Abstract It is challenging to relocate the embedded positions of the watermark when 3D models are subjected to rotation attacks, because the vertex data of 3D models has no implicit order. This paper proposes a zero-watermarking scheme for 3D models, using the Beamlet transform method. First, the area of the 1-ring neighbourhood of each vertex is calculated. Then, the 1-ring neighbourhoods of the vertices with moderate areas are projected onto the tangent planes; the X-axis of each tangent plane corresponds to the mesh line with the maximum curvature in the 1-ring neighbourhood. The Beamlet dictionary is constructed by recording the line segments on both sides of the X-axis; thus, a zero-watermarking image which protects the copyright of the model is produced. Simulations and experiments show that the proposed watermarking scheme exhibits a robust resistance to common signal-processing attacks, especially rotation attacks, and can protect the copyrights of 3D models reliably without distorting the content in any manner.

中文翻译：

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3D模型中抵抗旋转攻击的零水印方法

摘要 当3D模型受到旋转攻击时，重新定位水印嵌入位置具有挑战性，因为3D模型的顶点数据没有隐含的顺序。本文提出了一种用于 3D 模型的零水印方案，使用 Beamlet 变换方法。首先，计算每个顶点的 1 环邻域的面积。然后，将面积中等的顶点的1-环邻域投影到切平面上；每个切平面的 X 轴对应于 1 环邻域中曲率最大的网格线。Beamlet字典是通过记录X轴两侧的线段来构建的；因此，产生了保护模型版权的零水印图像。