Reflection Models IReflection ModelsTypes of Reflection FunctionsMaterialsThe Reflection EquationThe BRDFThe BSSRDFGonioreflectometerProperties of BRDF’sSlide 10Energy ConservationThe ReflectanceLaw of ReflectionIdeal Reflection (Mirror)Snell’s LawLaw of RefractionOptical ManholeFresnel ReflectanceExperimentCook-Torrance Model for MetalsIdeal Diffuse Reflection“Diffuse” ReflectionPhong ModelSlide 24Properties of the Phong ModelUniversity of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don FussellReflection Models ITodayTypes of reflection modelsThe BRDF and reflectanceThe reflection equationIdeal reflection and refractionFresnel effectIdeal diffuseNext lectureGlossy and specular reflection modelsRough surfaces and microfacetsSelf-shadowingAnisotropic reflection modelsUniversity of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don FussellReflection Models Definition: Reflection is the process by which light incident on a surface interacts with the surface such that it leaves on the incident side without change in frequency.PropertiesSpectra and Color [Moon Spectra]PolarizationDirectional distributionTheoriesPhenomenologicalPhysicalUniversity of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don FussellTypes of Reflection FunctionsIdeal SpecularReflection LawMirrorIdeal DiffuseLambert’s LawMatteSpecularGlossyDirectional diffuseUniversity of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don FussellMaterialsPlastic Metal MatteFrom Apodaca and Gritz, Advanced RenderManUniversity of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don FussellThe Reflection Equation2( , ) ( , ) ( , ) cosr r r i r i i i iHL x f x L x dω ω ω ω θ ω= →∫( , )r rL x ω( , )i iL x ωidωiθrθrφiφˆNUniversity of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don FussellThe BRDFBidirectional Reflectance-Distribution Function( )1( )r i rr i ridLfdE srω ωω ω→⎡ ⎤→ ≡⎢ ⎥⎣ ⎦( , )r rdL x ω( , )i iL x ωidωiθrθrφiφˆNUniversity of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don FussellThe BSSRDFBidirectional Surface Scattering Reflectance-Distribution Function( , , )( , , )r i i r ri i r ridL x xS x xdω ωω ω→→ ≡Φ( , )r rdL x ω( , )i iL x ωidωiθrθrφiφˆNrxixTranslucencyUniversity of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don FussellGonioreflectometerUniversity of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don FussellProperties of BRDF’s1. Linearity2. Reciprocity principle( ) ( )r r i r i rf fω ω ω ω→ = →From Sillion, Arvo, Westin, GreenbergUniversity of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don FussellProperties of BRDF’s3. Isotropic vs. anisotropic4. Energy conservation( , ; , ) ( , , )r i i r r r i r r if fθ ϕ θ ϕ θ θ ϕ ϕ= −( , , ) ( , , ) ( , , )r i r r i r r i i r r i r r if f fθ θ ϕ ϕ θ θ ϕ ϕ θ θ ϕ ϕ− = − = −Reciprocity and isotropyUniversity of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don FussellEnergy Conservation( )cos( ) cos( ) ( )cos cos( ) cos1rir iir r r rrii i i ir i r i i i i r ri i i iL dddL df L d dL dω θ ωω θ ωω ω ω θ ω θ ωω θ ωΩΩΩ ΩΩΦ=Φ→=≤∫∫∫∫∫iΩrΩUniversity of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don FussellThe ReflectanceDefinition: Reflectance is ratio of reflected to incident powerConservation of energy: 0 < < 13 by 3 set of possibilities:Units: [dimensionless], fr [1/steradians]( ) cos cos( )cos( ) ( )cos cos( ) cosr iir iir i r i i r ri ri ir i r i i i i r ri i i if d ddf L d dL dω ω θ ω θ ωρθ ωω ω ω θ ω θ ωω θ ωΩ ΩΩΩ ΩΩ→Ω → Ω ≡→≤∫∫∫∫∫∫iΩrΩ{ }{ }2 2, , , ,i i i r r rd H d Hω ωΩ × ΩUniversity of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don FussellLaw of Reflectionr iθ θ=rθiθiϕrϕ( )mod 2r iϕ ϕ π π= +NˆRˆIˆˆ ˆ ˆ ˆ ˆ ˆ( ) 2 co s 2( )θ+ − = =− •R I N I N Nˆ ˆ ˆ ˆ ˆ2( )= − •R I I N NUniversity of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don FussellIdeal Reflection (Mirror),(cos cos )( , ; , ) ( )cosi rr m i i r r i rifδ θ θθ ϕ θ ϕ δ ϕ ϕ πθ−= − ±,( , ) ( , )r m r r i r rL Lθ ϕ θ ϕ π= ±, ,( , ) ( , ; , ) ( , ) cos cos(cos cos )( ) ( , ) cos coscos( , )r m r r r m i i r r i i i i i ii ri r i i i i i iii r rL f L d dL d dLθ ϕ θ ϕ θ ϕ θ ϕ θ θ ϕδ θ θδ ϕ ϕ π θ ϕ θ θ ϕθθ ϕ π=−= − ±= ±∫∫rθiθ( , )i i iL θ ϕ( , )r r rL θ ϕUniversity of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don FussellSnell’s Lawsin sini i t tn nθ θ=ˆ ˆ ˆ ˆi tn n× = ×N I N TtθiθˆNˆTˆIiϕtϕt iϕ ϕ π= ±University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don FussellLaw of RefractiontθiθˆNˆTˆI/i tn nμ =ˆ ˆ ˆ ˆμ× = ×N T N Iˆ ˆ ˆ( ) 0μ× − =N T Iˆ ˆ ˆμ γ= +T I N2 2 2ˆ ˆ ˆ1 2μ γ μγ= = + + •T I N( )( ){ }{ }1212222 2ˆ ˆ ˆ ˆ1 1cos 1 sincos coscos c osi ii ti tγ μ μμ θ μ θμ θ θμ θ θ= − • ± − − •= ± −= ±= −I N I N1γ μ← = −( )22ˆ ˆ1 (1 ) 0μ− − • <I NTotal internal reflection:University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don FussellOptical ManholeFrom Livingston and LynchTotal internal reflection43wn =University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don FussellFresnel ReflectanceMetal (Aluminum) Dielectric (N=1.5)5( ) (0) (1 (0))(1 cos )F F Fθ θ= + − −Schlick ApproximationGlass n=1.5 F(0)=0.04Diamond n=2.4 F(0)=0.15Gold F(0)=0.82Silver F(0)=0.95University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don FussellExperimentReflections from a shiny floorFrom Lafortune, Foo, Torrance, Greenberg, SIGGRAPH 97University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don FussellCook-Torrance Model for MetalsMeasured ReflectanceReflectance of Copper as afunction of wavelength andangle of incidence2πλρLight spectraCopper spectraθApproximated Reflectance( ) (0)(0) ( / 2)( / 2) (0)F FR R RF Fθππ⎡ ⎤−= +⎢ ⎥−⎣ ⎦Cook-Torrance
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