A Construction Method of Fast and Accurate Orthogonal-Fourier-Mellin Moment Zero Watermark for Ternary Numbers

Abstract

The present invention discloses a method for constructing a ternary number fast and accurate orthogonal-Fourier-Mellin moment zero watermark, which relates to the technical field of watermark construction. The method is characterized by comprising the following steps: S1: obtaining an original color image; S2: performing initial orthogonal Fourier-Mellin moment feature extraction and calculation to obtain initial matrix features; S3: a data optimization method; S31: orthogonal Fourier-Mellin moment feature extraction and parameter optimization; S32: a multi-stage optimization framework; S321: global optimization, using particle swarm optimization to optimize singular values and Fourier coefficients. The technical problem to be solved by the present invention is to provide a method for constructing a ternary number fast and accurate orthogonal-Fourier-Mellin moment zero watermark. The present invention aims to solve the above technical challenges. By integrating an optimization method with zero watermark technology, the reconstruction quality is optimized during the image reconstruction process, while ensuring the integrity and security of the watermark information, so as to achieve more efficient and secure image reconstruction and copyright protection.

Description

Technical Field

[0001] The present invention relates to the technical field of watermark construction, and more specifically, to a method for constructing a ternary fast and accurate orthogonal-Fourier-Mellin moment zero watermark. Background Art

[0002] Existing image reconstruction techniques often adopt traditional optimization methods. Although these methods can achieve good reconstruction effects in theory, they face problems such as large computational amounts and low efficiency in practical applications. Especially when dealing with large-scale or high-resolution images, the computational complexity of traditional algorithms increases sharply, resulting in a slow reconstruction process and requiring a large amount of storage resources.

[0003] It is difficult to balance global and local optimization: Existing image reconstruction optimization methods often rely on a single optimization algorithm, making it difficult to effectively balance global search and local fine-tuning during the optimization process. Although many algorithms can find relatively accurate global solutions, they cannot further improve the reconstruction quality during local fine optimization, resulting in the inability to achieve the best reconstruction effect.

[0004] Watermark information is easily lost or attacked: During the image reconstruction process, traditional watermark techniques, especially visible watermarks, are easily affected by compression, transformation, and reconstruction operations during image processing, resulting in the loss or tampering of watermark information. Even for zero watermark techniques, during the low-rank approximation or matrix partitioning of images, there are also problems of watermark information being damaged or unextractable.

[0005] Difficulty in combining zero watermark with image reconstruction: Current zero watermark techniques are usually applied alone for image protection, but how to simultaneously optimize image quality and watermark information protection during image reconstruction remains a difficult problem. Especially in complex image reconstruction algorithms, the embedding and extraction processes of watermark information may be affected by reconstruction errors and optimization algorithms, resulting in the unextractability or extraction errors of watermark information. Summary of the Invention

[0006] The technical problem to be solved by the present invention is to provide a method for constructing a ternary fast and accurate orthogonal-Fourier-Mellin moment zero watermark. The present invention aims to address the above technical challenges by integrating an optimization method (Hybrid Optimization) with zero watermark technology to optimize the reconstruction quality during the image reconstruction process, while ensuring the integrity and security of watermark information, thereby achieving more efficient and secure image reconstruction and copyright protection.

[0007] The present invention adopts the following technical solutions to achieve the invention purpose:

[0008] A method for constructing a ternary fast and accurate orthogonal-Fourier-Mellin moment zero watermark, characterized by comprising the following steps:

[0009] S1: Obtain the original color image;

[0010] S2: Perform initial orthogonal Fourier-Mellin moment feature extraction calculation to obtain the initial matrix features;

[0011] S3: Data optimization method;

[0012] S31: Orthogonal Fourier-Mellin moment feature extraction and parameter optimization;

[0013] S32: Multi-stage optimization framework;

[0014] S321: Global optimization, using particle swarm optimization for singular values and Fourier coefficients;

[0015] S322: Local optimization, using simulated annealing to optimize matrix partitioning;

[0016] S323: Fine optimization, using genetic algorithm to optimize low-rank approximation and local solutions;

[0017] S33: Optimization process integration;

[0018] S4: Adaptive adjustment and dynamic optimization to design an efficient and reliable zero-watermark algorithm;

[0019] S41: Zero-watermark construction;

[0020] S42: Zero-watermark verification.

[0021] As a further limitation of this technical solution, the specific steps of S2 are as follows:

[0022] Optimized orthogonal Fourier-Mellin moment calculation: In the polar coordinate system of orthogonal Fourier-Mellin moments, is the angular component, and the image duplication is defined as follows:

[0023] (1);

[0024] Where: is a non-negative integer;

[0025] represents the imaginary unit;

[0026] The radial basis function is of The polynomial expression of order

[0027] (2);

[0028] Where: is the coefficient of the polynomial, is the coefficient of the polynomial;

[0029] is weighted orthogonal within the range of :

[0030] (3)

[0031] where: represents the radial basis function of order k ; is the Kronecker symbol; is the normalization factor;

[0032] According to the properties of the radial basis function and the angular circular harmonic factor the basis function of the orthogonal Fourier-Mellin moments is orthogonal within the unit circle:

[0033] (4);

[0034] where: represents the orthogonality condition between the radial basis function and ;

[0035] represents the orthogonality condition between the angular basis function and ;

[0036] According to the theory of orthogonal function systems, the original image is approximately reconstructed using a finite number of orthogonal Fourier-Mellin moments, known to have the highest order and the maximum repetition of the orthogonal Fourier-Mellin moments, then the image is reconstructed as follows:

[0037] (4);

[0038] A quaternion is usually represented as where is the real part, is the imaginary part;

[0039] The conjugate of a quaternion is defined as:

[0040] (5);

[0041] In color image processing, each color channel is represented by the corresponding function to define a quaternion image function to represent a color image:

[0042] (6)

[0043] wherein: respectively represent the red, green, and blue channels of the image.

[0044] As a further limitation of this technical solution, the specific steps of S31 are:

[0045] The original input color image generates a set of moment features, i.e., the image after being processed by orthogonal Fourier-Mellin moments, denoted as:

[0046] (7);

[0047] wherein: represents the n th m angular coefficient;

[0048] These moment coefficients are affected by multiple parameters. Optimize the parameters to minimize the reconstruction error of the image reconstructed by orthogonal Fourier-Mellin moments. Let the parameter vector be:

[0049] (8);

[0050] The optimization objective is defined as the reconstruction error between the reconstructed image and the original image:

[0051] (9);

[0052] wherein: represents the image reconstructed after being processed by orthogonal Fourier-Mellin moments according to the parameter vector .

[0053] As a further limitation of this technical solution, the specific steps of S321 are:

[0054] Use particle swarm optimization to globally search the parameter space. Each particle represents a parameter candidate solution , and its velocity update formula is:

[0055] (10);

[0056] wherein: is the inertia weight; is the acceleration constant; is a random number between [0, 1]; is the historical best position of particle ; g is the global best position;

[0057] The position of the particle is updated as:

[0058] (11).

[0059] As a further limitation of this technical solution, the specific steps of S322 are as follows: Introduce the simulated annealing algorithm to locally refine the parameters. Let the current state be , generate a new state , the change in the objective function is , and the probability of accepting the new state is:

[0060] (12);

[0061] Where: is the current temperature.

[0062] As a further limitation of this technical solution, the specific steps of S323 are as follows:

[0063] The genetic algorithm is suitable for dealing with complex combinatorial optimization problems;

[0064] The genetic algorithm uses the crossover operation to combine the parameters from different individuals to generate new solutions; the mutation operation can introduce randomness to avoid falling into local optimal solutions;

[0065] Crossover operation:

[0066] (13);

[0067] Where: and are the selected parents, is the mixing coefficient;

[0068] Mutation operation:

[0069] (14)

[0070] Where: is the random perturbation vector;

[0071] By continuously iterating the selection, crossover, and mutation processes of the genetic algorithm, a globally optimal or approximately globally optimal parameter combination is finally obtained .

[0072] As a further limitation of this technical solution, the specific steps of S33 are as follows:

[0073] After the integrated optimization in the above three stages, an optimal parameter combination is finally obtained , and the orthogonal Fourier-Mellin moment matrix feature extraction process at this time is:

[0074] (15).

[0075] As a further limitation of this technical solution, the specific steps of S41 are as follows:

[0076] Zero-watermark construction: Let be the original ternary Fourier-Mellin moment image, be the original binary Logo image, and the zero-watermark construction process is as follows:

[0077] S411: Calculate the orthogonal Fourier-Mellin moments of the ternary Fourier-Mellin moment image to obtain moment values;

[0078] S412: Copy and expand the obtained moment values multiple times to obtain moment values, and then calculate the amplitude of the moments to construct an amplitude sequence with a length of ;

[0079] represents the amplitude of the th moment value calculated from the orthogonal Fourier-Mellin moments;

[0080] S413: Binarize the amplitude sequence to obtain a binarized amplitude sequence :

[0081] (16);

[0082] Where: is the th element after binarization, taking a value of 0 or 1;

[0083] is the binarization threshold, which is taken as the mean value of here;

[0084] S414: Convert the binarized amplitude sequence into a binary feature image with rows and columns:

[0085] (17);

[0086] S415: Use the XOR operation to embed the Logo image into the binary feature image to obtain a zero-watermark image :

[0087] (18).

[0088] As a further limitation of this technical solution, the specific steps of S42 are as follows:

[0089] Zero-watermark verification: Zero-watermark verification is mainly a Logo image detection process. Detect the Logo image in the Fourier-Mellin moments of the ternary number to be verified, so as to perform copyright verification. The specific process is as follows:

[0090] S421: Calculate the QFMM of the original image to be verified to obtain moment values;

[0091] S422: Copy and expand the above moment values multiple times to obtain moment values, and then calculate the amplitude of the moments to construct an amplitude sequence of length ;

[0092] S423: Binarize the amplitude sequence to obtain a binarized amplitude sequence :

[0093] (19);

[0094] Where: is the binarization threshold, and here the mean value of is taken;

[0095] S424: Convert the binarized amplitude sequence into a row column binary feature image;

[0096] (20);

[0097] S425: Perform an exclusive OR operation on the zero-watermark image and the binary feature image to obtain the Logo image to be detected :

[0098] (21).

[0099] Compared with the prior art, the advantages and positive effects of the present invention are:

[0100] ​​1. Improve the efficiency and quality of image reconstruction: By integrating the Hybrid Optimization method, which combines Particle Swarm Optimization (PSO), Simulated Annealing (SA), and Genetic Algorithm (GA), this invention finely tunes each key parameter of the reconstruction process in multiple optimization stages, thus effectively improving the accuracy and efficiency of image reconstruction and solving the problems of high computational complexity and low efficiency in traditional optimization algorithms.

[0101] 2. Achieve an effective balance between global optimization and local optimization: Through a multi-stage optimization framework, PSO is used for global optimization to ensure the optimal selection of important global features (such as Fourier coefficients and singular values) during the reconstruction process; SA is used for local optimization to optimize the matrix block division and region selection of the image; GA further finely tunes the low-rank approximation to optimize local details. By this method, an effective balance can be found between global search and local optimization, thereby improving the quality of image reconstruction.

[0102] 3. Ensure the security and integrity of zero-watermark information: By integrating the optimization method, this invention embeds zero-watermark information during the image reconstruction process, ensuring that while optimizing the image quality, the watermark information is not affected and its security and extractability can be guaranteed during the post-image reconstruction process. By precisely controlling the embedding method of the watermark information, common problems such as loss or tampering in traditional watermarking techniques are avoided.

[0103] 4. Optimize the combination of zero-watermark and image reconstruction: Another objective of this invention is to achieve a deep integration of zero-watermark information and the image reconstruction process, such that when the image undergoes low-rank approximation and other optimization processes, the retention of watermark information and the optimization of image quality can be guaranteed simultaneously. By combining the integrated optimization method, the watermark information can be protected at different optimization stages, ensuring that the quality of the reconstructed image is not affected by the watermark information while guaranteeing the integrity and extractability of the watermark information. BRIEF DESCRIPTION OF THE DRAWINGS

[0104] Figure 1 It is a schematic diagram of the overall process of this invention.

[0105] Figure 2 It is a schematic diagram of the local three-dimensional structure of this invention Figure 1 . DETAILED DESCRIPTION OF THE INVENTION

[0106] The following combines the drawings to describe in detail a specific embodiment of this invention, but it should be understood that the protection scope of this invention is not limited by the specific embodiment.

[0107] This invention includes the following steps:

[0108] S1: Obtain the original color image.

[0109] S2: Calculate the initial Orthogonal Fourier Melling Moments (OFMM) features to obtain the initial matrix features.

[0110] The specific steps of S2 are as follows:

[0111] Optimized version of the Orthogonal Fourier Melling Moments calculation: In the polar coordinate system of the Orthogonal Fourier Melling Moments, is the angular component, representing the angle formed from the 0-degree position (usually the positive x-axis direction) of the polar coordinate system to a certain point, and its value range is , the image repetition degree is defined as follows:

[0112] (1);

[0113] Where: is a non-negative integer;

[0114] represents the imaginary unit;

[0115] The radial basis function is (is the radial component, representing the distance from the origin to a certain point, and its value range is usually [0, 1], because in image processing, the image is usually normalized to within the unit circle) The order polynomial expression of is:

[0116] (2);

[0117] Where: is the coefficient of the polynomial, is the coefficient of the polynomial; represents The weighted factor of the order polynomial at each point, and the coefficient determines the shape and size of the radial basis function;

[0118] Within the range of is weighted orthogonal:

[0119] (3)

[0120] Where: represents The radial basis function of order, which is a polynomial function about the radial component; is the Kronecker symbol; is the normalization factor;

[0121] According to the radial basis function and the angular circular harmonic factor The basis functions of the orthogonal Fourier-Mellin moments are orthogonal within the unit circle:

[0122] (4);

[0123] where: represents the orthogonality condition between the radial basis function and ; that is , when , ; when , , this condition indicates that the radial basis functions of different orders are orthogonal within the interval .

[0124] represents the orthogonality condition between the angular basis function and , that is , when ; ; when ; , this condition indicates that the angular basis functions of different angular frequencies are orthogonal within the interval . Only when and , the right side of the above equation is , otherwise it is 0, which indicates that the basis functions of the orthogonal Fourier-Mellin moments form a complete orthogonal basis function system within the unit circle.

[0125] According to the theory of orthogonal function systems, the original image is approximately reconstructed using a finite number of orthogonal Fourier-Mellin moments. Given the orthogonal Fourier-Mellin moments with the highest order and the maximum repetition , then the image is reconstructed as follows:

[0126] (4);

[0127] Trinomial is a simpler mathematical structure than complex numbers and quaternions, usually expressed as , where is the real part, is the imaginary part; different from quaternions, trinomial has only two imaginary units and , which makes its calculation more convenient. The multiplication of trinomial also follows specific rules, where the multiplication of imaginary units is non-commutative.

[0128] The conjugate of a ternion is defined as:

[0129] (5);

[0130] In color image processing, each color channel (such as red, green, blue) is represented by a corresponding function to represent, and a ternion image function is defined to represent a color image:

[0131] (6)

[0132] Where: respectively represent the red, green, and blue channels of the image.

[0133] To calculate the ternion Fourier-Mellin moments (TFOFM) of a color image, the ternion representation of each color channel will be used for the Fourier-Mellin transform. Compared with quaternions, the advantage of ternions in this calculation lies in their lower dimension and simpler multiplication rules, making the calculation process more efficient. By combining the Fourier-Mellin transform results of each channel, the Fourier-Mellin moments of the image in ternion form can be obtained.

[0134] Traditional factorial calculation methods are usually implemented by direct multiplication, that is, iterating from 1 to and multiplying one by one. However, for very large numbers, traditional calculation methods will consume a large amount of computing time and storage space, especially when the factorial number such as becomes very large.

[0135] The advantage of using ternions instead of quaternions is that the simplification of the ternion structure can reduce the requirements for computational volume and storage space. Since ternions have only two imaginary units, the multiplication rules involved in their calculation are relatively simple, and the required storage space and computing resources are significantly lower than those of quaternions. This enables ternions to provide higher efficiency and less resource consumption when dealing with large-scale calculations.

[0136] When dealing with large-scale calculations, especially those involving ternions, factorial and exponential operations may still face complexity problems. To solve this problem, an optimization algorithm is proposed, which combines the advantages of ternions, can effectively reduce the computing time, optimize the storage usage, and maintain high precision when dealing with large numbers. Through the ternion form of Fourier-Mellin moments (TFOFM), we can not only simplify the calculation process but also improve the stability and efficiency of the calculation.

[0137] Compared with quaternions, ternions offer the following main advantages:

[0138] The ternion only contains two imaginary parts. Compared with the quaternion (which contains three imaginary parts), its multiplication rule is relatively simple, reducing the computational complexity.

[0139] Due to the simpler structure of the ternion, less storage space is required during the calculation process. Especially when dealing with a large amount of data, this can significantly reduce the consumption of storage resources.

[0140] The simplified structure of the ternion can effectively control the precision loss during large number calculations, avoiding precision errors caused by insufficient computing resources.

[0141] In summary, the calculation method of Fourier-Mellin moments combined with the ternion provides higher computational efficiency and lower resource consumption, and is especially suitable for large-scale image processing and other fields that require efficient mathematical operations.

[0142] S3: Data optimization method.

[0143] The implementation idea of the Hybrid Optimization method in image reconstruction. The Hybrid Optimization method can combine the advantages of multiple optimization algorithms, play their respective strengths in different optimization stages, and achieve a balance between global optimality and local fine optimization. Specifically, by combining the applications of Particle Swarm Optimization (PSO), Simulated Annealing (SA), and Genetic Algorithm (GA) in image reconstruction, the following specific implementation stages can be achieved.

[0144] S31: Orthogonal Fourier-Mellin moment feature extraction and parameter optimization;

[0145] The specific steps of the above S31 are as follows:

[0146] The original input color image After being processed by the orthogonal Fourier-Mellin moment, a set of moment features, that is, an image, is generated , denoted as:

[0147] (7);

[0148] Where: represents the n - th order, m angular coefficient;

[0149] These moment coefficients are affected by multiple parameters, such as the scale factor of the Fourier basis function, the singular value truncation threshold, etc. To improve the stability and robustness of the orthogonal Fourier-Mellin moment matrix, these parameters need to be optimized so that the error of the image reconstructed by the orthogonal Fourier-Mellin moment is minimized. Let the parameter vector be:

[0150] (8);

[0151] The optimization objective is defined as the reconstruction error between the reconstructed image and the original image:

[0152] (9);

[0153] Where: represents the image reconstructed after orthogonal Fourier-Mellin moment processing according to the parameter vector .

[0154] In the integrated optimization framework, it is first necessary to clarify the optimization objective and constraints. The main objective of image reconstruction is usually to minimize the reconstruction error, that is, to optimize the difference between the reconstructed image and the original image, which may involve the following specific objectives:

[0155] Singular value optimization: In low-rank approximation, select the appropriate number and magnitude of singular values.

[0156] Fourier coefficient optimization: Optimize the frequency components of the image to ensure that the details and textures of the image are accurately retained.

[0157] Matrix block optimization: Image block optimization is used to improve computational efficiency and reduce storage requirements.

[0158] Low-rank approximation: Optimize the matrix rank to find the optimal rank that balances compression and accuracy.

[0159] Orthogonal Fourier-Mellin moment feature extraction and parameter optimization.

[0160] S32: Multi-stage optimization framework.

[0161] Divide the optimization problem into different stages, and use different optimization algorithms in each stage to optimize for specific optimization objectives. Specifically, it can be carried out according to the following steps.

[0162] S321: Global optimization, using Particle Swarm Optimization (PSO) to optimize singular values and Fourier coefficients.

[0163] The specific steps of the above S321 are as follows:

[0164] Initialize the particle swarm. Each particle represents a potential solution, and the dimension of the particle corresponds to the number of optimized singular values, Fourier coefficients, etc. The position of the particle represents different combinations of Fourier coefficients and singular values, and the velocity of the particle represents how to adjust these parameters to approach the global optimal solution. By calculating the fitness of each particle (such as reconstruction error, image quality, etc.), the particle continuously updates its position and velocity until the convergence condition is reached.

[0165] The particle swarm optimization (PSO) is used to globally search the parameter space, and each particle represents a candidate solution for the parameters. , and its velocity update formula is:

[0166] (10);

[0167] Where: is the inertia weight; are the acceleration constants; is a random number between [0, 1]; is the particle 's historical best position; g is the global best position;

[0168] The position of the particle is updated as:

[0169] (11).

[0170] The goal of the PSO stage is to globally search the parameter space as much as possible so that the objective function obtains a lower value, providing a better starting point for subsequent local optimization.

[0171] S322: Local optimization, using simulated annealing (SA) to optimize the matrix partitioning.

[0172] The specific steps of the said S322 are as follows: After obtaining a preliminary better solution by PSO, the simulated annealing algorithm is introduced to locally refine the parameters. Let the current state be , and a new state is generated, and the change in the objective function is . The probability of accepting the new state is:

[0173] (12);

[0174] Where: is the current temperature; by gradually reducing the temperature, it is possible to avoid falling into a local optimum and gradually converge to the optimal solution. SA mainly adjusts the local image blocks or regional parameters here, making the orthogonal Fourier-Mellin moment features of each region more refined and further reducing the reconstruction error. The SA algorithm is used to optimize the matrix partitioning of the image (for example, select which regions to process finely). For an image, it may be divided into multiple small blocks, and each block can be processed independently. SA optimization can select the most important regions for high-precision reconstruction, reducing the computational amount and storage requirements.

[0175] The annealing process of SA simulates the physical annealing process, accepting inferior solutions with a probability to avoid the algorithm falling into a local optimum. In this way, SA can find a better matrix partitioning strategy.

[0176] At this stage, SA can help adjust the structure of image blocks, select appropriate image blocks for key optimization, and reduce redundant calculations.

[0177] S323: Fine optimization, using the Genetic Algorithm (GA) to optimize low-rank approximation and local solutions.

[0178] The specific steps of S323 are as follows:

[0179] Advantages of GA: The genetic algorithm is suitable for dealing with complex combinatorial optimization problems; GA can search the solution space through population iteration, and locally refine the solution through operations such as crossover and mutation, so as to achieve the purpose of optimization.

[0180] Implementation method:

[0181] GA further optimizes the low-rank approximation at this stage, for example, selects the most appropriate combination of singular values, or adjusts the matrix rank (determines how many singular values to retain).

[0182] For different block partitioning schemes of the corresponding matrix, GA selects the optimal combination of singular values or the optimal matrix block partitioning strategy through global search and local fine operations.

[0183] The genetic algorithm uses the crossover operation to combine parameters from different individuals to generate new solutions; the mutation operation can introduce a certain degree of randomness to avoid falling into local optimal solutions;

[0184] Crossover operation:

[0185] (13);

[0186] Where: and are the selected parents, is the mixing coefficient;

[0187] Mutation operation:

[0188] (14)

[0189] Where: is the random perturbation vector;

[0190] By continuously iterating the selection, crossover, and mutation processes of the genetic algorithm, a globally optimal or approximately globally optimal parameter combination is finally obtained . GA can select appropriate combinations of singular values during the low-rank approximation process, optimize singular value decompositions of different orders, and thus improve the quality of image reconstruction.

[0191] S33: Optimization process integration.

[0192] The specific steps of S33 are as follows:

[0193] After the integration and optimization of the above three stages, an optimal parameter combination is finally obtained. At this time, the process of orthogonal Fourier-Mellin moment matrix feature extraction is as follows:

[0194] (15).

[0195] Using this optimal feature matrix, the embedding of zero-watermark information can be carried out. The specific process is as follows:

[0196] For Copy and expand it and perform amplitude binarization to construct a binary feature image;

[0197] Use the exclusive OR operation to combine the Logo information to be embedded with the binary feature image to generate a zero-watermark image;

[0198] In the extraction stage, the zero-watermark is restored through the same orthogonal Fourier-Mellin moment processing and optimized parameters, and then the watermark information is verified.

[0199] This integrated optimization method ensures that the orthogonal Fourier-Mellin moment feature extraction reaches the best state in both global parameter search and local detail adjustment, making the embedding and extraction processes more robust, efficient, and having high concealment and anti-attack capabilities.

[0200] S4: Adaptive adjustment and dynamic optimization to design an efficient and reliable zero-watermark algorithm.

[0201] In addition, in order to improve the flexibility and efficiency of the optimization process, an adaptive adjustment strategy can be considered. For example, in different optimization stages, adjust the parameters of each optimization algorithm (such as the number of particles in PSO, the temperature in SA, the population size in GA, etc.) to make the optimization process achieve the best effect under different conditions.

[0202] The integrated optimization method (Hybrid Optimization) can handle different types of optimization problems in the process of image reconstruction by combining the advantages of PSO, SA, and GA. PSO is used for global search to optimize singular values and Fourier coefficients, SA is used for local optimization of matrix partitioning, and GA is used for fine-tuning low-rank approximation and local details of the image. Such a multi-stage and multi-algorithm optimization framework can balance global and local optimization and ensure the quality and efficiency of image reconstruction.

[0203] Combined with the optimized TOFMM, an efficient and reliable zero-watermark algorithm is designed. This algorithm mainly includes two processes: the construction of zero-watermark and the verification of zero-watermark. The addition of the improved TOFMM makes the entire zero-watermark algorithm more efficient and reliable. The implementation process and specific steps of the algorithm will be introduced in detail below. Figure 2 Shows the overall flowchart.

[0204] S41: Zero-watermark construction.

[0205] The specific steps of S41 are as follows:

[0206] Zero-watermark construction: Let be the original quaternion Fourier-Mellin moment image, be the original binary Logo image, and the zero-watermark construction process is as follows:

[0207] S411: Calculate the orthogonal Fourier-Mellin moments of the quaternion Fourier-Mellin moment image to obtain moment values;

[0208] S412: Copy and expand the obtained moment values multiple times to obtain moment values, and then calculate the amplitude of the moments to construct an amplitude sequence with a length of ;

[0209] represents the amplitude (modulus) of the th moment value calculated from the orthogonal Fourier-Mellin moments; it reflects the spatial and frequency domain characteristics of the image. By calculating the amplitude sequence a ( i ), the continuously varying moment features can be transformed into an amplitude sequence form suitable for watermark embedding, facilitating subsequent amplitude binarization and zero-watermark construction.

[0210] S413: Binarize the amplitude sequence to obtain a binarized amplitude sequence :

[0211] (16);

[0212] where: is the th element after binarization, taking values of 0 or 1, and is used to construct a binary feature image;

[0213] is the binarization threshold, which is taken as the mean value of here;

[0214] S414: The binarized amplitude sequence become row binary feature image of columns:

[0215] (17);

[0216] S415: Embed the Logo image into the binary feature image using the exclusive OR operation to obtain the zero-watermark image :

[0217] (18).

[0218] S42: Zero-watermark verification.

[0219] The specific steps of S42 are as follows:

[0220] Zero-watermark verification: Zero-watermark verification is mainly the Logo image detection process. Detect the Logo image in the ternary Fourier-Mellin moments to be verified, so as to perform copyright verification. The specific process is as follows:

[0221] S421: Calculate the QFMM of the original image to be verified to obtain moment values;

[0222] S422: Copy and expand the above moment values multiple times to obtain moment values, and then calculate the amplitude of the moments to construct an amplitude sequence with a length of ;

[0223] S423: Binarize the amplitude sequence to obtain the binarized amplitude sequence :

[0224] (19);

[0225] where: is the binarization threshold, and here the mean value of is taken;

[0226] S424: Change the binarized amplitude sequence to row column binary feature image;

[0227] (20);

[0228] S425: The zero-watermark image ​​​Perform an exclusive OR operation with the binary feature image to obtain the Logo image to be detected :

[0229] (21).

[0230] The present invention proposes a method for constructing a ternary fast and accurate orthogonal - Fourier - Mellin moment zero - watermark, and its core innovations are reflected in the following aspects:

[0231] First, in terms of the optimization of OFMM moment feature extraction, by introducing optimization means such as the prime - factorization factorial algorithm, the calculation efficiency and stability of OFMM moments are greatly improved. The extracted matrix features not only meet strict orthogonality requirements but also remain invariant under geometric transformations such as rotation and scaling, which provides a solid foundation for subsequent watermark embedding and extraction. The protection of this part mainly focuses on the calculation process of OFMM moments, parameter settings, and their specific application methods in zero - watermark embedding.

[0232] Secondly, the present invention designs a feature expansion and binarization processing scheme. For the matrix data extracted from OFMM, a processing flow of replication expansion and amplitude binarization is adopted to convert the original matrix features into binary carriers suitable for watermark embedding and ensure high robustness under different image conditions. The innovation points and protection focuses of this part lie in the specific implementation of feature expansion, binarization strategy, and threshold - selection method.

[0233] In addition, in terms of the covert embedding and accurate extraction of zero - watermark, the present invention uses exclusive OR operation to covertly encode the Logo to be embedded, thus forming a zero - watermark image. When extracting the watermark, the information embedded is accurately restored by using the same steps as the initial OFMM processing flow, and through compensation processing for geometric transformations (such as rotation and scaling), the integrity of the watermark information is ensured. In this regard, the encoding, decoding strategies, and compensation methods in the embedding and extraction processes constitute important protected contents of the present invention.

[0234] Finally, the robustness design of the overall system is also a major highlight of the present invention. Utilizing the rotation and scaling invariance naturally possessed by the OFMM matrix, the constructed watermark system can still maintain a high watermark detection rate after the image undergoes common processing such as compression, noise, and filtering, while achieving high invisibility. The protection focus of this part lies in the overall architecture based on the combination of the OFMM matrix and zero - watermark and its technical strategies against various signal - processing attacks.

[0235] Through the comprehensive application of the above - mentioned key technologies, the present invention not only realizes the efficient protection of image copyright but also takes into account computational efficiency, robustness, and invisibility, providing a safe, reliable, and commercially promising image - watermark solution for practical applications.

[0236] The specific embodiments of the present invention disclosed above are only examples. However, the present invention is not limited thereto, and any variations that can be conceived by those skilled in the art shall fall within the protection scope of the present invention.

Claims

1. A ternary fast and accurate orthogonal-Fourier-Mellin moment zero watermark construction method, characterized in that: The following steps are involved: S1: Get the original color image; S2: Initial orthogonal Fourier Mellin moment feature extraction calculation to obtain initial matrix features; S3: Data optimization methods; S31: Orthogonal Fourier Mellin moment feature extraction and parameter optimization; S32: Multi-stage optimization framework; S321: Global optimization, using particle swarm optimization for singular values ​​and Fourier coefficients; S322: local optimization, using simulated annealing to optimize matrix partitioning; S323: Fine Optimization, using genetic algorithms to optimize low-rank approximations and local solutions; S33: Optimize process integration; S4: Adaptive adjustment and dynamic optimization, design of efficient and reliable zero watermark algorithm; S41: zero watermark construction; S42: Zero watermark verification.

2. The ternary fast and accurate orthogonal-Fourier-Mellin moment zero watermark construction method according to claim 1 is characterized by: The specific steps of S2 are: Optimized orthogonal Fourier-Mellin moment calculation: Orthogonal Fourier-Mellin moment polar coordinate system, is the angular component, image Repeatability is defined as follows: （1）； in: is a non-negative integer; represents an imaginary unit; Radial Basis Function yes of The polynomial expression is: （2）； in: are the coefficients of the polynomial, are the coefficients of the polynomial; exist is weighted orthogonal in the range: （3） in: express k Radial basis function of order; is the Kronecker symbol; is the normalization factor; According to radial basis function Angular harmonic factor The properties of the orthogonal Fourier-Mellin moments Orthogonal inside the unit circle: （4）； in: represents radial basis function and Orthogonality conditions between ; represents the angular basis function and Orthogonality conditions between ; According to the orthogonal function system theory, the original image Reconstructed approximately using a finite number of orthogonal Fourier Mellin moments, it is known that the highest order and maximum repetition The orthogonal Fourier-Mellin moments of Refactor as follows: （4）； Ternaries are usually represented as ,in is the real part, is the imaginary part; The conjugate of a ternary is defined as: （5）； In color image processing, each color channel is processed by a corresponding function To represent, define a ternary image function to represent the color image: （6） in: Represent the red, green and blue channels of the image respectively.

3. The ternary fast and accurate orthogonal-Fourier-Mellin moment zero watermark construction method according to claim 2 is characterized by: The specific steps of S31 are: Original input color image After processing by orthogonal Fourier-Mellin moments, a set of moment features, namely images, is generated. , recorded as: （7）； in: Indicates n Stage, m Angular coefficient; These moment coefficients are affected by multiple parameters, and the parameters are optimized so that the image error reconstructed by the orthogonal Fourier-Mellin moment is minimized. Let the parameter vector be: （8）； The optimization objective is defined as the reconstruction error between the reconstructed image and the original image: （9）； in: Represents the parameter vector The reconstructed image after processing with orthogonal Fourier-Mellin moments.

4. The ternary fast and accurate orthogonal-Fourier-Mellin moment zero watermark construction method according to claim 2 is characterized by: The specific steps of S321 are: Particle swarm optimization is used to perform a global search of the parameter space, where each particle represents a candidate solution for a parameter. , and its speed update formula is: （10）； in: is the inertia weight; is the acceleration constant; is a random number between [0,1]; It is a particle The best historical position of g; g is the global best position; The particle's position is updated as: （11）。 5. The ternary fast and accurate orthogonal-Fourier-Mellin moment zero watermark construction method according to claim 4 is characterized by: The specific steps of S322 are: The simulated annealing algorithm is introduced to locally refine the parameters. Suppose the current state is , generating a new state , the objective function changes to , the probability of accepting the new state is: （12）； in: is the current temperature.

6. The ternary fast and accurate orthogonal-Fourier-Mellin moment zero watermark construction method according to claim 4 is characterized by: The specific steps of S323 are: Genetic algorithms are suitable for dealing with complex combinatorial optimization problems; Genetic algorithms use crossover operations to combine parameters from different individuals to generate new solutions; mutation operations can introduce randomness to avoid falling into local optimal solutions; Crossover operation: （13）； in: and is the selected parent, is the mixing coefficient; Mutation operation: （14） in: is the random perturbation vector; Through the continuous iteration of the genetic algorithm selection, crossover and mutation process, a global optimal or approximately global optimal parameter combination is finally obtained. .

7. The ternary fast and accurate orthogonal-Fourier-Mellin moment zero watermark construction method according to claim 6 is characterized by: The specific steps of S33 are: After the above three stages of integrated optimization, an optimal parameter combination is finally obtained. , the orthogonal Fourier Mellin moment matrix feature extraction process at this time is: （15）。 8. The ternary fast and accurate orthogonal-Fourier-Mellin moment zero watermark construction method according to claim 7 is characterized by: The specific steps of S41 are: Zero watermark construction: Assume is the original ternary Fourier-Mellin moment image, For the original binary Logo image, the zero watermark construction process is as follows: S411: Calculate the ternary Fourier-Mellin moment image The orthogonal Fourier-Mellin moments of Moment values; S412: What you will get The moment values ​​are replicated and expanded multiple times to obtain moment values, and then calculate the amplitude of the moment to construct the length The amplitude sequence of ; represents the first The magnitude of the moment value; S413: The amplitude sequence Binarization, get the binary amplitude sequence : （16）； in: After binarization, elements, the value is 0 or 1; is the binarization threshold, here we take The mean of S414: Binarize the amplitude sequence becomes OK Binary feature image of the column: （17）； S415: Use XOR operation to convert the Logo image Embedded into binary feature image Get the zero watermark image : （18）。 9. The ternary fast and accurate orthogonal-Fourier-Mellin moment zero watermark construction method according to claim 8, characterized in that: The specific steps of S42 are: Zero watermark verification: Zero watermark verification is mainly a logo image detection process. Detect the Logo image in To verify copyright, the specific process is as follows: S421: Calculate the original image to be verified The QFMM gives Moment values; S422: The above The moment values ​​are replicated and expanded multiple times to obtain moment values, and then calculate the amplitude of the moment to construct the length The amplitude sequence of ; S423: The amplitude sequence Binarization, get the binary amplitude sequence : （19）； in: is the binarization threshold, here we take The mean of S424: Binarize the amplitude sequence becomes OK Binary feature image of the column; （20）； S425: Zero watermark image With binary feature image Perform XOR operation to obtain the Logo image that needs to be detected : （21）。

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