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## Monte Carlo Rendering

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### The study of random events. Random Variables. A random variable, x. Takes values from some ... Cast multiple rays per pixel, spread in time, to get motion blur ... – PowerPoint PPT presentation

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Title: Monte Carlo Rendering

1
Monte Carlo Rendering
• Central theme is sampling
• Determine what happens at a set of discrete
points, and extrapolate from there
• Central algorithm is ray tracing
• Rays from the eye or the light
• Theoretical basis is probability theory
• The study of random events

2
Random Variables
• A random variable, x
• Takes values from some domain, ?
• Has an associated probability density function,
p(x)
• A probability density function, p(x)
• Is positive over the domain, ?
• Integrates to 1 over ?

3
Expected Value
• The expected value of a random variable is
defined as
• The expected value of a function is defined as

4
Variance and Standard Deviation
• The variance of a random variable is defined as
• The standard deviation of a random variable is
defined as the square root of its variance

5
Sampling
• A process samples according to the distribution
p(x) if it randomly chooses a value for x such
that
• Weak Law of Large Numbers If xi are independent
samples from p(x), then

6
Algorithms for Sampling
• Psuedo-random number generators give independent
samples from the uniform distribution on 0,1),
p(x)1
• Transformation method take random samples from
the uniform distribution and convert them to
samples from another
• Rejection sampling sample from a different
distribution, but reject some samples

7
Distribution Functions
• Assume a density function f(y) defined on a,b
• Define the probability distribution function, or
cumulative distribution function, as
• Monotonically increasing function with F(a)0 and
F(b)1

8
Transformation Method for 1D
• Generate zi uniform on 0,1)
• Compute
• Then the xi are distributed according to f(x)
• To apply, must be able to compute and invert
distribution function F

9
Multi-Dimensional Transformation
• Assume density function f(x,y) defined on
a,b?c,d
• Distribution function
• Sample xi according to
• Sample yi according to

10
Rejection Sampling
• Say we wish to sample xi according to f(x)
• Find a function g(x) such that g(x)gtf(x) for all
x in the domain
• Geometric interpretation generate sample under
g, and accept if also under f
• Transform a weighted uniform sample according to
xiG-1(z), and generate yi uniform on 0,g(xi)
• Keep the sample if yiltf(x), otherwise reject

11
Important Example
• Consider uniformly sampling a point on a sphere
• Uniformly means that the probability that the
point is in a region depends only on the area of
the region, not its location on the sphere
• Generate points inside a cube -1,1x-1,1x-1,1
• Reject if the point lies outside the sphere
• Push accepted point onto surface
• Fraction of pts accepted ?/6
• Bad strategy in higher dimensions

12
Estimating Integrals
• Say we wish to estimate
• Write hgf, where f is something you choose
• If we sample xi according to f, then

13
Standard Deviation of the Estimate
• Expected error in the estimate after n samples is
measured by the standard deviation of the
estimate
• Note that error goes down with
• This technique is called importance sampling
• f should be as close as possible to g
• Same principle for higher dimensional integrals

14
Example Form Factor Integrals
• We wish to estimate
• Define
• Sample from f by sampling xi uniformly on Pi and
yi uniformly on Pj
• Estimate is

15
Basic Ray Tracing
• For each pixel in the image
• Shoot a ray from the eye through the pixel, to
determine what is seen through that pixel, and
what its intensity is
• Intensity takes contributions from
• Direct illumination (shadow rays, diffuse, Phong)
• Reflected rays (recurse on reflected direction)
• Transmitted rays (recurse of refraction direction)

16
Casting Rays
• Given a ray,
• Determine the first surface hit by the ray
(intersection with lowest t)
• Algorithms exist for most representations of
surfaces, including splines, fractals, height
fields, CSG, implicit surfaces
• Hence, algorithms based on ray tracing can be
very general with respect to the geometry

17
Distributed Ray Tracing
• Cook, Porter, Carpenter 1984
• Addresses the inability of ray tracing to
capture
• non-ideal reflection/transmission
• motion blur
• depth of field
• Basic idea Cast more than one ray for each
pixel, for each reflection, for each frame
• Rays are distributed, not the algorithm. Should
probably be called distribution ray tracing.

18
Specific Cases
• Sample several directions around the reflected
direction to get non-ideal reflection, and
specularities
• Send multiple rays to area light sources to get