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

Volumetric Zero-Variance-Based Path Guiding Sebastian Herholz 1 Yangyang Zhao 2 Oskar Elek 3 Jaroslav Křivánek 3 Derek Nowrouzezahrai 2 Hendrik P. A. Lensch 1 1 University Tübingen 2 McGill University Montreal 3 Charles University Prague S. Herholz: A Unified Manifold Framework for Efficient BRDF Sampling …

MOTIVATION 2 Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding

MOTIVATION 3 Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding

MOTIVATION 4 Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding

MOTIVATION • Introduction in Volumetric Light transport • Volumetric Path tracing • Samling decisions Volumetric Path Guiding 5 Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding

MONTE-CARLO 𝑔 𝑌 𝐽 = න 𝑔 𝑌 𝑒𝑌 p 𝑌 • Estimator: • Variance: 𝐽(𝑌 1 , … , 𝑌 𝑂 ) = 1 𝑂 ෍ 𝑔(𝑌 𝑗 ) 𝜏 2 = 𝑊 𝑔(𝑌) መ 𝑞(𝑌 𝑗 ) 𝑞(𝑌) 6 Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding

ZERO VARIANCE MONTE-CARLO 𝑔 𝑌 𝐽 = න 𝑔 𝑌 𝑒𝑌 p 𝑨𝑤 𝑌 (optimal) • Estimator: • Zero-Variance: 𝐽 𝑌 1 , … , 𝑌 𝑂 = 1 𝑂 ෍ 𝑔 𝑌 𝑗 𝑔(𝑌) 𝜏 2 = 𝑊 መ = 𝑑 = 𝐽 𝑞 𝑨𝑤 (𝑌) = 0 𝑞 𝑨𝑤 𝑌 𝑗 7 Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding

THE 4 SAMPLING DECISIONS: SCATTER 𝑄 𝑛 𝒚 𝑘 , 𝜕 𝑘 • Scatter: • Is the next path vertex inside or behind the volume? • Scatter probability: 𝑄 𝑛 𝒚 𝑘 , 𝜕 𝑘 8 Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding

THE 4 SAMPLING DECISIONS: DISTANCE 𝑞 𝑒 𝑒 𝑘+1 |𝒚 𝑘 , 𝜕 𝑘 • Distance: • The distance ( 𝑒 𝑘+1 ) the next scattering occurs • Distance PDF: 𝑞 𝑒 𝑒 𝑘+1 |𝒚 𝑘 , 𝜕 𝑘 9 Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding

THE 4 SAMPLING DECISIONS: DIRECTION 𝑞 𝜕 𝜕 𝑘+1 |𝒚 𝑘+1 , 𝜕 𝑘 • Direction: • In which direction ( 𝜕 𝑘+1 ) should the path continue? • Directional PDF: 𝑞 𝜕 𝜕 𝑘+1 |𝒚 𝑘+1 , 𝜕 𝑘 10 Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding

THE 4 SAMPLING DECISIONS: TERMINATION 𝑄 𝑆𝑆 𝒚 𝑘 , 𝜕 𝑘−1 • Russian Roulette Termination: • Should we continue generating the random path/walk? • Termination probability PDF: 𝑄 𝑆𝑆 𝒚 𝑘 , 𝜕 𝑘−1 11 Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding

VOLUMETRIC RANDOM WALK - DECISIONS 𝜕 𝑁−1 𝜕 𝑘+2 𝑦 𝑁−1 𝜕 𝑘 𝜕 0 𝑦 𝑘+2 𝑦 0 𝑦 𝑘 𝜕 𝑘+1 𝑦 𝑘+1 • Path-segment PDF: 𝑞 𝒚 𝑘+1 , 𝜕 𝑘+1 |𝒚 𝑘 , 𝜕 𝑘 = 𝑄 𝑛 … ∙ 𝑞 𝑒 … ∙ 𝑞 𝜕 … ∙ 1 − 𝑄 𝑆𝑆 … 12 Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding

VOLUMETRIC RANDOM WALK - DECISIONS 𝜕 𝑁−1 𝜕 𝑘+2 𝑦 𝑁−1 𝜕 𝑘 𝜕 0 𝑦 𝑘+2 𝑦 0 𝑦 𝑘 𝜕 𝑘+1 𝑦 𝑘+1 • Path-segment PDF: 𝑞 𝒚 𝑘+1 , 𝜕 𝑘+1 |𝒚 𝑘 , 𝜕 𝑘 = 𝑄 𝑛 … ∙ 𝑞 𝑒 … ∙ 𝑞 𝜕 … ∙ 1 − 𝑄 𝑆𝑆 … 𝑁−1 𝑞 𝒀 = ෑ 𝑞 𝒚 𝑘+1 , 𝜕 𝑘+1 |𝒚 𝑘 , 𝜕 𝑘 • Path PDF: 𝑘=1 13 Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding

VOLUMETRIC RANDOM WALK - DECISIONS 𝜕 𝑁−1 𝜕 𝑘+2 𝑦 𝑁−1 𝜕 𝑘 𝜕 0 𝑦 𝑘+2 𝑦 0 𝑦 𝑘 𝜕 𝑘+1 𝑦 𝑘+1 • Path-segment PDF: 𝑞 𝒚 𝑘+1 , 𝜕 𝑘+1 |𝒚 𝑘 , 𝜕 𝑘 = 𝑄 𝑛 … ∙ 𝑞 𝑒 … ∙ 𝑞 𝜕 … ∙ 1 − 𝑄 𝑆𝑆 … 𝑁−1 Source of variance 𝑞 𝒀 = ෑ 𝑞 𝒚 𝑘+1 , 𝜕 𝑘+1 |𝒚 𝑘 , 𝜕 𝑘 • Path PDF: 𝑘=1 14 Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding

VOLUME RENDERING EQUATION • Incident radiance: 𝑀 𝑦, 𝜕 = 𝑈 𝑦, 𝑦 𝑡 ⋅ 𝑀 𝑝 (𝑦 𝑡 , 𝜕) + න𝑈 𝑦, 𝑦 𝑒 ⋅ 𝜏 𝑡 (𝑦 𝑒 ) ⋅ 𝑀 𝑗 (x d , 𝜕)d𝑒 • In-scattered radiance: 𝑀 𝑗 x 𝑒 , 𝜕 = න𝑔 𝜕, 𝜕′ ⋅ 𝑀(x d , 𝜕′)d𝜕′ 15 Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding

VOLUME RENDERING EQUATION • Incident radiance (volume): 𝑀 𝑦, 𝜕 = 𝑈 𝑦, 𝑦 𝑡 ⋅ 𝑀 𝑝 (𝑦 𝑡 , 𝜕) + න𝑈 𝑦, 𝑦 𝑒 ⋅ 𝜏 𝑡 (𝑦 𝑒 ) ⋅ 𝑀 𝑗 (x d , 𝜕)d𝑒 Known Local Quantities • In-scattered radiance: 𝑀 𝑗 x 𝑒 , 𝜕 = න𝑔 𝜕, 𝜕′ ⋅ 𝑀(x d , 𝜕′)d𝜕′ 16 Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding

VOLUME RENDERING EQUATION • Incident radiance (volume): 𝑀 𝑦, 𝜕 = 𝑈 𝑦, 𝑦 𝑡 ⋅ 𝑀 𝑝 (𝑦 𝑡 , 𝜕) + න𝑈 𝑦, 𝑦 𝑒 ⋅ 𝜏 𝑡 (𝑦 𝑒 ) ⋅ 𝑀 𝑗 (x d , 𝜕)d𝑒 Known Local Unknown Light Quantities Transport Quantities • In-scattered radiance: 𝑀 𝑗 x 𝑒 , 𝜕 = න𝑔 𝜕, 𝜕′ ⋅ 𝑀(x d , 𝜕′)d𝜕′ 17 Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding

CHALLENGES FOR VOLUME SAMPLING 18

LIGHT SHAFTS 𝜕 𝑦 𝑘 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. 19 Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding

LIGHT SHAFTS 𝜕 𝑘 𝜕 𝑦 𝑘 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. 20 Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding

LIGHT SHAFTS 𝜕 𝑘 𝑦 𝑘 𝜕 𝑦 • Light shafts: • Specialized solutions: - 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. 21 Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding

SUB-SURFACE-SCATTERING 𝜕 𝑦 • Sub-Surface-Scattering: - We ‘often’ need stay close to the surface 22 Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding

SUB-SURFACE-SCATTERING 𝜕 𝑦 • Sub-Surface-Scattering: • Specialized solutions: - We ‘often’ need stay close to the surface - We need to leave the object with the right direction 23 Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding

DENSE MEDIA 𝜕 𝑦 • Specialized solutions: • 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). 24 Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding

NON-DENSE MEDIA 𝜕 𝑦 • Non-dense media: • Specialized solutions: - We may need to ‘ force ’ a scattering event even if the transmittance is high (e.g. no contribution from behind the volume). 25 Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding

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. 26 Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding

ZERO-VARIANCE RANDOM WALK THEORY • Theoretical framework for the optimal segment PDF • All 4 local decision have to be optimal: 𝑞 𝑨𝑤 … 𝑨𝑤 . . . 𝑨𝑤 . . . 𝑨𝑤 . . . 𝑨𝑤 . . . ) = 𝑄 ∙ 𝑞 𝑒 ∙ 𝑞 𝜕 ∙ (1 − 𝑄 𝑆𝑆 𝑛 27 Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding

ZERO-VARIANCE PDF EXAMPLES • Opt. distance PDF: 𝑨𝑤 𝑒 𝑘+1 |𝒚 𝑘 , 𝜕 𝑘 ∝ 𝑈(𝒚 𝑘 , 𝒚 𝑘+1 ) ⋅ 𝜏 𝑡 (𝒚 𝑘+1 ) ⋅ 𝑀 𝑗 (𝒚 𝑘+1 , 𝜕 𝑘 ) 𝑞 𝑒 Unknown Light Transport Quantities • Opt. direction PDF: 𝑨𝑤 𝜕 𝑘+1 |𝒚 𝑘+1 , 𝜕 𝑘 ∝ 𝑔 𝑦 𝑘+1 , 𝜕 𝑘 , 𝜕 𝑘+1 ∙ 𝑀(𝑦 𝑘+1 , 𝜕 𝑘+1 ) 𝑞 𝜕 28 Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding

FUN FACT: STD. VOLUME SAMPLING AND ZERO-VARIANCE • Std. volume sampling resolves to a zero-variance estimator if: ∀ 𝒚 , 𝜕 𝑀 𝒚, 𝜕 = 𝑑𝑝𝑜𝑡𝑢 𝑀 𝑗 𝒚, 𝜕 = 𝑑𝑝𝑜𝑡𝑢 • 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 !!! 29 Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding

ZERO-VARIANCE-BASED VOLUMETRIC PATH GUIDING 30

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 31 Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding

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: 32 Sebastian Herholz | Volumetric Zero-Variance-Based Path Guiding

Standard Sampling 45 min

Our Guided Sampling 45 min

Standard Sampling Our Guided Sampling 45 min