TheModeling-RenderingParadigm
EECS487:Interactive
ComputerGraphics
Renderer:
Modeler:
Modelingcomplexshapes
Vertexdata
•  noequationforachair,face,etc.
Fixedfunction
transformand
lighting
•  instead,achievecomplexityusing
•  polygons,parametricsurfaces,or
Lecture36:
•  Polygonalmeshsimplification
Vertexshader
simplepieces
implicitsurfaces
scenegraph
Clip,homogeneous
divideandviewport
Rasterize
•  witharbitraryprecision,inprinciple
Texturestages
Fragmentshader
Fragmentmerging:
stencil,depth
2D:RangeImage
3DGeometryRepresentations
Representdifferentkindsofinformation:
pointdata,surfacedata,volumetricdata
Imagewithdepthinformation
Points
Solids
•  2D:rangeimage
•  ConstructiveSolidGeometry
•  notacomplete3Ddescription:doesnotinclude
•  3D:pointcloud
•  Voxels
Surfaces
Procedural
•  Polygonalmesh
•  Particlesystem
•  Parametricsurfaces
•  Spring-masssystem
•  Subdivisionsurfaces
•  Fractals
•  Implicitsurfaces
•  acquiredfromrangescanner,incl.MicrosoftKinectand
GoogleTango
partofobjectoccludedfromviewpoint
Cyberware
Curless
Rangeimage Tessellation
Funkhouser
Rangesurface
Funkhouser,Ramamoorthi
Surfaces
3D:PointCloud
Unstructuredsetof3D
pointsamples
Boundaryrepresentation(B-reps)
•  sometimesweonlycareaboutthesurface,e.g.,when
renderingopaqueobjectsandperforminggeometric
computations
Acquiredfromrangefinder
Disadvantage:
nostructuralinfo
•  adjacency/connectivityhave
tousee.g.,k-nearest
neighborstocompute
Increasinglyhottopicin
graphics/visiontoday
3Dsurface
triangles
Microscribe3D
Funkhouser,Ramamoorthi,Ohtake
Chenney,Lozano
SolidModeling
ChoosingaRepresentation
Somerepresentationsarebestthoughtofas
definingthespacefilled
Efficiencyfordifferenttasks:
•  medicaldatawithinformationattachedtothespace
byfittingtomeasurements
•  interaction/manipulation:simplification,
compression,localcontrolofshapefor
modeling,animation,etc.
•  geometriccomputation:distance,intersection,
normalvectors,smoothnessandcontinuity,
abilitytoevaluatederivatives,curvature,
similaritycomparisons,indexing,search
•  storageandtransmission:compactness
•  rendering,e.g.,withhardwareaccelerator:
converttopolygon
•  creation/acquisition:byhand,procedurally,
•  transparentobjectswithinternalstructure
•  takingcutsoutofanobject,
“WhatwillIseeifIbreakthisobject?”
3Dvolume
volumeelement
“voxel”
Chenney,Lozano
Funkhouser,Chenney
AdvantagesofRepresentations
WhatdoPeopleUse?
Manipulation:
Trianglemeshesmostwidelyused
Subdivisionsurfacesusedalotinmovies
•  splineseasiestoriginally,butnow
manyalgorithmsforpolygonmeshes
Splinepatchesusedbymodelingprograms
ConstructiveSolidGeometry(CSG)usedfor
modelingmachineparts
Volumedatausedinmedicalimaging
Acquisitionandmodeling:
•  splines,CSGoriginallyusedformodeling
•  butincreasinglycomplexmeshes,rangeimages,
andpointcloudacquiredfromrealworld
Simplicity:meshes
Rangeimagesusedinimage-basedrendering
Pointcloudsbecomingincreasinglyrelevant,
especiallywithcomputervision
Efficienthardwarerendering:meshes
Ramamoorthi
ComparisonofB-Reps
Features
Polygonal Implicit Parametric Subdivision
Mesh
Surface Surface
Surface
Ramamoorthi
PolygonalMesh
Accurate
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Introductiontofivetopics:
Compact
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1.  howtodrawgoodlookingmeshes?
Intuitive
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2.  meshsimplification
Local
Affine
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3.  levelofdetail
Realobjects
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Continuity
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Parameterization
Rendering
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Intersections
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4.  meshrepresentation
5.  errorcheckingandmeshprocessing
Funkhouser
DelaunayTriangulation
DualMeshesandVoronoiDiagram
Howtocreatea“goodlooking”
trianglemeshfromasetofpoints?
Thedualofameshexchangesitsfacesandvertices
•  minimumverticesandtriangles
•  fewer,largertriangles
•  nosliver
Edgesindualcrossoriginaledges
•  placenewverticesatcentroidoffaces
•  atrightangles
•  atmidpoints
AtriangulationisDelaunayifffor
eachedgethecircumcircleofan
adjacenttriangledoesnotcontaintheoppositevertex
•  (butmightnotactually“cross”
ifnewvertexisatoriginaledge)
DualofaDelaunayTriangulation
isaVoronoiDiagram
Amongallpossibletriangulations,theDelaunay
triangulationmaximizesthesmallestangle
•  aVoronoifacedenotesaregioncloserto
agivenDelaunay(original)vertexthanto
anyotherDelaunayvertex
Rhymeswith“baloney”
[Hart,Bischoff&Kobbelt]
MeshSimplification
Hart,vanLaerhoven
MeshSimplification
Morepolygonsincreasesmodelaccuracy,but
requiresmorespaceandprocessing/renderingtime
Reducepolygoncount:
•  lessstorage
•  fasterrendering
•  simplermanipulation
•  needforaccuracydependsontheapplication
•  gamevs.medicalimaging
•  approximatesolutionsvs.finalsimulations
•  screenresolutionorviewingdistance
maynotcallforaveryaccuratemodel
Simplificationalgorithms:
•  vertexclustering
•  meshretiling
•  meshdecimation
•  meshoptimization
Acquisitionsystems(e.g.,3Dscanner)
oftenproducehugemodels
•  moredetailedthannecessary
Desiredproperties:
•  generalityintypesofmesh
•  efficiencyandscalability
•  qualityofapproximation
•  visual
•  geometric:topologicalmodifications
•  controlofapproximationquality
•  continuousLoD
•  smoothtransitionsbetweenmodels
•  millionsofpolygonsperobjectarecommon
•  billionsofpolygonsperobjectarestartingtohappen
•  300millionfaces=3.7GBaftergzipcompression
Yu
Funkhouser
VertexClustering
MeshRetiling
Resamplemeshwith
“uniformlyspaced”
randomvertices:
Method:
•  partitionspaceintocells
•  grids,spheres,octrees,...
•  mergeallverticeswithinthesamecell
•  triangleswithmultipleverticesinonecell
degenerateintolinesorpoints
•  generaterandom
verticesonsurface
•  spreadthemuniformly
usingdiffusion/repulsion
•  triangulatevertices
Properties:
•  generalandrobust
•  allowstopological
Properties:
changes
•  notbestquality
•  slow
•  blurssharpfeatures
Hart,Funkhouser
ManifoldSurface
MeshDecimation
LetMbeasurfacein3D
Simplificationalgorithm:
•  eachoperationsimplifiesthemodelbyasmallamount
•  applymanyoperationsinaniterative,greedyfashionto
Misamanifoldiffaneighborhood
ofanypointpinMistopologically
good
equivalenttotheunitopendisk
(interiorofaunitcircle)
graduallyreducecomplexityofmesh:
1.
2.
3.
4.
Funkhouser,Turk
measureerrorintroducedbypotentialdecimationoperations
placeoperationsinapriorityqueuesortedbyerror
performoperationsinqueuesuccessively
aftereachoperation,re-evaluateerrormetrics
Topologicalequivalence(or
homeomorphism)allows
deformationthatdoesnotrip,
tear,orpokeholes
Typesofoperations:
•  vertexremove
•  edgecollapse
•  vertexcluster(virtualedgecollapse)
Onamanifoldsurface:
First,sometopologicalpropertiesofmeshes...
TP3,Funkhouser
•  everyedgeissharedbyexactly2faces
•  aroundeachvertexexistsaclosedloopoffaces
bad
partof
manifold
surface
Hart,TP3
PiecewiseLinearManifolds
ManifoldswithBoundary
M’isamanifoldwithboundaryiffaneighborhood
ofanypointpinM’ishomeomorphictoahalfdisk
Closedmesh
Nodanglingfaces
Onamanifoldwithaboundary:
•  edgesontheboundarybelongto
exactlyoneface
•  aroundverticesontheboundary
theloopoffacesisopen
A3Dsurfacethatisamanifoldsurface
withoutboundaryisaclosedsurface
Edgesonlyboundtwofaces
Watertight
non-manifoldedge
•  noholesinthesurface
•  noboundary
•  orientable
amanifoldsurface
withboundary
TP3
OrientableSurface
Orientablesurfacehas2sides,
likeapieceofpaper
Byconvention,thenormalofa
closedorientablesurfacepoints“outwards”
TheMöbiusstripandKleinbottleare
non-orientablesurfaces:
dangling
face
non-manifold
boundaryvertex
Hart,TP3
VertexRemove
Method
•  removevertexandadjacentfaces
•  fillholewithnewtriangles
(2lesstriangles)
Properties
•  requiresmanifoldsurfacearoundvertex
•  preserveslocaltopologicalstructure
•  filingholewellmaynotbeeasy
TP3,EscherKarcher,Chu-Carroll,
Hart,Funkhouser,Ramamoothi
VirtualEdgeCollapse
EdgeCollapse
A.k.a.vertex-paircontractionorvertexcluster
Method
•  mergetwoedgeverticesintoone
•  deletetwodegeneratetriangles
•  joinspreviouslyunconnectedareas
•  allowstopologicalsimplification
•  usuallylimitedtosmalldistance
vertexsplit
edgecollapse
toavoidO(N2)virtualedges
•  notbestquality
Properties
•  requiresmanifoldsurfacearoundvertex
•  preserveslocaltopologicalstructure
•  allowssmoothtransition
Hart,Funkhouser
Hart,Manocha
IterativeEdgeCollapse
SimplicialComplexSurface
Properties
•  well-definedforanysimplicial
Asurfaceissimplicialcomplexiff
complexsurfaces
•  polygonsmeetonlyalongtheiredges
•  induceshierarchyonthesurface:
•  edgesintersectonlyattheirendpoints
allowssmoothtransition(geomorph)
•  currentlymostpopulartechnique
(standardfeatureinDirect3D)
yes
not
TP3
Hart,Funkhouser,Watt00
LevelofDetail(LoD)Pyramid
ErrorMetricsforSimplification
Singleresolutionnotenough
Usedtorankedgesduringsimplification
•  reflectsamountofgeometricerrorintroduced
•  applicationcontextdictatesrequireddetail
•  contextvariesovertime(andspace)
•  maindifferentiatingfeatureamongalgorithms
Levelofdetail:
•  replaceeachobjectinthescenegraphwithahierarchy
ofobjectsatdifferingresolutions
•  choosethemodelappropriateforcurrentcontext:
collisiondetectionvs.rendering,frameratevs.quality
Mustaddresstwointerrelatedproblems
•  whatisthebestcontractiontoperform?
•  whatisthebestpositionv’forremainingvertex?
•  canjustchooseoneoftheendpoints
•  butcanoftendobetterbyoptimizingpositionofv’
SeeTP3 §6.5fordetailsJ
Hart
RTR,
Hodgins
ContinuousLoD
DiscreteLoD
Needformulti-resolutionmeshes:
Givenamodel,buildasetofapproximations
•  toreduce“popping”whenswitchingmodels,
•  canbeproducedbyanysimplificationsystem
•  atruntime,simplyselectwhichtorender
•  fairlyefficient
geometricmorphbetweentwoLoDs
•  sometimescannotchooseasingleLOD,mayneeddifferent
amountsofdetailsonthesamesurface(seenextslide)
•  progressivetransmission(detailincreasesovertime)
•  storagerequired< 2xoriginal
•  costofchanginglevelofdetailwhilerenderingnotsignificant
•  imagepyramids(mip-maps)agoodexample
Inter-frameswitchingcauses“popping”
•  cansmoothtransitionwithimageblending
•  orgeometryblending(requirescontinuousLoD)
Supportedbyseveralscenegraphs:
•  RenderMan,OpenInventor,IRISPerformer,...
Hart
Funkhouser,Certainetal.
ProgressiveMeshing
ProgressiveMeshing
Iterativelydecimateameshusingedgecollapse
Storetheinversevertexsplitforeachcollapse
Themostsimplifiedmesh(basemesh)and
vertexsplitrecordsformtheprogressivemesh:
ecoln-1
ecoli
ecol0
M = Mn
…
M 175 …
M1
M0
13,546
3478
500
152
RatherthanafewdiscreteLODswehaveafullrange
•  vertexsplitdoesnotrequiremuchstorage
•  themeshesareflexibleandeasilyreversible
•  requiresoriginalpositionstobekeptwitheachedgecollapse
Supportforselectiverefinement
•  requiresmoreinfoonadjacentvertices
andfacesofeachcollapsededge
150
Cangeomorphsmoothly
betweenLODs
•  minimizes“popping”
Mn
…
vspln-1
Mi
…
M 175
…
M1
vspli-1
M0
=M
vpsl0
View-DependentSimplification
Commonlyusedforterraingeneration
TerrainclosetoviewerisshownwithagreaterLoD
Preserve:
•  silhouette
•  specularhighlights
Manocha,TP3
Hoppe96,Manocha,Funkhouse
r
cluster
PolygonalMesh
Introductiontofivetopics:
1.  howtodrawgoodlookingmeshes?
2.  meshsimplification
3.  levelofdetail
4.  meshrepresentation
5.  errorcheckingandmeshprocessing
Manocha,Hoppe
PolygonalMeshRepresentation
PolygonalMeshRepresentation
Extension:moretopologyinformation:inaddition
toalistofvertices,eachfacealso
Howwouldyourepresentapolygonalmesh?
(x , y , z ) (x , y , z )
Threealternatives:
3
3
3
4
4
•  pointstoitsadjacentfaces
•  andforeachadjacentface,theindexofthesharededge
4
1.  explicitmesh(facelist)
2.  vertexlist
3.  edgelist
Especiallyconvenientforsubdivisionandmultiresolution
hierarchies
(x1, y1, z1)
(x2, y2, z2)
Importantproperties:
•  efficienttraversaloftopology(fordrawing,e.g.)
•  efficientuseofmemory(compactness)
•  efficientupdates(UI,vertexremoval,computingpervertexnormals)
[Zorin,Bischoff&Kobbelt]
PolygonalMeshRepresentation
PolygonalMeshRepresentation
Adjacencyoperationsimportantinmesh
simplificationandmanyotherapplications:
Explicitmeshor“polygonalsoup”model:
alistofpolygonalfaces
•  givenface,finditsvertices
Example:P = {P1, P2, P3, …, Pn}
P1= ((x1, y1, z1), (x2, y2, z2), (x3, y3, z3)),
P2= ((x3, y3, z3), (x2, y2, z2), (x4, y4, z4))
Problems:
•  givenvertex,findfacestouchingit
•  givenface,findneighboringfaces
•  givenvertex,findneighboringvertices
•  givenedge,findverticesandfacesittouches
•  sharedverticesareduplicated
•  notopologyinformation(e.g.,whichvertices
andedgesareshared),mustsearchthewhole
listtomodifyavertex
•  roundofferrors:cracksandfailuretomatchvertices
Ramamoorthi
PolygonalMeshRepresentation
PolygonalMeshRepresentation
Vertexlistor“indexedfaceset”:
Edgelist:
Example:V = {(x1, y1, z1), (x2, y2, z2),
(x3, y3, z3), (x4, y4, z4)},
P1 = (v1, v2, v3), P2 = (v3, v2, v4)
Example:V = {(x1, y1, z1), (x2, y2, z2),
(x3, y3, z3), (x4, y4, z4)},
E1 = (v1, v2, P1, ø),
E2 = (v2, v3, P1, P2),
E3 = (v3, v1, P1, ø),
P1 = (E1, E2, E3), P2 = (E2, E4, E5)
Characteristics:
•  sharedverticesarestoredonlyonce
•  butstillnotopologyinformation:
Usedin
3dsmax,
X3DOBJ
fileformat
findingsharededgesstillrequiresasearch
Extension“wingededge”:
•  eachedgepointsto:
•  twoendpointvertices
•  twofacesthatshareedge
•  fouredgesemanatingfromitsendpoints
•  faces,verticescontainpointertooneedge
Extension:triangleneighborlist:
• vertexpointstoasingleneighboringtriangle
• trianglepointstoitsthreeneighboringtriangles
• canenumeratetrianglesaroundavertex
Marschner
PolygonMeshErrorChecking
Polygonmeshcanarriveinerror
duetohumanerrorsorscannerlimitations
Thingstocheck:
Marschner
PolygonMeshProcessing
Topologicalfixups:fixholes,cracks,self-intersection
•  everyvertexisanendpointtoatleasttwoedges
•  everyedgeispartofatleastonepolygon
•  everypolygonisclosed
•  everypolygonhasatleastonesharededge
Otherchecks:
•  vertexnormal
•  normaltotheplane
•  plane’simplicitfunction
•  convexornot
•  holes?surfacewatertight?
Podolak
Borodin
Funkhouser
TheProblemswithPolygons
Notaverycompactrepresentation
•  needsalotofflatelementstorepresent
smoothorhighlydetailedsurfaces
•  accuracy:exactnessofrepresentationcanonly
beapproximatedbyincreasingthenumberofpolygons
•  ifimageisenlarged,planarsurfacesagainbecomeobvious
Intersectiontest?Inside/outsidetest?
Hardtoedit
•  creatingpolygonalobjectsisstraightforward…
thoughlaboriousandtedious
•  howdoyoueditapolygonal-meshsurface?
•  don’twanttomoveindividualvertices…
•  difficulttodeformobject:aregionoflowcurvature,representedwithlow
polygoncount,cannotbedeformedintoahighcurvatureregion
•  itismoreamachinerepresentationthanaconvenientuserrepresentation