EECS487:InteractiveComputerGraphicsLecture41:IntroductiontoProceduralModelingandAnimation•  Fractals•  Dynamics•  Particlesystems•  Behavioralanimation

ProceduralModelingConstructs3Dmodelsusingalgorithms•  formodelsthataretoocomplex(ortedious)tocreatemanually,e.g.,•  landscapes,mountains,clouds,planets•  trees,plants,ecosystems•  buildings,cities• usuallydefinedbyasmallsetofdata,orrules,thatdescribestheoverallpropertiesofthemodel,e.g.,treesdefinedbybranchingpropertiesandleafshapes• modelisthenconstructedbyanalgorithm•  toaddvarietyandrealism,•  ofteninvolvesfractalgeometry•  oftenincludesrandomness•  e.g.,asingletreepatterncanbeusedtomodelanentireforest

Schulze

Self-SimilaritySelf-similar:•  viewingscalenotapparentfromobjectappearance• objectfeaturesarestatisticallysimilarbetweenobjectpartsandtheoverallobject•  forexample,wealwaysseeajaggedlinenomatterhowclosewelookatacoastline

•  infinitedetails:looksgoodandnaturalateveryresolution• achievedusingrandomnumbersandfractionaldimension

Usuallygeneratedbyrecursivelyapplyingthesameoperation(orsetofoperations)toanobject

Yu,Merrell

RandomnessMakesmodelsmoreinteresting,natural,andlessuniformand“clean”

Duetothediscreteandfinitenatureofcomputers,wecanonlygeneratepseudo-randomnumbers,basedonsomeinitialseedvalue

•  pseudo-randomsequencesarerepeatable,simplybyresettingtheseedvalue•  adifferentseedvaluegeneratesadifferentsequenceofpseudo-randomnumbers

•  forrepeatability,becarefulofdynamiceventseffectingtheuseofthepseudo-randomsequence

Schulze

FractalDimensionMeasuresthe“roughness”ofobject• morejaggedobjectshavelargerfractaldimension

UseHausdorffvarianttoapproximatefractaldimension:•  subdivideobjectintoself-similarpieceswithscalingfactors•  countnumberofpieces(n)coveredbyoriginalobject•  fractaldimensiond = log(n)/log(s)•  e.g.,SierpinskiTriangle•  takeatriangle•  scaledownto½•  make3copiesandarrangeintoatriangleoforiginalsize

•  repeatforever•  d = log(3)/log(2) = 1.58

Yu,Merrell

ProceduralTerrainModelingLandscapesareoftenconstructedasheightfields

Want:aheightfunctiony = h(x)Randommidpointdisplacement•  startwithsomeinitialfigure(e.g.,lineortriangle)•  splitatmidpointsandaddrandomdisplacement•  recurse,decreasingthemagnitudeofdisplacements(byafactor0 < f < 1)

1DExample:

Yu,Merrell

2DExample

Yu

L-Systems

DevelopedbyA.Lindenmayer,abiologist,in1968tomodelgrowthpatternsofalgaeandplantsGrammar-basedfractal-likemodels,a.k.a.,“graftals”Basedonparallelstring-rewritingrules•  describeanobjectbyastringofsymbols•  andasetofproduction/rewritingrules•  incorporatenotionssuchasbranching,pruning,…

Schulze,Yu

L-SystemsExtensions

Bracketed:save/restorestate(forbranches)

Parametric:productiongovernsbyparameter,e.g.,changecoloratcertainlevelofrecursion

Stochastic:randomlychooseoneofseveralproductionrulesprovidedforasymbol

Contextsensitive:productionbasedonneighboringsymbols

Hodgins

L-SystemsDefineagrammar:G = {V, S, ω, P}• V:alphabet,setofsymbolsthatcanbereplaced(variables)•  S:setofsymbolsthatremainfixed(constants)• ω:axiom,stringofsymbolsdefininginitialstate• P:production/rewritingrulesExample:• V:A,B •  S:•  +:turnleft•  -:turnright•  [:pushstateontostack•  ]:popstatefromstack• ω:B • P:B→A[+B][-B]

Schulze,Kuffner

ParametricL-SystemTurtlegraphics:•  F:drawforward•  f:moveforward•  +:turnleftbyangle//parameter•  -:turnrightbyangle//parameter•  [:pushcurrentstateontostack•  ]:popcurrentstatefromthestack

Koch’sSnowflake• angle(2π)/6 //parameter• axiomF • productionrule:F→F+F--F+F

Start:F Generation1:F+F--F+F Generation2:•  F+F--F+F+F+F--F+F--F+F--F+F+F+F--F+F

Generationn:

TurtleGraphicsKoch’sIsland:• angle(2π)/4 • axiomF+F+F+F • productionrule:F→F+F-F-FF+F+F-F

Tree:• angle(2π)/16• axiom++++FS• productionrule:S→+[FS]-[FS]-[FS]

L-SystemsforPlantsL-systemscancapturealargenumberofplantspecies,thoughdesigningrulesforspecificspeciesisnoteasySeealgorithmicbotany.org/papers/•  afree200pagesebook•  coversmanyvariantofL-systemsanddifferentplanttypes

Hodgins

L-SystemforCities

real generated

Streetsystem:•  startwithasinglestreet• branchandextendwithparametricL-system• parameters:goalsandconstraints•  goalscontrolstreetdirection,spacing•  constraintsallowforparks,bridges,roadloops

Hodgins

Parish01

L-SystemforCities

Buildings:• buildingshapesarerepresentedasCSGoperationsonsimpleshapes

Hodgins

Mülleretal.06 Wonkaetal.03

Parish&Müller01

Keyframing:interpolatesmotionfrom“key”positions+ perfectcontrol canbetedious norealism

Procedural:solvesequationstocomputemotion+  realisticmotion+  automaticgeneration•  onceyouhavetheprogram,youcangetlotsofmotion

 difficulttocontrol hardtotellastorywithpurelyproceduralmeans•  mostlyusedforsupportingbackground/effects

Motion

McMillan

ProceduralAnimation

Usingaprocesstocontroloranimatesomeattribute,includingshape(modeling),oftheobject

Steps:•  programsomerulesforhowthesystemwillbehave

•  choosesomeinitialconditionsfortheworld

•  runtheprogram,maybewithuserinputtoguidewhathappens

•  programoutputstheposition/shapeofthesceneovertime

Chenney

ProceduralAnimationPhysically-basedanimation• dynamics:movements

• pointmassparticlesystems

•  spring-mass:animatingtheshapeandreactionofcloth

• fluidflow:waterwaves

Behavioralanimation

• birdorfishflocking

•  crowdanimation

Physically-basedAnimation

Kinematic(keyframing):describesmotionwithoutconsideringcausesleadingtomotion•  considersonlyposes

Dynamics(procedural):considersunderlyingforces•  inversedynamics:•  givenprescribedmotion,whataretheforcesandtorquesrequired?

•  forwarddynamics:•  computemotionfrominitialconditionsandphysicallaws:•  givenforcesandtorques,whatisthemotion?

Funkhouser09,Hodgins,Funge99

InverseDynamicsGivenprescribedmotion,whataretheforcesandtorquesrequired?

Activecharacter:characterhasinternalsourceofenergy

Animatorspecifiesconstraints:•  character’sphysicalstructure•  e.g.,articulatedfigure

•  character’stask•  e.g.,jumpfromheretothereintimet

•  otherphysicalstructurespresent•  e.g.,floortopushoffandland

• motionrequirements•  e.g.,minimizeenergy

Funkhouser,Lasseter

InverseDynamicsComputersolvesforthe“best”physicalmotionsatisfyingconstraints

Example:objectwithjetpropulsion

•  x(t):positionofobjectattimet

•  f(t):propulsionforceattimet

•  equationofmotion:mx” − f − mg = 0

•  task:movefromatobfromt0tot1withminimumjetfuel,minimizewithx(t0) = aandx(t1) = b

•  solvewithiterativeoptimizationmethod

Funkhouser

f (t)2t0

t1

∫ dt

Witkin&Kass

InverseDynamicsAdvantages:•  animatorsdon’thavetospecifydetailsofphysicallyrealisticmotionwithsplinecurves•  easytovarymotionsduetonewparametersand/ornewconstraints

Forexample,adaptingmotion:

originaljump heavierbaseFunkhouser,Witkin&Kass

E.g.,withstatemachines

Challenges:•  specifyingconstraintsandobjectivefunctions•  avoidinglocalminimaduringoptimization•  retargetingmotiontonewcharacters•  needalargervarietyofmotions(notjustwell-definedsportmotions)•  real-timeperformance

ControllingMotion

Funkhouser,Hodgins

compression decompression flight1 flight2 landinghorse

compression

decompression

flight1

flight2

entry

E.g.,withstatemachines

Challenges:•  specifyingconstraintsandobjectivefunctions•  avoidinglocalminimaduringoptimization•  retargetingmotiontonewcharacters•  needalargervarietyofmotions(notjustwell-definedsportmotions)•  real-timeperformance

ControllingMotion

Funkhouser,Hodgins

compression decompression flight1 flight2 landinghorse

compression

decompression

flight1

flight2

entry

ForwardDynamics

Funkhouser,Hodgins,McAllister

Giveninitialconditionsandphysicallaws,whatisthemotion?•  initialconditions:mass,forces,torques

•  applyphysicallaws,e.g.,Newton’slaws,Hook’slaw,etc.

•  simulatephysicalphenomena:•  gravity,momentum(inertia),friction,collisions,elasticity,solidity,flexibility,fracture,explosion,fluidflow,aerodynamics(drag/viscosity,turbulence)

• motioncomputedusingnumericalsimulationmethods•  particlesystems•  rigidbodies•  soft-objects(spring-mass)•  fluid

HowtoRenderFire?

Gilles

Texturemappingpolygonsisfastandacceptableforshort-livedeffects•  overlayaflamepointwithaseriesoftexturedpolygons•  forenhancedrealism,wecouldintroducesecondarylightsourcesatthefire

Problems:• hardtosustainforlongperiods• hardtospreadorchangeshape• notranslucency

ParticleSystems

Gilles,Funkhouser

Approximateflame,andotheramorphousobjects,byadiscretesetofsmallparticlesSingleparticlesareverysimple,justpointmasswithattributes:• mass,position•  velocity,forces•  color,temperature,shape•  lifetime

Largegroupscanproduceinterestingeffects:•  rockets:fireworks•  clouds:waterdrops

x = (x, y, z)

v

ParticleSystems

Gilles,Funkhouser,Reeves

Foreachframe:•  createnewparticlesaccordingtoaprobabilitydistributionandassignattributes• deleteanyexpiredparticles• updateparticlesbasedonattributesandphysics•  numericalsolutionstoODE•  renderparticles:motionblur,compositing

Wheretocreateparticles?• predefinedsources•  surfaceofshape• whereparticledensityislow,etc.

ParticleSystems

Funkhouser,Gilles,McAllister

Whentodeleteparticles?• predefinedsink•  surfaceofshape• wheredensityishigh•  lifetime•  random

Example:water• newparticlescreatedeachframe•  thenumbercreatedperframeisnormallydistributed• eachparticlehasinitialdownwardvelocity•  again,normallydistributed• oneachsuccessiveframe,eachparticleisactedonby“wind”forcetotheright

Impact

Treuille

?

Davidhazy

Static:•  radius•  mass(m)•  racquetinfo

Dynamic:•  position(x)•  velocity(v)•  rotation?

Newtonianparticles:F = m a• Fandaarevectors• givenFandpositionattimet,x(t),howdoesthepositionchangetox(t+1)?

DifferentialEquations

Witkin&Baraff

Differentialequationsdescribetherelationbetweenanunknownfunctionanditsderivatives

Solvingadifferentialequationmeansfindingafunctionthatsatisfiestherelation,plussomeadditionalconstraints

OrdinaryDifferentialEquation(ODE):•  “ordinary”:functionofonevariable•  partialdifferentialequation(PDE):morevariables

Discretetime:x t +1( ) = f x t( ),t( )Continuoustime:

dx t( )dt

= f x t( ),t( )positionvelocity

x =

xyz!x!y!z

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

DifferentialEquations

Funkhouser,Durand

Computingparticlemotionrequiressolvinga2nd-orderdifferentialequation:

Instead,addvariablevtoformcoupled1st-orderdifferentialequations:

Foranimation,wantaseriesofvaluesx(ti), i = 0, 1, 2, …• samplesofthecontinuousfunctionx(t)

x = F(x, x,t)m

v = x, v = F(x,v,t)m

notation!x fordx / dt

f(x, t)isavectorfielddefinedeverywhere•  e.g.,avelocityfield

•  itmaychangebasedont•  x(t)isapaththroughthefield:•  usuallynoanalyticalsolution

PathThroughaField

x(t) = x0 + f (x,t)dtt0

t

∫

x(t)

howtocompute?

Sincewe’reworkinginfixed-frameintervals,wecanuseasimpleapproximation(Euler’smethod):• definestepsizeh

• givenx0 = x(t0),takestep:�t1 = t0 + h�x(t1) = x0 + h f(x0, t0)

• piecewise-linearapproximationofthecurve•  stepsizecontrolsaccuracy:smaller,moreaccurate•  mayneedtotakemanysmallstepsperframe

SolvingtheIntegration

Durand

f(x, t)

f(x0, t)

x0

SolvingtheIntegration

Durand

Euler’smethodisthesimplestnumericalmethod

ConsidertheTaylorexpansionofx(t):

theequationforEuler’smethodsimplydisregardshigher-ordertermsandreplacesthefirstderivativewiththeflowfieldfunction�theerrorisontheorderofO(h2)Consequences:Euler’smethodisinaccurateandunstable

x t + h( ) = x t( ) + hdxdt

+ h2

2d 2xdt 2

+

Euler’sMethod:InaccurateTomovealongacircle:Movingalongtangent,e.g.,Euler’smethodx(ti) = xi−1 + h f(xi−1, ti−1) spiralsoutward,canleavecurve,nomatterhowsmallhis•  smallerh willjustdivergemoreslowly

x = f (x,t) = −yx

⎛

⎝⎜⎞

⎠⎟

Durand

x(t) =r cos(t + h)r sin(t + h)

⎛

⎝⎜

⎞

⎠⎟

Euler’sMethod:UnstableConsiderthefollowingsystem:Exactsolutionisdecayingexponential:Limitedstepsize:Ifkisbig,hmustbesmall

Durand

x = −kx, x(0) = 1x = x0e

−kt

xi = xi−1 − h kxi−1( ) = (1− hk)xi−1

h ≤ 1/k, okh > 1/k, oscillates±h > 2/k, explodes

�

SolvingtheIntegration

HowcanweimproveuponEuler’smethod?

ByaddinganothertermfromtheTaylor’sexpansionofx(t):Considertwoalternatives:•  theTrapezoidmethod•  theMidpointmethodbothhaveerrorsontheorderofO(h3)

x t + h( ) = x t( ) + hdxdt

+h2

2d 2xdt 2

+

TrapezoidMethodx(ti) = xi−1 + h f(xi−1, ti−1)

Problem:approximatedfvariesfromactualfIdea:considerfatthearrivalofthestepand

compensateforvariationf1

a

b

Durand

Letf0 = f(x0, t0)f1 = f(x0 + hf0, t0 + h)

Thena = hf0, b = hf1, x(t0 + h) = x0+ (a + b)/2 + O(h3)x(t0 + h) = x0+ h(f0 + f1)/2 + O(h3) Thisisthetrapeziodmethod,a.k.a.improvedEuler’smethod

MidpointMethodx(ti) = xi−1 + h f(xi−1, ti−1)

Problem:approximatedfvariesfromactualfIdea:considerfathalfstepandcompensateforvariation

Durand

ChooseΔx = h/2 f(x0, t0)thenrearrangeasbefore,letf0 = f(x0, t0)

fmid = f(x0 + h/2 f0, t0 + h/2)thenx(t0 + h) = x0+ hfmid + O(h3)Thisisthemidpointmethod,a.k.a.,2nd-orderRunge-Kutta

fm

Comparison

Midpoint:• ½Eulerstep•  evaluatefmid•  computestepusingfmid

Trapezoid:•  Eulerstep(a)•  evaluatef1•  computestepusingf1(b)•  average(a)and(b)

Notexactlythesameresult,butsameorderofaccuracy

f1a

b

Durand

Higher-OrderRunge-Kutta4thorder:popularbecauseordershigherthan4thneedmorestagesthanorderstocompute

k1 = hf (x0 ,t0 )

k2 = hf (x0 +k12,t0 +

h2)

k3 = hf (x0 +k22,t0 +

h2)

k4 = hf (x0 + k3,t0 + h)

x(t0 + h) = x0 +16k1 +

13k2 +

13k3 +

16k4 +O(h

5 )

order 1 2 3 4 5 6 7 8stages 1 2 3 4 6 7 9 11

Treuille

Forcesactingontheparticlesystemisasumofanumberofthings

Forcefields:•  gravity:constantdownwardforceproportionaltomassf = −mg,�g: gravitationalconstant,9.78 m/sec2onEarth

•  otherparticles:particlesmutuallyattract/repell

•  attractiveforce:

•  repulsiveforce:

•  bewareO(n2)complexity!

MaytheForce...

O’Brien,Durand

v0

m g

f = Gm1m2dd 3

f = −krdd 3

pi

pj

Collisionswithenvironment,otherparticles• detection:potentiallargenumberneeded,usehierarchicalboundingvolumes•  response:particleshapeandsizeneeded• duetotemporalaliasing,sub-framecalculationrequiredsoasnottomisscollisiontime

Collisions

Durand

v

p

v’n

vvn

vt

vn = (v�n)nvt = v − vn

v’−krvn

vt

v’ = vt − krvnkr:coeficientof

restitution

MissedCollisionsWanttodetectcollisionwhen (x−p)•n < ε, ε ≥ 0 Oftencollisiondetectedwhenv�n < 0

Solution:backtrack•  computeintersectionpoint•  ray-objectintersection!•  computeresponsethere•  reconstructforremainingfractionaltimestep

Othersolution/hack:•  projecttosurfacepointclosesttoobject

fixing

backtracking

Durand

vp

nx

Simplerendering:•  projectparticlestoviewframe•  blendparticleprojectionwithframebuffercontent(translucency)

Particlecolor:• makecolorafunctionofage• makecolorafunctionoftemperature•  requiresotherODEstogoverntheseproperties

ParticleSystemsRendering

Gilles

SmokeParticleSystem

ConstantlycreateparticlesParticlesmoveupwards,withPerlinturbulenceaddedDrawthemaspartiallytransparentcirclesthatfadeovertime

Chenney

Pre-renderbitmapoffireball

Real-timerenderedglow

Animatedglowingparticles

Quake

Gilles

TreesinAndreandWallyB.

LotsoftreesEachtreebranchesrecursivelyLeavesasparticles• millionsofparticles• needasimplemodelforshadingeach

ShadingModelforTrees

Ambient• dependentonhowdeepinthetreeda•  independentoflightposition

Diffuse• dependentondistanceddinsidethetree,inthedirectionoflight

Shadowed•  ifbelowplane,onlyambientused

Spring-MassSystems

Modelobjectsassystemsofspringsandmasses

Thespringsexertforces,controlledbychangingtheirrestlength

Areasonable,butsimple,physicalmodelformuscles

Advantage:goodlookingmotionwhenitworks

Disadvantage:expensiveandhardtocontrol

Chenney

Manytypesofcloth•  verydifferentproperties•  notasimpleelasticsurface• wovenfabricstendtobeverystiff•  anisotropic

Resolutionofmeshiscritical

Computationofcollisionsisexpensive

Cloth

Hodgins,Breen,Popovic

stretchspringsshearspringsbendsprings

ChallengesforPhysically-BasedAnimation

ExpensiveandnotnecessarilyrealisticWhatpiecesofphysicsarenecessaryforappearance?Howtogiveanimatorcontrol?•  howtogiveartistsanddirectorstheresultstheywant?

Chenney,Hodgins

PerceptualHacksViewerscanbeprettyoblivioustothingslikeincorrectbouncesandcan’texactlypredicthowthingsbreakJustmakesureobjectsdon’tgothroughwallsShiftemphasisfromphysicalaccuracytofast-and-looks-good

Hodgins

BehavioralAnimation

Definerulesforthewayanobjectbehavesandinteracts•  programsimplementtherules•  objectsrespondtotheirchangingenvironment

Classicexample:“boids”(CraigReynolds)•  emergentbehavior:flocking•  reallyjustaparticlesystemwithabitofperceptionandabitofbrainpower

Hodgins

Flocking

Eachboidperceivesneighborsinaneighborhood

Eachboid’sbehaviorisasimplefunctionofnearbyboids:•  separationforce•  steertoavoidcrowdinglocalteammates,keepminimumdistance

•  alignmentforce•  alignvelocities

•  cohesionforce•  movetowardsaveragepositionofneighbors

http://www.red3d.com/cwr/boids/http://www.riversoftavg.com/downloads.htm

AnimationSummary(brief)

Technique ControlTimetoCreate

ComputationCost

Interactivity

Key-Framed Excellent Poor Low Low

MotionCapture

Goodattimeofcreation,afterthatpoor

Medium Medium Medium

Procedural Poor Poortocreateprogram

High High

Chenney

HowAreMoviesAnimated?

Keyframingmostly

Articulatedfigures,inversekinematics

Skinning•  complexdeformableskin,muscle,skinmotion

Hierarchicalcontrols•  smilecontrol,eyeblinking,etc.•  keyframesforthesehigher-levelcontrols

Ahugetimeisspentbuildingthe3Dmodels,itsskeletonanditscontrols

Durand

MixingTechniquesApplyphysicalsimulationofsecondarymotionontopofkey-framedprimarymotion•  particularlyappropriateforcloth,hair,water•  useparticlesystemsfor“fuzzy”objects

Mixmotion-captureandphysics:• motion-capturedpersonkicksaballwhichisthenphysicallysimulatedfortrajectory

Chenney