Shape watermarking based on minimizing the quadric error metric

IEEE INTERNATIONAL CONFERENCE ON SHAPE MODELING AND APPLICATIONS (SMI) 2009

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Shape watermarking based on minimizing the quadric error metric Ming Luo and Adrian G. Bors Department of Computer Science, University of York, YO10 5DD, UK E-mail: { ming , adrian.bors }@cs.york.ac.uk

Abstract—Blind and robust watermarking of 3D object aims to embed codes into a 3D object such that the object is not visually distorted from the original shape. An essential condition is that the message should be securely extracted even after the graphical object was processed. In this paper, we propose a novel blind and robust mesh watermarking method based on the quadric error metric. The vertices are firstly grouped into bins using a secret key according to their distances to the object center. The statistics of the distances in each bin is modified when embedding the message. A novel quadric selective vertex placement scheme is proposed for finding the best location of each vertex, following watermark embedding, such that the resulting shape distortion is minimal. Experimental results show that the proposed method reduces the distortion to a minimum in the 3D shape. Keywords—Digital Watermarking; Quadric Error Metric; Shape Distortion Analysis 1. I NTRODUCTION The application of watermarking methodology is extensive and includes copyright protection, authentication, digital fingerprinting, embedding information relevant to the graphical object for database indexing purposes, etc. [1]. Watermarking has two separate processing stages corresponding to the watermark embedding and retrieval, respectively. Depending on whether the original cover media is needed or not in the detection stage we have non-blind and blind watermarking. The methods from the first category usually have good robustness [2], [3], [4] but a limited range of applications. In this paper we consider robust and blind watermarking of 3-D graphical objects represented as meshes. Watermarking of graphics has been performed in the spatial domain and in the transform domain. Various watermarking methods use ratios of geometric measures for watermark embedding [5]. For authentication purposes, the watermark is supposed to be fragile and to disappear from the regions where the object was tampered [6], [7]. A copyright protection watermarking algorithm employing changes in the histograms of 3D object surface normals was used in [3]. The transform domain watermarking methods consider embedding the message in the mesh spectral domain [4], [8], [9],

[10] or the wavelet domain [11], [12], [13]. Most of the spectral domain watermarking algorithms are nonblind and usually require appropriate registration for watermark detection [4]. Although such algorithms have good robustness against most attacks, their application for practical purposes is limited. The regular wavelet decomposition requires the mesh to be a perfect manifold and with a rigorous regular connectivity [11], [13], while irregular wavelet decomposition [12] requires a well defined starting point. Watermarking methods can also be categorized as deterministic [14] or statistical [15], [16]. The methods from the first category employ a set of constraints for embedding messages while the second category extract the message by using a statistical test. Usually, deterministic methods allow a higher capacity of information embedding, making them suitable for steganography, but achieve lower robustness to attacks. On the other hand, the statistical methods are more robust but they achieve lower embedding capacity rates. Zafeiriou et al. [16] proposed a robust watermarking method using distributions of distances from surface vertices to the local symmetry axis. Cho et al. [15] proposed a similar approach as in [16] by considering vector norms. Sets of vertices are grouped into bins according to their distance to the object center. The embeddings are performed by changing the vertex locations according to two histogram mapping functions. The statistical algorithms proposed by Cho et al. show excellent robustness against most common mesh attacks. Moreover, these algorithms are blind and do not require the knowledge of additional information or specific object representation properties such as a regular connectivity. However, Cho’s algorithms produce visible artifacts in the 3D object surface. In this paper we propose a new methodology based on the Quadric Error Metric (QEM). QEM was used for mesh simplification by Garland et al. in [17] by properly defining the local shape similarity error. A novel Quadric Selective Vertex Placement scheme is proposed in order to embed the watermark. The proposed methodology significantly reduces the perceptual distortion in the watermarked shape while providing a similar robustness to that of Cho’s algorithms. The rest of the paper is organized as follows. Section II introduces the quadric error metric as a cost function in assessing shape distortion. Section III details the proposed method step by step. Section IV details the watermark detection procedure.

Experimental results are provided in Section V, while Section VI outlines the conclusions. 2. T HE Q UADRIC E RROR M ETRIC Quadric Error Metric (QEM), proposed by Garland and Heckbert [17], is used in mesh simplification based on edge collapsing for assessing the similarity between the simplified and the original local surface. The standard representation of a plane is the set of all points for which nT x + d = 0 where n = [nx ny nz ]T is a unit normal (i.e., n2x + n2y + n2z = 1), x = [x y z]T is a point in the 3D space and d is a constant. The quadric QF (x) is defined as the squared distance from a point x to a plane F in the 3D space as: QF (x) = xT Ax + 2bT x + c

(1)

where the quadric is described by the triplet QF = (A, b, c), where A = nnT , b = dnT and c = d2 . An important property of the quadrics is that the quadrics can be summed up easily for entire 3D surface regions and the resulting quadric will be a triplet whose components are the sums of each individual quadric triplet components. Let us assume Nvi as the number of neighbouring faces adjacent to the vertex vi of a mesh O and that each j 0 th face, j = 1, . . . , Nvi lies on the plane Fj . Then the quadric error with respect to the vertex vi given the location of a point x and Nvi can be defined as :   Nvi Nvi Nvi Nvi X X X X cj  bj , Aj , QFj (x) =  Qvi (x) = j

j

j

ρb =

(1 − 2ε)(ρmax − ρmin ) M

(4)

where ε ∈ [0, 0.15] is defined as a small percentage accounting for outliers. The value of ε is generated according to a secret key. This adds to the watermark security because without the knowledge of the secret key it would be impossible to retrieve the embedded watermark. Then all the vertices are grouped into M sets according to their distances to the object centre o as: Bi = {vj ∈ O |

ρmin + |O| ∗ ε + (i − 1)ρb ≤ ρj , ρj < ρmin + |O| ∗ ε + iρb } (5)

j

(2) where QFj = (Aj , bj , cj ). The quadric error of x with respect to vi is the sum of the squared distance from a point x to all the planes adjacent to vi . The quadric error metric is proved to have close relation to the local shape characteristics including its fundamental form, curvature and local moment. It is also shown that the metric represents a lower complexity implementation of the Hausdorff distance [18]. Moreover, QEM has low requirements for the mesh surface properties, i.e. it can be even applied on a non-manifold mesh. In this study, we employ the QEM as a measure in order to evaluate the local distortion error for selecting the location of the watermarked vertex. 3. Q UADRIC ERROR BASED WATERMARKING 3.1 Bin Generation Assume that we want to embed a code of M bits into the 3D object O. The vertices of the mesh object O are clustered into M bins such that each bin is used to hold one bit of message Bi , i = 1, . . . , M . In this paper, we cluster the vertices according to their distance from a vertex vj to the center of the object o, calculated as: X 1 area(vj )vj (3) o= area(O) vj ∈O

where area(vj ) is the sum of the areas of the triangular faces incident to the vertex vj and area(O) is the area of the entire surface of the graphical object O. The object center defined in this way is more robust than simply taking the average of all vertices particularly when considering attacks such as remeshing, simplification, etc. Note that there are other ways of evaluating the object center such as for example by using the object moments [19]. For a given vertex vj ∈ O let us denote ρj = kvj −ok as the distance from the vertex vj to the centre of the object o. After ranking these distances we find ρmin = min(ρj ) and ρmax = max(ρj ), where j = 1, . . . , |O| and |O| represents all vertices from O. The vertices from the shape surface are split into M sets, each containing a number of vertices equal to:

3.2 Histogram Mapping Function Assume we want to embed a message bit into the i0 th bin. We propose two methods for message bit embedding. The first one corresponds to changing the mean value of each bin and the second method embeds the bit by changing the bin variance, as proposed in [15]. The minimum and maximum distance for each bin are: ρi,min = ρmin + |O| ∗ ε + (i − 1)ρb

(6)

ρi,max = ρmin + |O| ∗ ε + iρb

(7)

For the mean embedding method, the distances are firstly normalized to the range [0, 1] by: ρ˜ij =

ρij − ρi,min ρi,max − ρi,min

(8)

As shown in [15], the statistical variable ρ˜ij is close to a uniform distribution. Thus the expected mean value of the statistical variable is 1/2. In order to embed one bit the mean value is changed as:  1 2 + α if Bi = 1 µ ˆi = (9) 1 2 − α if Bi = 0 where α is the watermark strength factor influencing the visual distortion and robustness and Bi , for i = 1, . . . , M , is the bit to be embedded in the i0 th bin. In

order to change the normalized distances for fulfilling (9), the first histogram mapping function is defined as:  β ∈ (0, 1) if Bi = 1 ρ˜0ij = ρ˜βij (10) β ∈ (1, ∞) if Bi = 0 ρ˜0ij is the resulting watermarked normalized vertex norm. Finally, the watermarked vertex norms are obtained by mapping ρ˜0ij back to the original interval as: ρˆij = ρ˜0ij (ρi,max − ρi,min ) + ρi,min

(11)

The second method embeds the message by changing the variance of the norms. In this case, ρij is normalized to the range [−1, 1] as: ρ˜ij = 2 ·

ρij − ρi,min −1 ρi,max − ρi,min

(12)

The expected variance in this case is 1/3 for a uniform distribution. We embed one bit of message by modifying the variance of ρ˜ij according to :  1 3 + α if Bi = 1 σ ˆi2 = (13) 1 3 − α if Bi = 0 The histogram mapping function for modifying each element from the set Bi is defined as:  β ∈ (0, 1) if Bi = 1 00 β ρ˜ij = sign(˜ ρij )|˜ ρij | β ∈ (1, ∞) if Bi = 0 (14) Accordingly, the watermarked vertex norms are obtained by the inverse normalization function: ρˆij =

1 · (˜ ρ00ij + 1) · (ρi,max − ρi,min ) + ρi,min (15) 2

3.3 Constraints for minimum distortion watermark embedding We have to evaluate the quadric Qvi for every vertex vi ∈ O as detailed in Section II. In the following, we use the capital letter e.g. Vi for labeling a vertex, while the lower bold letter e.g. vi represents its coordinates. The aim of this watermarking algorithm is to change the location of the vertex Vi to Vˆi such that we have the condition kOVˆi k = ρˆi satisfied, where O represents the object center and ρˆi is obtained from either equation (11) or (15). In [15] this change was performed along the direction from the object centre to the given vertex Vi −−→ as OV . Although the movement along the direction −−→ i OVi guarantees the minimum Euclidean distance, i.e. kVi Vˆi k = kˆ ρi − ρi k, this does not ensure a minimum distortion to the shape surface because it does not considers the surface curvature. However, any point on the sphere centered in O and of radius ρˆi satisfies the condition of kOVˆi k = ρˆi and thus can be used as the position of the watermarked vertex. In order to make sure that the distortion caused by the embedding process is minimum, we employ the quadric error metric for evaluating the error introduced by the vertex movement following watermark embedding. Consequently, we construct the following system

of equations representing constraints that should be fulfilled for minimum distortion watermark embedding:  Nvi  X   Qvi (ˆ vi ) = QFj (ˆ vi ) (16) j    kOVˆi k = ρˆi where Nvi represents the total number of triangle faces which are adjacent to the vertex vi . The system of ˆ i is equations means that under the constraint that v located on the sphere centered in O and of radius ρˆi , the point which minimises the quadric error function introduces the smallest distortion to the object surface. There is no unique solution as both equations are nonlinear while solving such an optimization system is computationally complex. In the following, we propose an efficient approach for finding an approximate solution for the equation system (16). 3.4 Distance Range Construction Let us consider that we have a triangle 4M N K ∈ O on the object surface. For a point O located outside 4M N K we can find two points P and Q, P, Q ∈ 4M N K such that we can define the minimal and maximal distances, i.e. kOP k = min(kOXk) and kOQk = max(kOXk), ∀X ∈ 4M N K, respectively. For each face of the given shape, Fj ∈ O, j ∈ 1 . . . Nvi adjacent to the vertex vi , there is a point Pj of the minimum distance and a point Qj of the maximum distance to a given external point location O. For each triangle face Fj , we can define the following range of distances from the location of O: RFj = [kOPj k, kOQj k]

(17)

The distance from any point in the triangle face Fj to O will be in the range RFj . For a vertex vi , by considering all its adjacent faces, we can define the following range of distances calculated from the location of O : Rvi = [min(kOPj k), max(kOQj k)], j = 1, . . . , Nvi (18) where Nvi represents the number of faces which are adjacent to vi . The distance range Rvi of vertex vi means that given a distance ρˆi and if ρˆi ∈ Rvi , there must be at least one point Wj on one of the neighbouring faces satisfying the distance condition that kOWj k = ρˆ. As the candidate point is always on the surface, it usually preserves well the surface, especially when considering a flat surface. However, this may not always be true. In the following section we introduce the Quadric Selective Placement scheme in order to ensure a minimal local distortion to the surface following watermark embedding. 3.5 Quadric Selective Placement Scheme Suppose that we have the new vertex distance following watermarking ρˆi , the distance range Rvi as in equation (18) and the QEM Qvi for a vertex vi as in

equation (2). We want to change the location of the ˆ i such that the distance of the updated vertex vi to v vertex to the center O fulfils our watermark constraint kOVˆi k = ρˆi and such a movement introduces the minimum distortion to the object surface. α

ρ

K=Q Wj

M

the range of RFj and the last two equations define the way of obtaining wj when ρˆi is out of the range RFj . Thus, for each face Fj we have one candidate vertex wj , j = 1, . . . , Nvi as the potential position of the ˆ i . For defining the location v ˆ i of watermarked vertex v the watermarked vertex we choose the one that has the minimum quadric error Qvi (wj ) from among all the candidates corresponding to the faces adjacent to vi : ˆ i = arg min Qvi (wj ), j = 1, . . . , Nvi v wj

P O’ N

O Fig. 1. The watermarked vertex Wj is located on the intersection between the sphere centered in O, of radius ρˆ, and 4M N K from the surface of O.

Let us assume that we have a given face Fj represented as 4M N K as shown in Figure 1. For each neighbouring face Fj , adjacent to the vertex vi , if the watermarked distance ρˆi ∈ RFj , then there must be a point Wj on the curve representing the intersection between the 4M N K and the sphere centered in O and of radius ρˆi . Let us consider wj as the candidate location ˆ i on the face Fj for embedding the constraint ρˆi . for v Depending on the local geometry, when comparing the watermark constraint ρˆi and the interval RFj , defined in equation (17), we have the following cases of possible point locations wj for a neighbouring face Fj , adjacent to the vertex vi :  −−→   V i Pj   vi + kVi Wj k ·   kVi Pj k     if ρˆi  ∈ [kOPj k, kOVi k]   −−−  →   V Q   vi + kVi Wj k · kVi Qj k i j wj =   −−→  if ρˆi ∈ [kOVi k, kOQj k]    OPj   if ρˆi < kOPj k  o + ρˆi kOPj k      − − →   OQj   o + ρˆi if ρˆi > kOQj k kOQj k

(19) where Pj and Qj represent points on a given face Fj corresponding to the minima and maxima of distances calculated from the center O. The first updating equation corresponds to the case shown in Figure 1, where we consider Vi ≡ K. In Figure 1 we also show the locations of the points P and Q, where the first is found using the theorem of the three perpendiculars and the second is identical to K. In Figure 1 it can be observed that kVi Wj k can be easily calculated using the cosine theorem from 4OKWj . The first two equations from (19) correspond to the calculations when ρˆi is within

(20)

The proposed selection scheme is a geometrically motivated approximation of the quadric error minimization problem as provided in equation (16). Although the algorithm makes sure that the new selected vertex introduces a minimum distortion over all the candidates with respect to the original object surface, it does not guarantee that the new triangle will not flip-over after the vertex movement. A further simple consistency check on the surface normals can be used in order to avoid such problems. However, such a constraint may limit the robustness of the algorithm as in some cases the vertex will not be able to move to the position satisfying the embedding condition. Each neighbour face is processed for a possible candidate wj thus resulting in the complexity of O(Nvi ) for finding the location of ˆi. a single watermarked vertex v 4. WATERMARK D ETECTION The watermark extraction algorithm is blind, i.e. it does not need the original object in the detection stage. We segment the vertices into bins using a secret key as we used in Section 3 for embedding. Then, the statistical variable of distances from the vertices to the object center is normalized according to either equation (8) or (12) for the mean or variance methods, respectively. The message is extracted by comparing the mean value of the histogram with 12 for the mean method as:  if µ ˆi > 12 then Bi =1 (21) if µ ˆi < 12 then Bi =0 And the variance is calculated and compared with 13 :  if σ ˆi2 > 13 then Bi =1 (22) if σ ˆi2 < 13 then Bi =0 5. E XPERIMENTAL RESULTS The proposed watermarking methodology is applied on three different 3D mesh objects of various sizes and characteristics. These objects are shown in Figure 2: Bunny with 35, 947 vertices, 69, 451 faces, Gear with 231, 703 vertices and 463, 430 faces and Dragon with 437, 645 vertices, 871, 414 faces. This set includes a CAD object. CAD objects are generally considered to be the most difficult objects for watermarking purpose because they often contain large flat regions and particularly requires very low distortion watermarking.

in the watermarked objects when using the proposed methodology.

(a) Bunny Fig. 2.

(c) Gear

Object QSPMean QSPVar ChoMean ChoVar

(d) Dragon

Mesh objects used in the experiments.

Bunny 0.25 0.17 0.77 0.37

Gear 447.58 164.53 1481.24 811.89

Dragon 0.23 0.11 0.85 0.44

TABLE 1 V ISUAL D ISTORTION MEASURED BY MRMS (×10−4 ).

In the following experiments, the objects are watermarked under the same conditions for all the methods by embedding a total of M = 64 bits considering a trade-off between the robustness and capacity, while the watermark strength factor is set as α = 0.1, unless stated otherwise. The robustness is measured by the detection ratio, i.e. the ratio of the number of bits correctly detected over the total number of embedded bits. The results correspond to the average for 50 test cases when using random messages and random keys. All four methods are perfectly robust against the similarity transformation and the vertex reordering attack. Figure 4 shows the effects of the watermarked Bunny after various attacks. In the experiments we vary the intensity of the attack up to the level where the resulting object becomes seriously degraded.

(a)Noise (ε = 0.5%)

(b) Smooth λ = 0.5, 10 iterations

(c) Simplified 90%

(d) Quantization 7 bits

Fig. 3. Visual effects of shape distortion caused by the watermark when applied on the given set of objects. The first row shows the original objects. The objects from the second row to the fifth row are watermarked by QSPMean, QSPVar, ChoMean and ChoVar, respectively.

In the rest of the paper, we will use the abbreviation QSPMean, QSPVar for the proposed methods of Quadric Selective Placement mean and variance, respectively, as proposed in this paper. We denote by ChoMean and ChoVar the mean change and variance change algorithms, respectively, as described in [15]. The visual quality of the watermarked objects is measured numerically by the method called MRMS proposed by Cignoni et al. in [20] which was shown to be a good evaluation measure for comparing graphical objects. As shown in Table 1, the proposed methods starting with QSP introduce much less distortion than Cho’s methods. Figure 3 shows the visual comparison of the watermark effects for the four methods. As shown in Figure 3, we hardly can observe any distortion

Fig. 4.

Object after attacks.

Additive noise is a common attack in mesh processing. In the following we consider additive random noise according to the following distortion equation: − ˜i = v ˆ i + εkˆ v vmax k→ p

(23)

˜ i represents the distorted watermarked vertex where v ˆ i , ε ∈ [0, 1] is the percentage of kˆ v vmax k, which corresponds to the largest Euclidean distance measured − from the object center to all object vertices, and → p is

a unitary vector of random direction. Figure 4(a) shows the watermarked Bunny corrupted with additive noise of ε = 0.5%. The plots from Figure 5 show the robustness against noise when varying ε = [0.1%, 1%] for all four methods. As it can be observed from these plots, the proposed mean and variance method possess similar robustness with the corresponding Cho’s methods. The watermarking method relying on shifting the mean of the distributions performs slightly better than the variance methods in all the tested objects. We use the Laplacian Smoothing proposed in [21] for the smoothing attack. A watermarked and smoothed Bunny when considering λ = 0.5 and 10 iterations is shown in Figure 4(b). The robustness of the watermarking methods against the Laplacian smoothing ranging from 1 to 20 iterations with λ = 0.5 are provided in Figure 6 for the four objects. Again, our methods show similar robustness with Cho’s methods. With the object size increasing from Bunny to Dragon, the robustness increases accordingly. The quadric metric simplification software described in [17] was used for testing the robustness at mesh simplification. Figure 4(c) shows the watermarked Bunny object after 90% simplification. Figure 7 shows the robustness to the simplification attack for the four methods. In these experiments, we test the robustness of the algorithms varying the mesh simplification from 5% to 95%. Again, as observed from Figure 7, the performance of the proposed methods is similar with that of Cho’s methods. Variance change methods perform slightly better than mean change methods to this attack. The experiments on the Gear and Dragon objects provide more than 90% watermark detection ratio even after 90% mesh simplification. Figure 4(d) shows the bunny object attacked after 7 bits quantization. As shown in Figure 8 all four algorithms are fairly robust up to applying 8 bits quantization attacks and the mean change methods perform better than variance change methods, similarly to the noise type attacks. We compare the robustness of all four methods against the remeshing attack after uniform sampling of 100%, 80%, 60%, 40% and 20% of vertices of the original object using the method proposed in [22]. The results are shown in Figure 9. In this attack, Cho’s algorithms perform slightly better than the proposed algorithms. We also test the robustness of all four algorithms against the subdivision attacks including midpoint scheme, butterfly scheme and the Loop scheme proposed in [23]. The results are shown in Table 2. Similar to the smoothing attack, the robustness of all four algorithms increases with the size of the objects. The Bunny object is also used for studying the relation between the numerical distortion and the watermark strength factor α as well as the numerical distortion when increasing the bit capacity embedded as shown in Figures 10(a) and 10(b), respectively. As the strength factor α increases we can observe that more distortion

Object Bunny

Gear

Dragon

Method QSPMean QSPVar ChoMean ChoVar QSPMean QSPVar ChoMean ChoVar QSPMean QSPVar ChoMean ChoVar

Butterfly 1 1 1 1 1 1 1 1 1 1 1 1

Loop 0.75 0.73 0.73 0.74 1 1 0.99 1 1 1 1 1

Midpoint 1 1 1 1 1 1 1 1 1 1 1 1

TABLE 2 ROBUSTNESS AGAINST THREE DIFFERENT KINDS OF SUBDIVISION ATTACKS

is introduced in the object. While the more bits are embedded, the less distortion is introduced. This is because the vertices in the object will be divided into more bins. Thus the statistics in each bin contains less variables and it is easier to modify its statistical characteristics. Clearly, according to the experiments described in this paper the proposed watermark methodology produces smaller distortion when compared to the Cho’s algorithm. Figures 11(a) and 11(b) demonstrate the relation between the robustness and capacity of the QSPMean and QSPVar methods, respectively. As the capacity of the message payload increases, the robustness of the proposed algorithms drops. 6. C ONCLUSION In this paper, we propose a novel robust and blind 3D watermarking method based on the Quadric Error Metric minimization scheme. The proposed method employs a statistical watermark embedding approach which has essential advantages such as high robustness while being blind, efficient computation complexity and no specific requirements from the cover object. The proposed method provides a significantly reduced object watermark distortion when compared to other methods. However, as our algorithm is a statistical based algorithm, small objects, e.g. less than 10, 000 vertices, are not suitable to be watermarked by the proposed methodology. Moreover, the watermarks can be removed by cropping the 3D object mesh because this would change the object center. The proposed watermarking methodology can be used in a large category of applications including copyright protection, fingerprinting and database management. R EFERENCES [1] J. L. Dugelay, A. Baskurt, M. Daoudi, and eds., 3D object processing: compression, indexing and watermarking, J. Wiley & Sons, 2008. [2] E. Praun, H. Hoppe, and A. Frinkelstein, “Robust mesh watermarking,” in Proc. SIGGGRAPH, 1999, pp. 69–76. [3] O. Benedens and C. Busch, “Towards blind detection of robust watermarks in polygonal models,” in Proc. Eurographics, Computer Graphics Forum 19(3):C199-C208, 2000. [4] R. Ohbuchi, A. Mukaiyama, and S. Takahashi, “A frequencydomain approach to watermarking 3-D shapes,” in Proc. Eurographics, Computer Graphics Forum, vol. 21, 2002, pp. 373–382. [5] O. Benedens, “Geometry-based watermarking of 3-D models,” IEEE Computer Graphics Applications, vol. 19, no. 1, pp. 46–55, 1999.

(a) Bunny Fig. 5.

Robustness against noise.

Fig. 6.

Robustness against smoothing.

Fig. 7.

Robustness against mesh simplification.

(a) Bunny

(a) Bunny

(b) Gear

(c) Dragon

(b) Gear

(c) Dragon

(b) Gear

(c) Dragon

[6] B.-L. Yeo and M. M. Yeung, “Watermarking 3D objects for verification,” IEEE Computer Graphics and Applications, vol. 19, no. 1, pp. 36–45, 1999. [7] C. M. Chou and D. C. Tseng, “A public fragile watermarking scheme for 3D model authentication,” Computer-Aided Design, 2006, vol. 22, no. 9-11, 2006. [8] R. Ohbuchi, S. Takahashi, T. Miyazawa, and A. Mukaiyama, “Watermarking 3D polygonal meshes in the mesh spectral domain,” in Proc. Graphics Interface, 2001, pp. 9–17. [9] Y. Liu, B. Prabhakaran, and X. Guo, “A robust spectral approach for blind watermarking of manifold surfaces,” in Proc. of ACM Multimedia and Security Workshop, 2008, pp. 43–52. [10] M. Luo and A. G. Bors, “Principal component analysis of spectral coefficients for mesh watermarking,” in Proc. IEEE Int. Conf. on Image Processing, San Diego, CA, 2008, pp. 441–444. [11] S. Kanai, H. Date, and T. Kishinami, “Digital watermarking for 3d polygons using multiresolution wavelet decomposotion,” in Proc. of Int. Workshop on Geometric Modeling: fundamentals and applications, Tokyo, Japan, 1998, pp. 296–307. [12] M.S. Kim, S. Valette, H.Y. Jung, and R. Prost, “Watermarking of 3D irregular meshes based on wavelet multiresolution analysis,” in Proc. of Int. Workshop on Digital Watermarking, LNCS vol 3710, 2005, pp. 313–324. [13] K. Wang, G. Lavoue, F. Denis, and A. Baskurt, “Hierarchical Watermarking of Semiregular Meshes Based on Wavelet Trans-

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

form,” IEEE Trans. on Information Forensics and Security, vol. 3, no. 4, pp. 620–634, 2008. F. Cayre and B. Macq, “Data hiding on 3-D triangle meshes,” IEEE Trans. Signal Processing, vol. 51, no. 4, pp. 939–949, 2003. J. W. Cho, R. Prost, and H. Y. Jung, “An oblivious watermarking for 3-D polygonal meshes using distribution of vertex norms,” IEEE Trans. Signal Processing, 2007, vol. 55, no. 1. S. Zafeiriou, A. Tefas, and I. Pitas, “Blind robust watermarking schemes for copyright protection of 3D mesh objects,” IEEE Trans. on Visualization and Computer Graphics, vol. 11, no. 5, pp. 496–607, 2005. M. Garland and P. Heckbert, “Surface simplification using quadric error metrics,” in Proc. SIGGRAPH, Graphical Models 66(6), 1997, pp. 370–397. P. Heckbert and M. Garland, “Optimal triangulation and quadricbased surface simplification,” Journal of Computational Geometry, vol. 14, no. 1-3, pp. 49–65, 1999. S. Sheynin A. Tuzikov and P. V. Vasiliev, “Computation of volume and surface body moments,” Pattern Recognition, vol. 36, no. 11, pp. 2521–2529, 2003. P. Cignoni, C. Rocchini, and R. Scopigno, “Metro: Measuring error on simplified surfaces,” Computer Graphics Forum, vol. 17, no. 2, pp. 167–174, 1998. G.Taubin, “Geometric signal processing on polygonal meshes,”

(a) Bunny Fig. 8.

Robustness against quantization.

Fig. 9.

Robustness against uniform resampling.

(a) Bunny

(b) Gear

(c) Dragon

(b) Gear

(c) Dragon

(a) Distortion when varying α

(a) QSPMean method.

(b) Distortion when increasing the capacity

(b) QSPVar method.

Fig. 10.

Distortion with respect to α and message length.

in Eurographics state of art report, 2000. [22] M. Attene and B. Falcidieno, “Remesh: An interactive environment to edit and repair triangle meshes,” in Proc. of the IEEE International Conference on Shape Modeling and Applications,

Fig. 11.

Robustness to noise when increasing the capacity.

2006, pp. 271–276. [23] D. Zorin and P. Schroder, “Subdivision for modeling and animation,” in Proc. of the ACM Siggraph Course Notes, 2000.