Reflecon Models Today Types of eflecon models The BRDF and eflecance The eflecon equaon Ideal eflecon and efacon Fesnel effec Ideal dffuse Thusday Glossy and specula eflecon models Rough sufaces and mcofaces Self-shadowng Ansoopc eflecon models Reflecon Models Defnon: Reflecon s he pocess by whch lgh ncden on a suface neacs wh he suface such ha leaves on he ncden sde whou change n fequency. Popees Speca and Colo [Moon Speca] Polazaon Deconal dsbuon Theoes Phenomenologcal Physcal Page 1
Types of Reflecon Funcons Ideal Specula Reflecon Law Mo Ideal Dffuse Lambe s Law Mae Specula Glossy Deconal dffuse Maeals Plasc Meal Mae Souce: Apodaca and Gz, Advanced RendeMan Page
The BRDF Bdeconal Reflecance-Dsbuon Funcon (BRDF dl( ω ω dl( ω ω 1 f( ω ω = de L ( ω cosθ dω s Gonoeflecomee Page 3
The Reflecon Equaon L ( x, ω L( x, ω θ dω L ( x, ω = f ( x, ω ω L( x, ω cosθ dω H Noe: Pon and dsan lgh souces ae dela funcons Popees of BRDF s 1. Lneay Souce: Sllon, Avo, Wesn, Geenbeg. Recpocy pncple f ( ω ω = f ( ω ω Page 4
Popees of BRDF s 3. Isoopc vs. ansoopc f ( θ, ϕ ; θ, ϕ = f ( θ, θ, ϕ ϕ Recpocy and soopy f ( θ, θ, ϕ ϕ = f ( θ, θ, ϕ ϕ = f ( θ, θ, ϕ ϕ 4. Enegy consevaon The Reflecance Defnon: A eflecance s a ao of efleced o ncden powe L( ωcosθ dω dφ Ω ρ( Ω Ω = dφ L( ω cosθ dω Devaon assumes unfom ncden adance Expemens measue eflecances Consevaon of enegy: 0 < <1vs.0< f < Ω Ω Ω Uns: ρ [dmensonless], f [1/seadans] = f ( ω ω cosθ dω cosθ dω Ω cosθ dω Page 5
Law of Reflecon Î θ Nˆ θ Rˆ ϕ ϕ θ = θ ϕ = ϕ ± π Rˆ + ( Iˆ = cosθ Nˆ = ( Iˆ Nˆ Nˆ Rˆ = Iˆ ( Iˆ Nˆ Nˆ Ideal Reflecon (Mo L( θ, ϕ L( θ, ϕ θ θ Lm, ( θ, ϕ = L( θ, ϕ ± π f δ(cosθ cos θ ( θ, ϕ ; θ, ϕ = δ ( ϕ ϕ ± π m, cosθ L ( θ, ϕ = f ( θ, ϕ ; θ, ϕ L( θ, ϕ cosθ dcosθ dϕ m, m, δ(cosθ cos θ = δ ( ϕ ϕ ± π L( θ, ϕ cos θ dcos θ d ϕ cosθ = L ( θ, ϕ ± π Page 6
Snell s Law Î ˆN θ ϕ ϕ θ ˆT ϕ = ϕ ± π n snθ = n snθ n Nˆ Tˆ = n Nˆ Iˆ Law of Refacon ˆN Î µ = n / n θ θ ˆT Toal nenal eflecon: ( I N 1 µ (1 ˆ ˆ < 0 Nˆ Tˆ = µ Nˆ Iˆ Nˆ ( Tˆ µ Iˆ = 0 Tˆ = µ Iˆ+ γ Nˆ ˆ ˆ ˆ T = 1= µ + γ + µγ I N { 1 ( 1 ( } 1 { } γ = µ Iˆ Nˆ ± µ Iˆ Nˆ = µ cosθ ± 1 µ sn θ = µ cosθ ± cosθ = µ cosθ cosθ γ = µ 1 1 Page 7
Expemen Reflecons fom a shny floo Souce: Lafoune, Foo, Toance, Geenbeg, SIGGRAPH 97 Fesnel Equaons Deleccs (Two polazaons R T n cosθ n cosθ n cosθ n cosθ = R = n n n n 1 1 1 1 1cosθ1 + cosθ 1cosθ + cosθ1 n cosθ n cosθ = T = n n n n 1 1 1 1 1cosθ1 + cosθ 1cosθ + cosθ1 Meals n+ κ a + b = n (1 κ sn θ a + b acosθ + cos θ R = a b + + a cosθ + cos θ a + b asnθ anθ + sn θ an θ T = R a + b + asnθ anθ + sn θ an θ Page 8
Fesnel Equaons Nomal ncdence Deleccs R n n 1 = n1+ n Meals ( n 1 ( n 1 + n κ R = + + n κ Glass: Damond: Slve: Gold: n=1.5 R=0.04 n=.4 R=0.15 n<1, κ=1 R=0.95 n<1, κ=1 R=0.8 Solve fo n gven R a nomal ncdence Cook-Toance Model fo Meals Lgh speca ρ Measued Reflecance π θ = Reflecance of Coppe as a funcon of wavelengh and angle of ncdence Coppe speca λ Appoxmaed Reflecance R( θ R(0 R = R(0 + R( π / R( π / R(0 Schlck appoxmae Fesnel F( θ = F + (1 F (1 cos θ 0 0 5 Page 9
Ideal Dffuse Reflecon Assume lgh s equally lkely o be efleced n any oupu decon (ndependen of npu decon. L ( ω = f L( ω cosθ dω d, d, = fd, L( ωcosθdω = f E d, B = L ( ω cosθ dω = L cosθ dω = π L B ρd = = π f E, d Lambe s Cosne Law B = ρ E = ρ E cosθ d d s s Dffuse Reflecon Theoecal Bougue - Specal mco-face dsbuon Seelge - Subsuface eflecon Mulple suface o subsuface eflecons Expemenal Pessed magnesum oxde powde Almos neve vald a hgh angles of ncdence Pan manufacues aemp o ceae deal dffuse Page 10
Phong Model R(L E N E N R(E L L ( Eˆ R Lˆ Recpocy: ( s ( ˆ ( ˆ s E R L = ( Lˆ R ( Eˆ ( Lˆ R Eˆ s ( s Dsbued lgh souce! Popees of he Phong Model Nomalze Phong Model ( Lˆ R Eˆ ρ( π ω = ( cosθ dω H ( Nˆ H ( Rˆ cos ( Lˆ R Eˆ ( cosθ dω π s+ 1 θdω = s H + s s Page 11