Operator Formulation of Light Transport II CS295, Spring 2017 - - PowerPoint PPT Presentation

Operator Formulation of Light Transport II

CS295, Spring 2017 Shuang Zhao

Computer Science Department University of California, Irvine

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Announcement

• The deadline for PA1 has been extended to

this Thursday (May 4)

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Last 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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Today’s Lecture

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

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Recap: 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

• Let

, then the throughput measure is given by

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Recap: Useful Operators

• For any function
• The local scattering operator K:
• The propagation operator G:

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Recap: Operator Formulation of the RE

• Given the two operators K and G, we can

rewrite the rendering equation (RE) as

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

where I denotes the identity operator:

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Operator Formulation of the RE

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=

L Le

…

Operator Formulation of the RE

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Direct illumination

= +

L Le TLe

…

Operator Formulation of the RE

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= + +

…

L Le TLe T2Le

2-bounce indirect illumination

Sensors and Measurements

• Up to this point, we have been considering

estimating individual radiance values L(x, ω) or Li(x, ω)

• In practice, we often need to estimate the

response of some sensor described by the measurement function

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Sensors and Measurements

• can be estimated using MC

integration: where p0 is a probability density over all rays

• In practice, it is desirable to have
• r
• After drawing r ~ p0, Li(r) can in turn be

estimated using path tracing

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Example: Irradiance Meter

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Example: Irradiance Meter

computeIrradiance(): x = uniformSampleSensor() ω = uniformRandomPSA(nx) return |A| * π * receivedRadiance(x, ω)

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Example: Pinhole Camera

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Center of projection Image plane

A pixel Pixel i

Example: Pinhole Camera

computePixelIntensity(i): x = uniformSamplePixel(i) ω = normalize(x - o) return receivedRadiance(x, ω)

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Center of projection Image plane

A pixel Pixel i

Adjoint Operators

• The adjoint of an operator H is denoted H*, and

is defined by the property that

• If H = H*, then H is called self-adjoint
• The propagation operator G is self-adjoint

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Adjoint Operators

• The local scattering operator K satisfies that
• K is self-adjoint if

for all

• This is the case for all BRDFs with reciprocity (which

holds for all BRDFs we have seen up to this point)

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Adjoint Operators

• For any operators A, B and C:
• A + B = B + A
• A(BC) = (AB)C
• (A + B)* = A* + B*, (A B)* = B* A*, (A-1)* = (A*)-1
• Based on these properties and the fact that

G is self-adjoint, we have

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Adjoint Operators

• G(I - KG)-1 is self-adjoint when K is self-adjoint
• Therefore,
• (I - KG)-1 Le gives the radiance L (as a function on

the ray space) satisfying the RE: L = KGL + Le

• (I - K*G)-1 We gives another function

satisfying W = K*GW + We

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The Importance Transport

• W is called the importance function
• Computing

W requires solving the important transport problem given by W = K*GW + We

• The sensor acts as the light source in the light

transport problem by emitting importance We

• Any algorithm that estimates L for the original light

transport problem (e.g., path tracing) can be adapted to estimate W

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Same results!

Light vs. Importance Transport

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Light transport

• Transport equation

L = KGL + Le

• Central quantity

L (radiance)

• Source term

Le (radiance emitted by light sources)

• Measurements

I = <We, GL> = <We, Li>

Importance transport

• Transport equation

W = K*GW + We

• Central quantity

W (importance)

• Source term

We (importance “emitted” by sensors)

• Measurements

I = <GW, Le> = <Wi, Le>

Adjoint Particle Tracing

• Goal: estimating
• Initial step:
• Draw x and ω from some probability density
• r
• Then,

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Known after drawing x and ω

Example: Sampling Area Lights

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Area light emitting radiance L0 into all directions

Similar to irradiance meters in the light transport problem

Example: Sampling Area Lights

computeMeasurement(): x = uniformSampleLight() ω = uniformRandomPSA(nx) return |A| * L0 * π * receivedImportance(x, ω)

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Area light emitting radiance L0 into all directions

Similar to irradiance meters in the light transport problem

Adjoint Particle Tracing

• Main step: estimating W(x, ω)
• This can be done using an algorithm equivalent to

path tracing since

• We call this algorithm adjoint particle tracing to

emphasize that it solves the adjoint (importance transfer) problem

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Path vs. Adjoint Particle Tracing

# Path tracing (version 0.1) radiance(x, ω): rad = emittedRadiance(x, ω) ωi = uniformRandomPSA(nx) y = RayTrace(x, ωi) rad += π * radiance(y, -ωi) * brdf(x, ωi, ω) return rad # Adjoint particle tracing (version 0.1) importance(x, ω): imp = emittedImportance(x, ω) ωo = uniformRandomPSA(nx) y = RayTrace(x, ωo) imp += π * importance(y, -ωo) * brdf(x, ω, ωo) return imp

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Next-Event Estimation

• Similar to path tracing, next-event estimation

can drastically improve the convergence rate

• Same idea: separating direct & indirect

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Path tracing: Adjoint particle tracing:

Direct and Indirect Importance

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Evaluating Direct Importance

• Sensor sampling
• psensor is usually picked as the uniform distribution
• ver the surface A of the sensor. Namely,

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Evaluating Direct Importance

• BRDF sampling
• Desirable to have
• r

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Evaluating Direct Importance

• Similar to evaluating direct illumination, the two

sampling methods can be combined using multiple importance sampling (MIS)

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Path vs. Adjoint Particle Tracing

• Since

, both algorithms lead to the same answer

• Which one to use depends on the light source

and measurement functions (namely, Le and We)

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Challenging Light Sources

• Collimated beam
• Problem:
• Neither light source nor

BRDF sampling can give a point y on A with precisely aligned with ωe

• Can never hit A precisely

from direction -ωe

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A A

Challenging Light Sources

• Collimated beam
• In this case, we should

solve the adjoint problem instead

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A A

Next Lecture

• Path integral formulation

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