Variance Reduction - Carnegie Mellon University

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Variance Reduction - Carnegie Mellon University

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Variance Reduction
Computer Graphics CMU 15-462/15-662

Last time: Monte Carlo Ray Tracing
Recursive description of incident illumination Difficult to integrate; tour de force of numerical integration Leads to lots of sophisticated integration strategies:
- sampling strategies - variance reduction - Markov chain methods - ...
Today: get a glimpse of these ideas Also valuable outside rendering!
- Monte Carlo one of the “Top 10 Algorithms of the 20th Century”!
CMU 15-462/662

Review: Monte Carlo Integration
any function*
Want to integrate:
any domain
General-purpose hammer: Monte-Carlo integration

volume of the domain
*Must of course have a well-defined integral!

uniformly random samples of domain

CMU 15-462/662

Review: Expected Value (DISCRETE)
A discrete random variable X has n possible outcomes xi, occuring w/ probabilities 0≤ pi ≤1, p1 + … + pn=1
probability of event i

expected value

value of event i
(just the “weighted average”!)

E.g., what’s the expected value for a fair coin toss?

p1 = 1/2 x1 = 1

p2 = 1/2 x2 = 0

CMU 15-462/662

Continuous Random Variables
A continuous random variable X takes values x anywhere in a set Ω

Probability density p gives probability x appears in a given region.

E.g., probability you fall asleep at time t in a 15-462 lecture:

cool motivating examples


professor is making dumb jokes

more theory class ends

probability you fall asleep exactly at any given time t is ZERO!

can only talk about chance of falling asleep in a given
interval of time

CMU 15-462/662

Review: Expected Value (CONTINUOUS)
Expected value of continuous random variable again just the “weighted average” with respect to probability p:
probability density at point x

expected value

sometimes just use “µ” (for “mean”)

E.g., expected time of falling asleep?
µ = 44.9 minutes

(is this result counter-intuitive?)
CMU 15-462/662

Flaw of Averages
CMU 15-462/662

Review: Variance
Expected value is the “average value” Variance is how far we are from the average, on average!



Standard deviation σ is just the square root of variance


µ = 44.9 minutes

σ = 15.8 minutes

(More intuitive perhaps?)
CMU 15-462/662

Variance Reduction in Rendering

higher variance

lower variance

CMU 15-462/662

Q: How do we reduce variance?
CMU 15-462/662
VarianceVariance ReductionDomainTimeFall