A novel watermarking algorithm for three-dimensional point-cloud models based on vertex curvature

Abstract

A new blind watermarking scheme for three-dimensional point-cloud models is proposed based on vertex curvature to achieve an appropriate trade-off between transparency and robustness. The root mean square curvature of local set of every vertex is first calculated for the three-dimensional point-cloud model and then the vertices with larger root mean square curvature are used to carry the watermarking information; the vertices with smaller root mean square curvature are exploited to establish the synchronization relation between the watermark embedding and extraction. The three-dimensional point-cloud model is divided into ball rings, and the watermarking information is inserted by modifying the radial radii of vertices within ball rings. Those vertices taking part in establishing the synchronization relation do not carry the watermarking information; therefore, the synchronization relation is not affected by the embedded watermark. Experimental results show the proposed method outperforms other well-known three-dimensional point-cloud model watermarking methods in terms of imperceptibility and robustness, especially for against geometric attack.

Introduction

Over the past few years, three-dimensional (3D) models are more and more important to business objectives. Greater amounts of sensitive data related to 3D models traversed both wired and wireless networks. These valuable data in business enterprises attract an increasing number of malware applications and hackers. ¹ As a way of preventing network from becoming a treasure trove for cybercriminals, the authentication and copyright of the 3D models have become an increased focus. Digital watermarking is considered as one of the effective solutions to protect the authentication and copyright for the 3D models. ²

The boundary representation of a 3D model has a point-cloud model, a mesh model, and a surface model. In recent years, there are many research works on digital watermark for 3D mesh models. Many of them are watermarking algorithms involving spatial domain. In spatial watermarking algorithms,^3–7 watermarking information is embedded by directly modifying topological or geometric features of mesh models. Liu et al. ³ presented a watermarking method for triangular mesh models; in this method, the multi-resolution adaptive parameterization of surface (MAPS) approach is used to partition the model, and the watermarking information is inserted into the vertices lying in the fine level. Zhan et al. ⁶ divided the model into bins in terms of the root mean square curvature (RMSC) value in the local neighborhood of every vertex; the mean curvature value of each bin was modified to embed watermark information. Jiang et al. ⁷ proposed a watermarking method that operates the least-significant bits of integer of the vertex coordinates to embed watermarking data. In frequency watermarking algorithms,^8–10 some frequency toolkits are used for to capture topological or geometric features of mesh models, and the watermarking information is embedded into the frequency coefficients representing the topological or geometric features. Wang et al. ⁸ proposed a high-capacity watermarking method for a semi-regular mesh model that the watermarking information is embedded into wavelet coefficients in different levels. Ai et al. ⁹ separated 3D mesh models into Voronoi patches, each of which was projected to a reference square plane. By applying the 2D discrete cosine transform (DCT) transformation to each square plane, a bit of the watermark information was repeatedly insert into the transform coefficients of each projection image. These watermarking algorithms for 3D mesh models have demonstrated a better robustness against geometric attack and have protected the copyright of 3D mesh models to some extent.

However, compared with the attention to the mesh model, research on digital watermarking for the 3D point-cloud models is very few. Cotting et al. ¹¹ separated a 3D point-cloud model into patches using a fast-hierarchical clustering approach; then, Laplacian operation was applied to each patch and the low-frequency components were selected from each Laplacian patch for watermark embedding. The 3D point-cloud model method proposed in Ohbuchi et al. ¹² was an extension from the watermarking methods of 3D mesh models. In this method, the disjoint clusters that were derived from the disordered points of the model first produced meshes and then Zachi’s spectral analysis was applied to the meshes; the watermarking information was embedded into the coefficients of the frequency domain. Feng ¹³ also presented a watermarking algorithm for 3D point-cloud models. In this scheme, the point-cloud model was split into a set of patches and watermarking information was inserted into these patches by modulating angle quantization index. By modifying vector length of points in each patch obtained from the partitioned the 3D point-cloud model, a self-similarity-based watermarking method for 3D point-cloud models was proposed in Ke et al., ¹⁴ in which each bit of the watermark information was repeatedly embedded to improve its robustness. Feng ¹⁵ split the model into several bins and embedded a watermarking bit by modulating the mean distance of the bin.

Overall, research on digital watermarking for 3D point-cloud models is still at early stage. 3D point-cloud models do not have any connection information between points, ¹⁶ which enables extracting the watermark to become very difficult. Therefore, the big challenge of robust point-cloud watermarking algorithms is how to find exactly the watermarked vertices at the extraction stage.

In this article, we propose a robust watermarking method for 3D point-cloud models. In order to be still able to locate the watermarked vertices in the case of the disorder of vertices of 3D point-cloud model, the synchronization relation is first established; then, the model is separated into ball rings in terms of radial radius and the watermark bit is repeatedly inserted into vertices of each ball ring. The vertices with larger RMSC are selected to carry watermarking information; the synchronization relation is established using the vertices with smaller RMSC. The main advantages of the proposed method are as follows: (1) the vertices, which are selected to carry the watermarking information, are located in the abrupt changing regions having a good visual masking effect. This helps to carry the large capacity watermarking information and enables the method to achieve an appropriate trade-off between transparency and robustness; (2) the watermarking information is not embedded into those vertices used for establishing the synchronization relation, which ensures that watermark detection and embedding are performed in the same coordinate system, thus improving the robustness of our method against geometric attack; and (3) each bit of watermarking information is repeatedly embedded by modifying radial radii of vertices of each ball ring. The ball ring embedding and repeated embedding enable the method to successfully overcome the problem of disorder of vertices for 3D point-cloud models. The flowchart of the proposed watermarking method of 3D point-cloud models is shown in Figure 1.

Figure 1.

Framework of the proposed method: (a) flowchart for watermark embedding and (b) flowchart for watermark extraction.

The article is organized as follows. Section “The proposed watermarking algorithm” describes the proposed method in detail. Section “Experiments and results” gives the simulation results and analyses about the performances of the proposed method. Finally, we state the conclusion and discuss future work in section “Conclusion.”

The proposed watermarking algorithm

Selection of the embedding positions

Whether the watermarking information can be embedded into the appropriate vertices plays an important role in achieving a good trade-off between imperceptibility and robustness. Theoretically, the embedding of the watermark into the 3D point-cloud models should not produce any visual changes. That the vertices of the flat regions of the 3D model are disturbed would result in perceptible distortions of the model, whereas the vertices of the regions with abrupt changes disturbed would not lead to marked visual distortions. Therefore, the vertices lying in the bumpy regions should be appropriate to carry the watermark information. We use the RMSC value of local set to find these vertices from the model, requiring the following two steps: (1) defining the local sets of vertices and (2) calculating the RMSC values of vertices.

Defining the local set

3D point-cloud model is a representation of a set of vertices lying in the exterior surfaces of an object and these vertices have no implicit order like pixels of 2D images (Figure 2(a)). To describe geometric feature of the neighborhood of each vertex, there is a need to define a geometric structure, the local set we called. Whether the vertex is suitable for carrying the watermark information depends on the feature parameter that indicates the geometry property of its local set. Delaunay triangulation scheme is exploited to define the local set of each vertex.

Figure 2.

(a) 3D point-cloud model for Pig; (b) the local set of vertex $V_i$ is formed by the vertices marked with white.

For every vertex of the 3D model, if there is a Delaunay edge connecting vertices $V_i$ and $V_k$ , vertex $V_k$ is a neighbor of vertex $V_i$ . All neighbors of vertex $V_i$ form the local set of vertex $V_i$ and is represented as

$$N\left(V_i\right)=\{V_j|0⩽\left|\overline{V_i{V}_j}\right|⩽1,j=0,1,\dots ,N\}$$

(1)

where $\left|\overline{V_i{V}_j}\right|$ expresses the number of vertices between vertices V_i and V_j, and N is the number of all the vertices of the 3D point-cloud model. Figure 2(b) shows the vertices within the local set of vertex of the point-cloud model.

Calculating RMSC

The vertices that are most appropriate to carry the watermarking information should be located in abrupt-change regions. In order to find these vertices, the Gaussian curvature and the mean curvature of each vertex should be first obtained. The Gaussian curvature G and the mean curvature H of vertex V_i can be approximately calculated from all the vertices within its local set; the calculation is shown in equations (2) and (3)

$$G=\frac{(2\pi -\sum _{j=1}^{N_i}{\alpha }_j)}{\sum _{j=1}^{N_i-1}{A}_j}$$

(2)

$$H=\frac{\sum _{j=1}^{N_i-1}{\beta }_j}{\sum _{j=1}^{N_i-1}{A}_j}$$

(3)

where N_i is the total number of local set of vertex V_i, α_j (j = 1, 2, …, N_i) denotes the interior angle adjacent to vertex V_i, β_j (j = 1, 2, …, N_i − 1) represents the angles between the normals of the two adjacent triangles, and A_j (j = 1, 2, …, N_i − 1) is the triangle area belonging to the local set of vertex V_i.

When the Gaussian curvature G and the mean curvature H of each vertex are obtained, the bumpy region of the 3D model can be measured according to the following equation

$$K_{\mathrm{rmsc}}\left(V_i\right)=\sqrt{2H^2-G}$$

(4)

The RMSC value is represented with $K_{\mathrm{rmsc}}(V_i)$ , which is regarded as a comprehensive evaluation of the abrupt-change characteristic of a surface determined by the vertices within the local set. Vertex V_i with a larger value $K_{\mathrm{rmsc}}(V_i)$ denotes that vertex V_i is situated in the regions with abrupt changes. That is, vertex V_i is probably appropriate to embed the watermark.

All the vertices of a 3D point-cloud model are ordered according to $K_{\mathrm{rmsc}}(V_i)$ values. If the threshold value is set to K_h and $K_{\mathrm{rmsc}}(V_i)$ value is not less than K_h, vertex V_i is regarded as a candidate vertex that may be appropriate to carry the watermarking information. The vertices whose $K_{\mathrm{rmsc}}(V_i)$ values are more than K_h form the candidate set S_c and the rest constitute reference set S_r (see Figure 3). The vertices of the candidate set S_c are used for the embedding of the watermark, whereas the vertices within the reference set S_r are used to establish the synchronization relation between the watermark embedding and extraction. Selection of threshold value K_h is closely related to the capacity of the watermarking information. Suppose that there are M bits of watermarking information, the 3D point-cloud model will be separated into M ball rings in terms of radial radius. Threshold value K_h is equal to the value $K_{\mathrm{rmsc}}(V_m)$ of ensuring that there is at least one candidate vertex in each ball ring. Notice that any two candidate vertices should not belong to the same local set; therefore, these candidate vertices into which the watermarking information is embedded should be irrelevant to each other.

Figure 3.

The candidate vertices (vertices marked with red) carry the watermarking information and the rest are used to establish the synchronization relation.

Establishing the synchronization information

Unlike pixels of an image, vertex data of 3D models do not have an implicit order. In order to find those watermarked vertices during the watermark extraction to improve the performance against geometric attacks, a synchronization relation needs to be established between the processes of watermark embedding and watermark extraction. In the method, we used the vertices of the reference set S_r to build a new coordinate system and embedded the watermarking information into the candidate vertices in the built coordinate system. During the watermark extraction, although the coordinate values of vertices have changed when the model is subjected to geometric attack, the relative positions between the watermarked vertices and the origin of the built coordinate system remain unchanged. This is because the embedded watermarking information has no any impacts on the vertices that are used to build the coordinate system. The steps of establishing the synchronization information are depicted as following:

• 1. The coordinate origin.

The origin of the established coordinate system is defined as equation (5)

$$V_c=(V_{c_x},V_{c_y},V_{c_z})=\frac{1}{N_l}\sum _i^{N_l}{V}_i$$

(5)

where $N_l$ represents the number of vertices of the reference set S_r, and vertices $V_i=(V_{i_x},V_{i_y},V_{i_z}),i=1,2,\dots ,N_l$ belong to the reference set S_r.

• 2. The coordinate axes.

Suppose that the coordinate axes of the established synchronization relation are u, v, and n axes, a covariance matrix Cov is first constructed using vertices of the reference set S_r to determine the directions of u, v, and n axes

$$\mathrm{Cov}=\left[\begin{array}{ccc}\sum _{i=0}^{N_l}{V}_{i_x}^2\hfill & \sum _{i=0}^{N_l}{V}_{i_x}{V}_{i_y}\hfill & \sum _{i=0}^{N_l}{V}_{i_x}{V}_{i_z}\hfill \\ \multicolumn{1}{c}{\sum _{i=0}^{N_l}{V}_{i_y}{V}_{i_x}}& \sum _{i=0}^{N_l}{V}_{i_y}^2\hfill & \sum _{i=0}^{N_l}{V}_{i_y}{V}_{i_z}\hfill \\ \multicolumn{1}{c}{\sum _{i=0}^{N_l}{V}_{i_z}{V}_{i_x}}& \sum _{i=0}^{N_l}{V}_{i_z}{V}_{i_y}\hfill & \sum _{i=0}^{N_l}{V}_{i_z}^2\hfill \end{array}\right]$$

(6)

The square matrix Cov is decomposed via eigen decomposition and can be represented as

$$\mathrm{Cov}=Q\Sigma Q^{-1}$$

(7)

where matrix $\mathrm{\Sigma }$ represents a diagonal matrix consisting of eigenvalues of matrix Cov and matrix Q is composed of the eigenvectors corresponding to the eigenvalues of matrix Cov. u, v, and n axes are decided by the eigenvectors that correspond to the three largest eigenvalues. After the new coordinate system is established, the spherical coordinates in the established coordinate space for the vertices of the model can be obtained by the transformations as following.

• 3. Translation transformation.

Translation matrix T_f

$$T_f=\left[\begin{array}{cc}\begin{array}{cc}1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \end{array}\hfill & \begin{array}{cc}0\hfill & -V_{c_x}\hfill \\ \multicolumn{1}{c}{0}& -V_{c_y}\hfill \end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{cc}0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}}& \begin{array}{cc}1\hfill & -V_{c_z}\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \end{array}\hfill \end{array}\right]$$

(8)

The corresponding coordinates of the transformed vertex are represented as

$$\begin{array}{c}\multicolumn{1}{c}{{\overline{V}}_i=({\overline{V}}_{i_x},{\overline{V}}_{i_y},{\overline{V}}_{i_z})=T_f·V_i}\\ \multicolumn{1}{c}{=(V_{i_x}-V_{c_x},V_{i_y}-V_{c_y},V_{i_z}-V_{c_z})}\\ \multicolumn{1}{c}{=\left[\begin{array}{cc}\begin{array}{cc}1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \end{array}\hfill & \begin{array}{cc}0\hfill & -V_{c_x}\hfill \\ \multicolumn{1}{c}{0}& -V_{c_y}\hfill \end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{cc}0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}}& \begin{array}{cc}1\hfill & -V_{c_z}\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \end{array}\hfill \end{array}\right]\left[\begin{array}{c}\begin{array}{c}{V}_{i_x}\hfill \\ \multicolumn{1}{c}{V_{i_y}}\end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{c}{V}_{i_z}\hfill \\ \multicolumn{1}{c}{1}\end{array}}\end{array}\right]}\end{array}$$

(9)

• 4. Rotation transformation.

Rotation matrix R_f

$$\begin{array}{c}{R}_f=R({\theta }_x,{\theta }_y,{\theta }_z)=\hfill \\ \left[\begin{array}{cccc}\cos {\theta }_y\cos {\theta }_z-\sin {\theta }_y\sin {\theta }_x\sin {\theta }_z\hfill & -\cos {\theta }_y\sin {\theta }_z-\sin {\theta }_y\sin {\theta }_x\cos {\theta }_z\hfill & -\sin {\theta }_y\cos {\theta }_x\hfill & 0\hfill \\ \multicolumn{1}{c}{\cos {\theta }_y\sin {\theta }_z}& \cos {\theta }_x\cos {\theta }_z\hfill & -\sin {\theta }_x\hfill & 0\hfill \\ \multicolumn{1}{c}{\sin {\theta }_y\cos {\theta }_z+\cos {\theta }_y\sin {\theta }_x\sin {\theta }_z}& -\sin {\theta }_y\sin {\theta }_z+\cos {\theta }_y\sin {\theta }_x\cos {\theta }_z\hfill & \cos {\theta }_y\cos {\theta }_x\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right]\hfill \end{array}$$

(10)

where ${\theta }_x,{\theta }_y,\mathrm{and}{\theta }_z$ represent the angles between the axis u (or v or n) and the plane XOZ (or YOZ or XOY). The corresponding coordinates of the transformed vertex are represented as

$${\tilde{V}}_i=({\tilde{V}}_{i_u},{\tilde{V}}_{i_v},{\tilde{V}}_{i_n})=R_f·{\overline{V}}_i$$

(11)

• 5. Conversion to spherical coordinates.

The vertices of the model need to be represented with spherical coordinates, and vertex V_i with Cartesian coordinates $({\tilde{V}}_{i_u},{\tilde{V}}_{i_v},{\tilde{V}}_{i_n})$ can be represented as vertex V_i with spherical coordinates (ρ_i, θ_i, ϕ_i) shown in Figure 4(a).

• 6. Scaling the model.

Figure 4.

(a) The spherical coordinates of vertex V_i, (b) dividing the model into ball rings, and (c) embedding a watermark bit into each ball ring.

In order to resist against scaling attacks, component ${\rho }_i$ of spherical coordinates of vertex V_i should be normalized using equation (13).

Scaling factor S_f

$$S_f=\max ({\rho }_1,{\rho }_2,\dots ,{\rho }_i,\dots ,{\rho }_N)$$

(12)

The final coordinates of the transformed vertex are represented as

$${\hat{V}}_i=({\hat{\rho }}_i,{\theta }_i,{\phi }_i)=\frac{{\rho }_i}{S_f}·{\tilde{V}}_i=(\frac{{\rho }_i}{S_f},{\theta }_i,{\phi }_i)$$

(13)

Watermark embedding

In addition to establishing the synchronization relation, the 3D point-cloud model has to be segmented into several units to locate the embedding positions of the watermark. We assume that the 3D point-cloud model is encircled by a sphere whose center is the coordinate origin of the established coordinate system, and radius is equal to the largest value of spherical coordinate of the 3D model. Then, the model is split into M (M represents the watermarking capacity) ball rings between the minimum radius ρ_min and maximum radius ρ_max using equation (14)

$$\begin{array}{c}\multicolumn{1}{c}{R_m=\{{\rho }_m|{\rho }_{\min }+\frac{{\rho }_{\max }-{\rho }_{\min }}{M}\times (m-1)⩽{\rho }_m⩽{\rho }_{\min }}\\ \multicolumn{1}{c}{+{\rho }_{\max }-\frac{{\rho }_{\min }}{M}\times m\},1⩽m⩽M}\end{array}$$

(14)

where R_m expresses all vertices belonging to the mth ball ring, and ${\rho }_m$ represents the radial radii of vertices within the mth ball ring. Figure 4(b) shows the projection of all vertices within the two ball rings. The red points represent the candidate vertices while the black points express the reference vertices that are used to build the synchronization relation.

The watermarking information is embedded by modulating the radial radius ${\hat{\rho }}_i$ of candidate vertex ${\hat{V}}_i$ while remaining θ_i and φ_i unchanged. The candidate vertices within each ball ring carry the same bit of watermarking information. If the radial interval between the two ball rings is $d=({\rho }_{\max }-{\rho }_{\min })/M$ , the parameter value $\mathrm{\epsilon }$ is set to $(1/2)\times d$ . Suppose that candidate vertex ${\hat{V}}_i$ ( ${\hat{\rho }}_i$ , θ_i, ϕ_i) carries a watermark bit and vertex ${\hat{V}}_i^{\prime }({\hat{\rho }}_i^{\prime },{\theta }_i,{\phi }_i)$ represents the corresponding watermarked vertex, component ${\hat{\rho }}_i^{\prime }$ of the watermarked vertex ${\hat{V}}_i^{\prime }$ is modulated by equation (15) as below

$${\hat{\rho }}_{\mathrm{mi}}^{\prime }=\{\begin{array}{cc}{\hat{\rho }}_{\mathrm{mi}}+\epsilon ,\hfill & \mathrm{if}{w}_m=1\hfill \\ \multicolumn{1}{c}{{\hat{\rho }}_{\mathrm{mi}}-\epsilon ,}& \mathrm{if}{w}_m=-1\hfill \end{array}$$

(15)

where ${\hat{\rho }}_{\mathrm{mi}}^{\prime }$ denotes the radial radius of the ith embedded vertex in the mth ball ring while ${\hat{\rho }}_{\mathrm{mi}}$ represents the radial radius of the ith candidate vertex before carrying the watermarking bit. The candidate vertex ${\hat{V}}_i$ that is chosen from each ball ring to carry watermarking information should satisfy the requirement of equation (16) to ensure that the watermarked vertex still situates in the same ring

$$\begin{array}{c}if{w}_m=1,\left(m-1\right)\times d⩽{\widehat{\rho }}_i<\left(m-½\right)\times d\hfill \\ if{w}_m=-1,\left(m-½\right)\times d⩽{\widehat{\rho }}_i<m\times d\hfill \end{array}$$

(16)

Figure 4(c) helps to explain the meaning of equation (16). Supposing that regions R₁ and R₂ on the largest cross section are the projections of the two ball rings and all the vertices of these two ball rings fall within region R₁ and region R₂. The black points express the vertices of the reference set S _r while the red points represent the vertices of the candidate set S_c. If the embedded bit of the watermark information is “1,” only the candidate vertices (points marked with red) within the inner half of region R₁ are modulated and moved to the outer half of region R₁ to become the watermarked vertices (points marked with green), whereas the candidate vertices (points marked with red) within the outer half of region R₁ keep unchanged because they do not meet the requirement of equation (16). If the embedded bit of the watermark information is “–1,” only the candidate vertices (points marked with red) within the outer half of region R₂ are modified and moved to the inner half of region R₂ to become the watermarked vertices (points marked with blue). Similarly, the candidate vertices (points marked with red) within the inner half of region R₂ should not be modified. We summarize the main steps for embedding a watermark as below.

The proposed framework for embedding a watermark
Input: the 3D point-cloud model, the watermarking information with M bits.Output: the watermarked 3D point-cloud model.1 Stage 1: distinguish the vertices of the 3D model2 for each vertex Vido3 Calculate RMSC $(K_{\mathrm{rmsc}}(V_i))$ value of its local set using equation (4).4 if $K_{\mathrm{rmsc}}(V_i)$ is not less than Kh, vertex Vi is a vertex of the candidate set Sfelse vertex Vi is assigned to the reference set Sr.5 end67 Stage 2: establish the synchronization relation8 Build a new coordinate space using the steps outlined in section “Establishing the synchronization information.”9 Cartesian coordinate of the model is converted to spherical coordinate.10 Out: spherical coordinates of each vertex relative to the origin of the established coordinate space.1112 Stage 3: separate the 3D point-cloud model13 surround the model with a sphere.14 split the model into M ball rings.1516 Stage 4: insert watermarking information17 for the bit $w_m$ of watermarking information do18 Modulate candidate vertices of each ball ring using equation (15).19 end2021 Spherical coordinates of the watermarked model is converted to Cartesian coordinates.22 Result: obtain the watermarked model.

Watermark extraction

In the proposed method, the watermark extracting does not need the original 3D point-cloud model, that is, it is a blind extraction. Like the process of the watermark embedding, the RMSCs $K_{\mathrm{rmsc}}(V)$ of all the vertices of the tested 3D model are first computed; then, all of vertices are divided into candidate vertices (candidate set S_c) and reference vertices (reference set S_r). The candidate vertices are used to extract the watermark sequence while the reference vertices are utilized to set up the synchronization information of extracting the watermark. After Cartesian coordinate of the tested model is converted to spherical coordinate and a sphere bound box is created for the model, the 3D tested point-cloud model is classified into M ball rings. Every bit of the watermarking information depends on the number of candidate vertices within the inner part and the outer part in the ball ring. The watermarking information hidden in the mth ball ring is determined using equation (17)

$$w_m^{\prime }=\{\begin{array}{cc}1,\hfill & \mathrm{if}\mathrm{num}1>\mathrm{num}2\hfill \\ \multicolumn{1}{c}{-1,}& \mathrm{if}\mathrm{num}1<\mathrm{num}2\hfill \end{array}$$

(17)

where num1 and num2 denote the number of candidate vertices within the inner part and the outer part in the mth ball ring, respectively. If the candidate vertex is located in the outer part, add to num1 with a value of 1; if the candidate vertex lies in the inner part, add value 1 to num2. The main steps of watermark extraction were summarized as below.

The proposed framework of extracting a watermark

Input: the tested 3D point-cloud model, the value M.Output: a binary sequence $w\prime$ .1 Stage 1: distinguish the vertices of the 3D model2 for each vertex Vido3 Calculate RMSC $(K_{\mathrm{rmsc}}(V_i))$ value of its local set using equation (4).4 if $K_{\mathrm{rmsc}}(V_i)$ is not less than Kh, vertex Vi is a vertex of the candidate set Sfelse vertex Vi is assigned to the reference set Sr.5 end67 Stage 2: establish the synchronization relation8 Build a new coordinate space using the steps outlined in section “Establishing the synchronization information.”9 Cartesian coordinate of the model is converted to spherical coordinate.10 Out: spherical coordinates of each vertex relative to the origin of the established coordinate space.1112 Stage 3: separate the 3D point-cloud model13 surround the model with a sphere.14 split the model into M ball rings.1516 Stage 4: extract watermarking information17 for each ball ring do18 Calculate the watermark bit $w_m^{\prime }$ using equation (17).19 end2021 Result: obtain the watermarking information $w\prime$ .

Experiments and results

We conducted the experiments to validate imperceptibility and robustness of the proposed method in this section, compared with the watermarking algorithms for 3D point-cloud models in Ke et al. ¹⁴ and Feng. ¹⁵ The experiments were carried out on some 3D models chosen from Stanford model library and the Princeton Shape Benchmark library, four models of which are shown in Figure 5: Bunny, M365, M355, and M285. The watermarking information with the size of $16\times 16\mathrm{bits}$ was embedded into each model. The tested attacks include common attack (noise and simplification) and geometric attack (cropping and rotation). The imperceptibility of the proposed method was evaluated using the root mean square error (RMSE). ¹⁷ The RMSE is defined by

$$\mathrm{RMSE}=\sqrt{\sum _{i=1}^N{(x_i^{\prime }-x_i)}^2+{(y_i^{\prime }-y_i)}^2+{(z_i^{\prime }-z_i)}^2}$$

(18)

Figure 5.

Original cloud-point models: (a) Bunny, (b) M365, (b) M355, and (d) M285.

In addition, the signal-to-noise ratio (SNR) was also used to evaluate the model degradation caused by the embedded watermark. The SNR is given by equation (19)

$$\mathrm{SNR}=10\mathrm{lo}{g}_{10}\left(\frac{\sum _{i=1}^N{x}_i^2+y_i^2+z_i^2}{\sum _{i=1}^N{(x_i^{\prime }-x_i)}^2+{(y_i^{\prime }-y_i)}^2+{(z_i^{\prime }-z_i)}^2}\right)$$

(19)

where N is the number of vertices of the 3D model, $(x_{i,}{y}_i,z_i)$ is the coordinates of vertex $V_i$ of the original model and $(x_i^{\prime },y_i^{\prime },z_i^{\prime })$ is the coordinates of vertex $V_i^{\prime }$ of the watermarked model. Lower RMSE value or higher SNR value illustrates that the embedded watermark makes the 3D model produce the smaller extent of distortion. The correlation coefficient Corr was used to evaluate the robustness of the proposed method against the various attacks. The Corr is a value that varies between 0 and 1, which can be calculated using equation (20)

$$\mathrm{Corr}(w,w\prime )=\frac{\sum _{i=1}^m{w}_i\times w_i^{\prime }}{\sqrt{\sum _{i=1}^m{(w_i^{\prime })}^2}\times \sqrt{\sum _{i=1}^m{(w_i)}^2}}$$

(20)

where $w_i$ and $w_{\prime }^i$ express the originally inserted watermarking bit and the extracted watermarking bit. The Corr value denotes the extent of similarity between the two watermarking sequences. The closer the Corr value is to 1, the more similar the two watermarking sequences are. In addition, we also used the receiver operating characteristics (ROC) curve to further verify the robustness performance of our watermarking method. The ROC curve is an assessment indicator of the watermarking benchmark that is suggested in Bors and Luo. ¹⁸

Imperceptibility evaluation

Figure 6 shows the four models that the watermarking information is embedded into using our proposed method. Some vertices with a larger RMSC value were chose from ball rings to carry the watermarking information. Radial radius ρ of the candidate vertex is modulated to embed the watermarking information while the other two components θ and φ are kept unchanged. These mechanisms are helpful for achieving a better imperceptibility for the watermark. It can also be seen from Figures 6 and 7 that the watermarked models have no significant difference from the original 3D models in visual perception. The two-objective metrics of RMSE and SNR also confirm the observation. In Table 1, the RMSE values are lower while the SNR values are higher obtained from our proposed method, when compared with the corresponding results obtained with the methods in Ke et al. ¹⁴ and Feng. ¹⁵ Beyond that, we observe that the imperceptibility of the watermark is associated with the proportion of the watermarked vertices that occupied in the 3D models. When each model carried the watermark with the same bits, the smaller the percentage of the watermarked vertices that occupied the 3D models is, that is, the model might contain plenty of vertices, the better the invisibility of the watermark is. Among the four models exhibited in Figure 5, the number of vertices of the 3D model Bunny is maximum while the 3D model M285 contains the minimum number of vertices. The objective values (RMSE and SNR) in Table 1 shows that, whether for our proposed method or the two methods in Ke et al. ¹⁴ and Feng, ¹⁵ the 3D model Bunny achieved the best results while the 3D model M285 got the worst results. However, whether for model M285 containing sparse vertices or model Bunny with dense vertices, our proposed method achieves the RMSE results less than 70 × 10⁻⁴ and the SNR results greater than 40 dB. This demonstrates that our proposed method can ensure the visual quality of the watermarked models under meeting the requirement of the capacity of the embedded watermark.

Figure 6.

Original illumination models: (a) Bunny, (b) M365, (b) M355, and (d) M285.

Figure 7.

Watermarked models: (a) Bunny, (b) M365, (b) M355, and (d) M285.

Table 1.

Invisibility comparison of the watermarking methods in terms of RMSE × 10⁻⁴ and SNR.

Models	Methods
	Method in Ke et al. 14		Method in Feng 15		Our proposed method
	RMSE	SNR (dB)	RMSE	SNR (dB)	RMSE	SNR (dB)
Bunny	48	45.2	46	47.1	43	48.2
M365	64	41.5	62	42.3	59	44.5
M355	70	40.8	68	40.6	65	42.9
M285	73	35.4	71	36.7	70	40.7

RMSE: root mean square error; SNR: signal-to-noise ratio.

Robustness evaluation

Robustness is an indispensable evaluation metric of 3D watermarking methods. The comparison of the robustness of the three methods was conducted on each model that the same watermarking information with the size of $16\times 16\mathrm{bits}$ was embedded into. The results indicating the robustness of the three methods against common attack and geometric attack are given in Table 2 and Figures 8 and 9. Among of them, Figures 8 and 9 give the graphic comparisons about the average values of the correlation coefficient Corr obtained from the tested models.

Table 2.

The values Corr of four 3D point-cloud models when subjected to different attacks.

Attacks		Method	Models				Average
			Bunny	M365	M355	M285
Additive noise	0.01%	Method in Ke et al. 14	0.9836	0.9718	0.9625	0.9539	0.9680
		Method in Feng 15	0.9892	0.9785	0.9716	0.9681	0.9769
		Proposed method	0.9921	0.9813	0.9797	0.9748	0.9820
	0.05%	Method in Ke et al. 14	0.8613	0.8545	0.8447	0.8334	0.8485
		Method in Feng 15	0.8692	0.8661	0.8514	0.8446	0.8578
		Proposed method	0.8802	0.8754	0.8741	0.8692	0.8747
	0.09%	Method in Ke et al. 14	0.5789	0.5681	0.5424	0.5262	0.5539
		Method in Feng 15	0.5892	0.5687	0.5593	0.5316	0.5622
		Proposed method	0.6015	0.5943	0.5739	0.5622	0.5830
Simplification	3%	Method in Ke et al. 14	0.9757	0.9717	0.9624	0.9641	0.9685
		Method in Feng 15	0.9801	0.9778	0.9658	0.9643	0.9720
		Proposed method	0.9976	0.9959	0.9907	0.9893	0.9934
	12%	Method in Ke et al. 14	0.8816	0.8729	0.8626	0.8578	0.8687
		Method in Feng 15	0.8887	0.8723	0.8698	0.8624	0.8733
		Proposed method	0.9384	0.9296	0.9193	0.9199	0.9268
	21%	Method in Ke et al. 14	0.6928	0.6889	0.6827	0.6804	0.6862
		Method in Feng 15	0.7272	0.7211	0.7194	0.7106	0.7196
		Proposed method	0.7731	0.7687	0.7513	0.7428	0.7590
	30%	Method in Ke et al. 14	0.5123	0.4945	0.4709	0.4607	0.4846
		Method in Feng 15	0.5218	0.4979	0.4827	0.4622	0.4912
		Proposed method	0.5654	0.5432	0.5299	0.5276	0.5415
Rotation	5º	Method in Ke et al. 14	0.6738	0.6671	0.6526	0.6547	0.6621
		Method in Feng 15	0.7838	0.7406	0.7026	0.6819	0.7272
		Proposed method	0.9015	0.8726	0.8503	0.8312	0.8639
	20º	Method in Ke et al. 14	0.5002	0.4856	0.4712	0.4634	0.4801
		Method in Feng 15	0.5293	0.5178	0.5049	0.4956	0.5119
		Proposed method	0.5884	0.5843	0.5768	0.5679	0.5794
	35º	Method in Ke et al. 14	0.4712	0.4654	0.4476	0.4338	0.4545
		Method in Feng 15	0.4936	0.4817	0.4589	0.4426	0.4692
		Proposed method	0.5293	0.5218	0.5196	0.5122	0.5207
Cropping	5%	Method in Ke et al. 14	0.8594	0.8519	0.8424	0.8397	0.8484
		Method in Feng 15	0.8618	0.8577	0.8428	0.8406	0.8507
		Proposed method	0.9473	0.9345	0.9263	0.9106	0.9297
	15%	Method in Ke et al. 14	0.5798	0.5695	0.5566	0.5474	0.5633
		Method in Feng 15	0.6106	0.6012	0.5939	0.5897	0.5989
		Proposed method	0.7154	0.7009	0.6925	0.6851	0.6985
	25%	Method in Ke et al. 14	0.4976	0.4865	0.4715	0.4609	0.4791
		Method in Feng 15	0.5188	0.5001	0.4996	0.4849	0.5009
		Proposed method	0.5564	0.5498	0.5347	0.5218	0.5407

Figure 8.

Comparison of average values of the correlation coefficients: robustness against (a) noise (σ × 10⁻³) and (b) simplification (%).

Figure 9.

Simplification attack on models (50% reduction): (a) Bunny, (b) M365, (c) M355, and (d) M285.

Additive noise attack

The Corr values of the second line in Table 2 were obtained in the case that the watermarked models were subjected to the attacks of Gaussian noise. Figure 8(a) shows the robustness of the three methods against noise when the strength $\sigma$ of additive noise changes from 0 to 0.001. It is observed that the three methods all displayed a better robustness when the watermarked models were subjected to a weak noise attack; however, as the noise strength increases, the results of our proposed method were superior to the results in Ke et al. ¹⁴ and Feng. ¹⁵ In our method, the candidate vertices to carry the watermarking information are selected according to the RMSCs of the local sets. The watermark embedding is implemented by modulating radial radii ρ of candidate vertices. The noise attack has no noticeable effect on the RMSC value of each vertex. Therefore, our method can still situate the watermarked positions in the event of a noise attack and improve the robustness of the watermark.

Simplification attack

The goal of vertex simplification is to generate a point-cloud model with as few vertices as possible on the premise of minimizing distortion from the original model. For a 3D model carrying the watermarking information, simplification operation is regarded as an attack that will remove some watermarked vertices from the model. The strength of simplification attack should not cause the perceived deformation of the model. If the attack intensity makes the tested 3D models severely degraded, the copyright of the tested models need not be protected because they have already lost their application value. Therefore, it is not important whether the watermark information can be extracted from the tested models. It can be observed from the third line in Table 2 and Figure 8(b) that, when simplification intensity of vertices does not cause any visual distortions, the values Corr obtained from our proposed method are all greater than 0.5. This illustrates that our watermarking method can detect the watermarking information successfully; comparatively speaking, the results of Corr obtained using the methods in Ke et al. ¹⁴ and Feng ¹⁵ are inferior to the results of our method when the number of the simplified vertices changes from 3%, 12%, to 21%. However, when the simplified vertices is greater than 30%, which results in significant visual distortions of the models (Figure 9), the performances of the three methods are not very well (Corr < 0.5). Our method is that more than one candidate vertex is selected from each ball ring to carry the same watermark bit, that is, redundant embedding. When a certain number of vertices within the ball ring are removed, we can still detect the watermark bit from the remained candidate vertices within the ball ring. Therefore, the performance of the proposed method to resist the simplification attack is relatively better than those of the methods in Ke et al. ¹⁴ and Feng. ¹⁵

Rotation attack

The performance to resist geometric attack is investigated, for example, rotation attack. In our method, the vertices with smaller RMSC value, that is, the reference vertices we called, are used to establish the synchronization relation between the watermark embedding and extraction; whereas, the vertices with larger RMSC value, that is, the candidate vertices we called, are exploited to carry the watermarking information. The reference vertices are not affected by the embedded watermarking information; as a result of which, the established synchronization relation is not going to change. If the synchronization relation between watermark detection and watermark embedding remains unchanged, the relative positions of the watermarked vertices with the origin of the established coordinate system are also unchanged; therefore, geometric attack such as a translation, scaling, and rotation attack does not weaken the robustness of the proposed method. The fourth line in Table 2 and Figure 10(a) illustrate that our proposed method is effective in resisting the rotation attacks.

Figure 10.

Comparison of average values of the correlation coefficients: robustness against (a) rotation angle around X-axis and (b) cropping percentage (%).

Cropping attack

Cropping is an operation that removes all the vertices from a certain part of 3D model, which is regarded as an attack difficult to improve robustness. The fifth line in Table 2 and Figure 10(b) display the evaluation results for robustness against cropping. The evaluation results indicate that the performance of our proposed method outperforms the methods in Ke et al ¹⁴ and Feng ¹⁵ when the tested models encounter the attacks of weak cropping; as the strength of the attack increases, the robustness of our method is close to those of the two methods. In our method, each bit of the watermark is embedded into multiply vertices which are irrelevant to each other in the same ball ring. This mechanism is helpful for successfully extracting the watermark information from the models attacked by a cropping operation. However, if the sheared part is large and the sheared part is close to 30% of the model, many vertices that carry watermark bits are lost, thus resulting in failing to detect the correct watermarking information. Therefore, the performance of the proposed method is not satisfied when encounter strong cropping attacks.

ROC evaluation

ROC curve is regarded as an important assessment indicator of the robustness performance of a watermarking method. ¹⁸ ROC curves describe the relationship between the probability of false negatives P_fn and the probability of false positives P_fp when correlation coefficient (Corr) value varies. The probability of false positives expresses the probability of extracting a watermark from a non-watermarked model or from a model that was embedded another watermark. The probability of false negative represents the probability of failing to extract the correct watermark from a watermarked model.

In this section, we verify the robustness of our proposed method by analyzing the ROC curves. The probabilities P_fn and P_fp that are evaluated with 200 correct watermarks and 200 wrong watermarks are plotted by varying the correlation coefficient (Corr) values and the correlation results are approximated by Gaussian functions. Equal error rate (EER) is the value of the point on the ROC curve when the probability of false negatives P_fn is equal to the probability of false positives P_fp and is considered as a quantitative evaluation index of robustness performance of the watermarking method. The smaller the EER value is, the better the robustness of the method can be.

Figure 11(a) and (b) shows the ROC curves of the three methods when the model Bunny is, respectively, subjected to attacks of the strength σ = 0.0001 of the additive noise and 10% vertices reduction. Figure 12(a) and (b) shows the ROC curves of the three methods when the model Bunny is, respectively, subjected to cropping attack (cutting off 5% of the model) and rotation attack (10° around X-axis). It can be seen from Figures 11 and 12 that the robustness of the proposed method to resist common attack and geometric attack is fairly well, especially for geometric attack. For example, when the model Bunny is subjected to the attack of a rotation 10° around X-axis, the EER value obtained using the proposed method is close to $1\times {10}^{-12}$ and is far smaller than the results of the methods in Ke et al. ¹⁴ and Feng. ¹⁵ This demonstrates that the robustness of the proposed method is better than other two methods Ke et al. ¹⁴ and Feng. ¹⁵

Figure 11.

ROC curves: (a) Gaussian noise (σ = 0.0001) attack and (b) simplification (10% reduction) attack.

Figure 12.

ROC curves: (a) rotation (10° around the X-axis) attack and (b) cropping (5% of the model) attack.

Conclusion

In this study, we present a blind and robust watermarking method for 3D point-cloud models. The RMSC of the local set of each vertex is first calculated; the vertices with smaller RMSC values are used to establish the synchronization relation between the watermark embedding and extraction; the vertices with RMSC values are considered to lie in the abrupt regions of the 3D point-cloud models and have a good visual masking effect for the embedded watermark. The model is divided into ball rings and a bit of the watermark is embedded by altering radial radii of candidate vertices of each ball ring. The embedding of the watermark does not have any impacts on the established synchronization relation. The highlights of our study are as follows: (1) the occlusion effect of the embedding positions improves the transparency of the embedded watermark; (2) the establishment of the synchronization relation between the watermark embedding and extraction, as well as its invariance to the embedding of the watermark, helps to enhance the robustness of the proposed method against geometric attack; and (3) the strategy that a watermark bit is repeatedly embedded into a ball ring of the model makes the watermark extraction irrelevant to connectivity of 3D point-cloud models. Experiments show that the proposed method has a good robustness against common attack and geometric attack while ensuring a minimal distortion. In future work, we will combine the proposed framework with the geometric properties of the 3D model^19,20 to further improve robustness against common attack and geometric attack, especially for a combination of common attack and geometric attack.