Path Guiding Sebastian Herholz 1 Yangyang Zhao 2 Oskar Elek 3 - - PowerPoint PPT Presentation

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Path Guiding Sebastian Herholz 1 Yangyang Zhao 2 Oskar Elek 3 - - PowerPoint PPT Presentation

Jan 20, 2024 506 likes •1.4k views

Volumetric Zero-Variance-Based Path Guiding Sebastian Herholz 1 Yangyang Zhao 2 Oskar Elek 3 Jaroslav Kivnek 3 Derek Nowrouzezahrai 2 Hendrik P. A. Lensch 1 1 University Tbingen 2 McGill University Montreal 3 Charles University Prague S.

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  • S. Herholz: A Unified Manifold Framework for Efficient BRDF Sampling …

Sebastian Herholz1 Yangyang Zhao2 Oskar Elek3 Derek Nowrouzezahrai 2 Hendrik P. A. Lensch1 Jaroslav Křivánek3

Volumetric Zero-Variance-Based Path Guiding

1University Tübingen 2McGill University Montreal 3Charles University Prague

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MOTIVATION

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MOTIVATION

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MOTIVATION

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MOTIVATION

  • Introduction in Volumetric Light transport
  • Volumetric Path tracing
  • Samling decisions

Volumetric Path Guiding

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MONTE-CARLO

𝑔 𝑌 p 𝑌

  • Variance:

𝜏2 = 𝑊 𝑔(𝑌) 𝑞(𝑌) 𝐽 = න 𝑔 𝑌 𝑒𝑌

  • Estimator:

መ 𝐽(𝑌1, … , 𝑌𝑂) = 1 𝑂 ෍ 𝑔(𝑌𝑗) 𝑞(𝑌𝑗)

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ZERO VARIANCE MONTE-CARLO

𝑔 𝑌 p𝑨𝑤 𝑌

(optimal)

  • Estimator:

መ 𝐽 𝑌1, … , 𝑌𝑂 = 1 𝑂 ෍ 𝑔 𝑌𝑗 𝑞𝑨𝑤 𝑌𝑗 = 𝑑 = 𝐽

  • Zero-Variance:

𝜏2 = 𝑊 𝑔(𝑌) 𝑞𝑨𝑤(𝑌) = 0 𝐽 = න 𝑔 𝑌 𝑒𝑌

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THE 4 SAMPLING DECISIONS: SCATTER

  • Scatter:
  • Is the next path vertex inside or behind the volume?
  • Scatter probability: 𝑄

𝑛 𝒚𝑘, 𝜕𝑘

𝑄

𝑛 𝒚𝑘, 𝜕𝑘

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THE 4 SAMPLING DECISIONS: DISTANCE

  • Distance:
  • The distance (𝑒𝑘+1) the next scattering occurs
  • Distance PDF: 𝑞𝑒 𝑒𝑘+1|𝒚𝑘, 𝜕𝑘

𝑞𝑒 𝑒𝑘+1|𝒚𝑘, 𝜕𝑘

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THE 4 SAMPLING DECISIONS: DIRECTION

  • Direction:
  • In which direction (𝜕𝑘+1) should the path continue?
  • Directional PDF: 𝑞𝜕 𝜕𝑘+1|𝒚𝑘+1, 𝜕𝑘

𝑞𝜕 𝜕𝑘+1|𝒚𝑘+1, 𝜕𝑘

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THE 4 SAMPLING DECISIONS: TERMINATION

  • Russian Roulette Termination:
  • Should we continue generating the random path/walk?
  • Termination probability PDF: 𝑄𝑆𝑆 𝒚𝑘, 𝜕𝑘−1

𝑄𝑆𝑆 𝒚𝑘, 𝜕𝑘−1

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VOLUMETRIC RANDOM WALK - DECISIONS

  • Path-segment PDF:

𝑞 𝒚𝑘+1, 𝜕𝑘+1|𝒚𝑘, 𝜕𝑘 = 𝑄

𝑛 …

∙ 𝑞𝑒 … ∙ 𝑞𝜕 … ∙ 1 − 𝑄𝑆𝑆 … 𝑦0 𝜕0 𝑦𝑘 𝜕𝑘 𝑦𝑘+1 𝜕𝑘+1 𝑦𝑁−1 𝜕𝑁−1 𝑦𝑘+2 𝜕𝑘+2

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VOLUMETRIC RANDOM WALK - DECISIONS

𝑦0 𝜕0 𝑦𝑘 𝜕𝑘 𝑦𝑘+1 𝜕𝑘+1 𝑦𝑁−1 𝜕𝑁−1 𝑦𝑘+2 𝜕𝑘+2

  • Path-segment PDF:

𝑞 𝒚𝑘+1, 𝜕𝑘+1|𝒚𝑘, 𝜕𝑘 = 𝑄

𝑛 …

∙ 𝑞𝑒 … ∙ 𝑞𝜕 … ∙ 1 − 𝑄𝑆𝑆 …

  • Path PDF:

𝑞 𝒀 = ෑ

𝑘=1 𝑁−1

𝑞 𝒚𝑘+1, 𝜕𝑘+1|𝒚𝑘, 𝜕𝑘

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VOLUMETRIC RANDOM WALK - DECISIONS

𝑦0 𝜕0 𝑦𝑘 𝜕𝑘 𝑦𝑘+1 𝜕𝑘+1 𝑦𝑁−1 𝜕𝑁−1 𝑦𝑘+2 𝜕𝑘+2

  • Path-segment PDF:

𝑞 𝒚𝑘+1, 𝜕𝑘+1|𝒚𝑘, 𝜕𝑘 = 𝑄

𝑛 …

∙ 𝑞𝑒 … ∙ 𝑞𝜕 … ∙ 1 − 𝑄𝑆𝑆 …

  • Path PDF:

𝑞 𝒀 = ෑ

𝑘=1 𝑁−1

𝑞 𝒚𝑘+1, 𝜕𝑘+1|𝒚𝑘, 𝜕𝑘 Source of variance

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VOLUME RENDERING EQUATION

  • Incident radiance:
  • In-scattered radiance:

𝑀 𝑦, 𝜕 = 𝑈 𝑦, 𝑦𝑡 ⋅ 𝑀𝑝(𝑦𝑡, 𝜕) + න𝑈 𝑦, 𝑦𝑒 ⋅ 𝜏𝑡(𝑦𝑒) ⋅ 𝑀𝑗(xd, 𝜕)d𝑒 𝑀𝑗 x𝑒, 𝜕 = න𝑔 𝜕, 𝜕′ ⋅ 𝑀(xd, 𝜕′)d𝜕′

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VOLUME RENDERING EQUATION

  • Incident radiance (volume):
  • In-scattered radiance:

𝑀 𝑦, 𝜕 = 𝑈 𝑦, 𝑦𝑡 ⋅ 𝑀𝑝(𝑦𝑡, 𝜕) + න𝑈 𝑦, 𝑦𝑒 ⋅ 𝜏𝑡(𝑦𝑒) ⋅ 𝑀𝑗(xd, 𝜕)d𝑒 𝑀𝑗 x𝑒, 𝜕 = න𝑔 𝜕, 𝜕′ ⋅ 𝑀(xd, 𝜕′)d𝜕′

Known Local Quantities

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VOLUME RENDERING EQUATION

  • Incident radiance (volume):
  • In-scattered radiance:

𝑀 𝑦, 𝜕 = 𝑈 𝑦, 𝑦𝑡 ⋅ 𝑀𝑝(𝑦𝑡, 𝜕) + න𝑈 𝑦, 𝑦𝑒 ⋅ 𝜏𝑡(𝑦𝑒) ⋅ 𝑀𝑗(xd, 𝜕)d𝑒 𝑀𝑗 x𝑒, 𝜕 = න𝑔 𝜕, 𝜕′ ⋅ 𝑀(xd, 𝜕′)d𝜕′

Known Local Quantities Unknown Light Transport Quantities

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CHALLENGES FOR VOLUME SAMPLING

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LIGHT SHAFTS

𝑦 𝜕 𝑦𝑘

  • Light shafts:
  • We need to scatter inside the light shaft.
  • We need to follow the direction of the light shaft.
  • We need to scatter towards the light shaft.

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LIGHT SHAFTS

𝑦 𝜕 𝑦𝑘 𝜕𝑘

  • Light shafts:
  • We need to scatter inside the light shaft.
  • We need to follow the direction of the light shaft.
  • We need to scatter towards the light shaft.

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LIGHT SHAFTS

𝑦 𝜕

  • Light shafts:
  • We need to scatter inside the light shaft.
  • We need to follow the direction of the light shaft.
  • We need to scatter towards the light shaft.

𝑦𝑘 𝜕𝑘

  • Specialized solutions:

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SUB-SURFACE-SCATTERING

𝑦 𝜕

  • Sub-Surface-Scattering:
  • We ‘often’ need stay close to the surface

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SUB-SURFACE-SCATTERING

𝑦 𝜕

  • Sub-Surface-Scattering:
  • We ‘often’ need stay close to the surface
  • We need to leave the object with the right direction
  • Specialized solutions:

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DENSE MEDIA

  • Dense media:
  • We may need to ‘avoid’ generating a scattering

event even if the transmittance is low (e.g. strong light source behind the volume).

𝑦 𝜕

  • Specialized solutions:

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NON-DENSE MEDIA

𝑦 𝜕

  • Non-dense media:
  • We may need to ‘force’ a scattering event

even if the transmittance is high (e.g. no contribution from behind the volume).

  • Specialized solutions:

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SPECIALIZED SOLUTIONS: SHORTCOMMINGS

  • Many individual solutions/algorithms:
  • Complicates the rendering code
  • Only considering special cases:
  • Surface-bounded volumes
  • Homogenous or isotropic volumes
  • Single scattering
  • Not intuitive (for an artist) to decided which feature helps when.

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ZERO-VARIANCE RANDOM WALK THEORY

  • Theoretical framework for the optimal segment PDF
  • All 4 local decision have to be optimal:

𝑞𝑨𝑤 … = 𝑄

𝑛 𝑨𝑤 . . .

∙ 𝑞𝑒

𝑨𝑤 . . .

∙ 𝑞𝜕

𝑨𝑤 . . .

∙ (1 − 𝑄𝑆𝑆

𝑨𝑤 . . . )

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ZERO-VARIANCE PDF EXAMPLES

  • Opt. distance PDF:

𝑞𝑒

𝑨𝑤 𝑒𝑘+1|𝒚𝑘, 𝜕𝑘 ∝ 𝑈(𝒚𝑘, 𝒚𝑘+1) ⋅ 𝜏𝑡(𝒚𝑘+1) ⋅ 𝑀𝑗 (𝒚𝑘+1, 𝜕𝑘)

  • Opt. direction PDF:

𝑞𝜕

𝑨𝑤 𝜕𝑘+1|𝒚𝑘+1, 𝜕𝑘 ∝ 𝑔 𝑦𝑘+1, 𝜕𝑘, 𝜕𝑘+1 ∙ 𝑀(𝑦𝑘+1, 𝜕𝑘+1)

Unknown Light Transport Quantities

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FUN FACT: STD. VOLUME SAMPLING AND ZERO-VARIANCE

  • Std. volume sampling resolves to a zero-variance estimator if:
  • Its variance depends on the deviation of the actual volumetric light transport

to this assumption!

  • Consequence:

Any conservative guiding towards the actual VLT results in a variance reduction !!!

∀ 𝒚, 𝜕

𝑀 𝒚, 𝜕 = 𝑑𝑝𝑜𝑡𝑢 𝑀𝑗 𝒚, 𝜕 = 𝑑𝑝𝑜𝑡𝑢

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ZERO-VARIANCE-BASED VOLUMETRIC PATH GUIDING

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ZV-BASED VOLUMETRIC PATH GUIDING: GOALS

  • Consider the complete volumetric light transport
  • No prior assumptions or special cases
  • Leverage success of local surface guiding methods
  • Extend the concept to volumes

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ZV-BASED VOLUMETRIC PATH GUIDING: CONTRIBUTIONS

  • Guiding all local sampling decisions:
  • 1+2 Guided product distance sampling:
  • 3 Guided product directional sampling:
  • 4 Guided Russian roulette and Splitting:

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Standard Sampling 45 min

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45 min Our Guided Sampling

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Standard Sampling 45 min Our Guided Sampling

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VOLUME RADIANCE ESTIMATES

  • Pre-processing step to fit estimates from photons (50M)
  • Spatial caches via BSP-tree: max. 2K photons per node

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VON MISES FISHER MIXTURE MODEL (VMM)

𝑊 𝜕|Θ = ෍

𝐿

𝜌𝑗𝑤(𝜕|𝜈𝑗, 𝜆𝑗) Θ = {𝜌0 … , 𝜈0 … , 𝜆0 … }

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VON MISES FISHER MIXTURE MODEL (VMM)

𝑊 𝜕|Θ = ෍

𝐿

𝜌𝑗𝑤(𝜕|𝜈𝑗, 𝜆𝑗) Θ = {𝜌0 … , 𝜈0 … , 𝜆0 … }

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VMM: PRODUCT

𝜌𝑗𝑤(𝜕|𝜈𝑗, 𝜆𝑗) ⋅ 𝜌𝑘𝑤(𝜕|𝜈𝑘, 𝜆𝑘) = 𝜌𝑗𝑘𝑤(𝜕|𝜈𝑗𝑘, 𝜆𝑗𝑘)

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VMM: PRODUCT

𝜌𝑗𝑤(𝜕|𝜈𝑗, 𝜆𝑗) ⋅ 𝜌𝑘𝑤(𝜕|𝜈𝑘, 𝜆𝑘) = 𝜌𝑗𝑘𝑤(𝜕|𝜈𝑗𝑘, 𝜆𝑗𝑘)

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VMM: CONVOLUTION

𝑤𝑗(… ) ∗ 𝑤𝑘(… ) = 𝑤𝑗𝑘(… )

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VMM: CONVOLUTION

𝑤𝑗(… ) ∗ 𝑤𝑘(… ) = 𝑤𝑗𝑘(… )

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INCIDENT RADIANCE ESTIMATES

  • Scaled Incident Radiance Distribution:
  • Fluence:

Φ 𝑦 = ׬

𝑇 𝑀 𝑦, 𝜕′ 𝑒𝜕′

෨ 𝑀 𝑦, 𝜕 = Φ 𝑦 ⋅ 𝑊 𝜕 Θ 𝑦

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INCIDENT RADIANCE ESTIMATES

  • Scaled Incident Radiance Distribution:
  • Fluence:

Φ 𝑦 = ׬

𝑇 𝑀 𝑦, 𝜕′ 𝑒𝜕′

෨ 𝑀 𝑦, 𝜕 = Φ 𝑦 ⋅ 𝑊 𝜕 Θ 𝑦

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INCIDENT RADIANCE ESTIMATES

Ground truth (2K spp) Our estimates (VMM)

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IN-SCATTERED RADIANCE ESTIMATES

  • Convolution between 𝑀 and phase function:

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IN-SCATTERED RADIANCE ESTIMATES

  • Convolution between 𝑀 and phase function:

𝑊

𝑀𝑗 𝜕 = (𝑊 𝑔 ∗ 𝑊 𝑀)(𝜕)

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IN-SCATTERED RADIANCE ESTIMATES

  • Convolution between 𝑀 and phase function:

𝑊

𝑀𝑗 𝜕 = (𝑊 𝑔 ∗ 𝑊 𝑀)(𝜕)

෨ 𝑀𝑗 𝑦, 𝜕 = Φ 𝑦 ⋅ 𝑊

𝑀𝑗(𝜕|Θ𝑀𝑗(𝑦))

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IN-SCATTERED RADIANCE ESTIMATES

Ground truth (2K spp) Our estimates (VMM)

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GUIDED SAMPLING DECISIONS

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GUIDED PRODUCT DISTANCE SAMPLING

  • 1. Scatter decision
  • 2. Scatter distance

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GUIDED PRODUCT DISTANCE SAMPLING

  • Optimal ZV-PDF:

𝑞𝑒

𝑨𝑤 𝑒𝑘+1|𝒚𝑘, 𝜕𝑘 = 𝑈(𝒚𝑘, 𝒚𝑘+1) ⋅ 𝜏𝑡(𝒚𝑘+1) ⋅ 𝑀𝑗 (𝒚𝑘+1, 𝜕𝑘)

𝑀 (𝒚𝑘, 𝜕𝑘)

  • Our guided PDF:

෤ 𝑞𝑒

𝑨𝑤 𝑒𝑘+1|𝒚𝑘, 𝜕𝑘 = 𝑈 𝒚𝑘, 𝒚𝑘+1 ⋅ 𝜏𝑡 𝒚𝑘+1 ⋅ ෨

𝑀𝑗(𝒚𝑘+1, 𝜕𝑘) ෨ 𝑀(𝒚𝑘, 𝜕𝑘) Our estimates

1+2. Event distance

Traditional distance sampling

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INCREMENTAL GUIDED DISTANCE SAMPLING

  • Incremental approach:
  • At each step make a local decision, if we scatter inside the bin.

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INCREMENTAL GUIDED DISTANCE SAMPLING

  • Local bin scatter probability:

𝑄𝑗 𝐸 ≤ 𝑒𝑗+1 ≈ 1 − 𝑈(𝑒𝑗, 𝑒𝑗+1) 𝜏𝑢(𝑒𝑗) ⋅ 𝜏𝑡 𝑒𝑗 ⋅ ෨ 𝑀𝑗(𝑒𝑗) ෨ 𝑀(𝑒𝑗, 𝜕)

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INCREMENTAL GUIDED DISTANCE SAMPLING

  • We only need to step until the scattering event happens

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No guiding (256 spp) Distance guiding (256 spp)

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No guiding (1024 spp) Distance guiding (1024 spp)

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45 min

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No guiding Spp: 960 relMSE: 1.342

45 min

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No guiding Distance guiding Spp: 960 relMSE: 1.342 Spp: 424 relMSE: 0.901

45 min

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GUIDED PRODUCT DIRECTIONAL SAMPLING

  • Optimal ZV-PDF:

𝑞𝜕

𝑨𝑤 𝜕𝑘+1|𝒚𝑘+1, 𝜕𝑘 ∝ 𝑔 𝑦𝑘+1, 𝜕𝑘, 𝜕𝑘+1 ∙ 𝑀(𝑦𝑘+1, 𝜕𝑘+1)

  • Our guided PDF:

෤ 𝑞𝜕

𝑨𝑤 𝜕𝑘+1|𝒚𝑘+1, 𝜕𝑘 ∝ ሚ

𝑔 𝑦𝑘+1, 𝜕𝑘, 𝜕𝑘+1 ∙ ෨ 𝑀(𝑦𝑘+1, 𝜕𝑘+1) Traditional directional sampling Our estimates

  • 3. Scatter direction

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GUIDED PRODUCT DIRECTIONAL SAMPLING

  • Product between two VMM is a VMM:

𝑊

𝑔 𝜕 𝑊 𝑀 𝜕 = 𝑊 𝑔 ⊗ 𝑊 𝑀

𝜕 = 𝑊

⊗(𝜕)

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GUIDED PRODUCT DIRECTIONAL SAMPLING

  • Product between two VMM is a VMM:

𝑊

𝑔 𝜕 𝑊 𝑀 𝜕 = 𝑊 𝑔 ⊗ 𝑊 𝑀

𝜕 = 𝑊

⊗(𝜕)

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GUIDED PRODUCT DIRECTIONAL SAMPLING

  • Product between two VMM is a VMM:

𝑊

𝑔 𝜕 𝑊 𝑀 𝜕 = 𝑊 𝑔 ⊗ 𝑊 𝑀

𝜕 = 𝑊

⊗(𝜕)

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30 min

65

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30 min

No guiding Spp: 2212 relMSE: 0.376

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Directional guiding Spp: 1756 relMSE: 0.048

30 min

No guiding Spp: 2212 relMSE: 0.376

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Directional guiding Dist + Direct Spp: 1756 relMSE: 0.048 Spp: 1228 relMSE: 0.034

30 min

No guiding Spp: 2212 relMSE: 0.376

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IMPORTANCE OF THE PRODUCT FOR DIRECTIONAL GUIDING

  • Phase function PDF:

𝑞𝜕

𝑔 …

∝ 𝑔 …

  • Incident Radiance PDF:

𝑞𝜕

𝑀 …

∝ 𝑀(𝑦𝑘+1, 𝜕𝑘+1)

  • Mixture PDF:

𝑞𝜕

𝑕𝑣𝑗𝑒𝑓 …

= 𝛽 ∙ 𝑞𝜕

𝑔 …

+ (1 − 𝛽) ∙ 𝑞𝜕

𝑀 …

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IMPORTANCE OF THE PRODUCT FOR DIRECTIONAL GUIDING

  • Mixture PDF:

𝑞𝜕

𝑛𝑗𝑦 …

= 𝛽 ∙ 𝑞𝜕

𝑔 …

+ (1 − 𝛽) ∙ 𝑞𝜕

𝑀 …

  • Product PDF:

෤ 𝑞𝜕

𝑨𝑤 …

∝ ሚ 𝑔 … ∙ ෨ 𝑀(… )

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GUIDED RUSSIAN ROULETTE AND SPLITTING

  • Post-sampling compensation strategies:
  • Identify, if we did a sub-optimal sampling decision
  • Terminate: to increase performance
  • Split: bound/reduce sample variance
  • 4a. Termination
  • 4b. Splitting

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GUIDED RUSSIAN ROULETTE AND SPLITTING

  • 4a. Termination
  • 4b. Splitting

Distance Directional

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GUIDED RUSSIAN ROULETTE AND SPLITTING

  • Path contribution: 𝐹[𝑌]
  • The expected contribution

if we continue the path

  • Reference solution: 𝐽
  • the final pixel value

𝑟 = 𝐹 𝑌 𝐽

Path contribution Reference solution survival prob / splitting factor

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GUIDED RUSSIAN ROULETTE AND SPLITTING

  • Path contribution: 𝐹[𝑌]
  • The expected contribution

if we continue the path

  • Reference solution: 𝐽
  • the final pixel value

𝑟 = 𝐹 𝑌 𝐽 = 1

Path contribution Reference solution survival prob / splitting factor

Zero-Variance Estimator

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GUIDED RUSSIAN ROULETTE AND SPLITTING

  • If q′ ≤ 1: Russian Roulette
  • Terminates low contributing paths
  • Survival probability: q′
  • If q′ > 1: Spitting
  • Splits an under sampled paths with

a potential high contribution (q′ times)

𝑟 = 𝐹 𝑌 𝐽

Path contribution Reference solution survival prob / splitting factor

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VOLUME RENDERING EQUATION

  • See course notes or paper for more details

෨ 𝑀𝑗 𝒀 𝐹 𝒀 = 𝑏′(𝒀) ⋅ ෨ 𝑀𝑗 𝑦𝑘, 𝜕𝑘−1

Path throughput: 𝑔(𝒀)/𝑞(𝒀) In-scattered radiance estimate 𝑦𝑘 𝜕𝑘−1

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GUIDED RUSSIAN ROULETTE AND SPLITTING: PIXEL ESTIMATE

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GUIDED RUSSIAN ROULETTE AND SPLITTING: PIXEL ESTIMATE

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No RR Spp: 468 relMSE: 0.454

45 min

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No RR Guided RR Spp: 468 relMSE: 0.454 Spp: 1500 relMSE: 0.174

45 min

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No RR Guided RR Spp: 468 relMSE: 0.454 Spp: 1500 relMSE: 0.174

45 min

+ Guided splitting Spp: 1340 relMSE: 0.066

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Guided RR + Guided splitting

Spp: 1500 relMSE: 0.174 Spp: 1340 relMSE: 0.066

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No guiding

Time: 60 min Spp: 10644 relMSE: 11.58

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Distance guiding

Time: 60 min Spp: 4624 relMSE: 3.520

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Distance + directional guiding

Time: 60 min Spp: 4448 relMSE: 0.468

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Distance + directional guiding + GRRS

Time: 60 min Spp: 3796 relMSE: 0.321

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ADDITIONAL RESULTS

Motivation Zero-Variance Theory Volume Guiding Guided Decisions Results Future Work

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