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

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

Mar 08, 2024 269 likes •773 views

CS580: Monte Carol Integration Sung-Eui Yoon ( ) Course URL: http://sglab.kaist.ac.kr/~sungeui/GCG Class Objectives Sampling approach for solving the rendering equation Monte Carlo integration Estimator and its variance

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CS580:

Monte Carol Integration

Sung-Eui Yoon (윤성의)

Course URL: http://sglab.kaist.ac.kr/~sungeui/GCG

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Class Objectives

  • Sampling approach for solving the

rendering equation

  • Monte Carlo integration
  • Estimator and its variance
  • Sampling according to the pdf
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Two Forms of the Rendering Equation

  • Hemisphere integration
  • Area integration
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Radiance Evaluation

  • Fundamental problem in GI algorithm
  • Evaluate radiance at a given surface point in a

given direction

  • I nvariance defines radiance everywhere else
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Radiance Evaluation

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Why Monte Carlo?

  • Radiace is hard to evaluate
  • Sample many paths
  • I ntegrate over all incoming directions
  • Analytical integration is difficult
  • Need numerical techniques

From kavita’s slides

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Monte Carlo Integration

  • Numerical tool to evaluate integrals
  • Use sampling
  • Stochastic errors
  • Unbiased
  • On average, we get the right answer
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  • Consider p(x) for estimate
  • We will study it as importance sampling later
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MC Integration - Example

  • I ntegral
  • Variance
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Advantages of MC

  • Convergence rate of
  • Simple
  • Sampling
  • Point evaluation
  • General
  • Works for high dimensions
  • Deals with discontinuities, crazy functions, etc.

) 1 ( N O

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Importance Sampling

  • Take more samples in important regions,

where the function is large

From kavita’s slides

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Sampling according to pdf

  • I nverse cumulative distribution function
  • Rejection sampling
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Inverse Cumulative Distribution Function – Discrete Case

, given uniform sampling

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Continuous Random Variable

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

 

y

dx x p y P ) ( ) (

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Rejection Method

  • Often not possible to compute the inverse
  • f cdf

From kavita’s slides

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Class Objectives were:

  • Sampling approach for solving the

rendering equation

  • Monte Carlo integration
  • Estimator and its variance
  • Sampling according to the pdf
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Next Time

  • Monte Carlo ray tracing