Constructing an interpolatory subdivision scheme from Doo-Sabin subdivision

Chongyang DengSchool of Science

Hangzhou Dianzi UniversityHangzhou 310018, China

dcy@hdu.edu.cn

Weiyin MaDepartment of MBE

City University of Hong KongHong Kong, China

mewma@cityu.edu.hk

Abstract

This paper presents an interpolatory subdivision schemederived from the Doo-Sabin subdivision scheme. We firstpresent the relations among three curve subdivision schemes,namely a four point interpolatory subdivision scheme, acubic B-spline curve subdivision scheme, and the Chaikin’salgorithm that generates uniform quadratic B-spline curves.By generalizing these relations to the surface case, we derivean interpolatory surface subdivision scheme from the Doo-Sabin subdivision scheme, a generalization of the Chaikin’salgorithm to surface subdivision. In the new subdivisionscheme, we also introduce a variable tension parameter thatis dependent to local control vertices. The variable tensionparameter can be used to effectively control the resultinglimit surface of the proposed subdivision scheme.

Index Terms

Interpolatory subdivision scheme; Doo-Sabin subdivision;Surface interpolation; Variable tension parameter

1. Introduction

Subdivision surfaces have emerged as a popular modelingtool in computer graphics, computer animation due to theirsimplicity and expressive power. There are two general class-es of subdivision schemes, i.e., approximatory subdivisionin which the limit surface usually does not go through itscontrol vertices and interpolatory subdivision in which thelimit surface exactly interpolates all control vertices.

Popular approximatory subdivision schemes includeCatmull-Clark subdivision[2], Doo-Sabin subdivision[7],Loop subdivision[21], and a

√3 subdivision scheme[13].

Catmull-Clark and Doo-Sabin subdivisions are further gener-alizations of bicubic and biquadratic B-splines, respectively,while the Loop subdivision scheme[21] is based on three-directional box splines. Typical examples of interpolato-ry subdivision schemes include butterfly subdivision forrefining triangular meshes[10] and Kobbelt interpolatorysubdivision for refining quadrilateral meshes[14]. Zorin[32]

also presented an improved version of butterfly subdivisionand Li et al[18] also reported an improved version of Kobbeltinterpolatory subdivision. All the above-mentioned interpo-latory subdivision schemes can be seen as generalizationsof the four point subdivision scheme[8], an interpolatoryscheme for curve design, to surface cases.

The control mesh of approximatory subdivision usuallyshrinks towards the final limit surface with the levels ofrefinement increasing and thus may lead to undesirable”popping effects” when switching between meshes at d-ifferent levels of resolution for graphics applications. Toaddress this issue, Maillot and Stam[23] propose subdivisionrules which blend approximatory spline based schemes withinterpolatory ones. Motivated by Maillot and Stam, Zhangand Wang[28] developed a framework for a class of semi-stationary subdivisions. Such an idea also leads to meth-ods for constructing interpolatory subdivision schemes. Byestablishing geometric rules of the associated interpolatorysubdivision through addition of further weighted averagingoperations to the approximatory subdivision, Li and Ma[17]presented a universal method for constructing interpolatorysubdivision schemes and blending subdivisions from approx-imatory subdivisions. Rossignac and Schaefer[26] studiedthe J-splines and extended J-splines to open curves and asmooth surface subdivision scheme for quadrilateral mesheswith arbitrary connectivity. Recently, Lin et al[20] alsoproposed a method for deducing interpolatory subdivisionschemes from approximatory subdivision schemes.

In this paper, we first explore the relations among threecurve subdivision schemes, namely the four point interpo-latory subdivision scheme, the cubic B-spline curve subdi-vision scheme, and the Chaikin’s algorithm that generatesuniform quadratic B-spline curves. By further generalizingthe relations to the surface case, we derive an interpolatorysubdivision scheme from the Doo-Sabin subdivision scheme.For regular vertices, if we set all shape parameters to 1, theproposed scheme reduces to the interpolatory scheme de-rived by the push-back operations[23]. For irregular verticeswith the selection of a variable shape parameter, however,the two schemes are different.

In general, interpolatory subdivision schemes are sensitiveto the presence of sharp features and may produce low

2011 12th International Conference on Computer-Aided Design and Computer Graphics

978-0-7695-4497-7/11 $26.00 © 2011 IEEE

DOI 10.1109/CAD/Graphics.2011.80

215

quality surfaces for many input meshes unless an initialmesh smoothing step is performed[31]. To interpolate aninitial mesh with more pleasing surfaces, many methods forinterpolating meshes by approximatory subdivision schemesare proposed. Hoppe et al[12] presented a modification ofthe Loop subdivision scheme and force the limit surfacesof Loop subdivision to go through a particular set ofcontrol points. Nasri[25] presented a modification of theDoo-Sabin subdivision and Brunet[1] introduced a set ofshape handles associated to the vertices for shape controlin Nasri’s approach. Halstead et al[11] proposed a schemefor interpolating meshes using Catmull-Clark surfaces withcertain fairness measure. Zheng and Cai[30] proposed a two-phase scheme for interpolating arbitrary topology meshesby Catmull-Clark surfaces. Similar interpolation method canalso be applied using Doo-Sabin subdivisions [29]. Maekawaet al[22] proposed an iterative interpolation technique similarto the one used in [19] for non-uniform B-spline surfaceto subdivision surfaces. Lai and Cheng proposed a methodfor interpolating meshes by Catmull-Clark surface basedon similarity[15]. Deng and Yang also proposed a methodfor constructing Doo-sabin surfaces[5] and Catmull-Clarksurfaces[6] by modifying the geometric rules of their firstsubdivision step for mesh interpolation.

The four point interpolatory subdivision scheme[8] forcurve design has a global tension parameter for adjusting theshape of limit curves. As an appropriate choice of a fixedtension parameter for an entire control polygon is not alwayspossible, a geometrically controlled four point subdivisionscheme[24] and some other geometry driven subdivisionschemes[4], [9], [27] are proposed in curve design. For thesurface case, although there are many interpolatory schemesgeneralized from the four point scheme, there are fewliteratures discussing how to adjust the tension parameterto control the shape of limit surfaces.

This paper is motivated by the concept of a displacementsafe scheme, a special geometrically controlled four pointinterpolatory scheme[24]. In this paper, we use a variabletension parameter for the interpolatory subdivision schemeto make it a displacement safe scheme with substantiallyreduced artifacts. Because the tension parameter has cleargeometric meaning, it is intuitive for its selection.

In next section, we first give insight on the curve caseand derive relations among the Chaikin’s scheme, the cubicB-spline scheme and the four point interpolatory scheme. InSection 3, we further describe the new interpolatory schemederived from the Doo-Sabin scheme. Some examples arepresented in section 4 and the conclusions are draw insection 5.

2. Curve scheme construction

In this section, we highlight the proposed scheme forthe curve case with the selection of the variable tension

parameter for shape control. The construction is based onthe relations among three subdivision schemes, namely theChaikin’s algorithm, B-spline curve subdivision and a fourpoint interpolatory curve subdivision scheme.

2.1. A four point interpolatory subdivision schemefrom Chaikin’s algorithm

Given an initial control polygon Γ0 defined by control ver-tices {P0

0, P01, · · · , P0

i , · · · }, the Chaikin’s algorithm is definedby the following iterative refinements:

(1) For each pair of control vertices Pki , P

ki+1 of Γk, where

k stands for the level of refinements, compute two refinedvertices Pk+1

2i+1, Pk+12i+2 as follows (see Fig. 1):

Pk+12i =

34

Pki +

14

Pki+1, (1)

Pk+12i+1 =

14

Pki +

34

Pki+1. (2)

(2) Connect the refined vertices {Pk+10 , P

k+11 , · · · , Pk+1

i , · · · }to form the control polygon Γk+1 after one step of refinement.

It is well known that by averaging two adjacent refinedvertices of the Chaikin’s scheme, we obtain the geometricrules for cubic B-spline curve subdivision as follows (seeFig. 1):

Pk+12i =

12

(Pk+12i−1 + Pk+1

2i ) =18

Pki−1 +

34

Pki +

18

Pki+1, (3)

Pk+12i+1 =

12

(Pk+12i + Pk+1

2i+1) =12

(Pki + Pk

i+1). (4)

k

iP

1

2

k

iP12 1

k

iP

1

2

k

iQ1

2 1

k

iQ

1

2 1

k

iP

1

2 2

k

iP

1

2 1

k

iP1

2

k

iP1

k

iP

1

2 1

k

iQ

1

2 2

k

iP

Figure 1. Relations among Chaikin’s algorithm, cubic B-spline curve subdivision, and a four point interpolatorysubdivision scheme.

Before introducing the four point scheme, we de-fine moved vertices Qk+1

2i ,Qk+12i+1, which corresponding to

Pk+12i , P

k+12i+1 of the Chaikin’s scheme, as

Qk+12i = Pk

i + 8ω(Pk+12i − Pk+1

2i−1) = Pki + 2ω(Pk

i+1 − Pki−1),

Qk+12i+1 = Pk

i+1 − 8ω(Pk+12i+2 − Pk+1

2i+1) = Pki+1 − 2ω(Pk

i − Pki+2),

where 0 < ω < 18 is a tensor parameter. By further averaging

two adjacent vertices of the vertex array {Qk+1i }i, we obtain

the four point interpolatory subdivision scheme as follows:

216

Qk+12i =

12

(Qk+12i−1 + Qk+1

2i ) = Pki , (5)

Qk+12i+1 =

12

(Qk+12i + Qk+1

2i+1)

=

(12+ ω

)(Pk

i + Pki+1) − ω(Pk

i−1 + Pki+2). (6)

Remark: We select 8ω to be in accord with the symbolsof the original four point interpolatory subdivision scheme.

From Eqs. (3)-(6) we can see that by averaging twoadjacent new vertices of the Chaikin’s scheme, we obtainthe cubic B-spline curve subdivision scheme. By furtheraveraging two adjacent vertices of the moved vertex ar-ray of the Chaikin’s scheme, we produce the four pointinterpolatory subdivision scheme. Doo-Sabin subdivisionand Catmull-Clark subdivision are the surface versions ofChaikin’s scheme and cubic B-spline scheme, respectively.Similar to the curve cases, the geometric rules for regularvertices of Catmull-Clark subdivision can also be seen asbeing produced by averaging the adjacent vertices of a newface of the Doo-Sabin subdivision. Guided by these facts, wederive an interpolatory subdivision scheme from Doo-Sabinsubdivision by averaging the moved vertices of new faces ofthe Doo-Sabin scheme, in the same way as that for producingthe four point scheme from the Chaikin’s scheme. We willgive details of such an interpolatory scheme in Section 3.

2.2. A variable tension parameter for shape control

We first define a displacement-safe scheme for the curvecase similar to that of Marinov et al.[24]:

Definition An interpolatory curve subdivision scheme isdisplacement-safe, if there exist 0 < C < 1

2 , such that dki =∥∥∥∥Pk+1

2i+1 −Pk

i +Pki+1

2

∥∥∥∥ ≤ C∥Pki+1 − Pk

i ∥ for every (i, k).While there is a tensor parameter in the four point

subdivision scheme to control the shape of the limit curve, itis however difficult to select an appropriate tensor parameterfor an entire control polygon[24]. To make the interpolatoryscheme displacement safe, we introduce a local variabletensor parameter that is dependent to local control verticesby defining the moved vertices Qk+1

2i ,Qk+12i+1 as

Qk+12i = Pk

i + 8ωki (Pk+1

2i − Pk+12i−1)

= Pki + 2ωk

i (Pki+1 − Pk

i−1),Qk+1

2i+1 = Pki+1 − 8ωk

i+1(Pk+12i+2 − Pk+1

2i+1)= Pk

i+1 − 2ωki+1(Pk

i − Pki+2),

where ωki , ω

ki+1 are local tension parameters attached to ver-

tices Pki , P

ki+1, respectively. We collectively call the tensions

parameters, ωki for all i, a local variable tension parameter

for the proposed scheme.

To remove artifacts on the limit curve, a simple selectionof the tension parameter ωk

i is

ωki = min

18,∥Pk

i+1 − Pki ∥

8∥Pki+1 − Pk

i−1∥,∥Pk

i−1 − Pki ∥

8∥Pki+1 − Pk

i−1∥

.In this case, we have

∥∥∥∥∥∥Qk+12i+1 −

Pki + Pk

i+1

2

∥∥∥∥∥∥ =∥∥∥∥∥∥Qk+1

2i + Qk+12i+1

2−

Pki + Pk

i+1

2

∥∥∥∥∥∥≤ ωk

i ∥Pki+1 − Pk

i−1∥ + ωki+1∥Pk

i − Pki+2∥ ≤

14∥Pk

i+1 − Pki ∥.

In other words, the interpolatory subdivision scheme isdisplacement safe and the limit curve should be artifact free.By extending this method to the surface case, we propose amethod for controlling the shape of the limit surfaces in thefollowing section.

3. An interpolatory surface subdivision schemeconstruction

3.1. Doo-Sabin subdivision

For the sake of integrity in presentation, we give a simpleintroduction of the Doo-Sabin subdivision scheme.

Given a control mesh defined by properly organized faces,edges, and vertices. Each vertex is a space point and is calleda control vertex. Each edge is a line segment bounded by twovertices. Each face is formed by a loop of edges. Each vertexis shared by a finite number of neighboring faces and eachedge is shared by exactly two faces. Refined meshes aregenerated through repeated subdivisions. Each refinementiteration includes the following main steps:

(1) For each face with m vertices V1, · · · ,Vm, a set of newvertices V ′1, · · · ,V ′m corresponding to the old face vertices arecomputed by

V ′i =m∑

j=1

αi jV j,

where

αi j =

{ m+54m i = j

3+2 cos[2(i− j)π/m]4m i , j

. (7)

(2) For each face, a new face created by connectingV ′1, · · · ,V ′m to replace the old one. Faces constructed in thisway are named as type F faces.

(3) For each edge, a new four-sided face is formed forevery edge of the old control mesh by connecting the imagesof the edge endpoints on each of the faces sharing the edge.Faces constructed in this way are named as type E faces.

(4) For each vertex, a new face is formed for every vertexof the old control mesh by connecting the images of thevertex on each of the faces surrounding the vertex. Facesconstructed in this way are named as type V faces.

217

type F type E type V

Figure 2. Three types of faces in the Doo-Sabin subdivi-sion scheme.

As mentioned above, by further averaging of the refinedvertices of each new face, we get the geometric rules ofCatmull-Clark subdivision (for regular case).

3.2. An interpolatory surface subdivision schemefrom Doo-Sabin subdivision

Similar to the curve case, we first define the movedvertex for each new vertex of the Doo-Sabin scheme. Foran old vertex P with valence m, suppose that faces sharingP are F1, · · · , Fm and the new vertices adjacent to P areQ1, · · · ,Qm (see Fig. 3). The moved vertices correspondingto Qi(i = 1, · · · ,m) are defined as

Qi = Pi + ωP(Qi −G), (8)

whereG =

Q1 + · · · + Qm

m, (9)

and 0 < ωP ≤ 1 is a tension parameter attached to localvertex P.

P

P2

F3F2Q3

Q2P3 GP1

Qm Q1

1Q

Fm F1Pm

Figure 3. Moved vertices of refined vertices of the Doo-Sabin subdivision scheme.

Just as the curve case, if we group all the moved verticesaccording to the topological rule of the Doo-Sabin scheme,and then average all the moved vertices of each face, weproduce an interpolatory surface subdivision scheme. Detailsfor each of the refinement iteration of the new interpolatorysubdivision scheme derived from the Doo-Sabin subdivisionscheme are as follows:

1) For each face F, compute a new face vertex Fk+1 as theaverage of all the moved vertices of the new face vertices(see Fig.4(a)).

2) For each edge E, compute a new edge vertex Ek+1 asthe average of four moved vertices which are adjacent to thetwo end vertices of E and topologically lie in the two faceswhich share E (see Fig.4(b)).

3) Create new edges by connecting each new face vertexto new edge vertices of the corresponding face, and connect-ing each old vertex to new edge vertices of correspondingedges incident to the old vertex (see Fig.4(c)).

4) Create new faces formed by individual loops of newedges (see Fig.4(c)).

Given a vertex P0 with valence N and surrounded withregular vertices, numerate its neighborhood vertices clock-wise as P1, P2, · · · , P2N+1 as shown in Fig. 5. We have thengeometric rules of the new face vertex and new edge vertexfor all ωP = 1(N ≥ 4) as follows:

Fk+1 =79256

P1 +85N − 24

256N(P4 + P6) +

20N − 164N

P5

−7(P2N+3 + P2N+4 + P2N+6 + P2N+7)256

− N + 24256N

(P2 + P8)

−P2N+2 + P2N+5 + P2N+8

256− 6N + 4

256N(P3 + P7)

− (P9 + · · · + P2N+1) + 6(P10 + · · · + P2N)64N

; (10)

Ek+1 =3564

P1 +19N − 6

32NP6 +

N − 264N

(P5 + P7)

+7N − 24

128N(P4 + P8) − 6P2N+4 + P2N+5 + P2N+3

128

−6(P2 + P10 + · · · + P2N)32N

− P3 + P9 + · · · + P2N+1

32N. (11)

When N = 4, the above two rules reduce to those ofthe interpolatory subdivision scheme derived from the push-back operations[23]. When N , 4, i.e., for an isolatedextraordinary corner patch, the resulting rules of the twosubdivision schemes are different.

P1 P2

2Q

1Q

Fk+1

4Q

3Q

P3P4

Ek+1

1Q

2Q

3Q4

Q

P1

P2

(a) (b) (c)

Figure 4. Geometric and topological rules of the newinterpolatory subdivision scheme: (a) new face vertex ⋆;(b) new edge vertex �; and (c) resulting new edges andnew faces.

3.3. A variable tension parameter for surface shapecontrol

Similar to the curve case, we also define a displacement-safe scheme for the surface case as follows:

218

P2N+1

P1

P2 P2N+8P3

P9P4

P2N+7

P8

P7

P6P5

P2N+2P2N+3P2N+4

P2N+5

P2N+6

Fk+1

Ek+1

Figure 5. A stencil with two masks for computing new facevertices and new edge vertices of the new interpolatorysubdivision scheme.

Definition An interpolatory surface subdivision schemeis displacement-safe, if there exist 0 < C < 1, such that forevery edge and face,

dkE =

∥∥∥∥∥Ek+1 − P1 + P2

2

∥∥∥∥∥ ≤ C∥P1 − P2∥,

where P1, P2 are the two end vertices of the edge; and

dkF =

∥∥∥∥∥Fk+1 − P1 + P2 + P3 + P4

4

∥∥∥∥∥ ≤ min{∥Pi − P(i+1)%4∥},

where P1, · · · , P4 are the vertices of the face.To make the interpolatory scheme introduced in Section

3 displacement safe, we recommend to set ωP as

ωP = min{

1,λ

max{m, 4}mini

{min{∥Pi − P∥, ∥P(i+1)%m − P∥}

∥Qi −G∥

}},

(12)where 0 < λ ≤ 1 is also a tension parameter for definingωP and for adjusting the shape of the limit surface, Pi is thevertex that shares edge with P and is a common vertex offaces Fi, F(i+1)%m, and m is the valence of vertex P (see Fig.3).

Under these conditions, we have

ωP∥Qi −G∥ ≤ λ

max{m, 4} min{∥Pi − P∥, ∥P(i+1)%m − P∥}.

This leads to∥∥∥∥∥Ek+1 − P1 + P2

2

∥∥∥∥∥ =

∥∥∥∥∥∥Q1 + Q2 + Q3 + Q4

4− P1 + P2

2

∥∥∥∥∥∥≤ ωP1 (∥Q1 −G1∥ + ∥Q2 −G2∥)

4

+ωP2 (∥Q3 −G2∥ + ∥Q4 −G2∥)

4≤ λ∥P1 − P2∥; (13)∥∥∥∥∥∥∥Fk+1 − 1

4

4∑l=1

Pl

∥∥∥∥∥∥∥ ≤ 14

4∑l=1

ωPl∥Ql −Gl∥

≤ λmini{∥Pi − P(i+1)%4∥}. (14)

From these equations, we can see that the interpolatorysubdivision scheme is displacement-safe, and the limit sur-face should be artifact free. Following Eqs. (12), (13) and

(14), we can see that for a large λ, the distance betweena new edge vertex and the midpoint of the correspondingedge should be large. The same conclusion also holds for anew face vertex. We can thus control the outshoot of the newvertices and avoid large deviation near short edges by settinga small λ, and thus obtaining an interpolatory subdivisionscheme of artifact free.

3.4. Smoothness conditions

We prove that the meshes produced by the new interpo-latory scheme defined in Section 3 converge to continuoussurfaces.

Let Lk = max{∥ei∥} be the maximum length of all edges ofa refined control mesh after the kth subdivision. Following(13), for all ωP = 1 we have

Ek+1 − P1 =916

(P6 − P1) +7(P4 − P1) + 7(P8 − P1)

128

+(P5 − P6) + (P7 − P6)

128+

23[(P1 − P4) + (P1 − P8)]128N

+6(P6 − P2N+4) + (P7 − P2N+5) + (P5 − P2N+3)

128

+24[(P2 − P1) + (P10 − P1) + · · · + (P2N − P1)]

128N

+4[(P3 − P1) + (P9 − P1) + · · · + (P2N+1 − P1)]

128N

+4[(P6 − P5) + (P6 − P7)]

128N+

32128N

(P1 − P6).

Combining with ωP ≤ 1, we have

∥Ek+1 − P1∥ ≤124N − 16

128NLk ≤

3132

Lk.

Similarly, we have also ∥Ek+1 − P2∥ ≤ 3132 Lk.

When ωP = 1, following (10) and (11), we have

Fk+1 − Ek+1 =69(P5 − P6) + 57(P4 − P1) + 15(P1 − P8)

256

+10(P6 − P7) + 8(P1 − P6) + 5(P2N+3 − P2N+4)

256

+7[(P4 − P3) + (P5 − P2N+6) + (P4 − P2N+7)]

256

+(P1 − P2) + (P3 − P2N+8) + (P2N+3 − P2N+2)

256

+4(P4 − P1) + 3[(P2 − P1) + (P8 − P1)]

32N

+(P5 − P4) + (P3 − P4) + (P7 − P6) + 7(P6 − P1)

64N

+(P9 − P1) + · · · + (P2N+1 − P1)

64N+

(P2N+5 − P2N+4)256

+3[(P10 − P1) + · · · + (P2N − P1)]

32N+

(P2N+4 − P2N+3)256

.

Combining with ωP ≤ 1, we have

∥Fk+1 − Ek+1∥ ≤ 6 + 109N128N

Lk ≤111128

Lk.

219

We have thenLk+1 ≤

111128

Lk.

For each edge and for each face, following (13) and (14),we have then∥∥∥∥∥Ek+1 − P1 + P2

2

∥∥∥∥∥ ≤ λLk ≤ λ(

3132

)k

L0;∥∥∥∥∥Ek+1 − P1 + P2 + P3 + P4

4

∥∥∥∥∥ ≤ λLk ≤ λ(

3132

)k

L0.

The above two equations imply that refined meshes of theproposed interpolatory subdivision scheme form a Cauchysequence and this sequence of meshes converges uniformly.We have also done some analysis about the eigen structureof the proposed subdivision scheme. At regular positionswith global ωP = 1, the leading four eigenvalues of thesubdivision matrix are λ1 = 1, λ2 = λ3 = 0.5, and λ4 =

0.25 < λ3. At extraordinary corner vertices with valencen = 5 and global ωP = 1, the leading four eigenvaluesof the subdivision matrix are λ1 = 1, λ2 = λ3 = 0.5497,and λ4 = 0.3603 < λ3. The eigen structures mentionedabove indicate that the limit surface of the resulting inter-polatory scheme meets the necessary condition of tangentplane continuity. Numerical examples of the next sectionalso show that the scheme produces smooth limit surfaces.We conjecture that the scheme produces C1 limit surfaces.However, a rigorous analysis on the continuity conditionsshould be further studied. The tension parameters λ and ωshould also be further optimized to improve the behavior ofthe limit surface at extraordinary positions.

4. Examples and discussions

In this section, we present some examples to demonstratethe effect of the resulting interpolatory subdivision schemediscussed in Section 3. In all these examples, the defaultselection of ωP is 1. In all other cases, we select ωP

according to (12) as a variable tension parameter. For clarity,we indicate the value of the tension parameters ωP or λ inthe captions of the respective figures.

In example 1 and example 2, we test the new interpolatorysubdivision scheme on two simple quadrilateral meshes andtheir limit surfaces are shown in Fig. 6 and 7, respectively.In example 3 and example 4, we test the new interpolatorysubdivision scheme using two simple triangular meshes andtheir limit surfaces are shown in Fig. 8 and 9, respectively.For these examples, there are some artifacts on the limitsurfaces, but if ωP is selected according to (12) as a variabletension parameter, the limit surfaces are then artifact free.For small λ, one may notice that the limit surfaces exhibitsemi-sharp features near very long edges. This effect mightbe used to generate creases on the limit surface.

In examples 5 to 7, we further test the new interpolatorysubdivision on three complex quadrilateral meshes. Theirlimit surfaces are shown in Figs. 10-12. From the illustra-tions of the limit surfaces, we can see that the new interpo-latory subdivision scheme produces smooth limit surfaces.Because the new edge vertex is obtained by blending morevertices than that of other interpolatory subdivision schemes,there are almost no artifacts on the limit surfaces, which istrue even for most tested triangular meshes.

(a) (b)

Figure 6. Interpolation of a box mesh (example 1): (a) o-riginal control mesh; (b) limit surface produced with defaultωP = 1.

(a) (b)

Figure 7. Interpolation of a 3D cross mesh (example 2):(a) original control mesh; (b) limit surface produced withdefault ωP = 1.

5. Conclusions

This paper introduces an interpolatory subdivision schemederived from Doo-Sabin subdivision. Numerical examplesshow that the scheme produces quality smooth limit surfaces.The main advantage of the proposed scheme lies in that itintroduces a variable tension parameter that can be used toeffectively control the shape of the resulting limit surface.As for future work, further investigations can be conductedto formulate a rigorous eigenanalysis of the proposed inter-polatory subdivision scheme. Modeling of crease features,such as crease curves and boundaries, is an important topicfor practical applications. Further investigation can also be

220

(a) (b)

(c) (d)

Figure 8. Interpolation of a simple triangular mesh (ex-ample 3): (a) original control mesh; (b) limit surface withdefault ωP = 1; (c) limit surface with variable ωP andλ = 0.75; and (d) limit surface with variable ωP and λ = 0.5.

conducted on the effect of the variable tension parameterin defining semi-sharp features. Another interesting topic iswhether there is a similar interpolatory subdivision schemethat is most suited for triangular meshes.

Acknowledgment

This work is supported by NSFC (10926058, 61003194,11026107), Research Grants Council of Hong Kong SAR,China (CityU 119208), and City University of Hong Kong(Strategic Research Grant No. 7002509).

References

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(c) (d)

Figure 9. Interpolation of a star shape triangular mesh(example 4): (a) original control mesh; (b) limit surfacewith default ωP = 1; (c) limit surface with variable ωP andλ = 0.75; and (d) limit surface with variable ωP and λ = 0.5.

(a) (b)

Figure 10. Interpolation of a mesh with many holes (ex-ample 5): (a) original control mesh; (b) limit surface withdefault ωP = 1.

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(a) (b)

Figure 11. Interpolation of a pipe mesh (example 6): (a)original control mesh; (b) limit surface with default ωP = 1.

(a) (b)

Figure 12. Interpolation of an axe shape mesh (example7): (a) original control mesh; (b) limit surface with defaultωP = 1.

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