FUNDAMENTALS J an Novk Scattering Disney Research Absorption - - PowerPoint PPT Presentation

FUNDAMENTALS

J an Novák


Disney Research

Absorption Scattering

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

FUNDAMENTALS

2 — FUNDAMENTALS

http://commons.wikimedia.org http://coclouds.com http://wikipedia.org

Absorption Scattering Emission

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

dz

x, ω)

x

x dz

RADIATIVE TRANSFER

3 — FUNDAMENTALS

L(x, ω)

Radiance

dA dz

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

ABSORPTION

4 — FUNDAMENTALS

dz

x, ω)

x −µa(x)L(x, ω)

µa(- absorption coefficient

dL dz =

x)L(x, ω)

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

OUT-SCATTERING

5 — FUNDAMENTALS

dz

x, ω)

x

x)L(x, ω)

dL dz = −µs(x)L(x, ω)

µs(- scattering coefficient

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

IN-SCATTERING

6 — FUNDAMENTALS

dz

x, ω)

x

x)L(x, ω)

dL dz = µs(x)Ls(x, ω)

In-scattered radiance

Ls(y, ω) = Z

S2 fp(ω, ¯

ω)L(y, ¯ ω)d¯ ω

µs(- scattering coefficient

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

EMISSION

7 — FUNDAMENTALS

dz

x, ω)

x dL dz = µa(x)Le(x, ω)

Le - emitted radiance

x)L(x, ω)

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

RADIATIVE TRANSFER

8 — FUNDAMENTALS

EQUATION

dL(x, ω) dz =

In-scattering Emission Out-scattering Absorption

−µa(x)L(x, ω) = −µs(x)L(x, ω) + = µa(x)Le(x, ω)+ = µs(x)Ls(x, ω)

[Chandrasekhar 1960]

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

RADIATIVE TRANSFER

9 — FUNDAMENTALS

EQUATION

Losses

dL(x, ω) dz =

µt(x) = µa(x) + µs(x)

Extinction coefficient

= −µt(x)L(x, ω)

What about a finite-length beam?

dz

Gains

+ = µa(x)Le(x, ω)+ = µs(x)Ls(x, ω)

[Chandrasekhar 1960]

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

Losses Gains

10 — FUNDAMENTALS

What about a finite-length beam?

dz

RADIATIVE TRANSFER EQUATION RTE — INTEGRAL FORM

dL(x, ω) dz =

+ = µs(x)Ls(x, ω) + = µa(x)Le(x, ω)

= −µt(x)L(x, ω)

L(x, ω) = Z z T(x, y) h µa(y)Le(y, ω) + µs(y)Ls(y, ω) i dy

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

L(x, ω) = Z z T(x, y) h µa(y)Le(y, ω) + µs(y)Ls(y, ω) i dy

11 — FUNDAMENTALS

dz

x y T(x, y)

is the fraction of light that makes
 it from y to x T ransmittance T(x, y) = e−

R y

0 µt(s)ds

RTE — INTEGRAL FORM

Optical thickness

τ(x, y) = Z y µt(s)ds

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION 12 — FUNDAMENTALS

RTE — INTEGRAL FORM

Emission

L(x, ω) = Z z T(x, y) h µa(y)Le(y, ω) + µs(y)Ls(y, ω) i dy

dz

x y

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION 13 — FUNDAMENTALS

RTE — INTEGRAL FORM

L(x, ω) = Z z T(x, y) h µa(y)Le(y, ω) + µs(y)Ls(y, ω) i dy

Phase function

¯)L(y, ¯ ω)d¯ ω Ls(y, ω) = Z

S2

2 fp(ω, ¯

ω)

dz

x y

In-scattering

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION 14 — FUNDAMENTALS

RTE — INTEGRAL FORM

Surface

z

+T(x, z)Lo(z, ω)

Background radiance

L(x, ω) = Z z T(x, y) h µa(y)Le(y, ω) + µs(y)Ls(y, ω) i dy

dz

x

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION 15 — FUNDAMENTALS

VOLUME RENDERING EQUATION

+T(x, z)Lo(z, ω) L(x, ω) = Z z T(x, y) h µa(y)Le(y, ω) + µs(y)Ls(y, ω) i dy

How do we solve it?

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION

Sphere

hFi = 1 N

N

X

i=1

Ray Path space

MONTE CARLO INTEGRATION

16 — FUNDAMENTALS

F = Z

D

f(x) dx f(xi)

=1 p(xi)

Probability density function (PDF)

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION 17 — FUNDAMENTALS

VRE ESTIMATOR

+T(x, z)Lo(z, ω) L(x, ω) = Z z T(x, y) h µa(y)Le(y, ω) + µs(y)Ls(y, ω) i dy

hF Fi T(x, z) P(z)

+T

T(x, y) p(y) p(y) - probability density of distance y P(z) - probability of exceeding distance z

MONTE CARLO METHODS FOR VOLUMETRIC LIGHT TRANSPORT SIMULATION 18 — FUNDAMENTALS

VRE ESTIMATOR

+T(x, z)Lo(z, ω) L(x, ω) = Z z T(x, y) h µa(y)Le(y, ω) + µs(y)Ls(y, ω) i dy

hF Fi T(x, z) P(z)

+T

T(x, y) p(y)

Distance sampling T ransmittance estimation