Zero watermarking algorithm for BIM data based on distance partitioning and local feature

ABSTRACT

Zero watermarking constructs the watermark information according to the characteristics of the original data, without changing the data structure and data accuracy. Maintaining high data accuracy is the premise of building information modelling (BIM) usability, so zero watermarking is a hotspot in the research of BIM data security protection. BIM model is a type of 3D model, however, most of the existing zero watermarking algorithms for 3D models are difficult to be better applied to BIM data due to data structure differences. To solve this problem, a zero-watermarking algorithm for BIM data based on distance partitioning and skewness measure is proposed. Firstly, after spatial partitioning based on element paradigm value, the mapping relationship between different partitions and watermarking bits is established. Then, the skewness of elements is calculated, and the skewness measure sign is used as the feature to obtain the binary sequence. Finally, the dissimilarity operation is performed on the binary sequence and the original watermarking sequence which was disordered to construct the zero watermark of the BIM data. The experimental results show that the zero watermarks constructed from different BIM data are unique and robust to translation, rotation, element deletion, element addition, and format conversion attacks. In addition, the superiority of this paper’s algorithm over the comparison algorithm in terms of robustness is compared. Therefore, the proposed algorithm can effectively provide technical support for BIM data copyright protection.

KEYWORDS:

• BIM data zero-watermarking
• distance partitioning
• skewness measure

1. Introduction

Building information modelling (BIM) data have become important data source for smart city construction due to the ability to express the full elemental information of the city, which helps in the description of urban geographic entities and micro-analysis (Chen et al. 2022; Kim and Kim 2023; Zhu et al. 2021). However, as the use of BIM data becomes more and more widespread, illegal use and leakage of data have been occurring (Lou and Lu 2022). Digital watermarking technology, as a cutting-edge technology, provides a solution for BIM data copyright protection by tightly linking copyright information to the original data (Lopez 2002; Zhang et al. 2020, 2023). Existing digital watermarking techniques can be divided into two categories: lossy and lossless watermarking. Lossy watermarking embeds copyright information into original data and inevitably causes a drop in data accuracy (Abulkasim, Jamjoom, and Abbas 2022; Delmotte et al. 2020; Hamidi et al. 2019; Sayahi et al. 2023; Zhang et al. 2023). This loss of accuracy can have a negative impact on the availability of BIM data. Lossless watermarking constructs the watermark sequence without changing the accuracy of the data (Hu and Xiang 2021; Xia et al. 2023; Yu et al. 2022). Therefore, for BIM data that requires high data accuracy, it is more advantageous to use lossless watermarking to protect data security.

As a lossless watermarking technology, zero watermarking uses the features of the original data to construct a watermarking sequence while completely maintaining the accuracy of the data (Tsai, Lai, and Lo 2013). It effectively balances the imperceptibility and robustness of digital watermarking, enabling copyright identification, infringement tracking, and content authentication throughout the data’s lifecycle (Liu et al. 2023; Wang et al. 2023; Xia et al. 2021). Therefore, zero watermarking is a hotspot in the research of BIM data security protection.

BIM model is a type of 3D model, so 3D model zero watermarking algorithms could provide a reference for BIM data zero watermarking research. Existing zero-watermarking algorithms for 3D models can be classified into two categories.

The first type is the global feature-based zero watermarking algorithm, which uses the geometric elements of the original data that reflect the overall features of the model as watermark construction primitives (Daoui et al. 2022; Li, Yang, and Jin 2023; Liu et al. 2020; Wang et al. 2019, 2011). This breed of algorithms harnesses the model’s overarching geometric attributes, employing maths methods, such as moment invariants (Daoui et al. 2022; Wang et al. 2011), wavelet transform (Li, Yang, and Jin 2023; Liu et al. 2020), etc. to distil pertinent features. For example, Daoui improved the computation of the Hahn moment invariant and used an optimized sine-cosine algorithm to compute the local parameters of the Hahn invariants. After the above processing, a zero watermark of the experimental 3D model was obtained. Experiments have demonstrated that this algorithm is resistant to translation, rotation, and other attacks. However, because the algorithm uses feature extraction that relies on the overall shape of the model, the zero watermark is not resistant to attacks that crop the shape of the model (Daoui et al. 2022). In addition, unlike 3D mesh models, BIM data are fine monolithic models with prominent hierarchical characteristics and low redundancy, making it difficult to perform invariant extraction and zero watermark construction based on the above algorithms. In summary, global feature-based zero watermarking algorithms are robust to some geometric attacks, but attacks that break the overall shape of models will result in watermark detection failure and have low applicability for BIM data.

The second type is the local feature-based zero watermarking algorithm, which uses local geometric features such as vertex distances of the original model as watermark construction primitives (Cui, Ni, and Zhao 2017; Jing et al. 2019; Lee et al. 2023; Liu et al. 2021; Wang and Zhan 2019). These approaches harness skewness measure (Lee et al. 2023), spherical integral invariants (Cui, Ni, and Zhao 2017), shape diameter function (Wang and Zhan 2019), etc. to glean local features and construct the zero watermarks. For example, Liu calculated the three-ring neighbourhood areas surrounding vertices and projected the one-ring neighbourhood of medium-area vertices onto the tangent plane. Through the utilization of the Beamlet transform, the mesh on both sides of the X-axis of the tangent plane was extracted and the zero watermark of the 3D mesh model was constructed. This algorithm is highly robust to attacks such as clipping (Liu et al. 2021). Compared to global features, local features are less affected by attacks that destroy the overall shape of the 3D models, which effectively improves the robustness of zero watermarks. However, most of the current algorithms of this type are used for data such as 3D mesh models and point cloud models, and BIM models have different data structures from them, so local features such as three-ring neighbourhood’s area do not work well with BIM data. At present, some scholars have proposed some zero watermarking algorithms for BIM data based on local features after in-depth research, and provide feasible solutions to the problem that existing 3D model watermarking algorithms cannot be used directly on BIM data. However, existing algorithms do not sufficiently consider the presence of element editing attacks, it is difficult to resist the impact of some operations in the use of BIM data on the correct extraction of watermarks (Jing et al. 2019).

In summation, global feature-based zero watermarking algorithms make full use of the overall structural features of 3D models, but such algorithms cannot resist attacks that destroy the overall shape of the models. In contrast, local features-based zero watermarking algorithms are less affected by such attacks, which can effectively improve the robustness of zero watermarking. However, most of the existing zero watermarking algorithms based on local features target 3D mesh models and point cloud models, while BIM data has different data structures, so existing 3D modelling algorithms cannot be directly used for BIM data. Therefore, it is necessary to analyse the structural characteristics of BIM data and select appropriate local features to construct robust watermarks.

Aiming at solving the above problems, this study introduces a novel zero watermarking algorithm based on BIM data characteristics, centred on the principles of distance partitioning and employing the statistical technique of skewness measure to construct watermark information, which solves the problem that existing zero watermarking algorithms for 3D models are difficult to be well applied to BIM data. The subsequent sections of this paper are organized as follows: Section II describes the scheme principle of the proposed algorithm. Section III provides an exposition of the algorithm’s implementation details. Section IV presents the experimental results and confirms the advantages of the algorithm over the comparison algorithm. In section V, an exploration of the algorithm’s extensibility is presented. Finally, in Section VI, we draw conclusions based on our research and suggest directions for future research.

2. Basic idea and preliminaries

2.1. Basic idea

The key to the zero-watermarking algorithm for BIM data is to ensure the uniqueness and robustness of the constructed watermark, so the local features of BIM data that are relatively stable should be selected to construct watermarks. Feature invariants such as curvature are commonly used in 3D model watermarking algorithms. However, although BIM data is also a type of visual 3D model, it has layered characteristics that make it difficult to design watermarking with feature invariants such as curvature. In the research process, a conspicuous observation made is that the utilization of BIM data tends to preserve structural stability, thereby preventing any proclivity towards tilting or distortion that might compromise the spatial structure of the building. This tendency emphasizes the important concept that various forms of attacks rarely change the distribution of important model components in the overall space. Consequently, the relative distances of the elements in relation to the model’s base point emerge as an obvious stabilizing feature. In addition, compared to the distance of individual elements, feature extraction methods based on groups of elements are more resistant to attacks such as deletion. Deletion or movement of an element does not change the characteristics of the entire group. For this reason, in this paper, this stable feature of BIM data is exploited for watermarking bit mapping and paradigm statistics analysis to finally obtain a unique and robust zero watermark sequence.

2.2. Spherical coordinate

Inherently composed of elements, BIM data distinguishes itself by variances in spatial dispositions, these spatial dispositions, fundamentally stemming from the distribution disparities among elements within the model’s coordinate framework, engender the essence of BIM data’s uniqueness and distinguishability. Thus, a logical approach entails extracting relatively stable attributes from these spatial arrangements of elements and subsequently engendering a zero-watermark reflective of these attributes. To extract such stable features, the distribution of elements in space can be quantitatively described by converting all the vertex coordinates of the model to spherical coordinates through Equation (1)(1) $\left\{\begin{array}{c}{r}_i=\sqrt{{\left(x_i-x_c\right)}^2+{\left(y_i-y_c\right)}^2+{\left(z_i-z_c\right)}^2}\\ {\theta }_i={tan}^{-1}\frac{y_i-y_c}{x_i-x_c}\\ {\varphi }_i={cos}^{-1}\frac{z_i-z_c}{\sqrt{{\left(x_i-x_c\right)}^2+{\left(y_i-y_c\right)}^2+{\left(z_i-z_c\right)}^2}}\end{array}\right.,\mathit{i}\in [1,\mathit{N}]$(1) .

(1) $\left\{\begin{array}{c}{r}_i=\sqrt{{\left(x_i-x_c\right)}^2+{\left(y_i-y_c\right)}^2+{\left(z_i-z_c\right)}^2}\\ {\theta }_i={tan}^{-1}\frac{y_i-y_c}{x_i-x_c}\\ {\varphi }_i={cos}^{-1}\frac{z_i-z_c}{\sqrt{{\left(x_i-x_c\right)}^2+{\left(y_i-y_c\right)}^2+{\left(z_i-z_c\right)}^2}}\end{array}\right.,\mathit{i}\in [1,\mathit{N}]$(1)

where N is the total number of elements, $V_i\left(x_i,y_i,z_i\right)$ is the original coordinate of the element, which is calculated as shown in Equation (2)(2) $\mathit{V}=\{(\frac{(bo{x}_i.X_{max}-\mathit{bo}{x}_i.X_{min})}{2},\frac{(bo{x}_i.Y_{max}-\mathit{bo}{x}_i.Y_{min})}{2},\text{break}\frac{(bo{x}_i.Z_{max}-\mathit{bo}{x}_i.Z_{min})}{2})|\mathit{i}\in [1,\mathit{N}]\}$(2) , ${V_i}^{\prime }\left(r_i,{\theta }_i,{\varphi }_i\right)$ is the spherical coordinate of the element, $V_c\left(x_c,y_c,z_c\right)$is the base point of the model, $r_i$ is the paradigm value, which is often used to describe the spatial distribution of vertices of a 3D model.

(2) $\mathit{V}=\{(\frac{(bo{x}_i.X_{max}-\mathit{bo}{x}_i.X_{min})}{2},\frac{(bo{x}_i.Y_{max}-\mathit{bo}{x}_i.Y_{min})}{2},\text{break}\frac{(bo{x}_i.Z_{max}-\mathit{bo}{x}_i.Z_{min})}{2})|\mathit{i}\in [1,\mathit{N}]\}$(2)

where $\mathit{bo}{x}_i$ is the cubic enclosure of the element.

2.3. Distance partitioning

Within the realm of BIM data’s utilization and transmission, changes in the coordinates of the elements and the update of the model constraint relationship caused by operations such as translation bear the potential to introduce errors into watermark bit mapping. In response, devising a watermark bit mapping approach characterized by a measure of stability can improve the ability of zero watermarking to resist attacks. A core premise of BIM data availability is its verticality and structural steadfastness, so the model will not be easily distorted and tilted, exemplified by the vertical orientation of walls and floors. This indicates that the distance from the element to the model base point is relatively stable. Therefore, relatively stable features can be extracted from the spatial distribution characteristics of the elements, and then a zero-watermark reflecting such features can be constructed. To extract such stable features, it is first necessary to quantitatively describe the distribution of elements in space. Mathematically, the distance characteristic of elements can be expressed in terms of the paradigm value. In this paper, this feature is fully considered in designing the watermark mapping rules. The elements’ paradigm value can be sorted and then partitioned to watermark bit mapping with the element group as the basic unit.

Figure 1 illustrates the process of converting the BIM data coordinate system to a spherical coordinate system and watermark mapping. The spherical coordinate system is centred on the project base point and spatially partitioned. This transformation engenders a spatial partitioning predicated on paradigm value, with each distinct spatial region corresponding to a watermark bit within the watermark sequence. On the right side of Figure 1, a watermark sequence of length n is shown. The movement and deletion of individual elements will not affect the overall watermark mapping relationship, so this mapping relationship is more secure.

Figure 1. Watermark bit mapping relationship.

2.4. Skewness measure

The statistical characteristics of data distribution can reflect the laws with stability in the data, which can be used for the extraction of local geometric characteristic invariants of BIM data. Skewness measure is a statistical analysis method to identify whether the group distribution is symmetric or not, if the data is symmetrically presented, the skewness measure value should be zero, and the positive or negative number of the asymmetric state indicates that the distribution of the values tends to be smaller and larger, respectively, and this positivity or negativity can be used to characterize the watermark information. The skewness measure is calculated according to Equation (3)(3) ${\alpha }_j=\frac{{\sum }_{i=1}^k{\left(r_i-\bar{r}\right)}^3}{\mathit{k}\times {\sigma }^3},\mathit{j}\in [1,\mathit{n}]$(3) .

(3) ${\alpha }_j=\frac{{\sum }_{i=1}^k{\left(r_i-\bar{r}\right)}^3}{\mathit{k}\times {\sigma }^3},\mathit{j}\in [1,\mathit{n}]$(3)

where ${\alpha }_j$ is the skewness measure for elements’ paradigm value in the partition, n is the watermark length, k is the number of elements in the partition, $r_i$ is the paradigm value, $\bar{r}$ is the average of elements’ paradigm value in the partition, and $\mathit{\sigma }$ is the standard deviation of elements’ paradigm value in the partition.

3. Proposed methods

3.1. Pre-processing and distance partitioning

Distance partitioning is to map the watermarked bits and then extract the local feature values within the partition. We fully consider the stability of the BIM data structure and partition the model space based on the paradigm values after transforming the spherical coordinate system, with each space indexing one bit of watermark information.

Step 1: Exclude annotations, combinatorial models and other less used elements, and filter the target types of elements. Then, each element coordinate $\mathit{V}$ is calculated according to Equation (2)(2) $\mathit{V}=\{(\frac{(bo{x}_i.X_{max}-\mathit{bo}{x}_i.X_{min})}{2},\frac{(bo{x}_i.Y_{max}-\mathit{bo}{x}_i.Y_{min})}{2},\text{break}\frac{(bo{x}_i.Z_{max}-\mathit{bo}{x}_i.Z_{min})}{2})|\mathit{i}\in [1,\mathit{N}]\}$(2) . In addition, obtain the model base point $V_c\left(x_c,y_c,z_c\right)$.

Step 2: Convert all coordinates to spherical coordinates according to Equation (1)(1) $\left\{\begin{array}{c}{r}_i=\sqrt{{\left(x_i-x_c\right)}^2+{\left(y_i-y_c\right)}^2+{\left(z_i-z_c\right)}^2}\\ {\theta }_i={tan}^{-1}\frac{y_i-y_c}{x_i-x_c}\\ {\varphi }_i={cos}^{-1}\frac{z_i-z_c}{\sqrt{{\left(x_i-x_c\right)}^2+{\left(y_i-y_c\right)}^2+{\left(z_i-z_c\right)}^2}}\end{array}\right.,\mathit{i}\in [1,\mathit{N}]$(1) , and sort all the paradigm value $r_i$ to obtain maximum paradigm value $r_{max}$ and minimum paradigm value $r_{min}$. If $r_{max}>2\times r_{max-1}$ or $r_{min}<2\times r_{min+1}$, it means that the mutability of the boundary element is too high and it is not suitable to be used as a boundary element, delete this element from the sequence. After screening, obtain the final $r_{max}$ and $r_{min}$.

Step 3: Distance partitioning. Partition the space region into n regions, partition is expressed as $C_j,\mathit{j}\in [1,\mathit{n}]$. The paradigm value $\mathit{r}$ of each element conforms to Equation (4)(4) $\left\{\begin{array}{c}{r}_{\mathit{min}}\le \mathit{r}<r_{\mathit{min}}+\frac{{\mathit{r}}_{\mathit{max}}-r_{\mathit{min}}}{\mathit{n}},\mathit{j}=1\\ r_{\mathit{min}}+\left(\mathit{j}-1\right)\times \frac{{\mathit{r}}_{\mathit{max}}-r_{\mathit{min}}}{\mathit{n}}\le \mathit{r}<r_{\mathit{min}}+\mathit{j}\times \frac{{\mathit{r}}_{\mathit{max}}-r_{\mathit{min}}}{\mathit{n}},\mathit{j}\in \left(1,\mathit{n}\right)\\ r_{\mathit{min}}+\left(\mathit{n}-1\right)\times \frac{{\mathit{r}}_{\mathit{max}}-r_{\mathit{min}}}{\mathit{n}}\le \mathit{r}\le r_{\mathit{max}},\mathit{j}=\mathit{n}\end{array}\right.$(4) .

(4) $\left\{\begin{array}{c}{r}_{\mathit{min}}\le \mathit{r}<r_{\mathit{min}}+\frac{{\mathit{r}}_{\mathit{max}}-r_{\mathit{min}}}{\mathit{n}},\mathit{j}=1\\ r_{\mathit{min}}+\left(\mathit{j}-1\right)\times \frac{{\mathit{r}}_{\mathit{max}}-r_{\mathit{min}}}{\mathit{n}}\le \mathit{r}<r_{\mathit{min}}+\mathit{j}\times \frac{{\mathit{r}}_{\mathit{max}}-r_{\mathit{min}}}{\mathit{n}},\mathit{j}\in \left(1,\mathit{n}\right)\\ r_{\mathit{min}}+\left(\mathit{n}-1\right)\times \frac{{\mathit{r}}_{\mathit{max}}-r_{\mathit{min}}}{\mathit{n}}\le \mathit{r}\le r_{\mathit{max}},\mathit{j}=\mathit{n}\end{array}\right.$(4)

where n is the watermark length, $r_{max}$ is the maximum paradigm value, and $r_{min}$ is the minimum paradigm value. After completing this step, zero watermark construction or zero watermark extraction can be performed.

3.2. Watermark construction

The zero-watermark construction process is shown in Figure 2.

Figure 2. The flowchart of the watermark construction process.

The watermark construction steps are as follows:

Step 1: Constructing the binary sequence. Based on the paradigm value of all the elements in the partition $C_j$, the skewness measure is calculated as in Equation (3)(3) ${\alpha }_j=\frac{{\sum }_{i=1}^k{\left(r_i-\bar{r}\right)}^3}{\mathit{k}\times {\sigma }^3},\mathit{j}\in [1,\mathit{n}]$(3) , and the binary sequence $W_j$ is obtained according to the positivity and negativity of the skewness measure values, as shown in Equation (5)(5) $W_j=\left\{\begin{array}{c}0,{\alpha }_j<0\\ 1,{\alpha }_j\ge 0\end{array}\right.,\mathit{j}\in \left[1,\mathit{n}\right]$(5) .

(5) $W_j=\left\{\begin{array}{c}0,{\alpha }_j<0\\ 1,{\alpha }_j\ge 0\end{array}\right.,\mathit{j}\in \left[1,\mathit{n}\right]$(5)

where $\mathit{j}(\mathit{j}=1,2,3\dots )$ represents the binary sequence index, and n is the watermark length. $W_j$ is the binary value, and ${\alpha }_j$ is the skewness measure value. When the skewness measure is 0, it indicates that the number of element paradigms is symmetric, and then the binary value is set to 1. When the skewness metrics of all partitions have been computed, the binary sequence $\mathit{W}$is obtained.

Step 2: Obtain the watermark sequence. Operate as in Equation (6)(6) $\mathit{W}{ }^{\mathrm{\prime }}=\mathit{W}\oplus W_0$(6) for the binary sequence $\mathit{W}$to obtain the zero-watermark sequence $W^{\prime }$.

(6) $\mathit{W}{ }^{\mathrm{\prime }}=\mathit{W}\oplus W_0$(6)

where $\mathit{W}$ is the binary sequence, $W_0$is the original watermark information being encrypted based on the secret key, and $W^{\prime }$ is the zero-watermark sequence. $\oplus$ is exclusive OR () operation.

Finally, the zero-watermark is registered to the IPR of the trusted third-party intellectual property protection centre, and the timestamp mechanism is introduced to achieve the copyright protection of the BIM data.

3.3. Watermark detection

The watermark detection process is similar to the zero-watermark construction, and the flow is shown in Figure 3.

Figure 3. The flowchart of the watermark detection process.

The detection steps are as follows:

Step 1: Constructing the binary sequence. This step is the same as step 1 of the zero-watermark construction. The binary sequence $\mathit{W}$is constructed by Calculating the paradigm skewness values.

Step 2: Obtain the original watermark. Obtain the registered watermark sequence $W^{\prime }$ in the IPR database and perform the exclusive OR () operation as in Equation (7)(7) ${W_0}^{\prime }=\mathit{W}\oplus W^{\prime }$(7) , the encrypted original watermark ${W_0}^{\prime }$ is generated.

(7) ${W_0}^{\prime }=\mathit{W}\oplus W^{\prime }$(7)

where ${W_0}^{\prime }$ is the encrypted original watermark, $\mathit{W}$ is the binary sequence extracted from the BIM data according to step 1, and $W^{\prime }$ is the watermark sequence obtained from the IPR database. After decrypting the ${W_0}^{\prime }$ using the secret key, the original watermark is obtained.

4. Experiments and results

4.1. Experimental data

To validate the applicability of the proposed zero watermarking algorithm across diverse BIM data variants, two distinct categories of data were employed as experimental samples: two architectural models and two structural models. As shown in Figure 4, the selected data sets are all BIM data in RVT format. Figure 4(a,b) are architectural models, with the main element types of walls, columns, floor slabs, etc. Figure 4(c,d) are structural models, with the main element types of structural columns, structural beams, and structural frames. The basic information of the original BIM data is shown in Table 1.

Figure 4. Experimental data.

Table 1. Basic information of the experimental data.

Experimental data	Data size/(MB)	Total number of elements
Architectural model 1	16.1	669
Architectural model 2	16.5	5824
Structural model 1	12.5	727
Structural model 2	5.55	849

4.2. Experimental design and evaluation indicators

To verify the effectiveness of the algorithm, this paper designs uniqueness experiments and various attack experiments to evaluate the usability and robustness of the algorithm. A binary sequence of length 64 is selected as the original watermark sequence.

The normalized correlation coefficient (NC) is a quantitative index that measures the similarity between the extracted zero watermark and the original copyright information. When BIM data is attacked or data traceability is required, the NC value can be calculated as according to Equation (8)(8) $\mathit{NC}=\frac{XOR(W_0,{W_0}^{\prime })}{\mathit{n}}$(8) to determine data copyright ownership.

(8) $\mathit{NC}=\frac{XOR(W_0,{W_0}^{\prime })}{\mathit{n}}$(8)

where $W_0$ is the original watermark sequence, ${W_0}^{\prime }$ is the watermark sequence detected from the BIM data, n is the watermark length, and XOR () is the exclusive OR () operation.

The higher the NC value is, the higher the similarity between the two sequences. If the NC value is greater than the threshold value, it can be determined that the testing party owns the copyright of the BIM data. In practice, the NC threshold is an empirical value, that should be set according to the specific application requirements and usage scenarios, and these requirements may vary from one scenario to another. In this paper, the experimental threshold is set to 0.75.

4.3. Uniqueness experiment

Uniqueness is an important metric to gauge the practicality of a zero watermark. A unique zero watermark ought to be constructed according to the data features and be significantly different from the zero watermarks extracted from other data. In light of this, we design uniqueness experiments, utilizing four constructed zero watermarks from distinct experimental datasets as reference points. These watermarks were then compared with the zero watermarks derived from other BIM datasets. The outcome of this operation was measured via the NC value. A value exceeding the threshold underscores the presence of congruent watermarking information between the datasets. Importantly, these evaluations were conducted without subjecting the BIM data to attacks. Table 2 shows the experimental results of zero watermark uniqueness for each experimental data.

Table 2. Uniqueness experiment results.

Experimental data	NC value
	Architectural model 1	Architectural model 2	Structural model 1	Structural model 2
Architectural model 1	1.00	0.39	0.55	0.53
Architectural model 2	0.49	1.00	0.59	0.47
Structural model 1	0.58	0.55	1.00	0.59
Structural model 2	0.53	0.47	0.59	1.00

As can be seen from Table 2, except for the experimental data with its own NC value of 1, the maximum NC value in the experiments was 0.59, which was lower than the set NC coefficient value of 0.75, indicating the heightened differentiation and inherent uniqueness characterizing the zero watermarks constructed by this algorithm across diverse BIM data. In summary, the zero-watermarking algorithm for BIM data proposed in this paper has uniqueness.

4.4. Robustness experiments

The unintentional or intentional attacks that may occur in BIM data usage and transmission scenarios are fully considered, and experiments were designed for translation, rotation, element attacks, and format conversion attacks. Jing et al.‘s zero watermarking algorithm for BIM data (Jing et al. 2019) was used as the comparison algorithm, which also partitions the distance of the elements, but extracts the feature values with a sequence of distance.

4.4.1. Translation attack

Translation is a common operation on BIM data, entailing changes in model coordinate information. There may exist horizontal translation attacks and vertical translation attacks. Vertical translation attack often occurs in the BIM data and terrain and other data suite of application scenarios, and subject to the constraints of the model as a whole translation, the model relative to the distance from the project base point remains unchanged, so the model local coordinate system and the coordinates of the elements do not change. Watermark extraction is not affected in this case. The horizontal translation attack is the relative translation of the project base point, the coordinates of the elements change. The spatial relationship of the elements of the BIM model subject to the constraints of the existence of spatial relationships and the mechanism of automatic updating, so when some of the elements find large translation changes, it may cause the automatic deletion, addition, and displacement of the elements, thus causing noise in the extraction of the watermark. As shown in Figure 5, horizontal translation attacks manifest as comprehensive movements across the entire model, potentially altering the morphology of select elements due to the inherent constraints embedded within the model structure.

Figure 5. Experimental data before and after translation.

In summary, to verify the robustness of the algorithm against horizontal translation attacks, a series of experiments is architected, spanning translation distances from 50 metres to 250 metres in 50-metre increments. The experimental results are shown in Table 3 and Figure 6.

Figure 6. Result of horizontal translation attacks.

Table 3. Result of translation attacks.

Experimental data	Translation distance/(m)	NC value
		The proposed	Jing’s
Architectural model 1	50	0.99	0.92
	100	0.96	0.88
	150	0.93	0.74
	200	0.93	0.71
	250	0.93	0.71
Architectural model 2	50	0.98	0.91
	100	0.97	0.91
	150	0.97	0.89
	200	0.96	0.87
	250	0.94	0.87
Structural model 1	50	1.00	1.00
	100	1.00	1.00
	150	1.00	1.00
	200	0.98	0.94
	250	0.98	0.92
Structural model 2	50	1.00	1.00
	100	1.00	1.00
	150	1.00	1.00
	200	1.00	1.00
	250	1.00	1.00

From the analysis of Table 3 and Figure 6, it can be seen that the horizontal translation attacks had a more substantial influence on the watermark extraction of the two architectural models. While both algorithms registered the NC value below 1.00, the proposed algorithm showcased the NC value consistently surpassing 0.90. By contrast, Jing’s algorithm faltered – particularly evidenced by architectural model 1’s failure to detect the watermark when translation surpassed 150 meters. Overall, the proposed algorithm resists translation attacks better than Jing’s, which is because the change of some elements does not affect the skewness measure sign. In conclusion, the experimental results are consistent with the theoretical analyses, the proposed algorithm is strongly robust to BIM data horizontal translation attacks and outperforms comparison algorithms.

4.4.2. Rotation attack

Rotation is a common type of geometric attack on zero watermarking, and under this type of attack, automatic element deletion, update, etc., may also occur in BIM data. Taking the architectural model 1 as an example, as shown in Figure 7, the original data has been rotated by 60° and 180° with different degrees of element deletion, which may affect the zero-watermark construction. To scrutinize the algorithm’s robustness in withstanding rotation-induced attacks, this paper designs rotational attack experiments with clockwise rotation direction and 60° interval intensity, and the experimental results are shown in Table 4 and Figure 8.

Figure 7. Experimental data before and after rotation.

Figure 8. Result of rotating attacks.

Table 4. Result of rotating attacks.

Experimental data	Rotation angle/(°)	NC value
		The proposed	Jing’s
Architectural model 1	60	0.98	0.89
	120	0.98	0.89
	180	0.98	0.86
	240	0.99	0.89
	300	1.00	0.89
Architectural model 2	60	1.00	1.00
	120	1.00	1.00
	180	1.00	1.00
	240	1.00	1.00
	300	1.00	1.00
Structural model 1	60	1.00	1.00
	120	1.00	1.00
	180	1.00	1.00
	240	1.00	1.00
	300	1.00	1.00
Structural model 2	60	1.00	1.00
	120	1.00	0.99
	180	1.00	0.99
	240	1.00	0.99
	300	1.00	0.99

From Table 4 and Figure 8, it can be seen that the proposed algorithm and Jing’s can correctly detect copyright information under different intensities of rotation attacks with Experimental data extracting zero watermark NC value above the threshold value of 0.75. This is because the two algorithms’ zero watermarking only relies on the elements’ paradigm value – features unaffected by the rotation of the model. However, the deletion and addition of elements caused by rotation caused some impact, the lowest NC value of architectural model 1 watermarking under Jing’s was only 0.86, compared with the NC value of 0.98 under the proposed with the same intensity of attack, which was obviously more capable of resisting changes such as element addition and deletion caused by rotation. In conclusion, the proposed scheme can effectively resist rotational attacks better than the comparison algorithms, which effectively improves the applicability of the algorithms.

4.4.3. Element attacks

In the realm of BIM data’s practical utilization, the addition and removal of elements emerge as common attacks, which not only make the watermark bit grouping change but also may cause the spatial location of neighbouring elements to change, leading to the zero-watermark construction error. Therefore, for the robustness of the algorithm to resist the element attack, taking into account the actual production and use of data requirements, this paper designs element deletion and addition experiments from 10% to 40% in 10% intervals. Part of the data attack results are shown in Figure 9, and the experimental results are shown in Table 5 and Figure 10.

Figure 9. Experimental data before and after element attacks.

Figure 10. Result of element attacks.

Table 5. Result of element attacks.

Attack type	Experimental data	Attack ratio/(%)	NC value
			The proposed	Jing’s
Element deletion attack	Architectural model 1	10	0.99	0.77
		20	0.95	0.73
		30	0.94	0.69
		40	0.89	0.62
	Architectural model 2	10	1.00	0.94
		20	1.00	0.90
		30	0.99	0.82
		40	0.98	0.75
	Structural model 1	10	1.00	0.84
		20	1.00	0.79
		30	1.00	0.62
		40	0.99	0.61
	Structural model 2	10	1.00	0.75
		20	0.99	0.71
		30	0.97	0.62
		40	0.95	0.59
Element addition attack	Architectural model 1	10	0.92	0.79
		20	0.89	0.75
		30	0.86	0.73
		40	0.86	0.73
	Architectural model 2	10	0.99	0.98
		20	0.89	0.85
		30	0.88	0.76
		40	0.85	0.75
	Structural model 1	10	1.00	1.00
		20	0.94	1.00
		30	0.91	1.00
		40	0.88	0.94
	Structural model 2	10	1.00	1.00
		20	0.94	0.99
		30	0.93	0.99
		40	0.91	0.94

From the analysis of Table 5 and Figure 10, it can be seen that both the proposed algorithm and Jing’s algorithm were affected by the element attack, but the NC value of the proposed algorithm was higher than the threshold value and significantly better than Jing’s. This is because the deletion or addition of elements will destroy the arrangement order of the distance feature values of Jing’s, which will make the zero-watermarked grouping not uniform with the original grouping, and ultimately make it difficult to confirm the copyright information. However, the skewness measure symbols remain relatively unaffected by this phenomenon. Drilling down into specific datasets, architectural model 1—owing to its modest number of elements – evinces considerable susceptibility to element attacks. However, the NC values of zero watermarks, derived from the proposed algorithm, consistently surpassed the 0.75 threshold, in stark contrast to Jing’s algorithm which falters in detecting copyright information under multiple intensities. In a parallel vein, both algorithms’ NC values for architectural model 2 exceeded the threshold, yet Jing’s algorithm’s watermark notably introduced significant noise at attack intensities of 20% and beyond. In structural model 1 and structural model 2, the NC values of the proposed algorithms were higher than 0.85, while Jing’s was difficult to resist the element deletion attack, with the lowest NC value of only 0.59. In conclusion, theoretical and experimental results are the same, the proposed algorithms are better than Jing’s in element attack, and they can resist the different intensity element attacks, which can satisfy the daily use.

4.4.4. Format conversion attack

To meet the diverse usage requirements, BIM data is frequently subjected to format conversion for integration into alternative scenarios. Among the common format conversions, BIM data is transmuted into formats like FBX or IFC, both of which subsist as 3D models. Figure 11 shows the experimental data after format conversion. In principle, this format transformation should have no bearing on the extraction of zero watermarks. In order to verify the ability of the proposed to resist format conversion attacks, this paper designs experiments, and the experimental results are shown in Table 6.

Figure 11. The experimental data after format conversion.

Table 6. Result of format conversion attacks.

Experimental data	Format converted	NC value
		The proposed	Jing’s
Architectural model 1	FBX	1.00	1.00
	IFC	1.00	1.00
Architectural model 2	FBX	1.00	1.00
	IFC	1.00	1.00
Structural model 1	FBX	1.00	1.00
	IFC	1.00	1.00
Structural model 2	FBX	1.00	1.00
	IFC	1.00	1.00

As can be seen from Table 6, neither the proposed algorithm nor Jing’s algorithm succumbed to the influence of format conversion attacks. This is because the format conversion does not affect the distribution relationship of the elements in the model space, the four experimental data can be completely detected with zero watermark information after the format conversion, and the theory is consistent with the experimental results.

5. Discussion

The algorithm proposed in this paper, to some extent, solves the problem that existing local feature-based algorithms cannot be well applied to BIM data and obtain better watermarking robustness. The experiments in Section Ⅳ prove the above point. To better understand the proposed watermarking algorithm, this section next gives more discussion from two aspects to consider the shortcomings of the algorithm and directions for further research.

5.1. Analysis of algorithmic applicability

In the watermarking algorithm proposed in this paper, the elements with similar values of paradigms are grouped into a zone, and the mathematical analysis of the value of paradigms in the zone is performed to obtain the watermark value. Although this approach improves the robustness, accordingly, BIM models with a small number of total elements may receive limitations. Some of these models may have multiple vertices in some partitions, while others have few or no vertices, which may result in the zero-watermark extracted from the model not being unique enough to resist attacks during use.

To delve into this matter further, we elected to conduct experimentation using architectural model 1—possession of the lowest total element count. The original number of vertices for the model is 669. Delete some vertices or introduce noise randomly to change the count of vertices. Subsequently, perform various types of attack experiments on the model to observe the zero-watermarking robustness. The results of the experiment are shown in Table 7.

Table 7. Experimental results of vertex number variation.

Vertex number	NC value
	Translation 50m	Rotate 60°	Delete 30% of elements	Add 30% of elements	Convert to FBX format	Convert to IFC format
500	0.99	0.98	0.92	0.81	1.00	1.00
669	0.99	0.98	0.94	0.86	1.00	1.00
1000	0.99	0.98	0.95	0.91	1.00	1.00
1500	0.99	0.99	0.98	0.97	1.00	1.00
2000	1.00	0.99	1.00	0.98	1.00	1.00
2500	1.00	1.00	1.00	1.00	1.00	1.00

From the analysis in Table 7, it can be seen that as the total count of vertices increased, the ability of the zero watermark to resist kinds of attacks increased. When the vertex number of points reached 2500, the zero watermark NC values were all 1.00. The watermark can be completely detected. In conclusion, a more uniform vertex distribution and a more total number of vertices mean stronger robustness. This illustrates the point that BIM data with too small several elements may cause a significant reduction in watermark robustness. This finding can provide some references for subsequent research. Such as finding ways to increase the number of watermark carrier elements in BIM data to improve the robustness of zero watermarking.

5.2. Analysis of watermark length

In the algorithm proposed in this paper, multiple elements of the model correspond to one bit of watermark information, so the length of the watermark sequence should be much smaller than the total number of points of the model, otherwise it may lead to the lack of uniqueness and robustness of the zero watermark. In the experiments of this paper, a 64-bit binary sequence is chosen to carry copyright information. To discuss the effect of watermark sequence length on watermark robustness, the experimental data building model 2 with more total vertices is chosen to carry out all kinds of attack experiments with watermark lengths of 32, 64, 128, 256, and 512, and the experimental results are shown in Table 8.

Table 8. Experimental results of watermark length variation.

Watermark length	NC value
	Translation 50m	Rotate 60°	Delete 30% of elements	Add 30% of elements	Convert to FBX format	Convert to IFC format
32	1.00	1.00	1.00	1.00	1.00	1.00
64	0.98	1.00	0.99	0.88	1.00	1.00
128	0.96	0.98	0.96	0.86	1.00	1.00
256	0.89	0.94	0.93	0.82	1.00	1.00
512	0.81	0.91	0.89	0.69	1.00	1.00

From Table 8, it can be seen that as the watermark length increased, the allotment of elements within each distance partition undergone a concomitant contraction. This reduction invariably engenders a decline in the resilience of locally extracted feature values – a pivotal facet contributing to watermark robustness. When the watermark length reached 512, the NC value of the extracted watermark after increasing 30% of the elements was only 0.69—a value falling below the established threshold of 0.75. Therefore, it can be concluded that in the algorithm of this paper, the watermark length shows a negative correlation with the zero watermark. That is to say, to ensure the ability of zero watermarks to resist attacks, it is impossible to choose too long watermark information. This problem is one of the limitations of this algorithm. How to improve the watermark capacity will be an important direction for scholars to study next.

6. Conclusion

Aiming at the demand for copyright protection of BIM data and the problem that zero watermarking of existing 3D models does not apply to BIM data, a zero watermarking algorithm based on distance partitioning and skewness measure of BIM data is proposed. This algorithm is based on the element spatial distribution characteristics of the model elements to partition, and establish the mapping relationship between the watermarking bits and the spatial location of the model, which improves the robustness of the algorithm. At the same time, using the skewness measure of the element paradigm value in each partition to construct the watermarking information, and this kind of local characteristics with a certain fault tolerance makes the algorithm resistant to element attacks. Experiments show that the constructed algorithm in this paper has uniqueness and the ability to resist translation, rotation, element deletion, element increase, and format conversion attacks are better than the comparison algorithm. The theory is consistent with the empirical evidence. In conclusion, the proposed algorithm provides an effective and feasible scheme for BIM data copyright protection, which has certain practical value for security protection. In the future, how to increase the watermark embedding capacity and improve the robustness will be the direction of deeper research on zero watermarking of BIM data.

Author contributions

Qianwen Zhou, Changqing Zhu, Na Ren, and Qifei Zhou performed the conceptualization; Qianwen Zhou, Changqing Zhu, and Na Ren performed the methodology; Qianwen Zhou and Na Ren performed the validation; Qianwen Zhou and Changqing Zhu performed the formal analysis; Qianwen Zhou performed the original writing; Qianwen Zhou and Na Ren performed the review & editing; Qianwen Zhou and Changqing Zhu performed the visualization; Changqing Zhu, Na Ren, and Qifei Zhou performed the supervision; Changqing Zhu, Na Ren, and Qifei Zhou performed the funding acquisition.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The data and codes that support the findings of this study are available from the corresponding author upon reasonable request.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [42071362] and [42301482]; the National Key Research and Development Program of China [2023YFB3907100]; and Open Project of Hunan Engineering Research Center of Geographic Information Security and Application [HNGISA2023002].