Reflection from Layered Surfaces due to Subsurface Scattering

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

1 Ref. & Trans.

2 Desc. of Materials

3 Light Trans. Eq.

4 Solving the Int. Eq.

4 Multiple Scattering

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

Reflected and Transmitted Radiances

Lr(θr, φr) = Lr,s(θr, φr) + Lr,v(θr, φr) (1) Lt(θt, φt) = Lri(θt, φt) + Lt,v(θt, φt) (2)

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

BRDF and BTDF

fr(θi, φi; θr, φr) ≡ Lr(θr, φr) Li(θi, φi) cos θidwi

(BRDF ) (3)

f_t(θ_i, φ_i; θ_t, φ_t) ≡ L_t(θ_t, φ_t) Li(θi, φi) cos θidwi

(BT DF ) (4)

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

Fresnel transmission and reflection

For planar surface

L_r(θ_r, φ_r) = R¹²(n_i, n_t; θ_i, φ_i → θ_r, φ_r)L_i(θ_i, φ_i) (5) L_t(θ_t, φ_t) = T¹²(n_i, n_t; θ_i, φ_i → θ_t, φ_t)L_i(θ_i, φ_i) (6)

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

Fresnel transmission and reflection

For planar surface

L_r(θ_r, φ_r) = R¹²(n_i, n_t; θ_i, φ_i → θ_r, φ_r)L_i(θ_i, φ_i) (5) L_t(θ_t, φ_t) = T¹²(n_i, n_t; θ_i, φ_i → θ_t, φ_t)L_i(θ_i, φ_i) (6) where

R¹²(ni, nt; θi, φi→ θ_r, φr) = R(ni, nt, cos θi, cos θt) T¹²(n_i, n_t; θ_i, φ_i→ θ_t, φ_t) = n²_t

n²_iT = n²_t

n²_i(1 − R)

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

Fresnel transmission and reflection

For planar surface

Lr(θr, φr) = R¹²(ni, nt; θi, φi → θ_r, φr)Li(θi, φi) (5) L_t(θ_t, φ_t) = T¹²(n_i, n_t; θ_i, φ_i → θ_t, φ_t)L_i(θ_i, φ_i) (6) In our model of reflection:

fr = Rfr,s+ T fr,v = Rfr,s+ (1 − R)fr,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 σt= σa+ σs

Scattering phase function Henyey-Greenstein p_HG(cos j) = 1

4π

1 − g²

(1 + g²− 2g cos j)^3/2

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 =

−σ_tL(~x, θ, φ) + σ_s Z

p(~x; θ, φ; θ⁰, φ⁰)L(~x, θ⁰, φ⁰)dθ⁰dφ⁰ (8)

cos θ∂L(θ, φ)

∂z =

−σ_tL(θ, φ) + σs

Z

p(θ, φ; θ⁰, φ⁰)L(θ⁰, φ⁰)dθ⁰dφ⁰ (9)

L(z; θ, φ) = (10)

Z z 0

e^{−R0z0σtcos θdz00} Z

σ_s(z⁰)p(z⁰; θ, φ; θ⁰, φ⁰)L(z⁰; θ⁰; φ⁰)dw⁰ dz⁰ cos θ

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

Light Transport Equations

∂L(~x, θ, φ)

∂s =

−σ_tL(~x, θ, φ) + σ_s Z

p(~x; θ, φ; θ⁰, φ⁰)L(~x, θ⁰, φ⁰)dθ⁰dφ⁰ (8)

cos θ∂L(θ, φ)

∂z =

−σ_tL(θ, φ) + σs

Z

p(θ, φ; θ⁰, φ⁰)L(θ⁰, φ⁰)dθ⁰dφ⁰ (9)

L(z; θ, φ) = (10)

Z z 0

e^{−R0z0σtcos θdz00} Z

σ_s(z⁰)p(z⁰; θ, φ; θ⁰, φ⁰)L(z⁰; θ⁰; φ⁰)dw⁰ dz⁰ cos θ

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

Light Transport Equations

∂L(~x, θ, φ)

∂s =

−σ_tL(~x, θ, φ) + σ_s Z

p(~x; θ, φ; θ⁰, φ⁰)L(~x, θ⁰, φ⁰)dθ⁰dφ⁰ (8)

cos θ∂L(θ, φ)

∂z =

−σ_tL(θ, φ) + σs

Z

p(θ, φ; θ⁰, φ⁰)L(θ⁰, φ⁰)dθ⁰dφ⁰ (9)

L(z; θ, φ) = (10)

Z z 0

e^{−R0z0σtcos θdz00} Z

σ_s(z⁰)p(z⁰; θ, φ; θ⁰, φ⁰)L(z⁰; θ⁰; φ⁰)dw⁰ dz⁰ cos θ

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; θ⁰, φ⁰) = Z

f_t,s(θ, φ; θ⁰, φ⁰)L_i(θ, φ)dw_i (12)

= T¹²(ni, nt; θi, φi → θ⁰, φ⁰)Li(θi, φi) (13)

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; θ⁰, φ⁰) =

Z

f_t,s(θ, φ; θ⁰, φ⁰)L_i(θ, φ)dw_i (12)

= T¹²(ni, nt; θi, φi → θ⁰, φ⁰)Li(θi, φi) (13)

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; θ⁰, φ⁰) =

Z

f_t,s(θ, φ; θ⁰, φ⁰)L_i(θ, φ)dw_i (12)

= T¹²(ni, nt; θi, φi → θ⁰, φ⁰)Li(θi, φi) (13)

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

L_r,v(θ_r, φ_r) = Z

f_t,s(θ, φ; θ_r, φ_r)L−(z = 0; θ, φ)dw (14)

= T²¹(n₂, n₁; θ, φ → θ_r, φ_r)L−(θ, φ) (15) L_t,v(θ_t, φ_t) =

Z

f_t,s(θ, φ; θ_t, φ_t)L₊(z = d; θ, φ)dw (16)

= T²³(n2, n3; θ, φ → θt, φt)L+(z = d; θ, φ)

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

L_r,v(θ_r, φ_r) = Z

f_t,s(θ, φ; θ_r, φ_r)L−(z = 0; θ, φ)dw (14)

= T²¹(n₂, n₁; θ, φ → θ_r, φ_r)L−(θ, φ) (15)

L_t,v(θ_t, φ_t) = Z

f_t,s(θ, φ; θ_t, φ_t)L₊(z = d; θ, φ)dw (16)

= T²³(n2, n3; θ, φ → θt, φt)L+(z = d; θ, φ)

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

L_r,v(θ_r, φ_r) = Z

f_t,s(θ, φ; θ_r, φ_r)L−(z = 0; θ, φ)dw (14)

= T²¹(n₂, n₁; θ, φ → θ_r, φ_r)L−(θ, φ) (15) L_t,v(θ_t, φ_t) =

Z

f_t,s(θ, φ; θ_t, φ_t)L₊(z = d; θ, φ)dw (16)

= T²³(n2, n3; θ, φ → θt, φt)L+(z = d; θ, φ)

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

L_r,v(θ_r, φ_r) = Z

f_t,s(θ, φ; θ_r, φ_r)L−(z = 0; θ, φ)dw (14)

= T²¹(n₂, n₁; θ, φ → θ_r, φ_r)L−(θ, φ) (15) L_t,v(θ_t, φ_t) =

Z

f_t,s(θ, φ; θ_t, φ_t)L₊(z = d; θ, φ)dw (16)

= T²³(n2, n3; θ, φ → θt, φt)L+(z = d; θ, φ)

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

Solving the Intergral Equation

L =

∞

X

i=0

L⁽ⁱ⁾

L⁽ⁱ⁺¹⁾(z; θ, φ) = (17)

Z z 0

e⁻

Rz0 0 σtdz00

cos θ

Z

σs(z⁰)p(z⁰; θ, φ; θ⁰, φ⁰)L⁽ⁱ⁾(z⁰; θ⁰; φ⁰)dw⁰ dz⁰ cos θ

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

τ (z) = Z z

0

σtdz

(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

τ (z) = Z z

0

σtdz

(19)

L⁽⁰⁾_t,v(θ_t, φ_t) = T²³(n₂, n₃; θ, φ → θ_t, φ_t)L⁽⁰⁾₊ (θ, φ)

= T¹²T²³e^−τdL_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

τ (z) = Z _z

0

σ_tdz

(19)

L⁽⁰⁾_t,v(θt, φt) = T²³(n2, n3; θ, φ → θt, φt)L⁽⁰⁾₊ (θ, φ)

= T¹²T²³e^−τdLi(θi, φi) (20)

L⁽¹⁾_r,v(θ_r, φ_r) = W T¹²T²¹p(φ − θ_r, φ_r; θ_i, φ_i) cos θi

cos θ_i+ cos θ_r (1 − e^{−τd(1/ cos θi+1/ cos θr)})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

Lr,v(θr, φr) = L⁽¹⁾(θr, φr) + L^m (22)