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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

367b44575446a2eccc58feea183c1475?s=47 shocker_0x15
September 06, 2014

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。
主な対象者:モンテカルロ積分の基礎を理解しており、簡単なパストレーシングなどの実装経験がある方。

紹介している手法:
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

Twitter: @Shocker_0x15

367b44575446a2eccc58feea183c1475?s=128

shocker_0x15

September 06, 2014
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  1. MONTE CARLO RAY TRACING! ΞϧΰϦζϜ௒ུ֓ ౉෦৺! Twitter: @Shocker_0x15! ϨΠτϨ߹॓ https://sites.google.com/site/raytracingcamp2/

  2. ϞϯςΧϧϩੵ෼

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

  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
  8. PATH TRACING

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

  10. NEXT EVENT ESTIMATION ޫݯ্ͷ఺Λ໌ࣔతʹαϯϓϧɺࢹઢܦ࿏ͱ઀ଓ͢Δ

  11. MULTIPLE IMPORTANCE SAMPLING
 ྫɿ௚઀র໌ͷਪఆ 

  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 
  16. ޫݯ໘ͷαϯϓϦϯά #4%'ͷαϯϓϦϯά

  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
  19. .*4ʹΑΔ΢ΣΠτ഑෼ .VMUJQMF*NQPSUBODF4BNQMJOH

  20. BIDIRECTIONAL PATH TRACING! VEACH-STYLE

  21. .*4ΛҰൠԽ 15ʹ͓͚Δ ௚઀র໌ͷ৔߹ʜ ೚ҙͷܦ࿏ʹҰൠԽʂ

  22. ࢹઢαϒύεͱޫݯαϒύεΛੜ੒ɺ֤௖఺Λ઀ଓ

  23. (2,2) (3,1) (1,3) ྫ௕͞ MIS ˎ    

     ΋͋ΓಘΔ
  24. #JEJSFDUJPOBM1BUI5SBDJOH 1BUI5SBDJOH ؒ઀র໌͕ࢧ഑తͳγʔϯʹ͓͍ͯ΋ϩόετ

  25. METROPOLIS LIGHT TRANSPORT

  26. ௨ৗͷ.$35ͷ໰୊఺ ྫɿ1BUI5SBDJOH  ͘͝Ұ෦ͷྖҬͷޫ༌ૹܦ࿏͕ॏཁͱͳΔγʔϯʹऑ͍! ྫɿগ͚ͩ͠։͍ͨυΞ͔Β࿙ΕΔޫɺίʔεςΟΫε͕ओཁͳޫݯ ຖճϥϯμϜʹܦ࿏ΛτϨʔεɿ໓ଟʹد༩͕ͱΕͳ͍ʂʂ!

  27. ௨ৗͷ.$35ͷ໰୊఺ ྫɿ1BUI5SBDJOH  ͘͝Ұ෦ͷྖҬͷޫ༌ૹܦ࿏͕ॏཁͱͳΔγʔϯʹऑ͍! ྫɿগ͚ͩ͠։͍ͨυΞ͔Β࿙ΕΔޫɺίʔεςΟΫε͕ओཁͳޫݯ ຖճϥϯμϜʹܦ࿏ΛτϨʔεɿ໓ଟʹد༩͕ͱΕͳ͍ʂʂ!

  28. ௨ৗͷ.$35ͷ໰୊఺ ྫɿ1BUI5SBDJOH  ͘͝Ұ෦ͷྖҬͷޫ༌ૹܦ࿏͕ॏཁͱͳΔγʔϯʹऑ͍! ྫɿগ͚ͩ͠։͍ͨυΞ͔Β࿙ΕΔޫɺίʔεςΟΫε͕ओཁͳޫݯ ຖճϥϯμϜʹܦ࿏ΛτϨʔεɿ໓ଟʹد༩͕ͱΕͳ͍ʂʂ!

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

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

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

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

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

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

  35. Bidirectional Path Tracing

  36. Metropolis Light Transport

  37. PRIMARY SAMPLE SPACE MLT

  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 
  41. PHOTON MAPPING

  42. None
  43. ϑΥτϯτϨʔγϯά

  44. ϑΥτϯτϨʔγϯά

  45. ϑΥτϯτϨʔγϯά

  46. ϑΥτϯτϨʔγϯά ີ౓ਪఆ

  47. ϑΥτϯτϨʔγϯά ີ౓ਪఆ

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

  49. PROGRESSIVE PHOTON MAPPING

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

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

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

  53. STOCHASTIC PPM

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

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

    શͯΛ·ͱΊΔ͜ͱͰ ฏۉً౓ͷਪఆ஋ΛϓϩάϨογϒʹਅ஋ʹ͚ۙͮΒΕΔ
  56. ً౓ܭଌ఺΋ຖճ࡞Γ௚͢ ϐΫηϧதͷҐஔ΍ɺϨϯζ্ͷҐஔɺ࣌ؒɺޫ୔൓ࣹํ޲ ͳͲΛຖճมߋ͢Δ ڞ༗౷ܭߋ৽ˍ൒ܘॖݮ

  57. Bidirectional Path Tracing

  58. Progressive Photon Mapping

  59. Stochastic PPM

  60. PPM: PROBABILISTIC APPROACH

  61. 411.ͷҰൠԽΛ͞Βʹਪ͠ਐΊͨख๏ ൒ܘΛঃʑʹখ͍ͯͬͨ͘͞͠ ΦϦδφϧͷϑΥτϯϚοϐϯάͷ݁ՌΛॏͶ߹ΘͤΔ͚ͩʂ ൒ܘॖݮ

  62. ADAPTIVE ! MARKOV CHAIN MONTE CARLO! PPM

  63. 11. 411. ͷ໰୊఺ ՄࢹྖҬ ෆՄࢹͳϑΥτϯܦ࿏
 ʹແବͳܭࢉ ༗ޮͳϑΥτϯܦ࿏

  64. AMCMCPPM = PPM + PSSMLT + α ॳظͷՄࢹܦ࿏

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

  66. AMCMCPPM = PPM + PSSMLT + α ॳظͷՄࢹܦ࿏

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

  68. AMCMCPPM = PPM + PSSMLT + α

  69. AMCMCPPM = PPM + PSSMLT + α Primary Sample Space


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


    தͷมҟΛ༻͍ͯ
 ܦ࿏Λੜ੒ มҟύϥϝλʔͷ ࣗಈௐ੔΋ߦ͏౳ આ໌ল͖·͢  + α
  71. ".$.$11. 411.

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

  73. UNIFIED PATH SAMPLING! (VERTEX CONNECTION AND MERGING)

  74. BPT ޫ୔໘ͷଟ͍γʔϯಘҙ 4%4ύεۤख PPM ޫ୔໘ͷଟ͍γʔϯۤख 4%4ύεಘҙ .*4 ͔͠͠໰୊͕͋Δ ྫɿ௕͞ͷܦ࿏ߏங BPT

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

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

    ࢹઢύεͷ୺఺ΛͣΒͯ͠ޫઢύεͷ୺఺Λ௥Ճ
 Ծ૝తʹPPMͱ࣍ݩΛ߹ΘͤΔ
  77. ֦ு#15ͱ11.ͷ.*4

  78. BATHROOM Bidirectional Path Tracing

  79. BATHROOM Progressive Photon Mapping

  80. BATHROOM Unified Path Sampling

  81. PATH SPACE REGULARIZATION

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

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

  84. AMCMCPPM

  85. VCM(UPS)

  86. Regularized MLT

  87. Regularized MLT + ME

  88. MULTIPLEXED MLT

  89. 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

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

  91. PSSMLT

  92. Original MLT

  93. Multiplexed MLT

  94. ͓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
  95. REFERENCES 1/3 n  [ERPT] CLINE, D., TALBOT, J., AND EGBERT,

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