Monte Carlo Path Tracing II CS295, Spring 2017 Shuang Zhao - - PowerPoint PPT Presentation

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Monte Carlo Path Tracing II CS295, Spring 2017 Shuang Zhao - - PowerPoint PPT Presentation

Oct 03, 2022 266 likes •620 views

Monte Carlo Path Tracing II CS295, Spring 2017 Shuang Zhao Computer Science Department University of California, Irvine CS295, Spring 2017 Shuang Zhao 1 Announcements PA1 base code updated to fix compilation issues under Mac OS You

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Monte Carlo Path Tracing II

CS295, Spring 2017 Shuang Zhao

Computer Science Department University of California, Irvine

CS295, Spring 2017 Shuang Zhao 1

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Announcements

  • PA1 base code updated to fix compilation

issues under Mac OS

  • You do NOT have to update yours if you already

had it compiled

CS295, Spring 2017 Shuang Zhao 2

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Last Lecture

  • Rendering equation
  • Describe the distribution of light at equilibrium
  • Monte Carlo Path Tracing I
  • An unbiased numerical solution to the rendering

equation

CS295, Spring 2017 Shuang Zhao 3

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Today’s Lecture

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

CS295, Spring 2017 Shuang Zhao 4

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Recap: Rendering Equation

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= +

= +

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Recap: Rendering Equation

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(Invariant of radiance along lines)

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Recap: Path Tracing (Version 1.0)

reflectedRadiance(x, ω, depth): [y1, pdf] = luminaireSample() ω1 = normalize(y1 - x) r2 = dot(y1 - x, y1 - x) reflRad = emittedRadiance(y1, -ω1) * brdf(x, ω1, ω) * visibility(x, y1) * dot(nx, ω1) * dot(ny, -ω1) / (r2 * pdf) if depth <= rrDepth: p = 1.0 else: p = survivalProbability if rand() < p: ω2 = uniformRandomPSA(nx) y2 = RayTrace(x, ω2) reflRad += π * reflectedRadiance(y2, -ω2, depth + 1) * brdf(x, ω2, ω) / p return reflRad

CS295, Spring 2017 Shuang Zhao 7

Direct illumination Indirect illumination

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Path Tracing (Version 1.0)

reflectedRadiance(x, ω, depth): [y1, pdf] = luminaireSample() ω1 = normalize(y1 - x) r2 = dot(y1 - x, y1 - x) reflRad = emittedRadiance(y1, -ω1) * brdf(x, ω1, ω) * visibility(x, y1) * dot(nx, ω1) * dot(ny, -ω1) / (r2 * pdf) if depth <= rrDepth: p = 1.0 else: p = survivalProbability if rand() < p: ω2 = uniformRandomPSA(nx) y2 = RayTrace(x, ω2) reflRad += π * reflectedRadiance(y2, -ω2, depth + 1) * brdf(x, ω2, ω) / p return reflRad

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inflexible Direct illumination Indirect illumination flexible

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Fixed Sampling of Incident Direction

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Uniform sampling Better sampling

[Lawrence et al. 2004]

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BRDF Sampling

Monte Carlo Path Tracing II

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Estimating Indirect Illumination

  • Path tracing version 1.1!

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MC Integration

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Path Tracing (Version 1.1)

reflectedRadiance(x, ω, depth): [y1, pdf] = luminaireSample() ω1 = normalize(y1 - x) r2 = dot(y1 - x, y1 - x) reflRad = emittedRadiance(y1, -ω1) * brdf(x, ω1, ω) * visibility(x, y1) * dot(nx, ω1) * dot(ny, -ω1) / (r2 * pdf) if depth <= rrDepth: p = 1.0 else: p = survivalProbability if rand() < p: [ω2, pdf2] = brdfSample(x) y2 = RayTrace(x, ω2) reflRad += reflectedRadiance(y2, -ω2, depth + 1) * brdf(x, ω2, ω) * dot(nx, ω2) / (pdf2 * p) return reflRad

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Direct illumination Indirect illumination

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Estimating Indirect Illumination

  • Ideally, we want
  • c is the normalization factor
  • Then,
  • Note: this is possible only for some BRDFs

(e.g., ideal diffuse, ideal specular)

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MC Integration

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Sampling Ideal Diffuse BRDF

  • In this case, ωi is drawn using cosine-weighted

sampling (i.e., uniform sampling of projected solid angle)

  • uniformRandomPSA() from the last lecture!

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Sampling Ideal Specular BRDF

  • In this case, the sampling of ωi is deterministic:

ωi is always set to ωmirrored, and the estimator becomes

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where

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Sampling the Phong BRDF

  • It is hard to sample according

to

  • Instead, sample according to
  • ver the hemi-

sphere around :

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Sampling the Phong BRDF

  • Obtaining c (the normalization factor):
  • To sample θ1 (inversion method):

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Sampling the Phong BRDF

  • Lastly,

where

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BRDF Sampling

  • Also applies to the estimation of the direct

illumination:

  • r the full radiance:

where

CS295, Spring 2017 Shuang Zhao 19

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BRDF Sampling

  • An active research area
  • Methods specialized to individual models
  • Ward BRDF [Walter 2005]
  • Micro-facet models [Walter et al. 2007]
  • …
  • General frameworks
  • Factorization-based sampling [Lawrence et al. 2004]
  • …

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Multiple Importance Sampling

Monte Carlo Path Tracing II

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Sampling Light Source vs. BRDF

  • Sampling y on the light source:

where and Ae is the surface area of all lights

  • Sampling ωi based on the BRDF at x:

where

  • Which one is better?

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Sampling Light Source vs. BRDF

  • It depends!

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What is Going On?

  • Consider the problem of estimating:
  • By randomly sampling x from PDF p, we have
  • Ideally, we would like

, but this can be difficult in practice

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What is Going On?

  • Assuming we have

and

  • Then, we should pick p = p1 if f(x) has higher

variance and p = p2 if otherwise (since we want to have low variance)

  • Can we combine multiple sampling strategies?

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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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Multiple Importance Sampling

  • Proof:

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Weighting Functions

  • How to choose the weighting functions wi(x)?
  • Method 1: constant
  • Let

for all x

  • Unfortunately, since

if one f(xi)/pi(xi) has high variance, so will

  • This is undesirable!

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The Balance Heuristic

  • Method 2:
  • Then,
  • It holds that

for any unbiased estimator

(with the n densities)

  • In other words, as long as there exists a “good”

estimator for , will also be “good”

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The Power Heuristic

  • Method 3:
  • Then,
  • It holds that

for any unbiased estimator

(with the n densities)

  • Sometimes works better than balance heuristic in

rendering!

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Example: MIS

  • Consider the problem of evaluating:

with two probability densities

  • f: normal distribution with mean 0 and variance σ2
  • g: uniform distribution between a and b

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denotes the indicator function

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Example: MIS

  • σ = 1; a = 1.9, b = 2.0

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Example: MIS

  • σ = 0.05; a = -3.0, b = 3.0

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Next Lecture

  • Monte Carlo Path Tracing III
  • Multiple importance sampling (con’t)
  • Operator formulation of light transport

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