Mixed domain statistical image watermarking method based on vector HMT modeling

Abstract

The invention discloses a mixed domain statistical image watermarking method based on vector HMT modeling, and the method comprises the steps: firstly completing image multi-scale transformation through a non-subsampled NSCT sink, and selecting a direction sub-band with the maximum scale two variance as a target sub-band; secondly, performing 8 * 8 non-overlapping partitioning on the selected target sub-band, and performing inverse transformation by using a linear multiplicative embedding function to obtain a watermark-containing image; then, a probability density function based on vector GPGWM-HMT is derived, other sub-bands without watermarks are selected for modeling, shape parameters and position parameters are obtained, and coefficients of the sub-bands without watermarks are estimated through a DEM parameter estimation method; and finally, according to a decoder designed based on a maximum likelihood criterion, extracting a watermark bit according to a decision threshold. Experimental results show that the method provided by the invention constructs vector-based GPGWM-HMT distribution to perform statistical modeling by fully utilizing various correlations among coefficients, so that the performance of a watermark decoder is better improved.

Description

Technical Field

[0001] The invention belongs to the technical field of information security and digital copyright protection, and in particular is a mixed domain statistical image watermarking method based on vector HMT modeling. Background Art

[0002] In today's modern life, as intelligent devices penetrate into people's lives, social activities conducted through digital images have become more frequent. Digital images play a vital role in all areas of society and are generally spread through images and audio, with images being the most common way of spreading. Therefore, the protection of image security is crucial. However, due to the easy-to-copy and spread attribute of digital content itself, piracy, illegal copying and unauthorized content dissemination have led to the demand for digital content protection technology. The hybrid domain statistical image watermarking method combines statistical modeling, signal processing and information hiding technology to provide an effective, robust and covert image watermarking scheme to meet the security and integrity challenges faced by digital media content.

[0003] The essence of image watermarking technology is to embed information (watermark) into the image. The embedded watermark is hidden and invisible to the image itself, but can be extracted by specific methods and algorithms to identify the source of the image, verify the integrity of the image, protect copyright, confirm ownership, etc. Image watermarking technology should have three basic requirements: imperceptibility, robustness and watermark capacity. Imperceptibility means that the watermark should be invisible or difficult to detect to the human eye to ensure that it does not affect the quality or visual perception of the image; robustness means that the watermark should have a certain resistance to some common image processing operations (such as compression, cropping, rotation, filtering, etc.) or malicious attacks (such as image tampering, adding noise, etc.), and be able to maintain stability while maintaining extractability; watermark capacity refers to the maximum amount of watermark information that can be embedded in an image or other media. The serious challenges currently faced are mainly to keep the watermark from being tampered with or deleted while keeping the sensory impact of the watermark on the image visual as small as possible.

[0004] In recent years, researchers have conducted extensive research on mixed domain statistical image watermarking, trying to improve the embedding capacity of watermarks, ensure the stability and robustness of watermarks, and reduce the impact on image quality. However, there are still the following shortcomings: first, the robustness of the modeled object is not ideal; second, the PDF of the edge distribution model is not able to describe the distribution characteristics of complex modeled objects; third, the correlation characteristics between modeled objects are not fully utilized; fourth, the parameter estimation method is not accurate enough, and there are problems such as high time complexity and low precision. Summary of the invention

[0005] The present invention aims to solve the above technical problems existing in the prior art and provides a mixed domain statistical image watermarking method based on vector HMT modeling.

[0006] The technical solution of the present invention is: a hybrid domain statistical image watermarking method based on vector HMT modeling, which is carried out in the following steps:

[0007] Convention: P represents the host image; P' represents the watermarked image; D represents the low-frequency subband obtained after the host image P is decomposed; H1 and H2 represent the high-frequency subbands of the first and second scales obtained after the host image is decomposed, respectively; L is the length of the watermark information; x i represents the original NSCT-FPCET amplitude coefficient; y i represents the amplitude coefficient of NSCT-FPCET containing watermark; W l is the watermark information to be embedded, x ij It refers to the amplitude without watermark information at the i-th position of the j-th scale; y ij It refers to the amplitude of the watermark information at the i-th position at the j-th scale; α represents the watermark embedding strength; w l Indicates the lth watermark bit; Indicates the amplitude x ij The probability of state m, and it satisfies represents the state transition probability from the parent scale ρ(j) to the child scale j in the i-th tree, where the hidden state is transferred from ρ(m) to m; and represents the shape parameter in the m state; and Indicates that when x ij The scale parameter when the hidden state is m; H0 represents the assumption when the watermark information -1 is embedded in the subband; H1 represents the assumption when the watermark information +1 is embedded in the subband;

[0008] a. Initial Setup

[0009] Get the host image P of size M×N and initialize the variables;

[0010] b. Watermark Embedding

[0011] b.1 Perform a two-level decomposition operation of non-subsampled contourlet transform (NSCT) on the host image P of size M×N, and generate a low-frequency subband D and a first-scale high-frequency subband H1 and a second-scale high-frequency subband H2 while ensuring good resolution and ideal reconstruction effect. Then, H1 and H2 are decomposed into multiple directions, that is, eight high-frequency subbands in the first scale and four high-frequency subbands in the second scale are obtained;

[0012] b.2 In the high-frequency sub-band obtained by decomposition, select the directional sub-band with the largest variance of the second scale amplitude as the target sub-band for embedding the carrier image of the watermark, divide the selected target sub-band into 8×8 blocks respectively, ensure that the blocks do not overlap, and calculate the entropy value of each block. Based on the length L of the watermark to be embedded, select the first l high entropy blocks with higher entropy values, record the position of each block and sort the entropy values;

[0013] b.3 Select l high entropy blocks with the same length as the watermark L and perform 5th-order fast extreme complex exponential transform (FPCET) decomposition respectively to calculate the NSCT-FPCET amplitude domain. For each block, the dimension of the block becomes 7×7 after 5th-order FPCET calculation;

[0014] b.4 Embed a +1bit or -1bit watermark for each block accordingly, and perform the watermark information embedding operation according to the multiplicative embedding rule of the watermark, and embed the watermark information according to the linear multiplicative rule:

[0015] y i =(1+aω l )x i ;

[0016] b.5 Perform 5th-order FPCET reconstruction on the L moment amplitudes containing watermark information, change the dimension of the target block from 7×7 back to the original target block dimension, and then put it back to the original position in the target subband with the largest variance of the second scale;

[0017] b.6 Perform inverse NSCT transformation on all coefficients in the target subband, and finally obtain the watermark image P′ which has embedded watermark information;

[0018] c. Generalized Power Generalized Weibull Mixture (GPGWM)-Hidden Markov Tree (HMT) Statistical Modeling

[0019] c.1 Perform [2,3] NSCT secondary decomposition on the watermarked image P' to obtain a low-frequency subband D1 and a first-scale high-frequency subband h1 and a second-scale high-frequency subband h2. Then decompose h1 and h2 into multiple directions, that is, obtain eight directions of the first scale and four directions of the second scale high-frequency subbands.

[0020] c.2 Select the high-frequency sub-bands with the largest variance of the second scale and the first scale respectively, implement the block operation according to the strategy of "same size and non-overlapping", sort each block from large to small according to the obtained entropy value, and select the first L high entropy blocks for watermark information extraction;

[0021] c.3 Perform 5th-order FPCET calculation on the selected L high-entropy blocks, and obtain a 7×7 NSCT-FPCET amplitude coefficient matrix for each block; similarly, perform 8×8 non-overlapping block division on the four directional subbands in the second scale, and calculate 5th-order FPCET for each block to obtain the corresponding NSCT-FPCET amplitude. At this time, the NSCT-FPCET amplitude between the second scale and the first scale presents a 1-to-2 correspondence;

[0022] c.4 According to the definition of the target position, the NSCT-FPCET amplitude between the first scale and the second scale presents a 2-to-1 relationship. The GPGWM-HMT hybrid model is constructed for the selected amplitude coefficients for statistical modeling, where GPGWM is used to capture the non-Gaussian distribution characteristics of the amplitude and HMT is used to capture the scale correlation of the amplitude. The probability density function of GPGWM-HMT is defined as:

[0023]

[0024] d. DEM parameter estimation

[0025] d.1 Construction of the vector GPGWM-HMT model in the NSCT-FPCET amplitude;

[0026] d.2 Use parameter estimation method DEM algorithm;

[0027] The training samples are selected from all high-frequency subbands obtained by NSCT decomposition (all aspects of high-frequency subbands of scale 1 and scale 2), including noise-free and noise-containing training samples, with a total of T groups of data, i.e., T trees; the vector set constructed by binding the NSCT-FPCET amplitude coefficients of the same position i frequency domain block in the i-th tree of the T trees with eight different direction subbands in the first scale is The vector set formed by binding the NSCT-FPCET amplitude coefficients of the frequency domain blocks in the same position of the four sub-bands in different directions of the second scale Calculate the GPGWM-HMT model parameter set;

[0028] e. Construct a maximum likelihood decoder to extract the watermark

[0029] e.1 Using the ML (Maximum likelihood) decision rule at the receiving end, a digital watermark decoder based on vector GPGWM-HMT is designed in the NSCT-FPCET amplitude to extract the specific watermark bit. Considering watermark detection as a binary hypothesis testing problem, we have:

[0030]

[0031] e.2 Using the ML decision criterion, the MLD decoder designed based on the maximum likelihood criterion is as follows:

[0032]

[0033] in: And f Y (y i H0)=f X (y i ), f X (x) represents the probability density function of the model used;

[0034] The e.3 decoder adopts a blind watermark extraction method. The watermarked image maintains the same statistical distribution characteristics as the original image. The GPGWM-HMT model is used to represent the amplitude distribution characteristics of the watermarked image. The ML detector based on GPGWM-HMT is constructed as follows:

[0035]

[0036] e.4 Judge by threshold, the following is the watermark bit decoding:

[0037]

[0038] The present invention firstly performs multi-scale transformation on the image through NSCT, and selects the directional subband with the largest scale bivariance as the target subband; secondly, the selected target subband is divided into 8×8 non-overlapping blocks, and then a linear multiplicative embedding function is used and an inverse transformation is performed to obtain a watermarked image; then, a probability density function based on vector GPGWM-HMT is derived, and other subbands without watermarks are selected for modeling, shape parameters and position parameters are obtained, and a DEM parameter estimation method is used; finally, a decoder designed based on the maximum likelihood criterion is used, and the watermark position is extracted according to the decision threshold. Experimental results show that the present invention improves the performance of the decoder by making full use of multiple correlations between coefficients and constructing a statistical modeling based on the vector GPGWM-HMT distribution.

[0039] Compared with the prior art, the present invention has the following advantages:

[0040] First, the robust NSCT-FPCET amplitude is used to embed watermarks and construct detectors / decoders. In the digital image watermarking scheme based on statistical models, a new robust modeling object, NSCT-FPCET amplitude, is generated by a reasonable combination of NSCT and FPCET;

[0041] Second, a non-Gaussian vector HMT model is established. A vector GPGWM-HMT model in the NSCT-FPCET amplitude is established, which can fully capture the distribution characteristics and inter-scale and inter-directional dependencies of the research object;

[0042] Third, using the DEM parameter estimation method, the parameter estimation is more accurate and the time complexity of the algorithm is greatly reduced;

[0043] Fourth, a vector GPGWM-HMT based ML decoder in the NSCT-FPCET magnitude is designed, which can extract the specific watermark bits at the receiving end. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 This is a result diagram of verifying the non-Gaussianity of the difference subband in an embodiment of the present invention.

[0045] Figure 2 This is a PDF fitting result diagram of the vector GPGWM-HMT model of an embodiment of the present invention.

[0046] Figure 3 This is a watermarked result image of a grayscale image hiding a 256-bit watermark according to an embodiment of the present invention.

[0047] Figure 4 This is a grayscale image in an embodiment of the present invention containing a 256-bit watermark image and a 20-fold difference result of the original image.

[0048] Figure 5 This is a graph showing the 1024-bit watermark extraction results under various attacks according to an embodiment of the present invention.

[0049] Figure 6 The following is a flow chart of watermark embedding according to an embodiment of the present invention.

[0050] Figure 7 This is a flow chart of watermark extraction according to an embodiment of the present invention. DETAILED DESCRIPTION

[0051] The method of the present invention includes four stages: linear multiplicative watermark embedding, vector GPGWM-HMT distribution modeling, DEM parameter estimation and construction of maximum likelihood criterion decoder to extract watermark. Figure 6 , Figure 7 As shown, follow the steps below:

[0052] Convention: P represents the host image; P' represents the watermarked image; D represents the low-frequency subband obtained after the host image P is decomposed; H1 and H2 represent the high-frequency subbands of the first and second scales obtained after the host image is decomposed, respectively; L is the length of the watermark information; x i represents the original NSCT-FPCET amplitude coefficient; y irepresents the amplitude coefficient of NSCT-FPCET containing watermark; W l is the watermark information to be embedded, x ij It refers to the amplitude without watermark information at the i-th position of the j-th scale; y ij It refers to the amplitude of the watermark information at the i-th position at the j-th scale; α represents the watermark embedding strength; w l Indicates the lth watermark bit; Indicates the amplitude x ij The probability of state m, and it satisfies represents the state transition probability from the parent scale ρ(j) to the child scale j in the i-th tree, where the hidden state is transferred from ρ(m) to m; and represents the shape parameter in the m state; and Indicates that when x ij The scale parameter when the hidden state is m; H0 represents the assumption when the watermark information -1 is embedded in the subband; H1 represents the assumption when the watermark information +1 is embedded in the subband;

[0053] a. Initial Setup

[0054] Get the host image P of size M×N and initialize the variables;

[0055] b. Watermark Embedding

[0056] b.1 Perform a two-level decomposition operation of non-subsampled contourlet transform (NSCT) on the host image P of size M×N, and generate a low-frequency subband D and a first-scale high-frequency subband H1 and a second-scale high-frequency subband H2 while ensuring good resolution and ideal reconstruction effect. Then, H1 and H2 are decomposed into multiple directions, that is, eight high-frequency subbands in the first scale and four high-frequency subbands in the second scale are obtained;

[0057] b.2 In the high-frequency sub-band obtained by decomposition, select the directional sub-band with the largest variance of the second scale amplitude as the target sub-band for embedding the carrier image of the watermark, divide the selected target sub-band into 8×8 blocks respectively, ensure that the blocks do not overlap, and calculate the entropy value of each block. Based on the length L of the watermark to be embedded, select the first l high entropy blocks with higher entropy values, record the position of each block and sort the entropy values;

[0058] b.3 Select l high entropy blocks with the same length as the watermark L and perform 5th-order fast extreme complex exponential transform (FPCET) decomposition respectively to calculate the NSCT-FPCET amplitude domain. For each block, the dimension of the block becomes 7×7 after 5th-order FPCET calculation;

[0059] b.4 Embed a +1bit or -1bit watermark for each block accordingly, and perform the watermark information embedding operation according to the multiplicative embedding rule of the watermark, and embed the watermark information according to the linear multiplicative rule:

[0060] y i =(1+aω l )x i ;

[0061] b.5 Perform 5th-order FPCET reconstruction on the L moment amplitudes containing watermark information, change the dimension of the target block from 7×7 back to the original target block dimension, and then put it back to the original position in the target subband with the largest variance of the second scale;

[0062] b.6 Perform inverse NSCT transformation on all coefficients in the target subband, and finally obtain the watermark image P′ which has embedded watermark information;

[0063] c. Generalized Power Generalized Weibull Mixture (GPGWM)-Hidden Markov Tree (HMT) Statistical Modeling

[0064] c.1 Perform [2,3] NSCT secondary decomposition on the watermarked image P' to obtain a low-frequency subband D1 and a first-scale high-frequency subband h1 and a second-scale high-frequency subband h2. Then decompose h1 and h2 into multiple directions, that is, obtain eight directions of the first scale and four directions of the second scale high-frequency subbands.

[0065] c.2 Select the high-frequency sub-bands with the largest variance of the second scale and the first scale respectively, implement the block operation according to the strategy of "same size and non-overlapping", sort each block from large to small according to the obtained entropy value, and select the first L high entropy blocks for watermark information extraction;

[0066] c.3 Perform 5th-order FPCET calculation on the selected L high-entropy blocks, and obtain a 7×7 NSCT-FPCET amplitude coefficient matrix for each block; similarly, perform 8×8 non-overlapping block division on the four directional subbands in the second scale, and calculate 5th-order FPCET for each block to obtain the corresponding NSCT-FPCET amplitude. At this time, the NSCT-FPCET amplitude between the second scale and the first scale presents a 1-to-2 correspondence;

[0067] c.4 According to the definition of the target position, the NSCT-FPCET amplitude between the first scale and the second scale presents a 2-to-1 relationship. The GPGWM-HMT hybrid model is constructed for the selected amplitude coefficients for statistical modeling, where GPGWM is used to capture the non-Gaussian distribution characteristics of the amplitude and HMT is used to capture the scale correlation of the amplitude. The probability density function of GPGWM-HMT is defined as:

[0068]

[0069] d. DEM parameter estimation

[0070] d.1 Construction of the vector GPGWM-HMT model in the NSCT-FPCET amplitude;

[0071] d.2 Use parameter estimation method DEM algorithm;

[0072] The training samples are selected from all high-frequency subbands obtained by NSCT decomposition (all aspects of high-frequency subbands of scale 1 and scale 2), including noise-free and noise-containing training samples, with a total of T groups of data, i.e., T trees; the vector set constructed by binding the NSCT-FPCET amplitude coefficients of the same position i frequency domain block in the i-th tree of the T trees with eight different direction subbands in the first scale is The vector set formed by binding the NSCT-FPCET amplitude coefficients of the frequency domain blocks in the same position of the four sub-bands in different directions of the second scale Calculate the GPGWM-HMT model parameter set;

[0073] e. Construct a maximum likelihood decoder to extract the watermark

[0074] e.1 Using the ML (Maximum likelihood) decision rule at the receiving end, a digital watermark decoder based on vector GPGWM-HMT is designed in the NSCT-FPCET amplitude to extract the specific watermark bit. Considering watermark detection as a binary hypothesis testing problem, we have:

[0075]

[0076] e.2 Using the ML decision criterion, the MLD decoder designed based on the maximum likelihood criterion is as follows:

[0077]

[0078] in: And f Y (y i H0)=f X (y i ), f X (x) represents the probability density function of the model used;

[0079] The e.3 decoder adopts a blind watermark extraction method. The watermarked image maintains the same statistical distribution characteristics as the original image. The GPGWM-HMT model is used to represent the amplitude distribution characteristics of the watermarked image. The ML detector based on GPGWM-HMT is constructed as follows:

[0080]

[0081] e.4 Judge by threshold, the following is the watermark bit decoding:

[0082]

[0083] Experimental test and parameter setting:

[0084] The environment of this experiment is MATLAB R2018a, and the grayscale images are all 512×512. Download address:

[0085] http: / / decsai.ugr.es / cvg / dbimagenes / index.php.

[0086] Figure 1 This is a result diagram of verifying the non-Gaussianity of the difference subband in an embodiment of the present invention.

[0087] Figure 2 This is a diagram of the PDF fitting results of the vector GPGWM-HMT distribution of an embodiment of the present invention.

[0088] Figure 3 This is a watermarked result image of a grayscale image hiding a 256-bit watermark according to an embodiment of the present invention.

[0089] Figure 3 (a) Original image Lena; (b) Original image Mandrill; (c) Original image Boat;

[0090] (d) Watermarked Lena image; (e) Watermarked Mandrill image; (f) Watermarked Boat image.

[0091] Figure 4 This is a grayscale image in an embodiment of the present invention containing a 256-bit watermark image and a 20-fold difference result of the original image.

[0092] Figure 4 (a) Lena - 20 times difference image; (b) Mandrill - 20 times difference image; (c) Boat - 20 times difference image.

[0093] Figure 5 This is a graph showing the 1024-bit watermark extraction results under various attacks according to an embodiment of the present invention.

[0094] Figure 5 (a) Median filtering; (b) JPEG compression; (c) Additive white Gaussian noise; (d) Rotation.

[0095] Figure 5 References used:

[0096] M Amini,M O Ahmad,M N S Swamy.A robust multibit multiplicativewatermark decoder using vector-based hidden Markov model in waveletdomain.IEEE Transactions on Circuits&Systems for Video Technology,2018,28(2):402-413.

[0097] Xiang-yang WANG,Xin SHEN,Jia-lin TIAN,Pan-pan NIU,Hong-yingYANG.Statistical image watermark decoder using high-order differencecoefficients and bounded generalized Gaussian mixtures-based HMT.SignalProcessing,2022,192:108371.

[0098] Xiangyang Wang,Yupan Lin,Yixuan Shen,Panpan Niu.UDTCWT-PHFMs domainstatistical image watermarking using vector BW-Type R distribution.IEEETrans.on Circuits and Systems for Video Technology.DOI:10.1109 / TCSVT.2023.3252042.

Claims

1. A hybrid domain statistical image watermarking method based on vector HMT modeling, characterized in that Follow these steps: Convention: P represents the host image; P' represents the watermarked image; D represents the low-frequency subband obtained after the host image P is decomposed; H1 and H2 represent the high-frequency subbands of the first and second scales obtained after the host image is decomposed, respectively; L is the length of the watermark information; x i represents the original NSCT-FPCET amplitude coefficient; y i represents the amplitude coefficient of NSCT-FPCET containing watermark; W l is the watermark information to be embedded, x ij It refers to the amplitude without watermark information at the i-th position of the j-th scale; y ij It refers to the amplitude of the watermark information at the i-th position at the j-th scale; α represents the watermark embedding strength; w l Indicates the lth watermark bit; Indicates the amplitude x ij The probability of state m, and it satisfies represents the state transition probability from the parent scale ρ(j) to the child scale j in the i-th tree, where the hidden state is transferred from ρ(m) to m; and represents the shape parameter in the m state; and Indicates that when x ij The scale parameter when the hidden state is m; H0 represents the assumption when the watermark information -1 is embedded in the subband; H1 represents the assumption when the watermark information +1 is embedded in the subband; a. Initial Setup Get the host image P of size M×N and initialize the variables; b. Watermark Embedding b.1 Perform a non-subsampled contourlet transform two-level decomposition operation on the host image P of size M×N to generate a low-frequency subband D and a first-scale high-frequency subband H1 and a second-scale high-frequency subband H2, and then decompose H1 and H2 into multiple directions, that is, obtain eight high-frequency subbands in the first scale and four high-frequency subbands in the second scale; b.2 In the high-frequency sub-band obtained by decomposition, select the directional sub-band with the largest variance of the second scale amplitude as the target sub-band for embedding the carrier image of the watermark, divide the selected target sub-band into 8×8 blocks respectively, ensure that the blocks do not overlap, and calculate the entropy value of each block. Based on the length L of the watermark to be embedded, select the first l high entropy blocks with higher entropy values, record the position of each block and sort the entropy values; b.3 Select l high entropy blocks with the same length as the watermark L and perform 5th-order fast extreme complex exponential transform decomposition respectively to calculate the NSCT-FPCET amplitude domain. For each block, the dimension of the block becomes 7×7 after 5th-order FPCET calculation; b.4 Embed a +1bit or -1bit watermark for each block accordingly, and perform the watermark information embedding operation according to the multiplicative embedding rule of the watermark, and embed the watermark information according to the linear multiplicative rule: and i =(1+aω l )x i ; b.5 Perform 5th-order FPCET reconstruction on the L moment amplitudes containing watermark information, change the dimension of the target block from 7×7 back to the original target block dimension, and then put it back to the original position in the target subband with the largest variance of the second scale; b.6 Perform inverse NSCT transformation on all coefficients in the target subband, and finally obtain the watermark image P′ which has embedded watermark information; c. Generalized Power Generalized Weibull Mixture-Hidden Markov Tree Statistical Modeling c.1 Perform [2,3] NSCT secondary decomposition on the watermarked image P' to obtain a low-frequency subband D1 and a first-scale high-frequency subband h1 and a second-scale high-frequency subband h2. Then decompose h1 and h2 into multiple directions, that is, obtain eight directions of the first scale and four directions of the second scale high-frequency subbands. c.2 Select the high-frequency subbands with the largest variance of the second scale and the first scale respectively, implement the block operation according to the strategy of "same size and non-overlapping", sort each block from large to small according to the obtained entropy value, and select the first L high entropy blocks for watermark information extraction; c.3 Perform 5th-order FPCET calculation on the selected L high-entropy blocks, and obtain a 7×7 NSCT-FPCET amplitude coefficient matrix for each block; similarly, perform 8×8 non-overlapping block division on the four directional subbands in the second scale, and calculate 5th-order FPCET for each block to obtain the corresponding NSCT-FPCET amplitude. At this time, the NSCT-FPCET amplitude between the second scale and the first scale presents a 1-to-2 correspondence; c.4 According to the definition of the target position, the NSCT-FPCET amplitude between the first scale and the second scale presents a 2-to-1 relationship. The GPGWM-HMT hybrid model is constructed for the selected amplitude coefficients for statistical modeling, where GPGWM is used to capture the non-Gaussian distribution characteristics of the amplitude and HMT is used to capture the scale correlation of the amplitude. The probability density function of GPGWM-HMT is defined as: d. DEM parameter estimation d.1 Construction of the vector GPGWM-HMT model in the NSCT-FPCET amplitude; d.2 Use parameter estimation method DEM algorithm; The training samples are selected from all high-frequency subbands obtained by NSCT decomposition, including noise-free and noise-containing training samples, with a total of T groups of data, i.e., T trees; the vector set constructed by binding the NSCT-FPCET amplitude coefficients of the same position i frequency domain block in the i-th tree of the T trees with eight different direction subbands in the first scale is The vector set formed by binding the NSCT-FPCET amplitude coefficients of the frequency domain blocks in the same position of the four sub-bands in different directions of the second scale Calculate the GPGWM-HMT model parameter set; e. Construct a maximum likelihood decoder to extract the watermark e.1 Using the ML decision rule at the receiving end, a digital watermark decoder based on vector GPGWM-HMT is designed in the NSCT-FPCET amplitude to extract the specific watermark bit. Considering the watermark detection as a binary hypothesis testing problem, we have: e.2 Using the ML decision criterion, the MLD decoder designed based on the maximum likelihood criterion is as follows: in: And f Y (y i |H0)=f X (y i ), f X (x) represents the probability density function of the model used; The e.3 decoder adopts a blind watermark extraction method. The watermarked image maintains the same statistical distribution characteristics as the original image. The GPGWM-HMT model is used to represent the amplitude distribution characteristics of the watermarked image. The ML detector based on GPGWM-HMT is constructed as follows: e.4 Judge by threshold, the following is the watermark bit decoding:

Images