EECS487:InteractiveComputerGraphicsLecture30:•  Environmentmapping•  Radiancemap•  Accumulationbuffer

EnvironmentMapping

Thekeytodepictingashiny-lookingmaterialistoprovidesomethingforittoreflect•  properreflectionrequiresraytracing,expensive•  canbesimulatedwithapre-renderedenvironment,storedasatexture•  imagineobjectisenclosedinaninfinitelylargesphereorcube•  raysarebouncedoffobjectintoenvironmenttodeterminecolor

EM*surfacecolor=reflectionmapping

Hanrahan

EnvironmentMappingSteps:•  loadenvironmentmap•  foreachreflectivepixel,computeitsnormal•  computethereflectionvectorfromthenormalandviewvectors•  usethereflectionvectortocomputeanindexintotheenvironmentmapinthereflectiondirection•  note:we’renotcomputingrayintersectionbetweenthereflectionvectorandtheenvironment!

•  usethetexelattheindextocolorthepixel

Shortcomings:•  nointer-objectreflection�workswellwhenthere’sjustasingleobject•  noself-reflection

Jensenetal.

EnvironmentMappingModel:•  environmentisinfinitelyfaraway

•  allreflectionsasseenfromthesame,farawayviewpoint:•  objectapproximatedasaninfinitelysmall,perfectlymirroringballconcentricwiththeobject

•  reflectedcolorcomputedonlyfromthedirectionofreflection•  notfromthepositiononsurface

•  determinedbysurfacenormal

•  noray-environmentintersectioncomputation

CubeMapMostpopularandfastest:easytoproducewithrenderingsystemorbyphotographyfromcenterofobject,onceforeachsideofcube

Simpletexture-coordinatescalculation

Texturecreationfromscene:•  viewindependent•  uniformsampling/resolution

Supportsbilinearfilteringandmipmapping

TP3

t

s

ts

t

s

y ComputingReflectionSteps:1. computereflectionvector,r•  efromeyetovertex•  nnormalineyecoordinates•  r = e – 2n(n•e)

2. reflectionisafunctionofonedirection:largestabsolutevalueofr’scomponentsdeterminesthecubefacetoreflect•  example:r = (5, −1, 2)gives+xasthereflectedface

3. dividerbythevalueofthe“reflection”coordinate(5)andmapto[0,1]�(s, t) = ((y + x)/2x, (z + x)/2x)�

= ((−1/5 + 1)/2, (2/5 + 1)/2)�= (0.4, 0.7)

+z +x -x -z

+y

-y

x

y

z

ne

r

CubeMap:DisadvantagesAngularsizeoftexelvariesacrossacubefaceUsuallydoesn’tinterpolateacrosscubefaces�cubeedgesarereflectedonobject

ToolssuchasAMD’sCubeMapGencanfixtheseproblems

RTR3

θθ

MethodstoCreateEM• Cubemap

•  Latitude/longitudeprojectionsmap•  createdbypainting•  oversamplingofpolescomparedtotheequator

•  Sphericalmap•  gazingball•  fisheyelens

• Parabolicmap

GazingBall(LightProbe)Createdbyphotographingareflectivesphere

Mapsalldirectionstoacircle

Reflectionindexedbynormal

Texturecreationfromscene:•  resolutionfunctionoforientation•  viewdependent:mustregenerateEMwhencameramovesorwillseethesamething

Hanrahan09

SphereMappingUseatexturemapofasphereviewedfrominfinityusertolookuptexel•  theeyevectorsareparallel•  rdeterminedonlybysurfacenormal

Want:computetexturecoordinates(s, t)fromr•  textureisnotreallypastedtotheinsideoftheenvironmentsphere,butprojected(nextslide)•  objectcanbeapproximatedasaninfinitelysmall,perfectlymirroringballconcentricwiththeobject• mapthenormalsofanobjecttothecorrespondingnormalsofasphere

ne r = (x, y, z)

environmentmaponasphere

viewpoint

object

h

Zhang08

Computing(s, t)fromrObservation:ncanbeexpressedintermsofrande:

r = e − 2n(n i e)r − e =αn

-e

s

RTR3Zhang08

αn = r − e =

xyz0

⎡

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⎥⎥⎥⎥

−

00−10

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

! where! αn = p = x 2 + y2 + (z +1)2

αnαn

=

x / py / p

(z +1) / p0

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⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

!since! r = 1,!p = 2(z +1)

texture

Computing(s, t)fromr

(1, 1)

(–1, –1)

h =

hx

hy

1− hx2 − hy

2

0

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⎤

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⎥⎥⎥⎥⎥

hx = x p hy = y p

s = x2p

+ 12

t = y2p

+ 12

!

αnαn

=

x / py / p

(z +1) / p0

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

where!p = 2(z +1)

Zhang08

!

scaled!to!texture!space![0,1]:

s = 12hx +

12!!!!!!!t = 1

2hy +

12

unitsphereineyespace,gazingdown–z:

-e

s

RTR3h

hx

hy

texture

ParabolicMap

RTR3

Usesz-componentofreflectedvectortodeterminetexel

Texturecreationfromscene:• viewindependent• uniformsampling• mapshardtocreate

Heidrich&Seidel

OpenGL2.1Spheremap: // insert where the texture is created glTexGeni(GL_S,GL_TEXTURE_GEN_MODE,GL_SPHERE_MAP); glTexGeni(GL_T,GL_TEXTURE_GEN_MODE,GL_SPHERE_MAP); glEnable(GL_TEXTURE_GEN_S); glEnable(GL_TEXTURE_GEN_T); Cubemap:•  loadsiximages,oneforeachfacewith:glTexImage2D(target)

•  texturecoordinatesgeneratedusingglTexGen*(…,GL_TEXTURE_GEN_MODE,GL_REFLECTION_MAP); glEnable(GL_TEXTURE_CUBE_MAP);

Functionsdeprecated

CubeMappingwithGLSL==== OpenGL app: initialize texture sampler to texture unit 0 ==== GLuint cubeid = glGetUniformLocation(myprog, “mycube”); glUniform1i(cubeid,0); // assign texure unit 0 to cubeid ==================== // vertex shader: compute r ================= varying vec3 r; void main() { gl_Position = gl_ModelViewProjectionMatrix * gl_Vertex; vec3 n = normalize(gl_NormalMatrix * gl_Normal); vec4 e = gl_ModelViewMatrix * gl_Vertex; r = reflect(-e,n); } ==================== // fragment shader =========================== varying vec3 r; uniform samplerCube mycube; void main() { gl_FragColor = textureCube(mycube, r); }

Angel09

LimitationofEM

Environmentmapassumesobjectinfinitesimallysmallandreflectionsinfinitelyfaraway

EMerrorshardtonoticeonnon-flatobjects,butdoesn’tworkwellforflat/planarsurfaces,•  reflectedraysusuallydonotvarybymorethanafewdegrees�asmallpartoftheEMisappliedtoalargearea• worsewithorthographicprojection:allorthographicreflectedvectorsareparallel

CansimulatereflectionfromstillwaterButnotwavywater(viabumpmap)

1.  reflectiondoesn’tmeetboat2.  reflectionbehindtheboat3.  environmentmapmagnified

LimitationsofEM

env.map

Hart08

BoatreflectedinwavywaterrenderedusinganenvironmentmapCanyoufindthreethingswrongwiththispicture?

DiffuseReflectanceWithEM,eachtexelisadirectionallightsource:•  forperfectlyspecularsurfaces,onlylightinginthereflecteddirectioncontributestolighting•  inthediffusecase,lightingisintegratedoverthehemisphereaboveapoint•  costofcomputingdiffusecolorofapoint(c)isontheorderofthenumberoftexelsintheEM!

•  kdirectionallights(texels)•  l(j): directionoflightj•  s(j):intensityoflightj•  n:surfacenormal•  m:materialreflectance

Schulze

c = m s( j ) max((l( j )in),0)j=1...k∑

IrradianceMapPrecomputationofdiffusereflection

Observations:•  irradianceatvariouspointsdiffersonlyonincomingdirectionsandthesurfacenormal•  allpointswiththesamenormalreflectthesameirradiance•  (methodlimitedtolightingcontributionfromadistantenvironment!)

Idea:•  precomputesumforallpossiblenormals•  storeresultsinasecondenvironmentmapcalledthediffuse(irradianceenvironment)map•  radiancemapindexedbysurfacenormal

r

n

IlluminationmapIrradiancemap

Schulze,Ramamoorthi

Moregenerally,giventheBRDFofthesurface,theReflectanceEquationis:needto(pre-)computeirradiance*BRDF

GlossySurfaces lv

ρωi

ωo

Ramamoorthi,Kautz

L(ωo ) = L(−π 2

π 2

∫ ωi )ρ(ωi ,ωo )dωi

Reflected Light (ρ)

InteractiveVisualFXAnti-aliasing:• accumulationbufferCameraeffects:• motionblur•  depthoffield•  accumulationbuffer

Shadows:• projected(soft)shadows•  stencilbuffer• depthbufferasshadowmap

Otherglobalilluminationeffects:•  reflection•  refraction•  colorbleed(onebounce)•  caustics

Environmentaleffects:•  participatingmediumandvolumerendering•  particlesystems•  fluiddynamics

LimitationsofGL_MULTISAMPLE

Nocontrolof:•  numberofsamples,can’thaveadaptivequality/performancetrade-off•  samplelocations:can’tdostochasticsamplingoradaptivesamplingorusedifferentsamplingpatterns(perhapsdifferentperpixel)•  averagingfunction(filtershapesandextents)WecanusetheaccumulationbuffertoaddressmostoftheshortcomingsofGL_MULTISAMPLE(exceptforper-pixelsamplingpattern,andatthecostofslowerperformance)

TheAccumulationBuffer

Samesizeasthecolorbuffer,usedtohold(accumulate)resultsfrompartialcomputationDeprecatedsinceOpenGL3.1

Instead,useframebufferobjectwithfloating-pointpixelformat• sameconceptasaccumulationbuffer

MultisamplingwiththeAccumulationBuffer

Akeley07

glutInitDisplayMode(… | GLUT_ACCUM); // set up desired rendering modes glAccum(GL_LOAD, 0.0); // or glClear(GL_ACCUM_BUFFER_BIT); for (int i=0; i<n; ++i) {

// specify sampling location for the i-th pass // by offsetting the frustrum // (google accpersp.c for the redbook source samples) render(scene); // to color buffer // accumulate the color buffer (multiplied by // a weight) to the accumulation buffer glAccum(GL_ACCUM, sampleweight[i]);

} // copy the accumulation buffer to color buffer glAccum(GL_RETURN, 1.0);

Hart,RTR3

Samplethescenektimes,placethemovingobject(s)atanewlocationeachtime

Eachsamplecontributes1/k-thofthefinalcolor:glAccum(GL_ACCUM, 1/k)

MotionBlurSamplethescenektimes,eachtimewithaslightlydifferenteyeposition,butsuchthatthefocalplaneboundedbythefrustrumisthesameineachsample

Eachsamplecontributes1/k-thofthefinalcolor:glAccum(GL_ACCUM, 1/k)

focalplaneHart

DepthofField

ShadowsforInteractiveRenderingLet’sstartwithhardshadowsPhongilluminationmodelwithhardshadows:(k:lightnumber,notexponentiation!)•  includesvisibilityterm(v(k) = 1),ifalightcan“see”thepoint•  ifpointisinshadow,onlyambienttermappliesHowtodetermineifpointisinshadow?

ct = cg + me + sspot(k )

k=1

n

∑ (ca(k ) + v(k ) f (d (k ) )(cd

(k ) + cs(k ) ))

Merrell

ComputingShadows

Planarreceiver•  projectedshadows

Non-planarreceiver•  shadowmaps•  projectivetexture•  shadowvolumes

Allperformedinreal-time/interactive(sortof)

ProjectedShadowsWaystothinkaboutshadows:• asadarkvolumeofspace• asplacesnotseenfromalightsourcelookingatthescene• asaseparateobject•  projectobjecttothereceiveranddrawitasecondtime

Akenine-Möller02,Durand

Worksonlywithpointlights Forprojectionontoy = 0:andplanarreceivers•  projectoccludersontoreceivers•  usingshadowprojectionmatrix

What’stheprojectionmatrixforplanarreceiveringeneral,otherthanfory = 0?

ProjectedShadows

x

y

l

v

p lx vx

p =Mv

pxpypzpw

⎡

⎣

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⎥⎥⎥⎥⎥

=

ly −lx 0 0

0 0 0 00 −lz ly 0

0 −1 0 ly

⎡

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⎥⎥⎥⎥⎥

vxvyvz1

⎡

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⎥⎥⎥⎥⎥

lylx − px

=vy

vx − pxlyvx − ly px = lxvy − pxvylyvx − lxvy = px (ly − vy )

px =lyvx − lxvyly − vy

Pointpisattheintersectionofrayandplaneif(mixingnotation):nip + D = 0, D = −niani L + t(v −L)( )+ D = 0

t = − D + niLni(v −L)

p = L − D + niLni(v −L)

⎛⎝⎜

⎞⎠⎟(v −L)

Shadowprojectionmatrix(M)cannowbecomputed:p =Mv

M =

niL + D − Lxnx −Lxny −Lxnz −LxD

−Lynx niL + D − Lyny −Lynz −LyD

−Lznx −Lzny niL + D − Lznz −LzD

−nx −ny −nz niL

⎡

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⎥⎥⎥⎥⎥

ProjectedShadows

L

vp

plane:n•p + D = 0, D = −n•a

ray:L + t(v − L)

Yu Hart

Samplethescenektimes,withobjectprojectedontothereceiver,eachtimewithaslightlydifferentlightposition

Eachsamplecontributes1/k-thofthefinalshadowcolor:glAccum(GL_ACCUM, 1/k)

SoftShadows