FUNDAMENTALS Absorption Scattering MONTE CARLO METHODS FOR - PowerPoint PPT Presentation

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 CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING — FUNDAMENTALS 2

RADIATIVE TRANSFER d z Radiance d z d A d z L ( x , ω ) x , ω ) x x MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING — FUNDAMENTALS 3

ABSORPTION d z x ) L ( x , ω ) x , ω ) x d L µ a ( - absorption coefficient d z = − µ a ( x ) L ( x , ω ) MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING — FUNDAMENTALS 4

OUT-SCATTERING d z x ) L ( x , ω ) x , ω ) x d L µ s ( - scattering coefficient d z = − µ s ( x ) L ( x , ω ) MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING — FUNDAMENTALS 5

IN-SCATTERING d z x ) L ( x , ω ) x , ω ) x In-scattered radiance Z L s ( y , ω ) = S 2 f p ( ω , ¯ ω ) L ( y , ¯ ω )d¯ ω d L µ s ( - scattering coefficient d z = µ s ( x ) L s ( x , ω ) MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING — FUNDAMENTALS 6

EMISSION d z x ) L ( x , ω ) x , ω ) x d L L e - emitted radiance d z = µ a ( x ) L e ( x , ω ) MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING — FUNDAMENTALS 7

RADIATIVE TRANSFER EQUATION Out-scattering Absorption d L ( x , ω ) Losses − µ a ( x ) L ( x , ω ) = − µ s ( x ) L ( x , ω ) = d z Gains = µ a ( x ) L e ( x , ω ) + = µ s ( x ) L s ( x , ω ) + In-scattering Emission [Chandrasekhar 1960] MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING — FUNDAMENTALS 8

RADIATIVE TRANSFER Extinction coefficient µ t ( x ) = µ a ( x ) + µ s ( x ) d L ( x , ω ) Losses = = − µ t ( x ) L ( x , ω ) d z Gains = µ a ( x ) L e ( x , ω ) + = µ s ( x ) L s ( x , ω ) + What about a finite-length beam? d z [Chandrasekhar 1960] MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING — FUNDAMENTALS 9

RTE — INTEGRAL FORM Z z h i L ( x , ω ) = T ( x , y ) µ a ( y ) L e ( y , ω ) + µ s ( y ) L s ( y , ω ) d y 0 d L ( x , ω ) Losses = = − µ t ( x ) L ( x , ω ) d z Gains = µ a ( x ) L e ( x , ω ) = µ s ( x ) L s ( x , ω ) + + What about a finite-length beam? d z MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING — FUNDAMENTALS 10

RTE — INTEGRAL FORM Z z h i L ( x , ω ) = T ( x , y ) µ a ( y ) L e ( y , ω ) + µ s ( y ) L s ( y , ω ) d y 0 R y is the fraction of light that makes 
 T ransmittance T ( x , y ) = e − 0 µ t ( s )d s it from y to x d z Optical thickness Z y T ( x , y ) τ ( x , y ) = µ t ( s )d s y x 0 MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING — FUNDAMENTALS 11

RTE — INTEGRAL FORM Z z h i L ( x , ω ) = T ( x , y ) µ a ( y ) L e ( y , ω ) + µ s ( y ) L s ( y , ω ) d y 0 Emission In-scattering d z y x MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING — FUNDAMENTALS 12

RTE — INTEGRAL FORM Z z h i L ( x , ω ) = T ( x , y ) µ a ( y ) L e ( y , ω ) + µ s ( y ) L s ( y , ω ) d y 0 Z L s ( y , ω ) = 2 f p ( ω , ¯ ω ) ¯) L ( y , ¯ ω )d¯ ω S 2 Phase function d z y x MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING — FUNDAMENTALS 13

PHASE FUNCTION 2 f p ( ω , ¯ ω ) ( ω , ( ω , ( ω , ( ω , ( ω , I sotropic Henyey-Greenstein Rayleigh Lorenz-Mie Lorenz-Mie small particles large particles MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING — FUNDAMENTALS 14

PHASE FUNCTION Backward scattering PF Forward scattering PF Smoke Steam MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING — FUNDAMENTALS 15

PHASE FUNCTION I sotropic PF Forward scattering PF MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING — FUNDAMENTALS 16

RTE — INTEGRAL FORM Z z h i L ( x , ω ) = T ( x , y ) µ a ( y ) L e ( y , ω ) + µ s ( y ) L s ( y , ω ) d y 0 + T ( x , z ) L o ( z , ω ) Background radiance d z z x Surface MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING — FUNDAMENTALS 17

VOLUME RENDERING EQUATION Z z h i L ( x , ω ) = T ( x , y ) µ a ( y ) L e ( y , ω ) + µ s ( y ) L s ( y , ω ) d y 0 + T ( x , z ) L o ( z , ω ) How do we solve it? MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING — FUNDAMENTALS 18

MONTE CARLO INTEGRATION Ray Sphere Path space Z F = f ( x ) d x D N h F i = 1 f ( x i ) X =1 p ( x i ) N i =1 Probability density function (PDF) MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING — FUNDAMENTALS 19

VRE ESTIMATOR Z z T ( x , y ) h i F i h F L ( x , ω ) = T ( x , y ) µ a ( y ) L e ( y , ω ) + µ s ( y ) L s ( y , ω ) d y p ( y ) 0 T ( x , z ) + T + T ( x , z ) L o ( z , ω ) P ( z ) p ( y ) - probability density of distance y P ( z ) - probability of exceeding distance z MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING — FUNDAMENTALS 20

VRE ESTIMATOR T ransmittance estimation Z z T ( x , y ) h i F i h F L ( x , ω ) = T ( x , y ) µ a ( y ) L e ( y , ω ) + µ s ( y ) L s ( y , ω ) d y p ( y ) 0 T ( x , z ) + T + T ( x , z ) L o ( z , ω ) P ( z ) Distance sampling MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING — FUNDAMENTALS 21