  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.
