MultiMark: Multi-Isosurface Embedded Watermarking for Signed-Distance Fields

Signed-Distance Fields are widely used in high-value applications owing to their continuity and topological flexibility, yet they remain highly vulnerable to replication and tampering. Existing watermarking techniques for these fields directly modify the zero level set, unavoidably introducing geometric distortions. We propose MultiMark, a watermarking framework that operates on multiple isosurfaces. It embeds watermark information for copyright protection into distance differences among nonzero-level sets by using periodic reference-watermark interval partitioning and gradient-guided perturbation, rather than directly adjusting the zero-level set. MultiMark supports embedding multiple bits both within individual isosurfaces via angular partitioning and across multiple isosurfaces, incorporating an automatic interval-scoring mechanism to identify optimal embedding regions. This framework ensures that fundamental geometric characteristics are preserved, offering a comprehensive solution for the secure distribution of implicit three-dimensional models.