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

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

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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

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SLIDE 1
  • 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

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SLIDE 2
  • 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

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SLIDE 3
  • 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)

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SLIDE 4
  • 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) ft(θi, φi; θt, φt) ≡ Lt(θt, φt) Li(θi, φi) cos θidwi (BTDF) (4)

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SLIDE 5
  • 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 → θr, φr)Li(θi, φi) (5) Lt(θt, φt) = T 12(ni, nt; θi, φi → θt, φt)Li(θi, φi) (6)

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SLIDE 6
  • 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 → θr, φr)Li(θi, φi) (5) Lt(θt, φt) = T 12(ni, nt; θi, φi → θt, φt)Li(θi, φi) (6) where R12(ni, nt; θi, φi → θr, φr) = R(ni, nt, cos θi, cos θt) T 12(ni, nt; θi, φi → θt, φt) = n2

t

n2

i

T = n2

t

n2

i

(1 − R)

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SLIDE 7
  • 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 → θr, φr)Li(θi, φi) (5) Lt(θt, φt) = T 12(ni, nt; θi, φi → θt, φt)Li(θi, φi) (6) In our model of reflection: fr = Rfr,s + Tfr,v = Rfr,s + (1 − R)fr,v (7)

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SLIDE 8
  • 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 pHG(cos j) = 1 4π 1 − g2 (1 + g2 − 2g cos j)3/2

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SLIDE 9
  • 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 = −σtL( x, θ, φ) + σs

  • p(

x; θ, φ; θ′, φ′)L( x, θ′, φ′)dθ′dφ′ (8)

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SLIDE 10
  • Ref. & Trans.
  • Desc. of Materials

Light Trans. Eq. Solving the Int. Eq. Multiple Scattering

Light Transport Equations

∂L( x, θ, φ) ∂s = −σtL( x, θ, φ) + σs

  • p(

x; θ, φ; θ′, φ′)L( x, θ′, φ′)dθ′dφ′ (8) cos θ∂L(θ, φ) ∂z = −σtL(θ, φ) + σs

  • p(θ, φ; θ′, φ′)L(θ′, φ′)dθ′dφ′

(9)

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SLIDE 11
  • Ref. & Trans.
  • Desc. of Materials

Light Trans. Eq. Solving the Int. Eq. Multiple Scattering

Light Transport Equations

∂L( x, θ, φ) ∂s = −σtL( x, θ, φ) + σs

  • p(

x; θ, φ; θ′, φ′)L( x, θ′, φ′)dθ′dφ′ (8) cos θ∂L(θ, φ) ∂z = −σtL(θ, φ) + σs

  • p(θ, φ; θ′, φ′)L(θ′, φ′)dθ′dφ′

(9) L(z; θ, φ) = (10) z e−

R z′ σt dz′′

cos θ

  • σs(z′)p(z′; θ, φ; θ′, φ′)L(z′; θ′; φ′)dw′ dz′

cos θ

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SLIDE 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)

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SLIDE 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; θ′, φ′) =

  • ft,s(θ, φ; θ′, φ′)Li(θ, φ)dwi

(12)

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SLIDE 14
  • 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′) =

  • ft,s(θ, φ; θ′, φ′)Li(θ, φ)dwi

(12) = T 12(ni, nt; θi, φi → θ′, φ′)Li(θi, φi)(13)

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SLIDE 15
  • Ref. & Trans.
  • Desc. of Materials

Light Trans. Eq. Solving the Int. Eq. Multiple Scattering

Lr,v(θr, φr) =

  • ft,s(θ, φ; θr, φr)L−(z = 0; θ, φ)dw

(14)

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SLIDE 16
  • Ref. & Trans.
  • Desc. of Materials

Light Trans. Eq. Solving the Int. Eq. Multiple Scattering

Lr,v(θr, φr) =

  • ft,s(θ, φ; θr, φr)L−(z = 0; θ, φ)dw

(14) = T 21(n2, n1; θ, φ → θr, φr)L−(θ, φ) (15)

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SLIDE 17
  • Ref. & Trans.
  • Desc. of Materials

Light Trans. Eq. Solving the Int. Eq. Multiple Scattering

Lr,v(θr, φr) =

  • ft,s(θ, φ; θr, φr)L−(z = 0; θ, φ)dw

(14) = T 21(n2, n1; θ, φ → θr, φr)L−(θ, φ) (15) Lt,v(θt, φt) =

  • ft,s(θ, φ; θt, φt)L+(z = d; θ, φ)dw

(16)

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SLIDE 18
  • Ref. & Trans.
  • Desc. of Materials

Light Trans. Eq. Solving the Int. Eq. Multiple Scattering

Lr,v(θr, φr) =

  • ft,s(θ, φ; θr, φr)L−(z = 0; θ, φ)dw

(14) = T 21(n2, n1; θ, φ → θr, φr)L−(θ, φ) (15) Lt,v(θt, φt) =

  • ft,s(θ, φ; θt, φt)L+(z = d; θ, φ)dw

(16) = T 23(n2, n3; θ, φ → θt, φt)L+(z = d; θ, φ)

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SLIDE 19
  • Ref. & Trans.
  • Desc. of Materials

Light Trans. Eq. Solving the Int. Eq. Multiple Scattering

Solving the Intergral Equation

L =

∞

  • i=0

L(i) L(i+1)(z; θ, φ) = (17) z e−

R z′ σt dz′′

cos θ

  • σs(z′)p(z′; θ, φ; θ′, φ′)L(i)(z′; θ′; φ′)dw′ dz′

cos θ

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SLIDE 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 σtdz (19)

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SLIDE 21
  • 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 σtdz (19) L(0)

t,v (θt, φt)

= T 23(n2, n3; θ, φ → θt, φt)L(0)

+ (θ, φ)

= T 12T 23e−τdLi(θi, φi) (20)

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SLIDE 22
  • 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 σtdz (19) L(0)

t,v (θt, φt)

= T 23(n2, n3; θ, φ → θt, φt)L(0)

+ (θ, φ)

= T 12T 23e−τdLi(θi, φi) (20) L(1)

r,v(θr, φr)

= WT 12T 21p(φ − θr, φr; θi, φi) cos θi cos θi + cos θr (1 − e−τd(1/ cos θi+1/ cos θr))Li(θi, φi) (21)

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SLIDE 23
  • Ref. & Trans.
  • Desc. of Materials

Light Trans. Eq. Solving the Int. Eq. Multiple Scattering

The reflection steadily increases as the layer becomes thicker.

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SLIDE 24
  • 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.

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SLIDE 25
  • Ref. & Trans.
  • Desc. of Materials

Light Trans. Eq. Solving the Int. Eq. Multiple Scattering

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SLIDE 26
  • 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:

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SLIDE 27
  • Ref. & Trans.
  • Desc. of Materials

Light Trans. Eq. Solving the Int. Eq. Multiple Scattering

Lr,v(θr, φr) = L(1)(θr, φr) + Lm (22)