Abstract. Most watermarking techniques for 3D mesh models have mainly focused on robustness against various attacks, such as adding noise, smoothing, simplification, remeshing, clipping, and so on. These attacks perceptually damage the stego model itself. Unlike watermarking of other multimedia data, serious attacks for 3D meshes includes similarity transform and vertex reordering. They can fatally destroy the watermark without any perceptual degradation of the stego model. In this paper, we propose a new watermarking technique for 3D polygonal mesh model, which modifies the distribution of vertex norms according to watermark bit to be embedded. In particular, the proposed employs blind watermark detection technique, which extracts the watermark without referring to cover meshes. The simulation results show that the proposed is remarkably robust against similarity transform and vertex reordering.
1. INTRODUCTION
With remarkable growth of network technology such as WWW (World Wide Web), digital media enables us to copy, modify, store, and distribute digital data without effort. As a result, it has become a new issue to research schemes for copyright protection. Watermarking is a copyright protection technique to embed information, socalled watermark, into cover data.
There are several challenges to develop robust watermarking technique in 3D polygonal mesh models. Unlike image data represented by brightness (or color) at pixel uniformly sampled over regular grid in dimension two, polygonal mesh model has no unique representation, i.e., no implicit order and connectivity of vertices [1]. For such a reason, most techniques developed for other types of multimedia are not effective for 3D meshes [2]. Besides it is hard to decide embedding primitives for robust watermark. And a variety of complex geometrical and topological operations are available to disturb the watermark extraction for assertion of ownership [3]. In particular, some distortionless attacks such as vertex reordering and similarity transform (including rotation, translation, and uniform scaling) are possible. They can fatally destroy the watermark without any perceptual change of the stego model.
In this paper, we propose a new watermarking technique for 3D polygonal mesh model, which modifies the distribution of vertex norms so as to shift mean value of vertex norms according to watermark bit to be inserted. Distribution of vertex norms is divided into several sections, called these bins later, so as to improve capacity and transparency of watermark. Clearly, the distribution of vertex norms is invariant to similarity transform and vertex reordering. In addition, the proposed employs blind scheme, which can extract the watermark without reference of cover mesh model.
This paper is organized as follows. In section 2, the proposed watermarking method is introduced. Section 3 shows the simulation results of the proposed against adding noise and clipping, as well as similarity transform and vertex reordering. Finally, we conclude this paper.
2. PROPOSED WATERMARKING SCHEME
The watermarking scheme proposed in this paper embeds watermark information into 3D polygonal mesh model by modifying the distribution of vertex norms. Fig. 1 shows main idea of the proposed watermarking scheme.
Fig. 1. Main idea of the proposed watermarking method modifying the distribution of vertex norm. Here, the mean value is denoted by ▲.
General embedding process of the proposed method is as follows. To begin with, Cartesian coordinates (x_{i}, y_{i}, z_{i}) of a vertex on cover mesh model V are converted to spherical coordinates (_{i}, _{i}, _{i}). Where, _{i} is ith vertex norm. Vertex norm is the distance between each vertex and center of the model. We modify the distribution of vertex norm, which is invariant to similarity transform and vertex reordering.
In the second step, the interval of the distribution is divided into N small sections. Here, each section is employed as watermark embedding unit, socalled bin. That is, vertices are classified into N bins according to their norm, and each bin is independently processed for watermark embedding. This allows enhancing both capacity and transparency of the watermark. For such purpose, maximum and minimum vertex norms, _{max} and _{min}, should be found in advance.
The third step is to calculate mean and reference values of vertex norms, _{n} and r_{n}, respectively in each bin. Where, middle value of the corresponding section is used as the reference to estimate the distribution by comparing with the mean. For the case of _{n} > r_{n}, it can be estimated that the distribution is centralized into rightside, and viceversa.
Next step is to alter the distribution of vertex norms, so that the mean and the reference values are changed according to watermark bit _{n}{–1, +1} to be embedded in each bin. For embedding watermark bit of +1, vertex norms are modified in order that the mean and the reference values satisfy _{n} > r_{n}. Here, the superscript prime means the value obtained from the modified vertex norm, _{n,j} , and r_{n} r_{n} is preserved since we do not modify both minimum and maximum norms, _{max} and _{min} . For embedding _{n }= –1, vertex norms are modified in order to be _{n} < r_{n}. In our proposed, the modified vertex norm is obtained by
,

(1)

where strength factor of the nth bin, _{n}, is determined so as to guarantee such that _{n} > r_{n} for _{n }= +1 and _{n} < r_{n } for _{n }= –1, respectively.

(2)

This means that the whole vertex norms in each bin are added by +_{n} or –_{n}. The relation between mean and reference values is retained, when the processing is performed in separate bin. Unfortunately, it leads to serious impact on robustness of watermark when the processing is performed in every each bin, because some modified vertices belong to certain bin can invade into other neighbor bins. To cope with these disadvantages, we propose advanced method. Mean value can be shifted by modifying only some part of vertex norms. In our approach, vertex norms that are smaller/greater than reference value are modified.
The final step is inverse transformation of vertex from spherical coordinates onto Cartesian coordinates. The Cartesian coordinates (x_{i}, y_{i}, z_{i}) of vertex v on stego mesh model V is given by this inverse transformation.
Note that the watermark embedding method utilizes only the distribution of vertex norms, which is invariant to similarity transform and vertex reordering.
Watermark extraction process is quite. Similar to embedding process, stego mesh model is first represented on spherical coordinates. After finding maximum and minimum vertex norms, vertex norms are classified into N bins. Mean and reference values, _{n} and r_{n}, in each bin are respectively calculated, and compared in order to extract the hidden watermark _{n}.

(3)

Note that the watermark extraction is blind scheme, which can extract the watermark without reference of cover mesh model.
3. SIMULATION RESULTS
The simulations are carried out on triangular mesh model of beethoven with 2655 vertices and 5028 cells. The quality of mesh model is measured by Metro [4], which is errormeasuring software for polygonal mesh models. In the simulation, maximum between forward and backward Hausdorff distances is measured. And the performance of watermark detection is evaluated using DR (Detection Ratio). To consider transparency of watermark, we embed watermark sequence of 55 bits.
To evaluate the robustness of the proposed, stego meshes suffer from several attacks such as adding random noise, clipping, similarity transform and vertex reordering. The simulation results are given in table 1. In case of adding random noise, DR decreases proportionally to error rate. The proposed has good watermark detection performance up to 1.00% of error rate. For the case of clipping attacks, we assumed that the center of gravity is known in watermark detection side. The proposed can extract correctly most watermark bits from some part of stego model, as it uses statistical approach. Vertex reordering attack is carried out repetitively 100 times, changing the seed value of random order generator. Similarity transform is carried out with various rotation, translation, and uniform scaling parameters. Simulation results demonstrate that the proposed is remarkably robust against both similarity transform and vertex reordering.
Table 1. Evaluation of the proposed, in terms of Hausdorff distance and DR, after various attacks
Attacks

Conditions

H(V,V)

DR

Adding
random noise

0.25%

0.00513

1.00

0.50%

0.00584

0.95

0.75%

0.00654

0.89

1.00%

0.00723

0.82

Clipping

5%

0.09995

1.00

20%

0.09996

1.00

50%

0.09999

0.98

Similarity transform

xaxis

1°

×0.8

+0.001

0.07634

1.00

yaxis

1°

×0.8

+0.001

zaxis

1°

×0.8

+0.001

xaxis

1°

×1.2

+0.010

0.06895

1.00

yaxis

3°

×1.2

+0.030

zaxis

5°

×1.2

+0.050

xaxis

15°

×1.5

+0.010

0.09999

1.00

yaxis

30°

×1.5

–0.030

zaxis

45°

×1.5

+0.080

Vertex reordering

Average of 100 times trials

0.00442

1.00

4. CONCLUSIONS
In this paper, we proposed a new blind watermarking technique for 3D polygonal mesh model. The proposed embeds watermark by using distribution of vertex norms. Through the simulations, we proved that the proposed is remarkably robust against similarity transform and vertex reordering, which are serious attacks on 3D watermarking field. In addition, the proposed has good watermark detection performance against adding random noise and clipping attacks. Moreover, the method is very simple in both embedding and detection processes. As results, the proposed presents a new possibility to solve the fundamental problem of 3D watermarking.
REFEREMCES
[1] Zhiqiang Yu, Horace H.S.Ip, and L.F.Kwok : A robust watermarking scheme for 3D triangular mesh models. Pattern Recognition, Vol.36, Issue.11, (2003) 2603–2614
[2] Satoshi Kanai, Hiroaki Date, and Takeshi Kishinami : Digital Watermarking for 3D Polygons using Multiresolution Wavelet Decomposition, Proceedings Sixth IFIP WG 5.2 GEO6, (1998) 296–307
[3] Oliver Benedens : Geometrybased watermarking of 3D models. IEEE Journal on Computer Graphics and Applications, Vol.19, Issue.1, (1999) 46–55
[4] P.Cignoni, C.Rocchini and R.Scopigno : Metro: measuring error on simplified surfaces. Computer Graphics Forum, Vol.17, No.2, (1998) 167–174