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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.
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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-
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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-
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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
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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
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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.
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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.