knp07-quadcover

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EUROGRAPHICS 2007 / D. Cohen-Or and P. Slavk(Guest Editors)

Volume 26 (2007 ), Number 3

QuadCover - Surface Parameterization

using Branched Coverings

Felix Klberer Matthias Nieser Konrad Polthier

Freie Universitt Berlin

Abstract

We introduce an algorithm for the automatic computation of global parameterizations on arbitrary simplicial 2-manifolds, whose parameter lines are guided by a given frame eld, for example, by principal curvature frames.The parameter lines are globally continuous and allow a remeshing of the surface into quadrilaterals.The algorithm converts a given frame eld into a single vector eld on a branched covering of the 2-manifold and generates an integrable vector eld by a Hodge decomposition on the covering space. Except for an optionalsmoothing and alignment of the initial frame eld, the algorithm is fully automatic and generates high qualityquadrilateral meshes.

Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Computational Geometryand Object Modeling, Hierarchy and geometric transformations, Geometric algorithms, languages, and systems

1. Introduction

The idea of local charts belongs to the most fruitful conceptsin the differential geometry of smooth manifolds. Originallyinvented by Riemann, each manifold is locally considered asa distorted Euclidean space where charts provide the math-ematical formalism to describe the relation, the deformationand the differences to Euclidean space. For example, chartson surfaces immediately provide a local coordinate systemwhich allows to transfer differential operators onto curvedsurfaces.

The eld of geometry processing investigates meshes,simplicial surfaces where no a priori parameterization isgiven. In order to reach a differential framework, beyond thepurely topological point of view, the eld of discrete differ-

ential geometry was invented. For the rst time it was thenpossible to transfer differential operators onto simplicial sur-faces and to work in a similar way like on smooth surfaces.But considering the strong similarities between image andgeometry processing, we still encounter many algorithmsinvented in image processing which donot have counterparts

Supported by the DFG Research Center MATHEON Mathemat-ics for key technologies and mental images GmbH.

Figure 1: QuadCover generates high quality parameter lines on simplicial surfaces. The automatic parameterizationis guided by a user-given frame eld, such as principal cur-vature directions, and is well suited for regular quadrilateralremeshing.

c The Eurographics Association and Blackwell Publishing 2007. Published by BlackwellPublishing, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden,MA 02148, USA.


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Klberer, Nieser, Polthier / QuadCover - Surface Parameterization using Branched Coverings

in the eld of geometry processing yet. For example, theefcient wavelet theory on images essentially requires a 2-dimensional coordinate system which has not been availableon general simplicial surfaces yet. In fact, having natural co-ordinates on a simplicial surface would pave the way fortransferring and adjusting many algorithms onto surfaces.

The present work focuses on the automatic constructionof a global surface parameterization. The parameterizationis guided by a user-dened frame eld, for example, by aeld of principal curvature directions, or by any other frameeld arising in applications.

This paper is driven by a conceptual clarity of the under-lying algorithm. Starting from a given frame eld, the al-gorithm rst constructs a locally integrable eld, and thena globally integrable eld taking the topology of the un-derlying surface into account. The issue of branch pointsof the frame eld is resolved by branched covering spaceson which the given frame eld naturally simplies to a sin-gle vector eld where standard Hodge decomposition tech-niques are used to assure global integrability. The paper hasclose contact to the fundamental paper of Ray, Lee, Levy,Sheffer, and Alliez [RLL06] and extends their initial ideasto a more general and automatic framework.

Finally, given a global parameterization on any simplicialsurface we will see a wealth of future applications. From tex-ture mapping to extension of image processing algorithms,from remeshing to the automatic construction of hierarchi-cal subdivision surfaces, all applications using natural co-ordinates will benet from the added structure of a globalparameterization.

1.1. Previous work

The research area of surface parameterization has a long andfruitful tradition. There already exists a wealth of differentprevious approaches to surface parameterization and, moregeneral, the generation of quad and quad dominant meshesfrom given triangle meshes.

One of the rst impressive works is by Gu and Yau[GY03] who construct global conformal parameterizationsof surfaces with arbitrary genus. The resulting parameterlines minimize angle distortion but may have a rather largemetric distortion. Furthermore, the space of global confor-mal parameterizations is too rigid to allow a local alignmentof the parameter lines at given surface features.

The method of Boier-Martin, Rushmeier, and Jin[BMRJ04] clusters the surface into macropatches and pa-rameterizes each surface patch. Kharevych, Springborn, andSchrder [KSS06] nd a conformal parameterization viacircle patterns. In contrast to Gu and Yau, they use cone-singularities to increase the exibility of purely conformalmappings. Dong et al. [DBG06] compute the Morse-Smalecomplex of eigenfunctions of the mesh Laplacian to com-

pute a patch layout. The nodes of the complex are then uti-lized similarly to the cone singularities in [KSS06].

Early approaches for quadrangular remeshing guidedby principal curvature directions are from Alliez et al.[ACSD03]. They were extended by Marinov and Kobbelt

[MK04], and base on the integration of curvature lines on thesurface. Dong, Kircher, and Garland [DKG05] presented analgorithm which traces isolines in two conjugate harmonicvector elds. Marinov and Kobbelt focus on creating coarsequad-dominant meshesin[MK06] by approximating thesur-face with very few patches, which are then individually sub-divided into quads.

Tong, Alliez, Cohen-Steiner, and Desbrun [TACSD06]use harmonic one-forms for surface parameterization. Theyenlarge the space of harmonic one-forms by allowing ad-ditional singular points on the surface. The extended cutgraph increases the homology group and thus the space of harmonic one-forms on the surface. As a consequence, the

user-dened choice of placing the singular points and thecut graph allows a controlled modeling of the harmonic one-forms. Still, the approach is constrained by the global natureof harmonic one-forms, in some sense, similar to the algo-rithm of Gu and Yau [GY03].

Ray et al. [RLL06] parameterize surfaces of arbitrarygenus with periodic potential functions guided by two or-thogonal input vector elds. This leads to a continuous pa-rameterization except in the vicinity of singular points onthe surface. These singular regions are detected and repa-rameterized afterwards. Our approach was strongly inspiredby their work.

1.2. ContributionsWe derive an algorithm to compute a global continuous pa-rameterization for an arbitrary given simplicial 2-manifold.The algorithm runs automatically and the parameter linesalign optimally with a user-dened frame eld, for example,the principal curvature directions.

We introduce a theoretical framework for frame elds onsurfaces and their relation to branched covering spaces of the surface. As a consequence, frame elds on the surfacesimplify to vector elds on the covering space, so that theproblem of parameterizing with frame elds reduces to theproblem of nding a proper integrable vector eld on thecovering surface.

The branch points of our covering space lie at the branchpoints of the given frame eld. They are conceptually sim-ilar to the singular points with fractional index used by[TACSD06] and [RVLL06].

Instead of restricting ourselves to harmonic frame elds,we allow an additional divergence part and seek the opti-mal parameterization in the larger space of locally integrableframe elds, enabling the parameter lines to better align withthe given input eld.

c The Eurographics Association and Blackwell Publishing 2007.


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a) b)

c) d)

Figure 2: A perfectly regular knot (a) was irregularly retri-angulated and disturbed by noise (b). Our parameterizationof the noisy model (c) provides a stable reconstruction (d).

2. Setting

We construct a parameterization of a 2-manifold, whose pa-

rameter lines divide the surface into quads. Akin to earlierquad-dominant remeshing algorithms [ACSD03,RLL06],a major aim of our method is to construct parameter lineswhich are close to the principal curvature directions. In thissection we will introduce the underlying concepts.

2.1. Matchings

Smooth setting. Given a smooth 2-manifold M with chartsi : U i M i R 2. The parameter grid on M is thepreimage under i of the unit grid lines Z R and R Z .A local parameterization can be easily dened in each chartby taking arbitrary map i; the non-trivial task is to assure

global continuity of the parameter lines on the surface.A globally continuous parameterization consists of a

set of charts{U i,i} for which the parameter lines coincidein all regions where two chartsU i, U j overlap. The transitionfunctions between adjacent charts of a global parameteriza-tion satisfy two conditions, see also [RLL06]:

First, the gradients of the parameterization functions haveto agree up to a rotation by multiples of 2 , because u- andv-lines should not be distinguished on the parameterized sur-

face. Thus, the Jacobians of the charts are related by

Di( p) = J r ij D j( p), J :=

0 1 1 0 , pU i U j (1)

with a constant integer r i j {0,1, 2, 3} on the intersectioni j. We call the values r i j matchings between chartsU iand U j. See Figure 3 for nontrivial matchings.

Second, the parameter values may differ only by integervalues in the u and v coordinate, since the unit grid is invari-ant under translations by integer values.

The transition functions j 1i of a parameterizationfullling the two conditions above are automorphisms of theunit grid. We call a linear function f : R 2 R 2 which meets

f ( z) = J r z+ t , r Z , t Z 2, zR 2 (2)

a grid automorphism .

U i

U j

0 1

2

U 0 U 1

U 2

i

Figure 3: Parameterization of two overlapping charts witha matching of r i j = 3 (left), and of a cube with matchingsr 01 = r 12 = 0 , r 20 = 1 (right).

Discretization. Just as [RLL

06], we consider each tri-angle as a chart on discrete triangle meshes. The transitionfunction between two adjacent triangles is fully determinedby the matching and the translation vector associated to theircommon edge.

The methods of [TACSD06] and [DBG06] divide thesurface into large scale quadrilateral patches. The patchesare used as charts and allow non-trivial matchings only atpatch boundaries. In contrast, triangle based charts allow tochoose matchings independently at every edge and permitmuch more exible topological structures of parameter lines.

2.2. Branched covering spaces

First, recall some denitions about Riemann surfaces, see[FK80], [Ful95], [Jos02].Denition 1 Let M be a Riemann surface. A (branched)covering M of M is a Riemann surface with a local homeo-morphism : M M , where for each p M there existsa neighborhood U p with local coordinates z : U C , z ( p ) = 0 and a neighborhoodU ( p ) with local coordi-nates z : U C , z(( p )) = 0 and an integer n p > 0, suchthat is given by z = ( z )n p in terms of local coordinates.



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If n p > 1, then p is called a branch point of M . The numbern p is usually referred to as the ramication index of the point p. If p M is no branch point, then there exists a neighbor-hoodV of p such that everyconnected componentof 1(V )is mapped by homeomorphically ontoV .

Denition 2 A trivial covering of a connected setU M isa coveringU where every connected component of 1(U )is mapped by homeomorphically ontoU .The components 1(U ) of a trivial covering are called lay-ers . They will be denoted byU l U , l {0, . . ., #layers 1}. Let lU = ( |U l ) 1 : U U l be the inverse of the pro- jection operator in the given layer l.

Here we consider coverings whose metric is induced fromthe surface M by 1. Thus, an n-sheeted trivial covering of M can be seen as just n copies of M , cf. Figure 5, top left.

One way of constructing a covering on M is to take cov-erings of its charts and glue them together at their intersec-tion: For each two charts U

i, U

j of M with U

iU

j = , let

: 1i (U i U j) 1 j (U i U j) be an isomorphism be-

tween trivial coverings of the charts. The patches can thenbe merged together by identifying the corresponding pointsof the two coverings, cf. Figure 5, top right.

The following construction shows, how the matchings r i jof a manifold M canonically induce a covering of M . Werestrict to 4-sheeted coverings as they naturally appear in thestudy of frame elds.

Denition 3 Let (U i ,i) be 4-sheeted trivial coverings of the charts U i. For two overlapping charts U i, U j, the map : 1i (U i U j)

1 j (U i U j), which maps a point p

lU i (U i U j) from layer l to

(r ij + l)mod 4U j U ( p) into layer(r i j + l)mod 4, is an isomorphism. Thus, by identifying the

two layers with , the trivial coverings of the charts can beglued together (Figure 5, top right). We call the resultingcovering M of the covering induced by r .

The covering M induced by the matchings has no branchpoints. To enlarge the space of possible frame elds on themanifold, the covering surface maypossess singlepoints p M , whose covering locally looks like z zn , nZ . For thesepoints, there is no neighborhood U , for which 1(U ) ismanifold (see Figure 5, bottom right).

Discretization. In the discrete setting, branch points arelocated at vertices. On a 4-sheeted covering they occur whenthe (oriented) sum of all matchings of outgoing edges differsfrom 0 (modulo 4). This means starting somewhere in theneighborhood of v and walking around the vertex, ends on adifferent layer in the covering than the start point. The fol-lowing notion is used fordescribing different types of branchpoints.

Denition 4 Letv beavertexof M , T 0, . . ., T n 1 the trianglesincident tov in counterclockwiseorder,T n = T 0 andr i,i+ 1 thematching at the edge between adjacent triangles. The layer

Figure 4: A set of four vectors given in each point whichcannot be described with global vector elds. Left: On a at torus an integral line meets itself perpendicularly. Right: Around the vertex of a cube exists no match of vectors.

shift around v is then given by:

ls (v) := n

i= 0

r i,i+ 1 mod 4 (3)

2.3. Frame eldsWe now introduce globally dened frame elds, which arethe guiding geometric structure for our parameterizations.Denition 5 Given a manifold M with charts U i and match-ings r . A frame eld on M is a collection of four vectorelds X i,0, X i,1, X i,2, X i,3, in each chart U i which satisfy inall overlapping charts U i U j:

X j,k = X i,(k r ij ) mod 4, k {0,1, 2, 3}.This means that the vectors X j,k are cyclically permuted to X i,k by a shift of r i j . If X i,2 = X i,0, X i,3 = X i,1 in alldomains i, the frame eld is called symmetric .

In the example of Figure 3 (left), X i,1 coincides with X j,0,corresponding to r i j = 3. Fig. 4 shows examples of a frameeld on a at torus and a cube, which cannot be expressed interms of 4 global vector elds.

Remark. The direction elds introduced in [RVLL06] area special kind of frame elds, which require the vectors X i,0and X i,1 to be perpendicular, and to have unit length. Theseelds may have single points with a fractional index. In ournotion, singular pointswith fractional indexn/ 4 can onlyoc-cur at branch points of the covering with layer shift n mod4.The integer part of the index is determined by the vector eldindex on the covering.

A global parameterization can be represented in differ-

ential form by a global frame eld on M using the followingtwo vector elds in each chartU i:

X i,0 := ( Di) 1(e1), X i,1 := ( Di) 1(e2) (4)

with the unit vectors e1, e2 inR 2. With e3 := e1, e4 := e2and X i,2 := ( Di) 1(e3), X i,3 := ( Di) 1(e4), we obtain in-deed a frame eld: for k {0, 1, 2,3}

X j,k = ( D j) 1(ek ) = ( D( J

r iji))

1(ek )

= ( Di)1( J r ij ek ) = X i,(k r ij ) mod 4 (5)



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U 0

U 1

0U 1U

U

U U V

U V

U V

U V

Figure 5: Top left: Trivial covering. Top right: Patching twocoverings together. Bottom left: A frame eld lifted to a vec-tor eld on the covering. Bottom right: Branch point.

Discretization. The frame elds are discretized to be con-stant on each triangle. For symmetric frame elds, only twoof the four vectors are stored. Together with the matchings,this denes discrete symmetric frame elds uniquely.Denition 6 A matching on a discrete manifold M is a map

r : {edges ei j | T i T j = ei j } {0, 1, 2,3},

which determines the matching r i j for two adjacent trianglesT i and T j. We denote the space of all those maps by R M .

2.4. Vector elds on covering spaces

In this section, we show how frame elds can be describedwith vector elds on a covering surface. This result allows usto apply the classical theory for vector elds to frame elds.

A frame eld X i,k on M with matchings r canonically liftsto a vector eld X on the covering: in each chart U i, lift X i,k to a vector eld X on a trivial 4-sheeted covering U i of U ias follows: For p lU (U ), set X ( p) = X i,l (i( p)) (Figure 5,bottom left). The result is a globally dened vector eld X

on the covering M induced by the matchings r .

When the coverings of the charts are patched together asdescribed in Def. 3, X becomes a well dened vector eldon M , since the layers of the covering are connected in thesame way as the vector elds permute when another chart ischosen.Denition 7 Let M be a manifold with matchings r and M the induced covering. A frame eld lifted to a vector eld X on M is called a covering eld of M .

Figure 6: Isolines of a symmetric scalar function at a branch point with layer shift 1 (left), 2 (middle), 3 (right).

2.5. Scalar functions on covering spaces

Parameterization functions are scalar functions on the cover-ing, which may be discontinuous: along the common edge of two adjacent triangles the function may differ by a constantvalue.

Denition 8 Given a surface M with matching r . Let G be acut graph of M , i. e., a set of edges in M , such that M \ G is atopological disk. We dene the function space

S r ( M ) := {piecewise linear functions on M

which are continuous except at G}

A function f S r ( M ) is called a symmetric covering func-tion , if for all charts U of M and pU , the values of f inthe preimages 1( p) satisfy:

f (0U ( p)) = f (2U ( p)) , f (

1U ( p)) = f (

3U ( p)) .

Given a function S r ( M ), its derivative isa piecewise constant and curl free vector eld on M and each curl free vector eld can be expressed as gra-dient eld of a function in S r ( M ), thus S r ( M ) ={curl free vector elds on M }, see [PP03].

Parameterization function. A symmetric covering func-tion can be projected to a function : M R 2 by taking thevalues of layer 0 and 1 in each point p U i:

i( p) = ( (0U i ( p)) ,

(1U i ( p)))T = : (ui( p), vi( p)) T . (6)

The components of this function i at the intersection of twoadjacent maps U i, U j then meet:

(ui, vi)T = i

1 j (u j, v j)

T = J r ij (u j, v j)T t i j . (7)

with the constant translation vector t i j R 2.

If is continuous up to jumps by integer values, all tran-sition functions of i are grid automorphisms (see Equa-tion 2) and therefore, the parameter lines are globally con-tinuous. Thus, we seek for a function in the space

S r ( M ) := {u S r ( M ) | adjacent T i, T j: t i j Z 2}.

Given a parameterization function S r ( M ), the func-tion values on any map U i may be translated by an arbitraryinteger offset without changing the gradient eld or the -nal parameter lines. Thus, we assume w.l.o.g. that all t i j = 0except for edges which lie on the given cut graph G.



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Discretization. In our algorithm, the covering surface isnot explicitly computed. It is only represented by the match-ings r . Symmetric scalar functions on the covering are storedas the vector | T ( p) = ( uT , p, vT , p) from equation (6) in eachtriangle T at each of its three corners p.

A cut graph can be obtained as the boundary of a span-ning disk of M , see Figure 8, middle. It is not of interestfor the algorithm, which cut graph is chosen or if it cuts thesurface into one or more connected components. The result-ing parameter lines are independent of this choice except forslightly different rounding results in Section 4.1.

Constructing a basis of S r ( M ). We now constructa basisof S r ( M ) which describes all scalar functions and possiblechoices of the translation valuest i j . At edges of the cut graphG, t i j cannot be chosen arbitrarily. At each vertex of M , the(oriented) sum of t i j at outgoing edges is always 0.

Let S h( M ) be the space of all continuous piecewise linear

functions on M

with basis functions i and let P = { i G | i {0, . . ., n p 1}} be a set of n p cut paths with P =G. For a genus g surface without boundaries, these n p = 2gpaths generate the homology group of M . If the surface hasnb > 0 boundary components, we additionally need paths,which connect the boundaries, thus we haven p = 2g + nb 1paths.

For each path i, let i S r ( M ) be the piecewise linearfunction on M , which is 1 on the right side at all verticesalong the path, 0 on the left side and 0 at all other vertices of the surface. This is a discontinuous function on M , however i is a curl free vector eld.

Theorem 1 The functions{i}{ j} are a basis of S r ( M ).

Proof For each path j P , the gradient eld of the cor-responding basis function j has periods different from0, i. e. the path integral i, ds along a path on M which crosses j at exactly one point pi is 1 or 1, de-pending on the orientation of and j. Since the new basisfunctions i are linear independent, the space S r ( M ) is of dimension nv + n p = nv + 2g + max(b 1, 0). This is exactlythe dimension of the space of curl free vector elds. Fur-ther, the gradients of all functions in S h are curl free, thus S h = {curl free elds}. This is exactly the space of har-monic vector elds plus gradient elds on M .

Denition 9 For a given function f S r ( M ), let i( f ) and j( f ) be the coefcients of f when written as linear combi-nation of basis vectors, that is, f = i ii + j j j.

We order the vertices of the covering, such that succes-sive vertices v2i, v2i+ 1 are associated to another, i.e.(v2i) =(v2i+ 1) and they differ by a layer shift of 2. For the algo-rithm, we also require that the paths are symmetric on thecovering, i.e., for each path 2 j the path 2 j+ 1 differs onlyby a layer shift of 2. How to nd cut paths with this propertyis described in Section 4.4. Thus, a function f S r ( M ) is a

symmetric covering function if and only if for all i

2i = 2i+ 1, 2 j = 2 j+ 1. (8)

Theorem 2 Let p be a branch point with layer shift ls( p) = 1or ls( p) = 3 (according to Def. 4). Then the value of anysymmetric scalar function (u, v)S r ( M ) at p is either a gridpoint (u( p), v( p)) Z 2 or a midpoint of a grid cell (u( p) +1/ 2, v( p) + 1/ 2) Z 2. If l s( p) = 2, then it is a grid point of grid size 1/ 2: 2(u( p), v( p)) Z 2.Let p be a vertex with l s( p) = 2. Then the value of any sym-metric scalar function (u, v) S h( M ) at p is a grid point of grid size 1/ 2: 2(u( p), v( p)) Z 2.

Proof Let T 0, . . ., T n be the triangles incident to p in counter-clockwise order. Looking at the values f = ( u, v) at vertex pin these triangles, they satisfy:

(uT 1,v, vT 1,v)T J r 01(uT 1,v, vT 1,v)

T + Z 2

(uT 1,v, vT 1,v)T J r 01 r 12(uT 1,v, vT 1,v)

T + Z 2...(uT 1,v, vT 1,v)

T J ls (v)(uT 1,v, vT 1,v)

T + Z 2

(uT 1,v, vT 1,v)T ( Id J ls (v) ) 1Z 2

Calculating the inverse matrix and applying it to all integergrid points leads to the claimed proposition.

In particular, this theorem holds for the resulting parame-terization map, since it is a symmetric scalar function on M .It follows, that all branch points get mapped to either a gridpoint or to the middle of a grid cell. Branch points with layershift 2 may also lie at the midpoint of an edge.

Remark. If a branch point is mapped to the middle of a

grid cell, this cell may become a triangle, pentagon or a poly-gon of higher degree. If the parameterization should gener-ate a pure quad mesh, all branch points should be locatedat grid points. According to Theorem 2, this can be assuredby parameterizing the surface half as ne (multiply all inputvectors with 1/ 2) and multiply the resulting parameteriza-tion function with 2.

3. Algorithm

Choice of input frame eld. The user input to our methodis a matching r R M and a symmetric frame eld K i,k , or,equivalently, a symmetric covering eld on a simplicial sur-face. This eld acts as a guidance eld for the nal parame-terization.

There are different ways of constructing such an inputeld. It is up to the user, if he species the matchings r orif he just provides two vectors per triangle (and their neg-atives). In the latter case, matchings can be automaticallygenerated by associating the four vectors in each triangle tothose in adjacent triangles. It is reasonable to identify thevectors with the smallest angle to each other in order to getstraight parameter lines.



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Figure 7: The Costa surface parameterized with different input elds. Middle: Input frames align with the edges of theinput mesh (left). Right: Input frames align with principalcurvature directions.

Constructing from curvatures. In most cases, we usedthe principal curvature directions as input. In general, it isnot possible to assemble the principal curvature directionsglobally in four separate vector elds. But locally in eachchart and away from umbilics, one can dene four vectorelds whose vectors have unit length and point into the fourprincipal curvature directions, sorted in a counterclockwiseorder. In regions where two chartsU i, U j overlap, these vec-tor elds then only differ by a cyclic permutation of the fourvectors. This shift is the matching r i j .

The resulting frame eld could also be extended to theumbilic regions of the surface, where no principal curvaturedirections are dened. In general, this is not possible with-out getting some branch points in the umbilic region. Fur-thermore, the calculation of principal curvature directionsin umbilic regions leads to vectors pointing in arbitrary di-rections in practice. Thus, we use a preprocessing step tosmooth the given frame eld in umbilic regions, such thatthey align with the curvature directions in their neighbor-hood (see Section 4.3).

Discrete principal curvature directions and values can becalculated as proposed in [CSM03] or [HP04]. Note that wedeal with curvatures given on triangles, not on vertices.

3.1. Algorithm overview

The algorithm is split into two parts, the preprocessing partand the main algorithm. In the preprocessing part, the guid-ance frame eld for the parameter lines is generated. Themethods applied there control the construction and improve-ment of the frame eld.

In the main part, a global parameterization function will

be generated. First, the given input frame eld is approxi-mated by the gradient of a scalar function. Second, the pa-rameterization will be adapted to make the parameter linesglobally continuous. The output is the function : M R 2,which can be used as a texture map to visualize the parame-ter grid or for quad mesh generation.

Preprocessing steps

1. Calculate principal curvature directions K (required).2. Smooth principal curvature eld, see Sect. 4.3 (opt.).

3. Dene the topology of the covering surface by calculat-ing matchings at the edges (required).

4. Optimize the topological structure, see Sect. 4.2 (opt.).Main steps

1. Find a potential function , whose gradient eld opti-mally aligns with the given input frames.

2. Add the harmonic scalar function with minimal L2norm such that the parameterization = + is glob-ally closed.Steps 1 and 2 are detailed in the following section.

4. Implementation details

Computing the potential function. The nal parameteriza-tion should align with the given input eld as well as possi-ble, i.e., the parameterization should minimize the energy

E () = M K 2

dA (9)

in the space S r ( M ), with xed matchings r .Recall, that K is a symmetric covering eld and the cov-

ering surface is symmetric due to a cyclic permutation of the layers by 2. Since the energy has a unique minimum, thesolution is also symmetric.

In the smooth setting, this algorithm can also be formu-lated using covering elds instead of the scalar function .The Hodge-Helmholtz decomposition of a vector eld K ona manifold M is a unique description of K as

K = PK + C K + H K (10)

with a gradient eld PK , a cogradient eld C K and a har-monic eld H K . Discarding the second term leads to a curlfree eld X := PK + H K whose integral is the minimizer of Energy (9). For the Hodge decomposition of discrete vectorelds see [PP03].

The Hodge decomposition does not need a cut graph of the surface, so the gradient of the optimal solution inEquation (9) must also be independent of the cut graph.

4.1. Global continuity

If we took the solution from the previous paragraph as pa-rameterization map, the parameter lines would not be contin-uous everywhere on the surface. They may have a mismatchat the cut graph G . With the representation of with basisfunctions (see Def. 9), the parameter lines are globally con-tinuous if and only if all j() Z .

In order to adapt to fulll the global continuity con-dition, we add another scalar covering function to suchthat := + satises j()Z . We determine by spec-ifying the coefcients j( ) := [ j()] j(), and calcu-late all other coefcients such that = 0. That means,



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Figure 8: Parameterized minimal surfaces: Trinoid (left),Schwarz P-surface with its cut graph, and the Hyperboloid parameterized using non-orthogonal frames.

that is required to be a harmonic vector eld, becausethese are the smallest vector elds with given periods in L2norm, so the distortion of the parameterization will be mini-mal. The Laplace equation can be written as a linear systemof equations for the unknown coefcients i() and solvedefciently.

Generating quads. If the function values at all branch

points are integers, and all triangles are positively orientedin R 2, then the parameterization produces only quads. Ac-cording to Theorem 2, if S r ( M ), then it may happen,that a branch point lies in the middle of a grid cell. As aresult, the parameterization would contain triangles or otherpolygons. A pure quad grid is guaranteed if

j() 2Z . (11)

Thus, rounding all j() coefcients to values in 2Z insteadof Z leads to a parameterization consisting of quads only.The rounding of the coefcients depends on whether a purequad grid is required.

Optimal solution. The minimal energy solution from

Section 4 is not necessarily optimal, since the values j()might not be rounded optimally. This is the only part in thealgorithm, where the parameterization results depend on thecut graph. The exact solution of the optimization problemdescribed below would lead to parameter lines which are in-dependent of the cut graph.

Let (h j) j= 1...#paths S r ( M ) be a basis of the harmonicfunctions on M , with k (h j) = 1, if k = j and k (h j) = 0, if k = j. Let H := j c jh j, c j R according to Def. 9. Consid-ering Condition (8), nding the symmetric harmonic eld H with minimal L2 norm is equivalent to nding the i, whichminimize

vAvT , such that j + q j( X ) Z , (12)

with v = ( 0, 0 . . . ,n , n), Ai j = M hi , h j dA.This is a quadratic discrete optimization problem. The

exact optimum is hard to obtain. For the generation of the matrix A, one has to compute a set of covering elds,which are a basis of the harmonic elds (consisting of 2genus( M )) + #singularities many minimization problems).Then, the quadratic integer problem has to be solved. Our

heuristic with just rounding the values j as described in thissection is in general not optimal but leads to good results.

4.2. Branch point optimization

As branch points often reduce the straightness of parameterlines, we employ a simple heuristic to nd nearby branchpoints, whose layer shiftsadd up to zero. Those points can bemerged by changing the matching at the edges in-between,resulting in a locally trivial covering.

LetC b be a cost function on vertices, which is 1 for branchpoints and 0 for all other vertices, thus C b penalizes the oc-currence of branchings. Let another cost function

C i j = ( X i,k , X j,(k + r ij ) mod 4)

on the edges of the mesh represent the parallelism betweenvectors in the incident triangles T i , T j.

If we change the matching consistently along an edge pathbetween two vertices, the layer shift of all but the two endpoints of the path remain constant. If one of the endpointsis a branch point, we can always change the matching alongthat path so the layer shift for one of the vertices becomes 0,possibly for both of them.

We use a greedy search strategy that nds paths betweensingularities, such that a change of the matching along thewhole path reduces the total costC = C b + i j C i j . Here, weights branch point reduction vs. vector eld straightness.

4.3. Aligning the frame elds

To smooth the frame eld, and to extend the principal curva-ture directions into umbilic regions, we adapted the smooth-ing method in [HZ00].

The frame X i in each triangle T i is rotated by an angle i,such that adjacent frames align best with respect to the fol-lowing smoothing energy. The energy operates on the anglesbetween frames of pairs of adjacent elements.

E smooth = i> j

area(T i) + area(T j)3 cos 4(i j + i j) ,

wherei j is the initial angle between the framesof trianglesiand j. Measuring angles between frames is not unique, sincethere are four vectors in each frame. However, since vectorsresulting from principal curvature directions differ by mul-tiples of 2 , this is irrelevant, due to the periodicity of thecosine.

The above energy will align all frames with each other,but will not retain the directions of principal curvature. Wedesire a balanced smoothing step, which xes angles wherethe curvature directions are very pronounced, but which alsoaligns frames in umbilic areas to the xed frames. For thatpurpose, we blend the smoothing energy with an alignment



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Figure 9: Parameterization of the hand model. Branch points are marked red.

term, which measures the distance to the original eld.

E = E smooth + ( 1 ) E align, where

E align = i

area(T i) i 2i .

The weights i indicate the stability of the principal curva-ture directions in the current triangle. As a heuristic, we use

i := | 1 2|/ 21 + 22and set i = 0 if 21 + 22 = 0. Here, 1 and 2 are the prin-cipal curvature values in the current triangle.

Minimization of E then yields a smooth vector eld whichis roughly aligned to the features of the mesh. The user-setfactor determines if there is more emphasis on smoothing( 1) or on principal curvature alignment ( 0).

4.4. Generation of symmetric cut paths

This section deals with the problem of nding paths i ona 4-sheeted symmetric covering : M M of a given dis-crete manifold M , such that M \ { i } is cut open to a topo-logical disk. This is done by rst cutting M open and thenlifting the paths to the covering. By comparing the Eulercharacteristic of M and M , one notices that the number of paths p which are necessary on M can be calculated fromthe number of paths p on M :

p = 4 p 3.

Our method constructs 4 p paths on M , thus we get 3 pathsmore than needed, so the covering gets cut into 4 simplyconnected components.

First, construct some paths i on M , which cut M open. If the surface has boundaries, there are n = 2g +numBoundaries 1 paths necessary. Choose one branchpoint of layer shift 1 as origin p0 of all paths, so all pathsbegin and end at p0. This could be done by extracting allcircles of a cut graph of M .

Figure 10: Comparison of the remeshed hand model with[TACSD06 ] (top). Bottom: constructed with QuadCover. Thehand model is courtesy of Pierre Alliez. Except from setting preprocessing parameters, no interaction was involved.

If there is no such branch point, one could also take anyother point on the manifold, but has to catch some specialcases in the following construction process. In practice, al-most all meshes have branch points of layer shift 1.

Now, for each branch point p i = p0, compute a cycle n+ istarting at p0, going around p i and ending at p0 again.

Each of these paths i can now be lifted to 4 paths ji , j = {0, 1, 2,3} on M . If some of the lifted paths are not

closed on M (since start and end point are on different lay-ers), extend them by walking around the root vertex p0 untilthe path gets closed (walking around the root vertex onceperforms a layer shift of 1). The collection of paths ob-tained by this construction cuts the covering manifold open.

5. Results and conclusion

Our algorithm delivered pleasing parameterizations on awide range of models, as shown in Figures 1, 9, 11, and10. Comparisons to state-of-the-art show that QuadCover

[RLL06] [TACSD06] [DBG06] QuadCoververtices 6355 6576 7202 6535irreg. vert. 314 34 26 37RSD edge 25.0% 28.3% 30.8% 18.2%RSD angle 10.7% 12.6% 7.8% 14.8%

Table 1: The number of total and irregular vertices of themodels shown in Figure 11 , as well as the relative standard deviation of their edge lengths and vertex angles.



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Figure 11: Comparison of remeshing results of the Stanford bunny. Models were produced by [ RLL06 ] and [ TACSD06 ](top), [ DBG06 ] and our algorithm (bottom). The upper his-togram next to each model shows the distribution of edgelengths, the lower histogram represents angle distribution.

produces very competitive results, see Fig. 11. As the g-ure illustrates, we roughly share the curvature alignmentof [RLL06], but managed to drastically reduce the oc-currence of irregular points, and we restricted ourselves toquadrilaterals. In contrast to the methods of [TACSD06] and[DBG06], our algorithm is suited to handle arbitrary loca-tions of branch points, as we do not restrict the branch pointsto be the corners of some coarse meta mesh.

Table 1 shows that QuadCover exhibits the smallest edgelength variation, at the cost of higher angular deviation.Discarding the curvature alignment term during smoothingsignicantly reduces angular deviation, but generally, morewrinkles in the nal quad mesh are introduced where param-eter lines dont follow high curvature.

We have identied frame elds with vector elds onbranched coverings. This theoretical foundation reduces theparameterization problem to well studied problems of vector

eld decompositions, once the set of branch points is xed.We hope that this simplifying conceptwill give rise to furtherfruitful ideas for surface parameterization and remeshing.

Acknowledgments

We thank Max Wardetzky for valuable discussions on cover-ing spaces. We thank Pierre Alliez, Timo Bremer, John Hart,Leif Kobbelt, and Bruno Lvy for generously supporting uswith comparison material.

References

[ACSD03] ALLIEZ P., COHEN-STEINER D., DEVILLERS O.,LVY B., DESBRUN M.: Anisotropic polygonal remeshing. ACM Trans. on Graph. (2003), 485493.

[BMRJ04] BOIER-MARTIN I., RUSHMEIER H., JIN J.: Parame-

terization of triangle meshes over quadrilateral domains. In Proc. Eurographics/ACM Symp. on Geom. Proc. (2004), pp. 193203.

[CSM03] COHEN-STEINER D., MORVAN J.-M.: Restricted de-launay triangulations and normal cycle. In Proc. of Symp. onComp. Geom. (2003), ACM Press, pp. 312321.

[DBG06] DONG S., BREMER P.-T., GARLAND M., PASCUCCIV., HART J.: Spectral surface quadrangulation. ACM SIG-GRAPH (2006).

[DKG05] DONG S., KIRCHER S., GARLAND M.: Harmonicfunctions for quadrilateral remeshing of arbitrary manifolds.Comp. Aided Design 22 , 4 (2005), 392423.

[FK80] FARKAS H. M., KRA I.: Riemann Surfaces . SpringerVerlag, 1980.

[Ful95] FULTON W.: Algebraic Topology, A rst course . SpringerVerlag, 1995.[GY03] GU X., YAU S.-T.: Global conformal surface parameter-

ization. In Symp. on Geom. Proc. (2003), pp. 127137.[HP04] HILDEBRANDT K., POLTHIER K.: Anisotropic ltering

of non-linear surface features. Computer Graphics Forum 23 , 3(2004), 391400.

[HZ00] HERTZMANN A., ZORIN D.: Illustrating smooth sur-faces. In ACM SIGGRAPH (2000), pp. 517526.

[Jos02] JOST J .: Compact Riemann Surfaces . Springer, 2002.[KSS06] KHAREVYCH L. , SPRINGBORN B., SCHRDER P.:

Discrete conformal mappings via circle patterns. ACM Trans-actions on Graphics 25 , 2 (2006).

[LRL06] LI W.-C., RAY N., LVY B.: Automatic and interactivemesh to T-spline conversion. In Proc. Eurographics/ACM Symp.on Geom. Proc. (2006), pp. 191200.

[MK04] MARINOV M., KOBBELT L.: Direct anisotropic quad-dominant remeshing. In Proc. of the 12th Pacic Conf. on Comp.Graph. and Appl. (2004), pp. 207216.

[MK06] MARINOV M., KOBBELT L.: A robust two-step proce-dure for quad-dominant remeshing. Computer Graphics Forum25 , 3 (2006), 537 U546.

[PP03] POLTHIER K., PREUSS E.: Identifying vector eld singu-larities using a discrete Hodge decomposition. In Visualizationand Mathematics III . Springer Verlag, 2003, pp. 113134.

[RLL06] RAY N., LI W. C., LVY B., SHEFFER A., ALLIEZP.: Periodic global parameterization. ACM Trans. Graph. 25 , 4(2006), 14601485.

[RVLL06] RAY N., VALLET B., LI W.-C., LVY B.: N-symmetry direction elds on surfaces of arbitrary genus . Tech.rep., January 2006.

[TACSD06] TONG Y., ALLIEZ P., COHEN-STEINER D., DES-BRUN M.: Designing quadrangulations with discrete harmonicforms. In Proc. Eurographics/ACM Symp. on Geom. Proc.(2006), pp. 201210.


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