Monte Carlo Path Tracing III & Operator Formulation of Light - - PowerPoint PPT Presentation

Monte Carlo Path Tracing III & Operator Formulation of Light Transport

CS295, Spring 2017 Shuang Zhao

Computer Science Department University of California, Irvine

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Announcement

• PA1 will be due next Tuesday
• PA2 to be released shortly after

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Last Lecture

• Monte Carlo Path Tracing II
• BRDF sampling
• Multiple importance sampling

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Today’s Lecture

• Monte Carlo Path Tracing III
• MIS for direct illumination
• Operator formulation of light transport
• Ray space, throughput measure
• Functions and operators
• Operator formulation of the RE

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Recap: Multiple Importance Sampling

• To estimate
• Assume there are n probability densities p1, p2,

…, pn to sample x. Then, is an unbiased estimator of as long as:

• for all x with
• whenever

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Recap: Weighting Functions

• The Balance Heuristic
• Then,
• It holds that

for any unbiased estimator

(with the n densities)

• In other words, as long as one provided density is

“good”, will also be “good”

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Recap: Weighting Functions

• The Power Heuristic
• Then,
• It holds that

for any unbiased estimator

(with the n densities)

• Sometimes works better than balance heuristic in

rendering

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Multiple Importance Sampling for Evaluating Direct Illumination

Monte Carlo Path Tracing III

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Recap: Direct Illumination

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Recap: Direct Illumination

• Measure:

solid angle

• Domain of integral:

the unit hemisphere Ωx around the normal nx at x

• Variable of integration:

ωi (which determines y)

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Light

Recap: Direct Illumination

• Measure:

surface area

• Domain of integral:

the union Ae of all light sources’ surfaces

• Variable of integration:

y (which determines ωi)

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Light

Ae Geometric term

Direct Illumination Estimators

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Light sampling BRDF sampling

MIS for Direct Illumination

• In the previous slide, the two density functions

pBRDF and plight have different underlying measures (i.e., solid angle vs. surface area)

• To apply MIS, we need to rewrite one of the

two densities so that both have the same measure

• We use solid angle as the unified measure

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MIS for Direct Illumination

• ωi1 generated by sampling a point y on the light source

and setting ωi1 = normalize(y - x)

• ωi2 generated by sampling the BRDF fr given x and ω
• The key is to evaluate the four density values
• Assume plight(y) and pBRDF(ω) are given
• We now need rewrite plight(ωi1) and plight(ωi2) as densities
• f directions instead of surface points

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Change of Measure

• Observation: the ray tracing operation

provides a one-to-one correspondence between all

• ωi with

and

• with

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Light

Change of Measure

• Observation: the ray tracing operation

provides a one-to-one correspondence between all

• ωi with

and

• with
• For all such ωi and y,

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Rewriting Density Functions

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MIS Direct Illumination Estimator

directRadianceMIS(x, ω): directRad = 0 [y, p1] = luminaireSample() ωi1 = normalize(y - x) if RayTrace(x, ωi1) == y: p1 *= dot(y - x, y - x) / dot(ny, -ωi1) p2 = pBRDF(ωi1) directRad += emittedRadiance(y, -ωi1) * brdf(x, ωi1, ω) * dot(nx, ωi1) / (p1 + p2)

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MIS Direct Illumination Estimator

# continue from last slide [ωi2, p2] = brdfSample() y = RayTrace(x, ωi2) if y lies on a light source: p1 = plight(y) * dot(y - x, y - x) / dot(ny, -ωi2) directRad += emittedRadiance(y, -ωi2) * brdf(x, ωi2, ω) * dot(nx, ωi2) / (p1 + p2) return directRad

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Example 1

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Light source BRDF (diffuse)

Example 1

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Light source BRDF (shiny)

Example 2

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Operator Formulation of Light Transport

CS295 Realistic Image Synthesis

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Ray Space

• Let be the set of surfaces in the scene,

then is the ray space consisting

• f all light rays
• riginating from all

surface points

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The Throughput Measure

• Let

, then the throughput measure is given by

• With this measure, we can then define

integrals over the ray space in the forms of

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Functions on the Ray Space

• Many quantities (e.g., L) we are interested in

can be interrupted as (real-valued) functions on the ray space

• For any two function f and g, their inner product

is defined to be

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Operators

• Let denote all real-valued functions on the

ray space . An operator then maps a function to another function

• Operators useful in our case
• Local scattering operator K
• Propagation operator G

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Useful Operators

• For any function

:

• The local scattering operator K:
• The propagation operator G:

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Operator Formulation of the RE

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= +

Operator Formulation of the RE

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(Invariant of radiance along lines)

Operator Formulation of the RE

• Given the results from the previous two slides,

we can rewrite the RE as

• Solving the RE is effectively inverting (I - T)

where I denotes the identity operator:

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Conditions for Invertibility

• When

, which translates to all BRDFs conserving energy in our case, (I - T) is invertible and

• It follows that

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Next Lecture

• Operator formulation of light transport II
• Sensors and measurements
• Adjoint operators and adjoint particle tracing

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