3D polygonal meshes watermarking using normal vector distributions Suk-Hawn Lee, Tae-su Kim, Byung-Ju Kim, Seong-Geun Kwon

The proposed 3D watermarking algorithm A block diagram of the proposed 3D watermark embedding algorithm.

Patch classification (1) The 3D mesh model is divided into 6 patches using a distance measure. By embedding the same watermark into each patch, the proposed algorithm is robust against partial geometric deformation, such as cropping.

Patch classification (2) The initial center points I as specified as the point with the maximum distance in the direction of 6 unit vector, (±x, ±y, ±z), from Origin and considered as the respective center point for each of the 6 patches. Which has minimum distance among the initial center points I.

Patch classification (3) I are updated to the new center point, after calculating the center of the clustered. Then, the clustering and updating process is iterated until the distortion measure D m satisfies the condition, (D m-1 -D m )/D m+1 ≤ th,where m is the number of iterations, the is the threshold value and

EGI per patch EGI (Extended Gaussian Image) of a 3D model is an orientation histogram that record the variation of surface area with surface orientation. The respective patch EGIs, PE={PE 1,PE 2,…,PE 6 } are obtained by mapping the mesh normal vector in P={p 1,p 2,…,p 6 } into the surface that the closest direction among the 240 surfaces.

To calculate normal of meshes The unit normal vector and area S ij for the j th the mesh surface in the i th patch P i are calculated as following: The normal vector for a mesh surface includes and - according to two directions, inward and outward. Between two vectors, the normal vector is determined as the one with the higher value of the dot product of, where is the mean vector of v 1, v 2, v 3.

Mapping each normal into bin The normal vector for each patch is mapped into bin B i with the smallest angle among the angles between and the unit normal vector of bin ;

The sampling of normal Half the angle between the of the neighborhood bin, A m, is the maximum value of, if is mapped into B i. around Am is excluded from the EGI. around Am is excluded from the EGI. Thus, only within a certain range is selected as following: Thus, only within a certain range is selected as following:

To determine the order of bit sequence The length of B i is the sum of area for all the meshes that are mapped into B i. There are 240 bins B j ={B ji | 0≤i<240} in the j-th patch EGI PE j. To sort length of B i arranges the order of embedded bit sequences.

Embedding watermark(1) The watermark, a 1-bit random sequence, is embedded in the ordered B of PE. The j-th watermark w j (1≤ j ≤ N) is embedded into the i-th ordered B ij (1≤ i ≤ 6) in each PE.

Embedding watermark(2) All the normal vectors in the selected bin are changed according to the watermark bit. To change a normal vector, the position of the vertex needs to be changed. The optimum position is identified for all the vertices that satisfy the cost measure for the condition of watermark embedding within the search region.

Embedding watermark(3) The search region for each coordinate of the current vertex v=(x,y,z) are x±Δx, y±Δy and z±Δz ; Δx=, Δy=, Δz=. v t is the t coordinate value of vertex which connected in v. in the 1-th step, 27 position for the vertex are selected for the initial search.

Embedding watermark(4) The current vertex v is updated to position v’. the minimizing cost function cost(v): where s(v) indicates the mesh surface connected to v and Ai is the angle between the normal vector of the i-th mesh surface and the of bin B i into which this vector is mapped among it patch EGI. the minimizing cost function cost(v): where s(v) indicates the mesh surface connected to v and Ai is the angle between the normal vector of the i-th mesh surface and the of bin B i into which this vector is mapped among it patch EGI. If the is included in the bins for watermark embedding, a i is1, otherwise, a i is 0. Plus, if the watermark bit of B i is 1,A w i is 0,otherwise, A w i is A max.

Extraction(1) The normal vector and its EGI distribution are calculated after dividing the original model into 6 patches based on the known center point I of each patch. the watermark can be extracted based on the average difference of the angle between all the normal vectors mapped into each bin and the bin center point.

Extraction(2) The watermark w ij of the B ij bin with a j-th length for the i-th patch is extracted as 1,if (0≤ A ij ≤ A th ), otherwise, it is extracted as 0. where N equals the number of all the mesh normal vectors mapped into the Bij bin. A watermark decision of 1 bit is performed based on the w ij obtained from the 6 patches, w ij is expressed as :

Experimental results (1) The 3D VRML data of the Stanford bunny model had 35,947 vertices and 69,451 faces. The experiment used a 1-bit watermark with 50 length W N=50 generated by a Gaussian random sequence.

Experimental results (2)