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3D model watermarking using surface integrals of generated random vector fields

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Abstract

We propose a new semi-blind semi-fragile watermarking algorithm for authenticating triangulated 3D models using the surface integrals of generated random vector fields. Watermark data is embedded into the flux of a vector field across the model’s surface and through gradient-based optimization techniques, the vertices are shifted to obtain the modified flux values. The watermark can be extracted through the recomputation of the surface integrals and compared using correlation measures. This algorithm is invariant to Euclidean transformations including rotations and translation, reduces distortion, and achieves improved robustness to additive noise.

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Data Availability

Source code and data has been made available on a published Code Capsule on Code Ocean with the following https://doi.org/10.24433/CO.0174131.v4.

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Authors and Affiliations

  1. School of Information Technology, Carleton University, 1125 Colonel By Dr, Ottawa, ON, K1S 5B6, Canada

    Luke Vandenberghe & Chris Joslin

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  1. Luke Vandenberghe
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  2. Chris Joslin
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Contributions

L.V. wrote the main manuscript text, generated the experimental results, and prepared the figures. C.J. was an advisor who reviewed and provided feedback to improve the manuscript.

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Correspondence to Luke Vandenberghe.

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The authors declare no competing interests.

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Communicated by Balakrishnan Prabhakaran.

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Appendix A: Gradient

Appendix A: Gradient

By chain rule, the gradient used in Eq. (28) is written as

$$\begin{aligned} \nabla _{{\varvec{t}}'}J({\varvec{t}}')&= \frac{\partial J}{\partial \phi ({\varvec{t}}')} \nabla _{{\varvec{t}}'}\phi ({\varvec{t}}') =\left( \phi ({\varvec{t}}') - \phi '({\varvec{t}}) \right) \nabla _{{\varvec{t}}'}\phi ({\varvec{t}}'), \end{aligned}$$

and the gradient of the flux with respect to the facet’s vertices is computed by

$$\begin{aligned} A_{n,1}&= ({\varvec{k}}_n \times {\varvec{\zeta }}_n) \cdot ({\varvec{p}}_i' \times {\varvec{p}}_j' - {\varvec{p}}_i' \times {\varvec{p}}_k' + {\varvec{p}}_j' \times {\varvec{p}}_k'),\\ A_{n,2}&= ({\varvec{k}}_n \times {\varvec{\xi }}_n) \cdot ({\varvec{p}}_i' \times {\varvec{p}}_j' - {\varvec{p}}_i' \times {\varvec{p}}_k' + {\varvec{p}}_j' \times {\varvec{p}}_k'),\\ A_{n,3}&= {\varvec{k}}_n \cdot ({\varvec{p}}_j' + {\varvec{p}}_k' - 2{\varvec{p}}_i'),\\ A_{n,4}&= {\varvec{k}}_n \cdot (2{\varvec{p}}_i' - {\varvec{p}}_k'),\\ A_{n,5}&= {\varvec{k}}_n \cdot {\varvec{p}}_j',\\ A_{n,6}&= {\varvec{k}}_n \cdot {\varvec{p}}_i',\\ A_{n,7}&= {\varvec{k}}_n \cdot ({\varvec{p}}_i' - {\varvec{p}}_k'),\\ A_{n,8}&= {\varvec{k}}_n \cdot ({\varvec{p}}_i' - {\varvec{p}}_j'),\\ B_{n,1} = \,&\frac{\cos (A_{n,6}) - \cos (A_{n,5})}{A_{n,7}^2A_{n,8}} + \frac{\cos (A_{n,6}) - \cos (A_{n,5})}{A_{n,7}A_{n,8}^2} \\&+ \frac{\cos (A_{n,4}) - \cos (A_{n,5})}{A_{n,3}A_{n,7}^2} - \frac{2(\cos (A_{n,4}) - \cos (A_{n,5}))}{A_{n,3}^2A_{n,7}} \\&+ \frac{\sin (A_{n,6})}{A_{n,7}A_{n,8}} + \frac{2\sin (A_{n,4})}{A_{n,7}A_{n,3}},\\ B_{n,2} = \,&\frac{\sin (A_{n,6}) - \sin (A_{n,5})}{A_{n,7}^2A_{n,8}} + \frac{\sin (A_{n,6}) - \sin (A_{n,5})}{A_{n,7}A_{n,8}^2} \\&+ \frac{\sin (A_{n,4}) - \sin (A_{n,5})}{A_{n,3}A_{n,7}^2} - \frac{2(\sin (A_{n,4}) - \sin (A_{n,5}))}{A_{n,3}^2A_{n,7}} \\&- \frac{\cos (A_{n,6})}{A_{n,7}A_{n,8}} - \frac{2\cos (A_{n,4})}{A_{n,7}A_{n,3}}, \\ B_{n,3} = \,&\frac{\cos (A_{n,6}) - \cos (A_{n,5})}{A_{n,7}A_{n,8}^2} - \frac{\cos (A_{n,4}) - \cos (A_{n,5})}{A_{n,3}^2A_{n,7}} \\&+ \frac{\sin (A_{n,5})}{A_{n,8}A_{n,7}} + \frac{\sin (A_{n,5})}{A_{n,7}A_{n,3}}, \\ B_{n,4} = \,&-\frac{\sin (A_{n,6}) - \sin (A_{n,5})}{A_{n,7}A_{n,8}^2} + \frac{\sin (A_{n,4}) - \sin (A_{n,5})}{A_{n,3}^2A_{n,7}} \\&+ \frac{\cos (A_{n,5})}{A_{n,8}A_{n,7}} + \frac{\cos (A_{n,5})}{A_{n,7}A_{n,3}}, \\ B_{n,5} = \,&\frac{\cos (A_{n,6}) - \cos (A_{n,5})}{A_{n,7}^2A_{n,8}} + \frac{\cos (A_{n,4}) - \cos (A_{n,5})}{A_{n,3}A_{n,7}^2} \\&- \frac{\cos (A_{n,4}) - \cos (A_{n,5})}{A_{n,3}^2A_{n,7}} + \frac{\sin (A_{n,4})}{A_{n,7}A_{n,3}}, \\ B_{n,6} = \,&\frac{\sin (A_{n,6}) - \sin (A_{n,5})}{A_{n,7}^2A_{n,8}} + \frac{\sin (A_{n,4}) - \sin (A_{n,5})}{A_{n,3}A_{n,7}^2} \\&- \frac{\sin (A_{n,4}) - \sin (A_{n,5})}{A_{n,3}^2A_{n,7}} - \frac{\cos (A_{n,4})}{A_{n,7}A_{n,3}}, \\ B_{n,7} = \,&\frac{\cos (A_{n,6}) - \cos (A_{n,5})}{A_{n,7}A_{n,8}} + \frac{\cos (A_{n,4}) - \cos (A_{n,5})}{A_{n,3}A_{n,7}},\\ B_{n,8} = \,&\frac{\sin (A_{n,6}) - \sin (A_{n,5})}{A_{n,7}A_{n,8}} + \frac{\sin (A_{n,4}) - \sin (A_{n,5})}{A_{n,3}A_{n,7}}, \end{aligned}$$
$$\begin{aligned} C_{n,1}&= \begin{vmatrix}0&z_k' - z_j'&y_j' - y_k'\\ k_{n,1}&k_{n,2}&k_{n,3}\\ \zeta _{n,1}&\zeta _{n,2}&\zeta _{n,3}\end{vmatrix}, \\ C_{n,2}&= \begin{vmatrix}z_k' - z_j'&0&x_j' - x_k'\\ k_{n,1}&k_{n,2}&k_{n,3}\\ \zeta _{n,1}&\zeta _{n,2}&\zeta _{n,3}\end{vmatrix},\\ C_{n,3}&= \begin{vmatrix}y_j' - y_k'&x_k'-x_j'&0\\ k_{n,1}&k_{n,2}&k_{n,3}\\ \zeta _{n,1}&\zeta _{n,2}&\zeta _{n,3}\end{vmatrix}, \\ C_{n,4}&= \begin{vmatrix}0&z_k' - z_j'&y_j' - y_k'\\ k_{n,1}&k_{n,2}&k_{n,3}\\ \xi _{n,1}&\xi _{n,2}&\xi _{n,3}\end{vmatrix},\\ C_{n,5}&= \begin{vmatrix}z_k' - z_j'&0&x_j' - x_k'\\ k_{n,1}&k_{n,2}&k_{n,3}\\ \xi _{n,1}&\xi _{n,2}&\xi _{n,3}\end{vmatrix},\\ C_{n,6}&= \begin{vmatrix}y_j' - y_k'&x_k'-x_j'&0\\ k_{n,1}&k_{n,2}&k_{n,3}\\ \xi _{n,1}&\xi _{n,2}&\xi _{n,3}\end{vmatrix},\\ C_{n,7}&= \begin{vmatrix}0&z_k' - z_i'&y_i' - y_k'\\ k_{n,1}&k_{n,2}&k_{n,3}\\ \zeta _{n,1}&\zeta _{n,2}&\zeta _{n,3}\end{vmatrix}, \\ C_{n,8}&= \begin{vmatrix}z_k' - z_i'&0&x_i' - x_k'\\ k_{n,1}&k_{n,2}&k_{n,3}\\ \zeta _{n,1}&\zeta _{n,2}&\zeta _{n,3}\end{vmatrix},\\ C_{n,9}&= \begin{vmatrix}y_i' - y_k'&x_k'-x_i'&0\\ k_{n,1}&k_{n,2}&k_{n,3}\\ \zeta _{n,1}&\zeta _{n,2}&\zeta _{n,3}\end{vmatrix},\\ C_{n,10}&= \begin{vmatrix}0&z_k' - z_i'&y_i' - y_k'\\ k_{n,1}&k_{n,2}&k_{n,3}\\ \xi _{n,1}&\xi _{n,2}&\xi _{n,3}\end{vmatrix},\\ C_{n,11}&= \begin{vmatrix}z_k' - z_i'&0&x_i' - x_k'\\ k_{n,1}&k_{n,2}&k_{n,3}\\ \xi _{n,1}&\xi _{n,2}&\xi _{n,3}\end{vmatrix},\\ C_{n,12}&= \begin{vmatrix}y_i' - y_k'&x_k'-x_i'&0\\ k_{n,1}&k_{n,2}&k_{n,3}\\ \xi _{n,1}&\xi _{n,2}&\xi _{n,3}\end{vmatrix},\\ C_{n,13}&= \begin{vmatrix}0&z_j' - z_i'&y_i' - y_j'\\ k_{n,1}&k_{n,2}&k_{n,3}\\ \zeta _{n,1}&\zeta _{n,2}&\zeta _{n,3}\end{vmatrix},\\ C_{n,14}&= \begin{vmatrix}z_j' - z_i'&0&x_i' - x_j'\\ k_{n,1}&k_{n,2}&k_{n,3}\\ \zeta _{n,1}&\zeta _{n,2}&\zeta _{n,3}\end{vmatrix},\\ C_{n,15}&= \begin{vmatrix}y_i' - y_j'&x_j'-x_i'&0\\ k_{n,1}&k_{n,2}&k_{n,3}\\ \zeta _{n,1}&\zeta _{n,2}&\zeta _{n,3}\end{vmatrix},\\ C_{n,16}&= \begin{vmatrix}0&z_j' - z_i'&y_i' - y_j'\\ k_{n,1}&k_{n,2}&k_{n,3}\\ \xi _{n,1}&\xi _{n,2}&\xi _{n,3}\end{vmatrix},\\ C_{n,17}&= \begin{vmatrix}z_j' - z_i'&0&x_i' - x_j'\\ k_{n,1}&k_{n,2}&k_{n,3}\\ \xi _{n,1}&\xi _{n,2}&\xi _{n,3}\end{vmatrix},\\ C_{n,18}&= \begin{vmatrix}y_i' - y_j'&x_j'-x_i'&0\\ k_{n,1}&k_{n,2}&k_{n,3}\\ \xi _{n,1}&\xi _{n,2}&\xi _{n,3}\end{vmatrix}, \end{aligned}$$
$$\begin{aligned} \frac{\partial \phi ({\varvec{t}}')}{\partial x_i'}&= \sum _{n=1}^N(k_{n,1}(A_{n,1}B_{n,1} + A_{n,2}B_{n,2}) - B_{n,7}C_{n,1} - B_{n,8}C_{n,4}),\\ \frac{\partial \phi ({\varvec{t}}')}{\partial y_i'}&= \sum _{n=1}^N(k_{n,2}(A_{n,1}B_{n,1} + A_{n,2}B_{n,2}) + B_{n,7}C_{n,2} + B_{n,8}C_{n,5}),\\ \frac{\partial \phi ({\varvec{t}}')}{\partial z_i'}&= \sum _{n=1}^N(k_{n,3}(A_{n,1}B_{n,1} + A_{n,2}B_{n,2}) + B_{n,7}C_{n,3} + B_{n,8}C_{n,6}),\\ \frac{\partial \phi ({\varvec{t}}')}{\partial x_j'}&= -\sum _{n=1}^N(k_{n,1}(A_{n,1}B_{n,3} + A_{n,2}B_{n,4}) - B_{n,7}C_{n,7} - B_{n,8}C_{n,10}),\\ \frac{\partial \phi ({\varvec{t}}')}{\partial y_j'}&= -\sum _{n=1}^N(k_{n,2}(A_{n,1}B_{n,3} + A_{n,2}B_{n,4}) + B_{n,7}C_{n,8} + B_{n,8}C_{n,11}),\\ \frac{\partial \phi ({\varvec{t}}')}{\partial z_j'}&= -\sum _{n=1}^N(k_{n,3}(A_{n,1}B_{n,3} + A_{n,2}B_{n,4}) + B_{n,7}C_{n,9} + B_{n,8}C_{n,12}),\\ \frac{\partial \phi ({\varvec{t}}')}{\partial x_k'}&= - \sum _{n=1}^N(k_{n,1}(A_{n,1}B_{n,5} + A_{n,2}B_{n,6}) + B_{n,7}C_{n,13} + B_{n,8}C_{n,16}),\\ \frac{\partial \phi ({\varvec{t}}')}{\partial y_k'}&= -\sum _{n=1}^N(k_{n,2}(A_{n,1}B_{n,5} + A_{n,2}B_{n,6}) - B_{n,7}C_{n,14} - B_{n,8}C_{n,17}),\\ \frac{\partial \phi ({\varvec{t}}')}{\partial z_k'}&= -\sum _{n=1}^N(k_{n,3}(A_{n,1}B_{n,5} + A_{n,2}B_{n,6}) - B_{n,7}C_{n,15} - B_{n,8}C_{n,18}), \end{aligned}$$
$$\begin{aligned} \nabla _{{\varvec{t}}'}\phi ({\varvec{t}}')&= \begin{pmatrix} {\partial \phi ({\varvec{t}}')}/{\partial x_i'}\\ {\partial \phi ({\varvec{t}}')}/{\partial y_i'}\\ {\partial \phi ({\varvec{t}}')}/{\partial z_i'}\\ {\partial \phi ({\varvec{t}}')}/{\partial x_j'}\\ {\partial \phi ({\varvec{t}}')}/{\partial y_j'}\\ {\partial \phi ({\varvec{t}}')}/{\partial z_j'}\\ {\partial \phi ({\varvec{t}}')}/{\partial x_k'}\\ {\partial \phi ({\varvec{t}}')}/{\partial y_k'}\\ {\partial \phi ({\varvec{t}}')}/{\partial z_k'} \end{pmatrix}. \end{aligned}$$

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Vandenberghe, L., Joslin, C. 3D model watermarking using surface integrals of generated random vector fields. Multimedia Systems 30, 253 (2024). https://doi.org/10.1007/s00530-024-01455-0

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