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

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MONTE CARLO RAY TRACING! ΞϧΰϦζϜ௒ུ֓ ౉෦৺! Twitter: @Shocker_0x15! ϨΠτϨ߹॓ https://sites.google.com/site/raytracingcamp2/

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ϞϯςΧϧϩੵ෼

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I = f (x)dx a b ∫ I ≈ 1 N f (xi ) p(xi ) i=1 N ∑ ਪఆ஋͸෼ࢄΛ࣋ͭ! ظ଴஋͸ਅ஋ʹҰக͢Δ ϞϯςΧϧϩਪఆؔ਺

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f (x) x a b p(x) x a b I ≈ 1 N f (xi ) p(xi ) i=1 N ∑ ೚ҙͷ1%'͕࢖༻Մೳ

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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೉

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άϩʔόϧΠϧϛωʔγϣϯ Χϝϥʗ؟ ޫݯ Χϝϥʹ౸ୡ͢Δ͋ΒΏΔޫܦ࿏ͷد༩Λੵ෼͢Δ I = f (x)dµ(x) Ω ∫

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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

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PATH TRACING

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ೖࣹํ޲Λ֬཰తʹαϯϓϧɺޫݯʹ౰ͨΕ͹د༩͕ͱΕΔ ࢹ఺͔Βޫ༌ૹܦ࿏ΛτϨʔε ͳ͔ͳ͔౰ͨΒͳ͍ʂ

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NEXT EVENT ESTIMATION ޫݯ্ͷ఺Λ໌ࣔతʹαϯϓϧɺࢹઢܦ࿏ͱ઀ଓ͢Δ

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MULTIPLE IMPORTANCE SAMPLING  ྫɿ௚઀র໌ͷਪఆ

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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͹

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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

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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͹

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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

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ޫݯ໘ͷαϯϓϦϯά #4%'ͷαϯϓϦϯά

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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 ∑

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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

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.*4ʹΑΔ΢ΣΠτ഑෼ .VMUJQMF*NQPSUBODF4BNQMJOH

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BIDIRECTIONAL PATH TRACING! VEACH-STYLE

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.*4ΛҰൠԽ 15ʹ͓͚Δ ௚઀র໌ͷ৔߹ʜ ೚ҙͷܦ࿏ʹҰൠԽʂ

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ࢹઢαϒύεͱޫݯαϒύεΛੜ੒ɺ֤௖఺Λ઀ଓ

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(2,2) (3,1) (1,3) ྫ௕͞ MIS ˎ ΋͋ΓಘΔ

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#JEJSFDUJPOBM1BUI5SBDJOH 1BUI5SBDJOH ؒ઀র໌͕ࢧ഑తͳγʔϯʹ͓͍ͯ΋ϩόετ

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METROPOLIS LIGHT TRANSPORT

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௨ৗͷ.$35ͷ໰୊఺ ྫɿ1BUI5SBDJOH ͘͝Ұ෦ͷྖҬͷޫ༌ૹܦ࿏͕ॏཁͱͳΔγʔϯʹऑ͍! ྫɿগ͚ͩ͠։͍ͨυΞ͔Β࿙ΕΔޫɺίʔεςΟΫε͕ओཁͳޫݯ ຖճϥϯμϜʹܦ࿏ΛτϨʔεɿ໓ଟʹد༩͕ͱΕͳ͍ʂʂ!

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ޫ༌ૹ΁ͷϝτϩϙϦεαϯϓϦϯάͷద༻ طଘͷ༗ޮͳύε΁มҟΛՃ͑ͯ৽ͨͳύεΛੜ੒! د༩͕খ͘͞ͳΔมҟ͸֬཰తʹغ٫͞ΕΔ

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ޫ༌ૹ΁ͷϝτϩϙϦεαϯϓϦϯάͷద༻ طଘͷ༗ޮͳύε΁มҟΛՃ͑ͯ৽ͨͳύεΛੜ੒! د༩͕খ͘͞ͳΔมҟ͸֬཰తʹغ٫͞ΕΔ

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ޫ༌ૹ΁ͷϝτϩϙϦεαϯϓϦϯάͷద༻ طଘͷ༗ޮͳύε΁มҟΛՃ͑ͯ৽ͨͳύεΛੜ੒! د༩͕খ͘͞ͳΔมҟ͸֬཰తʹغ٫͞ΕΔ

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ޫ༌ૹ΁ͷϝτϩϙϦεαϯϓϦϯάͷద༻ طଘͷ༗ޮͳύε΁มҟΛՃ͑ͯ৽ͨͳύεΛੜ੒! د༩͕খ͘͞ͳΔมҟ͸֬཰తʹغ٫͞ΕΔ

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ޫ༌ૹ΁ͷϝτϩϙϦεαϯϓϦϯάͷద༻ طଘͷ༗ޮͳύε΁มҟΛՃ͑ͯ৽ͨͳύεΛੜ੒! د༩͕খ͘͞ͳΔมҟ͸֬཰తʹغ٫͞ΕΔ د༩àغ٫

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ޫ༌ૹ΁ͷϝτϩϙϦεαϯϓϦϯάͷద༻ طଘͷ༗ޮͳύε΁มҟΛՃ͑ͯ৽ͨͳύεΛੜ੒! د༩͕খ͘͞ͳΔมҟ͸֬཰తʹغ٫͞ΕΔ ݩͷܦ࿏ʹ໭͢

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Bidirectional Path Tracing

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Metropolis Light Transport

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PRIMARY SAMPLE SPACE MLT

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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

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PHOTON MAPPING

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ϑΥτϯτϨʔγϯά

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ϑΥτϯτϨʔγϯά

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ϑΥτϯτϨʔγϯά

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ϑΥτϯτϨʔγϯά ີ౓ਪఆ

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ϑΥτϯτϨʔγϯά ີ౓ਪఆ

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ϑΥτϯτϨʔγϯά ີ౓ਪఆ ϑΥτϯϚοϐϯά͸ܦ࿏ΛΏΔ͘઀ଓ͢Δ͜ͱʹΑͬͯ! ܦ࿏Λ࠶ར༻ɺଟ༷ͳܦ࿏Λ·ͱΊͯܭࢉ

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PROGRESSIVE PHOTON MAPPING

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ϑΥτϯϚοϐϯάͷ໰୊఺ ਖ਼֬ͳً౓ਪఆʹ͸  ແݶখͷ୳ࡧ൒ܘʹແݶݸͷϑΥτϯͱ͍͏৚͕݅ඞཁ ϝϞϦ΍ܭࢉίετ໘ͰෆՄೳʂ

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PROGRESSIVE PHOTON MAPPING : PPM ϑΥτϯτϨʔγϯάΛ܁Γฦͯ͠౷ܭྔΛߋ৽ ͋Β͔͡Ίً౓ܭࢉ఺Λੜ੒͓ͯ͘͠ ౷ܭߋ৽ˍ൒ܘॖݮ

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ແݶখͷ൒ܘʹແݶݸͷϑΥτϯͱ͍͏৚݅ʹ  ϓϩάϨογϒʹۙͮ͘ ϑΥτϯͷ୳ࡧ൒ܘΛ൓෮͝ͱʹॖݮ ൒ܘॖݮʗ౷ܭྔͷߋ৽

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STOCHASTIC PPM

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

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SPPM ྖҬ಺Ͱ୳ࡧ൒ܘͳͲͷ౷ܭྔΛڞ༗ ޫ୔൓ࣹ ൓ࣹํ޲ ΞϯνΤΠϦΞε ϐΫηϧ Ϟʔγϣϯϒϥʔ γϟολʔ࣌ؒத ඃࣸքਂ౓ Ϩϯζ্ શͯΛ·ͱΊΔ͜ͱͰ ฏۉً౓ͷਪఆ஋ΛϓϩάϨογϒʹਅ஋ʹ͚ۙͮΒΕΔ

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ً౓ܭଌ఺΋ຖճ࡞Γ௚͢ ϐΫηϧதͷҐஔ΍ɺϨϯζ্ͷҐஔɺ࣌ؒɺޫ୔൓ࣹํ޲ ͳͲΛຖճมߋ͢Δ ڞ༗౷ܭߋ৽ˍ൒ܘॖݮ

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Bidirectional Path Tracing

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Progressive Photon Mapping

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Stochastic PPM

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PPM: PROBABILISTIC APPROACH

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411.ͷҰൠԽΛ͞Βʹਪ͠ਐΊͨख๏ ൒ܘΛঃʑʹখ͍ͯͬͨ͘͞͠ ΦϦδφϧͷϑΥτϯϚοϐϯάͷ݁ՌΛॏͶ߹ΘͤΔ͚ͩʂ ൒ܘॖݮ

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ADAPTIVE ! MARKOV CHAIN MONTE CARLO! PPM

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11. 411. ͷ໰୊఺ ՄࢹྖҬ ෆՄࢹͳϑΥτϯܦ࿏  ʹແବͳܭࢉ ༗ޮͳϑΥτϯܦ࿏

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AMCMCPPM = PPM + PSSMLT + α ॳظͷՄࢹܦ࿏

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AMCMCPPM = PPM + PSSMLT + α ॳظͷՄࢹܦ࿏ ෆՄࢹà غ٫

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AMCMCPPM = PPM + PSSMLT + α ॳظͷՄࢹܦ࿏

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AMCMCPPM = PPM + PSSMLT + α ॳظͷՄࢹܦ࿏ Մࢹà ࠾୒

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AMCMCPPM = PPM + PSSMLT + α

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AMCMCPPM = PPM + PSSMLT + α Primary Sample Space  தͷมҟΛ༻͍ͯ  ܦ࿏Λੜ੒

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AMCMCPPM = PPM + PSSMLT + α Primary Sample Space  தͷมҟΛ༻͍ͯ  ܦ࿏Λੜ੒ มҟύϥϝλʔͷ ࣗಈௐ੔΋ߦ͏౳ આ໌ল͖·͢ + α

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".$.$11. 411.

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SPPM AMCMCPPM ஫໨ྖҬ͕૬ରతʹখ͘͞ͳΔ΄Ͳ11.͸ഁ୼͢Δ .$.$ͱύϥϝλʔͷࣗಈௐ੔ʹΑΓ ".$.$11.͸શͯͷഒ཰Ͱ༏Εͨ݁Ռ

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UNIFIED PATH SAMPLING! (VERTEX CONNECTION AND MERGING)

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BPT ޫ୔໘ͷଟ͍γʔϯಘҙ 4%4ύεۤख PPM ޫ୔໘ͷଟ͍γʔϯۤख 4%4ύεಘҙ .*4 ͔͠͠໰୊͕͋Δ ྫɿ௕͞ͷܦ࿏ߏங BPT ܦ࿏ͷ࣍ݩ : A5 PPM ܦ࿏ͷ࣍ݩ : A6 ܦ࿏ߏஙͷ ࣍ݩ͕ҟͳΔ

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wBSDF (x) = pBSDF (x) pBSDF (x)+ plight (x) MIS΢ΣΠτͷܭࢉʹPDFͷՃࢉΛؚΉ ࣍ݩͷҟͳΔྔͷՃࢉ͸ޚ๏౓ ࠶ܝɿόϥϯεώϡʔϦεςΟοΫ

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BPT ܦ࿏ͷ࣍ݩ : A5 ֦ுBPT ܦ࿏ͷ࣍ݩ : A6 Vertex Perturbation ࢹઢύεͷ୺఺ΛͣΒͯ͠ޫઢύεͷ୺఺Λ௥Ճ  Ծ૝తʹPPMͱ࣍ݩΛ߹ΘͤΔ

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֦ு#15ͱ11.ͷ.*4

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BATHROOM Bidirectional Path Tracing

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BATHROOM Progressive Photon Mapping

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BATHROOM Uniﬁed Path Sampling

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PATH SPACE REGULARIZATION

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Specular BRDF Molliﬁed BRDF BSDF MOLLIFICATION #4%'Λ؇࿨ͯ͠د༩ΛऔΕΔΑ͏ʹ ͨͩ͠CJBTFE ൓෮͝ͱʹຊདྷͷ#4%'΁͚͍ۙͮͯ͘ ʹຊ࣭తʹ͸11.ͷ൒ܘॖݮͱಉ͡ σΟϑϡʔζ໘ʹ͸ద༻͠ͳ͍àඞཁ࠷௿ݶͷόΠΞε

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Original MLT

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AMCMCPPM

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VCM(UPS)

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Regularized MLT

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Regularized MLT + ME

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MULTIPLEXED MLT

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144.-5Ͱ͸ͭͷఏҊ෼෍ͱ࠾୒ɾغ٫Λ૊Έ߹Θͤͯ ໨ඪ෼෍ ܦ࿏ͷը૾΁ͷد༩ Λୡ੒͢Δ ఏҊ෼෍ɿ #15౳ʹΑΔͭͷϚοϐϯά #15౳Ͱ࣮ݱ͞ΕΔϚοϐϯά ఏҊ෼෍ ͕͋·Γྑ͘ͳ͍ àغ٫͕૿͑Δ

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ఏҊ෼෍ɿ ෳ਺ͷϚοϐϯάͷࠞ߹ 144಺ͷมҟʹՃ͑ͯϚοϐϯάͷมߋ΋ߦ͏ PRIMARY SPACE SERIAL TEMPERING

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PSSMLT

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Original MLT

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Multiplexed MLT

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͓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.*4BOEPS 144 .-5 ͷཧ࿦࢖͍ͬͯΔΠϝʔδ

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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.!

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