EECS487:Interactive

ComputerGraphicsLecture36:•  Polygonalmeshsimplification

TheModeling-RenderingParadigm

Modeler:

Modelingcomplexshapes

•  noequationforachair,face,etc.

•  instead,achievecomplexityusing

simplepieces

•  polygons,parametricsurfaces,or

implicitsurfaces

•  witharbitraryprecision,inprinciple

Renderer:

scenegraph

Vertexdata

Fixedfunction

transformand

lightingVertexshader

Clip,homogeneous

divideandviewport

Texturestages Fragmentshader

Rasterize

Fragmentmerging:

stencil,depth

3DGeometryRepresentations

Points

•  2D:rangeimage

•  3D:pointcloud

Surfaces

• Polygonalmesh

• Parametricsurfaces

•  Subdivisionsurfaces•  Implicitsurfaces

Solids

• ConstructiveSolidGeometry

• Voxels

Procedural

• Particlesystem

• Spring-masssystem

•  Fractals

Funkhouser

Representdifferentkindsofinformation:

pointdata,surfacedata,volumetricdata

2D:RangeImage

Imagewithdepthinformation

•  acquiredfromrangescanner,incl.MicrosoftKinectand

GoogleTango

•  notacomplete3Ddescription:doesnotinclude

partofobjectoccludedfromviewpoint

Rangeimage Tessellation Rangesurface

Funkhouser,Ramamoorthi

Curless

Cyberware

3D:PointCloud

Unstructuredsetof3D

pointsamples

Acquiredfromrangefinder

Disadvantage:

nostructuralinfo

•  adjacency/connectivityhavetousee.g.,k-nearestneighborstocompute

Increasinglyhottopicin

graphics/visiontoday

Microscribe3D

Funkhouser,Ramamoorthi,Ohtake

Surfaces

Boundaryrepresentation(B-reps)

•  sometimesweonlycareaboutthesurface,e.g.,when

renderingopaqueobjectsandperforminggeometric

computations

Chenney,Lozano

3Dsurface triangles

SolidModelingSomerepresentationsarebestthoughtofas

definingthespacefilled

• medicaldatawithinformationattachedtothespace

•  transparentobjectswithinternalstructure

•  takingcutsoutofanobject,“WhatwillIseeifIbreakthisobject?”

Chenney,Lozano

3Dvolume volumeelement

“voxel”

ChoosingaRepresentation

Efficiencyfordifferenttasks:

•  creation/acquisition:byhand,procedurally,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

Funkhouser,Chenney

AdvantagesofRepresentations

Manipulation:

•  splineseasiestoriginally,butnow

manyalgorithmsforpolygonmeshes

Acquisitionandmodeling:

•  splines,CSGoriginallyusedformodeling

•  butincreasinglycomplexmeshes,rangeimages,

andpointcloudacquiredfromrealworld

Simplicity:meshes

Efficienthardwarerendering:meshes

Ramamoorthi

WhatdoPeopleUse?

Trianglemeshesmostwidelyused

Subdivisionsurfacesusedalotinmovies

Splinepatchesusedbymodelingprograms

ConstructiveSolidGeometry(CSG)usedfor

modelingmachineparts

Volumedatausedinmedicalimaging

Rangeimagesusedinimage-basedrendering

Pointcloudsbecomingincreasinglyrelevant,

especiallywithcomputervision

Ramamoorthi

ComparisonofB-Reps

Funkhouser

Features Polygonal

Mesh

Implicit

Surface

Parametric

Surface

Subdivision

Surface

Accurate ✗ ✔ ✔ ✔

Compact ✗ ✔ ✔ ✔

Intuitive ✗ ✗ ✔ ✗

Local ✔ ✗ ✔ ✔

Affine ✔ ✔ ✔ ✔

Realobjects ✔ ✔ ✗ ✔

Continuity ✗ ✔ ✔ ✔

Parameterization ✗ ✗ ✔ ✗

Rendering ✔ ✗ ✔ ✔

Intersections ✗ ✔ ✗ ✗

Introductiontofivetopics:

1.  howtodrawgoodlookingmeshes?

2. meshsimplification

3.  levelofdetail

4. meshrepresentation

5.  errorcheckingandmeshprocessing

PolygonalMesh

DelaunayTriangulationHowtocreatea“goodlooking”

trianglemeshfromasetofpoints?

• minimumverticesandtriangles

•  fewer,largertriangles

•  nosliver

AtriangulationisDelaunayifffor

eachedgethecircumcircleofan

adjacenttriangledoesnotcontaintheoppositevertex

Amongallpossibletriangulations,theDelaunay

triangulationmaximizesthesmallestangle

Rhymeswith“baloney”[Hart,Bischoff&Kobbelt]

DualMeshesandVoronoiDiagram

Thedualofameshexchangesitsfacesandvertices

•  placenewverticesatcentroidoffaces

Edgesindualcrossoriginaledges

•  atrightangles•  atmidpoints

•  (butmightnotactually“cross”

ifnewvertexisatoriginaledge)

DualofaDelaunayTriangulation

isaVoronoiDiagram

• aVoronoifacedenotesaregionclosertoagivenDelaunay(original)vertexthanto

anyotherDelaunayvertex

Hart,vanLaerhoven

MeshSimplification

Morepolygonsincreasesmodelaccuracy,but

requiresmorespaceandprocessing/renderingtime

• needforaccuracydependsontheapplication•  gamevs.medicalimaging

•  approximatesolutionsvs.finalsimulations

•  screenresolutionorviewingdistance

maynotcallforaveryaccuratemodel

Acquisitionsystems(e.g.,3Dscanner)oftenproducehugemodels

• moredetailedthannecessary

• millionsofpolygonsperobjectarecommon

•  billionsofpolygonsperobjectarestartingtohappen•  300millionfaces=3.7GBaftergzipcompression

Yu

MeshSimplification

Reducepolygoncount:

•  lessstorage•  fasterrendering•  simplermanipulation

Simplificationalgorithms:

•  vertexclustering• meshretiling

• meshdecimation

• meshoptimization

Funkhouser

Desiredproperties:

•  generalityintypesofmesh

•  efficiencyandscalability

•  qualityofapproximation

•  visual•  geometric:topologicalmodifications

•  controlofapproximationquality•  continuousLoD•  smoothtransitionsbetweenmodels

VertexClustering

Method:

•  partitionspaceintocells•  grids,spheres,octrees,...

• mergeallverticeswithinthesamecell

•  triangleswithmultipleverticesinonecell

degenerateintolinesorpoints

Properties:

•  generalandrobust•  allowstopological

changes

•  notbestquality

Hart,Funkhouser

MeshRetiling

Resamplemeshwith

“uniformlyspaced”

randomvertices:

•  generaterandom

verticesonsurface

•  spreadthemuniformly

usingdiffusion/repulsion

•  triangulatevertices

Properties:

•  slow

•  blurssharpfeatures

Funkhouser,Turk

MeshDecimation

Simplificationalgorithm:

•  eachoperationsimplifiesthemodelbyasmallamount

•  applymanyoperationsinaniterative,greedyfashionto

graduallyreducecomplexityofmesh:1.  measureerrorintroducedbypotentialdecimationoperations

2.  placeoperationsinapriorityqueuesortedbyerror3.  performoperationsinqueuesuccessively

4.  aftereachoperation,re-evaluateerrormetrics

Typesofoperations:

•  vertexremove

•  edgecollapse•  vertexcluster(virtualedgecollapse)

First,sometopologicalpropertiesofmeshes...

TP3,Funkhouser

bad

ManifoldSurface

LetMbeasurfacein3D

Misamanifoldiffaneighborhood

ofanypointpinMistopologically

equivalenttotheunitopendisk

(interiorofaunitcircle)

Topologicalequivalence(or

homeomorphism)allows

deformationthatdoesnotrip,

tear,orpokeholes

Onamanifoldsurface:

•  everyedgeissharedbyexactly2faces•  aroundeachvertexexistsaclosedloopoffaces

good

Hart,TP3

partof

manifold

surface

M’isamanifoldwithboundaryiffaneighborhoodofanypointpinM’ishomeomorphictoahalfdisk

Onamanifoldwithaboundary:

•  edgesontheboundarybelongtoexactlyoneface

•  aroundverticesontheboundarytheloopoffacesisopen

A3Dsurfacethatisamanifoldsurface

withoutboundaryisaclosedsurface

ManifoldswithBoundary

amanifoldsurface

withboundary

TP3

PiecewiseLinearManifolds

Closedmesh

Nodanglingfaces

Edgesonlyboundtwofaces

Watertight

•  noholesinthesurface•  noboundary•  orientable

non-manifold

boundaryvertex

non-manifoldedge dangling

face

Hart,TP3

OrientableSurface

Orientablesurfacehas2sides,likeapieceofpaper

Byconvention,thenormalofa

closedorientablesurfacepoints“outwards”

TheMöbiusstripandKleinbottleare

non-orientablesurfaces:

TP3,EscherKarcher,Chu-Carroll,

VertexRemove

Hart,Funkhouser,Ramamoothi

Method

•  removevertexandadjacentfaces

•  fillholewithnewtriangles

(2lesstriangles)

Properties

•  requiresmanifoldsurfacearoundvertex

•  preserveslocaltopologicalstructure•  filingholewellmaynotbeeasy

EdgeCollapse

Method

• mergetwoedgeverticesintoone

•  deletetwodegeneratetriangles

Properties

•  requiresmanifoldsurfacearoundvertex

•  preserveslocaltopologicalstructure•  allowssmoothtransition

Hart,Funkhouser

vertexsplit

edgecollapse

VirtualEdgeCollapse

A.k.a.vertex-paircontractionorvertexcluster

•  joinspreviouslyunconnectedareas•  allowstopologicalsimplification

•  usuallylimitedtosmalldistance

toavoidO(N2)virtualedges•  notbestquality

Hart,Manocha

Asurfaceissimplicialcomplexiff

• polygonsmeetonlyalongtheiredges

•  edgesintersectonlyattheirendpoints

SimplicialComplexSurface

TP3

yes not

IterativeEdgeCollapse

Properties

• well-definedforanysimplicial

complexsurfaces

•  induceshierarchyonthesurface:allowssmoothtransition(geomorph)

•  currentlymostpopulartechnique

(standardfeatureinDirect3D)

Hart,Funkhouser,Watt00

ErrorMetricsforSimplification

Usedtorankedgesduringsimplification

•  reflectsamountofgeometricerrorintroduced

• maindifferentiatingfeatureamongalgorithms

Mustaddresstwointerrelatedproblems

• whatisthebestcontractiontoperform?

• whatisthebestpositionv’forremainingvertex?

•  canjustchooseoneoftheendpoints•  butcanoftendobetterbyoptimizingpositionofv’

SeeTP3 §6.5fordetailsJ

Hart

LevelofDetail(LoD)Pyramid

Singleresolutionnotenough

•  applicationcontextdictatesrequireddetail•  contextvariesovertime(andspace)

Levelofdetail:

•  replaceeachobjectinthescenegraphwithahierarchy

ofobjectsatdifferingresolutions

•  choosethemodelappropriateforcurrentcontext:

collisiondetectionvs.rendering,frameratevs.quality

RTR,

Hodgins

DiscreteLoD

Givenamodel,buildasetofapproximations

•  canbeproducedbyanysimplificationsystem

•  atruntime,simplyselectwhichtorender

•  fairlyefficient

•  storagerequired< 2xoriginal•  costofchanginglevelofdetailwhilerenderingnotsignificant

•  imagepyramids(mip-maps)agoodexample

Inter-frameswitchingcauses“popping”

•  cansmoothtransitionwithimageblending

•  orgeometryblending(requirescontinuousLoD)

Supportedbyseveralscenegraphs:

• RenderMan,OpenInventor,IRISPerformer,...

Hart

ContinuousLoD

Needformulti-resolutionmeshes:

•  toreduce“popping”whenswitchingmodels,

geometricmorphbetweentwoLoDs

•  sometimescannotchooseasingleLOD,mayneeddifferent

amountsofdetailsonthesamesurface(seenextslide)

•  progressivetransmission(detailincreasesovertime)

Funkhouser,Certainetal.

ProgressiveMeshing

Iterativelydecimateameshusingedgecollapse

Storetheinversevertexsplitforeachcollapse

Themostsimplifiedmesh(basemesh)and

vertexsplitrecordsformtheprogressivemesh:

M = M n � … � M 175 � … � M 1 � M 0

M n � … � M i � … � M 175 �… � M 1 � M 0 = M

� ecoln-1 ecoli ecol0

�

vspln-1 vspli-1 vpsl0

13,546 3478 500 152 150

Hoppe96,Manocha,Funkhouse

r

RatherthanafewdiscreteLODswehaveafullrange

•  vertexsplitdoesnotrequiremuchstorage

•  themeshesareflexibleandeasilyreversible

•  requiresoriginalpositionstobekeptwitheachedgecollapse

Supportforselectiverefinement

•  requiresmoreinfoonadjacentvertices

andfacesofeachcollapsededge

Cangeomorphsmoothly

betweenLODs

• minimizes“popping”

ProgressiveMeshing

Manocha,TP3

Commonlyusedforterraingeneration

TerrainclosetoviewerisshownwithagreaterLoD

Preserve:

•  silhouette•  specularhighlights

View-DependentSimplification

Manocha,Hoppe

cluster

Introductiontofivetopics:

1.  howtodrawgoodlookingmeshes?

2. meshsimplification

3.  levelofdetail

4. meshrepresentation

5.  errorcheckingandmeshprocessing

PolygonalMesh

PolygonalMeshRepresentation

Howwouldyourepresentapolygonalmesh?

Threealternatives:

1.  explicitmesh(facelist)

2.  vertexlist3.  edgelist

Importantproperties:

•  efficienttraversaloftopology(fordrawing,e.g.)

•  efficientuseofmemory(compactness)

•  efficientupdates(UI,vertexremoval,computingper-

vertexnormals)

(x1, y1, z1) (x2, y2, z2)

(x3, y3, z3) (x4, y4, z4)

Extension:moretopologyinformation:inaddition

toalistofvertices,eachfacealso

• pointstoitsadjacentfaces• andforeachadjacentface,theindexofthesharededge

Especiallyconvenientforsubdivisionandmultiresolution

hierarchies

PolygonalMeshRepresentation

[Zorin,Bischoff&Kobbelt]

PolygonalMeshRepresentation

Adjacencyoperationsimportantinmesh

simplificationandmanyotherapplications:

•  givenface,finditsvertices•  givenvertex,findfacestouchingit•  givenface,findneighboringfaces•  givenvertex,findneighboringvertices•  givenedge,findverticesandfacesittouches

Ramamoorthi

Explicitmeshor“polygonalsoup”model:

alistofpolygonalfaces

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:

•  sharedverticesareduplicated• notopologyinformation(e.g.,whichvertices

andedgesareshared),mustsearchthewhole

listtomodifyavertex

•  roundofferrors:cracksandfailuretomatchvertices

PolygonalMeshRepresentation

Vertexlistor“indexedfaceset”:

Example:V = {(x1, y1, z1), (x2, y2, z2),�(x3, y3, z3), (x4, y4, z4)},

P1 = (v1, v2, v3), P2 = (v3, v2, v4)Characteristics:

•  sharedverticesarestoredonlyonce•  butstillnotopologyinformation:

findingsharededgesstillrequiresasearch

Extension:triangleneighborlist:

• vertexpointstoasingleneighboringtriangle• trianglepointstoitsthreeneighboringtriangles• canenumeratetrianglesaroundavertex

PolygonalMeshRepresentation

Marschner

Usedin

3dsmax,

X3DOBJ

fileformat

Edgelist:

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)

Extension“wingededge”:

• eachedgepointsto:•  twoendpointvertices

•  twofacesthatshareedge

•  fouredgesemanatingfromitsendpoints

•  faces,verticescontainpointertooneedge

PolygonalMeshRepresentation

Marschner

Polygonmeshcanarriveinerror

duetohumanerrorsorscannerlimitations

Thingstocheck:

•  everyvertexisanendpointtoatleasttwoedges

•  everyedgeispartofatleastonepolygon•  everypolygonisclosed•  everypolygonhasatleastonesharededge

Otherchecks:

•  vertexnormal

•  normaltotheplane

•  plane’simplicitfunction

•  convexornot•  holes?surfacewatertight?

PolygonMeshErrorChecking PolygonMeshProcessing

Topologicalfixups:fixholes,cracks,self-intersection

Podolak Borodin

Funkhouser

Notaverycompactrepresentation

•  needsalotofflatelementstorepresent

smoothorhighlydetailedsurfaces

•  accuracy:exactnessofrepresentationcanonlybeapproximatedbyincreasingthenumberofpolygons

•  ifimageisenlarged,planarsurfacesagainbecomeobvious

Intersectiontest?Inside/outsidetest?

Hardtoedit

•  creatingpolygonalobjectsisstraightforward…

thoughlaboriousandtedious

•  howdoyoueditapolygonal-meshsurface?

•  don’twanttomoveindividualvertices…

•  difficulttodeformobject:aregionoflowcurvature,representedwithlow

polygoncount,cannotbedeformedintoahighcurvatureregion

•  itismoreamachinerepresentationthanaconvenientuserrepresentation

TheProblemswithPolygons