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Monte Carlo I

Monte Carlo I. Previous lecture Analytical illumination formula This lecture Numerical evaluation of illumination Review random variables and probability Monte Carlo integration Sampling from distributions Sampling from shapes Variance and efficiency. Lighting and Soft Shadows.

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Monte Carlo I

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  1. Monte Carlo I • Previous lecture • Analytical illumination formula • This lecture • Numerical evaluation of illumination • Review random variables and probability • Monte Carlo integration • Sampling from distributions • Sampling from shapes • Variance and efficiency

  2. Lighting and Soft Shadows • Challenges • Visibility and blockers • Varying light distribution • Complex source geometry Source: Agrawala. Ramamoorthi, Heirich, Moll, 2000

  3. Penumbras and Umbras

  4. Monte Carlo Lighting Fixed Random 1 eye ray per pixel 1 shadow ray per eye ray

  5. Monte Carlo Algorithms • Advantages • Easy to implement • Easy to think about (but be careful of statistical bias) • Robust when used with complex integrands and domains (shapes, lights, …) • Efficient for high dimensional integrals • Efficient solution method for a few selected points • Disadvantages • Noisy • Slow (many samples needed for convergence)

  6. Random Variables • is chosen by some random process • probability distribution (density) function

  7. Discrete Probability Distributions • Discrete events Xi with probability pi • Cumulative PDF (distribution) • Construction of samples • To randomly select an event, • Select Xi if Uniform random variable

  8. Continuous Probability Distributions • PDF (density) • CDF (distribution) Uniform

  9. Sampling Continuous Distributions • Cumulative probability distribution function • Construction of samples Solve for X=P-1(U) • Must know: 1. The integral of p(x) 2. The inverse function P-1(x)

  10. Trick Example: Power Function • Assume

  11. Sampling a Circle

  12. Sampling a Circle WRONG  Equi-Areal RIGHT = Equi-Areal

  13. Rejection Methods • Algorithm Pick U1 and U2 Accept U1 if U2< f(U1) • Wasteful? Efficiency = Area / Area of rectangle

  14. Sampling a Circle: Rejection do { X=1-2*U1 Y=1-2*U2 } while(X2 + Y2 > 1) May be used to pick random 2D directions Circle techniques may also be applied to the sphere

  15. Monte Carlo Integration • Definite integral • Expectation of f • Random variables • Estimator

  16. Unbiased Estimator Properties Assume uniform probability distribution for now

  17. a b Over Arbitrary Domains

  18. Non-Uniform Distributions

  19. Ray intersection Sample uniformly by Direct Lighting – Directional Sampling

  20. Ray direction Sample uniformly by Direct Lighting – Area Sampling

  21. Examples Fixed Random 4 eye rays per pixel 1 shadow ray per eye ray

  22. Examples Uniform grid Stratified random 4 eye rays per pixel 16 shadow rays per eye ray

  23. Examples Uniform grid Stratified random 4 eye rays per pixel 64 shadow rays per eye ray

  24. Examples Uniform grid Stratified random 4 eye rays per pixel 100 shadow rays per eye ray

  25. Examples 64 eye rays per pixel 1 shadow ray per eye ray 4 eye rays per pixel 16 shadow rays per eye ray

  26. Variance • Definition • Properties • Variance decreases with sample size

  27. Sample uniformly by Sample uniformly by Direct Lighting – Directional Sampling Ray intersection

  28. Sampling Projected Solid Angle • Generate cosine weighted distribution

  29. Examples Projected solid angle 4 eye rays per pixel 100 shadow rays Area 4 eye rays per pixel 100 shadow rays

  30. Variance Reduction • Efficiency measure • Techniques • Importance sampling • Sampling patterns: stratified, …

  31. Sampling a Triangle

  32. Sampling a Triangle • Here u and v are not independent! • Conditional probability

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