Towards Function Space Mesh Watermarking: Protecting the Copyright of Signed Distance Fields

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.