The Visual Computer manuscript No.(will be inserted by the editor)

Jianhua Wu · Leif Kobbelt

Efficient Spectral Watermarking of Large Mesheswith Orthogonal Basis Functions

Abstract Allowing for copyright protection and own-ership assertion, digital watermarking techniques, whichhave been successfully applied for classical media typeslike audio, images and videos, have recently been adaptedfor the newly emerged multimedia data type of 3D ge-ometry models. In particular, the widely used spread-spectrum methods can be generalized for 3D datasets bytransforming the original model to a frequency domainand perturbing the coefficients of the most dominant ba-sis functions. Previous approaches employing this kindof spectral watermarking are mainly based on multires-olution mesh analysis, wavelet domain transformationor spectral mesh analysis. Though they already exhibitgood resistance to many types of real-world attacks, theyare often far too slow to cope with very large meshes dueto their complicated numerical computations. In this pa-per, we present a novel spectral watermarking scheme us-ing new orthogonal basis functions based on radial basisfunctions. With our proposed fast basis function orthog-onalization, while observing similar persistence with re-spect to various attacks as other related approaches, ourscheme runs faster by two orders of magnitude and thuscan efficiently watermark very large models.

Keywords Digital Watermarking · Large MeshesWatermarking · Radial Basis Functions · SpectralDecomposition

1 Introduction

Watermarking is an established way to provide copy-right protection and ownership assertion in the area ofsteganography. Most availible digital watermarking tech-niques have been focusing mainly on classical media datatypes like audio, images and videos because of the dom-inancy of these data distributed on the Internet [15,8].

Computer Graphics GroupRWTH Aachen University, GermanyE-mail: {wu, kobbelt}@cs.rwth-aachen.de

Fig. 1 The Iphigenie model (left, 1.01M vertices, 2.02Mtriangles) is watermarked in 86 seconds (middle) using 100orthogonal basis functions. Although without perceptible vi-sual differences, these two meshes have a maximum Hausdorffdistance of 1.64% of the major bounding box diagonal length(right, distances with color coding - blue minimum and redmaximum).

Due to their regularly parametrized functional represen-tations, most watermarking schemes are based on spread-

spectrum methods with signal processing, i.e., the me-dia data have to be transformed into a spectral domain,then the coefficients corresponding to the most percep-tually salient basis functions will be modulated with wa-termarks to achieve robustness against possible attacks.

Extending the above spectral watermarking methodsto the newly emerged multimedia data type of 3D ge-

2 J. Wu, L. Kobbelt

ometry models is difficult mainly because of the lackof basic 3D signal processing tools like filtering, regu-lar parametrization and frequency analysis. On the otherhand, with a drastically increasing availability of 3D data-sets and practical geometric applications in recent years,the need for efficient watermarking schemes for 3D mod-els becomes more eminent.

Previous robust 3D spectral watermarking approacheshave been adapted and based mainly on multiresolutionmesh analysis, wavelet domain transformation or spec-tral mesh analysis. Although they already exhibit goodresistance and robustness to many types of real-world at-tacks, they are often far too slow to cope with nowadayslarge meshes due to the involved complicated numericalcomputations.

In this paper, we present a novel imperceptible spec-tral watermarking scheme to support ownership claimson triangle meshes of given 3D shapes (cf. Fig. 1). Tospan the spectral domain for watermarking, it uses anew set of orthogonal basis functions derived from ra-dial basis functions (RBFs), which can lead to optimalconcentration of the shape information to just a few (low-frequency) modes (cf. Fig. 4). Concluding from extensivetests on different models, our watermarking scheme ex-hibits almost the same watermarking quality and robust-ness against various real world attacks as other relatedspectral approaches. On the other side, by utilizing afast basis function orthogonalization algorithm, our wa-termarking scheme runs much faster by two orders ofmagnitude, hence can process and watermark very largemodels more efficiently.

1.1 Related Work

Most previous works to watermark 3D models have beentrying to mimic the common spectral approaches in somealternative ways, though early works on 3D watermark-ing even did not utilize the spectral idea. Watermarkswere embedded into 3D meshes by directly modifyingeither the geometry, the vertex positions, the topology,the vertex connectivity [18,19,12], or the surface nor-mals [3]. Simple enough, these kind of methods usuallycan not provide enough robustness with respect to manydifferent types of ordinary attacks.

Kanai et al. [13] decomposed the target mesh intoa spectral domain by applying the lazy wavelet trans-

form proposed by Lounsbery et al. [16]. Wavelet coef-ficients were then modified to embed watermarks. Ex-tending this work, a blind watermarking algorithm waspresented more recently [31] that can ignore the originalmesh information on the detector side. One constraint tothese methods is that the input mesh is limited to havea prerequisite semi-regular subdivision connectivity.

Multiresolution analysis is another way to constructthe spectral-like domains. While Praun et al. [23] usedstandard mesh simplification [10] to construct multires-olution hierarchies, Yin et al. [33] adopted the scheme

�����������

���������������

� ������������

�������������

� �����������

�����������

���������

���������������

� ���������������

��������

�������� ����������

�������

Fig. 2 A typical watermarking scenario.

in [11] to perform the multiresolution decomposition.Watermark information can be embedded into some spa-tial kernels of the low-frequency component of the shapecorresponding to the low-resolution representation in thegeometry hierarchy.

Recent spectral domain watermarking algorithms em-ployed the spectral mesh analysis first proposed by Karniand Gotsman [14]. Eigenvectors of the Laplace matrix tothe input mesh, the Laplace basis functions, can span anideal spectral space for robust watermarking, i.e., lead-ing coefficients corresponding to smallest eigenvalues canbe modulated with watermarks [20]. Later this idea wasgeneralized to watermark the topology-free point-basedgeometry in [7] where k-nearest neighbors have to beconstructed to compose substitute Laplace matrices.

Despite the variety of available mesh watermarkingalgorithms, to our knowledge, none of them has reportedto be able to watermark meshes with more than 104 ver-tices mainly because of their involved complicated nu-merical computations. We will present a fast yet robust

spectral watermarking algorithm based on the orthogo-nalization of a small set of radial basis functions that canefficiently handle large meshes even with more than 106

vertices (cf. Fig. 1).

Except for our geometry dependant basis functionsderived from RBFs, we note that other bases can havesimilar functionality as ours such as the harmonic func-tions computed with mesh Laplacian which were recentlyused for surface deformation and shape approximation[25,34]. Other than the necessary post orthogonaliza-tion step, they still have to solve sparse linear systemsof Laplace equations, which makes them less efficient asours. In addition, other than in digital watermarking,the fact that altering the low frequency components ofa shape remains nearly invisible to the human eyes hasalso been observed in mesh compression [26].

Spectral Watermarking of Large Meshes 3

2 Overview

Following the most successful spread-spectrum water-marking idea, we also watermark 3D meshes in the spec-tral domain. The whole watermarking scenario which istypically composed of the watermark embedding and thewatermark extraction steps, is illustrated in Figure 2similar to [20,7].

The major difference of our scheme compared to pre-vious work is that we use a new set of geometry de-pendant orthogonal basis functions derived from radialbasis functions (cf. Section 3) to span the spectral spacerather than using Laplace basis functions which emergefrom the Laplace matrix that depends only on the meshconnectivity. We will present a fast algorithm to gen-erate these orthogonal basis functions. Compared withthe time-consuming eigensolvers for Laplace matrices,our method runs faster by two orders of magnitude andthus can efficiently watermark very large meshes as theyare common today. Having the new orthogonal basisfunctions in hand, the remaining watermarking proce-dures are quite similar to other spectral watermarkingapproaches.

The watermark embedding phase (cf. Section 4) firstcomputes a small set of our new orthogonal basis func-tions. Then the geometry of the original mesh is pro-jected to these basis functions spanning the spectral do-main to acquire a set of corresponding spectral coeffi-cients. Watermarks will be encoded into the leading coef-ficients which can be used later to reconstruct the water-marked mesh together with the unmodified coefficients.Finally this watermarked mesh will be publized with li-censes if necessary and possibly receive some attacks.

To assert ownership, the watermark extraction step(cf. Section 5) has to be performed. The possibly at-tacked test mesh is first aligned by a registration routineand resampled by projections according to the originalwatermarked mesh. Then this modified attacked meshwill be transformed to the same spectral domain as inthe embedding phase. Watermarks can be extracted bycomparing the current and original coefficients. The cor-relation between the extracted and original watermarkswill also be found to draw the final ownership assertion.

3 New Orthogonal Basis Functions

The basic idea of spectral watermarking is to representthe geometric information of a mesh with respect to aspecial basis such that most of the information is cap-tured in just a few coefficients. These coefficients are thenused to embed the watermark.

More precisely, let M be a given mesh with vertexpositions p1 . . .pn ∈ IR3. Then we want to find a basisB1 . . .Bn ∈ IRn such that

(p1 . . .pn)T =

n∑

j=1

Bj cTj

-2

0

2

4

6

8

10

1 50 99

Singular ValuesSpectrum Energy

���

�

���

�

������ ���� ������� ������

����� ������� �����

Fig. 3 Singular values {wi} corresponding to the 100 or-thogonal basis functions of the Rocker arm mesh (Fig. 4,middle right). The respective spectrum energy Ei ∈ [0, 1] is

computed as Ei =√

(∑i

j=1w2

j )/(∑

100

j=1w2

j ). Note that most

of the energy is concentrated in the leading part.

with coefficients cj ∈ IR3. The particular basis shouldhave the property that the approximation error

Ek = ‖(p1 . . .pn)T −k

∑

j=1

Bj cTj ‖

decreases as quickly as possible. A very bad example isthe canonical basis Bj = (0 . . . 1 . . . 0)T for which the er-ror Ek decreases only linearly in k. A much better exam-ple is the set of basis functions which emerge as eigenvec-tors from the topological Laplace matrix defined throughthe connectivity of the given mesh. This basis has beenused in [14] where it has been shown empirically that theapproximation error Ek decreases so fast that only a fewhundreds to thousands of basis coefficients are sufficientto encode even fairly complex shapes [2].

Another important property of the basis is orthogo-nality since in this case the coefficients cj can simply becomputed by a dot product

cj = (p1 . . .pn) Bj .

Both, the canonical as well as the Laplace basis are or-thogonal. However, both bases have the important draw-back that their basis function definition completely ig-nores the geometry of the input shape. Although theLaplace basis does depend on the mesh connectivity, thegeometric shape of the input mesh only plays a minorrole under the assumption that the mesh quality (i.e.,the shape of the triangles) establishes a loose correlationbetween geometric and topological distance.

In this paper we are constructing a particular orthog-onal basis which, on the one hand is optimized for theshape of the input geometry and on the other hand con-centrates as much geometric information as possible inas few coefficients as possible. We start by defining apre-basis {Bj} and then augment it later.

We are using radial basis functions (RBFs) [29,6,17,30,21,22,28] to capture the mesh geometry information.Let q1 . . .qk be a set of 3D positions scattered in the

4 J. Wu, L. Kobbelt

Fig. 4 The Rocker arm mesh (left-most, 40K vertices) approximated with 100 orthogonal basis functions based on RBFsusing decimated centers (red dots, middle left) and the random uniform centers (middle right). The right-most image showsthe vertex-to-vertex deviation vectors from the random center RBF approximation to the original model where vectorspointing inside the object are flipped and rendered as green lines. Notice the good approximation quality of only 100 basisfunctions and the small differences between two different center placement strategies as well.

vicinity of the mesh surface. There are many possibilitiesto define these positions, e.g., by selecting the remain-ing points in pi after the mesh decimation with halfedgecollapses, or even randomly selecting a uniform subsetof the vertices pi. (See Figure 4 for a comparison, if nototherwise stated, we will always use random uniform se-lection for efficiency reasons). The qj are used as thecenters of a set of radial basis functions

φj(p) = φ(‖p − qj‖)

where we choose φ(·) to be a monotonically decreas-ing function with compact support, e.g. φ(r) = (1 −r)4+(4r +1) like in [32,22] with r = ‖p−qi‖/σ and σ itssupport size (usually half the length of main boundingbox diagonal to prevent singularity of the later composedmatrix B).

If we evaluate these radial basis functions at all ver-tex positions pi, we obtain our set of discrete pre-basisfunctions

Bj = [φj(p1) . . . φj(pn)]T .

Due to the compact support of φ(·), the discrete pre-basis functions are very likely to be linearly independentin IRn. Moreover if the radial basis functions φj(·) have asufficient overlap then it turns out that a relatively smallnumber k of pre-basis functions are already sufficient torepresent the geometric information of the input meshfairly accurately (cf. Fig. 4), i.e.,

(p1 . . .pn)T ≈

k∑

j=1

Bj cTj .

Moreover, if we choose the distribution of the centers qj

fairly uniform (and we assume that the vertices pi arealso distributed uniformly over the mesh surface) thenit turns out that the coefficients cj ’s magnitudes do notdiffer too much.

Notice that the choice of the actual number k of pre-basis functions has influence only on the computationtime of the subsequent orthogonalization step and noton the applicability of the approach in general.

Remember that our goal is to construct an orthogonalbasis which concentrates most of the geometric informa-tion in just a few coefficients. In order to obtain this basiswe compute a singular value decomposition (SVD [24]) ofthe matrix B ∈ Rn×k which has the pre-basis functionsBj as its columns. With this decomposition we find

(p1 . . .pn)T ≈ B (c1 . . . ck)T

= UWVT (c1 . . . ck)T

= U (c′1 . . . c′k)T .

Hence the orthogonal columns of U form a new set ofdiscrete basis functions. The corresponding coefficientsc′j are obtained by first multiplying the initial coefficients

cj by the orthogonal matrix VT and then multiplyingthe j-th coefficient with the j-th singular value of B.

With increasing overlap, i.e., with increasing radiusof the radial basis function φj(·), we observe a strongerand stronger decay of the singular values of B and hencea more and more pronounced concentration of the geo-metric information to just a few leading coefficients (cf.Fig. 3). In fact, suppressing the later coefficients c′l . . . c

′

k

(which have been multiplied by the smaller singular val-ues) causes only an additional squared approximationerror of

E′

l

2=

k∑

j=l

‖c′j‖2

due to the orthogonality of the columns of U.While the concentration effect observed with our or-

thogonalized basis is similar to the effect observed withthe eigenbasis of the Laplace operator, the advantageof our approach is that its computation is significantlyfaster. In fact, computing the eigenbasis of a very largesparse matrix is a numerically challenging task and usu-ally large meshes are split into smaller patches and pro-cessed separately in order to cope with this problem [14,20].

In our case, however, we do not have to analyze thelarge (n× k) matrix B directly. Instead it is sufficient to

Spectral Watermarking of Large Meshes 5

decompose the much smaller symmetric (k × k) matrixBT B into

BT B = VWUT UWVT = VW2 VT .

With this decomposition we can easily find the orthogo-nalized basis by

U = BVW−1.

Since k is usually much smaller than n, the time spentto compute the SVD is negligible and the computationtime is dominated by the multiplication of B and V.

4 Watermark Embedding

Similar to other spectral watermarking approaches, ourwatermarking scheme embeds the digital watermarks bymodifying the low-frequency components of a given shapein the spectral domain. As we have discussed in Sec-tion 3, the new orthogonal basis functions {Bi} derivedvia SVD will be used to decompose the input mesh into aspectral representation and the coefficients of the leadingpart of the spectrum which are more robust against at-tacks, then can be modulated. The watermarked mesh islater produced with an inverse transform using the samebasis functions and is ready to be distributed.

More specifically, given the original (large) input meshMo with n vertices, a set of k orthonormal basis func-tions {Bi} is first computed to span a spectral domain.Then spectral analysis is performed by projecting themesh geometry (all three x, y, z components) onto eachbasis function Bi to produce 3k mesh spectral coefficients

{α(x)i }, {α

(y)i } and {α

(z)i }, i.e.,

α(d)i =

n∑

j=1

p(d)j Bi,j , i = 1...k, d ∈ {x, y, z}, (1)

where Bi,j denotes the j-th entry of the i-th basis func-tion. Based on these, a spectral mesh representation M′

o

(cf. Fig. 4 middle) can in turn be assembled to approxi-mate the original mesh Mo with an inverse transform:

M′

o =

k∑

i=1

αiBi.

The approximation quality of M′

o is already quite good.But as we only use a small number of basis functions(k � n), we still observe small vertex-to-vertex differ-ences between the approximation M′

o and original Mo

(cf. Fig. 4 right). These differences ∆ will be recordedseparately to help the later reconstruction, i.e.,

∆ = Mo −M′

o = Mo −

k∑

i=1

αiBi.

The watermarks are a binary bit string {bi} withlength 3m and m < k. To embed watermarks into thespectral representation, we first convert {bi} to a sign

string {b′i} with b′i = −1 for bi = 0 and b′i = 1 for bi = 1.Then the first 3m spectral coefficients are modulated as

β(d)i = α

(d)i · (1 + b′3i+d · ρ), i = 1...m, d ∈ {x, y, z},

where ρ is the watermarking amplitude.Finally, these modulated coefficients {βi}, together

with the unchanged ones {αi} and the differences ∆, arecomposed to reconstruct the output watermarked meshMw,

Mw =

m∑

i=1

βiBi +

k∑

i=m+1

αiBi + ∆.

5 Watermark Extraction

The watermarked mesh Mw will be distributed with li-censes if necessary and will possibly receive some types ofreal world attacks. To assert ownership of this attackedtest mesh Mt, previous embedded watermarks have tobe extracted as we will discuss in the following.

To undo a possible similarity transform or translationattacks, the attacked mesh Mt first has to be alignedwith the watermarked mesh Mw. We have used a typi-cal registration toolbox developed in [1] to compute anaffine map for the final alignment. In principle it needsuser intervention to define three matching point pairsto compute initial absolute orientations followed by anautomatic iterative closest point (ICP) algorithm [4] toimprove the initial registration till a local error minimumis reached.

After registration, a resampling phase is usually nec-essary to deal with those attacks that may modify themesh topology like simplification or remeshing. The goalis to map the original topology of Mw to Mt as ourbasis functions {Bi} are defined in the vertex indices or-der. Only when the two meshes have the same vertex or-der, the comparisons between their spectral coefficientsto extract the watermarks will be reasonable. Becausethe registration has already minimized the distances be-tween Mt and Mw, we can use a simple nearest pointsearch strategy to obtain the resampled test mesh M′

t:the topology of M′

t is the same as Mw and each ver-tex v′

i of M′

t is fixed as the nearest point of the cor-responding vertex pi of Mw on the mesh surface Mt.Note that like [23] this resampling step also marks ver-tices as cropped if the nearest distances are larger thana user-defined threshold to account for possible croppingattacks.

Having the registered and resampled test mesh M′

t,we perform a similar spectral analysis to it as Equ. (1) to

get another set of 3m spectral coefficients {γ(x)i }, {γ

(y)i }

and {γ(z)i }. Then for each x, y or z component, water-

marks are extracted as follows,

c(d)i =

1, if γ(d)i > α

(d)i + ε,

0, if γ(d)i < α

(d)i − ε,

N/A, otherwise,

6 J. Wu, L. Kobbelt

0

20

40

60

80

100

0 3 6 9 12

����

�������

����

� ����������

����������

������

��������� � ����

��!� "� �

����#

Fig. 5 Computation times for embedding the watermarksinto various models.

where ε is the detection sensitivity and is set to 0.1ρ.Since various attacks might disturb some of the em-

bedded watermarks, bitwise comparisons have to be per-formed to create the correlation between embedded wa-termarks {bi} and extracted watermarks {c

(d)i }, i.e., the

correlation will be

R =1

3m·

m∑

i=1

z∑

d=x

( c(d)i == b(3i+d) ). (2)

The final ownership assertion can then be made as in [7].If the correlation R is larger than a specified threshold(e.g. 0.75), we affirm that the attacked test mesh Mt

contains the originally embedded watermarks.

6 Results

We have tested our watermarking algorithm on varioustypical 3D mesh datasets. The overall performance in-cluding the timing measurement and memory estima-tion is first to be discussed. The watermarking robust-ness of our scheme will be verified with a lot of diversereal world attacks. A comparison between our orthogo-nal basis functions based on RBFs and the Laplace basisfunctions will also be conducted. All experiments shownhere have been done on a commodity Linux PC with a3.2GHz P-IV CPU and 2GB main memory. Some im-portant parameters are usually set as following (if nototherwise specified): basis function number k = 100, wa-termark length 3m = 24 and watermarking amplitudeρ = 0.01.

6.1 Overall Performance

Computation times of our watermark embedding processare measured and summarized in Figure 5 as a functionof input model sizes. Other recent spectral watermarkingschemes like [20,7] need much longer running times thanours, as they have to solve large eigensystems which willbe impractical for meshes with more than 104 vertices.Even when taking the differences of computing hardware

Fig. 6 The Max model (left, 200K vertices), watermarkedmesh (middle) and the test mesh under both similarity trans-form and simplification attacks (right, 1% of original size).The extracted correlation is R = 1.

Fig. 7 The cropped Horse model (left, 51K vertices) and thecropped Igea model (right, 134K vertices). The correlation isR = 1 in both cases.

into account, we still find that (e.g. for the bunny modelreported in both papers) our algorithm runs more than100 times faster, which are two orders of magnitude.

Actually, as our watermarking method runs so fast,experiments show that the performance bottleneck arisesfrom the memory consumption. For example, our non-optimized watermarking proto-system needs more than1GB Linux process memory to deal with the Iphigeniemesh and the basis functions themselves require about400MB space already. However, this is just the realitythat all spectral-based approaches have to face as basisfunctions always need to be computed for both water-mark embedding and extraction phases.

In addition, as our watermarking scheme only modi-fies the “low-frequency” component of a given shape, thevisual differences between the original mesh and the wa-termarked mesh are almost imperceptible (cf. Fig. 1, 6and 10). And this will make the watermarking schememore robust to malicious attacks.

6.2 Robustness Against Attacks

We test our watermarking method with different meshmodels to verify the robustness of embedded watermarks

Spectral Watermarking of Large Meshes 7

under various types of real world attacks. The water-marks will be generated randomly many times for thesame testing object and the extracted correlation R (cf.Equa. 2) will be the mean value. If the correlation islarger than 0.75, the ownership will be claimed.

Similarity Transforms can be handled by the firstregistration process (cf. Fig. 6). As our registration re-sults are precise enough, the similarity transform attackwill have very few influences on the correlation R;Cropping is dissolved by the resampling process tomark the cropped vertices and neglect them in the ba-sis functions when performing the spectral analysis. Thecorrelation R is always 1 (cf. Fig. 7);Simplification is simulated with a standard QEM-based greedy decimation scheme [9]. The resampling pro-cess will equalize the mesh topology for watermark ex-traction (cf. Fig. 8). Note that even under extreme sim-plification, ownership still can be successfully asserted(cf. Fig. 6 and 8);Additive noise will randomly modify the vertex po-sitions in the normal directions. Figure 9 shows that ourwatermarking scheme is robust against strong noises;Remeshing attack tries to improve the shape qual-ity of mesh primitives where the initial topology will begreatly destroyed (cf. Fig. 10). Thus the resampling pro-cess is also necessary for watermark extraction.Smoothing will apply certain iterations of Laplaceoperators [27] to the watermarked mesh. Figure 11 com-pares the robustness effects of different iteration numbersto the extracted correlations.

The above testing scenarios show that we can cor-rectly assert the ownership of the watermarked mesh.We also test our watermarking scheme with false-positive

attacks, e.g., to perform the same watermark extractionstep to a mesh without actrually having embedded thedesignated watermarks or with other watermarks em-bedded. The extracted correlations are always below 0.5.Regarding the specified threshold, our watermarking al-gorithm will not incorrectly assert that a model is wa-termarked when it is not.

In summary, without perceptible differences betweenthe original and watermarked meshes, our watermarkingscheme is robust against a lot of different real world at-tacks, even the combined attacks as well (cf. Fig. 6), thuscan lead to accurate ownership assertion and copyrightprotection.

6.3 Comparing to Laplace Basis Functions

Figure 11 compares our new orthogonal basis functionsderived from RBFs to the Laplace basis functions (LBFs),the (leading) eigenvectors of the Laplace matrix of theinput mesh [14] adopted by recent spectral watermarkingschemes [20,7]. It is interesting to see that our basis func-tions can capture more shape details than LBFs with the

Fig. 8 The Rocker arm mesh (in Fig. 4) under simplifica-tion attack (left, 2% of original size) and the resampled mesh(right). The correlation is R = 0.96.

Fig. 9 The watermarked Dragon model (left, 438K ver-tices) and attacked by additive noise(right) with maximumdeviation of 1% to main bounding box diagonal length. Thecorrelation is R = 1.

same number of basis functions. This is because ours aregeometry-aware derived functions while LBFs are merelyderived from the mesh topology. On the other hand, it isstill easy to find that our watermarking scheme can havealmost the same robustness against attacks and water-marking quality as previous spectral methods based onLBFs.

7 Conclusions

In this paper, we have presented a novel spectral wa-termarking scheme using a new set of orthogonal ba-sis functions derived from radial basis functions. Whileobserving similar watermarking quality and robustnessagainst attacks as other related approaches, our methodruns much faster by two orders of magnitude thus leadingto efficient watermarking of very large models.

The proposed watermarking algorithm is fast, but itis also very flexible. In fact, we also find that our wa-termarking technique using RBFs can be easily appliedto point-sampled geometry by slightly changing the ex-traction process like adding registration and resamplingtools for point sets. Since RBFs are used, the method canbe extended for watermarking higher dimensional data,e.g. points with colors, meshes with texture coordinates,and even animated meshes.

8 J. Wu, L. Kobbelt

Fig. 10 The Isis mesh (left, 188K vert.), watermarked mesh(middle) and remeshed mesh (right, 8K vert.). The correla-tion is R = 1.

Future work can be extended in many directions: Re-lationships between our new orthogonal basis functionsand LBFs have to be more clearly understood. Also moretypes of attacks like free-form constrained modeling [5]need to be handled. Finally, out-of-core watermarkingimplementations are necessary as massive models areusually more precious and have higher demands on theprotection.

Acknowledgements We would like to thank the anony-mous reviewers for their insightful comments and StephanBischoff for proofreading the paper. The models in the paperare partially courtesy of the computer graphics group at theStanford University and the Cyberware website.

References

1. Alekseyev, S.: Volumetric surface reconstruction withlevel set methods. In: Diploma Thesis. RWTH AachenUniversity, Germany (2003)

2. Ben-Chen, M., Gotsman, C.: On the optimality of spec-tral compression of mesh data. ACM Transactions onGraphics 24(1), 60–80 (2005)

3. Benedens, O.: Geometry-based watermarking of 3d mod-els. IEEE Computer Graphics and Applications 19(1),46–55 (1999)

4. Besl, P., McKay, J.: A method for registration of 3-dshapes. IEEE Trans. on Pattern Recognition and Ma-chine Intelligence 14(2), 239–255 (1992)

5. Botsch, M., Kobbelt, L.: An intuitive framework for real-time freeform modeling. ACM Transactions on Graphics.Special issue for SIGGRAPH conference 23(3), 630–634(2004)

6. Carr, J.C., Beatson, R.K., Cherrie, J.B., Mitchell, T.J.,Fright, W.R., MacCallum, B.C., Evans, T.R.: Recon-struction and representation of 3d ojbects with radial

Fig. 11 The Bunny mesh (35K vertices) approximated infirst row with 100 LBFs (left) and RBFs (right) and water-marked in second row with corresponding basis functions.The two bottom rows show smoothing attacks on RBF wa-termarked mesh with 5, 15, 50 and 100 iterations of Laplaceoperators respectively from top to bottom and left to right.The corresponding extracted correlations are 1, 0.96, 0.83 and0.8 respectively. Under same attacks, the LBF watermarkingcorrelations are 1, 0.96, 0.83 and 0.83.

basis functions. In: Proceedings ACM SIGGRAPH 2001,pp. 67–76 (2001)

7. Cotting, D., Weyrich, T., Pauly, M., Gross, M.: Robustwatermarking of point-sampled geometry. In: Proc. ofThe International Conference on Shape Modeling andApplications 2004, pp. 233–242 (2004)

8. Cox, I.J., Miller, M.L., Bloom, J.A.: Digital Watermark-ing. Morgan Kaufmann (2001)

9. Garland, M., Heckbert, P.S.: Surface simplification us-ing quadric error metrics. In: Proceedings of ACM SIG-

Spectral Watermarking of Large Meshes 9

GRAPH 1997, pp. 209–216 (1997)10. Gotsman, C., Gumhold, S., Kobbelt, L.: Simplification

and compression of 3d-meshes. In: Tutorials on Mul-tiresolution in Geometric Modeling. Springer (2002)

11. Guskov, I., Sweldens, W., Schroder, P.: Multiresolutionsignal processing for meshes. In: Proceedings ACM SIG-GRAPH 1999, pp. 325–334 (1999)

12. Harte, T., Bors, A.: Watermarking 3d models. In: Inter-national Conference on Image Processing, pp. 661–664(2002)

13. Kanai, S., Date, H., Kishinami, T.: Digital watermarkingof 3d polygons using multiresolution wavelet decomposi-tion. In: Proc. Sixth IFIP WG 5.2 GEO-6, pp. 296–307(1998)

14. Karni, Z., Gotsman, C.: Spectral compression of meshgeometry. In: Proceedings ACM SIGGRAPH 2000, pp.279–286 (2000)

15. Katzenbeisser, S., Petitcolas, F.A.P.: Information HidingTechniques for Steganography and Digital Watermark-ing. Artech House Books (2000)

16. Lounsbery, M., DeRose, T.D., Warren, J.: Multiresolu-tion analysis for surfaces of arbitrary topological type.ACM Transactions on Graphics 16(1), 34–73 (1997)

17. Morse, B.S., Yoo, T.S., Chen, D.T., Rheingans, P.,Subramanian, K.R.: Interpolating implicit surfaces fromscattered surface data using compactly supported radialbasis functions. In: Proc. of Shape Modeling and Appli-cations 2001, pp. 89–98 (2001)

18. Ohbuchi, R., Masuda, H., Aono, M.: Watermarkingthree-dimensional polygonal meshes. In: Proc. ACMMultimedia ’97, pp. 261–272 (1997)

19. Ohbuchi, R., Masuda, H., Aono, M.: Watermarkingthree-dimensional polygonal models through geometricand topological modifications. IEEE Journal on SelectedAreas in Communications 16(4), 551–559 (1998)

20. Ohbuchi, R., Mukaiyama, A., Takahashi, S.: A frequency-domain approach to watermarking 3d shapes. ComputerGraphics Forum 21(3), 373–382 (2002). (Proc. Euro-graphics 2002)

21. Ohtake, Y., Belyaev, A., Seidel, H.P.: A multiscale ap-proach to 3d scattered data interpolation with compactlysupported basis functions. In: Proc. of the InternationalConference on Shape Modeling and Applications, pp.153–161 (2003)

22. Ohtake, Y., Belyaev, A., Seidel, H.P.: 3d scattered dataapproximation with adaptive compactly supported radialbasis functions. In: Proc. of the International Conferenceon Shape Modeling and Applications, pp. 31–39 (2004)

23. Praun, E., Hoppe, H., Finkelstein, A.: Robust mesh wa-termarking. In: Proceedings ACM SIGGRAPH 1999, pp.49–56 (1999)

24. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flan-nery, B.P.: Numerical Recipes in C: The Art of ScientificComputing, Second Edition. Cambridge University Press(1995)

25. Sorkine, O., Cohen-Or, D., Irony, D., Toledo, S.:Geometry-aware bases for shape approximation. IEEETransactions on Visualization and Computer Graphics11(2), 171–180 (2005)

26. Sorkine, O., Cohen-Or, D., Toledo, S.: High-pass quanti-zation for mesh encoding. In: Proceedings of the Euro-graphics Symposium on Geometry Processing, pp. 42–51.Eurographics Association (2003)

27. Taubin, G.: A signal processing approach to fair sur-face design. Computer Graphics (SIGGRAPH 1995)29(Annual Conference Series), 351–358 (1995)

28. Tobor, I., Reuter, P., Schlick, C.: Reconstruction of im-plicit surfaces with attributes from large unorganizedpoint sets. In: Proc. of the International Conference onShape Modeling and Applications, pp. 19–30 (2004)

29. Turk, G., O’Brien, J.F.: Variational implicit surfaces. In:Techical Reports GIT-GVU-99-15 (1999)

30. Turk, G., O’Brien, J.F.: Modeling with implicit surfacesthat interpolate. ACM Transactions on Graphics 21(4),855–873 (2002)

31. Uccheddu, F., Corsini, M., Barni, M.: Wavelet-basedblind watermarking of 3d models. In: Proc. Multime-dia and Security Workshop on Multimedia and Security,pp. 143–154 (2004)

32. Wendland, H.: Piecewise polynomial, positive definiteand compactly supported radial functions of minimaldegree. Advances in Computational Mathematics 4(4),389–396 (1995)

33. Yin, K., Pan, Z., Shi, J., Zhang, D.: Robust mesh water-marking based on multiresolution processing. Computerand Graphics 25(3), 409–420 (2001)

34. Zayer, R., Rossl, C., Karni, Z., Seidel, H.P.: Harmonicguidance for surface deformation. In: Proceedings of Eu-rographics 2005. Blackwell (2005). To appear

Leif Kobbelt is a full pro-fessor and the head of theComputer Graphics Group atthe RWTH Aachen University,Germany. His research inter-ests include all areas of Com-puter Graphics and Geome-try Processing with a focuson multiresolution and free-form modeling as well as theefficient handling of polygo-nal mesh data. He was a se-nior researcher at the Max-Planck-Institute for ComputerSciences in Saarbrucken, Ger-many from 1999 to 2000 andreceived his Habilitation degree

from the University of Erlangen, Germany where he workedfrom 1996 to 1999. In 1995/96 he spent a post-doc year at theUniversity of Wisconsin, Madison. He received his Master’s(1992) and Ph.D. (1994) degrees from the University of Karl-sruhe, Germany. Over the last years he has authored manyresearch papers in top journals and conferences and servedon several program committees.

Jianhua Wu graduated in2002 with a master’s degreein computer science from theTsinghua University in Bei-jing, China. He then joined theComputer Graphics Group atthe RWTH Aachen University,Germany, where he is currentlypursuing his Ph.D.’s degree.His research interests mainlyfocus on efficient surface rep-resentations for geometry pro-cessing and efficient geometricdata structures for distributedand mobile multimedia com-munications.