Reflection from Layered Surfaces due to Subsurface Scattering Pat - PowerPoint PPT Presentation

Ref. & Trans. Desc. of Materials Light Trans. Eq. Solving the Int. Eq. Multiple Scattering Reflection from Layered Surfaces due to Subsurface Scattering Pat Hanrahan Wolfgang Krueger SIGGRAPH 1993

Ref. & Trans. Desc. of Materials Light Trans. Eq. Solving the Int. Eq. Multiple Scattering Outlines Ref. & Trans. 1 Desc. of Materials 2 Light Trans. Eq. 3 Solving the Int. Eq. 4 Multiple Scattering 4

Ref. & Trans. Desc. of Materials Light Trans. Eq. Solving the Int. Eq. Multiple Scattering Reflected and Transmitted Radiances L r ( θ r , φ r ) = L r,s ( θ r , φ r ) + L r,v ( θ r , φ r ) (1) L t ( θ t , φ t ) = L ri ( θ t , φ t ) + L t,v ( θ t , φ t ) (2)

Ref. & Trans. Desc. of Materials Light Trans. Eq. Solving the Int. Eq. Multiple Scattering BRDF and BTDF L r ( θ r , φ r ) f r ( θ i , φ i ; θ r , φ r ) ≡ ( BRDF ) (3) L i ( θ i , φ i ) cos θ i dw i L t ( θ t , φ t ) f t ( θ i , φ i ; θ t , φ t ) ≡ ( BTDF ) (4) L i ( θ i , φ i ) cos θ i dw i

Ref. & Trans. Desc. of Materials Light Trans. Eq. Solving the Int. Eq. Multiple Scattering Fresnel transmission and reflection For planar surface R 12 ( n i , n t ; θ i , φ i → θ r , φ r ) L i ( θ i , φ i ) L r ( θ r , φ r ) = (5) T 12 ( n i , n t ; θ i , φ i → θ t , φ t ) L i ( θ i , φ i ) L t ( θ t , φ t ) = (6)

Ref. & Trans. Desc. of Materials Light Trans. Eq. Solving the Int. Eq. Multiple Scattering Fresnel transmission and reflection For planar surface R 12 ( n i , n t ; θ i , φ i → θ r , φ r ) L i ( θ i , φ i ) L r ( θ r , φ r ) = (5) T 12 ( n i , n t ; θ i , φ i → θ t , φ t ) L i ( θ i , φ i ) L t ( θ t , φ t ) = (6) where R 12 ( n i , n t ; θ i , φ i → θ r , φ r ) = R ( n i , n t , cos θ i , cos θ t ) n 2 T = n 2 t t T 12 ( n i , n t ; θ i , φ i → θ t , φ t ) = (1 − R ) n 2 n 2 i i

Ref. & Trans. Desc. of Materials Light Trans. Eq. Solving the Int. Eq. Multiple Scattering Fresnel transmission and reflection For planar surface R 12 ( n i , n t ; θ i , φ i → θ r , φ r ) L i ( θ i , φ i ) L r ( θ r , φ r ) = (5) T 12 ( n i , n t ; θ i , φ i → θ t , φ t ) L i ( θ i , φ i ) L t ( θ t , φ t ) = (6) In our model of reflection: f r = Rf r,s + Tf r,v = Rf r,s + (1 − R ) f r,v (7)

Ref. & Trans. Desc. of Materials Light Trans. Eq. Solving the Int. Eq. Multiple Scattering Description of Materials Index of Refraction Absorption and scattering cross section σ t = σ a + σ s Scattering phase function Henyey-Greenstein 1 − g 2 p HG (cos j ) = 1 (1 + g 2 − 2 g cos j ) 3 / 2 4 π

Ref. & Trans. Desc. of Materials Light Trans. Eq. Solving the Int. Eq. Multiple Scattering Light Transport Equations Transport theory models the distribution of light in a volume by ∂L ( � x, θ, φ ) = ∂s � x ; θ, φ ; θ ′ , φ ′ ) L ( � x, θ ′ , φ ′ ) dθ ′ dφ ′ − σ t L ( � x, θ, φ ) + σ s p ( � (8)

Ref. & Trans. Desc. of Materials Light Trans. Eq. Solving the Int. Eq. Multiple Scattering Light Transport Equations ∂L ( � x, θ, φ ) = ∂s � x ; θ, φ ; θ ′ , φ ′ ) L ( � x, θ ′ , φ ′ ) dθ ′ dφ ′ − σ t L ( � x, θ, φ ) + σ s p ( � (8) cos θ∂L ( θ, φ ) = ∂z � p ( θ, φ ; θ ′ , φ ′ ) L ( θ ′ , φ ′ ) dθ ′ dφ ′ − σ t L ( θ, φ ) + σ s (9)

Ref. & Trans. Desc. of Materials Light Trans. Eq. Solving the Int. Eq. Multiple Scattering Light Transport Equations ∂L ( � x, θ, φ ) = ∂s � x ; θ, φ ; θ ′ , φ ′ ) L ( � x, θ ′ , φ ′ ) dθ ′ dφ ′ − σ t L ( � x, θ, φ ) + σ s p ( � (8) cos θ∂L ( θ, φ ) = ∂z � p ( θ, φ ; θ ′ , φ ′ ) L ( θ ′ , φ ′ ) dθ ′ dφ ′ − σ t L ( θ, φ ) + σ s (9) L ( z ; θ, φ ) = (10) � z σ s ( z ′ ) p ( z ′ ; θ, φ ; θ ′ , φ ′ ) L ( z ′ ; θ ′ ; φ ′ ) dw ′ dz ′ R z ′ � σ t dz ′′ e − 0 cos θ cos θ 0

Ref. & Trans. Desc. of Materials Light Trans. Eq. Solving the Int. Eq. Multiple Scattering L ( θ, φ ) = L + ( θ, φ ) + L − ( π − θ, φ ) (11)

Ref. & Trans. Desc. of Materials Light Trans. Eq. Solving the Int. Eq. Multiple Scattering L ( θ, φ ) = L + ( θ, φ ) + L − ( π − θ, φ ) (11) � L + ( z = 0; θ ′ , φ ′ ) f t,s ( θ, φ ; θ ′ , φ ′ ) L i ( θ, φ ) dw i = (12)

Ref. & Trans. Desc. of Materials Light Trans. Eq. Solving the Int. Eq. Multiple Scattering L ( θ, φ ) = L + ( θ, φ ) + L − ( π − θ, φ ) (11) � L + ( z = 0; θ ′ , φ ′ ) f t,s ( θ, φ ; θ ′ , φ ′ ) L i ( θ, φ ) dw i = (12) T 12 ( n i , n t ; θ i , φ i → θ ′ , φ ′ ) L i ( θ i , φ i ) (13) =

Ref. & Trans. Desc. of Materials Light Trans. Eq. Solving the Int. Eq. Multiple Scattering � L r,v ( θ r , φ r ) = f t,s ( θ, φ ; θ r , φ r ) L − ( z = 0; θ, φ ) dw (14)

Ref. & Trans. Desc. of Materials Light Trans. Eq. Solving the Int. Eq. Multiple Scattering � L r,v ( θ r , φ r ) = f t,s ( θ, φ ; θ r , φ r ) L − ( z = 0; θ, φ ) dw (14) T 21 ( n 2 , n 1 ; θ, φ → θ r , φ r ) L − ( θ, φ ) = (15)

Ref. & Trans. Desc. of Materials Light Trans. Eq. Solving the Int. Eq. Multiple Scattering � L r,v ( θ r , φ r ) = f t,s ( θ, φ ; θ r , φ r ) L − ( z = 0; θ, φ ) dw (14) T 21 ( n 2 , n 1 ; θ, φ → θ r , φ r ) L − ( θ, φ ) = (15) � L t,v ( θ t , φ t ) = f t,s ( θ, φ ; θ t , φ t ) L + ( z = d ; θ, φ ) dw (16)

Ref. & Trans. Desc. of Materials Light Trans. Eq. Solving the Int. Eq. Multiple Scattering � L r,v ( θ r , φ r ) = f t,s ( θ, φ ; θ r , φ r ) L − ( z = 0; θ, φ ) dw (14) T 21 ( n 2 , n 1 ; θ, φ → θ r , φ r ) L − ( θ, φ ) = (15) � L t,v ( θ t , φ t ) = f t,s ( θ, φ ; θ t , φ t ) L + ( z = d ; θ, φ ) dw (16) T 23 ( n 2 , n 3 ; θ, φ → θ t , φ t ) L + ( z = d ; θ, φ ) =

Ref. & Trans. Desc. of Materials Light Trans. Eq. Solving the Int. Eq. Multiple Scattering Solving the Intergral Equation ∞ � L ( i ) L = i =0 L ( i +1) ( z ; θ, φ ) = (17) � z σ s ( z ′ ) p ( z ′ ; θ, φ ; θ ′ , φ ′ ) L ( i ) ( z ′ ; θ ′ ; φ ′ ) dw ′ dz ′ R z ′ � σ t dz ′′ e − 0 cos θ cos θ 0

Ref. & Trans. Desc. of Materials Light Trans. Eq. Solving the Int. Eq. Multiple Scattering First-Order Approximation L (0) L + ( z = 0) e − τ/ cos θ = (18) + where � z τ ( z ) = σ t dz 0 (19)

Ref. & Trans. Desc. of Materials Light Trans. Eq. Solving the Int. Eq. Multiple Scattering First-Order Approximation L (0) L + ( z = 0) e − τ/ cos θ = (18) + where � z τ ( z ) = σ t dz 0 (19) L (0) T 23 ( n 2 , n 3 ; θ, φ → θ t , φ t ) L (0) t,v ( θ t , φ t ) = + ( θ, φ ) T 12 T 23 e − τ d L i ( θ i , φ i ) = (20)

Ref. & Trans. Desc. of Materials Light Trans. Eq. Solving the Int. Eq. Multiple Scattering First-Order Approximation L (0) L + ( z = 0) e − τ/ cos θ = (18) + where � z τ ( z ) = σ t dz 0 (19) L (0) T 23 ( n 2 , n 3 ; θ, φ → θ t , φ t ) L (0) t,v ( θ t , φ t ) = + ( θ, φ ) T 12 T 23 e − τ d L i ( θ i , φ i ) = (20) cos θ i L (1) WT 12 T 21 p ( φ − θ r , φ r ; θ i , φ i ) r,v ( θ r , φ r ) = cos θ i + cos θ r (1 − e − τ d (1 / cos θ i +1 / cos θ r ) ) L i ( θ i , φ i ) (21)

Ref. & Trans. Desc. of Materials Light Trans. Eq. Solving the Int. Eq. Multiple Scattering The reflection steadily increases as the layer becomes thicker.

Ref. & Trans. Desc. of Materials Light Trans. Eq. Solving the Int. Eq. Multiple Scattering The distributions vary as a function of reflection direction. Lambert’s Law predicts a constant reflectance in all directions.

Ref. & Trans. Desc. of Materials Light Trans. Eq. Solving the Int. Eq. Multiple Scattering

Ref. & Trans. Desc. of Materials Light Trans. Eq. Solving the Int. Eq. Multiple Scattering Multiple Scattering An Monte Carlo Algorithm: Initialize: Events: Step: Scatter: Score:

Ref. & Trans. Desc. of Materials Light Trans. Eq. Solving the Int. Eq. Multiple Scattering L r,v ( θ r , φ r ) = L (1) ( θ r , φ r ) + L m (22)