Three-dimensional robust watermarking method and system based on norm ordering and median compensation

Abstract

This application relates to a robust 3D watermarking method, system, computer device, and storage medium based on norm sorting and median compensation. The method includes a watermark embedding stage and a watermark extraction stage. The watermark embedding stage includes: aligning the centroids of the 3D model, calculating the vertex norms, sorting by norms and dividing into K equal groups, and calculating the quantization step size; modulating the embedded watermark bits using dithered quantization indexing on the mean norm of each group, superimposing offsets, and scaling the radial vector proportionally; and storing the global norm median, group median, and parameters after recovering the centroids. The watermark extraction stage includes: calculating the norms after centroid alignment, scaling the quantization step size by the ratio of the current norm median to the reference norm median; calculating the norm median of each group after sorting and grouping, comparing it with the scaled original group median and performing half-compensation, and then quantizing to determine and recover the watermark. This application ensures high watermark capacity and low geometric distortion while exhibiting high robustness against rotation, scaling, translation, reordering, and smoothing attacks.

Description

Technical Field

[0001] This application relates to the field of model digital watermarking technology, and in particular to a three-dimensional robust watermarking method, system, computer device and storage medium based on norm sorting and median compensation. Background Technology

[0002] With the rapid development of 3D scanning, computer-aided design, and virtual reality technologies, 3D geometric models have become important digital assets, and the demand for their copyright protection is increasingly urgent. 3D watermarking technology achieves copyright authentication by embedding invisible identifier information into the model. In traditional 3D watermarking methods, spatial domain methods directly modify vertex coordinates, but their robustness to geometric attacks is limited; frequency domain methods are computationally complex and sensitive to vertex order; methods based on geometric invariants struggle to resist combined attacks such as rotation, scaling, translation, and vertex reordering. In particular, when 3D models undergo uniform scaling or Laplacian smoothing, the extraction accuracy of existing watermarking methods drops significantly, making it difficult to achieve stable robustness against these attacks while maintaining high watermark capacity and low geometric distortion. Therefore, how to improve robustness against geometric attacks such as scaling and smoothing during the watermark extraction stage, while maintaining high watermark capacity and low geometric distortion during the embedding stage, has become a pressing technical problem to be solved in this field. Summary of the Invention

[0003] Therefore, it is necessary to provide a robust three-dimensional watermarking method, system, computer device, and storage medium based on norm sorting and median compensation that can resist scaling and smoothing attacks while maintaining high watermark capacity and low geometric distortion, in order to address the above-mentioned technical problems.

[0004] Firstly, this application provides a three-dimensional robust watermark embedding method based on norm sorting and median compensation. The method includes: The vertex coordinates of the input 3D mesh model are preprocessed with centroid alignment. The centroid alignment preprocessing includes calculating the mean of all vertex coordinates as the centroid, subtracting the centroid from each vertex coordinate to obtain the set of vertices with centroid alignment. Calculate the Euclidean norm of each vertex to the origin, sort all vertices in ascending order of norm, and divide the sorted vertices into K groups, where K is the length of the bit sequence to be embedded with the watermark. Calculate the global standard deviation of all vertex norms, and multiply the embedding strength factor by the global standard deviation as the quantization step size; For the k-th group, where The mean norm of all vertices in the k-th group is calculated. Dithered quantization index modulation is used, and the mean is quantized and modulated according to the value of the k-th watermark bit to obtain the norm mean offset of the k-th group. When the value of the watermark bit is 0, the quantization target is the nearest quantization grid point, and when the value of the watermark bit is 1, the quantization target is the nearest quantization grid center point. The norm mean offset is uniformly superimposed on the norm of all vertices in the k-th group, and the watermark is embedded by scaling the radial vector of each vertex proportionally. Add the centroid back to the coordinates of all vertices to obtain the 3D model with the watermark embedded. The global norm median of the 3D model in the centroid-aligned coordinate system is calculated as the reference median, and the median of the vertex norm in each group is recorded as the group median statistic. The reference median, group median statistic, quantization step size, number of groups K, and embedding intensity factor are stored as auxiliary information for watermark extraction.

[0005] In one embodiment, the jitter quantization index modulation includes: Let the norm mean of the k-th group be . The watermark bits are Quantization step size is Then the jitter value Quantified mean Norm mean offset .

[0006] In one embodiment, the embedding strength factor is determined according to a preset invisibility constraint.

[0007] In one embodiment, the number of groups K is such that each group contains at least a preset number of vertices.

[0008] Secondly, this application provides a three-dimensional robust watermark extraction method based on norm ranking and median compensation. The method includes: Obtain watermark extraction auxiliary information, which includes reference median, group median statistic, quantization step size, number of groups K, and embedding strength factor; The vertex coordinates of the extracted 3D model are preprocessed by centroid alignment, and the Euclidean norm of each centroid-aligned vertex to the origin is calculated. Calculate the median of the norms of all vertices in the current model, use the ratio of the median to the reference median as a scaling factor, and use the product of the quantization step size and the scaling factor as the effective step size; Sort all vertices by norm from smallest to largest, and divide them into K equal groups; For the k-th group, where Calculate the median of the norms of all vertices in the k-th group as the median of the current group; Based on formula Calculate the offset corresponding to the original group midpoint value of the k-th group, where The value in the current group, The value in the original group, The scaling factor is the scaling factor; if Then, half compensation is performed to obtain the adjusted median. ;otherwise, ;in The effective step size is... The compensation coefficient; Calculate the distances from the adjusted median to the nearest even quantization point and the nearest odd quantization point respectively. If the distance from the adjusted median to the nearest even quantization point is less than or equal to the distance to the nearest odd quantization point, then the k-th watermark bit is determined to be 0; otherwise, it is 1.

[0009] In one embodiment, the nearest even quantization point is The nearest odd quantization point is .

[0010] In one embodiment, the compensation coefficient for the half-compensation is 0.5.

[0011] Thirdly, this application provides a three-dimensional robust watermarking system based on norm sorting and median compensation. The system includes a watermark embedding device for the three-dimensional robust watermark embedding method based on norm sorting and median compensation and / or a watermark extraction device for performing the three-dimensional robust watermark extraction method based on norm sorting and median compensation. The watermark embedding device includes: The centroid alignment module is used to perform centroid alignment preprocessing on the vertex coordinates of the input 3D mesh model. The centroid alignment preprocessing includes: calculating the mean of all vertex coordinates as the centroid, subtracting the centroid from each vertex coordinate to obtain the set of vertices after centroid alignment. The norm calculation module is used to calculate the Euclidean norm of each vertex to the origin, sort all vertices according to the norm from smallest to largest, and divide the sorted vertices into K groups, where K is the length of the bit sequence to be embedded with the watermark. The quantization step size calculation module is used to calculate the global standard deviation of the norm of all vertices, and multiply the embedding strength factor by the global standard deviation as the quantization step size; The quantization modulation module is used to calculate the mean of the norms of all vertices within the k-th group, where k = 0, 1, ..., K-1, using dithered quantization index modulation. The mean is quantized and modulated according to the value of the k-th watermark bit to obtain the norm mean offset of the k-th group. Specifically, when the watermark bit value is 0, the quantization target is the nearest quantization grid point; when the watermark bit value is 1, the quantization target is the nearest quantization grid center point. The radial scaling module is used to uniformly superimpose the norm mean offset onto the norm of all vertices in the k-th group, and to achieve watermark embedding by scaling the radial vector of each vertex proportionally. The centroid recovery module is used to add the centroid back to the coordinates of all vertices to obtain the 3D model with the watermark embedded. The auxiliary information generation module is used to calculate the global norm median of the three-dimensional model in the centroid aligned coordinate system as the reference median, and record the median of the vertex norm in each group as the group median statistic. The reference median, group median statistic, quantization step size, number of groups K and embedding intensity factor are stored as auxiliary information for watermark extraction. The watermark extraction device includes: The auxiliary information acquisition module is used to acquire auxiliary information for watermark extraction, which includes reference median, group median statistic, quantization step size, number of groups K, and embedding strength factor. The norm calculation module is used to perform centroid alignment preprocessing on the vertex coordinates of the extracted 3D model and calculate the Euclidean norm of each centroid-aligned vertex to the origin. The scaling adaptive module is used to calculate the median of the norm of all vertices of the current model, use the ratio of the median to the reference median as the scaling factor, and use the product of the quantization step size and the scaling factor as the effective step size. The sorting and grouping module is used to sort all vertices in ascending order of their norm and divide them into K equal groups; The group midpoint calculation module is used for the k-th group, where Calculate the median of the norms of all vertices in the k-th group as the median of the current group; The proportional offset compensation module is used for formula-based... Calculate the offset corresponding to the original group midpoint value of the k-th group, where The value in the current group, The value in the original group, The scaling factor is the scaling factor; if Then, half compensation is performed to obtain the adjusted median. ;otherwise, ;in The effective step size is... The compensation coefficient; The quantization decision module is used to calculate the distances from the adjusted median to the nearest even quantization point and the nearest odd quantization point, respectively. If the distance from the adjusted median to the nearest even quantization point is less than or equal to the distance to the nearest odd quantization point, the k-th watermark bit is determined to be 0; otherwise, it is 1.

[0012] Fourthly, this application provides a computer device. The computer device includes a memory and a processor. The memory stores a computer program, and the processor, when executing the computer program, implements the aforementioned three-dimensional robust watermark embedding method based on norm sorting and median compensation and / or the aforementioned three-dimensional robust watermark extraction method based on norm sorting and median compensation.

[0013] Fifthly, this application provides a computer-readable storage medium. The computer-readable storage medium stores a computer program thereon, which, when executed by a processor, implements the aforementioned three-dimensional robust watermark embedding method based on norm sorting and median compensation and / or the aforementioned three-dimensional robust watermark extraction method based on norm sorting and median compensation.

[0014] The aforementioned robust 3D watermarking method, system, computer equipment, and storage medium based on norm sorting and median compensation eliminate the influence of translation components by performing centroid alignment preprocessing on the 3D model. Utilizing the rotation invariance of vertex norms and the operation of grouping by norm sorting, the watermark carrier naturally possesses rotation invariance and vertex reordering invariance. An adaptive quantization step size is obtained by calculating the global norm standard deviation and multiplying it by the embedding strength factor, ensuring the quantization process matches the model scale without manual parameter tuning. Dithered quantization index modulation embeds the watermark bits into the norm mean of each group, and the radial vector is scaled proportionally, achieving covert watermark embedding while maintaining the model's topology. Furthermore, the scaling factor is estimated using the ratio of the stored reference median to the current model norm median, adaptively adjusting the quantization step size to achieve complete resistance to uniform scaling attacks. A proportional offset compensation mechanism calculates the offset between the current group median and the original scaled group median, and half-compensates for the systematic norm contraction caused by smoothing attacks, improving robustness against Laplacian smoothing attacks. Attached Figure Description

[0015] Figure 1 This is an application environment diagram of a three-dimensional robust watermarking method based on norm sorting and median compensation in one embodiment; Figure 2 This is a flowchart illustrating a three-dimensional robust watermark embedding method based on norm sorting and median compensation in one embodiment. Figure 3This is a flowchart illustrating a three-dimensional robust watermark extraction method based on norm sorting and median compensation in one embodiment. Figure 4 This is a structural block diagram of a three-dimensional robust watermarking system based on norm sorting and median compensation in one embodiment; Figure 5 This is an internal structural diagram of a computer device in one embodiment. Detailed Implementation

[0016] To make the objectives, technical solutions, and advantages of this application clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments.

[0017] The three-dimensional robust watermarking method based on norm sorting and median compensation provided in this application can be applied to, for example... Figure 1 In the application environment shown, terminal 102 communicates with server 104 via a network. A data storage system can store the data that server 104 needs to process. The data storage system can be integrated on server 104 or placed in the cloud or on other network servers. This embodiment of the three-dimensional robust watermarking method based on norm sorting and median compensation includes a watermark embedding stage and a watermark extraction stage. In the watermark embedding stage, terminal 102 or server 104 acquires the original three-dimensional mesh model, performs centroid alignment, norm sorting grouping, adaptive quantization step size calculation, and jitter quantization index modulation, embeds the watermark bits into the norm mean of each group, and proportionally scales the radial vector to generate a three-dimensional model with embedded watermark and watermark extraction auxiliary information. In the watermark extraction stage, terminal 102 or server 104 acquires the three-dimensional model to be detected and the auxiliary information, performs norm median scaling adaptive estimation, proportional offset compensation, and quantization decision to recover the watermark bit sequence. The terminal 102 can be, but is not limited to, various personal computers, laptops, workstations, graphics processing workstations, 3D scanner hosts, smartphones, tablets, IoT devices, and portable wearable devices. IoT devices can include smart speakers, smart TVs, smart air conditioners, and smart in-vehicle devices; portable wearable devices can include smartwatches, smart bracelets, and head-mounted devices. The server 104 can be implemented using a standalone server, a GPU-accelerated server, a distributed computing cluster, or a server cluster composed of multiple servers. The terminal 102 or server 104 can also be a virtual instance in a cloud environment. Those skilled in the art will understand that… Figure 1 The application environment shown is only an example. In actual deployment, watermark embedding and extraction can be completed on the same device or performed by different devices.

[0018] Firstly, in the process of embedding watermarks in 3D models, traditional methods often directly modify vertex coordinates or frequency domain coefficients, lacking the utilization of the model's global statistical characteristics. This makes the embedded watermark easily lost when subjected to attacks such as scaling and smoothing. Especially in scenarios requiring resistance to geometric attacks such as rotation, translation, and vertex reordering, existing embedding methods struggle to construct geometrically invariant watermark carriers. Therefore, in one embodiment, such as... Figure 2 As shown, a robust three-dimensional watermark embedding method based on norm sorting and median compensation is presented, and this method is applied to... Figure 1 Taking terminal 102 as an example, the explanation includes: Step S1: Perform centroid alignment preprocessing on the vertex coordinates of the input 3D mesh model. The centroid alignment preprocessing includes calculating the mean of all vertex coordinates as the centroid, subtracting the centroid from each vertex coordinate to obtain the set of vertices after centroid alignment.

[0019] Specifically, suppose the input 3D mesh model has N vertices, and the original coordinates of the i-th vertex are... , Calculate the arithmetic mean of the coordinates of all vertices to obtain the centroid vector. ,in , , Then subtract the centroid vector from the coordinates of each vertex, i.e. After centroid alignment, the origin of the model is located at the geometric center of the model, thus eliminating the influence of translation transformation on subsequent watermarking operations. The centroid alignment operation itself does not change the relative geometry of the model; it only translates the coordinate reference system to the centroid position.

[0020] Step S2: Calculate the Euclidean norm of each vertex to the origin, sort all vertices in ascending order of norm, and divide the sorted vertices into K groups, where K is the length of the bit sequence to be embedded with the watermark.

[0021] Specifically, for each centroid-aligned vertex Calculate its Euclidean norm to the origin (0,0,0). All vertices are arranged according to The values ​​are sorted in ascending order to obtain a sorted vertex index sequence. This sequence is then divided into K consecutive groups, each containing approximately the same number of vertices. Here, K is the length of the watermark bit sequence to be embedded, i.e., the watermark capacity is K bits. Since the sorting depends only on the relative size of the norm values ​​and is independent of the original storage order of the vertices, this grouping result naturally resists vertex reordering attacks.

[0022] Step S3: Calculate the global standard deviation of all vertex norms, and multiply the embedding strength factor by the global standard deviation as the quantization step size.

[0023] Specifically, the global standard deviation is defined as First, calculate the mean of the norms of all vertices. , then calculate ,in To calculate the square root function. The global standard deviation reflects the overall scale characteristics of the model. An embedding strength factor α is set to control the intensity of the watermark embedding; its specific value can be determined based on invisibility constraints (e.g., ensuring the normalized root mean square error after embedding is less than 0.001). Quantization step size. .because Related to model scale, It will automatically adapt to models of different sizes, without the need for manual parameter adjustment.

[0024] Step S4: For the k-th group, where The mean norm of all vertices in the k-th group is calculated. Dithered quantization index modulation is used to quantize and modulate the mean according to the value of the k-th watermark bit to obtain the norm mean offset of the k-th group. When the value of the watermark bit is 0, the quantization target is the nearest quantization grid point. When the value of the watermark bit is 1, the quantization target is the nearest quantization grid midpoint.

[0025] Specifically, the set of vertex indices contained in the k-th group is The norm values ​​of all vertices in this group are Calculate the norm mean of this group. ,in This indicates the number of vertices contained in the group. The watermark bit sequence to be embedded is... Each The value is either 0 or 1. Dither-quantization index modulation is used: when... At that time, the quantified target is distance. The most recent even-quantized grid points, i.e., integer multiples The point; when At that time, the quantified target is distance. The most recent odd-numbered grid midpoints, i.e., half-integer multiples The point. The quantized mean is denoted as Then the offset This offset is the adjustment that needs to be superimposed on the norm of all vertices in this group.

[0026] Step S5: The norm mean offset is uniformly superimposed on the norm of all vertices in the k-th group, and the watermark is embedded by scaling the radial vector of each vertex proportionally.

[0027] Specifically, for each vertex j within the k-th group, its current radial length is The new radial length after watermark embedding is set to... To ensure that the vertex direction remains unchanged and the radial length is positive, it is necessary to ensure (in practice) Compared to (Very small, usually meets the conditions). Then the vertex coordinates are from... Updated to This proportional radial scaling operation maintains the vertex's orientation relative to the origin, changing only its distance from the origin. Due to the offset... Compared to The watermark is very small, and its embedding causes only a slight change to the model geometry, making it highly invisible.

[0028] Step S6: Add the centroid back to the coordinates of all vertices to obtain the 3D model with the embedded watermark.

[0029] Specifically, after radial scaling of all groups, all vertex coordinates are in a centroid-aligned coordinate system. To restore the model's original spatial position, the previously subtracted centroid *c* needs to be added back to each vertex coordinate, i.e. The resulting set of vertices forms the complete 3D mesh model after the watermark is embedded.

[0030] Step S7: Calculate the global norm median of the 3D model in the centroid aligned coordinate system as the reference median, and record the median of the vertex norm in each group as the group median statistic. Store the reference median, group median statistic, quantization step size, number of groups K, and embedding strength factor as auxiliary information for watermark extraction.

[0031] Specifically, the centroids of the watermarked model are aligned again (or using an existing centroid-aligned coordinate system), the norms of all vertices to the origin are calculated, and the median of these norms is taken as denoted as . Then, for each group k, calculate the median of the norms of all vertices within that group, denoted as . The median is more robust to extreme values ​​(such as vertex norm anomalies near group boundaries due to attacks) than the mean because the median depends on only one or two values ​​in the middle. The median will be used as a reference. , group midpoint array (Length is K), quantization step size The number of groups K and the embedding strength factor α are saved as auxiliary information for subsequent watermark extraction.

[0032] Based on the above steps, high-capacity, low-distortion watermark embedding of 3D models is achieved. Centroid alignment eliminates the effects of translation; norm sorting and grouping ensures invariance to rotation and vertex reordering; adaptive quantization step size makes the method applicable to models of different scales; and the recording of median statistics provides a necessary benchmark for the compensation mechanism in the extraction stage, thus laying the foundation for resisting geometric attacks and degradation attacks.

[0033] To ensure the quantization process can be accurately reproduced, in one embodiment, the dithering quantization index modulation includes: Let the norm mean of the k-th group be . The watermark bits are Quantization step size is Then the jitter value Quantified mean Norm mean offset .

[0034] Specifically, The function performs a rounding operation, when... hour, The quantification formula is simplified to That is, quantized to the nearest integer multiple. Grid points; when hour, The quantification formula becomes That is, quantized to the nearest half-integer multiple. The grid points. This dithering quantization method makes the quantized subsets of bits 0 and 1 evenly interleaved on the number line, improving the robustness of quantization to noise.

[0035] Based on the above, the norm mean offset of each group can be accurately calculated, providing a reliable numerical basis for subsequent radial scaling and ensuring the repeatability and stability of the embedding process.

[0036] The selection of the watermark embedding strength factor α directly affects the watermark's invisibility and robustness. Traditional methods often use fixed empirical values, leading to significant differences in performance across different models. To maximize quantization discriminability while ensuring invisibility, in one embodiment, the embedding strength factor is determined based on a preset invisibility constraint.

[0037] Specifically, the invisibility of watermark embedding is typically quantified using the normalized root mean square error (RMSE). RMSE is calculated by taking the root mean square of the Euclidean distance between corresponding vertices of the original model and the watermarked model, then normalizing it by dividing by the diagonal length of the model's bounding box. A preset invisibility constraint can be a specific upper limit for RMSE, such as 0.001. A larger embedding strength factor α and a larger quantization step size Δ result in stronger watermark robustness, but also a higher RMSE.

[0038] To determine the optimal value of α and its acceptable range, a systematic parametric scanning experiment was conducted on 14 three-dimensional geometric models of different sizes. The tested models included aobin (119978 vertices), armadillo (90000 vertices), bird (5384 vertices), bunny (30000 vertices), curvy_container (4067 vertices), fish (17362 vertices), hat (18601 vertices), ironman (599373 vertices), leaf (8905 vertices), long (300727 vertices), pillow (8362 vertices), seashell (4549 vertices), shuitun (285368 vertices), and wedding_dress (6625 vertices). The number of groups was fixed at K=256 during testing. For each model, five different random seeds were used to generate the watermark, and the average of the five results was taken.

[0039] When α=0.06, the average normalized root mean square error of the 14 models is 0.00085. Among them, 12 models meet the strict invisibility constraint of RMSE<0.001, achieving a compliance rate of 85.7%. The RMSEs of the non-compliant Bird and Ironman models are 0.001026 and 0.001062, respectively, slightly exceeding the threshold and still falling within the visually imperceptible range in practical applications. Meanwhile, the average bit accuracy for a 30-degree rotation attack is 96.9% (range 96.2%-97.3%), the average bit accuracy for a 2x scaling attack is 96.9%, the average bit accuracy for a 0.5x translation attack is 96.9%, and the average bit accuracy for a vertex reordering attack is 96.9%. The average bit accuracy for a Laplacian smoothing attack (low smoothness, 5 iterations) is 92.1%, and the average bit accuracy for a Gaussian noise attack (moderate noise, standard deviation of 0.3% of the bounding box diagonal length) is 94.2%. These data indicate that α=0.06 provides sufficient quantization discrimination without sacrificing invisibility, making it an optimal value that balances invisibility and robustness.

[0040] When α=0.03, the quantization step size is half that of α=0.06, resulting in reduced quantization discrimination and a significant decrease in the robustness of the watermark against attacks. When α=0.10, the quantization step size increases, further improving the robustness of the watermark, but geometric distortion increases, and the RMSE of some models may exceed 0.001. Therefore, the acceptable range for α is 0.03 to 0.10. Below 0.03, robustness cannot meet the needs of practical applications, and above 0.10, invisibility may not meet strict constraints. Users can choose α within this range according to specific application scenarios: if invisibility requirements are extremely high, α=0.03 can be selected; if robustness requirements are higher, α=0.10 can be selected; generally, α=0.06 can obtain the best overall performance. The specific value of α can be determined using any optimization method in the existing technology, and this embodiment does not limit it.

[0041] In this way, the embedding strength factor can adaptively meet the invisibility requirement and provides a clear preferred value and acceptable range, avoiding performance loss caused by blind setting.

[0042] The selection of the number of groups K directly affects the watermark capacity and the statistical stability within each group. If the number of groups is too large, the number of vertices in each group is too small, and the statistics are easily affected by attacks; if the number of groups is too small, the watermark capacity is insufficient. Therefore, in one embodiment, the number of groups K is such that each group contains at least a preset number of vertices.

[0043] Specifically, let the total number of vertices in the input model be N, and the number of groups be K. Then, the average number of vertices contained in each group is... To ensure the robustness of within-group statistics (mean and quantiles), it is necessary to... The number of vertices should be no less than a preset lower limit, such as 100. This ensures that even after a moderate degradation attack, there are still enough vertices in the group to reflect the overall statistical characteristics.

[0044] Based on test data from 14 models, when K=256, for models with N>25600 vertices, each group contains an average of at least 100 vertices, providing a sufficient statistical sample size. The statistics (mean and median) exhibit good robustness when the sample size is greater than 100, with individual vertex variations having a relatively small impact on the group statistics. When K increases to 512, the watermark capacity increases to 512 bits, but the number of vertices per group is halved, resulting in a slight decrease in statistical stability. This method is only recommended for models with an exceptionally large number of vertices (N>100000). When K decreases to 128, the number of vertices per group increases, leading to better statistical robustness, but the watermark capacity decreases to 128 bits.

[0045] The acceptable range for K is 128 to 512. K=256 is the optimal choice, ensuring at least 100 vertices per group for most models (N>25600) with a 256-bit watermark capacity, providing sufficient statistics. For models with a particularly large number of vertices (N>100000), K can be appropriately increased to 512 to obtain higher capacity; for models with a small number of vertices (N<25600), K should be appropriately reduced to 128 to ensure sufficient vertices per group. In this case, the robustness loss caused by the reduction in statistics can be compensated by increasing the embedding strength factor α.

[0046] Therefore, the selection rule for K is: the number of groups K ensures that each group contains at least a preset number of vertices, which is usually set to 100. Users can determine K based on the total number of vertices N in the model and the target watermark capacity: first determine the minimum number of vertices threshold T for each group (e.g., 100), then set K to be no greater than... The largest integer, or directly set K as the target capacity (e.g., 256), and then check. If the value is greater than or equal to T, then consider increasing the model size or decreasing K. The specific value of K can be determined using any grouping strategy in the prior art, and this embodiment does not limit it.

[0047] By using the above method, the number of groups K ensures that each group has a sufficient statistical sample size, thereby improving the robustness of the watermark against local attacks, while providing guidance on parameter adjustment that is appropriate for the model size.

[0048] Secondly, during 3D model watermark extraction, the model to be detected may be subjected to attacks such as uniform scaling and Laplacian smoothing, leading to overall scale changes or systematic shrinkage of vertex norms. Traditional extraction methods cannot adaptively correct these changes, causing a sharp drop in watermark extraction accuracy. Therefore, in one embodiment, a robust 3D watermark extraction method based on norm ranking and median compensation is proposed, such as... Figure 3 As shown, it includes: Step T1: Obtain watermark extraction auxiliary information, which includes reference median, group median statistic, quantization step size, number of groups K, and embedding strength factor.

[0049] Specifically, auxiliary information is extracted from the watermark by loading it from the storage location, including: reference median. (Global norm median of the model in the centroid-aligned coordinate system after watermarking), group midpoint statistics (An array of length K, where each element is the median norm of the corresponding group), quantization step size Δ, number of groups K, and embedding strength factor α. This information is the baseline data required for the extraction process.

[0050] Step T2: Perform centroid alignment preprocessing on the vertex coordinates of the extracted 3D model, and calculate the Euclidean norm of each centroid-aligned vertex to the origin.

[0051] Specifically, the centroid alignment preprocessing involves calculating the mean coordinates of all vertices as the centroid, subtracting this centroid from the coordinates of each vertex to obtain the centroid-aligned vertex coordinates. Then, the Euclidean norm of each vertex to the origin is calculated, resulting in a norm sequence. The purpose of centroid alignment is to maintain consistency with the coordinate system used in the embedding stage, thereby eliminating the impact of translation attacks.

[0052] Step T3: Calculate the median of the norms of all vertices in the current model, use the ratio of the median to the reference median as the scaling factor, and use the product of the quantization step size and the scaling factor as the effective step size.

[0053] Specifically, the median of all vertex norms is denoted as... Since the uniform scaling attack multiplies the norm of all vertices by the same factor s, therefore Therefore, the scaling factor Then, the original quantization step size Δ used in the embedding stage is multiplied by this scaling factor to obtain the effective step size. In this way, even if the model undergoes arbitrary uniform scaling, the effective step size can adaptively align to the current model's scale, ensuring the correctness of subsequent quantization decisions.

[0054] Step T4: Sort all vertices by norm from smallest to largest and divide them into K equal groups.

[0055] Specifically, the sorting and grouping operations are completely consistent with the embedding phase. Since sorting depends only on the magnitude of the norm, vertex reordering attacks will not affect the grouping results.

[0056] Step T5: For the k-th group, where The median of the norms of all vertices in the k-th group is calculated as the median of the current group.

[0057] Specifically, the midpoint of the current group in the k-th group is denoted as... The reason for using the median instead of the mean is that the median is not sensitive to outliers. Even when an attack causes a large norm shift in a few vertices near the group boundaries, the median can still stably reflect the dominant norm characteristics of the group. (Original group median) It is obtained from auxiliary information; it is the median norm of the group after embedding the watermark without any attacks. Step T6: Based on the formula Calculate the offset corresponding to the original group midpoint value of the k-th group, where The value in the current group, The value in the original group, The scaling factor is the scaling factor; if Then, half compensation is performed to obtain the adjusted median. ;otherwise, ;in The effective step size is... This is the compensation coefficient.

[0058] Specifically, taking into account overall scaling, the expected group midpoint is calculated. The midpoint of the group currently being measured. Difference from expected value The offset `shift` may originate from systematic norm shrinkage caused by a smoothing attack, or from quantization noise. If |shift| is less than twice the effective step size (i.e., ... If the offset is considered to originate primarily from systemic contraction, then half-compensation is applied: adjusted median. If |shift| is large, it indicates that the attack strength exceeds the compensation capacity or belongs to another type. In this case, no compensation will be performed, and the original measurement value will be used directly. Threshold These are empirical values ​​determined through numerous experiments, which can effectively distinguish between systematic bias and quantization noise.

[0059] Step T7: Calculate the distances from the adjusted median to the nearest even quantization point and the nearest odd quantization point respectively. If the distance from the adjusted median to the nearest even quantization point is less than or equal to the distance to the nearest odd quantization point, then determine that the k-th watermark bit is 0; otherwise, it is 1.

[0060] Specifically, the quantization grid is determined based on the effective step size effective_Δ: even quantization points are integer multiples of each other. The points (…, -2Δ, -Δ, 0, Δ, 2Δ, …) are quantized as odd numbers that are half-integer multiples. The points are (…, -1.5Δ, -0.5Δ, 0.5Δ, 1.5Δ, …). Calculate the adjusted median. The distance d0 to the nearest even-quantized point and the distance d1 to the nearest odd-quantized point are used. If d0 ≤ d1, the watermark bit to be extracted is 0; otherwise, it is 1. This decision rule strictly corresponds to the dithered quantization index modulation used during embedding.

[0061] Through the above steps, this implementation can accurately extract the watermark bit sequence from the attacked 3D model. The scaling adaptive mechanism eliminates the influence of uniform scaling, and the proportional offset compensation mechanism improves the extraction accuracy under Laplacian smoothing attacks. Experimental verification shows that when α=0.06 and K=256, the bit accuracy against rotation, scaling, translation, and vertex reordering attacks can reach 96.2% to 97.3%; the average bit accuracy against Laplacian smoothing attacks (low smoothing, 5 iterations) is 92.1%, an improvement of 42 percentage points compared to approximately 50% without compensation; and the average bit accuracy against Gaussian noise attacks (medium noise) is 94.2%.

[0062] To accurately execute the quantization decision, in one embodiment, the nearest even quantization point is: The nearest odd quantization point is .

[0063] Specifically, for even-quantized points, the adjusted median is divided by the effective step size, rounded, and then multiplied by the effective step size to obtain the nearest point that is an integer multiple of the effective step size. For odd-quantized points, since they are located at half-integer multiples of the effective step size, the adjusted median is first subtracted by half the step size, then the same rounding is performed, and finally half the step size is added back to obtain the nearest point that is a half-integer multiple of the effective step size. These two formulas can accurately calculate the two candidate quantized points required for the decision, providing a clear numerical basis for distance comparison.

[0064] The above formula provides a reproducible mathematical basis for the quantitative decision-making process, ensuring the consistency of the extracted results.

[0065] In the proportional offset compensation mechanism, the value of the compensation coefficient directly affects the compensation effect. Traditional methods typically employ full compensation or fixed offset, but these are prone to over-correction or under-correction. Therefore, in one embodiment, the compensation coefficient for the half-compensation is 0.5.

[0066] Specifically, in the proportional offset compensation step, when the offset |shift| is determined to be less than twice the effective step size, the compensation operation performed is as follows: β is the compensation coefficient. This implementation specifies that the compensation coefficient is 0.5. The reason for using 0.5 is that the systematic norm shrinkage caused by smoothing attacks (such as Laplacian smoothing) is usually slow and uniform, and the offset includes both the true bias caused by the attack and the quantization error. By compensating only half of the offset, most of the systematic shrinkage caused by the attack can be effectively offset without completely eliminating quantization information, thus making the adjusted median closer to the original quantization target.

[0067] Experimental data shows that, when testing Laplacian smoothing attacks (low-level smoothing: 5 iterations, λ=0.3) on 14 models, the average bit accuracy without compensation is approximately 50%. After applying half-compensation (compensation coefficient 0.5), the average bit accuracy improves to 92.1%, reaching 99.2% for the fish model, 98.3% for the long model, and 98.2% for the shuitun model. When the compensation coefficient is 0.3 or 0.7, the improvement is less than that of 0.5. Therefore, a compensation coefficient of 0.5 is the optimal and universally applicable choice, as verified by experiments. In practical applications, this coefficient can be fine-tuned according to the attack type, but 0.5 provides the best general robustness.

[0068] Through the aforementioned compensation coefficients, the proportional offset compensation mechanism achieves a balance between attack offset and quantization information, enhancing the robustness of the watermark extraction method against smoothing attacks.

[0069] Finally, to verify the technical effectiveness of this application, a comprehensive robustness test was conducted on 14 three-dimensional geometric models of different sizes. The results are detailed below. The test parameters were set as follows: embedding strength factor α = 0.06, number of groups K = 256 (watermark capacity 256 bits), and watermarks were generated using 5 different random seeds for each model, with the average of 5 results taken. The tested models covered a variety of model types, ranging from small-scale (4067 vertices) to very large-scale (599373 vertices).

[0070] (a) Results of the Invisibility Test Table 1 shows the invisibility test results for 14 models, where RMSE is the normalized root mean square error, and the standard is RMSE < 0.001.

[0071] Table 1 As shown in Table 1, the average RMSE of the 14 models is 0.000798, of which 12 models (85.7%) meet the invisibility requirement of RMSE < 0.001. The RMSEs of the non-compliant Bird and Ironman models are 0.001026 and 0.001062, respectively, slightly exceeding the threshold and still falling within the visually imperceptible range in practical applications. This result demonstrates that the method proposed in this application can satisfy the strict invisibility constraint on most models.

[0072] (II) Robustness test results of RST and vertex reordering Table 2 shows the bit accuracy of 14 models against attacks involving 30-degree rotation, 2x scaling, 0.5x translation (relative to the coordinate range), and vertex reordering.

[0073] Table 2 As shown in Table 2, the bit accuracy of all 14 models under rotation, scaling, translation, and vertex reordering attacks ranged from 96.2% to 97.3%, with an average of 96.9%. This verifies the high robustness of this application to distortionless geometric transformation attacks.

[0074] (III) Results of robustness test against degradation attacks Table 3 shows the bit accuracy of 14 models against simplification, smoothing, noise, and pruning degradation attacks. Among them: Simplified attack: Low (retain 80% of vertices), Medium (retain 60% of vertices), High (retain 40% of vertices); Smoothing attack: Laplacian smoothing is used, with 5 iterations for low degree, 15 iterations for medium degree, and 30 iterations for high degree (λ=0.3). Noise attack: Gaussian noise, with a standard deviation of 0.1% (low), 0.3% (medium), and 0.5% (high) of the model bounding box diagonal length. Pruning attack: Randomly prunes 20% (low-degree), 50% (medium-degree), and 80% (height) of consecutive vertices.

[0075] Table 3 As shown in Table 3, this application exhibits the following robustness characteristics against various degradation attacks: Simplified attacks: Low-level simplification (retaining 80% of vertices) achieved an average accuracy of 82.4%, with large models reaching 95.9% for Ironman, 93.5% for Long, 86.9% for Aobin, and 91.2% for Shuitun. Smoothing attack: Low-smoothing achieves an average accuracy of 92.1%, outperforming traditional methods. The fish model achieves 99.2% accuracy with low smoothing, the long model 98.3%, and the shuitun model 98.2%. Even with high smoothing, it maintains an average accuracy of 80.7%. Noise attack: The average accuracy rates for low and medium noise levels were 94.4% and 94.2%, respectively, with all models meeting the standard; the average accuracy rate for high noise levels was 86.2%. Pruning attacks: The average accuracy is about 52% to 55%, which is an inherent limitation of norm sorting grouping methods. In practical applications, this can be mitigated by increasing group redundancy or using error correction coding.

[0076] (iv) Analysis of the relationship between model size and robustness The 14 models were divided into three groups based on the number of vertices: Large models (>100K vertices): aobin(119978), ironman(599373), long(300727), shuitun(285368), average number of vertices 326117; Medium-sized models (10,000-100,000 vertices): armadillo (90,000), bunny (30,000), fish (17,362), hat (18,601), with an average of 38,991 vertices; Small models (<10000 vertices): bird(5384), curvy_container(4067), leaf(8905), pillow(8362), seashell(4549), wedding_dress(6625), average number of vertices 6378; The average bit accuracy of each group under low simplification, low smoothing, and medium noise attack was statistically analyzed, and the results are shown in Table 4.

[0077] Table 4 As shown in Table 4, larger models perform better under degradation attacks because each group in a larger model contains more vertices, resulting in more stable statistics. For models with fewer vertices, robustness can be improved by increasing the embedding strength factor α or decreasing the number of groups K.

[0078] It should be understood that although the steps in the flowcharts of the above embodiments are shown sequentially according to the arrows, these steps are not necessarily executed in the order indicated by the arrows. Unless explicitly stated herein, there is no strict order restriction on the execution of these steps, and they can be executed in other orders. Moreover, at least some steps in the flowcharts of the above embodiments may include multiple steps or multiple stages. These steps or stages are not necessarily completed at the same time, but can be executed at different times. The execution order of these steps or stages is not necessarily sequential, but can be performed alternately or in turn with other steps or at least some of the steps or stages in other steps.

[0079] Thirdly, based on the same inventive concept, this application also provides a system for implementing the aforementioned three-dimensional robust watermark embedding and extraction methods based on norm sorting and median compensation. The solution provided by this system is similar to the implementation scheme described in the above methods; therefore, the specific limitations in one or more system embodiments provided below can be found in the limitations of the three-dimensional robust watermarking method based on norm sorting and median compensation described above, and will not be repeated here.

[0080] In one embodiment, such as Figure 4 As shown, the system includes: a watermark embedding device for performing a three-dimensional robust watermark embedding method based on norm ranking and median compensation, and / or a watermark extraction device for performing a three-dimensional robust watermark extraction method based on norm ranking and median compensation. It should be noted that the system includes a watermark embedding device and / or a watermark extraction device, meaning the system may contain only a watermark embedding device, only a watermark extraction device, or both. The watermark embedding device includes: The centroid alignment module is used to perform centroid alignment preprocessing on the vertex coordinates of the input 3D mesh model. The centroid alignment preprocessing includes: calculating the mean of all vertex coordinates as the centroid, subtracting the centroid from each vertex coordinate to obtain the set of vertices after centroid alignment. The norm calculation module is used to calculate the Euclidean norm of each vertex to the origin, sort all vertices according to the norm from smallest to largest, and divide the sorted vertices into K groups, where K is the length of the bit sequence to be embedded with the watermark. The quantization step size calculation module is used to calculate the global standard deviation of the norm of all vertices, and multiply the embedding strength factor by the global standard deviation as the quantization step size; The quantization modulation module is used to calculate the mean of the norms of all vertices within the k-th group, where k = 0, 1, ..., K-1, using dithered quantization index modulation. The mean is quantized and modulated according to the value of the k-th watermark bit to obtain the norm mean offset of the k-th group. Specifically, when the watermark bit value is 0, the quantization target is the nearest quantization grid point; when the watermark bit value is 1, the quantization target is the nearest quantization grid center point. The radial scaling module is used to uniformly superimpose the norm mean offset onto the norm of all vertices in the k-th group, and to achieve watermark embedding by scaling the radial vector of each vertex proportionally. The centroid recovery module is used to add the centroid back to the coordinates of all vertices to obtain the 3D model with the watermark embedded. The auxiliary information generation module is used to calculate the global norm median of the three-dimensional model in the centroid aligned coordinate system as the reference median, and record the median of the vertex norm in each group as the group median statistic. The reference median, group median statistic, quantization step size, number of groups K and embedding intensity factor are stored as auxiliary information for watermark extraction.

[0081] The watermark extraction device includes: The auxiliary information acquisition module is used to acquire auxiliary information for watermark extraction, which includes reference median, group median statistic, quantization step size, number of groups K, and embedding strength factor. The norm calculation module is used to perform centroid alignment preprocessing on the vertex coordinates of the extracted 3D model and calculate the Euclidean norm of each centroid-aligned vertex to the origin. The scaling adaptive module is used to calculate the median of the norm of all vertices of the current model, use the ratio of the median to the reference median as the scaling factor, and use the product of the quantization step size and the scaling factor as the effective step size. The sorting and grouping module is used to sort all vertices in ascending order of their norm and divide them into K equal groups; The group midpoint calculation module is used for the k-th group, where Calculate the median of the norms of all vertices in the k-th group as the median of the current group; The proportional offset compensation module is used for formula-based... Calculate the offset corresponding to the original group midpoint value of the k-th group, where The value in the current group, The value in the original group, The scaling factor is the scaling factor; if Then, half compensation is performed to obtain the adjusted median. ;otherwise, ;in The effective step size is... The compensation coefficient; The quantization decision module is used to calculate the distances from the adjusted median to the nearest even quantization point and the nearest odd quantization point, respectively. If the distance from the adjusted median to the nearest even quantization point is less than or equal to the distance to the nearest odd quantization point, the k-th watermark bit is determined to be 0; otherwise, it is 1.

[0082] In one embodiment, the quantization modulation module includes a quantization index modulation unit. The quantization index modulation unit is used to set the norm mean of the k-th group to be... The watermark bits are Quantization step size is Then the jitter value Quantified mean Norm mean offset .

[0083] In one embodiment, the quantization step size calculation module includes an embedding strength factor setting unit. The embedding strength factor setting unit is used to determine the embedding strength factor based on a preset invisibility constraint.

[0084] In one embodiment, the norm calculation module includes a group number limiting unit. The group number limiting unit is used to determine the group number K such that each group contains at least a preset number of vertices.

[0085] In one embodiment, the quantization decision module includes a quantization point setting unit. The quantization point setting unit is used to determine the quantization point based on a formula. Determine the nearest even quantization point, and use it according to the formula Identify the nearest odd quantization point.

[0086] In one embodiment, the proportional offset compensation module includes a compensation coefficient determination unit. This compensation coefficient determination unit is used to set the compensation coefficient for half-amount compensation to 0.5.

[0087] The modules in the aforementioned norm-based and median-compensated 3D robust watermarking system can be implemented entirely or partially through software, hardware, or a combination thereof. These modules can be embedded in or independent of the processor in a computer device, or stored in the computer device's memory as software, allowing the processor to call and execute the corresponding operations of each module.

[0088] In one embodiment, a computer device is provided, which may be a server, and its internal structure diagram may be as follows: Figure 5 As shown, the computer device includes a processor, memory, and network interface connected via a system bus. The processor provides computational and control capabilities. The memory includes a non-volatile storage medium and internal memory. The non-volatile storage medium stores the operating system, computer programs, and a database. The internal memory provides an environment for the operation of the operating system and computer programs in the non-volatile storage medium. The database stores all data required to execute the corresponding method. The network interface communicates with external terminals via a network connection. When executed by the processor, the computer program implements a three-dimensional robust watermark embedding and / or extraction method based on norm sorting and median compensation. It should be noted that when executed by the processor, the computer program may implement only the embedding method, only the extraction method, or both.

[0089] Those skilled in the art will understand that Figure 5 The structure shown is merely a block diagram of a portion of the structure related to the present application and does not constitute a limitation on the computer device to which the present application is applied. Specific computer devices may include more or fewer components than those shown in the figure, or combine certain components, or have different component arrangements.

[0090] Fourthly, a computer device is provided, including a memory and a processor. The memory stores a computer program, and when the processor executes the computer program, it implements any of the method steps in the embodiments of the three-dimensional robust watermarking method based on norm sorting and median compensation and / or the three-dimensional robust watermark extraction method based on norm sorting and median compensation. It should be noted that when the computer program is executed by the processor, it may implement only the embedding method, only the extraction method, or both the embedding method and the extraction method.

[0091] Fifthly, a computer-readable storage medium is provided, on which a computer program is stored. When executed by a processor, the computer program implements any of the method steps in the embodiments of the three-dimensional robust watermarking method based on norm sorting and median compensation and / or the three-dimensional robust watermark extraction method based on norm sorting and median compensation. It should be noted that when the computer program is executed by a processor, it may implement only the embedding method, only the extraction method, or both the embedding method and the extraction method.

[0092] Those skilled in the art will understand that all or part of the processes in the above embodiments can be implemented by a computer program instructing related hardware. The computer program can be stored in a non-volatile computer-readable storage medium. When executed, the computer program can include the processes of the embodiments described above. Any references to memory, databases, or other media used in the embodiments provided in this application can include at least one of non-volatile and volatile memory. Non-volatile memory can include read-only memory (ROM), magnetic tape, floppy disk, flash memory, optical memory, high-density embedded non-volatile memory, resistive random access memory (ReRAM), magnetic random access memory (MRAM), ferroelectric random access memory (FRAM), phase change memory (PCM), graphene memory, etc. Volatile memory can include random access memory (RAM) or external cache memory, etc. By way of illustration and not limitation, RAM can take many forms, such as Static Random Access Memory (SRAM) or Dynamic Random Access Memory (DRAM). The databases involved in the embodiments provided in this application may include at least one type of relational database and non-relational database. Non-relational databases may include, but are not limited to, blockchain-based distributed databases. The processors involved in the embodiments provided in this application may be general-purpose processors, central processing units, graphics processing units, digital signal processors, programmable logic devices, quantum computing-based data processing logic devices, etc., and are not limited to these.

[0093] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0094] The embodiments described above are merely illustrative of several implementation methods of this application, and while the descriptions are specific and detailed, they should not be construed as limiting the scope of this patent application. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of this application, and these all fall within the protection scope of this application. Therefore, the protection scope of this application should be determined by the appended claims.

Claims

1. A three-dimensional robust watermark embedding method based on norm ordering and median compensation, characterized in that, The method includes: The vertex coordinates of the input 3D mesh model are preprocessed with centroid alignment. The centroid alignment preprocessing includes calculating the mean of all vertex coordinates as the centroid, subtracting the centroid from each vertex coordinate to obtain the set of vertices with centroid alignment. Calculate the Euclidean norm of each vertex to the origin, sort all vertices in ascending order of norm, and divide the sorted vertices into K groups, where K is the length of the bit sequence to be embedded with the watermark. Calculate the global standard deviation of all vertex norms, and multiply the embedding strength factor by the global standard deviation as the quantization step size; For the k-th group, where The mean norm of all vertices in the k-th group is calculated. Dithered quantization index modulation is used, and the mean is quantized and modulated according to the value of the k-th watermark bit to obtain the norm mean offset of the k-th group. When the value of the watermark bit is 0, the quantization target is the nearest quantization grid point, and when the value of the watermark bit is 1, the quantization target is the nearest quantization grid center point. The norm mean offset is uniformly superimposed on the norm of all vertices in the k-th group, and the watermark is embedded by scaling the radial vector of each vertex proportionally. Add the centroid back to the coordinates of all vertices to obtain the 3D model with the watermark embedded. The global norm median of the 3D model in the centroid-aligned coordinate system is calculated as the reference median, and the median of the vertex norm in each group is recorded as the group median statistic. The reference median, group median statistic, quantization step size, number of groups K, and embedding intensity factor are stored as auxiliary information for watermark extraction.

2. The method of claim 1, wherein, The jitter quantization index modulation includes: Let the norm mean value of the kth group be , the watermark bit be , and the quantization step be , then the jitter value is , the quantized mean value is , and the norm mean value offset is .

3. The method of claim 1, wherein, The embedding strength factor is determined according to a preset invisibility constraint.

4. The method according to claim 1, characterized in that, The number of groups K is such that each group contains at least a preset number of vertices.

5. A method for extracting a three-dimensional robust watermark based on norm ordering and median compensation, characterized in that, The method includes: Obtain watermark extraction auxiliary information, which includes reference median, group median statistic, quantization step size, number of groups K, and embedding strength factor; The vertex coordinates of the extracted 3D model are preprocessed by centroid alignment, and the Euclidean norm of each centroid-aligned vertex to the origin is calculated. Calculate the median of the norms of all vertices in the current model, use the ratio of the median to the reference median as a scaling factor, and use the product of the quantization step size and the scaling factor as the effective step size; Sort all vertices by norm from smallest to largest, and divide them into K equal groups; For the kth group, wherein , the median of the norm of all vertices in the kth group is calculated as the current group median; Based on formula Calculate the offset corresponding to the original group midpoint value of the k-th group, where The value in the current group, The value in the original group, The scaling factor is the scaling factor; if Then, half compensation is performed to obtain the adjusted median. ;otherwise, ;in The effective step size is... The compensation coefficient; Calculate the distances from the adjusted median to the nearest even quantization point and the nearest odd quantization point respectively. If the distance from the adjusted median to the nearest even quantization point is less than or equal to the distance to the nearest odd quantization point, then the k-th watermark bit is determined to be 0; otherwise, it is 1.

6. The method of claim 5, wherein, The nearest even quantization point is , and the nearest odd quantization point is .

7. The method of claim 5, wherein, The compensation coefficient for the half-compensation is 0.

5.

8. A three-dimensional robust watermarking system based on norm ordering and median compensation, characterized in that, The system includes a watermark embedding device for performing the method according to any one of claims 1 to 4 and / or a watermark extraction device for performing the watermark extraction device according to any one of claims 5 to 7; The watermark embedding device includes: The centroid alignment module is used to perform centroid alignment preprocessing on the vertex coordinates of the input 3D mesh model. The centroid alignment preprocessing includes: calculating the mean of all vertex coordinates as the centroid, subtracting the centroid from each vertex coordinate to obtain the set of vertices after centroid alignment. The norm calculation module is used to calculate the Euclidean norm of each vertex to the origin, sort all vertices according to the norm from smallest to largest, and divide the sorted vertices into K groups, where K is the length of the bit sequence to be embedded with the watermark. The quantization step size calculation module is used to calculate the global standard deviation of the norm of all vertices, and multiply the embedding strength factor by the global standard deviation as the quantization step size; The quantization modulation module is used to calculate the mean of the norms of all vertices within the k-th group, where k = 0, 1, ..., K-1, using dithered quantization index modulation. The mean is quantized and modulated according to the value of the k-th watermark bit to obtain the norm mean offset of the k-th group. Specifically, when the watermark bit value is 0, the quantization target is the nearest quantization grid point; when the watermark bit value is 1, the quantization target is the nearest quantization grid center point. The radial scaling module is used to uniformly superimpose the norm mean offset onto the norm of all vertices in the k-th group, and to achieve watermark embedding by scaling the radial vector of each vertex proportionally. The centroid recovery module is used to add the centroid back to the coordinates of all vertices to obtain the 3D model with the watermark embedded. The auxiliary information generation module is used to calculate the global norm median of the three-dimensional model in the centroid aligned coordinate system as the reference median, and record the median of the vertex norm in each group as the group median statistic. The reference median, group median statistic, quantization step size, number of groups K and embedding intensity factor are stored as auxiliary information for watermark extraction. The watermark extraction device includes: The auxiliary information acquisition module is used to acquire auxiliary information for watermark extraction, which includes reference median, group median statistic, quantization step size, number of groups K, and embedding strength factor. The norm calculation module is used to perform centroid alignment preprocessing on the vertex coordinates of the extracted 3D model and calculate the Euclidean norm of each centroid-aligned vertex to the origin. The scaling adaptive module is used to calculate the median of the norm of all vertices of the current model, use the ratio of the median to the reference median as the scaling factor, and use the product of the quantization step size and the scaling factor as the effective step size. The sorting and grouping module is used to sort all vertices in ascending order of their norm and divide them into K equal groups; The group midpoint calculation module is used for the k-th group, where Calculate the median of the norms of all vertices in the k-th group as the median of the current group; The proportional offset compensation module is used for formula-based... Calculate the offset corresponding to the original group midpoint value of the k-th group, where The value in the current group, The value in the original group, The scaling factor is the scaling factor; if Then, half compensation is performed to obtain the adjusted median. ;otherwise, ;in The effective step size is... This is the compensation coefficient; The quantization decision module is used to calculate the distances from the adjusted median to the nearest even quantization point and the nearest odd quantization point, respectively. If the distance from the adjusted median to the nearest even quantization point is less than or equal to the distance to the nearest odd quantization point, the k-th watermark bit is determined to be 0; otherwise, it is 1.

9. A computer device comprising a memory and a processor, wherein the memory stores a computer program, characterized in that, When the processor executes the computer program, it implements the method of any one of claims 1 to 4, and / or implements the method of any one of claims 5 to 7.

10. A computer-readable storage medium having stored thereon a computer program, characterized in that, When the computer program is executed by a processor, it implements the method of any one of claims 1 to 4, and / or the method of any one of claims 5 to 7.

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