On Geometric Evolution and Cascadic Multigrid in Subdivision

U. Diewald�, S. Morigi

�

, M. Rumpf�

�Bonn University,

�

Bologna University,�

Duisburg University

Abstract

A new approach to subdivision based on the evo-lution of surfaces under curvature motion is pre-sented. Such an evolution can be understood as anatural geometric filter process where time corre-sponds to the filter width. Thus, subdivision can beinterpreted as the application of a geometric filter onan initial surface. The approach closely connectssubdivision to surface fairing concerning the geo-metric smoothing and to cascadic multigrid meth-ods with respect to the actual numerical procedure.The derived method does not distinguish betweendifferent valences of nodes nor between differentmesh refinement types.Keywords: Variational Subdivision, Surface Fair-ing, Curves & Surfaces, Geometric Modeling, Im-age Processing

1 Introduction

Multiresolution mesh representations are a key toolused in computer graphics to achieve real-time in-teraction with large and complex object models. Ina multiresolution modeling environment we are ableto deal with global shape and structural details ofthe same object managing meshes of it at dif-

Figure 1: Starting from a coarse mesh (left) andconsidering a spatially varying filter width we ob-tain a limit surface with locally different smooth-ness modulus.

Figure 2: Increasing a varying filter width one ob-tains a scale of subdivision surfaces ranging fromsmoothed sharp edges up to a smoothing of thecomplete geometry. On the right the correspondingmean curvatures are color coded on the surfaces,especially showing the boundedness of the curva-ture and giving an indication of bounded secondderivatives.

ferent level of refinement. Starting from coarserepresentations of surfaces we can generate appeal-ing smooth representation by iterative applicationsof refinement steps while keeping the connectiv-ity of the original mesh. We propose a refinementapproach that combines the advantages of subdi-vision (arbitrary topology, local control and effi-ciency) with those of variational design (high qual-ity surfaces) [25].

VMV 2001 Stuttgart, Germany, November 21–23, 2001

Subdivision surface modeling is a lively area ofresearch and a promising approach to the efficientdesign of surfaces with complex geometry. The ba-sic idea behind subdivision is to refine and smootha given coarse mesh until a smooth surface is ob-tained. Suppose our surface is represented as a tri-angular mesh � embedded in � � � . Starting withan initial coarse mesh � � , successive meshes aredetermined iteratively by the equation

� � � � � � �

where is the subdivision operator at the � th levelwhich takes the points from level � to points on thefiner level � � � . Assuming that the subdivision con-verges, the actual subdivision surface is defined asthe limit of this sequence of successive refinements:

� � � ! # % & (

* + * � � + / / / + * � � � 2

The subdivision schemes for arbitrary topologycontrol meshes come in two classes: approximat-ing and interpolating. Many variants of the approxi-mating schemes have been considered; the classicalones are due to Catmull-Clark [2], and Doo-Sabin[9], which considered an extension of quadratic andcubic B-splines on rectangular meshes respectively,while a scheme based on quartic box splines on tri-angular meshes was presented by Loop [20]. Anextension of Loop’s scheme, introduced by Hoppeet al. [14] incorporates sharp edges on the final limitsurface.

Dyn, Gregory and Levin [10] introduced the but-terfly scheme, a simple interpolating subdivision al-gorithm applicable to arbitrary triangular meshes.Since it only leads to 4 � surfaces in the regular set-ting (all vertices of the mesh have valence 6), animproved butterfly scheme, the so-called modifiedbutterfly, resulting in smoother surfaces, has beenintroduced in [28].

Most known stationary subdivision schemes gen-erate at least 4 � -continuity surfaces on arbitrarymeshes in the regular setting [22, 26]. Recently, thesmoothness of the subdivision surfaces in irregularsetting (that is near extraordinary vertices) has beenrigorously proved in [22, 27].

In [16, 19] an approach to mesh refinementbased on variational methods has been proposedin order to define univariate variational subdivisionschemes. Kobbelt [18] considered energies whichinvolve curvature quantities. The corresponding

evolution problems would lead to Willmore flowand surface diffusion respectively, which are fourthorder parabolic problems. Whereas we here restrictto second order mean curvature flow.

In this paper the smoothing step is based on ageometric diffusion and filtering approach relatedto mean curvature motion, which has already beenproved to be very promising for surface fairing pur-poses [6, 7]. Actually, we consider a single fullyimplicit timestep of mean curvature motion as oursmoothing method. In an iteration we successivelyrefine the surface mesh - which turns this approachinto a subdivision scheme - and solve a semiimplicitproblem to approximate the fully implicit step. Thissemiimplicit scheme is explicit with respect to thegiven metric from the last step and implicit con-cerning the new positions of the nodes. Thus, ourmethod can be regarded as a fixed point iteration,where we simultaneously expect to improve themetric and the resolution. This leads to a suitablegeometric smoothing filter on the initial mesh. Ourapproach is a usual subdivision scheme, but nowfounded on tools from the theory of geometric evo-lution problems (mean curvature motion) and nu-merical analysis (cascadic multigrid). We try to out-role this new perspective which we believe to offerstrong provisions concerning the theoretical analy-sis of subdivision schemes as well as the range ofapplications. Furthermore, as already mentionedthis approach bridges the gap between subdivisionand surface fairing on a rigorous basis. Not verysurprisingly, several important question within thisnew perspective remain open and require further in-vestigation. Concerning the regularity and conver-gence we only state conjectures here.

The resulting method is closely related to stan-dard subdivision schemes concerning the computa-tional complexity. Our approach however has thefollowing advantages:

- Many qualitative properties of the approachcan be studied already on the continuous leveland do not require a detailed analysis of thediscretization.

- The model is independent of the type ofmeshes, especially of the valences of the meshnodes, and the considered refinement rules.These characteristics are naturally incorpo-rated in the finite element matrices and donot influence the method’s performance signif-icantly (cf. Fig. 5).

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- The resulting scheme can be easily adapted todifferent applications, e. g. to spatially varia-tion of the corresponding filter width (cf. Fig.1) or to solely smoothing the edges of coarsepolygonal models (cf. Fig. 2).

Furthermore, we expect the resulting limit surfacesto be 4 5 7 8 -smooth for every 9 : < > ? � � .

2 Curvature motion as a naturalsmoothing process

In this section we will outline evolutionary smooth-ing methods and their application in surface fairingto motivate our subdivision scheme (cf. Section 6)which generates new surfaces instead of improvingthe quality of given surfaces reviewed in this sec-tion. Simultaneously we will introduce the basicnotation of geometry and geometric differential op-erators. For details we refer to [8] and [3, Chap-ter 1]. Let us consider a smooth compact embed-ded manifold � A � � � without boundary. LetC � D E � H J LE C � J � be some coordinate mapfrom an atlas. For each point C on � the tangentspace P Q � is spanned by the basis R S Q

S T U ? S QS T X

Z.

By P � we denote the tangent bundle. Measur-ing length on � requires the definition of a met-ric [ � / ? / � � P Q � _ P Q � E � � . As the corre-sponding matrix notation we obtain [ � � [ e f � e fwith [ e f � S Q

S T i / S QS T j , where / indicates the scalar

product in � � � . The inverse of [ is denoted by[ � � � � [ e f � e f . The gradient l m o of a function

o is defined as the representation of q o with respectto the metric [ . In coordinates we obtain

l m o � � se 7 f

[ e f u � o + C �u J f

uu J e 2

We define the divergence div m w of a vector fieldw : P � as the dual operator of the gradient withrespect to the y 5 -product on � and obtain in coor-dinates

div m w � � se

uu J e � � w e + C � { | } � [ � �� | } � [ 2

Finally, the Laplace Beltrami operator � m is givenby � m � � � div m l m � . With this operator athand we can define geometric diffusion in anal-ogy to the linear diffusion problem in the Euclideanspace. Furthermore, we can consider a diffusion of

the manifold geometry itself. I. e., we seek a one pa-rameter family of embedded manifolds R � � � � Z � �

�and corresponding parametrizations C � � � , such that

u � C � � � � � m � � � C � � � � > ?� � > � � � � 2

Already in ’91 Dziuk [11] presented a semi implicitfinite element scheme for geometric diffusion basedon this formulation. The fundamental observation isthat this geometric diffusion of the coordinate map-ping itself coincides with the motion by mean cur-vature ( � 4 � ) [15]; in fact for any manifold �we have � m C � � � � C � � � C � , and thus we ob-tain u � C � � � � C � � � C � , where � � C � is the corre-sponding mean curvature (here defined as the sumof the two principal curvatures), and � � C � is thenormal on the surface at each point C . For the sakeof simplicity we define � 4 � � � � � � � � � � � � ,where � � � � is the solution surface for time � . Thus

� 4 � � � 5 � � � � can be regarded as the applica-tion of a “geometric ” Gaussian filter of width � to

� . The mean curvature motion model is known asthe gradient flow with respect to surface area. Thisis one indication for the strong regularizing effectof � 4 � .

Previous work on surface fairing has already in-volved the concept of curvature motion. Taubin[23] and Kobbelt [17] considered an umbrella op-erator, which is a “spring force type” implementa-tion of the Laplace Beltrami operator. Desbrun et al.[6] discussed an implicit discretization of geometricdiffusion closely related to Dziuk’s approach to ob-tain strongly stable numerical smoothing schemes.Here we will apply Dziuks method [11] as the ba-sic discrete smoothing scheme. Hence, in the MCMcase in each step of a mean curvature evolution oneasks for a family of triangular surfaces R � Z

�

�and corresponding parametrizations � : � � � � ,where � denotes the affine finite element space onthe grid � , such that

� � � � �� � � m   ¡ U � � > ?where the discrete Laplace Beltrami op-erator is defined by � � m   £ ? ¤ � ¦m   � �

� � l m   £ ? l m   ¤ � § m   for all ¤ : � usingthe scalar product � w ? ª � § m   � � « m   [ � w ? ª � d Con P � and the lumped mass y 5 product

� / ? / � ¦m   , which is defined by � £ ? � � ¦m   � �« m   ® ¦   � £ � � d C for two discrete functions

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£ ? ¯ : � . Here ® ¦   � 4 � � � � E � denotesthe nodal projection operator (cf. [24]).

3 The concept of cascadic iterations

Multigrid methods [13] are known to be efficientsolvers for systems of linear equations ° ±� � ´characterized by an intrinsic hierarchical structurefor example resulting from a finite element dis-cretization. Throughout this paper we alway in-dicate vectors, especially the vector collecting allnodal coordinates by a bar on top. Typically an un-derlying grid hierarchy induces a hierarchical struc-ture on the corresponding discrete function spaces.Let us consider a sequence of finite element spacesR � f Z

, with � � A � � A � 5 A / / / A � f A / / / A� f max corresponding to a hierarchy of nested grids� � ? / / / ? � f max . In the solution process one typ-ically iterates over ¸ , solving an appropriately re-stricted system ° f ±� f � ´ f on level ¸ and thenrefines the grid. Here ° f and ´ f are restrictions of° � ° f max and ´ � ´ f max respectively.

Thus, the previously calculated solution ±� f � �can be considered as good initial data for an iter-ative solver on level ¸ . Somewhat surprising thisnaive strategy turns out to be theoretically wellfounded and robust [1] as long as an error control inthe energy norm is considered. Indeed, the numberof required iterations or smoothing steps ¼ f on level¸ can be fixed a priori. On coarse levels more iter-ations are required than on finer levels, where veryfew iterations are sufficient to ensure a required ap-proximation quality on the finest grid level. Evenbetter, Bornemann and Deuflhard [1] proved opti-mality for this cascadic scheme in the sense thatthe overall cost of the solution process is ½ � ¾ �where ¾ is the number of unknowns on the finestgrid level. Hence under these circumstances onecan save more complex nested iterations in a gen-eral multigrid solver.

4 The function graph case

At first, let us consider the case of surfaces � ,which are graphs in the C 5 direction over a polygo-nal domain D in the C � ? C � plane. Here we will out-line a very simple but effective subdivision schemeas a first model case. We denote by � the corre-sponding graph function. Given an initial graph � �- which will later be our coarse polygonal mesh -

we can ask for a solution � �of the elliptic partial

differential equation

� � � ¿ � � � � � � �with natural boundary conditions on u D . This prob-lem corresponds to a single approximate timestepfor the heat equation with timestep size � � ¿or the approximation of a Gaussian filter of width� � � , respectively. For Lipschitz continuous initialdata � � it is known [12] that the solution is uniqueand 4 5 7 8 regular for any 9 : < > ? � � . Now we ap-proximate � �

by a sequence of linear finite elementsolutions R £ Z

 � 7 à à à on successively refined reg-ular grids D for the parameter domain D . Due tothe convergence properties of linear finite elements[4] we know that for � E Ä and correspondingvanishing grid size Æ on D the sequence £ con-verges to � �

in the energy as well as in the y (norm. In each step of this scheme we have to solvea linear system of equations of the type

� � � ¿ y � ±£ � � È ¦   � �where � , y are the mass and stiffness matrix re-spectively, and È ¦   � � is the linear interpolation of� � on D . The solution £ minimizes the energy

É � £ � � � Ê � £ � È ¦   � � � 5 � ¿ Ë l £ Ë 5over all admissible functions £ in the � th finite el-ement space. Hence, if � � is the function corre-sponding to a polygonal surface graph over D , wecan interprete this approach as a variational sub-division approach. As usual the linear systemsof equations are solved applying iterative solvers.If we consider a cascadic multigrid method, wecan reduce the required number of iterations enor-mously without effecting the convergence proper-ties (cf. Section 3). Given a final level of resolution� Ì Í Î we a priori fix the required number of smooth-ing steps within the cascadic algorithm on all levelsfollowing the recipe given by Bornemann and Deu-flhard [1]. Finally we end up with an subdivisionscheme of optimal complexity, i. e. a cost propor-tional to the number of vertices on the finest gridlevel. Still we now that the resulting sequence ofsolutions R £ Ï Ð Ñ

Zconvergences to � �

. Thus, oursimple iteration leads to 4 5 7 8 regularity in the limit,independent of the grid type and the applied refine-ment scheme.

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Unfortunatly, most surfaces in computational ge-ometry are not graphs, except form a local perspec-tive (cf. Mandal et al. [21] for a finite element ap-proach to subdivision which deals with patches ofthe subdivision surfaces as graphs over the initial,coarse surface polygons). Furthermore, the selec-tion of a parameter domain introduces the metricon that domain as the valid metric. Thus subdivi-sion results significantly depend on this metric andthereby on the proper choice of the parameter do-main. It would be much more natural to apply thesame concept based on a single diffusion time stepbut taking into account the metric on the limit sur-face or suitable approximations, respectively. Thiswill be what we are going to investigate in the fol-lowing section.

5 Subdivision via cascadic filtering

Figure 3: Comparison of different subdivision re-sults. From left to right the images correspond toexact solution of the linear systems in each sub-division step, cascadic cg-iterations, and cascadicJacobi-iterations. Visually there is more or less nodifference except closed to the boundary in the neckregion. This clearly reflects the convergence prop-erties of the cascadic scheme.

As initial surface we consider any discrete, typ-ically triangulated surface � � and denote its pa-rameterization by C � . To underline the geometricorigin and to straighten the presentation we deriveour final method in several steps:

Step 1. As it has already been mentioned meancurvature motion is the geometric counterpart ofGaussian filtering and solving the heat equation re-spectively. Then under reasonable assumptions onthe initial surface and for short times a unique so-lution exists and it is 4 ( -regular. Thus we apply

the mean curvature motion semigroup ( � 4 � ) asa geometric filter of width � to � � and obtain fora time step � � Ò X

5� � � � 4 � � � � � �

where the corresponding parametrization is definedby C � � C � � � . A suitable choice for the filter widthon a polygonal surface with grid size Æ � appears tobe � Õ Æ � . At first, we assume the triangulationof our initial surfaces to be uniform. In Section 7we will generalize our method to nonuniform filterwidth. Alternatively, we can incorporate the filterwidth in the diffusion coefficient and confine to atime step � � � . That is we consider for a spatiallyfixed filter width �

u � C � � � � ¿ � m � � � C � � � � >with ¿ � � Ò X

5 and evaluate the evolution at time� � � . As one advantage of this rewriting wenow can consider a spatially varying filter width �(cf. Fig. 1).

Step 2. We can replace the continuous nonlinearsemigroup by a time discrete evolution and focuson the first step. In the resulting implicit schemewe have to select a metric (cf. Section 2). We ap-proximate a fully implicit scheme, where the metricis evaluated on the unknown surface, by a sequenceof semi implicit schemes. Hence, in each iterationwe consider the metric from the previous step andcalculate parameterizations C of surfaces � for

� × > solving the linear problem

� C � C � � � ¿ � m   ¡ U C � > 2Let us emphasize that the index � does not indi-cate a curvature motion timestep but only succes-sively improved approximations of the fully im-plicit scheme � C � � C � � � ¿ � m Ù C � � > . For

� E Ä we expect this iteration to be a fix point it-eration with a convergence of the parametrizationsC at least pointwise to a parametrization C �

of aunique fix point surface � �

. Our numerical re-sults give a strong indication for this convergence.In Section 4 we have alread mentioned that such aregularity result holds in the simplified case. GivenLipschitz continuous initial data � � the solution � �of � Id � ¿ � � � � � � � is 4 5 7 8 -continuous for any

9 : < > ? � � . Our conjecture is that an analogous reg-ularity result holds for the implicit MCM timestepproblem and the Laplace Beltrami operator. Thus,

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we expect our limit surface to be 4 5 7 8 -continuousfor a triangular initial surface (cf. Fig. 5). Experi-mentally we have verified at least bounded discretesecond derivatives (cf. Fig. 2). Rigorous proofs forboth the convergence and the smoothness of thelimit surface are still missing. We abbreviate thenotation and write

� � � � � � � � �where the argument of the time step operator � / �indicates the corresponding metric. The limit sur-faces � �

turns out to be a fix point of the mapping � / � � � , i. e. � � � � � � � � � . So far we havederived a geometrically natural smoothing method,which results in solving a sequence of spatially con-tinuous but linear problems. Still the surfaces arecontinuous in space.

Step 3. Now we discretize in space, consideringtriangular surfaces � � of different grid size Æ andcorresponding linear finite element spaces � ¦ andask for parametrizations � : � � ¦ � � of surfaces

� ¦ (cf. Section 2). Thus, we consider

� ¦ � ¦ � � � �¦ � � �where ¦ � / � denotes the corresponding spatiallydiscrete time step operator. I. e. for the parame-terization � of � ¦ we get

� � � ® ¦ C � � � ¿ � m   ¡ Uß � � > 2Here ® ¦ again indicates the nodal projection onto

� ¦ . We expect � ¦ to converge to � �for Æ E >

(cf. the convergence result by Deckelnick and Dz-iuk [5]).

Step 4. Furthermore, we consider sequences ofnested grids generated by any recursive and regularrefinement rule and apply a diagonalization argu-ment with respect to the grid level and the above it-eration. After each iteration we refine the grid onceand obtain the following subdivision scheme:

� ¦   � ¦   � � � �¦   ¡ U � � � 2This corresponds to the operator equation

� � ¦   � ® ¦   C � � � ¿ � m   ¡ Uß   ¡ U� ¦   � >

for the parameterizations � ¦   of � ¦   . For thesake of simplicity we write � ,

* , and � in-

stead of C ¦   ,*

¦   , and � ¦   respectively. We sup-pose geometric decay of the sequence of grid sizes

Æ , i. e. à � Æ á Æ â � á à â Æ . In case of a stan-dard butterfly like refinement rule without smooth-ing, which results in the splitting of each triangleinto four up to scaling identical children, we obtainà â � à � � �5 .

Step 5. In each step of the above subdivisionscheme the solution of a system of linear equations(cf. Section 2) is required. As usual in finite el-ement calculus this system is sparse and iterativesolvers can be applied. In each subdivision step wemodify the metric and refine the underlying grid.This obviously is a cascadic strategy (cf. Section3) and we know that for increasing iteration indicesa decreasing number of smoothing steps ¼ in thelinear solver has to be performed if we consider ap-propriately prolongated solutions from the previouslevel as initial data. Let us indicate the number ofsmoothing steps ¼ on grid level � by an upperindex. Bornemann and Deuflhard proved that therequired number of iteration decays geometrically,i. e. in case of conjugate gradient iterations (CG)

¼ � ¼ Ï Ð Ñ � å X � Ï Ð Ñ � �

and for the damped Ja-cobi iteration ¼ � ¼ Ï Ð Ñ � 5 � Ï Ð Ñ �

�. Given an

error tolerance for the algebraic error we can pre-set the required number of smoothing steps ¼ Ï Ð Ñon the finest grid level � Ì Í Î . In our application wealways set ¼ Ï Ð Ñ � � . Thus given a final levelof refinement � Ì Í Î up to which we want to iteratethe overall cost of the resulting algorithm has opti-mal complexity for CG in case of a quatering typerefinement and nearly optimal complexity for thedamped Jacobi iterations. Optimal here means thecost is proportional to the number of fine grid nodes.If the goal is only to ensure appropriate smoothingresults, based on our experience one can further re-duce the number of iterations ¼ especially on finegrid levels � and confine to a fixed number of iter-ations ¼ . Thus we obtain a suitable approximationof our original model by the iteration

� � ç   � � � � � � � � �(cf. 3 and Fig. 4). As initial data and for the evalua-tion of the metric we consider the coordinate vectorof � � � � � prolongated to level � by trivial interpo-lation. The number of considered smoothing stepswill correspond to the stencil width of our schemeas it operates on nodal coordinates of a triangulargrid. Different from the case of linear problem,where convergence of the cascadic multigrid has

72

been proven, here the dependence on the metric in-troduces a nonlinearity. Nevertheless on sufficientlyfine grids we expect a neglectable impact of this ef-fect on the performance of a cascadic iteration.

Figure 4: A sequence of flat shaded subdivision sur-faces subdividing each triangle into four children ineach refinement step.

6 Algorithmical aspects

Let us give some details on the non exact solutionof the linear system in each step of our subdivi-sion scheme at least in case of damped Jacobi it-erations. In fact, we confine to a few smoothingsteps. We define ±� è � � * ç   � ±� � where

*is a suit-

able smoothing operator and the exponent ¼ in-dicates the number of considered smoothing steps,i. e.

* ç â � � ±� � � � * + * ç � ±� � . The damped Jacobiiteration

* êis defined by

* ê ±� � � ±� � í î � � � � � � � ±� � ±� � � � ¿ y � � ±� �where î is the matrix representing the diagonal partof � � � � � ¿ y � � � and í is the damping factor.Here � � � and y � � denote the mass and stiffnessmatrix on the grid � � � . We always have set í �

�ð .

So far we have not specified the refinementmethod to be applied in every iteration of the pre-sented subdivision schemes. As long as a regularrefinement rule is considered which guarantees suit-able upper and lower bounds for the angles of the

Figure 5: A coarse torus (top left) is processedby our subdivision method leading to a smoothlimit surface (top right). We have used a quater-ing scheme (left column) and a bisection refinementstrategy (right column).

generated triangles our approach in principle is in-dependent of the concrete scheme. Figure 5 depictstwo frequently used refinement schemes, the qua-tering scheme and the recursive bisection scheme.

Finally, we can write our subdivision scheme inpseudo code notation as follows:

Define � � � � as the initial mesh; � � > ;do R

� � � � � ;� � Refinement � � � ;Compute � � MassMatrix( � )and y � StiffnessMatrix( � );

±� � nodal vector of � ;±� � � ® m � � ;

for( ¼ � > H ¼ ò ¼ H ¼ � ¼ � � )±� � * � ±� � ? � ? y � ±� ;

� � surface � with updated nodes ±� ;Zwhile ( � á � Ì Í Î );

Here the operator Refinement( / ) denotes any

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Figure 6: A sequence of flatshaded subdivision sur-faces is generated using the local filter width al-gorithm. The starting surface comes along witha very irregular triangular grid (different valencesof the nodes, thin triangles, nonhomogeneous gridsize), which we keep without changes. The proposedmethod is able to effectively deal with such surfaces.

regular refinement scheme, MassMatrix( / ) andStiffnessMatrix( / ) evaluate the compute the corre-sponding matrices on a given surface and ® m rep-resents trivial interpolation of the nodal positions of

� � on � . By trivial interpolation we mean thebuild in recursive interpolation due to the appliedrefinement rules without any smoothing operations.

7 A local filter width expansion

In many applications the initially coarse mesh willbe characterized by considerable variations in thelocal grid size. We can take care of this by adap-tation of the filter width in our implicit time stepscheme to the local grid size (cf. Fig. 6). Here theidea is to consider a smoothed local grid size of theinitial grid as filter width and modify the diffusion

coefficient with respect to this filter width. Figure 7shows a comparison between the expanded modelincorporating a smooth local filter width functionand the fixed filter width problem studied so far.

Figure 7: A comparison of limit surfaces based ona small and spatially fixed filter width (left) and thelocal filter width expansion (right) for a coarse ini-tial grid with considerable variation in the grid size.

In our iterative scheme we apply the samesmoothing operators in every step to this grid sizefunction to ensure still 4 5 7 8 -smoothness of thelimit surface. Hence, we consider the followingcontinuous problem:

� C � C � � � div m   ¡ U ó ¿ � � l m   ¡ U C ô � >� Æ � Æ � � � div m   ¡ U ó ¿ � � l m   ¡ U Æ ô � > 2where ¿ � � 4 5 � ¦   � X

5 and Æ � is the initial gridsize function. We expect the sequence R C ? Æ Z

toconverge to a solution C �

, Æ �of the fully implicit

problem:

� C � � C � � � div m Ù � ¿ � l m Ù C � � � >� Æ � � Æ � � � div m Ù � ¿ � l m Ù Æ � � � > 2

where ¿ � � � 4 5 � ¦ Ù � X5 and � �

is the resultingmesh with parameterization C �

. Furthermore, weexpect the same regularity result as in the case withfixed filter width, i. e. 4 5 7 8 -smoothness of the limitsurface � �

and of the smoothed filter width func-tion Æ �

.

AcknowledgementThe authors thank Leif Kobbelt for comments on anearly draft of this manuscript.

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