loop subdivision surface based progressive...

Cheng FH (F), Fan FT, Lai SH et al. Loop subdivision surface based progressive interpolation. JOURNAL OF COMPUTER

SCIENCE AND TECHNOLOGY 24(1): 39–46 Jan. 2009

Loop Subdivision Surface Based Progressive Interpolation

Fu-Hua (Frank) Cheng1, Feng-Tao Fan1, Shu-Hua Lai2, Cong-Lin Huang1, Jia-Xi Wang1

and Jun-Hai Yong3 (雍俊海)

1Department of Computer Science, University of Kentucky, Lexington, KY 40506, U.S.A.2Department of Mathematics and Computer Science, Virginia State University, Petersburg, VA 23806, U.S.A.3School of Software, Tsinghua University, Beijing 100084, China

E-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected];

[email protected]

Received June 4, 2008; revised December 11, 2008.

Abstract A new method for constructing interpolating Loop subdivision surfaces is presented. The new method is anextension of the progressive interpolation technique for B-splines. Given a triangular mesh M , the idea is to iterativelyupgrade the vertices of M to generate a new control mesh M such that limit surface of M would interpolate M . It canbe shown that the iterative process is convergent for Loop subdivision surfaces. Hence, the method is well-defined. Thenew method has the advantages of both a local method and a global method, i.e., it can handle meshes of any size and anytopology while generating smooth interpolating subdivision surfaces that faithfully resemble the shape of the given meshes.The meshes considered here can be open or closed.

Keywords geometric modeling, Loop subdivision surface, progressive interpolation

1 Introduction

Subdivision surfaces are becoming popular in manyareas such as animation, geometric modeling and gamesbecause of their capability in representing any shapewith only one surface. A subdivision surface is gen-erated by repeatedly refining a control mesh to get alimit surface. Hence, a subdivision surface is deter-mined by the way the control mesh is refined, i.e., thesubdivision scheme. A subdivision scheme is called aninterpolating scheme if the limit surface interpolatesthe given control mesh. Otherwise, it is called an ap-proximating scheme. Popular subdivision schemes suchas Catmull-Clark scheme[1], Doo-Sabin scheme[2], andLoop scheme[3] are approximating schemes while theButterfly scheme[4], the improved Butterfly scheme[5]

and the Kobbelt scheme[6] are interpolating schemes.An interpolating subdivision scheme generates new

vertices by performing local affine combinations onnearby vertices. This approach is simple and easy toimplement. Because of its local property, it can handlemeshes with a large number of vertices. However, sinceno vertex is ever moved once it is computed, any dis-tortion in the early stage of the subdivision will persist.

This makes interpolating subdivision schemes very sen-sitive to irregularity in the given mesh. In addition, itis difficult for this approach to interpolate normals orderivatives.

On the other hand, even though subdivision surfacesgenerated by approximating subdivision schemes do notinterpolate their control meshes, it is possible to use thisapproach to generate a subdivision surface to interpo-late the vertices of a given mesh. One method, calledglobal optimization, does the work by building a globallinear system with some fairness constraints to avoidundesired undulations[7,8]. The solution to the globallinear system is a control mesh whose limit surface in-terpolates the vertices of the given mesh. Because of itsglobal property, this method generates smooth interpo-lating subdivision surfaces that resemble the shape ofthe given meshes well. But, for the same reason, it isdifficult for this method to handle meshes with a largenumber of vertices.

To avoid the computational cost of solving a largesystem of linear equations, several other methods havebeen proposed. A two-phase subdivision method thatworks for meshes of any size was presented by Zhengand Cai for Catmull-Clark scheme[9]. A method

Regular PaperResearch work presented here is supported by NSF of USA under Grant No. DMI-0422126. The last author is supported by the

National Natural Science Foundation of China under Grant Nos. 60625202, 60533070.

40 J. Comput. Sci. & Technol., Jan. 2009, Vol.24, No.1

proposed by Lai and Cheng[10] for Catmull-Clark sub-division scheme avoids the need of solving a system oflinear equations by utilizing the concept of similarityin the construction process. Litke, Levin and Schroderavoid the need of solving a system of linear equationby quasi-interpolating the given mesh[11]. However, amethod that has the advantages of both a local methodand a global method is not available yet.

In this paper a new method for constructing asmooth Loop subdivision surface that interpolates thevertices of a given triangular mesh is presented. Thenew method is an extension of the progressive interpo-lation technique for B-splines[12−16]. The idea is to iter-atively upgrade the locations of the given mesh verticesuntil a control mesh whose limit surface interpolatesthe given mesh is obtained. It can be proved that theiterative interpolation process is convergent for Loopsubdivision surfaces. Hence, the method is well-definedfor Loop subdivision surfaces. The limit of the iterativeinterpolation process has the form of a global method.But the control points of the limit surface can be com-puted using a local approach. Therefore, the new tech-nique enjoys the advantages of both a local methodand a global method, i.e., it can handle meshes of anysize and any topology while generating smooth interpo-lating subdivision surfaces that faithfully resemble theshape of the given meshes. The meshes considered herecan be open or closed.

The remaining part of the paper is arranged as fol-lows. In Section 2, we present the concept of progressiveinterpolation for Loop subdivision surfaces on closedmeshes. In Section 3, we prove the convergence of thisiterative interpolation process. Extension of this tech-nique to open meshes is considered in Section 4. Im-plementation issues and test results are presented inSection 5. Concluding remarks are given in Section 6.

2 Progressive Interpolation Using LoopSubdivision Surfaces for Closed Meshes

The concept of Loop subdivision surface based pro-gressive interpolation for closed meshes can be de-scribed as follows.

Given a closed 3D triangular mesh M = M0. Tointerpolate the vertices of M0 with a Loop subdivi-sion surface, one needs to find a closed control meshM whose Loop surface passes through all the verticesof M0. Instead of finding the relationship between thevertices of M and the vertices of M0 directly, we usean iterative process to do the job.

First, we consider the Loop surface S0 of M0.For each vertex V 0 of M0, we compute the distance

between this vertex and its limit point V 0∞ on S0,

D0 = V 0 − V 0∞,

and add this distance to V 0 to get a new vertex calledV 1 as follows:

V 1 = V 0 + D0.

The set of all the new vertices is called M1. We thenconsider the Loop surface S1 of M1 and repeat thesame process.

In general, if V k is the new location of V 0 after kiterations of the above process and Mk is the set of allthe new V k’s, then we consider the Loop surface Sk

of Mk. We first compute the distance between V 0 andthe limit point V k

∞ of V k on Sk

Dk = V 0 − V k∞. (1)

We then add this distance to V k to get V k+1 as follows:

V k+1 = V k + Dk. (2)

The set of new vertices is called Mk+1.This process generates a sequence of control meshes

Mk and a sequence of corresponding Loop surfaces Sk.Sk converges to an interpolating surface of M0 if thedistance between Sk and M0 converges to zero. There-fore the key task here is to prove that Dk converges tozero when k tends to infinity. This will be done in thenext section.

Note that for each iteration in the above process, themain cost is the computation of the limit point V k

∞ ofV k on Sk. For a Loop surface, the limit point of acontrol vertex V with valence n can be calculated asfollows:

V ∞ = βnV + (1− βn)Q (3)

where

βn =3

11− 8×(3

8+

(38

+14

cos2π

n

)2) (3)

and

Q =1n

n∑

i=1

Qi.

Qi are adjacent vertices of V . This computation in-volves nearby vertices only. Hence the progressive in-terpolation process is a local method and, consequently,can handle meshes of any size.

Another point that should be pointed out is, eventhough this is an iterative process, one does not haveto repeat each step strictly. By finding out when thedistance between M0 and Sk would be smaller thanthe given tolerance, one can go directly from M0 toMk, skipping the testing steps in between.

Fuhua (Frank) Cheng et al.: Loop Subdivision Surface Based Progressive Interpolation 41

3 Convergence of the IterativeInterpolation Process for Closed Meshes

The proof needs a fact about the eigenvalues of theproduct of positive definite matrices. This fact is pre-sented in the following lemma.

Lemma 1. Eigenvalues of the product of positivedefinite matrices are positive.

Proof. The proof of Lemma 1 follows immediatelyfrom the fact that if P and Q are square matrices ofthe same dimension, then PQ and QP have the sameeigenvalues (see, e.g., [17], p.14).

To prove the convergence of the iterative interpola-tion process for Loop subdivision surfaces, note that atthe (k + 1)-st step, the difference Dk+1 can be writtenas:

Dk+1 =V 0 − V k+1∞

=V 0 − (βnV k+1 + (1− βn)Qk+1)

where Qk+1 is the average of the n adjacent vertices ofV k+1

Qk+1 =1n

n∑

i=1

Qk+1i .

By applying (2) to V k+1 and each Qk+1i ,

Qk+1i = Qk

i + DkQk

i,

we get

Dk+1 =V 0 − (βnV k + (1− βn)Qk)

−(βnDk +

1− βn

n

n∑

i=1

DkQi

)

=Dk −(βnDk +

1− βn

n

n∑

i=1

DkQi

)

where Qk is the average of the n adjacent vertices ofV k. In matrix form, we have

[Dk+11 ,Dk+1

2 , . . . ,Dk+1m ]T =(I −B)

Dk1

Dk2

...Dk

m

=(I −B)k+1

D01

D02

...D0

m

where m is the number of vertices in the given matrix, Iis an identity matrix and B is a matrix of the following

form:

βn1 . . .1− βn1

n1. . .

.... . .

1− βni

ni. . . βni . . .

... · · · βnm

.

The matrix B has the following properties:1) bij > 0, and

∑nj=1 bij = 1 (hence, ‖B‖∞ = 1);

2) there are ni + 1 positive elements in the i-th row,and the positive elements in each row are equal exceptthe element on the diagonal line;

3) if bij = 0 , then bji = 0.Properties 1) and 2) follow immediately from the

formula of Dk+1 in (3). Property 3) is true becauseif a vertex V i is an adjacent vertex to V j then V j

is obviously an adjacent vertex to V i. Due to theseproperties, we can write the matrix B as

B = DS

where D is a diagonal matrix

D =

1− βn1

n10 . . . 0

01− βn2

n2. . . 0

.... . .

01− βnm

nm

and S is a symmetric matrix of the following form:

S =

n1βn1

1− βn1

. . . 1 . . .

.... . .

1 . . .niβni

1− βni

. . .

...nmβnm

1− βnm

.

D is obviously positive definite. We will show thatthe matrix S is also positive definite, a key point in theconvergence proof. ¤

Theorem 1. The matrix S is positive definite.Proof. To prove S is positive definite, we have to

show the quadric form

f(x1, x2, . . . , xm) = XTSX

is positive for any non-zero X = (x1, x2, . . . , xm)T.Note that if vertices V i and V j are the endpoints

of an edge eij in the mesh, then sij = sji = 1 in the


matrix S. Hence, it is easy to see that

f(x1, x2, . . . , xn) =∑eij

2xixj +m∑

i=1

niβni

1− βni

x2i

where eij in the first term ranges through all edges ofthe given mesh. On the other hand, if we use fijr torepresent a face with vertices V i, V j and V r in themesh, then since an edge in a closed triangular mesh isshared by exactly two faces, the following relationshipholds:

∑

fijr

(xi + xj + xr)2 =∑eij

4xixj +m∑

i=1

nix2i

where fijr on the left hand side ranges through all facesof the given mesh. The last term in the above equationfollows from the fact that a vertex with valence n isshared by n faces of the mesh.

Hence, f(x1, x2, . . . , xn) can be expressed as

f(x1, x2, . . . , xn) =∑

fijr

12(xi + xj + xr)2

+m∑

i=1

( niβni

1− βni

− ni

2

)x2

i .

From (3), it is easy to see that nβn

1−βn> 3

5n for n > 3.Hence, f(x1, x2, . . . , xm) is positive for any none zeroX and, consequently, S is positive definite. ¤

Based on the above lemma and theorem, it is easyto conclude that the iterative interpolation process forLoop subdivision is convergent.

Theorem 2. The iterative interpolation process forLoop subdivision surface is convergent.

Proof. The iterative process is convergent if andonly if absolute value of the eigenvalues of the matrixP = I − B are all less than 1, or all eigenvalues λi,1 6 i 6 m, of B are 0 < λi 6 1.

Since ‖B‖∞ = 1, we have λi 6 1. On the otherhand, since B is the product of two positive definite

matrices D and S, following Lemma 1, all its eigen-values must be positive. Hence, the iterative process isconvergent. ¤

4 Extension to Open Meshes

Loop subdivision surface based progressive interpo-lation technique can be used for open meshes as well.Actually the same advantages hold for open meshes too.Before we present Loop subdivision surface based pro-gressive interpolation technique for open meshes, weneed to review subdivision rules for the boundaries ofan open mesh first.

Two kinds of boundary rules have been presentedfor Loop subdivision in the literature[18−21]. In thispaper, we follow the rules presented in [18, 19]. Theserules, together with the Loop subdivision schemes, gen-erate a smooth surface that is C1 continuous at theboundaries[18,19].

For these rules to work, the vertices on the bound-ary are divided into two categories: regular vertices andextraordinary vertices. A boundary vertex is called aregular vertex if its valence is 4, as the one shown inFig.1(a). Otherwise, a boundary vertex is called an ex-traordinary vertex.

For each existing boundary vertex, a new vertex iscomputed as a linear combination of the existing ver-tex and its two neighbors with weights 3/4, 1/8 and1/8, respectively. This vertex formula applies to bothregular vertices and extraordinary vertices.

For each boundary edge, a new edge vertex is gen-erated in two ways. If the endpoints of the edge areboth regular or both extraordinary, then the new ver-tex is just the average of the endpoints. If one of themis regular and the other one is extraordinary, then thenew vertex is a linear combination of the regular vertexand the extraordinary vertex with weights 5/8 and 3/8,respectively, as in Fig.1(c).

Fig.1. (a) Regular vertex. (b)∼(d) Boundary subdivision rules. (e)∼(j) Limit point forumlas. The solid circular points are regular

vertices and rectangle points are extraordinary vertices.


Limit points are computed using two formulas, onefor regular vertices and one for extraordinary vertices,as in Figs. 1(e) and 1(f). These formulas require bothneighbors to be regular vertices. On boundaries of theinitial mesh, one vertex could have either regular or ex-traordinary vertices. There are totally 6 different con-figurations. For each configuration, we can get a newlimit point formula by combining the boundary subdi-vision rules with the standard limit points. These 6limit point formulas are shown in Figs. 1(e)∼1(j).

Note that boundary subdivision involves only theboundary vertices. Therefore interpolation can be per-formed for boundary vertices first. Once we have allthe boundary vertices, interpolation of interior verticesis then performed without making any changes to theboundary vertices. The final mesh will interpolate boththe boundary vertices and the interior vertices exactly.Interpolation of the boundary vertices is done using ourprogressive interpolation technique. The convergenceof the interpolation is guaranteed. A mesh could haveseveral disconnected closed boundaries, such as 6 in thepipe model in Fig.3(a). For each closed boundary, a lin-ear system of equations can be built based on the limitpoint formulas for boundary vertices.

V ∞B = EVB.

Since every row is from one of the 6 limit point for-mulas, then E must be a strictly diagonally dominantmatrix which means

∑nj=1,j 6=i eij < eii. The eigenval-

ues λi of E satisfy |λi| 6 1 for∑n

j=1 eij = 1. Thusthe eigenvalues of E are in (0, 1]. The advantage of ourtechnique is very desirable. No matter how many dis-connected boundaries there are, interpolation is donefor all boundaries at the same time just through thelocal geometric operations. It avoids explicitly solvingseveral linear equations separately.

Interpolation of interior vertices also uses the pro-gressive interpolation technique. Its convergence is alsoprovable. Let V = VB ∪VI be the vertex set of the ini-tial mesh M , where VB and VI are the set of boundaryvertices and interior vertices, respectively. If we addone extra vertex q to M and connects every boundaryvertices with q, we get an closed mesh M ′ with vertexset VC = V ∪ q.

Applying the vertex limit point formula (3) to inte-rior vertices, we get a linear equation:

V ∞I = WV = WIVI + WBVB.

WI is a submatrix of W consisting of |VI | columnsof W corresponding to the interior vertices. WB is asimilar submatrix corresponding to the boundary ver-tices. WI is similar to B for closed meshes. It can be

decomposed into one diagonal matrix DI and a sym-metric matrix SI .

For this new closed mesh M ′, it is okay to applythe progressive interpolation technique developed in theprevious sections. Therefore, the following equationshold.

V ∞C = FVC

That is,

V ∞I

V ∞B

q∞

=

WI 0WB 0wq αq

VI

VB

q

.

It is clear that W is just a submatrix of F . F is de-composed into DC and SC . SC depends only on thetopology of the mesh. M is part of M ′. Therefore, SI

is just a minor of a positive definite matrix SC . Now itis clear that WI satisfies the convergent condition forprogressive interpolation. The examples in Fig.3 showinterpolation results of open meshes.

5 Results

The progressive interpolation process is implementedfor Loop subdivision surfaces on a Windows plat-form using OpenGL as the supporting graphics system.Quite a few cases have been tested. Some of the closedcases (a hog, a rabbit, a tiger, a statue, a boy, a turtleand a bird) are presented in Fig.2. All the data sets arenormalized, so that the bounding box of each data setis a unit cube. For each closed case, the given mesh andthe constructed interpolating Loop surface are shown.The sizes of the data meshes, numbers of iterations per-formed, maximum and average errors of these cases arecollected in Table 1.

Table 1. Loop Surface Based Progressive

Interpolation: Test Results

Model No. of No. of Max Error Ave Error

Vertices Iterations

Hog 606 10 0.000 870 799 0.000 175 255

Rabbit 453 13 0.000 932 857 0.000 111 197

Tiger 956 9 0.000 720 453 0.000 141 480

Statue 711 11 0.000 890 806 0.000 109 163

Boy 17 342 6 0.000 913 795 0.000 095 615

Turtle 445 10 0.000 955 765 0.000 172 600

Bird 1 129 9 0.000 766 811 0.000 088 345

Open mesh examples are shown in Fig.3. The per-formance is about the same as the closed mesh exam-ples. For instance, for the face model (299 vertices)shown in Fig.3(c), it takes 10 iterations to reach an er-ror of 0.000 998 516 for boundary vertex interpolationand also 10 iterations to reach an error of 0.000 896 328


Fig.2. Examples of progressive interpolation using Loop subdivision surfaces (G-mesh ≡ given mesh; I-surface ≡ interpolating Loop

surface). (a) (c) (e) (g) (i) (k) (m) G-mesh. (b) (d) (f) (h) (j) (l) (n) I-surface.


Fig.3. Open mesh examples (I-surface ≡ interpolating Loop sur-

face). Red solid dots are boundary vertices of the given mesh.

(a) (c) Given mesh. (b) (d) I-surface. (e) Different view.

for interior vertex interpolation by the new pogressiveinterpolation technique.

From these results, it is easy to see that the pro-gressive interpolation process is very efficient and canhandle large meshes with ease. This is so because ofthe expotential convergence rate of the iterative pro-cess. Another point that can be made here is, althoughno fairness control factor is added in the progressiveiterative interpolation, the results show that it can pro-duce visually pleasing surface easily.

6 Concluding Remarks

A progressive interpolation technique for Loop sub-division surfaces is presented and its convergence isproved. The limit of the iterative interpolation pro-cess has the form of a global method. Therefore, thenew method enjoys the strength of a global method. Onthe other hand, since control points of the interpolat-ing surface can be computed using a local approach, thenew method also enjoys the strength of a local method.Consequently, we have a subdivision surface based in-terpolation technique that has the advantages of both alocal method and a global method. The new techniqueworks for both open and closed meshes. Our next jobis to investigate progressive interpolation for Catmull-Clark and Doo-Sabin subdivision surfaces.

Acknowledgement Triangular meshes used inthis paper are downloaded from the Princeton ShapeBenchmark[22].

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Fuhua (Frank) Cheng is pro-fessor of computer science and di-rector of the Graphics & Geomet-ric Modeling Lab at the Universityof Kentucky. He holds a Ph.D. de-gree from the Ohio State University,1982. His research interests includecomputer aided geometric modeling,computer graphics, parallel comput-ing in geometric modeling and com-

puter graphics, approximation theory, and collaborativeCAD. He is on the editorial board of Computer Aided De-sign & Applications and Journal of Computer Aided Design& Computer Graphics.

Feng-Tao Fan is currently aPh.D. candidate in the Departmentof Computer Science at the Univer-sity of Kentucky. His research inter-ests include computer graphics, ge-ometric modeling, GPU techniquesand computer vision.

Shu-Hua Lai is currently an as-sistant professor of computer scienceat the Virginia State University. Hereceived his Ph.D. degree in com-puter science from the University ofKentucky in 2006. His research inter-ests include computer graphics andcomputer aided geometric modeling.

Cong-Lin Huang is currentlya Master candidate in the Depart-ment of Computer Science, the Uni-versity of Kentucky. He receiveda B.S. degree in computer sciencefrom Sichuan University, China. Hisresearch interests include computergraphics, solid modeling and com-puter vision.

Jia-Xi Wang is currently work-ing for Avnet, Inc. She received herB.E. degree of computer science fromNorth China Electric Power Univer-sity in 2005, and her M.S. degreeof computer science from Universityof Kentucky in 2008. Her researchinterests include computer graphicsand computer networks.

Jun-Hai Yong is currently aprofessor in School of Software atTsinghua University, China. He re-ceived his B.S. and Ph.D. degreesin computer science from TsinghuaUniversity, China, in 1996 and 2001,respectively. He held a visiting re-searcher position in the Departmentof Computer Science at Hong KongUniversity of Science & Technology

in 2000. He was a post doctoral fellow in the Department ofComputer Science at the University of Kentucky from 2000to 2002. His research interests include computer-aided de-sign, computer graphics, computer animation, and softwareengineering.

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