The signed distance field (SDF) represents 3D geometries in continuous function space. Due to its continuous nature, explicit 3D models (e.g., meshes) can be extracted from it at arbitrary resolution, which means losing the SDF is equivalent to losing the mesh. Recent research has shown meshes can also be extracted from SDF-enhanced neural radiance fields (NeRF). Such a signal raises an alarm that any implicit neural representation with SDF enhancement can extract the original mesh, which indicates identifying the SDF's intellectual property becomes an urgent issue. This paper proposes FuncMark, a robust and invisible watermarking method to protect the copyright of signed distance fields by leveraging analytic on-surface deformations to embed binary watermark messages. Such deformation can survive isosurfacing and thus be inherited by the extracted meshes for further watermark message decoding. Our method can recover the message with high-resolution meshes extracted from SDFs and detect the watermark even when mesh vertices are extremely sparse. Furthermore, our method is robust even when various distortions (including remeshing) are encountered. Extensive experiments demonstrate that our \tool significantly outperforms state-of-the-art approaches and the message is still detectable even when only 50 vertex samples are given.

翻译：符号距离场（SDF）将三维几何体表示于连续函数空间中。由于其连续性，可从中以任意分辨率提取显式三维模型（例如网格），这意味着丢失SDF等同于丢失网格。最新研究表明，网格亦可从SDF增强型神经辐射场（NeRF）中提取。此类信号发出警示：任何具备SDF增强的隐式神经表征均可提取原始网格，因此识别SDF的知识产权已成为紧迫问题。本文提出FuncMark——一种鲁棒且不可见的水印方法，通过利用解析表面形变嵌入二进制水印消息，以保护符号距离场的版权。此类形变能够承受等值面提取，并由此被提取后的网格继承，以进一步解码水印信息。我们的方法能通过从SDF提取的高分辨率网格恢复消息，即便网格顶点极度稀疏仍能检测到水印。此外，本方法在面对包括重网格化在内的多种失真时依然保持鲁棒性。广泛实验证明，我们的工具显著优于现有方法，即使在仅提供50个顶点样本的情况下，水印仍可被检测到。