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 IlluminationCS295 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 ICS295 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 forcomputing π:

• Generating n independent samples of Y based on ni.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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