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Sampling

Sampling. Attila Gyulassy Image Synthesis. Overview. Overview Problem Statement Random Number Generators Quasi-Random Number Generation Uniform sampling of Disks, Triangles, Spheres Stratified Sampling Importance Sampling of General Functions. Problem Statement. What is sampling?

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Sampling

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  1. Sampling Attila Gyulassy Image Synthesis

  2. Overview • Overview • Problem Statement • Random Number Generators • Quasi-Random Number Generation • Uniform sampling of Disks, Triangles, Spheres • Stratified Sampling • Importance Sampling of General Functions

  3. Problem Statement • What is sampling? • Want to Take a function f and recreate it using only certain values • e.g. data points used in interpolation • where to pick those points? • Sometimes don’t know f but can evaluate it • would like to choose data points used to reconstruct function in an optimal way

  4. Problem Statement (ctd) • Monte Carlo Integration • 2 ways to improve • improve estimation method • carefully selecting samples****** • Use filtering to recreate original function • covered next time • important to know necessary sampling frequency

  5. Example

  6. Overview of Sampling • Over some domain • Sometimes parametrizable • Some sample density • Random / Regular

  7. Random Numbers • Would like to get uniformly distributed random numbers over a range [a,b] • Problems • large open spaces • slow convergence • nondeterministic

  8. RNG methods • Non-linear additive feedback • Linear congruence methods • Mersenne Twister algorithm • many more...

  9. Quasi-Monte Carlo • Use deterministic roughly uniform aperiodic distribution through domain • I.e. pseudo-random numbers • Want low discrepancy • small = evenly distributed • large = clustering • causes clumping and sparse regions • Want high speed

  10. Quasi-Random Generators

  11. Halton Sequence • N-dimensional points xi xm = (2(m), 3(m),…, PN-1(m), PN(m)) PI = ith prime number (2,3,5,7,…) r(m) is the radical-inverse function of m to the base r. The value is obtained by writing m in base r and then reflecting the digits around the decimal point. 2610 = 110102 reflecting 0.010112 = 11/2710

  12. Halton Sequence • N-dimensional points xi xm = (2(m), 3(m),…, PN-1(m), PN(m)) PI = ith prime number (2,3,5,7,…) m = a0r0 + a0r1 + a0r2 + a0r3 + ... r(m) = a0r-1 + a0r-2 + a0r-3 + a0r-4 + ...

  13. Halton Sequence • Starting at (1,1,…,1) better than starting at (0,0,…,0) 1 = 1.0 => 0.1 = 1/2 2 = 10.0 => 0.01 = 1/4 3 = 11.0 => 0.11 = 3/4 4 = 100.0 => 0.001 = 1/8 5 = 101.0 => 0.101 = 5/8 6 = 110.0 => 0.011 = 3/8 7 = 111.0 => 0.111 = 7/8 Notice even distribution

  14. Hammersley Sequence • Similar to Halton xm = (m/N,2(m), 3(m),…, PN-1(m)) PI = ith prime number (2,3,5,7,…) m = a0r0 + a0r1 + a0r2 + a0r3 + ... r(m) = a0r-1 + a0r-2 + a0r-3 + a0r-4 + ... Where N is number of total samples

  15. Hammersley Sequence

  16. Hammersley Sequence

  17. Poisson Random Numbers • Generate random numbers according to the Poisson distribution function This turns out to be the same as just “throwing darts”**

  18. Result of RNGs • Basically, now we have random numbers in [0,1] • what do we do with these? • How does this relate to sampling?

  19. Uniform Sampling of a Disk • Want Subdivision into equal area regions

  20. Uniform Sampling Over a Sphere • demo

  21. Uniform Sampling - Disk vs Sphere • Sampling of disk and projecting onto hemisphere = sampling on 1/2 of sphere

  22. Uniform Sampling of Triangles • Compute probability density function for triangles

  23. Uniform Sampling of Triangles • The u and v are not independent

  24. Stratified Sampling • Alternative to uniform • break domain into strata • fills in gaps faster

  25. Importance Sampling • Basic Idea • sample at important locations to decrease variance

  26. Importance Sampling ctd. • As seen last time, use a probability density function f to pick samples • properties

  27. Importance Sampling ctd. • Then, our approximation becomes (here g(x) is prob. Dens. Funciton, not f(x))

  28. Importance Sampling ctd. • How do we pick f? • want to minimize variance • where G is integral of original function g(x) • … after much math we get • which is great!! Except, G is what we are trying to find G2 f(x) = |g(x)| / G

  29. Importance Sampling ctd. • If we don’t know G, how can we pick f • If we apply a filter to g, so integral is of form Then if the filter is clamped [0,1] the filter itself becomes a reasonable estimate for f Problems with this method?

  30. Importance Sampling ctd. • Remember f(x) = |g(x)| / G gives least variance • motivation for adaptive sampling • build f from first few samples

  31. Conclusion • Multiple ways to generate “random” numbers • have to pick best method for each application • Many sampling techniques, with pros and cons • uniform • stratified • importance • adaptive

  32. References • http://www.math.iastate.edu/reu/2001/voronoi/halton_sequence.html • http://www.cse.cuhk.edu.hk/~ttwong/papers/udpoint/udpoint.pdf • http://www.fz-juelich.de/nic-series/volume10/janke1.pdf • http://graphics.stanford.edu/courses/cs348b-00/lectures/lecture13/montecarlo.1.pdf • Principles of Digital Image Synthesis, Glassner

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