Monte Carlo 1: Integration

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1 Monte Carlo : Integraton Prevous lecture: Analytcal llumnaton formula Ths lecture: Monte Carlo Integraton Revew random varables and probablty Samplng from dstrbutons Samplng from shapes Numercal calculaton of llumnaton CS348b Lecture 6 Pat Hanrahan / Matt Pharr, Sprng 207

2 Irradance from the Envronment 2 d Φ ( x, ω) = L ( x, ω)cosθ da dω de( x, ω) = L ( x, ω)cosθ dω θ dω L ( x, ω) Lght meter E( x) = L ( x, ω)cosθ dω CS348b Lecture 6 Pat Hanrahan / Matt Pharr, Sprng 207 H 2 da

3 Irradance from a Unform Area Source A E( x) = L cosθ dω H = L 2 Ω = LΩ! cosθ dω Ω! Ω Drect Illumnaton CS348b Lecture 6 Pat Hanrahan / Matt Pharr, Sprng 207

4 Unform Trangle Lght Source ~N ~N edge n n!! A = γ N N = = CS348b Lecture 6 Pat Hanrahan / Matt Pharr, Sprng 207

5 Penumbras and Umbras CS348b Lecture 6 Pat Hanrahan / Matt Pharr, Sprng 207

6 Lghtng and Soft Shadows E( x) = L ( x, ω)cosθ dω H Challenges Occluders - Complex geometry - Number of occluders 2 Non-unform lght sources 2 Source: Agrawala. Ramamoorth, Herch, Moll, 2000 CS348b Lecture 6 Pat Hanrahan / Matt Pharr, Sprng 207

7 Monte Carlo Illumnaton Calculaton Center Random shadow ray per eye ray CS348b Lecture 6 Pat Hanrahan / Matt Pharr, Sprng 207

8 Monte Carlo Algorthms Advantages Easy to mplement Easy to thnk about (but be careful of subtletes) Robust when used wth complex ntegrands (lghts, BRDFs) and domans (shapes) Effcent for hgh dmensonal ntegrals Effcent when only need soluton at a few ponts Dsadvantages Nosy Slow (many samples needed for convergence) CS348b Lecture 6 Pat Hanrahan / Matt Pharr, Sprng 207

9 Random Varables X s a random varable A random varable takes on dfferent values (represents a dstrbuton of potental values) X ~ p( x) probablty dstrbuton functon (PDF) CS348b Lecture 6 Pat Hanrahan / Matt Pharr, Sprng 207

10 Dscrete Probablty Dstrbutons Dscrete values wth probablty x p p p 0 x n = p = CS348b Lecture 6 Pat Hanrahan / Matt Pharr, Sprng 207

11 Dscrete Probablty Dstrbutons Cumulatve PDF P j P j j = = p 0 P P n = P j 0 CS348b Lecture 6 Pat Hanrahan / Matt Pharr, Sprng 207

12 Dscrete Probablty Dstrbutons Constructon of samples To randomly select an event, Select x f P U P < U Unform random varable CS348b Lecture 6 x 3 0 Pat Hanrahan / Matt Pharr, Sprng 207

13 Contnuous Probablty Dstrbutons PDF p( x) p( x) 0 Unform CDF P( x) CS348b Lecture 6 x P( x) = p( x) dx 0 P( x) = Pr( X < x) Pr( α X β ) = p( x) dx β α P () = = P( β ) P( α) 0 0 Pat Hanrahan / Matt Pharr, Sprng 207

14 Samplng Contnuous Dstrbutons Cumulatve probablty dstrbuton functon P( x) = Pr( X < x) Constructon of samples Solve for X = P (U) U Must know the formula for:. The ntegral of p(x) 2. The nverse functon P (x) 0 X CS348b Lecture 6 Pat Hanrahan / Matt Pharr, Sprng 207

15 Samplng a Crcle 2π 2π 2 # r $ 2π A = r dr dθ = r dr dθ = % & θ = π ' ( 0 r p( r, θ ) dr dθ = r dr dθ p( r, θ ) = π π p( r, θ ) = p( r) p( θ ) p( θ ) = 2π P( θ ) = 2π p( r) = 2r P( r) = r 2 θ θ = 2π U r = U 2 CS348b Lecture 6 Pat Hanrahan / Matt Pharr, Sprng 207 rdθ dr

16 Samplng a Crcle WRONG Equ-Areal RIGHT = Equ-Areal θ r = = 2πU U 2 θ r = = 2πU U 2 CS348b Lecture 6 Pat Hanrahan / Matt Pharr, Sprng 207

17 Rejecton Samplng do { X=Unform(-,) Y=Unform(-,) } whle (X*X+Y*Y>); Effcency? Area of crcle / Area of square CS348b Lecture 6 Pat Hanrahan / Matt Pharr, Sprng 207

18 Samplng 2D Drectons do { X=Unform(-,); Y=Unform(-,); } whle (X*X+Y*Y>); R = sqrt(x*x+y*y) dx = X/R dy = Y/R CS348b Lecture 6 Pat Hanrahan / Matt Pharr, Sprng 207

19 Computng Area of a Crcle A = 0 for( =0; <N; ++ ) { X=Unform(-,); Y=Unform(-,); f(x*x+y*y < ) A += ; } A = 4*A/N CS348b Lecture 6 Pat Hanrahan / Matt Pharr, Sprng 207

20 Monte Carlo Integraton Defnte ntegral Expectaton of f Random varables I( f ) f ( x) dx 0 E[ f ] f ( x) p( x) dx X Y 0 ~ p( x) = f ( X ) Estmator CS348b Lecture 6 F N N = Y N = Pat Hanrahan / Matt Pharr, Sprng 207

21 Unbased Estmator E[ F ] = I( f ) N Propertes E[ Y ] = E[ Y ] E[ ay] = ae[ Y] N E[ FN ] = E[ Y ] N CS348b Lecture 6 Pat Hanrahan / Matt Pharr, Sprng = N N = E[ Y ] = E[ f ( X)] N N = = = N N = = N = 0 N = 0 f ( x) dx f ( x) p( x) dx f ( x) dx Assume unform probablty dstrbuton for now

22 Drect Lghtng: Hemsphercal Integral L ( x, ω) E( x) = L( x, ω)cosθ dω H 2 θ dω da CS348b Lecture 6 Pat Hanrahan / Matt Pharr, Sprng 207

23 Drect Lghtng: Sold Angle Samplng Sample hemsphere unformly L ( x, ω) H 2 p( ω) dω = θ dω da CS348b Lecture 6 Pat Hanrahan / Matt Pharr, Sprng 207

24 Drect Lghtng: Sold Angle Samplng L ( x, ω) θ dω Estmator da CS348b Lecture 6 Pat Hanrahan / Matt Pharr, Sprng 207

25 Drect Lghtng: Hemsphere Samplng Hemsphere 6 shadow rays CS348b Lecture 6 Pat Hanrahan / Matt Pharr, Sprng 207

26 Drect Lghtng: Area Integral A! x! Integral x θ θ" Vsblty! 0 blocked V ( x, x" ) = # $ vsble Radance CS348b Lecture 6 Pat Hanrahan / Matt Pharr, Sprng 207

27 Drect Lghtng: Area Integral A! x! θ θ" x CS348b Lecture 6 Pat Hanrahan / Matt Pharr, Sprng 207

28 Drect Lghtng: Area Samplng A! x! θ θ" x Sample shape unformly by area CS348b Lecture 6 Pat Hanrahan / Matt Pharr, Sprng 207

29 Drect Lghtng: Area Samplng A! x! θ θ" Estmator ω" x CS348b Lecture 6 Pat Hanrahan / Matt Pharr, Sprng 207

30 Drect Lghtng: Area Samplng Area 6 shadow rays CS348b Lecture 6 Pat Hanrahan / Matt Pharr, Sprng 207

31 Random Samplng Introduces Nose Center Random shadow ray per eye ray CS348b Lecture 6 Pat Hanrahan / Matt Pharr, Sprng 207

32 Qualty Improves wth More Rays Area Area shadow ray 6 shadow rays CS348b Lecture 6 Pat Hanrahan / Matt Pharr, Sprng 207

33 Why s Area Better than Hemsphere? Hemsphere 6 shadow rays Area 6 shadow rays CS348b Lecture 6 Pat Hanrahan / Matt Pharr, Sprng 207

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