Direct Illumination & Monte Carlo Integration CS295, Spring - - PowerPoint PPT Presentation

Direct Illumination & Monte Carlo Integration

CS295, Spring 2017 Shuang Zhao

Computer Science Department University of California, Irvine

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Announcement

• Homework 1 will be out on Thursday, Apr 13
• Check course website for additional readings

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Last Lecture

• Radiometry
• Physics of light
• BRDFs
• How materials reflects light

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Today’s Lecture

• Direct illumination
• Monte Carlo integration I

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Direct Illumination

CS295 Realistic Image Synthesis

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Direct and Indirect Illumination

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[scratchpixel.com]

Today’s focus

Recap: BRDF

• The Bidirectional Reflectance Distribution

Function (BRDF)

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Convention: all vectors point away from x

Reflected Radiance

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Convention: all vectors point away from x

Exitant Incident

Incident and Exitant Radiance

• Lr(x, ωo): all reflected light leaving x (exitant)
• Li(x, ωi): all light entering x (incident)
• Other exitant quantities
• Le(x, ωo): all emitted light leaving x

(nonzero for light sources)

• L(x, ωo): all light leaving x

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Incident and Exitant Radiance

• In general:
• y is computed by tracing a ray from x in direction ω
• Recap: invariant of radiance (in vacuum)

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Direct Illumination

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Example: Uniform Spherical Light

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Diffuse surface fr(ωi↔ωo) ≡ kd/π Spherical light source with radius r and Le(x,ω) ≡ L0

Recall:

Example: Uniform Spherical Light

• For a spherical light source with fixed radiant

power Ф0 (i.e., ):

• Further, when r goes to zero (point source):

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Recall:

Computing Direct Illumination

• Challenges
• y changes discontinuously with ωi due to occlusion
• fr can be complicated (e.g., microfacet BRDFs)

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Object 1 Object 2

Monte Carlo Integration I

CS295 Realistic Image Synthesis

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Monte Carlo Integration

• A powerful framework for computing integrals
• Numerical
• Nondeterministic (i.e., using randomness)
• Scalable to high-dimensional problems

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Recap: Random Variables

• (Discrete) random variable X
• Possible outcomes: x1, x2, …, xn
• with probabilities p1, p2, …, pn such that
• E.g., “fair” coin
• Outcomes: x1 = “head”, x2 = “tail”
• Probabilities: p1 = p2 = ½

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Recap: Random Variables

• (Continuous) random variable X
• Possible outcomes: [a, b]
• with probability density p(x) such that

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Recap: Expected Value & Variance

• Expected value:
• Variance:

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Recap: Expected Value & Variance

Let X and Y be two random variables, then:

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Today’s focus

Recap: Strong Law of Large Numbers

Let x1, x2, …, xn be n independent observations (aka. samples) of X

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Sample mean “Actual” mean

Example: Evaluating π

• Let X be a point uniformly

distributed in the square

• Probability for X inside S:

π/4

• Let
• Then

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Unit circle S Circle area = π Square area = 4

Example: Evaluating π

• (Simple) solution for

computing π:

• Generating n independent

samples of Y based on n i.i.d. X samples

• Computing their mean

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Unit circle S Circle area = π Square area = 4

Example: Evaluating π

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Unit circle S Circle area = π Square area = 4

Integral

• f(x): one-dimensional function

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Deterministic Integration

• Quadrature rules:
• Problem:
• Scales poorly with high

dimensionality (more on this

in the next lecture)

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Monte Carlo Integration: Overview

• Goal: Estimating
• Idea: Constructing random variable
• Such that
• is called an unbiased estimator of
• But how?

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Monte Carlo Integration

• Let p() be any probability density function over

[a, b] and X be a random variable with density p

• Let

, then:

• To estimate

: strong law of large numbers

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Monte Carlo Integration

• Goal: to estimate
• Pick a probability density function
• Generate n independent samples:
• Evaluate

for j = 1, 2, …, n

• Return sample mean:

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How to pick density function p()?

• In theory
• (Almost) anything
• In practice:
• Uniform distributions (almost) always work
• As long as the domain is bounded
• Choice of p() greatly affects the effectiveness (i.e.,

convergence rate) of the resulting estimator

• More in later lectures

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Monte Carlo Integration “Hello World”

• Estimating
• Algorithm:
• Draw x1, x2, …, xn from U[0, 1) independently
• Return

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Monte Carlo Integration “Hello World”

• Estimating

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