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Watermarking 3D Polygonal Meshes

Watermarking 3D Polygonal Meshes. 報告者：梁晉坤 指導教授：楊士萱博士 日期：2002.4.24. Outline. 3D Mesh Object And VRML 3D Mesh Watermarking Attacks Spatial Domain Watermarks Frequency Domain Watermarks Future Works Reference . 3D Mesh Object And VRML. The 3D Mesh Object is a 3D Polygonal model.

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Watermarking 3D Polygonal Meshes

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  1. Watermarking 3D Polygonal Meshes 報告者：梁晉坤 指導教授：楊士萱博士 日期：2002.4.24

  2. Outline • 3D Mesh Object And VRML • 3D Mesh Watermarking Attacks • Spatial Domain Watermarks • Frequency Domain Watermarks • Future Works • Reference

  3. 3D Mesh Object And VRML • The 3D Mesh Object is a 3D Polygonal model. • VRML(Virtual-reality modeling language) • VRML allows to create “Virtual Worlds” networked via the Internet and hyperlink with the World Wild Web. • 3D mesh object is represented in VRML by IndexFaceSet nodes.

  4. Simple VRML Example

  5. 3D Mesh Watermarking Attacks • Rotation, translation, and uniform scaling. • Polygon smoothing and simplification • Randomization of points • Re-meshing (re-triangulation)-generating equal shaped patches with equal angles and surface • Sectioning-removing parts of the model.

  6. Spatial Domain Watermarks • Triangle Similarity Quadruple (TSQ) • The algorithm uses a quadruple of adjacent triangles that share edges as a Macro-Embedding-Primitive (MEP). • Each MEP stores a quadruple of values {Marker, Subscript, Data1, Data2}.

  7. TSQ(Cont.)

  8. TSQ(Cont.) • This algorithm is vulnerable to more powerful watermark attacks, including geometrical transformations and re-meshing.

  9. Tetrahedral Volume Ratio(TVR) • This algorithm is similar to the TSQ algorithms. • This algorithm does not withstand re-meshing and point randomization.

  10. Frequency Domain Watermarks • 3D Watermarking using Multi-resolution Wavelet Decomposition proposed by Kanai et al. • This method requires the mesh to have 1-to-4 subdivision connectivity.

  11. Frequency Domain Watermarks(Cont.) • Watermarking 3D Polygonal Meshes in the Mesh Spectral Domain • This algorithm is robust against geometric transformation , mesh smoothing, random noise added to vertex coordinates, and resection.

  12. Eigenvalue and Eigenvector • Ax=u*x,in which u is eigenvalue, and x is eigenvector • Char A(x)=det(A-uI) • V(u)=ker(A-uI)

  13. L: Laplacian matrix • H:a diagonal matrix whose diagonal element Hii=1/di • L=I-HA, in which I is an identity matrix • Eigenvalue decomposition of Laplacian matrix will produces a sequence of eigenvalues and a corresponding sequence of eigenvectors of the matrix L • Smaller eigenvalues correspond to lower spatial frequencies, and larger eigenvalues correspond to higher spatial frequencies

  14. K: Krichhoff matrix(n*n), in which n is vertex numbers. • D: A diagonal matrix whose diagonal element Dii=di is a degree of vertex i • A: An adjacent matrix of the polygonal mesh whose aij are defined as follow; • If vertex i and j are adjacent aij=1;otherwise aij=0 • K = D-A, K is a symmetric matrix and easy to compute eigenvalue decomposition

  15. A polygonal mesh M having n vertices produces a K Matrix(n*n), whose eigenvalue decomposition produce n eigenvalues ui(1<=i<=n) and n n-dimensional eigenvector wi (1<=i<=n) • The i-th normalized eigenvectors ei=wi/norm(wi) • To make spectral transformation as follow; • (x1,x2,…,xn)T =rs,1e1+rs,2e2+…+rs,nen • (y1,y2,…,yn)T =rt,1e1+rt,2e2+…+rt,nen • (z1,z2,…,zn)T =ru,1e1+ru,2e2+…+ru,nen

  16. Embedding Watermark • a=(a1,a2,…am):watermark bit vector, in which ai takes values{0,1} • b=(b1,b2,…bmc ):chip rate = c ,bi takes values{0,1} bi=aj, j*c<=i<(j+1)*c • bi’=(b1’,b2’,…bmc’): if bi=0 bi’=-1,if bi=1,bi’=1 • p=(p1,p2,…pmc):pi takes values{-1,1} according to key kw • rs,i’=rs,i+bi’*pi*α,ri’=(rs,i’, rt,i’, ru,i’) • vi’=(xi’, yi’, zi’)

  17. Extracting Watermark

  18. A Frequency-Domain Approach to Watermarking 3D Shapes: • This algorithm is based on previous algorithm, and that improved by • (1)Much large meshes can be watermarking within a reasonable time • (2)Robust against connectivity alteration • (3)Robust against mesh smoothing and simplification

  19. Future Works • Replace previous algorithm from spectral transformation to wavelet transformation, and then compare performance between them. • Blind detection is another issue.

  20. Reference • Digital Watermarking of 3D Polygonal Models Andrew Morrow acm@cs.brown.edu December 19, 1999 • Watermarking 3D Polygonal Meshes in the Mesh Spectral Domain Ryutarou Ohbuchi, Shigeo Takahashi, Takahiko Miyazawa, Akio Mukaiyama • A Frequency-Domain Approach to Watermarking 3D Shapes Ryutarou Ohbuchi, Akio Mukaiyama, Shigeo Takahashi 2002

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