Pattern Oriented Remeshing For Celtic Decoration

Matthew Kaplan Emil Praun Elaine CohenUniversity of Utah

Abstract

Decorating arbitrary meshes with Celtic knots requirespolygonal meshes with regular connectivity and close toregular face geometry. Because these properties are oftenirregular, especially in scanned or reconstructed models,the Celtic knots produced may be erratic and undesireable.In this paper we remesh models based on planar tilings de-fined by the user. Such pattern-oriented surfaces allow usto decorate models with attractive Celtic knots in a consis-tent fashion and may be applicable to a large number of al-gorithms that are sensitive to mesh structure.

1. Introduction

Object decoration has been a primary outlet for artisticexpression in many cultures yet the research towards com-puter generated decorations has not been proportional to itsinfluence. Computer graphics scientists have typically con-centrated their efforts on more traditional media such aspen-and-ink and painting. The decoration of objects withCeltic knotwork (a form of artwork that consists of rhyth-mically interwoven threads) has become widespread in re-cent years. It is seen frequently in jewelry, tattoos and as or-nament on many everyday items.

The goal of this work is to decorate the surfaces of 3Dmeshes with Celtic knotwork. Kaplan and Cohen [12] de-scribe methods to produce Celtic artwork in both 2D and3D. The edges of a planar graph (in 2D) and the 2D mani-fold mesh (in 3D) define the pattern of the output knots. Inearlier work, allowing the user to design repetitive tilingsof geometric shapes was shown to be the key to produc-ing attractive knots. Designing tilings is easy for a planarsurface but is extremely difficult for surfaces which are 2Dmanifold. Therefore, in 3D, the edges of the original in-put meshes were used to compute the knots. In such in-put meshes the geometric properties such as face structureand edge connectivity are typically arbitrary and the resul-tant knots seemed random and lacked the repetitive stylizedquality that makes Celtic knotwork attractive, as seen inFigure 1. Therefore, we desire a process by which the user

can design infinite planar tilings in 2D and transfer suchtilings onto 3D meshes for the calculation of Celtic orna-ment.

We propose a remeshing technique that imposes shapeand connectivity contraints on the resulting mesh. Our tech-nique samples the original mesh with a pattern defined by aplanar tiling thus allowing us to transfer designs in a planeto the mesh surface. The Celtic knots computed on suchmeshes accurately transfer the original artistic intent of theuser in a way not previously possible. Furthermore, theremay be a large class of artistic algorithms that can benefitfrom such meshes. The resulting meshes may be more at-tractive and artistic, due to the patterns embedded on thesurface, than meshes produced using other techniques.

1.1. Approach Overview

Given a surface mesh we obtain a spherical parameteri-zation of the surface as described by Praun and Hoppe [15].We unfold the spherical parameterization into a geometryimage. We define rules that allow the user to tile the planewith an infinite lattice of a specific pattern of interlockingpolygons. Since the geometry image exists in a 2D plane,we construct such a lattice within the geometry image do-main. The vertices of this lattice define the sampling patternused to construct a new mesh from the geometry image. Fi-nally, we compute Celtic knots from the new surface mesh.

Figure 1. An example of the knots producedon the original bunny model is shown. Noticethat the thread pattern seems essentially ran-dom.

2. Related Work

2.1. Surface Parameterization

Associating a surface with a planar domain has tradi-tionally consisted of partitioning the surface into charts andpacking the charts into a texture atlas. Other approacheshave used the connectivity of the surface charts to forma domain complex (semi-regular remeshing), such as thatdescribed by Khodakovsky et al. [13]. Because we are at-tempting to transfer an infinite tiling of a planar domain,chart based methods which do not result in a planar param-eterization may not be appropriate. Gu et al. [7] introducedgeometry images, in which the geometry is resampled intoa completely regular 2D grid. Praun and Hoppe [15] andGotsman et al. [6] both describe methods for mapping genuszero objects to a sphere to produce a spherical parameteri-zation.

2.2. Remeshing

A great number of remeshing schemes have been pro-posed. A good review is given by Alliez et al. [2]. Mostof these combine vertex optimization with mesh simplifica-tion but place no constraint on the local shape of mesh ele-ments. Alliez et al. [1] have proposed a remeshing schemethat samples the surface anistropically to produce a meshwhose edge connectivity follows the lines of maximal andminimal curvature. While this method is the closest in spiritto ours, it shares very little in common procedurally withour algorithm. Gu et al. [7] and Praun and Hoppe [15] haveboth shown how to remesh from geometry images. Whilethey remesh with quadrilaterals, we will demonstate how to

remesh arbitrary geometries taking into account the bound-ary rules they describe.

2.3. Ornament and Tilings

Kaplan and others have examined the generation and ap-plication of shapes in several contexts such as Islamic art-work [9], symettrohedra [10], and Escherized shapes [11].Both the Escherization and Islamic art systems producetilings of interesting polygonal shapes in a 2D plane butare too specific to their stated problems to fully character-ize the set of patterns we might wish to produce. The sym-metrohedra produce polyhedra by embedding polygons ona sphere. While this seems ideal for the spherical param-eterization, it does not encompass a large enough class ofpatterns to be sufficient for our needs and offers no abilityfor higher level control over the patterns produced. Neyretand Cani [14] create pattern based ornament by construct-ing a tiling of equilateral triangles on the original mesh.Their method was limited to triangular primitives and re-quired extensive user design to achieve texture primitivesthat matched along triangle boundaries.

2.4. Celtic Knotwork

The need for quality meshes has been noted extensivelyin the context of computer generated Celtic knots whichmost often have relied on grid meshes to achieve inter-esting knots, largely due to the regularity of the struc-ture [5, 12]. Browne [4] attempts to construct semi-regularmeshes around 2D shapes in order to impose order on theresulting knots and notes the importance of quality meshstructure for font generation. We use the method describedby Kaplan and Cohen [12] due to its ability to constructknots around arbitrary mesh configurations and to constructcoherent knots in 3D.

3. Parameterization

Since we are attempting to transfer a planar tiling pat-tern onto a 3D surface we require a parameterization thatassociates a planar domain with the 3D mesh. While ob-taining a parameterization is not the goal of this work it isa critical component of our technique. The infinite planartiling imposes the constraints that the choice of parameteri-zation model should be continuous in the plane and shouldbe able to expand to an infinite plane. If a parameteriza-tion does not expand to the potentially infinite plane, therewill be seams over which the continuity of the tiling pat-tern will be broken.

These constraints rule out the use of chart based meth-ods such as Khodakovsky et al. [13] which have the desire-able properties of low distortion and the ability to handle ar-

bitrary genus. They evaluate to a planar domain but their at-lases are not continuous. Traditional texture atlases also liein the plane but are not continuous. Global conformal maps[8] don’t work well for genus zero objects since they intro-duce cuts and do double covering, changing the object toa higher genus. They would work for higher genus mod-els producing several infinitely tiled planes with transitioncuts between them. These transitions make them difficult towork with but might be a useful avenue for future work. Theoriginal geometry images described by Gu et al [7] wouldnot allow us to tile the infinite plane and have irregularboundary conditions which it would make it difficult to ap-ply the tiling across the seams. We use the spherical param-eterization described by Praun and Hoppe [15]. Their pa-rameterization method transforms the original mesh into ageometry image which defines a square planar domain withdimensions [0→1] in both x, y. This meets our requirementsof planar continuity and the ability to expand to an arbitrar-ily large plane. This method does constrain us to use genuszero surfaces but this describes a large enough class of mod-els to be considered useful. The boundary conditions of thegeometry image may introduce seams but they are regularand predictable and can be handled as a special case.

We will describe the basics of the spherical parameter-ization but refer the reader to [15] for more information.Given a surface mesh M , a spherical parameterization of thesurface is created, (S→M) by forming a single continousinvertible map from the unit sphere to the mesh. Each meshedge is mapped to a great circle arc and each mesh trian-gle is mapped to a spherical triangle bounded by these arcs.Next, a spherical parameterization (S→D) of a domainpolyhedron, D, is created from either a tetrahedron, octa-hedron or cube. Then the domain is unfolded into the ge-ometry image (D→I). Because the domain of the polyhe-dron parameterization matches that of the spherical param-eterization, the spherical parameterization of M can nowbe mapped to the geometry image. The geometry image (asshown in Figure 2), is essentially a 2D square representa-tion of the original object. This method is quite robust, butwill introduce distortion near narrow extremities.

We use an unfolded octahedron as the polyhedron tospecify the domain map with. This unfolds cleanly into asquare domain map and permits simple extension rules thatallow traversal over the boundaries in a continous manner.The key is to extend the grid by rotating it 180◦ around themidpoints of the boundaries, leading to the topology shownin Figure 3.

4. Tiling Patterns

Prior to the remeshing operation, we tile the plane with auser-defined pattern of specific polygonal shapes within thedomain of the geometry image.

Figure 2. The bunny and gargoyle models andtheir corresponding geometry images.

Figure 3. Left: Boundary conditions for aspherically parameterized geometry imageusing an octahedral unfolding. The geometryimage is represented by the central square.Right: The boundaries of the geometry imageare shown in blue on the bunny model.

The tiling patterns we show are periodic and can be rep-resented as translational units and translation vectors. Atranslational unit is a set of interlocking polygons that rep-resents the basic pattern. Translation vectors define how totranslate “copies” of each translational unit to tile the planewith the pattern. To perform the tiling, we draw an initialtranslational unit within the geometry image domain. Thenwe create additional copies of the translational units andtranslate them by the translation vectors. We continue thisuntil the domain space is completely occupied. Any poly-gons whose edges fall outside the domain boundary are notadded. In order to align the pattern appropriately with thegeometry image the bounds of the translational unit shouldbe integral divisors of the domain dimensions.

For examples of patterns laid over geometry images seeFigure 9.

4.1. Boundary Conditions

The octahedral unfolding that produces the geometry im-age creates special boundary conditions. Regard Figure 5a.The unfolding process essentially “slices open” the octagonalong the edges from one corner vertex. The result of thisis that the corner vertex, E, actually occurs at all four cor-ners of the geometry image and each edge adjacent to ver-tex E is “split” and occurs symmetrically about the mid-

Figure 4. At left is a simple translational unitwith translation vectors in each of the fourcardinal directions. The planar tiling of thetranslatioal unit is shown in the middle. Amore complex pattern created with our sys-tem is shown at right.

point of each geometry image boundary edge.Because of this, any vertex or edge crossing that oc-

curs along the domain boundary edge along the segmentAE must also occur symmetrically about vertex A alongthe segment AE′, as seen in Figure 5b. If this conditiondoes not hold, there will be gaps in the new mesh where theedges AE and AE′ connect since the the new geometry willnot approach the edge uniformly. Because the tilings mustmaintain this 180◦ rotational symmetry about the midpointsof the boundary edges, our patterns tend to have lengths andwidths that are integral divisors of the domain dimensionsand expand uniformly in the x and y directions. Rotating thetiling patterns by angles that are not multiples of 90◦ maynot be appropriate in many cases since changing the axis ofexpansion won’t preserve the rotational symmetry.

Additionally, there are several cases in which pat-terns may preserve the above properties may still evaluateto a non-manifold surface. The symmetry about the mid-point along any boundary edge means that vertices thatare equidistant from the midpoint along the bound-ary edge evaluate to the same position, as we would expect.Examine the case shown in Figure 5c. In this case, both ver-tices that compose edge BB′ are really the same vertex.Therefore, this edge evaluates to a single vertex so it is re-moved from the polygonal face.

When a figure has a similar topology to Figure 5c butthere is a vertex at the midpoint, as seen in Figure 5d, theedges on either side of the midpoint A are really the sameedge. Geometrically, this evaluates to a polygon that has aset of edges that move towards the interior and double backon each other. To make this face valid, we remove both ofthese edges from the polygon.

In Figures 5c-d, edge removal left us with a polygon withonly two edges which cannot be a closed polygonal face.We remove such faces from the output mesh and the result-ing boundary seam on either side will be BC.

Another problem that may occur is when two shapes aregeometrically identical in 3D despite occupying differentareas of the domain in 2D. In Figure 5e, both faces have

E

A

A

E’

B

B’E’

E

(a)

E

B’

B

E’

A

(b)

B’

B

AC

(c)

B’

B

AC

(d)

C

B’

B

A

(e)

Figure 5. a)Unfolding an octahedral domaininto a planar domain. b) Domain boundarysymmetry requirements. The red lines repre-sent the boundary of the geometry image do-main. A is the midpoint of the left edge of thegeometry image. Any vertex B along a do-main boundary edge must be matched by anequivalent vertex 180◦ about A. Edges suchas AB and AB′ are geometrically identical. c-e) Error conditions near boundary midpoints.

matching boundary vertices and identical interior vertices.Since both of these shapes are identical, we end up with ageometrically flat area on the remeshed model, where bothfaces end up being essentially flip sides of a coin. We de-tect such situations and subdivide the non-boundary edgesCB and CB′ to differentiate between the two faces.

We apply polygons across domain boundaries so thatthe pattern fully tiles the remeshed model. In some in-stances, however, polygons applied across the boundary donot match up with the available space on the opposing sideof the boundary. This may occur when patterns do not main-tain the required rotational symmetry or are drawn off-axisor at the wrong scale. Such cases should still be handledsince there may be many patterns that we would like to usethat do not meet the previously stated requirements.

When this does occur, it leaves holes in the new meshas seen in Figure 6. While there may be no perfect solu-tion to this, besides reorienting the pattern so that it is situ-ated correctly, there are several methods that will allow usto obtain useable meshes. First, we can pull all vertices nearthe domain boundary to actually lie on the domain bound-ary. This has the effect of filling the remaining space andpreserving the topology of the pattern though shapes maybecome larger than they would normally be and the transi-tions between shapes along the boundary may not be pre-served. Second, we can fill the remaining open space nearthe boundaries by inserting polygons in that space. These

Figure 6. If a pattern does not approach theboundary correctly (as in the two leftmost im-ages), a variety of techniques may be used tohandle the boundaries appropriately. In thiscase, the vertices near the boundary havebeen dragged to the boundary and adjacentpolygonal faces across the domain bound-ary have been stiched together. Here, the leftedges of the rectangular images are the leftboundary edges of the geometry image.

polygons are formed by walking along the remaining openedges of the pattern and creating polygons between verticesthat touch the domain boundary. This minimizes distortionof the pattern but does not preserve the pattern topology.

Because the domain boundaries require rotational sym-metry about their midpoints, if the pattern of vertices andedge crossings between segments AE and AE ′, as seen inFigure 5b, do not match, then there will be discontinuities inthe new mesh. To handle this, for each vertex in AE that isnot matched in AE′ we insert a new vertex at the appropri-ate location and vice versa. The edges and polygons alongAE and AE′ are remapped to take the new vertices into ac-count. This method does not preserve the topology of thepattern but is important for producing closed models.

5. Remeshing

The vertices of the tiling pattern are the locations atwhich we sample the geometry image and the edges of thepattern define the connectivity of the new mesh. For eachsample vertex, we find the barycentric coordinates of thetriangle that the sample location lies within in the geometryimage. The new vertex position is the location defined bythe barycentric coordinates within the triangle in the orig-inal model that the geometry image maps to. The result ofthis algorithm is a 2D manifold mesh with the pattern ofshapes embedded into its structure.

We have shown simple periodic tilings here since that isthe class of patterns that work well with the Celtic knots al-gorithms. There is no reason that other patterns that fill thedomain space could not be used to perform the remeshing.

At this point, we have essentially remeshed twice; onceto obtain the geometry image and a second time to embedthe pattern. It is possible to instead transform the patternvertices back through each of the mappings (parametriza-tion and geometry image) to the original model instead ofdoing a linear approximation directly on the geometry im-age with the trade off of slightly more complex code. Wechose not to do this since the geometry images we workedwith were extremely high resolution and the results we ob-tained worked well with the Celtic Knots program and werevisually appealing.

6. Knotwork

Celtic knotwork is a type of art that consists of inter-twined threads that maintain a characteristic over-under pat-tern at each corssing. The method of producing Celtic knotsdescribed by Kaplan and Cohen [12], uses the midpointsof each edge in the mesh to define the crossing positionsand connectivity of the threads in the knot. The thread pathsconnect adjacent crossing positions which is the reason thatthe shape of the edges is of critical importance in design-ing Celtic knots (see Figure 7).

An important question to ask is: if we have a planar pa-rameterization of the mesh and can calculate planar knotseasily, why remesh at all? Why not use the parameteriza-tion as texture coordinates and texture map the ornament?Why should the mesh structure bear this load? The answeris that the knots program constructs splines paths for thethreads based on the midpoints of each edge. If we texturemap a planar knot onto the surface, the knot threads will ap-pear disjoint and “painted” onto the surface of the objectwhereas calculating knots based on the new mesh will re-sult in smooth knots that approximate the surface but do notexactly lie on it.

The crossing patterns of the Celtic knots can be changedby associating a breakpoint value at every edge. Thesebreakpoint values (0,1 or 2) are used to change the threadpaths of the knot and give interesting variations on knot pat-terns. With large meshes, it is very time consuming to inputthese breakpoints by hand. We associate breakpoint valueswith each edge of each polygon defined in the pattern ap-plication system. We save the breakpoint information alongwith the model file and they are read as a pair into the Celticknots program which computes the knots automatically, asseen in Figure 9 bottom.

7. Results

The results of this algorithm may be evaluated from twoperspectives: the quality of the remeshed models and thequality of the knots produced from these meshes though it

Figure 7. An example of a 2D Celtic knotproduced on a graph of 2x2 squares. Noticethat crossings occur at each edge midpointand connect adjacent edge midpoints exceptat the breakpoint edges. Regular edges arecolored black. Breakpoint edges are coloredblue and yellow.

must be stressed that the remeshed models are a pleasantby-product rather than our goal.

A variety of remeshed examples appear in Figures 6- 9.The class of tilings that we can successfully create and em-bed that maintain the boundary constraints is large and en-compasses all of the patterns we transferred from the 2DCeltic knot program. With the techniques we have presentedfor handling boundary conditions, we are able to handle all2D patterns, though the behavior near the boundaries is notguaranteed to preserve the topology or minimize distortion.

The quality of the approximation can be affected forthree reasons: parameterization, pattern scale (i.e. samplevolume) and pattern shape. We do not measure the distor-tion of the geometric properties of the new mesh since thatis a problem with parameterization and as parameterizationtechniques improve, so will the results of our method. Dis-tortion does occur on narrow extremities consistent with thedistortion in the spherically parameterized geometry image.

The images presented here were done with relativelycoarse patterns. To minimize the error between the origi-nal and new meshes, the tiling patterns can be drawn at asfine a scale as desired. Because the sampling pattern pro-duced by the tiling may not be uniform, different parts ofthe remeshed model may approximate the original modelless well than others depending on the density of samples.Also, because the polygons of each face are not tesselatedand may be arbitrarily complex (depending on user prefer-ence), the faces are most probably non-planar for any non-triangular faces. Fortunately, this is not a problem for ussince the Celtic knots program does not require planar faces.

The 3D knotwork produced with this technique is a sig-nificant improvement over previous results. An example ofknots produced with the previous method is shown in Fig-ure 1 at left. Note that the threads are essentially random andcontain no repetitive patterns or motifs (a hallmark of Celticdesign). Examples of the type of results possible using thethe remeshed models are shown in Figure 9. The new im-

ages show a variety motifs taken from classical Celtic knot-work repeated over the surface of the models. The threadsare coherent and create repeated figures that traverse the en-tire surface in a consistent fashion.

Because the output of the Celtic knots program is sodependent on the quality of the mesh, prior work onlyshowed quality knots calculated over uniform polyhedrasince the quality of more interesting models was so poor.The remeshing procedure presented here has made knotdecoration of 3D meshes a viable tool for animators and de-signers.

8. Conclusion and Future Work

We have presented techniques for creating 2D tilingsof user-defined patterns of polygons, for remeshing mod-els with those patterns and for applying the results of thesemethods to the automated construction of Celtic knotwork.

Our remeshed models are 2D manifold and correctly em-bed the desired pattern of shapes yet there are several natu-ral avenues for further research. Currently, our system onlyhandles genus zero models. We would like to extend thisto include other methods of parameterizations that includemodels of higher genus. Remeshing has been a widely re-searched area in recent years and we believe that the ap-plications relevent to pattern-oriented remeshing will growas such meshes become available. Our technique naturallycompresses the mesh but is almost definitely not an opti-mal compression.

This area of research has a number of natural extensionsto texture synthesis techniques. Each vertex of the remeshedmodel can be used as coordinates in a texture parameteriza-tion with texture coordinates could be generated automati-cally from the tiling application.

The 3D Celtic knotwork produced in this framework issignificantly better than 3D results from previously pub-lished work. The ability to embed arbitrary patterns ofshapes into the meshes allows us to transfer previously de-signed motifs and designs from 2D onto models allowingus a large measure of control in designing specific deco-rations. The extension of this system to include non-genuszero models would allow us to use models of all the charac-ter symbols. This would allow us to decorate an entire fontset ( a traditional Celtic decoration motif) automatically byremeshing all the character models with a single pattern andcomputing knots on the remeshed models automatically.

Pattern-oriented remeshing has further implications fortransferring decorative toolsets from 2D to 3D. It seems nat-ural to extend the work of Wong et al. [16] in floral decora-tion, Kaplan and Salesin [11] in escherization and others tocompute 3D decorations using such meshes. This may havefar reaching implications for creating a wide variety of or-

nament directly on 3D models which could be a useful toolfor artists and animators.

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Figure 8. Examples of pattern-orientedremeshed models.

Figure 9. Each row represents: At left are the geometry images with tiling patterns overlaid. Insetare larger views of the tiling patterns. Blue areas near the borders are edges that overlap the do-main boundary. Next are the results of a remeshing performed with the tiling pattern. At right areexamples of Celtic knots computed on the remeshed models.