A survey of applications of an affine invariant norm
Mathematical methods in computer aided geometric design
SIGGRAPH '96 Proceedings of the 23rd annual conference on Computer graphics and interactive techniques
The NURBS book (2nd ed.)
Watermaking three-dimensional polygonal models
MULTIMEDIA '97 Proceedings of the fifth ACM international conference on Multimedia
Proceedings of the 26th annual conference on Computer graphics and interactive techniques
A Shape-Preserving Data Embedding Algorithm for NURBS Curves and Surfaces
CGI '99 Proceedings of the International Conference on Computer Graphics
Robust Watermarking of Polygonal Meshes
GMP '00 Proceedings of the Geometric Modeling and Processing 2000
Watermarking three-dimensional polygonal models through geometric and topological modifications
IEEE Journal on Selected Areas in Communications
A robust high-capacity affine-transformation-invariant scheme for watermarking 3D geometric models
ACM Transactions on Multimedia Computing, Communications, and Applications (TOMCCAP) - Special Issue on Multimedia Security
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We present a scheme for embedding secret or public readable watermarks into 3D models consisting of polygonal or NURBS surfaces. The scheme realizes affine invariant watermarks by displacing vertices (control points) and satisfies constraints regarding maximum tolerated vertex movements or, in the NURBS case, differences of original and watermarked surfaces. The algorithm uses the volume of two tetrahedrons as an embedding feature. The scheme described can be stacked on a more robust scheme allowing transmission of labeling information to the user or increasing blind detection capabilities of the underlying scheme. The paper makes two major contributions, both driven by real world requirements: The first one is a technique to cope with reduced precisions of vertex coordinates. Real world modeling applications represent vertex coordinates with single floating point precision. Vertex coordinates in VRML scenes are represented by 6 decimal digits or even less. Mesh compression schemes may quantize vertex coordinates to even below precision of 4 decimal digits. The second contribution of this paper is a general technique for reducing processing time of watermark (label) extraction satisfying impatient users and enhancing robustness with respect to affine transformations and, in particular, vertex randomization attacks. The technique is based on simplifying the mesh applying edge collapses prior to watermark embedding and retrieval. The technique depends on a consistent order of vertices in embedding and retrieval process. We sketch possible extensions of the proposed scheme.