1. INTRODUCTION:

Since the 3D watermarking was firstly introduced by Ohbuchi it is becoming an active research area during the last decade. 3D watermarking is inspired from the image watermarking and video watermarking. The techniques in 2D watermarking cannot be directly applied to 3D watermarking. Generally speaking, the 3D watermarking can be classified into transformed domain watermarking and the spatial domain watermarking from the perspective of the embedding domain. Then the transformed domain methods can be further split into spectral methods and multiresolution methods. In this paper, I will firstly comprehensively survey the transformed domain methods followed by that of the spatial domain methods. We focus mainly on the robust methods and briefly mention the others. And we introduce the assessment methodology of the 3D watermarking algorithms.

2. Spectral domain algorithms:

The methods of mesh spectral analysis are inspired by the development of spectral graph theory, signal processing and the kernel principal component analysis and spectral clustering in the computer vision and machine learning. The mesh spectral analysis of a given mesh object O with N vertices generally has the following three steps in common:

1) A square Laplacian matrix L of size N × N is constructed. The Laplacian matrix which is a discretization of a continuous operator represents a discrete linear operator based on the connectivity of the input mesh.

2) The second step is almost identical for all methods. This consists of to eigen decomposing the matrix L.

3) Process the calculated eigen values usually by embedding constraints or by adding noise, i.e. frequency coefficients, and the eigenvectors, i.e. the Ortho-normalizing space.

The Laplacian matrix L is a square matrix which characterizes the pair wise information (also called affinity in the literature) between any two vertices on the mesh O, e.g., Li, j reflecting the weight between the i^th vertex and the j^th vertex. The Laplacian operator has a strong physical meaning which is equivalent to a second order differential operator in physics in the study of wave propagation, heat diffusion, electrostatics and fluid mechanics. Because the matrix encodes the one ring neighbourhood information of the mesh, it can also be considered as a convolution kernel from the signal processing perspective. According to the different requirements, the Laplacian matrix can be used to simulate different continuous operators. Not only the connectivity information can be considered but also the geometric information can be embedded in the matrix as well. Because the Laplacian matrix L is square and positive semi-definite, it means the eigen-decomposition produces a set non-negative eigen values and a set of ortho normal eigenvectors. Chung stated that Laplacian eigen values are closely related to almost all major graph invariants. In other words, the eigen values contain most of the information about the characteristics of the shape. Inspired by such properties, the graph spectra are used for shape matching and retrieval in computer vision and for indexing On the other hand, eigenvectors provide a more refined shape characterization. Furthermore, the eigenvectors have much wider applications including object segmentation clustering parameterization and shape matching.

Instead of using the eigenvalues and eigenvectors directly, the eigenvectors can be used similarly to Fourier descriptors. And the spectral coefficients can be obtained by projecting the mesh geometry, i.e. the vertex coordinates, onto the orthonormal eigenspace defined by the eigenvectors. The coefficients also contain the energy and global information of the mesh global information. They can be used in various ways such as geometry compression, mesh watermarking and as Fourier descriptors.

In the following of this section, the watermarking methods are classified according to the type of basis functions used in the spectral analysis. Methods based on Combinatorial Laplacian are firstly introduced. Most of the spectral 3D watermarking methods belongs to this branch, Methods based on manifold harmonics is followed and lastly the other types of spectral methods.

2.1. Combinatorial Laplacian methods: A combinatorial Laplacian is a matrix operator that solely depends on the connectivity of the mesh. It treats the pairwise relation as a binary delta function, i.e. if vi is connected with vj, the corresponding entry is 1 otherwise, is 0. The idea was firstly introduced by Taubin to approximate low pass filters. Kaini et al compress the mesh geometry making use of the eigen projections. Zhang studies several variants of combinatorial Laplacian and their properties for spectral geometry processing and JPEG-like mesh compression. Most of the spectral watermarking methods so far tend to embed the message in the spectral coefficients called eigen projections in some papers. This is because the basis functions, i.e. eigenvectors, of the combinatorial Laplacian operator are stable and insensitive to the geometry changes since only the connectivity is considered in the matrix. Thus, after watermarking, the connectivity is not changed so the watermarked coefficients can always be detected. Some of the watermarking methods tend to remesh the mesh object ensuring the connectivity is consistent.

2.1.1 Theoretical background .We first briefly review the theoretical background of spectral analysis using the combinatorial Laplacian based on the work proposed by Karni et al .Given a mesh object O containing N vertices, the Laplacian matrix of dimension N × N is built according to its connectivity as follows:

Where, |Nvi | represents the valence of the vertex vi, i.e. the number of its neighbours directly connected to it. Then, the Laplacian matrix is eigen-decomposed as:

Where Ω is the diagonal matrix containing the eigen values and q is the matrix consisting of the eigenvectors. The eigenvector matrix q is sorted in the ascending order according to the magnitude of its corresponding eigen values in the diagonal matrix Ω. While the eigen values in Ω are considered as frequencies, q constitutes an orthonormal basis of the mesh O. The spectral coefficients are calculated by projecting the vertex coordinates on the basis functions defined by the eigenvectors q:

C = qV

where V is the matrix containing the geometry of the vertex coordinates. The spectral coefficients of low frequencies, i.e. the coefficients correspond to the small eigen values in Ω, reflects the general shape or the large scale information of the mesh. In contrast, the high frequency coefficients indicate the details or the small scale information of the mesh. Figure 1 shows a set of spectral coefficients. 90% of the mesh energy is contained in the low frequency, while the energy in the high frequency is much lower. To reverse the transformation process, the geometry can be recovered as:

V = q^TC

Figure 1: A plot of spectral coefficients

Non-blind methods

Ohbuchi et al proposed a non-blind method in 2001 based on Karni’s analysis. This is the first 3D watermarking method based on the spectral domain. The method applies the spectral analysis employing the basis functions of the combinatorial Laplacian. The message is embedded by slightly modifying the low frequency and medium frequency coefficients. In the detection stage, both the original object and the watermarked object need to be spectrally decomposed. The embedded information is retrieved by comparing the difference of the spectral coefficients between the original and the watermarked ones. However, any modification to the combinatorial Laplacian would result in different eigenvectors. So the algorithm is sensitive to any of the attacks that modifies the connectivity of the mesh. Furthermore, the method is computational expensive. The matrix eigen decomposition requires O(N3) complexity. Thus, although any mesh can be spectral decomposed theoretically, it is not feasible to do so in a large mesh in practice.

In 2002, Ohbuchi et al extended their previous work in three directions. The mesh size was reduced by splitting it into several patches. Each patch is used to carry a set of bits. A more efficient numerical method called Arnoldi is employed to eigen-decompose the Laplacian matrix. The Arnoldi method can calculate the leading spectral coefficients as required, instead of calculating the full set of the eigenvectors. Finally, the 3D object is remeshed before detection in order to recover the original connectivity such that the Laplacian Matrix is identical to the original one. The method proposed in 2002 is resistant to the connectivity alteration attacks like mesh simplification and cropping because the connectivity is enforced to be the same in the detection stage. This method is computationally more efficient as not only the matrix size is reduced but also the numerical routine for eigen decomposition is improved. In 2004, Ohbuchi et al and Cotting et al proposed two similar methods to extend the ideas from the other papers to the point sampled object. Although the object is in point cloud format, the intrinsic connectivity is built before constructing the Laplacian Matrix.

Lavou´e et al proposed a similar method as Ohbuchi’s to watermark subdivision surfaces. The message is embedded in the control mesh (called also base mesh) of the subdivided mesh. Control mesh is the lower-resolution version of the original mesh after the wavelet decomposition. In the message retrieval, the control mesh synchronization need to be done on the attacked model so as to detect the message. There are two improvement of Lavou´e’s method over Ohbuchi’s methods. Firstly, Lavou´e proposed a so-called Low Frequency Favoring (LFF) modulation scheme. The full range of spectrum can be used for embedding by employing the LFF scheme. The LFF takes the magnitude of the spectral coefficients into account and adaptively embed the watermark. For the high frequency coefficients, i.e. a small numerical value, the embedding strength is adjusted to a smaller value. Moreover, the capacity of the watermark and the imperceptibility is optimized using error correcting codes. A large message can be encoded using a relatively small number of bits. The method claimed a 20% improvement of the watermark robustness over Ohbuchi’s method.

All these methods are non-blind and the bit carriers are the low frequency and medium frequency coefficients. The main strength of these methods is the relatively high robustness. Nevertheless, the premises is made that the original object must be present in the message retrieval stage. There are three disadvantages. Firstly, the original object is required to recover the original connectivity. This involves extra steps and computational cost. Secondly, the computational cost is higher than spatial domain methods in general. Thirdly, it is hard to control the distortion. Although there are embedding strength parameters in order to control the visual distortion, there is no explicit relation between the coefficients and the vertex co ordinates. Therefore, decreasing the embedding strength is the only way to reduce the visual distortion. Furthermore, the distortion is large because the change of low frequency will change the general shape of the object.

Blind methods:

Cayre and Alface et al proposed a blind algorithm based on the spectral domain in 2003. A mesh object can be considered as a three dimensional signal, i.e. (vx, vy, vz), we can have the corresponding spectral coefficient triplet (Cx, Cy, Cz). Every triplet is considered as an embedding primitive. The triplet is sorted in the ascending order and the maximum Cmax = max(Cx,Cy,Cz) and minimum value Cmin = min(Cx,Cy,Cz) are regarded as the modulation range. The mean value Mean = (Cmax + Cmin)/2 is used to distinguish the bits 1 and 0 intervals. When embedding a 1 bit, the medium coefficient is moved into the interval of values corresponding to the bit 1 and vice versa. Figure 2 shows an example of the triplet embedding. The embedding message is inserted repetitively into the low and medium frequency to ensure the robustness. The method is the first blind algorithm based on the spectral domain, but its robustness is very limited. Alface et al in 2005 proposed to segment the 3D object into patches for reducing the embedding complexity while the core embedding method is still the same as Cayre’s method. Firstly, the feature points are automatically selected through a multi-scale estimation of the curvature tensor field. Then, the algorithm proceeds by partitioning the mesh shape using a geodesic Delaunay triangulation of the detected feature points. Each of these geodesic triangle patches is then parametrized and remeshed by a subdivision strategy to obtain a robust base mesh. The remeshed patches are watermarked in the spectral domain and original mesh points are finally projected on the corresponding watermarked patches. The automatic feature point detection and the patch generation are the main contribution of Alface’s method. The core watermarking process is basically identical with Cayre’s method. Thus, it suffers from the low robustness problem as well.

All these methods are blind the main embedding idea is to encode information into the coefficient triplet. A lot of efforts are made on the preprocessing steps such as the robust feature points detection and the patch generation. And the spectral decomposition is made on the remeshed and parametrized model. The robustness of those methods somehow depend on the robustness of the pre-processing stages more than the embedding itself. Same as in Ohbuchi’s methods mentioned in the previous section, Alface’s methods embed the message in the low and medium frequencies. In other words, it embeds the message into the “shape” of the object. The algorithms show certain robustness. And lastly, the methods strongly rely on the pre-processing of the robust feature points, patch generation, parametrization and remeshing.

Figure 2: Cinter is moved into the 1 bit interval when embedding 1 bit. Figure is taken from Cayre’s method

2.1.2 Manifold harmonics: Although the combinatorial Laplacian has the perfect reversibility and it is simple to implement, the lack of the geometry information makes it inadequate to describe the feature of an object. There is another kind of discrete Laplacian which deals with the geometry properties of the mesh, called Manifold Harmonics, proposed by Vallet. Its transformation is called Manifold Harmonics Transform (MHT).

The Manifold Harmonics injects the geometry information by calculating the cotangent (cotan) weights of the one ring neighbourhood. The weight between vi and vj is measured by the cotan angle opposite to the edge formed by the two vertices. The cotan weight derived from Finite Element Modeling has been proved a close relationship with the surface curvature. They converge to the continuous Laplacian under certain conditions as explained. Nonetheless, the cotan weights are calculated by the dual cell area of each vertex, which is nonsymmetric. Thus, the cotan weights cannot be used for the spectral analysis directly. L´evy tried empirical symmetrization. Vallet et al clarify these issues based on a rigorous Discrete Exterior Calculus (DEC) formulation and recover symmetry by expressing the operator in a proper basis. The symmetry property ensures its eigenfunctions are both geometry aware and orthogonal as well. Theory background

In this section, we clarify the theoretical issues of the Manifold Harmonic Transform.

Similar to the Laplace operator in Euclidean space, the Laplace-Beltrami operator

∆ is defined as the divergence of the gradient for functions defined over a manifold

O with its metric tensor. The eigenfunction and the eigenvalue pair (Hk, ƛk) of ∆

on manifold O satisfy:

The above eigen-problem is then discretized and simplified within the finite element modeling framework as the following matrix equation:

where hk = [H^k₁, H^k₂, . . . , H^k_n]^T, the N × N matrix D is diagonal and called lumped mass matrix as:

where NFi is the number of neighbouring faces of vertex vi. t is a neighbour of vertex vi. |t| gives the area of the triangle. The matrix Q called stiffness matrix is also of size N × N:

where αi,j and βi, j are the two angles opposite to the edge ViVj. The Manifold Harmonics Basis can be calculated by eigen-decomposing the matrix Q in equation. The frequencies are represented by the corresponding eigenvalues. Let us define vector x = (x1, . . . , xN) (respective y and z) containing the x coordinates of the mesh. With the Manifold Harmonics Basis, the kth spectral coefficient can be calculated as:

Thus, the amplitude of the spectral coefficients is defined as:

The object can be exactly reconstructed by using the inverse manifold harmonics transform. For coordinates x (resp. y, z), we have

With the geometry information embedded in the operator, the spectrum obtained from the MHT nicely captures shape characteristics of the object. However, on the other hand, the side effect is that when the geometry of the mesh is changed, e.g. watermarked, the approximation matrix Q will be changed. Thus, if we apply the MHT again on the modified mesh, we can no longer retrieve the watermarked coefficients again. The causality problem is the major obstacle of using the MHT to design a watermarking method. People tend to use the iteration methods to recheck the coefficients to ensure a successful embedding.

Another major contribution of Vallet’s work is a band-by-band spectrum computation algorithm and an out-of-core implementation that can compute thousands of eigenvectors for meshes with up to a million vertices. These make the spectral analysis directly usable in practice on a large mesh object, besides its common use as a theoretical tool.

Blind methods:

Since the Manifold Harmonics Basis incorporates more geometry information of the mesh object, it captures more shape information rather than when considering topology only. The spectrum obtained from the MHT is very stable and consistent for the other object representations. It means that the attacks like mesh simplification, resampling and remeshing, which do not alter the shape of the object, will not affect the spectrum very much. Because this feature of the MHT, it becomes a popular transformation technique to devise robust watermarking schemes. In this section, I will briefly introduce two recent robust and blind algorithm based on the manifold harmonics transform proposed by Vallet et al. Liu et al proposed a robust and blind algorithm based on the manifold harmonics in 2008. The method takes the medium frequency coefficients as the embedding domain. The authors experimentally show that the medium frequency changes affect the spectrum very little and can be accepted for watermarking purpose.

Every ten coefficients are grouped as a embedding primitive used to carry one bit of message. Two embedding methods were proposed, one is called progressive embedding and the other is non-progressive embedding. The progress embedding picks one coefficient magnitude from the primitive and calculate the mean value of the other nine coefficient magnitudes. The selected embedding candidate magnitude is moved more than the mean value for embedding a 1 bit and less than the mean value for embedding a 0 bit. In order to overcome the causality problem, i.e. the watermarked coefficients cannot be exactly recovered from the watermarked model, the method iteratively check the coefficients until it satisfies the embedding condition. While the assumption is made that a small change on the medium range spectrum will not affect the shape of the spectral coefficients significantly. The experiments show certain robustness against various attacks when 5 bits of message are embedded.

Wang et al proposed another robust and blind algorithm based on the MHT in 2009. Unlike Liu’s method where only 5 bits of message a embedded, Wang’s method is able to carry 16 bits of message. The scalar Costa scheme which is a quantization algorithm is used to modulate the low frequency coefficients to embed the message. The unique code-book generated from the Costa scheme ensures a good security of the algorithm. The spectral coefficients are repetitively embedded and iteratively checked to avoid the causality problem. The author argued that although the low frequency changes introduce a large numerical error on the mesh object, human eyes are not sensitive with respect to distortion of the large scale changes. The method was compared with Liu’s method and Cho’s method and shows a good result on both visual quality and robustness.

Other transformed methods:

Other transformed domain methods includes Singular Spectrum Analysis (SSA), Discrete Cosine Transform (DCT), spherical parametrization, Oblate Spheroidal Harmonics.

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 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 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 cannot 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.

Li et al proposed a non-blind method based on the spherical parametriza 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 embedding watermarks. The algorithm shows a good robustness against the additive noise attack.

CONCLUSION:

In 2005, Wu et al 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 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. 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 simplification 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 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 introduced a method that firstly finds 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.

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Received on 21.11.2010 Accepted on 26. 02.2011

Research J. Engineering and Tech. 2(1): Jan.-Mar. 2011 page 42-47