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

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

Aug 03, 2023 126 likes •640 views

CS680: CS680: Monte Carol Integration Sung-Eui Yoon ( ) ( ) C Course URL: URL http://jupiter.kaist.ac.kr/~sungeui/SGA/ Course Administration Course Administration Due is this Thur. HW HW 2 Rendering

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CS680: CS680:

Monte Carol Integration

Sung-Eui Yoon (윤성의) (윤성의)

C URL Course URL: http://jupiter.kaist.ac.kr/~sungeui/SGA/

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Course Administration Course Administration

HW

  • HW
  • Due is this Thur.

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Previous Time Previous Time

Radiometry

  • Radiometry
  • Rendering equation

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Two Forms of the Rendering Equation Equation

Hemisphere integration

  • Hemisphere integration
  • Area integration

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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

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Radiance Evaluation Radiance Evaluation

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Why Monte Carlo? Why Monte Carlo?

Radiace is hard to evaluate

  • Radiace is hard to evaluate

From kavita’s slides

  • Sample many paths
  • I ntegrate over all incoming directions

From kavita s slides

g g

  • Analytical integration is difficult
  • Need numerical techniques

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q

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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

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C ( ) f

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

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MC Integration Example MC Integration - Example

I ntegral

  • I ntegral
  • Variance

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Advantages of MC Advantages of MC

Convergence rate of

) 1 ( O

  • Convergence rate of

) ( N O

  • Simple
  • Sampling

P i t l ti

  • Point evaluation

G l

  • General
  • Works for high dimensions
  • Deals with discontinuities crazy functions etc
  • Deals with discontinuities, crazy functions, etc.

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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

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Sampling according to pdf Sampling according to pdf

I nverse cumulative distribution function

  • I nverse cumulative distribution function
  • Rejection sampling

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Inverse Cumulative Distribution Function Discrete Case Function – Discrete Case

, given uniform sampling

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Continuous Random Variable Continuous Random Variable

Algorithm

  • Algorithm
  • Pick u uniformly from [0, 1)
  • Output y = P-1(u) where

y

d P ) ( ) (

  • Output y = P 1(u), where

 

 dx x p y P ) ( ) (

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Rejection Method Rejection Method

Often not possible to compute the inverse

  • Often not possible to compute the inverse
  • f cdf

From kavita’s slides

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Summary Summary

Monte Carlo integration

  • Monte Carlo integration
  • Estimators

f

  • Sampling non-uniform distribution

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Next Time Next Time

Monte Carlo ray tracing

  • Monte Carlo ray tracing

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