Other transformed methods

Other transformed domain methods includes Singular Spectrum Analysis (SSA) [94], Discrete Cosine Transform (DCT) [65], spherical parametrization [83], Oblate Spheroidal Harmonics [75].

The Singular Spectrum Analysis is assuming a time series of the object geometry. The time series is virtually the vertex order in the object file. Murotani et al [94] proposed a non-blind algorithm based on the SSA in 2003. The spectrum is then computed using the SSA for the trajectory matrix derived from the vertex series and used for watermarking. The original object is required in the detection stage to retrieve the watermark. The experiments show the algorithm is robust against the similarity transforms and the additive noise. The algorithm is a spectral domain method but obviously the assumption of the time series in SSA is not robust and the watermark can be easily destroyed. Any attack that modifies the vertex order, for instance a vertex reordering, will fail the algorithm.

Jeon et al [65] applied the Discrete Cosine Transform (DCT) to devise a 3D watermarking algorithm. The algorithm generates a set of triangle strips according to a secret key. The strips are then transformed into the spectral domain using DCT. The mid-frequency band of AC coefficients are used to carry the watermark in order to balance the trade-off between the robustness and the imperceptibility. The authors claim three advantages of using triangle strips. 1. The user who doesn’t know the starting face for creating triangle strips can not distinguish a watermark pattern. 2. The triangle strips also have the property of mesh partition, it can be considered as a subset of the mesh object. 3. Finally, inserting the message into multiple strips strengthen the robustness. As proved experimentally, the method is rather robust against the additive noise attack and geometry compression. However, it is not resistant against any attacks that alter the connectivity of the mesh.

tion in 2004. The geometry of a 3D object is transformed into spherical signals using a global spherical parametrization and an evenly sampling scheme. Spherical harmonic transformation is then applied to generate frequency coefficients for em- bedding watermarks. The algorithm shows a good robustness against the additive noise attack.

In 2005, Wu et al [139] argued that the Combinatorial Laplacian spectral method does not encode any geometric information in the discrete operator. In addition, the inverse of a large matrix is computationally unfeasible. Wu et al introduced a new set of geometry dependent orthonormal basis functions derived from the Radial Basis Functions. By using the scheme, the main features of the mesh object can be recovered by using just a few spectral coefficients. The advantage of the new basis functions is that its computation is significant faster than the Laplacian based functions. However, the same as the other non-blind spectral methods, the message detection relies on the mesh registration, resampling and remeshing. As a consequence, the robustness benefits from those extra steps.

In 2009, Konstantinides et al [75] proposed a blind and robust method based on the Oblate Spheroidal Harmonics. The transform is based on the use of one of the many variants of oblate spheroidal harmonics; namely the Jacobi ellipsoidal coordinates [54, 120]. The algorithm realigns the mesh object by translating the object onto the mass centre, uniformly normalization and PCA rotation. However, the robustness of these traditional alignment methods can be severely affected by attacks. Thus, a smoothing scheme is proposed prior to the alignment. This is based on the observation that attacks like noise, resampling, remeshing and mesh simpli- fication tends to alter the high frequency properties, while the smoothing tends to eliminate the high-frequency attributes, the smoothed versions of the attacked mesh and the intact one converge to roughly the same one. Patches are then generated on the smoothed surface while the patch centre is established as the intersection between an randomly-generated ray and the object surface. The radius of the patch is defined according to the geodesic distance. While the patch is generated on the smoothed surface, the points are sampled on the original object by projection from the smoothed version to the original one. When the preprocessing steps are all completed, the patches are spectral transformed and the watermark is embedded

in the spectral coefficients. The algorithm is compared with the state of the art algorithm proposed by Cho et al [31] and the results show a better robustness and better visual quality. However, the algorithm involves too many preprocessing steps like reorientation, patch generation and sampling etc. Moreover, the capacity of the algorithm is quite low and it is tested for embedding only 7 bits of message is tested in the experiments.

Ai et al [1] introduced a method that firstly find out the feature points from the rapid changing regions. The mesh is uniquely segmented into Voronoi patches using those feature points. Each patch is used to generate a range image. A Discrete Cosine Transform is then applied on the range image and the bit message is inserted into the high frequency of the image. The algorithm is robust against various attacks including mesh simplification, additive noise and cropping etc. This method directly applies 2D image watermark techniques to the 3D methods by generating the range image of the mesh object.