additive white Gaussian noise.

[0130] Fig. 9 Comparative literature used in:

[0131] [1]Pan-pan Niu, Li Wang, Jia-lin Tian, ​​Si-yu Zhang, and Xiang-yang Wang.Astatistical color image watermarking scheme using local QPCET and Cauchy-Rayleigh distribution,Circuits Systems and Signal Processing 40(2021):4516-4545.

[0132] [2] Xiang-yang Wang, Pan-pan Niu, Jing Tian, ​​and Jia-lin Tian. A new statistical image watermark detector in RHFMs domain using beta-exponentialdistribution, Soft Computing 26(2022):9707-9727.

[0133] [3]MarziehAmini,Hamidreza Sadreazami,M.OmairAhmad,and M.N.S.Swamy.Achannel-dependent statistical watermark detector for color images,IEEETransactions on Multimedia 21(2019):65-73.

[0134] [4]Hamidreza Sadreazami,M.Omair Ahmad,and M.N.Shanmukha Swamy.Arobust multiplicative watermark detector for color images in sparse domain,IEEE Transactions on Circuits and Systems II:Express Briefs 62(2015):1159-1163。

Claims

1. A digital watermark detection method based on vector binary conditional Weibull distribution, characterized in that Follow these steps: Convention: I represents the original host image; I′ represents the watermarked image; L represents the length of the embedded watermark information; x i is the original UDTCWT-FACHFMs amplitude coefficient; y i is the amplitude coefficient of UDTCWT-FACHFMs containing watermark; ω l is the L-bit watermark information to be embedded; λ is the strength of the embedded watermark; m and n represent the rows and columns of the matrix respectively; f(x) represents the probability density function of the vector binary conditional Weibull distribution; C1 and C2 are the covariance matrices of scale one and scale two respectively, det(C1) is the determinant of C1; γ and β are the shape parameters of scale one and scale two respectively; F(x) represents the cumulative distribution function of the vector binary conditional Weibull distribution; PWM1 1,0,s Represents the overall probability weighted moment; PWM2 1,0,s Represents the probability weighted moment of the Weibull distribution; PWM3 1,0,s represents the probability weighted moment formula corresponding to the vector binary conditional Weibull distribution; s is a positive integer used to specify the weight of the probability distribution in the upper tail; Γ(.) is the gamma function; k represents the shape parameter; H0 refers to the assumption when no watermark is embedded; H1 is the assumption when the watermark is embedded; Λ(y) represents the likelihood ratio; η is the decision threshold; τ is ln(η), which also represents the decision threshold; is the coefficient sample of binding subbands in different directions into one subband at the first scale, is the coefficient sample of binding subbands in different directions into one subband under the second scale, y 1i T and 2i T Represents y 1i and 2i The transposed matrix of ; g(y) is the inverse function of the embedding function; a. Initial Setup Get the original host image I and initialize the variables; b. Watermark Embedding b.1 Perform a two-level undecimated dual complex wavelet transform on the original host image I, and obtain two scales after the transform. Each scale contains 6 real high-frequency sub-bands and 6 imaginary high-frequency sub-bands, and each sub-band has the same size as the original host image I; b.2 Select the subband with the largest energy as the target subband, divide the target subband into equal-sized non-overlapping blocks, and calculate the entropy value of each block, and sort them from large to small. Select the first L high-entropy blocks with the same length as the watermark according to the order of entropy values; b.3 Perform 3rd-order fast and accurate Chebyshev Fourier moment decomposition on the selected L blocks of high entropy. Each block is transformed from the original 8×8 UDTCWT amplitude coefficient matrix into a 4×7 UDTCWT-FACHFMs amplitude coefficient matrix. In each matrix, select the corresponding 1≤m≤4,1≤n≤3 as the watermark embedding position; b.4 Use the multiplication rule to modify the UDTCWT-FACHFMs amplitude coefficient, embed the watermark information into each high entropy block, and obtain the UDTCWT-FACHFMs amplitude block containing the watermark information. The multiplication rule is as follows: y i =(1+lō) l )x i ; b.5 Perform 3rd-order FACHFMs on the unmodified and modified amplitude coefficients, map the high entropy block containing the watermark information back to the original position, and obtain the UDTCWT subband containing the watermark; b.6 Perform a 2nd-level UDTCWT inverse transform on the UDTCWT subband containing the watermark and the unchanged subband to obtain the watermarked image I′; c. Vector Bivariate Conditional Weibull Distribution Modeling c.1 Perform a two-level UDTCWT on the image to obtain 6 real high-frequency sub-bands and 6 imaginary high-frequency sub-bands at each scale; c.2 Divide the obtained UDTCWT high-frequency amplitude subband into uniform 8×8 amplitude coefficient blocks, and select the first L high entropy blocks in each subband; c.3 Perform 3rd order FACHFMs on each high entropy block, and obtain L UDTCWT-FACHFMs amplitude coefficient blocks of size 4×7 in each subband, and select accurate moment coefficients from each block to form the training samples of the subband; c.4 Bind the amplitude coefficients at the same position on different subbands at the same scale by vectors to obtain training samples at two scales. Input the two sets of training samples as two vector variables into the probability density function of vector BCWD for statistical modeling. The probability density function of vector BCWD is expressed as follows: d. Probability weighted moment parameter estimation method based on power density d.1 Based on the probability weighted moment method, the sample PWM is equal to the overall PWM to obtain a scale shape parameter. The overall PWM1 and sample PWM2 are: PWM1 1,0,s =E[X(1-F(X)) s ]; d.2 Substitute the obtained parameter value into the following formula to obtain the shape parameter on the second scale: PWM3 1,0,s =E[XY(1-F(X,Y)) s ]; d.3 Parameters C1 and C2 are obtained from the UDTCWT-FACHFMs amplitude coefficient matrix at the first and second scales of the image, respectively; e. Constructing a local optimal watermark detector based on vector binary conditional Weibull distribution e.1 regards the image watermark detection problem as a weak signal detection problem. If there is a watermark in the image, it is expressed as H1, otherwise it is considered that there is no watermark in the image, expressed as H0. The expressions in the two cases are as follows: H0:y i =x i ; H1:y i =x i (1+l l ); e.2 The likelihood ratio Λ(y) of the watermark detector designed according to the Niemann-Pearson criterion is expressed as follows: In actual hypothesis testing, the log-likelihood ratio is used instead of the likelihood ratio, and the expression of the log-likelihood ratio is: Where g(y) is y i =(1+λω l )x i The inverse function of e.3 Using the LMP detection criterion, the expression is expanded by Taylor series at λ = 0, ignoring the second order and higher orders, and the LO detector is obtained: Among them, h LO (y 1i ,y 2i ) represents "local optimal nonlinearity" and is expressed as follows according to the vector binary conditional Weibull distribution model: e.4 Substitute the above formula into l(y i ), the final statistical decision formula of the local optimal detector of the vector binary conditional Weibull distribution is obtained as: in, e.5 When the statistic l LOD When (y) is greater than the threshold τ, the detector at the receiving end considers that there is a watermark signal. On the contrary, if the statistic l LOD When (y) is less than the threshold τ, the detector at the receiving end considers that there is no watermark signal.

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