Subdivision as a fundamental building block of digital geometry processing algorithms

Journal of Computational and Applied Mathematics 149 (2002) 207–219www.elsevier.com/locate/cam

Subdivision as a fundamental building block of digitalgeometry processing algorithms

Peter Schr(oder

Department of Computer Science, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125,USA

Received 6 June 2001; received in revised form 19 January 2002

Abstract

Digital Geometry Processing (DGP) is a newly emerging discipline which aims to build the mathematicaland algorithmic foundations for the next generation of digital multimedia. Digital signal processing washighly successful for earlier generations of digital multimedia such as sound, image, and video. Now that 3Dscanning technologies make sampled surfaces broadly available a similar apparatus is needed to deal withthis new multimedia datatype. Due to the inherent curvature of such surfaces though traditional tools are notimmediately applicable. In this paper we give a brief overview of some recent developments in subdivisionsurfaces and how these help to build a foundation for DGP algorithms.c© 2002 Elsevier Science B.V. All rights reserved.

Keywords: Subdivision; Digital signal processing; Approximation theory; Geometric modeling; Multiscale algorithms

1. Introduction

Multimedia data types such as digital sound, images, and video are now ubiquitous in all areasof computing and daily life. This wide impact was made possible by a number of factors. A keyfactor in the wide use of a given data type is the ease and economy of acquiring it. Using arough time line one can observe that this was true for sound in the 1970s, images in the 1980s,and :nally video in the 1990s, roughly following the development of computing hardware withits ever increasing cpu and memory resources (Fig. 1). Another key factor in the wide use of agiven data type is the existence of e<cient algorithms for creation, storage, transmission, editing andother manipulations of the data. The mathematical foundation for these algorithms has for a verylong time rested on sampling and associated Fourier techniques. Even more recent developments,such as the use of wavelets for image and video compression still rest upon the foundation laid by

E-mail address: [email protected] (P. Schr(oder).

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208 P. Schr)oder / Journal of Computational and Applied Mathematics 149 (2002) 207–219

70 80 90 00sound images video geometry

Fig. 1. The development of earlier generations of digital multimedia can be seen as an increase in the dimensionality ofthe data, enabled by simultaneous advances in acquisition, computation and underlying theory. Digital geometry is nowentering a similar phase. ? Peter Schr(oder, 2001.

Fig. 2. Examples of digital geometry from many diBerent application areas. ? Peter Schr(oder, 2001.

Fourier analysis. As such, these methods now codi:ed as “digital signal processing” (DSP) havebeen extraordinarily successful impacting areas ranging from cheap consumer devices such as cellphones and MP3 players to high end scienti:c computing applications solving some of today’s mostdemanding PDEs, for example.

We are now witnessing the arrival of a new multimedia data type: digital geometry. As withthe earlier waves of multimedia this development parallels increasing cpu and memory resourceson modern PCs as well as the availability of increasingly cheap and reliable acquisition devices.The latter includes laser range scanners, 3D photography systems based on stereo matching, contactdigitizers, as well as volumetric imaging techniques such as industrial and medical MRI and CTscanners (Fig. 2). These systems range from low-cost consumer level devices all the way to extremely

P. Schr)oder / Journal of Computational and Applied Mathematics 149 (2002) 207–219 209

high-resolution military and scienti:c systems. Once again, we need corresponding developmentsin algorithms and computational representations to help us realize the potential this data presents.Unfortunately, earlier DSP techniques cannot be immediately brought to bear on this new datatype. Instead, we need to develop a new toolbox of mathematics, computational representations andalgorithms for digital geometry processing (DGP) [37].

2. The need for new tools

Earlier generations of digital multimedia can all be treated as functions of a Euclidian do-main. In the case of sound we have a univariate function of time, for images a bivariate func-tion of a section of the image plane, and video may be thought of as a trivariate function ofthe image plane and time. This is not true anymore for surfaces: They carry intrinsic curvatureand their essential 2-manifold (with boundary) structure cannot be ignored. For example, in theEuclidian setting it is possible to use regular sampling to uniquely represent bandlimited func-tions. There are no comparable uniform, i.e., equispaced, sampling patterns on general 2-manifolds.Since the entire mathematical machinery for digital signal processing is based on translation in-variance, none of the DSP tools developed so far are immediately applicable. A possible solu-tion to this problem is the use of local parameterizations on a given 2-manifold for which onecould then use regular sampling and apply (windowed) Fourier techniques. However, since thereare many possible parameterizations for a given smooth manifold it is not clear which one touse and, in general, the result of the computation will depend on the particular parameterizationchosen. Instead, we seek to :nd sampling patterns for geometry which are as regular as possible(Fig. 3).

As an additional ingredient we are also looking for representations which are hierarchical. Hi-erarchical methods in the classical setting add the idea of scale invariance to the already present

Fig. 3. A surface is typically represented by samples (points) and their connectivity (often triangles). Many samplingtechnologies result in irregular meshes, i.e., those in which each vertex may have any number of neighbors and trianglesizes do not vary smoothly. Much more attractive are smooth, semi-regular samplings such as the one on the right. Thereeach point has six neighbors almost everywhere and sample spacing varies smoothly. In both cases the geometry itself isthe same (leftmost and rightmost). ? Peter Schr(oder, 2001.


Fig. 4. Subdivision surfaces are constructed through a process of repeated re:nement applied to a coarse control mesh.Topologic re:nement splits each face in four while the geometric part of re:nement assigns spatial positions to allnew vertices as well as moves old vertices to new positions. A (piecewise) smooth surface results in the limit. Whilequadrilaterals are used in this case other schemes based on triangles or even hexagons are possible. ? Peter Schr(oder,2001.

translation invariance of the Fourier setting and lead to wavelet approaches. Multiresolution of thistype, be it called wavelets, or multigrid, or :lterbanks, has become a critical ingredient in buildinghighly e<cient, scalable algorithms for many processing and numerical computingtasks.

The task then is to rebuild a signal processing like toolbox of mathematical theory and numer-ical algorithms which replaces regular sampling with semi-regular sampling and enables multires-olution algorithms. The latter requires three ingredients: up/down sampling, smoothing, and detailcomputations. The surface representation, commonly called Subdivision, provides these elementsand can be used to build the foundations of Digital Geometry Processing algorithms and theory(Fig. 4).

3. Subdivision surfaces

Subdivision surfaces were :rst introduced by Catmull and Clark [4] and Doo and Sabin [12]to address the problem of constructing free-form, smooth surfaces of arbitrary topology. Instead of“glueing” together individual tensor product spline patches, they used the fact that a spline patchcan be produced in the limit of repeated uniform knot insertion (“control mesh re:nement”) andgeneralized these re:nement rules from the regular setting to the arbitrary topology setting (Fig. 5).

For quadrilateral schemes vertices with valence k = 4 are regular—corresponding to the regularquadrangulation of the plane—while those with valence k �= 4 are called irregular. In the caseof triangle schemes a regular vertex has k = 6. The Euler–PoincarNe formula for 2-manifolds (withboundary) implies that only a topologic torus, in:nite plane, or in:nite cylinder admit entirely regularcontrol meshes, indicating that the problem of irregular control points is fundamental and cannot becircumvented.

Early subdivision methods were based on generalization of spline knot insertion [23] and theso-called “Oslo algorithm” [9]. Later on, subdivision was studied in its own right [5]. However, onlyrecently have subdivision surfaces received broad attention and development in the computer graphicscommunity (see [46] and the references therein). In particular, a broad theoretical understandingof the analytic properties of subdivision surfaces was missing for a long time. Early attempts atanalysis of the convergence of surface subdivision schemes and the smoothness of the resulting


Fig. 5. Re:nement rules of Catmull and Clark showing the weights (“stencils”) used to compute new control pointsas averages of control points in the coarser mesh. These rules—face, edge, vertex—are indexed based on the topologicassociation of the new vertex with the coarser mesh. On the right the boundary rules including one special rule for edgesadjacent to a boundary vertex. Here, k denotes the number of faces incident on a vertex, � = 3=(2k); � = 1=(4k), and�= 3

8 − 14 cos(�=k) where the latter k is taken to be the valence of the boundary vertex. For more details see [3]. ? Peter

Schr(oder, 2001.

limit functions was based on spectral analysis around irregular vertices [2], but was incomplete.Only very recently did this situation change when an essentially complete theoretical understandingof subdivision surfaces was achieved in the works of Reif [34,36] and Zorin [50,44,43].On the practical side, many recent algorithmic developments have moved applications of subdivi-

sion surfaces rapidly forward. Examples include, interactive multiresolution editing [49], reconstruc-tion of sampled data with subdivision surfaces [18], direct evaluation at arbitrary parameter values[38,45], inclusion of boundary conditions and smoothness constraints [3,24], trimming [26], approx-imation with subdivision surfaces [25], simulation of the mechanics of surfaces [7] for engineeringdesign [8], nonmanifold subdivision [42], and many more. In fact, techniques are mature enough fordeployment in movie production [10] and many geometric modeling packages (e.g., Mirai, Maya,3DMax, etc.).

A prototypical example of a DGP application enabled by subdivision is the compression andprogressive transmission of geometry. Fig. 6 illustrates this idea. A surface is encoded into a bitstreamwhich allows partial reconstructions of the original geometry as soon as bits arrive at the receiver.Initially, the geometry is “blurry” but improves rapidly. Such constructions are well known from theimage setting where wavelets together with zero-tree coders are used for progressive encodings andtransmission. Using subdivision basis functions as scaling functions and building associated wavelets[28] one can build compression algorithms for surfaces [19,6], which are very similar in constructionto the image case.

4. The �avors of subdivision

As mentioned above, subdivision schemes were originally derived from B-spline knot re:ne-ment rules [9] to address the challenge of building smooth surfaces of arbitrary topology [4,12].


Fig. 6. Partial bit-stream reconstructions from a progressive encoding of the Venus head model. File sizes are given inbytes and relative L2 reconstruction error in multiples of 10−4. The rightmost reconstruction is indistinguishable from theoriginal. ? Peter Schr(oder, 2001.

Fig. 7. Examples of subdivision based on triangles and quadrilaterals respectively. On the left, application of Loop’sscheme and on the right, application of the scheme of Catmull and Clark. ? Peter Schr(oder, 2001.

Beginning with an arbitrary connectivity control mesh, which forms a topologic 2-manifold possiblywith boundary, the surface is constructed through a limiting process of repeated re:nement. Fig. 7shows examples for both quadrilateral and triangle based schemes, giving the control mesh and theresulting limit surface.

Subdivision consists of two components, a topologic split rule describing how the connectivity ofthe control mesh is re:ned, and a geometric rule which determines the new control point positionsfrom the old positions. For the latter, most constructions use smoothing :lters of (small) :nitesupport and local de:nition with constant coe<cients depending only on the valence of vertices orfaces in the support of the :lter. These schemes can be grouped according to a number of basic


Table 1Classi:cation of major subdivision schemes

Scheme Approx. Interpol. Quad Triangle Primal Dual

Midedge [17,32] ∗ ∗ ∗Doo–Sabin [12] ∗ ∗ ∗Catmull–Clark [4] ∗ ∗ ∗Loop [27] ∗ ∗ ∗ButterSy [14,48] ∗ ∗ ∗Kobbelt quad.[20] ∗ ∗ ∗Kobbelt

√3 [21] ∗ ∗ ∗

Oswald√3 [30] ∗ ∗ ∗

(a) (b) (c)

(d) (e) (f)

Fig. 8. Topologic styles of subdivision. For each pair the coarser level is shown on the left with the :ner level on theright. (a) Primal quadrilateral quadrisecion. (b) Primal triangle quadrisection. (c) Primal

√3 trisection subdivision. (d)

Dual quadrilateral quadrisection. (e) Dual triangle quadrisection. (f) Dual√3 subdivision. ? Peter Schr(oder, 2001.

criteria (Table 1) indicating whether they are

• approximating or interpolating with respect to the original control mesh point positions;• based on quadrilateral, triangle, or hexagon faces as basic primitive;• of primal or dual type depending on the split rule (faces or vertices, respectively).

Among approximating subdivision schemes, those based on quadrilaterals assume a special rolesince there exist both primal and dual schemes based on quadrilaterals [12,4,17,32,40,41] (in thecase of triangles, dual schemes are based on hexagons). Typically, the topologic step of a primalscheme is described as a face split while dual schemes employ vertex splits. Primal schemes whichquadrisect are shown in Fig. 8(a) for quadrilaterals and Fig. 8(b) for triangles. The topologic split


Fig. 9. A more complex example showing the initial control mesh on the left followed by the results of applying subdivisionof degree two (Doo–Sabin), three (Catmull–Clark), four, and nine. ? Peter Schr(oder, 2001.

step for dual schemes which quadrisect, is illustrated in Fig. 8(d) for quadrilaterals and Fig. 8(e)for dual triangles, i.e., hexagons. The Doo–Sabin scheme [12], for example, is a dual quadrilateralscheme, while half box splines [33]—being dual triangle schemes based on vertex quadrisection—lead to hexagonal tilings.

Given these basic schemes one may ask for interrelationships between them both in terms ofgeometric and topologic rules. In the case of univariate splines, for example, the Lane–Riesenfeldalgorithm [23] establishes an interlaced ladder of increasing order B-splines through repeated aver-aging. Here a single geometric rule, which averages back and forth between primal and dual meshesafter upsampling, generates B-splines of arbitrary order. For quadrilateral schemes this relationshipcontinues to hold in the arbitrary topology setting [47,39] (see Fig. 9).

Even more interesting are schemes in which the topologic subdivision rule calls for joining primaland dual meshes in some fashion. An example are the primal

√3-schemes [15,21,22] in which the

re:ned mesh contains all vertices of the primal and dual mesh (Fig. 8(c)). Dual schemes of thistype are possible as well (Fig. 8(f)). Just as in the case of quadrilateral subdivision, primal and dual√3-schemes can also be built based on repeated averaging [30]. Examples of primal

√3-schemes

are shown in Fig. 10. Dual schemes, i.e., those based on hexagons and trisection are illustrated inFig. 11.

Many more constructions are possible and the area is still quite active. However, in practice,Catmull–Clark and Loop subdivision are the most frequently deployed methods.

5. Open research questions

The last few years have seen rapid development of fairly comprehensive theory and algorithmsfor basic surface subdivision, reaching a certain level of maturity. However, the area remains activeand fertile for ongoing research. In the following paragraphs we lay out a few of the interestingopen questions in the area of subdivision and more generally in DGP.


Fig. 10. Examples of repeated averaging based√3 subdivision with two respectively four averaging cycles for the

mannequin and venus models with the respective control meshes on the left. ? Peter Schr(oder, 2001.

Higher order smoothness at irregular points: While C1 smoothness at irregular points is fairlyeasy to achieve, no practical subdivision schemes which achieve (Sexible) C2 at irregular pointsare currently known. Lower bound estimates [35] suggest that such schemes with small stencils areunlikely. However, it may be possible to achieve higher smoothness through nonstationary schemes,i.e., schemes which change their stencil (number of coe<cients, or coe<cient values with :xedstencil) at each level of subdivision.Uni6ed treatment of subdivision schemes: Repeated averaging which alternates between primal

and dual meshes has recently emerged as a general primitive for the construction of broad classes ofsubdivision. This is true for quadrilateral schemes [39,47] as well as certain triangle based schemes[39,30], although the pattern is not as elegant in the latter case. An open question is whether in thecase of repeated averaging there exist simple su<cient conditions which ensure the smoothness ofthe resulting method at irregular points.Subdivision in higher dimensions: So far subdivision has been mostly employed for curves and

surfaces. Many application domains, especially in physical modeling, require treatment of higherdimensional domains, in particular 3D. Some generalizations of subdivision to 3D [29,1,13] andhigher dimensions [31] have been pursued, but little is as yet known about them. It appears that thenumber of cases to distinguish when dealing with the smoothness analysis of such schemes is muchlarger, considerably complicating the picture in this setting.Shape insensitive and irregular subdivision: An annoying problem in practical applications of

subdivision schemes is the fact that all established schemes are speci:c to a particular element type.


Fig. 11. Dual control mesh and surface after four levels of dual√3 subdivision. ? Peter Schr(oder, 2001.

For example, the Loop scheme for triangle meshes and the Catmull–Clark scheme for quadrilaterals.While the latter can be applied to triangle meshes, the resulting surfaces exhibit subtle artifactssuch as undulations. In practice, it is desirable to work with surfaces which locally have two pre-ferred directions, favoring schemes such as Catmull–Clark. However, when building control meshesfor such surfaces it is generally di<cult to ensure that they contain quadrilaterals only. Finding asubdivision scheme which performs well, i.e., produces “fair” surfaces independent of the controlmesh face types would be highly desirable. A much more general instance of this problem is thedesign of fully irregular subdivision schemes [16,11], i.e., those which admit arbitrary control pointinsertion locations in meshes with irregular connectivity everywhere. Very little is known aboutsuch schemes.


6. Summary

Digital geometry processing is a newly emerging research area which aims to make manipulationof geometry as simple and e<cient as methods currently in place for image processing, for example.Subdivision surfaces and their underlying machinery and algorithms are a key component to makingsuch algorithms a reality.

Here, we aimed to give a short overview of the variety of methods available and provide pointersto the literature for readers interested in more details.

Acknowledgements

This work was supported in part by NSF (DMS-9874082, ACI-9721349, DMS-9872890, ACI-9982273), Toyota, Intel, Alias|Wavefront, Pixar, Microsoft, and the Packard Foundation.

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Subdivision as a fundamental building block of digital geometry processing algorithms