FUNDAMENTALS Absorption Scattering MONTE CARLO METHODS FOR - - PowerPoint PPT Presentation

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FUNDAMENTALS Absorption Scattering MONTE CARLO METHODS FOR - - PowerPoint PPT Presentation

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FUNDAMENTALS Absorption Scattering MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING FUNDAMENTALS 1 FUNDAMENTALS Absorption Scattering Emission http://commons.wikimedia.org http://coclouds.com http://wikipedia.org MONTE

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SLIDE 1

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING

FUNDAMENTALS

1 — FUNDAMENTALS

Absorption Scattering

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SLIDE 2

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING

FUNDAMENTALS

2 — FUNDAMENTALS

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

Absorption Scattering Emission

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SLIDE 3

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING

dz

x, ω)

x

x dz

RADIATIVE TRANSFER

3 — FUNDAMENTALS

L(x, ω)

Radiance

dA dz

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SLIDE 4

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING

ABSORPTION

4 — FUNDAMENTALS

dz

x, ω)

x −µa(x)L(x, ω)

µa(- absorption coefficient

dL dz =

x)L(x, ω)

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SLIDE 5

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING

OUT-SCATTERING

5 — FUNDAMENTALS

dz

x, ω)

x

x)L(x, ω)

dL dz = −µs(x)L(x, ω)

µs(- scattering coefficient

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SLIDE 6

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING

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

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SLIDE 7

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING

EMISSION

7 — FUNDAMENTALS

dz

x, ω)

x dL dz = µa(x)Le(x, ω)

Le - emitted radiance

x)L(x, ω)

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SLIDE 8

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING

RADIATIVE TRANSFER EQUATION

8 — FUNDAMENTALS

Losses

dL(x, ω) dz =

In-scattering Emission Out-scattering Absorption Gains

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

[Chandrasekhar 1960]

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SLIDE 9

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING

RADIATIVE TRANSFER

9 — FUNDAMENTALS

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]

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SLIDE 10

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING

Losses Gains

10 — FUNDAMENTALS

What about a finite-length beam?

dz

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

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SLIDE 11

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING

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

Optical thickness

τ(x, y) = Z y µt(s)ds

RTE — INTEGRAL FORM

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SLIDE 12

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING 12 — FUNDAMENTALS

RTE — INTEGRAL FORM

In-scattering 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

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SLIDE 13

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING 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

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SLIDE 14

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING 14 — FUNDAMENTALS

PHASE FUNCTION

I sotropic

(ω,

Henyey-Greenstein

(ω,

Rayleigh

(ω,

Lorenz-Mie small particles

(ω,

Lorenz-Mie large particles

(ω,

2 fp(ω, ¯

ω)

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SLIDE 15

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING 15 — FUNDAMENTALS

PHASE FUNCTION

Smoke Steam Backward scattering PF Forward scattering PF

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SLIDE 16

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING 16 — FUNDAMENTALS

PHASE FUNCTION

I sotropic PF Forward scattering PF

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SLIDE 17

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING 17 — 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

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SLIDE 18

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING 18 — 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?

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SLIDE 19

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING

Sphere

hFi = 1 N

N

X

i=1

Ray Path space

MONTE CARLO INTEGRATION

19 — FUNDAMENTALS

F = Z

D

f(x) dx f(xi)

=1 p(xi)

Probability density function (PDF)

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SLIDE 20

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING 20 — 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

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SLIDE 21

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING 21 — 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