Sampling

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# Sampling - PowerPoint PPT Presentation

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

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?
• 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)
• 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
• 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
• 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
• 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