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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

• Problem Statement

• Random Number Generators

• Quasi-Random Number Generation

• Uniform sampling of Disks, Triangles, Spheres

• Stratified Sampling

• Importance Sampling of General Functions

• 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

• 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

• Over some domain

• Sometimes parametrizable

• Some sample density

• Random / Regular

• Would like to get uniformly distributed random numbers over a range [a,b]

• Problems

• large open spaces

• slow convergence

• nondeterministic

• Linear congruence methods

• Mersenne Twister algorithm

• many more...

• 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

• 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

• 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 + ...

• 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

• 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

• Generate random numbers according to the Poisson distribution function

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

• Basically, now we have random numbers in [0,1]

• what do we do with these?

• How does this relate to sampling?

• Want Subdivision into equal area regions

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

• Compute probability density function for triangles

• The u and v are not independent

• Alternative to uniform

• break domain into strata

• fills in gaps faster

• Basic Idea

• sample at important locations to decrease variance

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

• properties

• Then, our approximation becomes

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

• 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

• 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?

• Remember f(x) = |g(x)| / G gives least variance

• build f from first few samples

• Multiple ways to generate “random” numbers

• have to pick best method for each application

• Many sampling techniques, with pros and cons

• uniform

• stratified

• importance