shocker_0x15
September 06, 2014
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# モンテカルロレイトレーシング：アルゴリズム超概略 / Super Simple Overview of Monte Carlo Ray Tracing Algorithms

レイトレ合宿2!!のセミナーで使用した資料です。
スライドの趣旨：各手法の理解ではなく、どんな手法が存在するかを知ってもらうことと、その概要。

Path Tracing, Next Event Estimation, Multiple Importance Sampling, Bidirectional Path Tracing, Metropolis Light Transport, Primary Sample Space MLT, Photon Mapping, PPM, SPPM, PPPM, AMCMCPPM, Unified Path Sampling (VCM), Path Space Regularization, Multiplexed MLT

## shocker_0x15

September 06, 2014

## Transcript

3. ### I = f (x)dx a b ∫ I ≈ 1

N f (xi ) p(xi ) i=1 N ∑ ਪఆ஋͸෼ࢄΛ࣋ͭ! ظ଴஋͸ਅ஋ʹҰக͢Δ ϞϯςΧϧϩਪఆؔ਺
4. ### f (x) x a b p(x) x a b I

≈ 1 N f (xi ) p(xi ) i=1 N ∑ ೚ҙͷ1%'͕࢖༻Մೳ
5. ### I ≈ 1 N f (xi ) p(xi ) i=1

N ∑ ॏ఺తαϯϓϦϯά f (x) x a b p(x) f (x) x a b p(x) ଎͍ऩଋ(௿෼ࢄ) ஗͍ऩଋ(ߴ෼ࢄ) ཧ૝తͳPDFΛٻΊΔ͜ͱ͸ࠔ೉

7. ### Χϝϥʗ؟ ޫݯ ܦ࿏ I ≈ 1 N f (xi )

p(xi ) i=1 N ∑ f (xi ): ܦ࿏ʹԊͬͨد༩ xi : ϥϯμϜͳܦ࿏ ϞϯςΧϧϩੵ෼Λ࢖ͬͯղ͘ p(x i ): ϥϯμϜͳܦ࿏Λੜ੒͢ΔPDF

12. ### Light ޫ୔BSDF I ≈ 1 N f (xBSDF, i )

pBSDF (xBSDF, i ) i=1 N ∑ #4%'ͷد༩ʹԊͬͨॏ఺తαϯϓϦϯά #4%'د༩ʹԊͬͯೖࣹํ޲αϯϓϧɿߴ͍֬཰Ͱߴ͍د༩ à௿͍෼ࢄ ޫݯ͕ྑ͍৔ॴʹ͋Ε͹ 
13. ### Light ֦ࢄBSDF I ≈ 1 N f (xBSDF, i )

pBSDF (xBSDF, i ) i=1 N ∑ #4%'ͷد༩ʹԊͬͨॏ఺తαϯϓϦϯά #4%'د༩ʹԊͬͯೖࣹํ޲αϯϓϧɿ௿͍֬཰Ͱߴ͍د༩ àߴ͍෼ࢄ ͨ·ʹ͔͠౰ͨΒͳ͍ͨΊ 
14. ### Light ֦ࢄBSDF I ≈ 1 N f (x light, i

) p light (x light, i ) i=1 N ∑ ޫݯ্ͷҐஔͷॏ఺తαϯϓϦϯά ޫݯ্ͷҐஔΛαϯϓϧͯ͠઀ଓɿߴ͍֬཰Ͱߴ͍د༩ à௿͍෼ࢄ #4%'ͷ஋͕ൺֱతҰ༷Ͱ͋Ε͹ 
15. ### Light ޫ୔BSDF I ≈ 1 N f (x light, i

) p light (x light, i ) i=1 N ∑ ޫݯ্ͷҐஔͷॏ఺తαϯϓϦϯά ޫݯ্ͷҐஔΛαϯϓϧͯ͠઀ଓɿ௿͍֬཰Ͱߴ͍د༩ àߴ͍෼ࢄ #4%'ͷ஋͕ඇҰ༷ͳͨΊ 

17. ### Multiple Importance Sampling I ≈ 1 N f (xBSDF, i

) pBSDF (xBSDF, i ) i=1 N ∑ I ≈ 1 N f (xlight, i ) plight (xlight, i ) i=1 N ∑ I ≈ 1 N wBSDF (xBSDF, i ) f (xBSDF, i ) pBSDF (xBSDF, i ) + wlight (xlight, i ) f (xlight, i ) plight (xlight, i ) " # \$ % & ' i=1 N ∑
18. ### w BSDF (x) = p BSDF (x) p BSDF (x)+

p light (x) wlight (x) = plight (x) pBSDF (x)+ plight (x) I ≈ 1 N wBSDF (xBSDF, i ) f (xBSDF, i ) pBSDF (xBSDF, i ) + wlight (xlight, i ) f (xlight, i ) plight (xlight, i ) " # \$ % & ' i=1 N ∑ .*4΢ΣΠτ όϥϯεώϡʔϦεςΟοΫ

 ΋͋ΓಘΔ

38. ### n࣍ݩͷ0 ~ 1ཚ਺! 㱨 Primary Sample Space n࣍ݩ௒ཱํମ 0 1

0 1 ܦ࿏ͷૉʹͳΔཚ਺ϨϕϧͰมҟΛՃ͑Δ 144ͷ࠲ඪͱܦ࿏͸ ҰରҰରԠ 15΍#15ʹΑΔϚοϐϯά  ΦϦδφϧ.-5ΑΓ࣮૷͕؆୯͔ͭϩόετ ͱظ଴͞ΕΔ 
39. ### n࣍ݩͷ0 ~ 1ཚ਺! 㱨 Primary Sample Space n࣍ݩ௒ཱํମ 0 1

0 1 ܦ࿏ͷૉʹͳΔཚ਺ϨϕϧͰมҟΛՃ͑Δ 144ͷ࠲ඪͱܦ࿏͸ ҰରҰରԠ 15΍#15ʹΑΔϚοϐϯά  ΦϦδφϧ.-5ΑΓ࣮૷͕؆୯͔ͭϩόετ ͱظ଴͞ΕΔ 
40. ### n࣍ݩͷ0 ~ 1ཚ਺! 㱨 Primary Sample Space n࣍ݩ௒ཱํମ 0 1

0 1 ܦ࿏ͷૉʹͳΔཚ਺ϨϕϧͰมҟΛՃ͑Δ 144ͷ࠲ඪͱܦ࿏͸ ҰରҰରԠ 15΍#15ʹΑΔϚοϐϯά  ΦϦδφϧ.-5ΑΓ࣮૷͕؆୯͔ͭϩόετ ͱظ଴͞ΕΔ 

53. ### 11.ͷ໰୊఺ ޫ୔൓ࣹ ΞϯνΤΠϦΞε Ϟʔγϣϯϒϥʔ ඃࣸքਂ౓ ͜ΕΒͷޮՌ͸ฏۉ์ًࣹ౓ਪఆΛඞཁͱ͢Δ ਖ਼֬ͳਪఆʹ͸ແݶͷً౓ਪఆ఺͕ඞཁ ྫɿΞϯνΤΠϦΞε ɹɹϐΫηϧ಺ͷαϯϓϧ఺ ྫɿඃࣸքਂ౓

ɹɹϨϯζ্ͷαϯϓϧ఺
54. ### SPPM ྖҬ಺Ͱ୳ࡧ൒ܘͳͲͷ౷ܭྔΛڞ༗ ޫ୔൓ࣹ ൓ࣹํ޲ ΞϯνΤΠϦΞε ϐΫηϧ Ϟʔγϣϯϒϥʔ γϟολʔ࣌ؒத ඃࣸքਂ౓ Ϩϯζ্

શͯΛ·ͱΊΔ͜ͱͰ ฏۉً౓ͷਪఆ஋ΛϓϩάϨογϒʹਅ஋ʹ͚ۙͮΒΕΔ

68. ### AMCMCPPM = PPM + PSSMLT + α Primary Sample Space

தͷมҟΛ༻͍ͯ  ܦ࿏Λੜ੒
69. ### AMCMCPPM = PPM + PSSMLT + α Primary Sample Space

தͷมҟΛ༻͍ͯ  ܦ࿏Λੜ੒ มҟύϥϝλʔͷ ࣗಈௐ੔΋ߦ͏౳ આ໌ল͖·͢  + α

73. ### BPT ޫ୔໘ͷଟ͍γʔϯಘҙ 4%4ύεۤख PPM ޫ୔໘ͷଟ͍γʔϯۤख 4%4ύεಘҙ .*4 ͔͠͠໰୊͕͋Δ ྫɿ௕͞ͷܦ࿏ߏங BPT

ܦ࿏ͷ࣍ݩ : A5 PPM ܦ࿏ͷ࣍ݩ : A6 ܦ࿏ߏஙͷ ࣍ݩ͕ҟͳΔ
74. ### wBSDF (x) = pBSDF (x) pBSDF (x)+ plight (x) MIS΢ΣΠτͷܭࢉʹPDFͷՃࢉΛؚΉ

࣍ݩͷҟͳΔྔͷՃࢉ͸ޚ๏౓ ࠶ܝɿόϥϯεώϡʔϦεςΟοΫ
75. ### BPT ܦ࿏ͷ࣍ݩ : A5 ֦ுBPT ܦ࿏ͷ࣍ݩ : A6 Vertex Perturbation

ࢹઢύεͷ୺఺ΛͣΒͯ͠ޫઢύεͷ୺఺Λ௥Ճ  Ծ૝తʹPPMͱ࣍ݩΛ߹ΘͤΔ

81. ### Specular BRDF Molliﬁed BRDF BSDF MOLLIFICATION #4%'Λ؇࿨ͯ͠د༩ΛऔΕΔΑ͏ʹ ͨͩ͠CJBTFE  ൓෮͝ͱʹຊདྷͷ#4%'΁͚͍ۙͮͯ͘

ʹຊ࣭తʹ͸11.ͷ൒ܘॖݮͱಉ͡ σΟϑϡʔζ໘ʹ͸ద༻͠ͳ͍àඞཁ࠷௿ݶͷόΠΞε

93. ### ͓ΘΓʹ ຊεϥΠυͰ৮Εͨͷ͸਺͋Δख๏ͷҰ෦ ϘϦϡʔϜϨϯμϦϯάʹؔͯ͠͸Ұ੾৮Εͯͳ͍ Energy Redistribution Path Tracing / Bidirectional Photon

Mapping / ! Manifold Exploration Path Tracing / Replica Exchange Light Transport / ! Population Monte Carlo - ER / Noise Aware MLT / ! Bidirectional Light Cuts / Gradient-domain MLT … ࠷৽ख๏͸جຊతʹ.*4BOEPS 144 .-5 ͷཧ࿦࢖͍ͬͯΔΠϝʔδ
94. ### REFERENCES 1/3 n  [ERPT] CLINE, D., TALBOT, J., AND EGBERT,

P. 2005. Energy redistribution path tracing. ACM Trans. Graph. (SIGGRAPH Proceedings) 24, 3, 1186–1195.! n  [VCM] GEORGIEV, I., KŘIVÁNEK, J., AND SLUSALLEK, P. 2011. Bidirectional light transport with vertex merging. In ACM SIGGRAPH Asia 2011 Sketches, 27:1–27:2.! n  [SPPM] HACHISUKA, T., AND JENSEN, H. W. 2009. Stochastic progressive photon mapping. In ACM SIGGRAPH Asia Papers. ACM, New York, 1–8.! n  [AMCMCPPM] HACHISUKA, T., AND JENSEN, H. W. 2011. Robust adaptive photon tracing using photon path visibility. ACM Transaction on Graphics 30 (October), 114:1–114:11.! n  [MMLT] HACHISUKA, T., KAPLANYAN, A. S., AND DACHSBACHER, C. 2014. Multiplexed Metropolis light transport. ACM Trans. Graph. (Proc. of SIGGRAPH 2014) 33, 4.! n  [PPM] HACHISUKA, T., OGAKI, S., AND JENSEN, H. W. 2008. Progressive photon mapping. ACM Trans. Graph. (Proc. of SIGGRAPH Asia) 27, 5.! n  [UPS] HACHISUKA, T., PANTALEONI, J., AND JENSEN, H. W. 2012. A path space extension for robust light transport simulation. ACM Trans. Graph. (Proc. of SIGGRAPH Asia) 31, 6 (Nov.).! n  [Noise Aware MLT] HOBEROCK, J., AND HART, J. C. 2010. Arbitrary importance functions for Metropolis light transport. Comput. Graph. Forum 29, 6, 1993–2003.! n  [MEPT] JAKOB, W., AND MARSCHNER, S. 2012. Manifold exploration: a Markov chain Monte Carlo technique for rendering scenes with difﬁcult specular transport. ACM Transactions on Graphics (Proc. SIGGRAPH) 31, 4, 58:1–58:13.!
95. ### REFERENCES 2/3 n  [PM] JENSEN, H. W. 1996. Global illumination

using photon maps. In Proceedings of the Eurographics Workshop on Rendering Techniques ’96, Springer-Verlag, London, UK, 21–30.! n  [PT] KAJIYA, J. T. 1986. The rendering equation. In Computer Graphics (Proc. of SIGGRAPH).! n  [Regularization] KAPLANYAN, A. S., AND DACHSBACHER, C. 2013. Path space regularization for holistic and robust light transport. Computer Graphics Forum (Proc. of Eurographics) 32, 2.! n  [PSSMLT] KELEMEN, C., SZIRMAY-KALOS, L., ANTAL, G., AND CSONKA, F. 2002. A simple and robust mutation strategy for the metropolis light transport algorithm. In Eurographics 2002, vol. 21, 531–540.! n  [RELT] KITAOKA, S., KITAMURA, Y., AND KISHINO, F. 2009. Replica exchange light transport. Computer Graphics Forum 28, 8, 2330–2342.! n  [PPPM] KNAUS, C., AND ZWICKER, M. 2011. Progressive photon mapping: A probabilistic approach. ACM Transaction on Graphics 30 (May), 25:1–25:13.! n  [PMC-ER] LAI, Y.-C., FAN, S. H., CHENNEY, S., AND DYER, C. 2007. Photorealistic image rendering with population Monte Carlo energy redistribution. In In Rendering Techniques 2007 (Proceedings of the Eurographics Symposium on Rendering), 287–295.! n  [Gradient-domain MLT] LEHTINEN, J., KARRAS, T., LAINE, S., AITTALA, M., DURAND, F., AND AILA, T. 2013. Gradient-domain Metropolis light transport. ACM Transactions on Graphics (Proc. SIGGRAPH) 32, 4.! n  [MIS, BPT] VEACH, E. 1997. Robust Monte Carlo methods for light transport simulation. PhD thesis, Stanford, CA, USA.!
96. ### REFERENCES 3/3 n  [BPM] VORBA, J. 2011. Bidirectional photon mapping.

In Proc. of the Central European Seminar on Computer Graphics (CESCG ‘11).! n  [BLC] WALTER, B., KHUNGURN, P., AND BALA, K. 2012. Bidirectional lightcuts. ACM Transactions on Graphics (Proc. SIGGRAPH) 31, 4, 59:1–59:11.!