EECS487:InteractiveComputerGraphicsLecture26:• Bumpmapping• Solidandproceduraltexture
BumpMapping2Dtexturemaplooksunrealisticallysmoothacrossdifferentmaterial,especiallyatlowviewingangle
Foolthehumanviewer:• perceptionofshapeisdeterminedbyshading,whichisdeterminedbysurfacenormal• usetexturemaptoperturbthesurfacenormalperfragment• doesnotactuallyalterthegeometryofthesurface• shadeeachfragmentusingtheperturbednormalasifthesurfacewereadifferentshape
Spherew/DiffuseTexture SwirlyBumpMap Spherew/DiffuseTexture&BumpMap
BumpMapping
t
TP3
s
Treatthetextureasasingle-valuedheightfunction(heightmap)• grayscaleimagestoresheight:black,higharea;white,low(orviceversa)
• differenceinheightsdetermineshowmuchtoperturbninthe(u, v)directionsofaparametricsurface• ∂b/∂u = bu = (h[s+1, t] − h[s−1, t])/ds• ∂b/∂v = bv = (h[s, t+1] − h[s, t−1])/dt• computeanew,perturbednormalfrom(bu,bv)
ComputingPerturbedNormal
!
n = pu × pv
pu =∂∂up(u,v), !pv =
∂∂vp(u,v)
p ' = p(u,v)+ b(u,v)n!⇐ !perturbed!surface
p 'u = ∂∂u
p(u,v)+ b(u,v)n( ) = pu + bun+ b(u,v)nu = pu + bun
p ' v = ∂∂vp(u,v)+ b(u,v)n( ) = pv + bvn+ b(u,v)nv = pv + bvn
0: n ⊥ p
wrinkledsurfacep’(u,v)
wrinklefunction
smoothsurfacep(u,v)
np p
p’ p’p n’
n’
TP3
Blinn
!
n ' = p 'u × p ' v ! ⇐ !normal!of!perturbed!surface
p 'u = pu + bunp ' v = pv + bvnn ' = (pu + bun)× (pv + bvn)
= pu × pv + bu (n× pv )+ bv (pu × n)+ bubv (n× n)= n+ bu (n× pv )− bv (n× pu )
PerturbedNormal
Recall:
a × (kb + c) = k(a × b)+ (a × c)a × b = −b × a
pv
�
n'
�
n
pu
n�pv
–n�pu
bu
bv
BumpMapvs.NormalMapComputingn’requirestheheightsamplesfrom4neighbors• eachsamplebyitselfdoesn’tperturbthenormal• anall-whiteheightmaprendersexactlythesameasanall-blackheightmap
Insteadofencodingonlytheheightofapoint,anormalmapencodesthenormalofthedesiredsurface,inthetangentspaceofthesurfaceatthepoint• canbeobtainedfrom:
• ahigh-resolution3Dmodel• photos(http://zarria.net/nrmphoto/nrmphoto.html)• aheight-map(withmorecomplexofflinecomputationofperturbednormals)• filteredcolortexture(Photoshop,Blender,Gimp,etc.withplugin)
[Hastings-Trew]
NormalMap
Hanrahan09
heightmap normalmap(nx, ny, nz) = (r, g, b)
InterprettheRGBvaluespertexelastheperturbednormal,notheightvalue
[Hastings-Trew]
NormalMapCreationonGimpOnMacOSX,runGimp-2.6.11(not2.8.4)• loadRGBfile,thenselectFilters�Map�Normalmap
ForWindows,seehttp://code.google.com/p/gimp-normalmap/
NormalMapping:Complications
1. Normalizednormalsrange[-1, 1],butRGBvaluesrange[0,1],convertnormalsbyn’= (n+1)/2Valuesinnormalmapmustbeconvertedbackbeforeuse:n = n’*2–1
2. Normalsareinobjectspace,sonormalsmustbetransformedwheneverobjectistransformed
Instead,mostimplementationsstorenormalsintangentspace,butthenlightandviewvectorsmustbetransformedtotangentspace
TangentSpaceIsacoordinatesystemattachedtothelocalsurfacewithbasisvectorscomprisingthenormalvector(N),perpendiculartothesurface,andtwovectorstangenttothesurface:thetangent(T)andbitangent(B)WewantTandBtospanourtexture:
[Premecz]N
texture/tangentspace objectspace
[s0, t0] [s1, t1]
[s2, t2]
p0
p1p1
p2
Apointp(s, t) = p0 + (s−s0)T + (t−t0)B3Dvectorsp1−p0 = (s1−s0)T + (t1−t0)B,
andp2−p0 = (s2−s0)T + (t2−t0)BLetΔsi = (si−s0)andΔti = (ti−t0),thenintexturespace,T,B,Nareorthonormal,butnotnecessarilysoinobjectspace,useGram-SchmidtOrthogonalization:B’= N×T;T’= B’×N
TangentSpace
objectspace
p1
p2
p0
[Premecz,Lengyel]
p1 − p0p2 − p0
⎡
⎣⎢⎢
⎤
⎦⎥⎥=
Δs1 Δt1Δs2 Δt2
⎡
⎣⎢⎢
⎤
⎦⎥⎥TB
⎡
⎣⎢
⎤
⎦⎥
TB
⎡
⎣⎢
⎤
⎦⎥ =
1Δs1Δt2 − Δs2Δt1
Δt2 −Δt1−Δs2 Δs1
⎡
⎣⎢⎢
⎤
⎦⎥⎥
p1 − p0p2 − p0
⎡
⎣⎢⎢
⎤
⎦⎥⎥
Ntexture/tangentspace
[s0, t0] [s1, t1]
[s2, t2]
p(s, t) p0
GivenT’,B’,Northonormal,[T’B’N]matrixtransformsfromtangenttoobjectspace([T’B’N]TmatrixinDirect3D)[WeassumeTBNorthonormalinthefigureandinsubsequentslidesandwedropthe“prime”sign]Bitangentissometimescalledbinormal,secondnormal,whichisapplicabletocurves,butnottosurfaces(seelectureonFrenetframe)Seealso:http://www.terathon.com/code/tangent.html
TangenttoObjectSpace
[Lengyel,DreijerMadsen]
ModelViewmatrix
Projectionmatrix
TBNmatrix
Whatwehave:• per-texelnintangentspacestoredinnormalmap• T,Ninobjectspace• l,vineyespaceTocomputelightingwithnormalmapintangentspace:1. transformlandvtotangentspace(how?)2. sampleper-texelnfromnormalmap3. computelightingintangentspace
LightingComputation
[Lengyel,DreijerMadsen]
ModelViewmatrix
Projectionmatrix
TBNmatrix
l,v
T,B,N
n
Tangent-SpaceLightingInapp:• loadnormalmapintoitsowntextureunit• computeTpertriangleandassignittoallthreevertices
• averageoutTofsharedverticesforcurvedsurface• passpertriangleNandTtovertexshader
Invertexshader:• transformNandTfromobjecttoeyespace(how?)• computeBandorthonormaleye-to-tangentmatrix(how?)
• transformlandvtotangentspace,normalize,andpassthem,interpolated,tothefragmentshader
Infragmentshader:• samplenormalmappertexel(n)• computelightingintangentspaceusingnormalizedn,l,andv
Bump/NormalMappingLimitations
SmoothsilhouetteSmoothwhenviewedatlowviewingangleNoself-shadowing/self-occlusion
O’Brien08
Parallax/ReliefMapping
idealparallaxmap
offsetapproximation,badforgrazingangle
offsetlimiting*a.k.a.ParallaxOcclusionMappingorSteepParallaxMapping RTR3
ParallaxMappingandReliefMapping*:• useheightfieldandviewvectortocomputewhich“bump”isvisible• howtoavoidcomputingviewvectorandheightfieldintersection?
reliefsampling:linearsearchtofirstpointinsidesurface,thenbinarysearchbetweenthispointandlastpointoutsidesurface
DisplacementMappingInterprettexelasoffsetvectortoactuallydisplacefragments:p’ = p + h(p)n• correctsilhouettesandshadows• mustbedonebeforevisibilitydetermination• complicatescollisiondetection,e.g.,ifdoneinvertexshader
TP3
VertexTextureFetch
Marschner08
Traditionally,duringvertexshading,theonlytexture-relatedcomputationiscomputingtexturecoordinatespervertexWithShaderModel3.0,vertexshadercanusetexturemaptoprocessvertices,e.g.,fordisplacementmapping,fluidsimulation,particlesystems
pervertex
texCoordsperfragment
TextureMapping
Alternativedefinition:ageneraltechniqueforstoringandevaluatingfunctionsTexturesarenotjustforshadingparametersanymore!
Marschner
SolidTexturesSolidtextures:• createa3Dparameterization 2Dmapping 3D“carving”
(s, t, r)forthetexture• mapthisontotheobject• theeasiestparameterizationistousethemodel-spacecoordinatestoindexintoa3Dtexture(s, t, r) = (x, y, z)
• like“carving”theobjectfromthematerial
Solidproceduraltextures:• moregenerally,insteadofusingthetexturecoordinatesasanindex,usethemtocomputeafunctionthatdefinesthetexture
Wolfe
Perlin
SolidProceduralTextureExampleInsteadofanimage,useafunction// vertex shader varying vec3 pos; ... pos = gl_Position.xyz; ... // fragment shader varying vec3 pos; ... color = sin(pos.x)*sin(pos.y); ...
Advantagesoverimagetexture:• infiniteresolutionandsize• morecompactthantexturemaps
• f(x, y, z)maybeasubroutineinthefragmentshader• noneedtoparameterizesurface• noworriesaboutdistortionanddeformation• objectsappearsculptedoutofsolidsubstance• cananimatetextures
Disadvantages:• difficulttomatchexistingtexture• notalwayspredictable• moredifficulttocodeanddebug• perhapsslower• aliasingcanbeaproblem
ProceduralTextures
Peachey
Stripe:coloreachpointoneortheothercolordependingonwherefloor(z)(orfloor(x)orfloor(y))isevenoroddRings:coloreachpointoneortheothercolordependingonwhetherthefloor(distancefromobjectcenteralongtwocoordinates)isevenorodd
SimpleProceduralTextures
Wolfe
Rampfunctions:• ramp(x, y, z, a) = ((float) mod(x, a))/a• mod(3.75, 2.0)/2.0 = 1.75/2.0 = .875
• ramp(x, y, z) = (sin(x)+1)/2Combination:proceduralcolortablelookup:f(x, y, z) computesanindexintoacolortable,e.g.,usingtherampfunctiontocomputeanindex
SimpleProceduralTextures
Wolfe
WoodTextureClassifytexturespaceintocylindricalshells
f(s, t, r) = (s2 + t2)
Outerringsclosertogether,whichsimulatesthegrowthrateofrealtreesWoodcoloredcolortable• woodmap(0) = brown“earlywood”• woodmap(1) = tan“latewood”
wood(p) = woodmap( f(p) mod 1)p = (s, t, r)
f (s, t, r) = (s2 + t2)
wood(p)
woodmap( f (p))0 1
Hart08
AddingNoiseAddnoisetocylinderstowarpwood:wood(x2 + y2 + A*�
noise( fx *x + ϕx, fy *y + ϕy, fz *z + ϕz))controls:• frequency( f ):coarsevs.finedetail,numberandthicknessofnoisepeaks,noise( fx x, fy y, fz z)• phase(ϕ):locationofnoisepeaksnoise(x + ϕx, y + ϕy, z + ϕz)• amplitude(A):controlsdistortionduetonoiseeffect
connectedpixel.com/blog/texture/woodHart08
noise(p):pseudo-randomnumbergeneratorwiththefollowingcharacteristics: • memoryless whitenoise Perlinnoise• repeatable• isotropic• bandlimited(coherent):differenceinvaluesisafunctionofdistance
• noobviousperiodicity• translationandrotationinvariant(butnotscaleinvariant)
• knownrange[-1, 1]• scaleto[0,1]using0.5(noise()+1)
• abs(noise())createsdarkveinsatzerocrossings
PerlinNoise
Hodgins07
Fractal:• summultiplecallstonoise:
1-8octavesofturbulence
• eachadditionaltermaddsfinerdetail,withdiminishingreturn
Turbulence
Hodgins
turbulance(p) = 12i f
noise(2i f ⋅p)i=1
octaves
∑
MarbleTextureUseasinefunctiontocreatethestripes:marble = sin( f * x + A*turbulence(x, y, z))
• thefrequency( f )ofthesinefunctioncontrolsthenumberandthicknessoftheveins
• theamplitude(A)oftheturbulencecontrolsthedistortionoftheveins
legakis.net/justin/MarbleApplet/
