Monte Carlo Rendering

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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

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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 ?

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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

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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

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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

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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

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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

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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

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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

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Estimating Integrals

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

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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

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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

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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)

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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

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Distributed Ray Tracing

• Cook, Porter, Carpenter 1984
• Addresses the inability of ray tracing to
  capture
• non-ideal reflection/transmission
• soft shadows
• 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.

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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
  soft shadows
• Cast multiple rays per pixel, spread in time, to
  get motion blur
• Cast multiple rays per pixel, through different
  lens paths, to get depth-of-field

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Whats Good? Whats Bad?

• Easy to implement - standard ray tracer plus
  simple sampling
• Which L(DS)E paths does it get?
• Which previous method could it be used with to
  good effect?