1. Robust Watermarking of 3D Mesh Models

2. Introduction in this paper, it proposes an algorithm that extracts 2D image from the 3D model and embed watermark to the 2D image. The 2D imagine extracted from the object is the virtual range data, and the DCT based 2D watermarking is employed.

3. Virtual Ranging (A) The virtual scanner consists of cylinder and three reference axes. The cylinder is positioned around the 3D model and is set to fit the model tightly. It employs the eigen-vector of covariance matrix of vertex coordinates as the reference axes.

4. Virtual Ranging (A) mv is the mean of vertices s 1 is the eigen-vector of covariance matrix corresponding to the largest eigen-value, s 2 and s 3 are the eigen-vectors corresponding to the second and third eigen-value.

5. Virtual Ranging (A) Where V is vertex points The radius r and height h of the virtual scanner are also related with the eigen-vectors as :

7. Virtual Ranging (B1) In this section, we describe how to make the range image from the above virtual scanning system. First, we make N v * N u grid on the side face of the cylinder. The grid point where the ranges are calculated, are denoted by (u r,v r ). The vertical line (v r =0) lies in the direction of s 3 and v r increase along the counterclockwise direction.

8. Virtual Ranging (B2) On the grid which corresponds to q’ in (x,y,z) coordinate, we draw a line towards the point q on the opposite side of the cylinder. The triangle abc represents one of the triangular faces which intersects the line segment q’q. The l is the range value that we wish to find, and the l computed for every (u r,v r ) point on the grid.

9. Virtual Ranging (B3) To be more precise on the computation of l, it first finds the intersecting triangles. After finding the faces, it defines two vectors as: Where e v (x) is a unit vector and a function of x. x is the arbitrary v-coordinate of the grid in the range of Hence, e v (x) is the radial direction vector of the cylinder which points to the line whose v-coordinate is x.

10. Virtual Ranging (B4) In terms of eu and ev the coordinates of the point q and q’ can be expressed as: The range is calculated by solving the simultaneous equation of intersecting point x which is on the line qq’ and also on the triangle abc as: The solution of the above equation is the range as given by:

11. Watermark Embedding (1) For applying the DCT domain watermarking to the virtual range imagine, the range values are first linearly normalized to [0,255]. Then, then range image is divided into 8*8 DCT blocks and the watermark is added to AC components as c’=c+αw. Watermark is a uniform noise which is -1 or 1, and the seed value of the uniform noise generator is also another key of watermark. α is decided by the human visual system model.

12. Watermark Embedding (2) Show the model surface in the u-direction, where the horizontal line is the model surface and the arrow is the ranging ray. In order to find the vertices which correspond to the new range value, all the vertices are first projected to the u-v coordinates along the radial direction of the cylinder.

13. Watermark Embedding (3) We have the knowledge of u-v coordinates of all vertices. The u-v coordinate of an arbitrary vertex v is denoted as (u v,v v ). Hence we can define a set of vertices, B u r, v r, which contains the vertices whose the coordinates (u v,v v ) satisfy |u v – u r | < 0.5 and |v v – v r | < 0.5. B u r, v r is a set of vertices which are projected to 1*1 rectangular region in the u-v grid and the center of the region is (ur,vr). Each B u r, v r corresponds to a range value, that is, l(u r,v r ). We can express the modified vertices as:

14. Watermark Embedding (4) It multiply weighting function to Δl and smooth the surface. The 1D weighting function employed here is: The weighting function influences eight neighboring regions, and it adds the overlapped values.

15. Extraction Procedure For the extraction of watermark, two keys and watermarked model are needed. One the keys is the seed number and the other is the corner points of the rectangular region where the watermarked is embedded. N v and N u are also needed. In order to extract the watermark, the range image of the watermarked model is obtained using the virtual ranging process. Then, the watermark is extracted from the range imagine.