Color image watermarking method based on vector binary statistical modeling
By combining DNST and QIPHFMs for color image watermarking, and utilizing the vector BIUPWD model and MLE method, a novel watermark decoder is constructed. This solves the problems of insufficient robustness of embedding position and low detection accuracy in existing technologies, achieving efficient embedding and decoding of color image watermarks and improving robustness and imperceptibility.
Patent Information
- Application Number
- CN202511459402.2
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-10-13
- Publication Date
- 2025-12-23
AI Technical Summary
Existing digital image watermarking methods suffer from insufficient robustness at the embedding location, limited statistical model fitting ability, and low watermark detection and decoding accuracy. Furthermore, there is limited research on color image watermarking algorithms, making it difficult to maintain a good balance between robustness, imperceptibility, and watermark capacity.
A color image watermarking method based on vector binary statistical modeling is adopted. By fusing DNST and QIPHFMs, and combining the vector BIUPWD model and MLE method, a novel watermark decoder is constructed to achieve effective embedding, detection and decoding of watermark information.
It improves the robustness and imperceptibility of watermark embedding, enhances algorithm accuracy, expands the application range of color images, adapts to more practical application scenarios, and improves practical value.
Smart Images
Figure CN121190291A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of information hiding and digital watermarking technology, and particularly relates to a color image watermarking method based on vector binary statistical modeling. Background Technology
[0002] Digital image watermarking technology is a key technology in the field of information security, widely used in scenarios such as copyright protection, content authentication, and anti-counterfeiting and traceability. The performance of watermarking algorithms is mainly measured by three core indicators: robustness, imperceptibility, and watermark capacity. However, these indicators often have interdependent relationships, and how to effectively maintain a good balance between the robustness, imperceptibility, and capacity of digital watermarks is a hot research topic in this field.
[0003] Digital image watermarking methods based on statistical models have demonstrated unique advantages in this research process. By establishing accurate statistical models, they can effectively regulate the balance among the three factors, providing new research ideas and breakthrough directions for the innovative development of watermarking technology. However, existing research methods still face many challenges, such as insufficient robustness of embedding location, limited fitting ability of statistical models, low accuracy of watermark detection and decoding, and limited research on color image watermarking algorithms. Summary of the Invention
[0004] The purpose of this invention is to provide a color image watermarking method based on vector binary statistical modeling, which aims to solve the problems mentioned in the background art.
[0005] The present invention is implemented as follows: a color image watermarking method based on vector binary statistical modeling includes the following steps: Step 1: Construct the amplitude coefficient domain of DNST-QIPHFMs; The three color channels of the original color image are subjected to a second-order non-subsampled shear wave transform (DNST). The resulting high-frequency subbands are combined and divided into blocks. The fifth-order quaternion improved polar harmonic Fourier moments (QIPHFMs) are calculated for each coefficient block to generate DNST-QIPHFMs amplitude coefficient blocks. Step 2: Watermark embedding; The three color channels of the original color image are decomposed into two-level DNST, specific sub-bands are selected and divided into blocks, the block entropy values are calculated and sorted, and the high-entropy blocks are selected for QIPHFMs calculation to obtain amplitude coefficient blocks. The watermark information is embedded into the amplitude coefficient blocks using the multiplicative rule. Then, QIPHFMs reconstruction and DNST inverse transformation are performed to obtain the watermarked color image. Step 3: Modeling the vector binary unit power-Weiboll distribution (BIUPWD); The three color channels of the watermarked color image are decomposed into two-level DNST, the high-frequency subbands are combined and divided into blocks, the block entropy value is calculated, the high-entropy blocks are selected for QIPHFMs decomposition to obtain amplitude coefficient blocks, the amplitude coefficient blocks are bound by scale and direction to construct vector training samples, and the parameters of the training sample set are estimated using the vector BIUPWD model and the maximum likelihood estimation (MLE) method to complete the modeling. Step 4: Parameter estimation and optimization; Based on the probability density function of the vector BIUPWD model, the likelihood function is established using the MLE method, and the model parameters are iteratively optimized using the gradient descent method. Step 5: Watermark Decoding and Decision; Based on the vector BIUPWD model and the estimated model parameters, and in accordance with the binary hypothesis theory and the maximum likelihood decision criterion, a watermark decoder expression is constructed and the decision variables are calculated, and finally the embedded watermark information is extracted.
[0006] In a further technical solution, step 1 includes the following specific steps: Step 1.1: Extract the original color image Divided into , and The three color channels are subjected to two-level DNST transformations, resulting in 8 high-frequency subbands at the first scale and 4 high-frequency subbands at the second scale. Step 1.2: Arrange all the DNST high-frequency subbands obtained in the above steps according to... , and The channels are combined and divided into 4096 non-overlapping coefficient blocks of size 8×8; Step 1.3: Perform fifth-order QIPHFMs calculations on each coefficient block to generate a 6×11 size DNST-QIPHFMs amplitude coefficient block.
[0007] In a further technical solution, step 2 includes the following specific steps: Step 2.1: Process the original color image of , and The three color channels are decomposed into two-level DNST, each yielding 8 high-frequency subbands at the first scale and 4 high-frequency subbands at the second scale. Step 2.2: Divide the high-frequency subbands after DNST decomposition into... , and The channels are combined, and the DNST subband with two scales and third-order orientation is selected as the target subband. The target subband is divided into uniform and non-overlapping blocks. The entropy value of each block is calculated and sorted in descending order according to the entropy value. The high-entropy block is selected according to the watermark sequence length L. Step 2.3: Perform fifth-order QIPHFMs calculations on the selected high-entropy blocks to obtain L DNST-QIPHFMs amplitude coefficient blocks. Modify the DNST-QIPHFMs amplitude coefficient blocks using the multiplicative rule, embedding the watermark information into the high-entropy blocks to obtain DNST-QIPHFMs amplitude coefficient blocks containing watermark information. ; in, The amplitude coefficient without embedded watermark. The amplitude coefficient after embedding the watermark; For the first to be embedded The watermark information is in bits, and the watermark length is L. For watermark embedding strength, It is a sequence of equally probable random numbers consisting of "1" and "-1"; Step 2.4: Perform QIPHFMs reconstruction on the watermarked DNST-QIPHFMs amplitude coefficient block to obtain the watermarked DNST high-entropy block and restore it to its position in the original high-frequency subband; Step 2.5: Divide the watermarked DNST amplitude subbands into... , and The three channels are split and each undergoes an inverse DNST transform to obtain a watermarked color image. .
[0008] In a further technical solution, step 3 includes the following specific steps: Step 3.1: Process the watermarked color image The three color channels are decomposed into two-level DNST, each yielding 8 high-frequency subbands at the first scale and 4 high-frequency subbands at the second scale. Step 3.2: Combine all the DNST high-frequency subbands obtained in the above steps according to the three color channels, divide them into 4096 non-overlapping coefficient blocks of size 8×8, and calculate the entropy value of each block; Step 3.3: Perform fifth-order QIPHFMs decomposition on the selected high-entropy blocks to obtain L 6×11 size DNST-QIPHFMs amplitude coefficient blocks; Step 3.4: Bind the obtained DNST-QIPHFMs amplitude coefficient block partition scale to construct vector training samples, the first set of vector samples The second group of vector samples is bound together with the DNST-QIPHFMs amplitude coefficient blocks at the same positions in the 1st, 3rd, 5th, and 7th directions of the first scale. The amplitude coefficients of DNST-QIPHFMs at the same positions in the four directions of the second scale (1st, 3rd, 5th, and 7th) are bound together. Step 3.5: Estimate the organized vector training sample set using the vector BIUPWD model and the MLE method to model the amplitude coefficients of DNST-QIPHFMs. The probability density function corresponding to the vector BIUPWD model is... The calculation formula is as follows: ; in, and Indicates shape parameters, Represents the correlation parameter. and These represent vector samples of different scales and directions, respectively. and The covariance matrix formed.
[0009] In a further technical solution, step 4 includes the following specific steps: Step 4.1: Estimate the parameters of the probability density function corresponding to the vector BIUPWD model using the MLE method, and determine the likelihood function using its PDF formula: ; The log-likelihood function is: ; Step 4.2: Gradient calculation and optimization. First, define the negative log-likelihood function as the objective function: ; Then, take the partial derivative of the log-likelihood function with respect to each parameter: ; Step 4.3: Iteratively update the parameters using gradient descent. Repeat the iteration to update the parameters until the log-likelihood function converges.
[0010] In a further technical solution, step 5 includes the following specific steps: Step 5.1: For each location where a watermark is embedded, determine the binary hypothesis theory. and These represent the embedded watermark information as "+1" and "-1" respectively, that is: ; Step 5.2: Based on the maximum likelihood (ML) decision criterion, design the watermark decoder as follows: ; in, Represents the log-likelihood ratio. This represents the probability density function of the vector BIUPWD model; Step 5.3: Combining the vector binary condition and the parameters of the vector BIUPWD model obtained in Step 4, input them into the PDF formula of the vector BIUPWD model to obtain the expression for the watermark decoder: ; in, and This represents two sets of vector magnitude coefficient samples. , , and They represent and derivative , ; Step 5.4: Construct the decision variables; ; ; in, Represents the expression for the decision variable. This represents the expression for the decision threshold. Step 5.5: Based on the maximum likelihood principle, make the following decision for each bit value: ; in, The extracted digital watermark information.
[0011] The color image watermarking method based on vector binary statistical modeling provided in this invention has the following beneficial effects: (1) Integration of DNST and QIPHFMs: By combining the advantages of DNST and QIPHFMs, the robustness and imperceptibility of watermark embedding are improved, while the overall accuracy of the algorithm is enhanced.
[0012] (2) Vector BIUPWD model: Based on the statistical characteristics of the amplitude coefficients of DNST-QIPHFMs, the vector BIUPWD model is used for modeling. By constructing vectors, the dependency relationship between coefficients is effectively solved and the modeling accuracy is improved.
[0013] (3) Constructing a new watermark decoder: Based on the vector BIUPWD model and ML decision rules, a new watermark decoder was designed, which can effectively extract watermark information.
[0014] (4) Application for color images: Taking color images as the research object, the application scope of the algorithm is expanded. Compared with the traditional color image watermarking method, this method can effectively decode color images, adapt to a wider range of practical application scenarios, and further enhance its practical value. Attached Figure Description
[0015] Figure 1 The figure shows the robustness test results for the amplitude coefficient domain of DNST-QIPHFMs. Figure 2 The watermarked result image with a 1024-bit watermark embedded in a standard color test image; Figure 3 This is an image showing the difference between the original color image and the image after embedding a 1024-bit watermark, multiplied by a factor of 20. Figure 4 Images showing the results of 1024-bit watermark extraction under various attacks. Detailed Implementation
[0016] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.
[0017] The specific implementation of the present invention will be described in detail below with reference to specific embodiments.
[0018] An embodiment of the present invention provides a color image watermarking method based on vector binary statistical modeling, comprising the following steps: Step 1: Construct the amplitude coefficient domain of DNST-QIPHFMs; Step 1.1: Extract the original color image Divided into , and The three color channels are subjected to two-stage non-subsampled shear wave transform (DNST) to obtain 8 high-frequency subbands at the first scale and 4 high-frequency subbands at the second scale. Step 1.2: Arrange all the DNST high-frequency subbands obtained in the above steps according to... , and The channels are combined and divided into 4096 non-overlapping coefficient blocks of size 8×8; Step 1.3: Perform fifth-order quaternion-modified polar harmonic Fourier moments (QIPHFMs) calculations on each coefficient block to generate a 6×11 size DNST-QIPHFMs amplitude coefficient block.
[0019] Step 2: Watermark embedding; Step 2.1: Process the original color image of , and The three color channels are decomposed into two-level DNST, each yielding 8 high-frequency subbands at the first scale and 4 high-frequency subbands at the second scale. Step 2.2: Divide the high-frequency subbands after DNST decomposition into... , and The channels are combined, and the DNST subband with two scales and third-order orientation is selected as the target subband. The target subband is divided into uniform and non-overlapping blocks. The entropy value of each block is calculated and sorted in descending order according to the entropy value. The high-entropy block is selected according to the watermark sequence length L. Step 2.3: Perform fifth-order QIPHFMs calculations on the selected high-entropy blocks to obtain L DNST-QIPHFMs amplitude coefficient blocks. Modify the DNST-QIPHFMs amplitude coefficient blocks using the multiplicative rule, embedding the watermark information into the high-entropy blocks to obtain DNST-QIPHFMs amplitude coefficient blocks containing watermark information. ; in, The amplitude coefficient without embedded watermark. The amplitude coefficient after embedding the watermark; For the first to be embedded The watermark information is in bits, and the watermark length is L. For watermark embedding strength, It is a sequence of equally probable random numbers consisting of "1" and "-1"; Step 2.4: Perform QIPHFMs reconstruction on the watermarked DNST-QIPHFMs amplitude coefficient block to obtain the watermarked DNST high-entropy block and restore it to its position in the original high-frequency subband; Step 2.5: Divide the watermarked DNST amplitude subbands into... , and The three channels are split and each undergoes an inverse DNST transform to obtain a watermarked color image. .
[0020] Step 3: Modeling the vector binary unit power-Weiboll distribution (BIUPWD); Step 3.1: Process the watermarked color image The three color channels are decomposed into two-level DNST, each yielding 8 high-frequency subbands at the first scale and 4 high-frequency subbands at the second scale. Step 3.2: Combine all the DNST high-frequency subbands obtained in the above steps according to the three color channels, divide them into 4096 non-overlapping coefficient blocks of size 8×8, and calculate the entropy value of each block; Step 3.3: Perform fifth-order QIPHFMs decomposition on the selected high-entropy blocks to obtain L 6×11 size DNST-QIPHFMs amplitude coefficient blocks; Step 3.4: Bind the obtained DNST-QIPHFMs amplitude coefficient block partition scale to construct vector training samples, the first set of vector samples The second group of vector samples is bound together with the DNST-QIPHFMs amplitude coefficient blocks at the same positions in the 1st, 3rd, 5th, and 7th directions of the first scale. The amplitude coefficients of DNST-QIPHFMs at the same positions in the four directions of the second scale (1st, 3rd, 5th, and 7th) are bound together. Step 3.5: Estimate the organized vector training sample set using the vector BIUPWD model and the MLE method to model the amplitude coefficients of DNST-QIPHFMs. The probability density function corresponding to the vector BIUPWD model is... The calculation formula is as follows: ; in, and Indicates shape parameters, Represents the correlation parameter. and These represent vector samples of different scales and directions, respectively. and The covariance matrix formed.
[0021] Step 4: Parameter estimation and optimization; Step 4.1: Estimate the parameters of the probability density function corresponding to the vector BIUPWD model using the MLE method, and determine the likelihood function using its PDF formula: ; The log-likelihood function is: ; Step 4.2: Gradient calculation and optimization. First, define the negative log-likelihood function as the objective function: ; Then, take the partial derivative of the log-likelihood function with respect to each parameter: ; Step 4.3: Iteratively update the parameters using gradient descent. Repeat the iteration to update the parameters until the log-likelihood function converges.
[0022] Step 5: Decoding and Decision; Step 5.1: For each location where a watermark is embedded, determine the binary hypothesis theory. and These represent the embedded watermark information as "+1" and "-1" respectively, that is: ; Step 5.2: Based on the maximum likelihood (ML) decision criterion, design the watermark decoder as follows: ; in, Represents the log-likelihood ratio. This represents the probability density function of the vector BIUPWD model; Step 5.3: Combining the vector binary condition, and based on the vector BIUPWD model parameters obtained in Step 4, substituting them into the PDF formula of the vector BIUPWD model, we obtain the expression for the watermark decoder: ; in, and This represents two sets of vector magnitude coefficient samples. , , and They represent and derivative , ; Step 5.4: Construct the decision variables; ; ; in, Represents the expression for the decision variable. This represents the expression for the decision threshold. Step 5.5: Based on the maximum likelihood principle, make the following decision for each bit value: ; in, The extracted digital watermark information refers to the digital watermark information successfully extracted from the image by this method.
[0023] As a preferred embodiment of the present invention, relevant experimental tests were conducted, and the experimental parameters were set as follows: The experimental tests were conducted in the Matlab R2014a environment. The test images were from public image libraries and were all 512×512 color images, including commonly used standard images such as Barbara, Boat, Horse and Landscape, which are representative.
[0024] Figure 1 The graph shows the robustness test results for the amplitude coefficient domain of DNST-QIPHFMs. Figure 1 b and Figure 1 The conventional attack shown in c, and Figure 1 d、 Figure 1 e and Figure 1 Under the geometric attack conditions shown in f, the statistical distribution characteristics of the amplitude coefficients are similar to... Figure 1 The overall characteristics remain consistent under no-attack conditions, demonstrating that the amplitude coefficients of DNST-QIPHFMs have good stability.
[0025] Figure 2 The watermarked result image of a standard color test image with a 1024-bit watermark embedded. Figure 3 This is an image showing the difference between the original color image and the image after embedding a 1024-bit watermark, multiplied by a factor of 20. (Through...) Figure 2 and Figure 3 It can be intuitively seen that there is no obvious difference between the images before and after the watermark is embedded, which verifies the imperceptibility of this method from a subjective visual perspective.
[0026] Figure 4 Images showing the results of extracting 1024-bit watermarks under various attacks. Figure 4 Experimental results show that the watermark decoder designed in this invention can effectively extract the target watermark information, and the extracted flower image watermark can maintain high integrity. Under different types of attacks, the algorithm yields low BER values, further verifying that the proposed watermark algorithm has good robustness.
[0027] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.
Claims
1. A color image watermarking method based on vector binary statistical modeling, characterized in that, Includes the following steps: Step 1: Construct the amplitude coefficient domain of DNST-QIPHFMs; The three color channels of the original color image are subjected to a second-order DNST transformation. The resulting high-frequency subbands are combined and divided into blocks. A fifth-order QIPHFMs calculation is performed on each coefficient block to generate DNST-QIPHFMs amplitude coefficient blocks. Step 2: Watermark embedding; The three color channels of the original color image are decomposed into two-level DNST, specific sub-bands are selected and divided into blocks, the block entropy values are calculated and sorted, and the high-entropy blocks are selected for QIPHFMs calculation to obtain amplitude coefficient blocks. The watermark information is embedded into the amplitude coefficient blocks using the multiplicative rule. Then, QIPHFMs reconstruction and DNST inverse transformation are performed to obtain the watermarked color image. Step 3: Vector BIUPWD modeling; The three color channels of the watermarked color image are decomposed into two-level DNST, the high-frequency subbands are combined and divided into blocks, the block entropy value is calculated, the high-entropy blocks are selected for QIPHFMs decomposition to obtain amplitude coefficient blocks, the amplitude coefficient blocks are bound by scale and direction to construct vector training samples, and the parameters of the training sample set are estimated using the vector BIUPWD model and MLE method to complete the modeling. Step 4: Parameter estimation and optimization; Based on the probability density function of the vector BIUPWD model, the likelihood function is established using the MLE method, and the model parameters are iteratively optimized using the gradient descent method. Step 5: Watermark Decoding and Decision; Based on the vector BIUPWD model and the estimated model parameters, and in accordance with the binary hypothesis theory and the maximum likelihood decision criterion, a watermark decoder expression is constructed and the decision variables are calculated, and finally the embedded watermark information is extracted.
2. The color image watermarking method based on vector binary statistical modeling according to claim 1, characterized in that, Step 1 includes the following specific steps: Step 1.1: Extract the original color image Divided into , and The three color channels are subjected to two-level DNST transformations, resulting in 8 high-frequency subbands at the first scale and 4 high-frequency subbands at the second scale. Step 1.2: Arrange all the DNST high-frequency subbands obtained in the above steps according to... , and The channels are combined and divided into 4096 non-overlapping coefficient blocks of size 8×8; Step 1.3: Perform fifth-order QIPHFMs calculations on each coefficient block to generate a 6×11 size DNST-QIPHFMs amplitude coefficient block.
3. The color image watermarking method based on vector binary statistical modeling according to claim 2, characterized in that, Step 2 includes the following specific steps: Step 2.1: Process the original color image of , and The three color channels are decomposed into two-level DNST, each yielding 8 high-frequency subbands at the first scale and 4 high-frequency subbands at the second scale. Step 2.2: Divide the high-frequency subbands after DNST decomposition into... , and The channels are combined, and the DNST subband with two scales and third-order orientation is selected as the target subband. The target subband is divided into uniform and non-overlapping blocks. The entropy value of each block is calculated and sorted in descending order according to the entropy value. The high-entropy block is selected according to the watermark sequence length L. Step 2.3: Perform fifth-order QIPHFMs calculations on the selected high-entropy blocks to obtain L DNST-QIPHFMs amplitude coefficient blocks. Modify the DNST-QIPHFMs amplitude coefficient blocks using the multiplicative rule, embedding the watermark information into the high-entropy blocks to obtain DNST-QIPHFMs amplitude coefficient blocks containing watermark information. ； in, The amplitude coefficient without embedded watermark. The amplitude coefficient after embedding the watermark; For the first to be embedded The watermark information is in bits, and the watermark length is L. For watermark embedding strength, It is a sequence of equally probable random numbers consisting of "1" and "-1"; Step 2.4: Perform QIPHFMs reconstruction on the watermarked DNST-QIPHFMs amplitude coefficient block to obtain the watermarked DNST high-entropy block and restore it to its position in the original high-frequency subband; Step 2.5: Divide the watermarked DNST amplitude subbands into... , and The three channels are split and each undergoes an inverse DNST transform to obtain a watermarked color image. .
4. The color image watermarking method based on vector binary statistical modeling according to claim 3, characterized in that, Step 3 includes the following specific steps: Step 3.1: Process the watermarked color image The three color channels are decomposed into two-level DNST, each yielding 8 high-frequency subbands at the first scale and 4 high-frequency subbands at the second scale. Step 3.2: Combine all the DNST high-frequency subbands obtained in the above steps according to the three color channels, divide them into 4096 non-overlapping coefficient blocks of size 8×8, and calculate the entropy value of each block; Step 3.3: Perform fifth-order QIPHFMs decomposition on the selected high-entropy blocks to obtain L 6×11 size DNST-QIPHFMs amplitude coefficient blocks; Step 3.4: Bind the obtained DNST-QIPHFMs amplitude coefficient block partition scale to construct vector training samples, the first set of vector samples The second group of vector samples is bound together with the DNST-QIPHFMs amplitude coefficient blocks at the same positions in the 1st, 3rd, 5th, and 7th directions of the first scale. The amplitude coefficients of DNST-QIPHFMs at the same positions in the four directions of the second scale (1st, 3rd, 5th, and 7th) are bound together. Step 3.5: Estimate the organized vector training sample set using the vector BIUPWD model and the MLE method to model the amplitude coefficients of DNST-QIPHFMs. The probability density function corresponding to the vector BIUPWD model is... The calculation formula is as follows: ； in, and Indicates shape parameters, Represents the correlation parameter. and These represent vector samples of different scales and directions, respectively. and The covariance matrix formed.
5. The color image watermarking method based on vector binary statistical modeling according to claim 4, characterized in that, Step 4 includes the following specific steps: Step 4.1: Estimate the parameters of the probability density function corresponding to the vector BIUPWD model using the MLE method, and determine the likelihood function using its PDF formula: ； The log-likelihood function is: ； Step 4.2: Gradient calculation and optimization. First, define the negative log-likelihood function as the objective function: ； Then, take the partial derivative of the log-likelihood function with respect to each parameter: ； Step 4.3: Iteratively update the parameters using gradient descent. Repeat the iteration to update the parameters until the log-likelihood function converges.
6. The color image watermarking method based on vector binary statistical modeling according to claim 5, characterized in that, Step 5 includes the following specific steps: Step 5.1: For each location where a watermark is embedded, determine the binary hypothesis theory. and These represent the embedded watermark information as "+1" and "-1" respectively, i.e.: ； Step 5.2: Based on the maximum likelihood decision criterion, design the watermark decoder as follows: ； in, Represents the log-likelihood ratio. This represents the probability density function of the vector BIUPWD model; Step 5.3: Combining the vector binary condition, and based on the vector BIUPWD model parameters obtained in Step 4, substituting them into the PDF formula of the vector BIUPWD model, we obtain the expression for the watermark decoder: ； in, and This represents two sets of vector magnitude coefficient samples. , , and They represent and derivative , ; Step 5.4: Construct the decision variables; ； ； in, Represents the expression for the decision variable. This represents the expression for the decision threshold. Step 5.5: Based on the maximum likelihood principle, make the following decision for each bit value: ； in, The extracted digital watermark information.