Monte Carol Integration Sung-Eui Yoon ( ) ( ) C Course URL: URL - PowerPoint PPT Presentation

CS680: CS680: Monte Carol Integration Sung-Eui Yoon ( 윤성의 ) ( 윤성의 ) C Course URL: URL http://jupiter.kaist.ac.kr/~sungeui/SGA/

Course Administration Course Administration ● Due is this Thur. ● HW HW 2

● Rendering equation Previous Time Previous Time ● Radiometry Radiometry 3

Two Forms of the Rendering Equation Equation ● Hemisphere integration Hemisphere integration ● Area integration 4

Radiance Evaluation Radiance Evaluation ● Fundamental problem in GI algorithm Fundamental problem in GI algorithm ● Evaluate radiance at a given surface point in a given direction given direction ● I nvariance defines radiance everywhere else 5

Radiance Evaluation Radiance Evaluation 6

Why Monte Carlo? Why Monte Carlo? ● Radiace is hard to evaluate Radiace is hard to evaluate From kavita’s slides From kavita s slides ● Sample many paths ● I ntegrate over all incoming directions g g ● Analytical integration is difficult ● Need numerical techniques q 7

Monte Carlo Integration Monte Carlo Integration ● Numerical tool to evaluate integrals Numerical tool to evaluate integrals ● Use sampling ● Stochastic errors ● Unbiased ● On average, we get the right answer 8

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• Consider p(x) for estimate C ( ) f • We will study it as importance sampling later 24

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Example MC Integration - Example MC Integration ● Variance ● I ntegral I ntegral 29

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Advantages of MC Advantages of MC 1 ● Convergence rate of Convergence rate of O O ( ( ) ) N ● Simple ● Sampling ● Point evaluation P i t l ti ● General G l ● Works for high dimensions ● Deals with discontinuities crazy functions etc ● Deals with discontinuities, crazy functions, etc. 33

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Importance Sampling Importance Sampling ● Take more samples in important regions, Take more samples in important regions where the function is large From kavita’s slides 36

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Sampling according to pdf Sampling according to pdf ● I nverse cumulative distribution function I nverse cumulative distribution function ● Rejection sampling 41

Inverse Cumulative Distribution Function Function – Discrete Case Discrete Case , given uniform sampling 42

Continuous Random Variable Continuous Random Variable ● Algorithm Algorithm ● Pick u uniformly from [0, 1)    y  ● Output y = P 1 (u), where ● Output y = P -1 (u) where P P ( ( y ) ) p ( ( x ) ) dx d  43

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Rejection Method Rejection Method ● Often not possible to compute the inverse Often not possible to compute the inverse of cdf From kavita’s slides 1 48

Summary Summary ● Monte Carlo integration Monte Carlo integration ● Estimators ● Sampling non-uniform distribution f 49

● Monte Carlo ray tracing Monte Carlo ray tracing Next Time Next Time 50