ACM Transactions on Graphics (January)

179 Pages·2005·4.976 MB·English

Feature-Based Surface Parameterization and
Texture Mapping
EUGENEZHANG,KONSTANTINMISCHAIKOW,andGREGTURK
GeorgiaInstituteofTechnology
Surfaceparameterizationisnecessaryformanygraphicstasks:texture-preservingsimplification,remeshing,surfacepainting,
and precomputation of solid textures. The stretch caused by a given parameterization determines the sampling rate on the
surface.Inthisarticle,wepresentanautomaticparameterizationmethodforsegmentingasurfaceintopatchesthatarethen
flattenedwithlittlestretch.
Many objects consist of regions of relatively simple shapes, each of which has a natural parameterization. Based on this
observation,wedescribeathree-stagefeature-basedpatchcreationmethodformanifoldsurfaces.Thefirsttwostages,genus
reductionandfeatureidentification,areperformedwiththehelpofdistance-basedsurfacefunctions.Inthelaststage,wecreate
oneortwopatchesforeachfeatureregionbasedonacovariancematrixofthefeature’ssurfacepoints.
Toreducestretchduringpatchunfolding,wenoticethatstretchisa2×2tensor,whichinidealsituationsistheidentity.
Therefore, we use the Green-Lagrange tensor to measure and to guide the optimization process. Furthermore, we allow the
boundaryverticesofapatchtobeoptimizedbyaddingscaffoldtriangles.Wedemonstrateourfeature-basedpatchcreationand
patchunfoldingmethodsforseveraltexturedmodels.
Finally,toevaluatethequalityofagivenparameterization,wedescribeanimage-basederrormeasurethattakesintoaccount
stretch,seams,smoothness,packingefficiency,andsurfacevisibility.
CategoriesandSubjectDescriptors:I.3.5[ComputerGraphics]:ComputationalGeometryandObjectModeling—Geometric
algorithms,languages,andsystems
GeneralTerms:Algorithms
AdditionalKeyWordsandPhrases:Surfaceparameterization,segmentation,texturemapping,topology
1. INTRODUCTION
Surfaceparameterizationisawell-studiedproblemincomputergraphics.Ingeneral,surfaceparame-
terizationreferstosegmentinga3Dsurfaceintooneormorepatchesandunfoldingthemontoaplane
withoutanyoverlap.Borrowingterminologyfrommathematics,thisisoftenreferredtoascreatingan
atlas of charts for a given surface. Surface parameterization is necessary for many graphics applica-
tionsinwhichpropertiesofa3Dsurface(colors,normal)aresampledandstoredinatexturemap.The
qualityoftheparameterizationgreatlyaffectsthequalityofsubsequentapplications.Oneofthemost
ThisworkissupportedbyNSFgrantsACI-0083836,DMS-0138420,andDMS-0107396.
Authors’addresses:E.Zhang,SchoolofElectricalEngineeringandComputerScience,OregonStateUniversity,102Dearborn
Hall, Corvallis, OR 97331-3202; email: [email protected]; K. Mischaikow, Center for Dynamical Systems and Nonlinear
Studies,SchoolofMathematics,GeorgiaInstituteofTechnology,Atlanta,GA30332;email:[email protected];G.Turk,
CollegeofComputing/GVUCenter,GeorgiaInstituteofTechnology,Atlanta,GA30332;email:[email protected].
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ACMTransactionsonGraphics,Vol.24,No.1,January2005,Pages1–27.
2 • E.Zhangetal.
Fig.1. Thefeatureregions(left)andtheunfoldedpatches(right,colorsareusedtoencodesurfacenormal)forthebunnysurface
usingouralgorithm.
importantqualitymeasurementsisstretch.Whenunfoldingasurfaceontoaplane,stretchingoccursif
thesurfacecontainshighlysphericalorhyperbolicregions.Highstretchinaparameterizationresults
inanunevensamplingrateacrossthesurface.
Weobservethatmanyobjectscanbedecomposedintoasetof“simple”shapesthatroughlyapprox-
imate cylinders, cones, flat disks, and spheres. Cylinders, cones, and planes are developable surfaces,
which are Euclidean by nature. Unfolding them results in little stretch, without any overlap. In this
article,wemakeuseofsomedistance-basedsurfacefunctionstodivideamanifoldsurfaceintofeature
regions, each of which is similar to one of the simple shapes. In Figure 1 (left), the bunny surface is
decomposedintofourfeatureregions(ears,head,andbody)usingoursegmentationalgorithm.These
regions are converted into patches and unfolded with little stretch (right, colors are used to encode
surfacenormal).
Existing patch unfolding techniques are often carried out in two stages: an initial patch layout to
achievesomeobjectivesuchasconformalmapping,followedbyaninteriorvertexoptimizationbasedon
somestretchmetric.Weobservethatanidealsurfaceparameterizationbetweenapatchanditstextural
imageisanisometry,thatis,abijectivemapthatpreservesdistances.TheGreen-Lagrangedeformation
tensor has the property that it measures anisotropic stretch faithfully and penalizes undersampling
more severely than oversampling. In addition, it can be seen as a balance between area-preserving
mappings and conformal mappings. We use this metric to guide the vertex optimization process for
patch unfolding. In addition, we use what we call scaffold triangles to convert the original boundary
verticesinto“interior”vertices,whichcanthenbefreelymovedaroundwithinthesameoptimization
framework.Thisisanewwayofcreatingnonconvexpatchesthatmayevenhaveholes.
Inthisarticle,wepresentanautomaticsurfaceparameterizationtechniquethatconsistsofseveral
newideasandimprovesuponexistingtechniquesinthefollowingaspects.Forpatchcreation,instead
of relying on local curvature information for feature detection as in the case of most previous param-
eterization methods, we extract and segment large protrusions based on the topological analysis of
some distance-based surface functions that are global in nature. This results in a small number of
largepatchesthatcanbeunfoldedwithrelativelylittlestretch.Forpatchunfolding,weusetheGreen-
Lagrangetensortomeasurestretchandtoguidethestretchoptimizationprocess.Inaddition,wecreate
a“virtualboundary”toallowthepatchboundariestobeoptimized,withouttheneedtocheckforglobal
self-intersections.Finally,wedescribeanovelimage-basedqualitymetricforsurfaceparameterization
thatimplicitlytakesintoaccountstretch,seams,packingefficiency,smoothness,andsurfacevisibility.
ACMTransactionsonGraphics,Vol.24,No.1,January2005.
Feature-BasedSurfaceParameterizationandTextureMapping • 3
The remainder of the article is organized as follows. In Section 2, we review existing surface pa-
rameterization techniques. Then, we present our feature-based patch creation method in Section 3,
followed by our new balanced stretch metric in Section 4.1, boundary vertex optimization technique
in Section 4.2, and our packing algorithm in Section 5. In Section 6, we show the results of applying
ourtechniquetovarious3Dmodelsanddescribeourimage-basedqualitymetricforparameterization
techniques.Section7providesasummaryofourcontributionsandadiscussionofsomepossiblefuture
work.
2. PREVIOUSWORK
Therehasbeenaconsiderableamountofrecentworkinthegraphicscommunityonbuildingasurface
parameterizationbyunfoldingapolygonalsurfaceintoplanarpatches.Muchofthemotivationforthis
isfortexturemapping,themappingofpixelsfromarectangulardomain(thetexturemap)ontoasurface
thatisdescribedbyacollectionofpolygons.Thesurfaceparameterizationproblemistosubdividethe
given surface into a (hopefully small) number of patches that are then flattened onto a plane and
arrangedinatexturemap.Usesforsurfaceparameterizationincludesurfacepainting[Hanrahanand
Haeberli1990],fastrenderingofproceduraltextures[Perlin1985;Turk2001;WeiandLevoy2001;Carr
and Hart 2002], applying photographed color variations onto digitized surfaces [Cignoni et al. 1998],
andcreatingnormalmapsfromdetailedgeometry[Sanderetal.2001].Thesesameparameterization
methodsmayalsobeusedforremeshing,thatis,forcreatinganewmeshfromtheoriginalsurface[Alliez
etal.2002].Remeshingcanbeusedtoimprovethetriangleshapes,tovarythetrianglesizeaccording
tocurvaturedetails,andtoinducesemi-regulartessellations.Recently,octreeshavebeenusedtostore
colorsin3Dforsurfacetexturingwithoutanyparameterization[BensonandDavis2002;DeBryetal.
2002]. Although octree techniques are supported with programmable GPU’s, they are not yet directly
supportedbygraphicshardware.
2.1 PatchCreation
There are two common approaches to the patch creation problem. The first of these is to find a single
cut for the surface that makes the modified surface topologically equivalent to a disk [Piponi and
Borshukov2000;Guetal.2002;ShefferandHart2002;EricksonandHar-Peled2002;Nietal.2004].
Thisapproachhasthevirtueofcreatingasfewseamsaspossible,butwilloftenintroducelargestretch
between the patch and the surface. Such stretching is undesirable because different portions of the
surface are represented using quite different amounts of color detail, as measured in pixel resolution
inthetexturemap.
The other major approach is to divide the surface into a collection of patches that can be unfolded
withlittlestretch[Ecketal.1995;Leeetal.1998;Sanderetal.2001;Alliezetal.2002;Le´vyetal.2002;
Sorkine et al. 2002]. Though stretch is minimized, this approach creates seams between the patches.
Theseseamscauseproblemswhencreatingtexturedimagesofthesurfacebecausethecolorvariation
across the seams must be treated with extreme care or the seams will be noticeable. Some methods
create small disk-like patches [Eck et al. 1995; Lee et al. 1998; Sander et al. 2001; Alliez et al. 2002],
whileothersattempttocreatelargepatchesthatmatchthefeaturescontainedintheobject[Le´vyetal.
2002; Sorkine et al. 2002; Katz and Tal 2003]. Our own work takes this latter approach. We cut the
surfaceintomultiplepatches,butaccordingtothelargegeometricfeaturesofthesurface.Forexample,
wewouldliketorecognizetheheadandlimbsofananimalasimportantfeaturesandtocreatepatches
thatrespectthesefeatures.TheworkofLe´vyetal.[2002]andKatzandTal[2003]havesimilargoals,
althoughtheirfeature-basedpatchcreationmethodsarequitedifferentthanourown.
Thedefinitionoftheterm“geometricfeature”variesindifferentcontexts.Forsurfacereconstruction
and mesh simplification, features are often defined in terms of local curvature. This is reasonable
ACMTransactionsonGraphics,Vol.24,No.1,January2005.
4 • E.Zhangetal.
becausehighcurvatureregionsareexactlywhattheseapplicationsaretryingtopreserve.Ontheother
hand, surface parameterization algorithms incur higher stretch on a smooth surface with long thin
protrusionsthananoisysurfacewithsmallprotrusions.Inthiswork,wedefinegeometricfeaturesas
largeprotrusions,andouralgorithmsegmentsasurfacebasedonitsfeaturesbyperformingtopological
analysisofsomedistance-basedsurfacefunctions.
2.2 PatchUnfolding
There have been many patch unfolding techniques. The classical approach treats the patch unfolding
problemasfindingtheminimumofsomefunctionalthatmeasuresthedifferencebetweenaparameter-
izationwithisometry[Ecketal.1995;Floater1997].First,theboundaryverticesareassignedinitial
positions(usuallyonacircleorasquare).Thentheparameterizationfortheinteriorverticesisdeter-
minedbysolvingalargelinearsystemorthroughanonlinearoptimizationprocess.Othershaveused
stretchmeasuressuchastheGreen-Lagrangedeformationtensor[Maillotetal.1993]andavariantof
Dirichlet energy [Hormann and Greiner 1999]. Sander et al. [2001] define a geometric stretch metric
thatisbasedontheaverageandmaximalstretchinalldirectionsofatriangle.Sorkineetal.[2002]and
Khodakovsky et al. [2003] have devised stretch metrics based on the maximum and minimum eigen-
valuesofthestretchtensor.Sanderetal.[2001]alsoproposeapost-processingvertexoptimizationstep
thatimprovestheirgeometricstretch.Aswedescribelater,Sander’spatchoptimizationapproachwas
aninspirationforourownwork.Toallowtheboundaryverticesofapatchtobefreefromthearbitrary
initialassignment,Le´vyetal.[2002]usealeast-squaresconformalmapping,andDesbrunetal.[2002]
proposeanequivalentformulation,Discrete,NaturalConformalParameterization.Sanderetal.[2002]
allowboundaryverticestomove,whilecheckingforglobalintersections.Leeetal.[2002]addlayersof
“virtualboundaries”aspartoftheedgespringstoallowthepatchboundariestohavenaturalshapes.
Recently,ShefferanddeSturler[2001;2002]proposetouseanangle-basedflatteningapproachforpatch
unfolding.Thisapproachmeasuresstretchintermoftheanglesdeficitsbetweenthetrianglesonthe
surfaceandtheirtexturalimages,anditremovestheneedtocheckforglobalself-intersections.
3. FEATURE-BASEDPATCHCREATION
Ourfeature-basedpatchcreationmethodiscarriedoutinthreestages:
(1) genusreduction:asurfacewithhandles(nonzerogenus)isconvertedintoagenuszerosurface.
(2) featureidentification:agenuszerosurfaceisdividedintoanumberofrelativelysimpleshapes.
(3) patchcreation:everysimpleshapeiscutintooneortwotopologicaldisks.
Forbothgenusreductionandfeatureidentification,webuildasurface-basedReebgraphbasedonthe
averagegeodesicdistanceintroducedbyHilagaetal.[2001].Thisgraphconsistsofverticesandedges
in the mesh surface, and we call it an embedded Reeb graph. When properly constructed, this graph
reveals the location of the handles and protrusions in the surface. Since our goal is to create patches
thataretopologicaldisks,weneedtoperformouroperationsinatopologicallyconsistentmanner.For
thispurpose,weusesurfaceregiongrowingforallthreestages:genusreduction,featureidentification
andpatchcreation.Startingfromaninitialtriangle,wegrowaregionbyaddingonetriangleatatime
untilthewholesurfacehasbeencoveredoruntilsomeotherstoppingcriterionhasbeenmet.Wewill
nowdescribetheaveragegeodesicdistancefunctionandtheembeddedReebgraphthatitinduces.
3.1 TheAverageGeodesicDistanceFunction
TheaveragegeodesicdistancefunctionwasintroducedbyHilagaetal.[2001]forthepurposeofshape
matching. This is a function A(p) that takes on a scalar value at each point p on the surface S. Let
g(p,q)bethegeodesicdistancebetweentwopointspandqon S.Thentheaveragegeodesicdistance
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Feature-BasedSurfaceParameterizationandTextureMapping • 5
Fig.2. Theaveragegeodesicdistancefunction(AGD)onthedinosaurmodeliscolor-codedintheleftofthisfigure.Theglobal
minimumislocatedunderneaththebelly,coloredinred.Levelsetsarepaintedinrepeatedpatternsofred,green,andblue.
Noticethatthetipsoflargeprotrusions(horns,legs,tail)arelocalmaximaofAGD.Themiddle,figureshowstheembeddedReeb
graphcreatedbysurface-growingbasedonAGD.Localmaximaareindicatedbyredspheres,andsaddlepointsarehighlighted
bybluespheres.Successivecriticalpointsareconnectedbysurfacepathsshowninsolidyellow(visible)anddashgreen(hidden).
Thefinalsurfacesegmentationresultbasedonouralgorithmisshownontheright.
Fig.3. ComparisonamongthreeAGDfunctionsforthedragon:AGD1(left),AGD2(middle),andAGD∞(right).
ofpisdefinedasfollows:
(cid:1)
g(p,q)dq
A(p)= q∈S . (1)
Area(S)
A(p)isamemberofthefollowingsetoffunctions:
(cid:2)(cid:1)
gn(p,q)dq
A (p)= n q∈S (2)
n
Area(S)
with n = 1. When n → ∞, A∞(p) := limn→∞ An(p) = maxq∈S g(p,q), which measures the maximal
distancebetweenpandanypointon S.Wedefineforn=1,2,...,∞
A (p)
AGD (p):= n . (3)
n
minq∈S An(q)
Foranyn≥1, AGD hasseveralusefulproperties.First,itsvaluemeasureshow“isolated”apoint
n
isfromtherestofthesurface.Second,itslocalmaximacoincidewiththetipsofthegeometricfeatures
containedinthemodel.Third,itisscale-invariantandcanbeusedtocomparefeaturesfromdifferent
shapes. Figure 2 (left) shows a polygonal model of a dinosaur, color-coded according to AGD . The red
2
regiononthedinosaur’sbellysignifiesthatpointsinthisregionhavelowvaluesofAGD .Highervalues
2
adjacent to this middle region are colored in green and then in blue. The colors then cycle repeatedly
throughred,green,andblue.Notethatthetipsofthelargefeaturesofthisobject(legs,horns,tail)are
markedbylocalmaximaofAGD .(InsubsequentsectionswewillusethetermlocalmaximaofAGD
2
andtipsinterchangeably.)Inpractice,weuseAGD sinceitseemstoproducesmoothresults.Figure3
2
ACMTransactionsonGraphics,Vol.24,No.1,January2005.
6 • E.Zhangetal.
comparesthelevelsetsofthefollowingthreefunctionsonthedragon:AGD (left),AGD (middle),and
1 2
AGD∞ (right).Fromnowon,wewillusethetermAGDtomeanAGD2.
For a genus zero surface, we use AGD to identify and measure its geometric features. The tip of
a protrusion is a local maximum. Larger values at local maxima signify larger protrusions. Notice
thatanoisysurfaceoftencontainsmanysmallbumpsthatcorrespondtolocalmaximawithrelatively
smallAGDvalues.Creatingpatchesbasedonthesebumpswillincreasetheamountofseams,without
significantlyreducingstretch.Therefore,weonlyconsiderlocalmaximawhoseAGDvaluesareabove
athreshold.Inpractice,wefindchoosinganynumberin[1.3,1.5]asthethresholdforminimalfeature
sizeproducesreasonableresults,andweuse1.4forallourtestmodels.
ComputingAGDexactlywouldbequitecostly.WecloselyfollowthealgorithmofHilagaetal.[2001]to
quicklycomputeasatisfactoryapproximationofAGD.Briefly,thegeodesicdistancesarenotcalculated
from all the surface points, but rather from a small number of evenly spaced points on the surface.
We find the geodesic distances from each of these points to all other points efficiently using the fast-
marchingmethodforsurfaces[KimmelandSethian1998].
3.2 BuildinganEmbeddedReebGraph
Tofindhandlesandlargeprotrusionsinamodel,weperformtopologicalanalysisofAGDandconstruct
anembeddedReebgraph(cid:1)thatisinducedbyAGD.Theleafnodesof(cid:1)aresituatedatthetipsofprotru-
sions,andtheloopsin(cid:1) revealtheexistenceofhandles.Weconstruct(cid:1) byperformingregion-growing
in the increasing order of AGD and tracking the topological changes in the wavefront. This is based
uponideasfromMorseTheory[Milnor1963]andReebgraphs[Reeb1946],whichwewillreviewhere.
Let f beasmoothfunctiondefinedonasmoothsurface S ⊂R3.Foranypointp ∈ S,letµ=(u,v)
0
be a parameterization of some neighborhood of p in S such that µ(0,0) = p . The gradient ∇f and
0 0
theHessian Hf aredefinedasfollows:
   
∂f/∂u ∂2f/∂u2 ∂2f/∂u∂v
∇f =  , Hf = . (4)
∂f/∂v ∂2f/∂u∂v ∂2f/∂v2
p is a critical point of f if ∇f(0,0) = 0. Otherwise, p is regular. A critical point p is said to be
0 0 0
nondegenerate if Hf(0,0) does not have any zero eigenvalues. In this case, p can be classified as a
0
mimimum/saddle/maximum if Hf(0,0) has zero/one/two negative eigenvalues. f is Morse over S if
f possesses no degenerate critical points. Morse theory relates the critical points of Morse functions
to the topology of the underlying surface. For instance, when S is a closed orientable 2-manifold with
Euler characteristic χ(S) (twice the number of handles plus two), the following is true for any Morse
function f definedon S withα maxima,β saddles,andγ minima.
α−β+γ =χ(S). (5)
BanchoffextendsMorsetheorytotriangularmeshes[1970].
Acontinuousfunction f : S →RinducesaReebgraph(cid:1) ,whichcanbeusedtorevealthetopological
f
andskeletalstructuresofS[Hilagaetal.2001].Formally,wedefineanequivalencerelationship∼ on
f
S asfollows.Letp,q∈ S betwopoints,thenp∼ qifandonlyif f(p)= f(q)andpandqbelongto
f
the same connected component of f−1(f(p)). Notice f does not have to be Morse. Figure 4 illustrates
anexampleReebgraphthatcorrespondstotheverticalheightfunction,definedona3Dsurface.Many
applications make use of Reeb graphs, such as shape-matching [Hilaga et al. 2001] and topological
simplification[Woodetal.2004].AGDis,ingeneral,notMorse.Forinstance,itisaconstantfunction
on a sphere, in which case every point is a degenerate critical point. Axen and Edelsbrunner [1998]
show that a function can be perturbed into a Morse function with surface wave traversal, provided
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Feature-BasedSurfaceParameterizationandTextureMapping • 7
Fig.4. AnexampleofaReebgraph(right)foraverticalheightfunction,definedona3Dsurfaceofgenusone(left).Thecritical
pointsarehighlightedbycoloredspheres(redformaxima,greenforminima,andblueforsaddles).Thenumberofloopsinthe
graphequalsthenumberofhandlesinthesurface.
thatthemeshisproperlysubdivided.Weuseasimilarstrategyexceptthatwerecordcriticaltriangles
insteadofcriticalvertices.
OuralgorithmforbuildinganembeddedReebgraph(cid:1) startswithcomputingAGDforeveryvertex.
ForatriangleT ={v ,v ,v },wedefineAGD(T)=min{AGD(v ), AGD(v ), AGD(v )}.Startingwitha
1 2 3 1 2 3
trianglewhoseAGDvalueequalstheglobalminimum,weaddonetriangleatatimeintheincreasing
orderoftheAGDuntilthesurfaceiscovered.Theboundaryofthevisitedregionconsistsofanumber
ofloops.Welabelatriangle,whenitisadded,accordingtooneofthefollowingfivecriteria:
(1) Minimum:whereonenewboundaryloopstarts.Forourapplication,thereisonlyonesuchtriangle,
oneoftheglobalminimaoftheAGD.
(2) Maximum:whereoneboundaryloopvanishes.Thisisthetipofaprotrusion.
(3) Splittingsaddle:whereoneboundaryloopintersectsitselfandsplitsintotwo.
(4) Mergingsaddle:wheretwoboundaryloopsintersectandmergeintoone.Thissignifiestheformation
ofahandle.
(5) Regular:wherethenumberofboundaryloopsdoesnotchange.
A triangle that is not regular is a critical triangle. Let n be the genus of the surface, and let N ,
max
N , N , and N be the number of the triangles that are maxima, minima, splitting saddles, and
min ss ms
mergingsaddles.Thenwehave,
N =n (6)
ms
N −N +N +N =2. (7)
max ss ms min
Equation 7 corresponds to the handle-body decomposition of a closed and orientable piecewise linear
2-manifold[RourkeandSanderson1972].InterestedreadersmayrefertoLopesetal.[2003]formore
details. For our application, N = 1. Furthermore, we mark the center of a critical triangle as the
min
positionofthecorrespondingcriticalpoint.Theregiononthesurfacesweptoutbetweenapairofcritical
triangles(notincludingthesecriticaltriangles)ishomeomorphictoacylinderwithoutcaps.Let Aand
Bbeapairofcriticaltriangles,andassumethat Aisvisitedearlierthan B.Wereferto Aastheparent
criticaltriangleandBasthechildcriticaltriangle.Foragenuszerosurface,everychildcriticaltriangle
hasasingleparent.Forsurfaceswithagenusgreaterthanzero,achildcriticaltrianglemayhaveone
or two parents. Let R be the connecting region between Aand B, which consists of a set of regular
AB
triangles={T1,...,Tk}intheorderofwhichthey(cid:7)arev(cid:7)isited.Thereisashortestpaththatconnects A
and B usingtheedgesofthesetoftriangles{A} {B} R .WeconstructtheembeddedReebgraph
AB
byfindingtheshortestpathsbetweeneverypairofparent/childcriticaltriangles.
As mentioned earlier, the embedded Reeb graph (cid:1) is much like a Reeb graph that corresponds to
AGD. It reveals the distribution of the geometric features over the surface. The middle of Figure 2
showstheembeddedReebgraphofthedinosaur.Localmaximaarehighlightedwithredspheres,while
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8 • E.Zhangetal.
Fig.5. EmbeddedReebgraphsforthebunnysurfacewithdifferentfilteringconstantsα:1.01(left),1.1(middle),and1.5(right).
Weuse1.1asthefilteringconstantforallthetestmodels.
bluespheresindicatethelocationofsplittingsaddlepoints.Theglobalminimumismarkedwithalight
blue sphere on the belly. Successive critical points are connected by paths on the surface, which are
drawninsolidyellow(visible)anddashgreen(hidden).Notethatlocalmaximacoincidewiththetips
ofthegeometricfeatures(horns,feet,andtail).
Since complex surfaces often contain many small protrusions (bumps), the embedded Reeb graph
can contain an excessive number of local maxima and saddle points. This increases the subsequent
processingtimesincethenumberoffeaturesismuchmorethanwhatweconsiderlarge(or“persistent”
as described in Edelsbrunner et al. [2003]). We use the following filtering scheme to weed out extra
localmaximaandsplittingsaddlepoints.Duringsurface-growing,wealtertheorderinwhichtriangles
areadded.Tobemorespecific,lettbetheunvisitedtrianglewiththesmallestAGDvalue.Ifaddingt
causesaboundarytosplit,welookforothertrianglesthatcouldbeaddedwithoutcausingaboundary
split.Ifoneofthesetriangles,t(cid:8) satisfies:
AGD(t(cid:8))<αAGD(t) (8)
where α is a global filtering constant, then we add t(cid:8) instead of t. When there are multiple choices,
we choose the triangle with the smallest AGD value. Our filtering process is related to the concept
of topological persistence and simplification [Edelsbrunner et al. 2003], but with a different scalar
functionandadifferentmeasureforpersistence.Also,thesimplificationprocessisimplicit.Weapply
thefilteringschemetothebunnysurface(Figure5)withthreedifferentα’s:1.01(left),1.1(middle),1.5
(right).Noticetheexcessivesaddlepointsandmaximaappearintheheadandthepawswhenα =1.01
(left). When α = 1.1 (middle), the local maxima that reveal large geometric structures are kept (the
tips of ears and the center of tail). Excessive filtering may result in a trivial embedded Reeb graph,
forexample,α = 1.5(right).Thisbecomesaclassicaltradeoffbetweendenoisingandoverblurring.In
practice,wefindα ∈[1.1,1.3]workswell,andweuseα =1.1forallthetestmodelsshowninthearticle.
For a genus n > 0 surface, there are n loops in the embedded Reeb graph (cid:1) that are homologically
inequivalentandformthebasesofallloopsin(cid:1).InSection3.4,wedescribehowweusetheseloopsfor
genusreduction,thatis,convertingagenusnsurfaceintoagenuszerosurface.
3.3 FeatureIdentification
Oncethetipofaprotrusionislocated,weconstructaclosedcurveγ onthesurfacethatseparatesthe
feature from the remaining body. Using the terminology from Erickson and Har-Peled [2002], γ is a
separatingcycle.Wecomputeγ intwosteps.First,wefindaseparatingregion R correspondingtothe
tipoftheprotrusion.Next,weconstructγ from R.
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Feature-BasedSurfaceParameterizationandTextureMapping • 9
Fig.6. Inthisfigure,wefindaseparatingregionforthebunny’sear.IntheleftwegraphA(x),theareaoftheregionsgivenby
evenlyspaceddistancesfromtheear’stip(red),andthesmoothedareas(blue).Wethencalculate A(cid:8)(cid:8)(x),thesecondderivative
(middle).Themaximumof A(cid:8)(cid:8)(x)correspondstotheplacewheretheearjoinsthehead(right,thegreenband).
To find a separating region for the tip point p of a feature, we first calculate the function f (q) =
p
g(p,q), the surface geodesic distance function with respect to p. f is normalized to take on values
p
in [0,1]. We consider the regions bounded by iso-value curves of this function. Specifically, we divide
the interval [0,1] into k equal sections. Next, by performing region-growing from p, we partition the
surfaceintolevelsetbandsbasedonthevaluesof f intheseintervals:
p
(cid:8) (cid:9)
i−1 i
M := q∈ S| ≤ f (q)≤ (9)
i p
k k
A :=Area(M ). (10)
i i
TheconstructionoflevelsetsinEquation9isinspiredbyMorsetheory.Theareaofthissequenceof
bandschangesslowlyalongaprotrusion,butitchangesabruptlywherethefeaturejoinstotherestof
the surface.We findthe separatingregion byanalyzing {A},which wetreat asa continuousfunction
i
A(x). Along a perfect cylindrical feature, A(x) is constant. In the case of a cone, the function grows
linearly. At places where a protrusion joins the main body, A(x) will have a sudden increase, and this
will be the boundary of the feature. We find these increases by looking for the maxima in A(cid:8)(cid:8)(x), the
second derivative of A(x). To eliminate small undulations in A(x), we first low-pass filter A(x) using
a Gaussian function for N times. Both k and N affect the efficiency of separating region detection.
The larger k is, the more samples are used to discretize A(x), and the more likely small noise will be
consideredaspotentialplacesfortheseparatingregion.Similarly,if N istoolarge,thelocationforthe
separatingregionmaybeshiftedorevenlost.Inpractice,weusek = 100and N = 30forallourtest
models. These choices seem to produce reasonable results. Figure 6 illustrates the process of feature
identificationforthebunny’sear.
Let m be the location where A(cid:8)(cid:8)(m) takes on its maximum value. We define the separating region
R := {q ∈ S | m−(cid:7) ≤ f (q) ≤ m+(cid:7)}, where we typically use (cid:7) = 0.02. By choosing a positive (cid:7), we
p
intentionallymakeRlargefortworeasons.First,poortriangulationsmaycauseRtobenonseparating,
that is, it does not contain a separating cycle. Second, we would like more flexibility to allow the
separatingcycle(withinthisregion)tobeshortandsmooth.
The topology of the separating region R can be rather complex if there are other features that join
the surface in nearby places. The only guarantee R provides is that it indeed separates the feature
ACMTransactionsonGraphics,Vol.24,No.1,January2005.
10 • E.Zhangetal.
from the rest of the surface. We produce a separating cycle γ from R as follows. First, we reduce R
into its skeleton, that is, a collection of edges in the surface that separate the feature from the rest
of the surface. Dangling edges are removed as well. Gu et al. [2002] perform a similar operation to
producegeometryimagesfrommeshes.Next,wefindaseparatingcycleρ fromthisskeleton.Finally,
weconstructanotherseparatingcycleγ thatisbasedonρ,butthatisingeneralshorterandsmoother.
Theseoperationsareeasytoimplementonmeshesandwedescribethemindetailnext.
(1) Reduce a separating region R into its skeleton and remove dangling edges. This is achieved
by treating R as a 2-complex (with boundary edges) and repeatedly performing “elementary col-
lapses”[Kaczynskietal.2004],inwhichatrianglewithatleastoneboundaryedgeisremovedfromthe
complexalongwithoneoftheboundaryedges.Intheend,all2-cells(triangles)areremoved,andthe
2-complexisreducedtoa1-complex.Whentherearemultiplechoicesofboundaryedgestocollapse,we
selecttheedgewiththelargestAGDvalue,whichtendstobeclosertothefeaturetippthantheother
edges in the 2-complex. The resulting graph is a skeleton of R with dangling edges. We remove the
danglingedgesthroughelementarycollapsesonthe1-complex.Thisresultsinacollectionofloops,one
ofwhichmeetsourrequirementastheseparatingcycle.Theothersfallintotwocategories:separating
cyclesforsomegeometricfeaturesinsidethefeatureregion,andseparatingcyclesforsomegeometric
featuresoutsidethefeatureregion.
(2)Eliminatefromthe1-complexseparatingcyclesthatareeitherinsideoroutsidethefeatureregion.
Toremovetheloopsoutsidethefeatureregion,weperformregion-growingfromthefeaturetippwith
theconstraintthatnotrianglescanbeaddedthatcrossanedgeintheloopscomputedfromthestepsin
(1).Thismakestheloopsoutsidethefeatureregionunreachablefromp.Forloopsinsidethefeaturere-
gion,theaverageAGDvaluesoftheirverticesare,ingeneral,greaterthanthatontheseparatingcycle.
Therefore,theseloopscanbeeasilyidentifiedanddiscarded.Thisstepproducesaseparatingcycleρ.
(3) Shorten and smooth the separating cycle ρ. We choose two vertices t and t on ρ that are the
1 2
closesttothefeaturetipp.Wefindtwopathsthatconnectt andt top,respectively.Thetwopaths
1 2
dividethefeatureregionintotwodisjointregions.Withineachregion,thereisashortestpathbetween
t and t . Together, they form a separating cycle, which tends to be shorter and smoother than ρ. By
1 2
repeatingthisprocesstwice,weobtainadesiredseparatingcycleγ.
Figure 7 illustrates this process. For a feature point p and a separating region R (a, the shaded
region),wereduce R toitsskeletonthroughelementarycollapses(b).Next,loopsthatareeitherinside
oroutsidethefeatureregionofpareeliminated(c).Inthebottomrow,theseparatingcycleρisshortened
andsmoothedtoproduceγ,throughstep3(d-f).
A separating cycle divides S into two surfaces with boundaries. We eliminate these boundaries by
“filling in” the holes with triangles. Basically, we compute c, the average position of the boundary
vertices, and make c a new vertex. We then triangulate the hole by connecting c to the vertices on
theboundary.Thefillertrianglesaresubdividedtwice,followedbyLaplaciansmoothingonthenewly
created vertices. Some filler triangles can be seen where the head has been separated from the neck
of the bunny in Figure 1. These filler triangles are flagged so that they have minimal effect on patch
unfolding.Theybecomewhatwecallscaffoldtriangles,tobedescribedlater.
We repeat the feature identification process for the resulting surfaces until the original surface is
dividedintoasetoffeatureregionsandtherearenomorefeatureregionstobefound.Figure1shows
theresultofthisprocessonthebunny,inwhichfourregionswerecreated.
Our feature identification algorithm assumes that a single loop always divides the surface into two
disjointregions,whichisnotnecessarilytrueforsurfaceswithhandles.Forthesesurfaces,thetopology
oftheseparatingregioncanbearbitrarilycomplexwheremanysmallhandlesareclusteredtogether.
To avoid dealing with this situation, we perform genus reduction before feature identification. Genus
reductionconvertsagenusn>0surfaceintoagenuszerosurface.
ACMTransactionsonGraphics,Vol.24,No.1,January2005.