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

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. 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.

Simple VRML Example

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.

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}.

TSQ(Cont.)

This algorithm is vulnerable to more powerful watermark attacks, including geometrical transformations and re- meshing.

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

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.

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.

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)

L: Laplacian matrix H:a diagonal matrix whose diagonal element H ii =1/d i 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

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

A polygonal mesh M having n vertices produces a K Matrix(n*n), whose eigenvalue decomposition produce n eigenvalues u i (1<=i<=n) and n n-dimensional eigenvector w i (1<=i<=n) The i-th normalized eigenvectors e i =w i /norm(w i ) To make spectral transformation as follow; (x 1,x 2, …,x n ) T =r s,1 e 1 +r s,2 e 2 + … +r s,n e n (y 1,y 2, …,y n ) T =r t,1 e 1 +r t,2 e 2 + … +r t,n e n (z 1,z 2, …,z n ) T =r u,1 e 1 +r u,2 e 2 + … +r u,n e n

Embedding Watermark a=(a 1,a 2, … a m ):watermark bit vector, in which a i takes values{0,1} b=(b 1,b 2, … b mc ):chip rate = c,b i takes values{0,1} b i =a j, j*c<=i<(j+1)*c b i ’ =(b 1 ’,b 2 ’, … b mc ’ ): if b i =0 b i ’ =-1,if bi=1,b i ’ =1 p=(p 1,p 2, … p mc ):p i takes values{-1,1} according to key k w r s,i ’ =r s,i +b i ’ *p i *α,r i ’ =(r s,i ’, r t,i ’, r u,i ’ ) v i ’ =(x i ’, y i ’, z i ’ )

Extracting Watermark

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

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

Reference Digital Watermarking of 3D Polygonal Models Andrew Morrow December 19, 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