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Presentation Transcript

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?
- 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

- Want to Take a function f and recreate it using only certain values

Problem Statement (ctd)

- Monte Carlo Integration
- 2 ways to improve
- improve estimation method
- carefully selecting samples******

- 2 ways to improve
- Use filtering to recreate original function
- covered next time
- important to know necessary sampling frequency

Overview of Sampling

- Over some domain
- Sometimes parametrizable

- Some sample density
- Random / Regular

Random Numbers

- Would like to get uniformly distributed random numbers over a range [a,b]
- Problems
- large open spaces
- slow convergence
- nondeterministic

RNG methods

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

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

- 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 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 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

- 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

- Generate random numbers according to the Poisson distribution function

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

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

- Want Subdivision into equal area regions

Uniform Sampling - Disk vs Sphere

- Sampling of disk and projecting onto hemisphere = sampling on 1/2 of sphere

Uniform Sampling of Triangles

- Compute probability density function for triangles

Uniform Sampling of Triangles

- The u and v are not independent

Stratified Sampling

- Alternative to uniform
- break domain into strata
- fills in gaps faster

Importance Sampling

- Basic Idea
- sample at important locations to decrease variance

Importance Sampling ctd.

- As seen last time, use a probability density function f to pick samples
- properties

Importance Sampling ctd.

- Then, our approximation becomes

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

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 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 ctd.

- Remember f(x) = |g(x)| / G gives least variance
- motivation for adaptive sampling
- build f from first few samples

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

- 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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