1
Blind Robust 3DMesh Watermarking Based onOblate Spheroidal Harmonics
John M. Konstantinides
1
, Athanasios Mademlis
1
, Petros Daras
2
,
Member, IEEE
,Pericles A. Mitkas
1
,
Senior Member, IEEE
, and Michael G. Strintzis
1
,
2
,
Fellow, IEEE
Abstract
—In this paper, a novel transformbased, blind and robust 3D mesh watermarking scheme is presented. The 3D surfaceof the mesh is ﬁrstly divided into a number of discrete continuousregions, each of which is successively sampled and mappedonto oblate spheroids, using a novel surface parameterizationscheme. The embedding is performed in the spheroidal harmoniccoefﬁcients of the spheroids, using a novel embedding scheme.Changes made to the transform domain are then reversed backto the spatial domain, thus forming the watermarked 3D mesh.The embedding scheme presented herein resembles, in principal,the ones using the multiplicative embedding rule (inherentlyproviding high imperceptibility). The watermark detection isblind and by far more powerful than the various correlatorstypically incorporated by multiplicative schemes. Experimentalresults have shown that the proposed blind watermarking schemeis competitively robust against similarity transformations, connectivity attacks, mesh simpliﬁcation and reﬁnement, unbalancedresampling, smoothing and noise addition, even when juxtaposedto the informed ones.
Index Terms
—3D watermarking, blind detection, copyrightprotection, mesh watermarking, spheroidal harmonics.
I. I
NTRODUCTION
In recent years, as an enormous number of multimediadata becomes publicly available (such as digital photographs,video, audio, etc.), the dire necessity for an effective protectionof the ownership rights of the developers as well as of theowners of multimedia content emerges. 3D virtual models donot constitute an exception, in as much as they are widelyutilized in applications such as gaming, simulation, industrialdesign, advertising, and the like. Watermarking is a concertedapproach to address this need.Although the application of watermarking for copyrightprotection of images, video and audio is already reachingmaturity level, yet the 3D mesh watermarking continues toremain a challenging issue due to particularities so much tothe representation as to the rendering of 3D models. In thisway, initially there are numerous alternative representationstructures for 3D models, such as pointsampled surfaces,polygonal meshes, parametric surfaces (e.g. through BezierSplines, NonUniform Rational BSplines (NURBS), etc.).
This work was supported in part by the VICTORY EU project and by thePENED project (coﬁnanced 75% by the EU and 25% by the Greek Secretariatfor Research and Technology).
1
J. M. Konstantinides, A. Mademlis, P. A. Mitkas and M. G. Strintzisare with the Information Processing Laboratory, Electrical and ComputerEngineering Department, Aristotle University of Thessaloniki, 540 06 Thessaloniki, Greece.
2
P. Daras and M. G. Strintzis are with the Informatics and TelematicsInstitute, P.O. Box 60361, 57001 ThermiThessaloniki, Greece.
Regarding the rendering of 3D models, depending on theapplication area, 3D models are frequently combined withattributes that carry additional information such as texture,color, reﬂective characteristics, etc., yet reducing, in an indirectway, the signiﬁcance of the level of details in the 3D structureof the model (as part of this information can be conveyed bythese attributes).In a typical watermarking scheme, a signal (the watermark) is cast into the srcinal content being protected. Therequirements which must be met by a watermarking schemedepend on the area of application. Typical application areas of watermarking include:
•
Data hiding,
where watermarks are used to carry hiddeninformation within the srcinal content.
•
Authentication and integrity checking,
where watermarksare used to verify the srcinality of the protected contentor even to locate potentially altered parts of the content.
•
Copyright protection,
where watermarks are used to carryinformation about content ownership, in a way that isrobust against malicious modiﬁcations of the protectedcontent (called watermark attacks).Furthermore, if, in a watermarking scheme, the detectionof the presence of a watermark requires the mere knowledgeof a private key, then the scheme is said to have a blinddetection. Should further information be required, such asspeciﬁcities of the watermark embedding procedure or eventhe knowledge of the srcinal content, then this scheme is saidto have an informed detection. Blind detection, obviously, hasadvantages over informed detection, since in the former thereis no necessity of additional information to hold the detection,the knowledge of which constitutes a security issue. For moredetails on the advantages of blind detection, the interestedreader can refer to [1].Given that the present work focuses on copyright protectionapplications, henceforth we will be solely occupied with thisapplication area, when referring to watermarking schemes. Inthe sequel, there will be a concise presentation of the publishedwork on 3D watermarking. A detailed presentation of mostexisting 3D watermarking algorithms and watermark attackscan be found in [2] and [3].II. R
ELATED
W
ORK
Since the ﬁrst work published on 3D watermarking byOhbuchi
et al.
[4], a considerable number of 3D watermarkingschemes have been proposed. From those performing thewatermark embedding in the spatial domain, [5], [6], [7], and
2
[8] use informed detection, while [9], [10], [11], [12], [13],and [14] use blind detection. Likewise, from those performingthe embedding in the transform domain, informed detection isthe one used in [15], [16], [17], [18], and [19], while blinddetection is used in [20], [21], [22], and [23].Informed schemes robust against most known watermark attacks can be found in both spatial and transformbasedalgorithms. For example, in the spatial domain, Benedens[8] embeds a watermark in properly selected feature pointsof the 3D mesh, using Free Form Deformations (FFDs), ina scheme robust against afﬁne transformations, smoothing,cropping, noise addition and resampling. In the transformdomain, Praun
et al.
[18] use wavelet coefﬁcients and radialbasis functions (RBFs) to embed the watermark, resulting ina scheme robust against similarity and connectivity attacks,noise addition, compression, croppingand intense resampling.On the contrary, the existing blind schemes to date fallappreciably short, in terms of robustness, from their counterpart informed ones. Nevertheless, blind detection constitutesan appealing property for any copyright protection system. Inthe spatial domain embedding, Wagner [9] proposes a blindscheme which embeds the watermark in the point normals,estimated by the application of the Laplacian operator inthe neighborhood of any watermarked point. This schemefails under connectivity attacks. Harte
et al.
[10] embed thewatermark via alterations of the position of the mesh points.The watermarked points are chosen based on their distanceto their neighboring points center, while the watermark bitembedded to them depends on their relative position to abounding volume of their neighboring points. Though thescheme is robust against similarity attacks, smoothing andnoise addition, it fails under connectivity attacks.Zafeiriou
et al.
[14] use vertex deformations along the radius
r
of the spherical coordinates
(
r,θ,φ
)
computed on the surfaceof the 3D object, parameterized through continuous NURBSpatches. The proposed scheme is robust against similarityand meshsimpliﬁcation attacks. Spherical coordinates are alsoused in the blind scheme of Cho
et al.
[11], which embeds thewatermark by modifying the mean or variance of the distribution of vetex norms, trough histogram mapping functions. Bothschemes fail under cropping.In a recent publication by RondaoAlface
et al.
[13], a method to withstand cropping attacksin schemes similar with that of [14] and [11] is proposed.They employ geodesics and protrusion functions for the robustfeature points extraction on the surface of the mesh, andpropose a modiﬁcation of the method presented in [11] tobetter withstand resampling attacks.Concluding with the blind schemes in the spatial domain,A. G. Bors [12] uses controlled nonlinear perturbations onproperly selected and ordered vertices of the mesh. Theselection and ordering of the vertices to be watermarked isbased on a visibility criterion, applied to their neighborhood.The algorithm resists similarity attacks, noise addition andcropping.In the transformbased blind schemes, Cayre
et al.
[20]propose a method, which, based on spectral decomposition,embeds watermarks in the middle and high pseudofrequenciesby ﬂipping spectral coefﬁcient triplets. The scheme is robustagainst smoothing and noise addition, but fails under connectivity attacks. An attempt to decrease the vulnerabilityof this scheme to connectivity attacks was performed byRondao Alface
et al.
[21], by using feature points to regainsynchronization.Finally, Uccheddu
et al.
[22] propose a blind waveletbasedwatermarking scheme, applicable to meshes with semiregularsubdivision connectivity. This limitation is vanished in thescheme of Valette
et al.
[23], by employing lazy wavelets.Still, both schemes suffer from fragility against connectivityattacks.Typically, the minimum requirements that any robust 3Dwatermarking scheme must fulﬁll are robustness against similarity transformations and connectivity attacks, as both leavethe geometry of the 3D mesh unaffected. The most robust blindschemes presently available fail under connectivity attacks.The rest of the blind schemes are only capable of handlinga limited number of attacks each.Motivated by the need for a blind 3D watermarking methodrobust against combined attacks, in this paper, a novel approach to blind 3D mesh watermarking, suitable for copyrightprotection applications, is proposed. The mesh is ﬁrstly divided into a number of locally selected continuous regions(patches), each of which is sampled, based on an intensivelysmoothed version of the mesh. The watermark is embedded inthe transform domain, by incorporating spheroidal harmonics,using a novel robust multiplicative embedding scheme. A compatible detector that comes along with the proposed embeddingscheme is capable of robustly performing even under a verylimited number of input coefﬁcients. In this way, the detectioncan be separately held for the high and low pseudofrequencycoefﬁcients respectively.The proposed watermarking scheme presents some similarities with the work of Zafeiriou
et al.
[14], Cho
et al.
[11]and Rondao Alface
et al.
[13]. In this way, the proposedmethod partitions the surface of the mesh into patches of properly selected geodesic length, just as the method presentedin [13] does. Likewise [14] and [11], which use the radius
r
of spherical coordinates to embed the watermark, the proposedmethod uses the spheroidal height
u
of the Jacobi ellipsoidalcoordinates.The key differences between our method and the above threelie in the introduction of a robust sampling technique over asmoothed version of the srcinal mesh, the execution of theembedding in a transformed domain, and the use of a novelembeddingand detection scheme. The new sampling techniquepresents high tolerance against noise and smoothing attacks,thus signiﬁcantly enhancing the overall robustness of thewatermarking scheme against these attacks. The invertibilityof the transform guarantees that all the available data will beused to embed the watermark, while the orthonormality (andhence statistical independence) of the spheroidal coefﬁcientsmakes it highly unlikely for the energy of a certain watermark attack to be properly distributed with equal severity into allwatermarked coefﬁcients. The less affected coefﬁcients canthen be used for a reliable decision extraction from the sideof the detector, in regard to the presence of the watermark under inspection.
3
The resulting scheme presents high embedding capacityand robustness against similarity transformations, connectivityattacks, mesh simpliﬁcation and reﬁnement, unbalanced resampling, smoothing and noise addition, or a combination of these very attacks. The scheme fails under cropping.The rest of the paper is organized as follows: In SectionIII the spheroidal harmonics series expansion is presented.The details of the embedding and detection procedures of theproposed watermarking scheme are described in Section IV,while the experimental results verifying the robustness of thewatermarking algorithm against various watermark attacks aregiven in Section V. Finally, conclusions are drawn in closingSection VI.III. S
PHEROIDAL
H
ARMONICS
The watermark embedding and detection procedures of theproposed scheme are both held in a transform domain, basedon the use of one of the many variants of oblate spheroidalharmonics; namely the Jacobi ellipsoidal coordinates. A detailed presentation of oblate spheroidal harmonics and Jacobiellipsoidal coordinates can be found in [24] and [25]. A brief overview of both follows, for the sake of completeness.
A. Oblate Spheroidal Coordinates
An oblate spheroid created by the revolution of an ellipsearound the zaxis in 3D space is deﬁned by
x
2
+
y
2
α
2
+
z
2
b
2
= 1
, α
2
> b
2
(1)where
α
and
b
refer to the semimajor and semiminor axis of the spheroid respectively.There exists a number of variants of the oblate spheroidalcoordinates. The most commonly used variant, called Jacobiellipsoidal coordinates
{
λ,φ,u
}
, is chosen, by the use of which, a point in
R
3
is uniquely described by the intersectionof families of (Fig.1)
•
Half planes
P
2cos
λ,
sin
λ
:=
x
∈
R
3

y
=
x
tan
λ for λ
∈
[0
,
2
π
)
,λ
=
π
±
π
2
,x
= 0
, y
∈
[0
,
sgn
λ
·∞
)
for λ
=
π
±
π
2
, z
∈
(
−∞
,
+
∞
)
(2)
•
Confocal oblate spheroids
E
2
√
u
2
+
ε
2
,u
:=
x
∈
R
3

x
2
+
y
2
ε
2
+
u
2
+
z
2
u
2
= 1
,u
∈
(0
,
+
∞
)
(3)
•
Confocal half hyperboloids of revolution
H
2
ε
cos
φ,ε
sin
φ
:=
x
∈
R
3

x
2
+
y
2
ε
2
cos
2
φ
−
z
2
ε
2
sin
2
φ
= 1
,φ
∈
(
−
π
2
, π
2)
, φ
= 0
,
sgn
z
= sgn
φ
(4)where
ε
:=
√
α
2
−
b
2
is the absolute eccentricity.By intersecting the aforementioned families of 3D surfaces,the forward transformation of spheroidal coordinates
{
λ,φ,u
}
into Cartesian coordinates
{
x,y,z
}
can be derived:
x
=
u
2
+
ε
2
cos
φ
cos
λ
(5)
y
=
u
2
+
ε
2
cos
φ
sin
λ
(6)
z
=
u
sin
φ
(7)and hence the backward (inverse) transformation of Cartesiancoordinates
{
x,y,z
}
into spheroidal coordinates
{
λ,φ,u
}
becomes
λ
= arctan(
y,x
)
(8)
φ
=
z

z

arcsin
12
ε
2
(
C
−
A
)
(9)
u
=
12 (
A
+
C
)
(10)where
A
=
x
2
+
y
2
+
z
2
−
ε
2
and
C
=
√
4
ε
2
z
2
+
A
2
.
B. Spheroidal Harmonic Expansion
Thong and Grafarend [24] have shown that Jacobi ellipsoidal coordinates decompose the three dimensional Laplacedifferential equation into separable ordinary differential equations. Thus, a function
U
(
λ,φ,u
)
deﬁned in the orthogonalcurvilinear coordinates
{
λ,φ,u
}
, and such that it satisﬁes thethreedimensional Laplacian, can be expressed through
U
(
λ,φ,u
) =
∞
n
=0
m
=
n
m
=
−
n
u
nm
Q
∗
n

m

(
uε
)
Q
∗
n

m

(
bε
)
e
nm
(
λ,φ
)
(11)where
e
nm
(
λ,φ
) =
P
∗
n

m

(sin
φ
)cos(
mλ
)
∀
m
≥
0
P
∗
n

m

(sin
φ
)sin(
−
mλ
)
∀
m <
0
(12)are the surface spheroidal harmonics,
P
∗
nm
(
.
)
and
Q
∗
nm
(
.
)
are the normalized associated Legendre functions of ﬁrst andsecond kind respectively (both orthonormal functions), and
u
nm
are the spheroidal harmonics coefﬁcients of degree
n
and order
m
.The values of
U
(
·
,
·
,
·
)
onto the surface of the referencespheroid are given, as depicted in (11), through
U
(
λ,φ,u
=
b
) =
∞
n
=0
m
=
n
m
=
−
n
u
nm
e
nm
(
λ,φ
)
(13)The weight function
w
(
φ
) :=
α
b
2
+
ε
2
sin
2
φ
12 +
b
2
4
αε
ln
α
+
εα
−
ε
(14)sets the surface harmonic functions orthonormal with respectto their weighted scalar product, namely
e
nm
(
λ,φ
)

e
kl
(
λ,φ
)
w
:= 1
S
E
2
√
u
2+
ε
2
,u
w
(
φ
)
e
nm
(
λ,φ
)
e
kl
(
λ,φ
)
dS
=
δ
kn
δ
lm
(15)
4
(a) (b) (c) (d)
Fig. 1. Visual Representation of Jacobi Ellipsoidal Coordinates. A representative surface for each family of 3D surfaces is shown: (a) the half plane
P
2cos
π
3
,
sin
π
3
, (b) the reference oblate spheroid
E
2
a,b
, (c) the half hyperboloid
H
2
ε
cos
π
4
,ε
sin
π
4
, and (d) the intersection of all three surfaces into a uniquepoint, namely (in Jacobi ellipsoidal coordinates)
x
u
=
{
b,
π
3
,
π
4
}
.
where we employed the standard Kronecker delta
δ
ij
, theglobal area of the reference spheroid
E
2
α,b
S
= 4
πα
·
12 +
b
2
4
αε
ln
α
+
εα
−
ε
(16)and the inﬁnitesimal surface element
dS
=
α
·
b
2
+
ε
2
sin
2
φ
cos
φdλdφ
(17)Using the weighted orthonormality property depicted in(15), the spheroidal harmonic coefﬁcients can be computedas follows:
u
nm
=
∞
k
=0
m
l
=
−
m
u
kl
δ
kn
δ
lm
=
∞
k
=0
m
l
=
−
m
u
kl
Q
∗
k

l

(
bε
)
Q
∗
k

l

(
bε
)
e
nm
(
λ,φ
)

e
kl
(
λ,φ
)
w
=
U
(
λ,φ,u
=
b
)

e
kl
(
λ,φ
)
w
=
α
4
π
2
π
0
dλ
+
π
2
−
π
2
Λ(
λ,φ,u
=
b
)
dφ
(18)where
Λ(
λ,φ,u
) := cos
φ e
nm
(
λ,φ
)
U
(
λ,φ,u
)
(19)Thus, every
u
nm
coefﬁcient can be computed through (18),provided that the values that the
U
(
·
,
·
,
·
)
function takes ontothe surface of the reference spheroid
E
2
α,b
are known. It isobvious that any other valid oblate spheroid (i.e. confocalwith
E
2
α,b
) could be used instead. Clearly, the knowledge of
U
(
·
,
·
,
·
)
onto the surface of any oblate spheroid
E
2
√
u
2
+
ε
2
,u
sufﬁces for the calculation of the spheroidal coefﬁcients
u
nm
of any degree and order. Still, the choice of
E
2
α,b
is advantageous for it simpliﬁes (11) into (13).Let us now consider the special case of a function
U
(
·
,
·
,
·
)
symmetric with respect to
φ
, namely
U
(
λ,φ,u
) =
U
(
λ,
−
φ,u
)
(20)In this case, using the following property of the normalizedassociated Legendre functions of the ﬁrst kind ([26]):
P
∗
nm
(
−
x
) = (
−
1)
(
n
+
m
)
P
∗
nm
(
x
)
(21)(18) simpliﬁes into
u
nm
=
α
2
π
2
π
0
dλ
+
π
2
0
Λ(
λ,φ,b
)
dφ
(
−
1)
(
n
+
m
)
= 10 (
−
1)
(
n
+
m
)
=
−
1
(22)In this paper, all formed
U
(
·
,
·
,
·
)
functions are symmetricwith respect to
φ
. Furthermore, the values that those functionstake outside the surface of the reference spheroid are notof interest. In this context, (22) can be used to go into thetransform domain of spheroidal harmonics, while (13) can beused to return back into the spatial domain.IV. 3D W
ATERMARKING VIA
S
PHEROIDAL
H
ARMONICS
In this section, the process of watermark embedding anddetection in the transform domain is fully analyzed. Forconvenience, the presentation of the proposed watermarkingscheme is broken down into the following discrete steps:1)
Preprocessing:
The purpose of this step is to provide aunique normalization of the targeted 3D object (i.e. rotation, translation and uniform scaling). The robustnessof the normalization process against various watermark attacks is of high signiﬁcance, as it drastically affects theoverall robustness of the entire watermarking scheme.2)
Patch Generation And Surface Sampling:
The normalized mesh is divided into a number of continuous regions(patches), each of which is properly sampled, thusleading to a number of 2D functions (equal in numberwith that of the patches).3)
Transform Domain Analysis:
Each of the 2D functionsproduced in the preceding step is mapped onto thesurface of the reference spheroid
E
2
α,b
and subsequentlyanalyzed into spheroidal coefﬁcients, up to a certaindegree and order.4)
Watermark Embedding:
The watermark is embeddedin the spheroidal coefﬁcients of each patch, using anovel embedding scheme, which, in principle, resemblesthe ones using the multiplicative embedding rule. Thewatermark embedding strengths are different for thecoefﬁcients in low and middle pseudofrequencies thanare for those in high pseudofrequencies. The imperceptibility of the watermark is inherent, due both to thecontinuity of the spheroidal harmonic functions as wellas to the multiplicative embedding rule adopted herein.
5
(a) (b) (c)
Fig. 2. Visual Representation of the preprocessing stage. Column description:(a) the srcinal mesh
O
or
prior to normalization, (b) the smoothed normalizedmesh
O
sn
, and (c) the srcinal normalized mesh
O
on
. Row description:First row corresponds to the model being watermarked, while the second rowcorresponds to the same model attacked by 1.5% uniform noise addition anda similarity transformation. Note that the differences between the smoothednormalized versions of both cases are relatively small.
5)
Watermark Detection:
The input of the detection stepis the 3D mesh under suspicion of copyright violation.The ﬁrst three aforementionedsteps are repeated here forthis mesh, prior to holding the detection. The spheroidalcoefﬁcients of the produced patches are grouped intotwo sets: the set of low and middle pseudofrequenciesand the set of high pseudofrequencies. The detection isheld separately for each set, by forming a weighted sumfrom the elements of each set and by comparing it witha properly chosen threshold. Essentially, the two sets aretreated as independent watermarks.A detailed analysis of each of these steps is presented in thefollowing subsections. Both embedding and detection requirethe knowledge of the owner’s private key. This private key isused to seed (ﬁnd a starting point) a random number generator,in order to produce all needed pseudorandom entities.
A. Preprocessing
One fundamental type of watermark attack, that any robust watermarking scheme should be able to withstand, isa basic afﬁne (or similarity) transform comprised of 3Dmesh translation, uniform scaling, rotation or a combinationof the above. Thus, any scheme that does not embed anddetect the watermark in an afﬁneinvariant manner, includingthe one proposed herein, has to be provided with a uniqueorientation and scaling of the input mesh. This process iscalled normalization. A typical normalization procedure is thefollowing [14]:
•
Translation:
The 3D mesh is translated in such a way thatits center of mass becomes the srcin of the coordinatesystem. A common mistake in this, rather trivial, processis to use the points of the 3D mesh, instead of the area of its faces, in order to compute the object’s center of mass.
•
Uniform Scaling:
The 3D mesh is uniformly scaled so asto ﬁt, in its entirety, within a bounding volume. A typicalbounding volume used is the unit sphere, centered aroundthe mass center of the mesh.
•
Rotation:
The 3D mesh is reoriented by means of Principal Component Analysis (PCA). Again, the computationof the covariance matrix needed for the PCA is performedupon the barycenters of the faces of the 3D mesh, eachweighted by the surface area of its corresponding face.The robustness of the normalization scheme against variousattacks deﬁnes, to a considerable degree, the correspondingrobustness of the overall watermarking scheme against theseattacks. The reason for this is that an incorrect normalizationat the detection stage would desynchronize the entire process,by forcing the detector to search for the watermark in differentregions than the ones actually used at the embedding stage.The normalization method previously described is indeedrobust against basic afﬁne transformations. Its robustness,however, signiﬁcantly decreases if afﬁne transformations arecombined with attacks that offset the center of mass or desynchronize the PCA (e.g. unbalanced resampling, smoothing, noise addition, etc.). In order to improve the robustnessof the normalization procedure against this kind of combinedattacks, the mesh is properly processed prior to holding thenormalization.The preprocessing stage we propose is based on the simpleobservation that an intensively smoothed version of the srcinal mesh adequately converges to the corresponding smoothedversion of the attacked mesh. The latter holds for attacks likenoise addition, smoothing, connectivity attacks, unbalancedresampling or mesh simpliﬁcation, that mainly alter highfrequency attributes of the mesh. Thus, since smoothing tendsto eliminate highfrequency attributes, the smoothed versionsof the former and the latter mesh converge to a rough representation of the bulk of the 3D object under consideration.In this context, let us ﬁrst assume that the polyhedral surfaceof the srcinal 3D object
O
or
is expressed through a setof vertices
V
or
and a list of polygonal faces
L
or
. At thebeginning of the preprocessing stage (Fig.2), the srcinal mesh
O
or
is intensively smoothed. The normalization procedurealready described is subsequently applied to the smoothedmesh, leading to the normalized smoothed version of the mesh
O
sn
. The exact same translation, rotation and scaling that thesmoothed mesh has undergone is ﬁnally applied to the srcinalmesh, resulting in the normalized srcinal mesh
O
on
. Thatends the preprocessing stage.With regard to the mesh smoothing, it can be performed byrepetitive applications of one or more linear smoothing ﬁlters.A fairly general form of such ﬁlters follows [27]:
x
newj
=
x
oldj
+
λ
x
i
∈
N
j
w
ij
(
x
i
−
x
j
)
(23)where
N
j
is the neighborhood of point
x
j
and
w
ij
positiveweights adding up to one. In case of connectivity or remeshing attacks, the rate of convergence of repetitive smoothing iterations towards the desired smoothed mesh
O
sn
isstrongly affected. An effective solution to this problem is tochoose neighborhoods
N
j
of constant area.In really sparse meshes, a mesh reﬁnement by means of mesh subdivision is necessitated (surface lengths are going tobe computed upon it). Classical subdivision schemes are the