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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
  • 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.

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
    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

19
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?
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