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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

Sampling

Attila Gyulassy

Image Synthesis

overview
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
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
problem statement ctd
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
overview of sampling
Overview of Sampling
  • Over some domain
    • Sometimes parametrizable
  • Some sample density
  • Random / Regular
random numbers
Random Numbers
  • Would like to get uniformly distributed random numbers over a range [a,b]
  • Problems
    • large open spaces
    • slow convergence
    • nondeterministic
rng methods
RNG methods
  • Non-linear additive feedback
  • Linear congruence methods
  • Mersenne Twister algorithm
  • many more...
quasi monte carlo
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
halton sequence
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

halton sequence12
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 + ...

halton sequence13
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

hammersley sequence
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

poisson random numbers
Poisson Random Numbers
  • Generate random numbers according to the Poisson distribution function

This turns out to be the same as just “throwing darts”**

result of rngs
Result of RNGs
  • Basically, now we have random numbers in [0,1]
    • what do we do with these?
    • How does this relate to sampling?
uniform sampling of a disk
Uniform Sampling of a Disk
  • Want Subdivision into equal area regions
uniform sampling disk vs sphere
Uniform Sampling - Disk vs Sphere
  • Sampling of disk and projecting onto hemisphere = sampling on 1/2 of sphere
uniform sampling of triangles
Uniform Sampling of Triangles
  • Compute probability density function for triangles
uniform sampling of triangles24
Uniform Sampling of Triangles
  • The u and v are not independent
stratified sampling
Stratified Sampling
  • Alternative to uniform
    • break domain into strata
    • fills in gaps faster
importance sampling
Importance Sampling
  • Basic Idea
    • sample at important locations to decrease variance
importance sampling ctd
Importance Sampling ctd.
  • As seen last time, use a probability density function f to pick samples
    • properties
importance sampling ctd28
Importance Sampling ctd.
  • Then, our approximation becomes

(here g(x) is prob. Dens. Funciton, not f(x))

importance sampling ctd29
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

importance sampling ctd30
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?

importance sampling ctd31
Importance Sampling ctd.
  • Remember f(x) = |g(x)| / G gives least variance
  • motivation for adaptive sampling
  • build f from first few samples
conclusion
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
references
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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