Jonathan Palacios and Eugene Zhang- Rotational Symmetry Field Design on Surfaces

8/3/2019 Jonathan Palacios and Eugene Zhang- Rotational Symmetry Field Design on Surfaces

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Rotational Symmetry Field Design on Surfaces

Jonathan Palacios

Oregon State UniversityEugene Zhang

Oregon State University

(a ) (b) (c) (d )

Figure 1: Pen-and-ink sketching of the Venus using four different elds: (a) the curvature tensor smoothed as a 2-RoSy eld, (b) the curvaturetensor smoothed as a 4-RoSy eld, (c) topological editing operations applied to (b), and (d) more global smoothing performed on (b). Noticethat treating the curvature tensor as a 4-RoSy eld (b) leads to fewer unnatural singularities and therefore less visual artifacts than as a 2-RoSyeld (a). In addition, both topological editing (c) and global smoothing (d) can be used to remove more singularities from (b). However,topological editing (c) provides local control while excessive global smoothing (d) can cause hatch directions to deviate from their naturalorientations (neck and chest).

Abstract

Designing rotational symmetries on surfaces is a necessary task fora wide variety of graphics applications, such as surface parameteri-zation and remeshing, painterly rendering and pen-and-ink sketch-ing, and texture synthesis. In these applications, the topology of arotational symmetry eld such as singularities and separatrices canhave a direct impact on the quality of the results. In this paper, wepresent a design system that provides control over the topology of rotational symmetry elds on surfaces.

As the foundation of our system, we provide comprehensive analy-sis for rotational symmetry elds on surfaces and present efcientalgorithms to identify singularities and separatrices. We also de-scribe design operations that allow a rotational symmetry eld tobe created and modied in an intuitive fashion by using the idea of basis elds and relaxation. In particular, we provide control overthe topology of a rotational symmetry eld by allowing the user toremove singularities from the eld or to move them to more desir-able locations.

e-mail:{

palacijo|zhange}

@eecs.oregonstate.edu

At the core of our analysis and design implementations is the obser-vation that N -way rotational symmetries can be described by sym-metric N -th order tensors, which allows an efcient vector-basedrepresentation that not only supports coherent denitions of arith-metic operations on rotational symmetries but also enables manyanalysis and design operations for vector elds to be adapted to ro-tational symmetry elds.

To demonstrate the effectiveness of our approach, we apply our de-sign system to pen-and-ink sketching and geometry remeshing.

CR Categories: I.3.5 [Computer Graphics]: ComputationalGeometry and Object ModelingGeometric algorithms, lan-guages, and systems;

Keywords: rotational symmetry, eld design, eld analysis, sur-faces, topology, non-photorealistic rendering, remeshing.

1 Introduction

Many objects in computer graphics can be described by rotationalsymmetries , such as brush strokes and hatches in non-photorealisticrendering, regular patterns in texture synthesis, and principle cur-vature directions in surface parameterization and geometry remesh-ing. Intuitively, an N-way rotational symmetry ( N -RoSy) representsphenomena that are invariant under rotations of an integer multipleof 2 N . Example N -RoSys include a vector ( N = 1), an eigenvectorof a symmetric matrix ( N = 2), and a cross ( N = 4).

Symmetries naturally appear in surfaces, such as the ve Platonicshapes (Figure 2). Notice that the order of the symmetry N is equalto the ratio between 2 and the angle of decit at a vertex. In sur-


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Figure 2: N -way rotational symmetries appear naturally in the Platonic solids: tetrahedron ( N = 2), octahedron ( N = 3), cube ( N = 4),icosahedron ( N = 6), and dodecahedron ( N = 10).

faces where global symmetry is lacking, local symmetries can stilloccur, such as the singularities (the vertices). In Figure 1 (c), sin-gularities of a 4-RoSy eld appear in natural places such as thecorner of the shoulder (not visible due to the highlight) and underthe armpit.

The ability to design and control N -RoSy elds on surfaces is es-sential in many applications. For example, in non-photorealisticrendering, the orientation of brush strokes and hatches are usu-ally guided by an N -RoSy eld. Different artistic effects can be

achieved by using guiding elds with different characteristics. Inaddition, singularities in the guiding eld can lead to visual arti-facts such as brush strokes and hatches with unnatural orientations.Singularities also present difculties in the construction of a globalsurface parameterization, where a signicant amount of stretch canoccur in nearby regions. Similarly, it is difcult to produce ideal tri-angular and quad elements near singularities in geometry remesh-ing. For these applications, a design system can be used to createa wide variety of N -RoSy elds on surfaces, to add desirable fea-tures in an existing eld, and to control the number and locationof the singularities in the eld. Most existing design systems focuson vectors ( N = 1) [Praun et al. 2000; Turk 2001; Wei and Levoy2001; Theisel 2002; Stam 2003; Zhang et al. 2006] and tensors( N = 2) [Zhang et al. 2007]. A number of difculties must be ad-dressed before a general design system can be developed for N 3.

First, there has been a lack of a mathematical representation forrotational symmetries, which is required to dene algebraic oper-ations on N -RoSys such as sums and scalar multiples as well asimportant concepts of N -RoSy elds such as singularities, continu-ity, and differentiability. We overcome this difculty by embedding N -way rotational symmetries in the space of N -th order tensors,which allows algebraic operations to be carried over from N -th or-der tensors to N -RoSys. We further derive a vector-based repre-sentation of rotational symmetries based on the embedding, whichenables efcient analysis of N -RoSy elds on surfaces as well asallows vector eld design functionalities to be adapted to N -RoSyelds. Furthermore, the concepts of singularities, continuity, anddifferentiability are well-dened.

Second, there is relatively little understanding of the topological

structure in an N -RoSy eld. While the concepts of singularitieshave been used before, a proper denition of separatrices is miss-ing when N 3. To address this, we present efcient algorithms onextracting the singularities and separatrices in a eld. In particular,we adopt the approach of Zhang et al. [2006] that allows contin-uous N -RoSy elds to be generated on mesh surfaces despite thediscontinuity in the surface normal.

With the above issues addressed, we present a design system for N -RoSy elds on surfaces that not only allows a wide variety of N -RoSy elds to be generated but also provides explicit control overthe number and location of the singularities in the eld. The main

functionalities of our work is reminiscent of that for vector eld de-sign [Zhang et al. 2006]. However, the implementations are ratherdifferent. For instance, we can create an initial N -RoSy eld on asurface using relaxation techniques [Turk 2001] which do not re-quire a global surface parameterization. This greatly increases theinteractivity of our system. We also reuse algorithms for singularitypair cancellation and movement in vector elds through the afore-mentioned vector-based representation.

We have applied our design system to graphics applications such as

pen-and-ink sketching and quad-dominant remeshing.In this paper, we make the following contributions.

1. We provide coherent denitions for algebraic operations on N -RoSys by observing the link between N -RoSys and N -thorder symmetric tensors. This link also enables the denitionsof analytic properties of N -RoSy elds such as continuity, dif-ferentiability, and singularity.

2. We present a vector-based representation for N -RoSys thatsupports compact storage of discrete N -RoSy elds on meshsurfaces and facilitates the analysis and design of N -RoSyelds.

3. We describe the topology of N -RoSy elds on surfaces andprovide efcient algorithms to extract singularities and sepa-ratrices.

4. We develop a design system in which N -RoSy elds canbe in-teractively created and modied on surfaces. In particular, oursystem provides explicit control over the number and locationof the singularities in the eld. We demonstrate the effective-ness of our system with pen-and-ink sketching of surfaces andquad-dominant remeshing.

The remainder of the paper is organized as follows. We rst reviewrelated work on N -RoSy elds in Section 2 and present our vector-based representation of N -RoSys in Section 3. Next, we discuss theanalysis and design of N -RoSy elds in Sections 4 and 5, respec-tively. In Section 6, we show the results of applying our analysisand design system to pen-and-ink sketching and geometry remesh-ing. Finally, we summarize our contributions and discuss some pos-sible future work in Section 7.

2 Previous Work

There has been a signicant amount of work in the analysis anddesign of vector and tensor elds. In contrast, relatively little isknown about N -RoSy elds when N 3.

N -RoSy Analysis and Design: To the best of our knowledge,Hertzmann and Zorin [2000] are the rst to propose that hatchesshould follow a cross eld (4-RoSy). They provide a smoothing


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operation on 4-RoSy elds that is based on a non-linear optimiza-tion. Ray et al. [2006a] construct a periodic global parameterizationthat facilitates quad remeshing. At the heart of their approach is theuse of 4-RoSy elds, which allows more natural-shaped quads to begenerated near singularities. They also develop a framework thatallows the optimization to be performed on the sines and cosinesof the parameterization, which turns a non-linear optimization intoa linear one. They later provide the analysis of singularities on N -RoSys by extending the Poincar e-Hopf theorem as well as describe

an algorithm in which a eld with a minimal number of singulari-ties can be constructed based on user-specied constraints and theEuler characteristic of the underlying surface [Ray et al. 2006b].

Vector and Tensor Field Design: There have been a number of vector eld design systems for surfaces. Most of them are gen-erated for a particular graphics application such as texture synthe-sis [Praun et al. 2000; Turk 2001; Wei and Levoy 2001], uid sim-ulation [Stam 2003], and vector eld visualization [van Wijk 2002;van Wijk 2003]. These systems do not address topological controlin the eld. Systems providing topological control include [Theisel2002; Zhang et al. 2006]. The system of Zhang et al. has also beenextended to create periodic orbits [Chen et al. 2007] and to designtensor elds [Zhang et al. 2007]. Our system is also reminiscent of the vector eld design system of Zhang et al. [2006] in terms of thefunctionality. However, the implementation is rather different.

Vector and Tensor Field Analysis: There has been much work in vector and tensor eld analysis. To review all the work is be-yond the scope of this paper. Here we refer to the most relevantwork. Helman and Hesselink [1991] propose vector eld visu-alization techniques based on topological analysis. Delmarcelleand Hesselink provide analysis of second-order symmetric tensorelds [1994].

3 Vector-Based Representation

The analysis and design of N -RoSy elds require coherent den-itions of the following concepts: summation and scalar multipli-cation of N -RoSys as well as continuity, singularity, and differen-tiability. However, it is not immediately clear how to dene theseconcepts in a consistent manner given that a non-zero N -RoSy sconsists of N directions. For convenience, we will refer to thesedirections the member vectors of s. Consider the case when N = 2,i.e., lines that can be modeled by eigenvectors of a symmetric ma-trix with a zero trace [Zhang et al. 2007]. Given two 2-RoSys

s1 = {10 } and s2 = {

01 }, dening s1 + s2 as the sum of

member vectors can lead to inconsistent results: (1) { 11 }, or

(2) { 1 1 }. As demonstrated by Zhang et al. [2007], treating

a line eld as a vector eld results in discontinuities and inconsis-tencies. This is also true for N -RoSy elds when N 3.

To overcome this problem, we describe a representation for N -RoSys that is free of directional ambiguity. For an N -RoSy

s = { Rcos ( +2k N )

Rsin ( + 2k N )| 0 k N 1} (1)

where R 0 is the strength of s and is the angular component of one of the member vectors, we dene the representation vector of

s as Rcos N Rsin N . Notice that the representation vector is indepen-

dent of the choice of member vectors since

Figure 3: A comparison between a 4-RoSy eld (left) and its rep-resentation vector eld (right). Notice that the sets of points with azero value are the same for both elds (colored dots). Representa-tion vectors remove the directional ambiguity in an N -RoSy eld.

N ( +2k N

) N mod 2 (2)

for any integer k . Consequently, directional ambiguity no longerexists under this representation. We will dene a map which mapsan N -RoSy s to its representation vector (s). Given two N -RoSyss1 and s2 and a real number , we dene

s1 + s2 = 1

( (s1)+ (s2)) , s1 = 1

( (s1)) (3)These denitions are coherent as they do not rely on the choice of member vectors, which allow us to dene important concepts on N -RoSy elds, such as continuity, singularity, and differentiabilityin a consistent manner.

A representation vector and a member vector differ in how theircomponents change under a change of basis. Consider the casewhere the transformation matrix from the new basis to the original

basis has the form Q = cos sin sin cos . While a member vector

v will have the form Qv under the new basis, a representation vec-

tor w will be of the form Q w where Q = cos N sin N sin N cos N .

Therefore, a representation vector is not a vector since its compo-nents do not change like a vector under changes of basis. In theAppendix, we show that N -RoSys can be represented by symmet-ric N -th order covariant tensors of a special form and such tensorscan be compactly represented by a vector (representation vector)under any given basis. Consequently, the components of a repre-sentation vector transform differently than a vector under a changeof basis.

As will become apparent soon, not only provides a coherent andcompact representation of N -RoSys, but also allows efcient im-plementations of key operations on N -RoSy elds by borrowingcorresponding algorithms for vector elds, such as interpolationsand singularity extraction (Section 4.3), blending (Section 5.1), andtopological editing (Section 5.2). Figure 3 shows a 4-RoSy eld(left) and its representation vector eld (right). Notice that they

have the same set of singularities. Next, we discuss the analysisand design of N -RoSy elds on surfaces.

4 Topological Analysis of N -RoSy Fields

In this section, we describe important topological properties of N -RoSy elds on manifold surfaces, such as singularities and separa-trices . We will also present efcient algorithms to compute thesequantities on mesh surfaces.

Singularity identication is necessary for providing explicit con-trol over the number and location of singularities, which is needed


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Figure 4: This gure illustrates the importance of singularity andseparatrix extraction in quad-dominant remeshing. Given a 3Dmodel (left), the singularities and separatrices are highlighted bycolored dots and curves, respectively. Including separatrices duringremeshing can lead to better meshes near the singularities (right)than not including them (middle).

in pen-and-ink sketching of 3D surfaces as undesirable singulari-ties cause visual artifacts [Hertzmann and Zorin 2000] (Figures 1and 7). Singularity and separatrix extraction allow better meshingnear singularities during the remeshing process (Section 6.2). Fig-ure 4 illustrates this with a 3D surfaceobtained by joining two cubeswith rounded corners. In the left, the singularities are highlightedby colored dots and the separatrices are the colored curves emanat-

ing from the singularities. Notice that singularities appear in nat-ural places (corners and joints) and separatrices indicate importantdirections near the singularities. Utilizing separatrices during theremeshing process produces meshes that can better maintain fea-tures in the original mesh (Figure 4, right) than disregarding them(Figure 4, middle).

The concepts of singularities and separatrices are well dened forvector and tensor elds, i.e., when N = 1, 2. Next, we will extendthem to N -Rosy elds when N 3.

4.1 Singularities

For simplicity, consider a planar vector eld V ( x, y) = F ( x, y)G( x, y) .

A singularity p 0 is a point in the domain where V (p 0) = 0. p 0is isolated if there exists an open neighborhood U of p 0 with theproperty that p0 is the unique singularity in the interior of U andthere are no singularities on the boundary of U . An isolated singu-

larity p0 can be characterized by its Jacobian DV (p 0) =a bc d ,

where a = F x (p 0), b = F y (p 0), c =

G x (p 0), and d =

G y (p 0).

The local linearization of V at a point p 0 is a function LV (p ) = DV (p0)(p p 0).

We now consider how the angular component of a vector eldchanges on an innitesimal circle centered at p 0 . Given a

point p , we have p = p 0 +r cos r sin where (r , ) are the

polar coordinates. The local linearization at p 0 is LV (p ) =

r a cos + b sin c cos + d sin . The angular component is

tan 1(c cos + d sin a cos + b sin

) (4)

which has a derivative

ad bc(a cos + b sin )2 + ( c cos + d sin )2

(5)

When ad bc = 0, the sign of the quantity ad bc is related tothe Poincar e index of p 0 , which is dened in terms of the windingnumber for the Gauss map .

Let V be a continuous planar vector eld and D0 R 2 be thezero set for V . The Gauss map : R 2 \ D0 S1 is dened as ( x) = V ( x)|V ( x)| . is continuous in R

2 \ D0 . In particular, it introducesa continuous map | on any simple loop that does not containany singularity. When traveling along in the positive directiononce, the image under | necessarily covers the unit circle S1 aninteger number of times counting orientation. This integer is thewinding number of V along . The Poincar e index of an isolatedsingularity p 0 is the winding number of any simple loop that en-closes p 0 and contains no other singularities either in its interior oron the boundary. Denote this number as (V ; p 0). The Poincar e in-dex is + 1 for sources, sinks, centers, and foci. It is 1 for saddles,and 0 for regular points.

The Poincar e-Hopf theorem links the topology of a vector eld tothat of the underlying domain in the following way. Let M be aclosed orientable manifold with an Euler characteristic ( M ). Fur-thermore, let V be a continuous vector eld dened on M with onlyisolated singularities p 1 , ..., p n . Then

n

i= 1

(V ; p i) = ( M ) (6)

.

We now extend the concepts of singularities to N -RoSy elds where N 2. Given a planar N -RoSy eld S, we dene V S(p ) = (S(p)) asthe representation vector eld of S. Then, a point p 0 is an isolatedsingularity of S if and only if it is also an isolated singularity of V S.We dene the index of a singularity p0 with respect to S as

(S; p 0) = (V S; p 0)

N (7)

A singularity p 0 is of L-th order if its index is | L N | . p 0 is positive ,negative , or regular if (S; p0) > 0, (S; p 0) < 0, or (S; p 0) = 0,respectively. As in the case of vector elds and tensor elds, an L-th order singularity can be constructed by clustering L rst-ordersingularities [Scheuermann et al. 1998; Delmarcelle and Hesselink 1994]. Assuming an N -RoSy eld has rst-order singularities only, N | ( M )| provides a lower bound on the number of singularitiesin the eld. While more singularities can occur, their signed sumremains constant, i.e., ( M ).

Figure 2 shows natural symmetries on the Platonic solids. To un-derstand what symmetry is natural in a shape, let us consider thetotal angle around a vertex of the cube, which is 3 2 . In order to ad-mit continuity across such a corner, one must accept a rotation of 2 ,which is essentially the case when N = 4. Similarly, an octahedronnaturally admits 3-RoSys, an icosahedron admits 6-RoSys, a do-decahedron admits 10-RoSys, and a tetrahedron admits 2-RoSys(tensors). In all these cases, there are K rst-order positive singular-ities where K is the number of vertices in the shape. Furthermore,the total signed index sum is 2, which is the Euler characteristic of agenus-zero surface. Let be the angle of decit of a vertex, whichequals 2 minus the total angle around the vertex. = 0 impliesthat the neighborhood of the vertex is likely to admit an N = 2 | |symmetry. When > 0, the vertex is a positive singularity, andwhen < 0, the vertex is a negative singularity. Next, we discussthe separatrices in an N -RoSy eld.


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4.2 Separatrices

A separatrix in a vector eld is a trajectory that passes through asaddle. The topology of a vector eld on a two-dimensional surfaceconsists of singularities and separatrices. Extending the denitionof separatrices to N -RoSy elds when N 2 is more difcult be-cause separatrices can emanate from singularities with a positiveindex. To address this, Delmarcelle and Hesselink [1994] deneseparatrices for a tensor eld (2-RoSy) as the boundaries of a hy-

perbolic sector , which is a region in the vicinity of a singularityinside which trajectories sweep pass the singularity. As an exam-ple, there are four hyperbolic sectors for every saddle.

We adopt this approach and dene separatricesof an N -RoSy eld Sas the boundary of a hyperbolic sector for a singularity. Figure 5 (a-d) show the four hyperbolic sectors for a positively-indexed singu-larity in a 3-RoSy eld, and (e) their composition. The red lines areincoming separatrices and the green lines are outgoing separatri-ces . In (f) we show the separatrices of a negatively-index singularityin a 4-RoSy eld. When N is even, separatrices do not have direc-tions. Notice when N 3, hyperbolic sectors can overlap, which isa reection of the fact that there are N member vectors. When N is even, there are N 2 streamlines passing through every non-singularpoint. When N is odd, there are N such streamlines.

To extract separatrices, we make use of the following observation:a separatrix approaches a singularity in a radial direction. In otherwords, given an isolated singularity p 0 , we consider its local lin-

earization DS = a bc d , which is dened in terms of the lin-

earization of the representation vector eld. On an innitesimalcircle centered at p0 , we consider directions v = p p 0 such that vis also a member vector at p . Let be the angular coordinate of oneof the member vectors. The aforementioned statement is equivalent

to nding directions v = cos sin such that

N N mod 2 (8)

which can be used to nd a separatrix that passes through the singu-

larity. Note that this is consistent with the denitions of separatricesin vector elds [Tricoche 2002] and tensor elds [Delmarcelle andHesselink 1994]. The condition can be rewritten as

sin N cos N

=c cos + d sin a cos + b sin

(9)

Recall that

cos N = N

i= 0

cosi 2

N i cos

N i sin i (10)

sin N = N

i= 0

sini

2

N

icos N i sin i (11)

.

Consequently, solving Equation 9 amounts to nding the roots of an ( N + 1)-th order polynomial.

A rst-order negative singularity has N + 1 separatrices while arst-order positive singularity has at most N + 1 separatrices. Notethat the above denition extracts all the separatrices in an N -RoSyeld S only when N is even. In case N is odd, the solutions to Equa-tion 9 correspond to the outgoing separatrices only. To capture theincoming separatrices, we need to compute the solutions to

(a ) (b) (e)

(c) (d ) ( f )

Figure 5: The separatrices of an N -RoSy eld are the boundariesof the hyperbolic sectors in the vicinity of a singularity. In (a)-(d),we show the four hyperbolic sectors for a positive singularity of a 3-RoSy eld. These sectors overlap and each of them has an in-coming separatrix (red line) and an outgoing separatrix (green line).

Together, they describe the topology of the eld near the singularity(e). In (f), we show the separatrices for a negative singularity in a4-RoSy eld. Notice that separatrices do not have directions when N is even.

N N + mod 2 (12)

which is equivalent to computing the separatrices of S. Next, wedescribe how we represent an N -RoSy eld on a triangular meshsurface and how to extract the topological features in the discretesetting.

4.3 Discrete Representation

On a triangular mesh, we use a vertex-based representation for an N -RoSy eld S. In such a setting, the values of S are dened atthe vertices and interpolations are used to obtain values on edgesand inside triangles. Note that in practice we use the representationvector eld V S instead of S itself.

When the triangular mesh represents a planar domain, we use thepopular piecewise linear interpolation scheme of vector elds [Tric-oche 2002]. Basically, inside every triangle the representa-tion vector eld V S is linear and therefore can be expressed

asax + by + ecx + dy + f under the global coordinate systems. Here,

a bc d is the Jacobian of the representation vector eld, and it

is constant inside every triangle. This representation supports ef-cient singularity and separatrix extraction. We perform the follow-ing steps in computing the topology of an N -RoSy eld.

1. We locate the singularities of the representation vector eldusing the method in [Tricoche 2002] and compute the lin-earization, which is constant for each triangle.

2. We extract separatrix directions by solving Equation 9 forevery singularity.

3. For each separatrix direction w, we perform streamline tracing


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from a point sufciently close to the singularity in the direc-tion of w.

On meshes that represent curved surfaces, the above piecewise lin-ear scheme no longer produces continuous N -RoSy elds. We re-fer the readers to [Zhang et al. 2006] for an example when N = 1.Furthermore, a curved surface in general lacks a global parameter-ization and consistent local frames. To overcome these problems,Zhang et al. [2006] develop a non-linear interpolation scheme thatproduces continuous vectors elds and supports efcient singular-ity and separatrix extraction. Their scheme is based on the ideas of geodesic polar maps and parallel transport from differential geom-etry. Extending this scheme to N -RoSy elds is straightforward,and it has been done when N = 2 [Zhang et al. 2007]. Recently,Wang et al. [2006] have proposed another scheme based on edgesubdivision and discrete differential forms. Adapting their schemeto N -RoSy elds is also promising.

5 Design of N -RoSy Fields

In this section, we describe our design system for N -RoSy eldson 3D surfaces. In our approach, the user can create a eld eitherfrom scratch or by modifying an existing eld, such as the curva-ture tensor. Our system allows a user to add features to the eld,to remove unwanted singularities, and to relocate singularities tomore natural locations. These functionalities can be used in pen-and-ink sketching to reduce visual artifacts caused by singularities(Section 6.1) and in geometry remeshing to maintain geometry de-tails (Section 6.2).

Our design system employs a two-stage pipeline: initialization andediting. In the rst stage, the user quickly creates or modies an N -RoSy eld through a set of constraints called design elements .Design elements can be in the forms of desired orientations or sin-gularities at a given point. The eld obtained this way often con-tains unwanted or misplaced singularities, which can be handledin the second stage through editing operations such as singularitypair cancellation and movement. This pipeline is consistent withthe design systems for vector elds [Zhang et al. 2006] and ten-sor elds [Zhang et al. 2007] when N = 1 and N = 2, respectively.

Given the intrinsic connection between an N -RoSy and its repre-sentation vector, our design system can be adapted from a vectoreld design system such as [Zhang et al. 2006]. Next, we describeour implementations for both stages.

5.1 Initialization

There have been a number of techniques for creating a vector eldon a 3D surface, such as blending basis elds on surfaces [Zhanget al. 2006] or in 3D [van Wijk 2003], relaxation [Turk 2001; Weiand Levoy 2001], and propagation [Praun et al. 2000]. These ap-proaches provide tradeoffs among a number of factors such as con-trollability, interactivity, and ease of use. For planar domains, theidea of using basis elds is highly desirable due to its simplicity andintuitiveness [van Wijk 2002]. We employ this approach to create

an N -RoSy eld S in the plane by creating its representation vectoreld V S. Basically, given a set of constraints C = {c1 , ..., ck }, wedene V S = k i= 1 V Si where

V Si ( , ) = e d 2 cos N ( L N + 0)

sin N ( L N + 0)(13)

In the above equation, ( , ) are the polar coordinates of a point( x, y) with the center of ci being the origin. 0 is the phase shiftconstant that can turn a source into a sink or a center when N = 1. L is the order of the design element. When L = 0, Equation 13 leads

to a eld of a constant direction. When L = 0, the design elementis either a positive or negative L -th order element. Finally, d is aconstant that is used to describe the falloff speed in the strength of V Si . Such a falloff function enables fast blending of basis elds sothat a user-desired pattern is not affected by other basis elds [vanWijk 2002].

While our system can be used to generate singularities of arbitraryorders, we are primarily interested in the cases when L = 0 or L =

1 since an L-th order element can be simulated by combining Lrst-order elements.

On surfaces, the idea of blending basis vector elds encounters a se-rious difculty, i.e., Equation 13 requires a global parameterization,which is often unavailable for a 3D surface. Zhang et al. [2006]address this problem by computing a global parameterization withrespect to each design element. However, this approach is ratherexpensive. Van Wijk [2003] creates volume basis vector elds be-fore projecting them onto the tangent planes. While this approachis fast, it is difcult to achieve local patterns such as sources, sinks,and saddles due to surface curvature.

Relaxation techniques such as [Turk 2001; Wei and Levoy 2001]provide a nice balance between controllability and computationalcost. In this approach, vector values are dened at a (small) num-ber of vertices. Then, values elsewhere are obtained by solvinga Laplace equation on each of the three components of the vectoreld with the specied vector values being the boundary conditions.While being fast and intuitive, two issues must be addressed beforewe can use such an approach to create N -RoSy elds on surfaces.First, how do we automatically generate vector values so that theresulting N -RoSy eld contains the desired singularities. Second,directly performing relaxation on the representation vector eld V Sis likely to produce undesired results as V S depends on the localframe. Yet, without a global parameterization, local frames at thevertices are typically uncorrelated.

The rst issue can be resolved relatively easily. Given a designelement whose L = 1, 0, 1, we can compute the N -RoSy valuesat the three vertices of the triangle containing the design element.

When | L| > 1, the support of the element is larger than one triangle.In practice, we nd it sufcient to only specify zeroth- and rst-order design elements as an L-th order element can be created bydesigning L rst-order elements.

The second issue is more challenging, and it is related to the con-cept of parallel transport from differential geometry. Consider twopoints p and q on the surface M and a geodesic : [0, 1] M suchthat (0) = p and (1) = q . Given two vectors vp and vq that aredened at p and q respectively, vp is said to be equivalent to vq withrespect to if the angle between vp and (p) is equal to the anglebetween vq and (q ). Recall that the shortest geodesic betweenthe two incident vertices is the edge connecting them. This allowsus to set up the Laplace equation 2V = 0 and its discrete form ona mesh surface,

wi = j J

i j T i j (w j) (14)

where wi is the representation vector value at vertex vi , i j is themean value coordinate [Floater 2003] and T i j is the transport func-

tion from the tangent plane at vertex v j to vertex vi . LetF iG i

be

the coordinates of wi under the local frame at vertex vi . Equation 14has the following more explicit form:


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F iG i

= j J

i jcos N i j sin N i jsin N i j cos N i j

F jG j

(15)

where i j is the difference between the angle from the X -axis tothe geodesic at q and that from the X -axis to the geodesic at p .

Equation 15 is a sparse linear system that can be solved efcientlyby a bi-conjugate gradient solver.

There have been a number of other relaxation methods for smooth-ing a 4-RoSy eld that are similar to our formulation in spirit.Hertzmann and Zorin [2000] dene a non-linear functional. Whilehigh-quality, it is more time-consuming than linear optimization ap-proaches. Ray et al. [2006a] set up a linear system by performingsmoothing on member vectors. As discussed in Section 3, smooth-ing (adding) member vectors may lead to incoherent results.

Note that none of the aforementioned techniques including relax-ation provide explicit control over the number and location of thesingularities in the eld. Consequently, unwanted or misplaced sin-gularities can occur. We address this issue through topological edit-ing operations such as singularity pair cancellation and movement,to be described next.

5.2 Editing

Topological control over an N -RoSy eld requires the ability to per-form local modication to the eld near a singularity. Our systemoffers two topological editing operations: singularity pair cancel-lation and singularity movement . Singularity pair cancellation al-lows a rst-order positive singularity to be cancelled with a nearbyrst-order negative singularity. Note that a singularity cannot beremoved by itself due to the topological constraints imposed by thesurface. Singularity movement allows a singularity to be moved toa more desirable location. Given the vector-based representation of an N -RoSy eld, we perform both operations on its representationvector eld, which allows us to reuse corresponding vector eldediting algorithms such as [Zhang et al. 2006].

In their framework, singularity pair cancellation and movement op-erations are implemented in a unied fashion that is based on Con-ley index theory from dynamical systems [Mischaikow and Mrozek 2002]. According to this theory, to cancel a singularity pair in avector eld that have opposite Poincar e indices, one can systemati-cally nd a region that encloses the singularity pair without cover-ing any other singularities. By construction, this region will havea trivial Conley index, which means that it is possible to alter theow inside the region such that no singularity exists after the mod-ication. Singularity movement is performed similarly. Zhang etal. [2006] provide practical algorithms to compute the region andperform vector-valued Laplacian smoothing in order to modify theow inside. We refer the readers to [Mischaikow and Mrozek 2002]for details on Conley index theory and to [Zhang et al. 2006] for theimplementation details on topological editing on vector elds. Fig-ure 6 illustrates topological editing operations on a 4-RoSy eld

that contains two positive singularities and one negative singular-ity (left). First, the negative singularity is cancelled with a posi-tive singularity (middle). Next, the remaining singularity is moved(right). The actual pair cancellation and movement operations areconducted on the representation vector elds.

Performing singularity pair cancellation and movement on surfacesrequires the ability to conduct Laplacian smoothing on the repre-sentational vector eld inside a region. To avoid the use of surfaceparameterization that is computationally expensive, we reuse theidea of parallel transport discussed in Section 5.1, which allows re-laxation to be carried out on surfaces.

Figure 6: This gure illustrates the topological editing operations of our system. The elds shown in the images are: a 4-Rosy eld withthree singularities (left), after singularity pair cancellation (middle),and after singularity movement (right). The actual editing was per-formed on the representation vector elds, which allows us to reusecorresponding vector eld editing operations.

6 Applications

We have applied our design system to pen-and-ink sketching andquad-dominant remeshing.

6.1 Pen-and-ink Sketching of Surfaces

Pen-and-ink sketching of surfaces is a non-photorealistic style of shape visualization. The efciency of the visualization as well asthe artistic appearance depend on a number of factors, one of whichis the direction of hatches. Girshick et al. [2000] show that 3Dshapes are best illustrated if hatches follow principle curvature di-rections. However, curvature estimation on discrete surfaces is achallenging problem. While there have been several algorithmsthat are theoretically sound and produce high-quality results [Hertz-mann and Zorin 2000; Meyer et al. 2002; Cohen-Steiner and Mor-van 2003; Rusinkiewicz 2004], most of them still rely on smooth-ing to reduce the noise in the curvature estimate. Consequently,these methods do not provide control over the singularities in theeld. Hertzmann and Zorin [2000] propose the concept of crosselds, which are 4-RoSy elds obtained from the curvature tensor(a 2-RoSy eld) by removing the distinction between the major and

minor principle directions. They demonstrate that smoothing onthe cross eld tends to produce more natural hatch directions thansmoothing directly on the curvature tensor. They also point out theneed to control the number and location of the singularities in theeld. Zhang et al. [2007] address this issue by providing singularitypair cancellation and movement operations on the curvature tensoreld. However, their technique cannot handle a 4-RoSy eld.

In this paper, we follow Hertzmann and Zorin [2000] by treatinghatch directions as a 4-RoSy eld and use topological editing oper-ations to control the number and location of the singularities. Fig-ure 7 illustrates the utility of topological editing operations with theBimba model. The original 4-RoSy eld (left) was obtained fromthe curvature tensor, which we computed using the algorithm of Meyer et al. [2002]. This eld contains three singularities on thevisible side of the face, which cause visual artifacts in the result.

Two of them (on the lower side of the face near the neck) were re-moved through singularity pair cancellation (middle), and the thirdone (near the corner of the right eye) was moved within the high-light region (right).

Figure 1 provides the following comparisons with the Venus model:2-RoSy (a) versus 4-RoSy (b), and topological editing (c) versusglobal smoothing (d). Representing the curvature tensor as a 4-RoSy eld leads to smoother results. Notice the visual artifactscaused by the singularities on the chest in (a). The hatch directionsin those regions are more natural with 4-RoSys (b). In (c) and (d),we compare topological editing and global smoothing that can both


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Figure 7: Topological editing operations were applied to pen-and-ink sketching of the Bimba model in order to remove visual artifacts causedby undesirable singularities. The original 4-RoSy eld (left) contains a number of such singularities on the visible side of the face (left). Twoof them (on the lower side of the face near the neck) were removed through singularity pair cancellation (middle). Next, a singularity nearthe corner of the right eye was moved to reduce the amount of discontinuity in the hatching directions near the eye.

be used to further reduce the visual artifacts caused by unwantedsingularities. Compare these images to (b) near the chest and un-der the armpit. Furthermore, topological editing operations providelocal control that is lacking in global smoothing. Notice that ex-cessive global smoothing can lead to signicant deviations in thehatch directions. Compare (b) and (d) around the neck and on thechest. Topological editing operations, on the other hand, preservecurvature directions in regions where topological editing was notperformed. See the same regions in (c).

6.2 Quad-Dominant Remeshing

The problem of quad-dominant remeshing, i.e., constructing aquad-dominant mesh from an input mesh, has been a well-studiedproblem in computer graphics. The key observation is that a nicequad-mesh can be generated if the orientations of the mesh ele-ments follow the principle curvature directions [Alliez et al. 2003].This observation has led to a number of efcient remeshing algo-rithms that are based on streamline tracing [Alliez et al. 2003; Mari-nov and Kobbelt 2004; Dong et al. 2005]. Ray et al. [2006a] notethat better meshes can be generated if the elements are guided by a4-RoSy eld. They also develop an energy functional that can beused to generate a periodic global parameterization and to performquad-based remeshing. The connection between quad-dominantremeshing and 4-Rosy elds has also inspired Tong et al. [2006]to generate quad meshes by letting the user design a singularitygraph that resembles the behavior of the topological skeleton of a4-RoSy eld. On the other hand, Dong et al. [2006] perform quad-remeshing using spectral analysis, which produces quad meshesthat in general do not align with the curvature directions.

We apply our design system to a 4-RoSy eld that is obtained fromprinciple curvature directions by not distinguishing between themajor and minor directions. Our method is based on streamlinetracing. In contrast to most existing approaches, we rst trace allthe separatrices for a certain distance. This allows singularities tobe the vertices in the new mesh and that the nearby regions con-sist of nice quad elements (Figure 8). Notice that this would nothave been possible without the analysis of N -RoSy elds. In addi-tion, 4-RoSy eld design enhances the chance of obtaining a bettermesh by removing noise and placing singularities in natural loca-tions. Figure 8 illustrates this with two examples elds, both wereobtained by editing a 4-RoSy eld corresponding to the curvature

tensor. The top eld corresponds to a sequence of global smooth-ing, which tends to lose sharp features in the model and no longerfollows the principle curvature directions. The bottom eld, on theother hand, was obtained through a sequence of singularity editingoperations that allow singularities to be edited in an isolated fash-ion, thus maintain sharp features and natural singularities.

We wish to emphasize that the focus of our work is on the analy-sis and design of N -RoSy elds. While we have employed astreamline-based remeshing approach [Alliez et al. 2003] to demon-strate the capabilities of our system, elds designed using our sys-tem can potentially be input to other and more recent remeshingtechniques such as [Ray et al. 2006a].

7 Conclusion

In this paper, we have developed a design system for N -RoSy eldson surfaces with explicit control over the number and location of singularities in the elds. We demonstrate the effectiveness of oursystem with applications in non-photorealistic rendering and quad-dominant remeshing.

As part of our system, we describe a mathematically sound rep-resentation for N -RoSys that is based on higher-order symmetrictensors. This link allows us to dene algebraic operations as wellas analytic characteristics such as singularities, differentiability, andcontinuity. Furthermore, we present a compact vector-based repre-sentation of N -RoSy elds on mesh surfaces that makes it possibleto perform eld smoothing using linear systems.

We also provide topological analysis of N -RoSy elds and efcientalgorithms with which singularities and separatrices can be identi-ed.

In the future, we plan to investigate the use of 6-RoSy eld de-sign to obtain optimal triangulations from an input mesh. Figure 9demonstrates this potential with a user-guided 6-RoSy eld on thedragon. Notice that the streamlines according to eld, while notaligned perfectly, intersect at angles that are multiples of 6 . Adap-tation of quadrangulation methods such as [Ray et al. 2006a; Tonget al. 2006] has the potential of generating high-quality triangularmeshes.

In many applications, multiple, coexisting symmetries of differ-


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Figure 8: 4-RoSy eld design was applied to quad-dominantremeshing. The eld in the top was obtained by performingsmoothing on a 4-RoSy eld derived from the curvature tensor,whilethe eld in the bottom was obtained by performing a sequenceof topological editing operations on the same tensor eld. Noticethat while both elds are smooth, the eld in the bottom containssingularities that situate in natural locations such as near the leg. Incontrast, the eld in the top is overly smooth and does not followthe curvature directions in many parts of the model.

ent N s are of interest, such as pen-and-ink sketching and texturesynthesis. We plan to investigate a mathematical framework withwhich such objects can be analyzed and designed.

Appendix: N -RoSys and N -th Order Tensors

In this appendix, we point out the link between N -RoSys and N -thorder covariant symmetric tensors, which justies the way in whichthe components of a representation vector transform under a changeof basis (Section 3).

Recall that when N = 1 and 2, an N -RoSy can be modeled as avector or a symmetric traceless matrix [Zhang et al. 2007], respec-tively. When N 3, we turn to N -th order tensors and dene aninjective map from the set of two-dimensional N -RoSys into theset of two-dimensional N -th order symmetric tensors. allows al-gebraic operations on N -RoSys to be dened in terms of corre-sponding operations on N -th order tensors. Before describing ,we briey review relevant facts about N -th order tensors. Given anothonormal basis for R 2 , an N -th order covariant tensor t = t i1 ... i N (1 i1 , ..., i N 2) is a multi-linear form

t (w1 , ..., w N ) = t i1 ... i N wi11 ... w

i N N (16)

where w js (1 j N ) are two-dimensional vectors and wik k refersto the ik -th component of wk . Here we are using the Einstein con-vention in which the summation signs are omitted. t is symmetricif t i1 ... i N = t i p(1) ... i p( N ) for any permutation p over the set {1, ..., N }.

Figure 9: A 6-RoSy eld designed using our system. Notice thatthe network of streamlines resembles a highly regular triangulation.

In the remainder of the discussion we will only focus on covariant

and symmetric tensors and therefore omit the words symmetric andcovariant for convenience.

Given an N -RoSy

s = { R cos ( +2k N )

sin ( + 2k N )| 0 k N 1} (17)

we consider the following N -th order tensor

t i1 ... i N =

Rcos ( N ) if i1 + i2 + .. + i N 0 mod 4 Rsin ( N ) if i1 + i2 + .. + i N 1 mod 4

Rcos ( N ) if i1 + i2 + .. + i N 2 mod 4 Rsin ( N ) if i1 + i2 + .. + i N 3 mod 4

(18).

Given a vector w = cos sin , it is straightforward to verify that

t (w, ..., w) = R cos N ( ). Furthermore, it can be shown that t isthe only tensor that has this property. t (w, ..., w) achieves its max-imum on the unit circle S1 when 0 mod 2 N . There are N such vectors, which are exactly the member vectors of s. Thisallows us to map s to t = (s) as dened in Equation 18. When N = 1, s is a vector and t = s is also a vector. When N = 2, t is thetraceless symmetric matrix whose major eigenvalue is R and majoreigenvectors are given by the member vectors of s.

Notice that tensors dened in Equation 18 form a two-dimensionallinear subspace of the set of N -th order tensors, which leads to the

following construction of a bijective map from the subspace to atwo-dimensional vector space such that

(t i1 ... i N ) =t 00 ... 0t 10 ... 0

= Rcos N Rsin N (19)

The composite map = has been used to map an N -RoSy toits representation vector as described in Section 3. As a compactrepresentation of an N -th order tensor, the components of a repre-sentation vector transform differently than a vector under a changeof basis.


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Acknowledgements

We would like to thank the following people and groups for the3D models they provided: Mark Levoy and the Stanford GraphicsGroup, Cyberware, and the AIM@SHAPE Shape Repository. Thediscussions with Greg Turk have been valuable. We are also grate-ful for the help of Patrick Neill and Guoning Chen in preparing thevideo production. Finally, we wish to thank our reviewers for theirexcellent comments and suggestions. This work is funded by NSF

grant CCF-0546881.

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