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

Transcript

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

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2. ϞϯςΧϧϩੵ෼
   

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3. 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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4. 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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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೉
   

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

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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
   

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8. PATH TRACING
   

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

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

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11. MULTIPLE IMPORTANCE SAMPLING

    ྫɿ௚઀র໌ͷਪఆ
    
    

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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͹
    
    

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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
    
    

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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͹
    
    

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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
    
    

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16. ޫݯ໘ͷαϯϓϦϯά
    #4%'ͷαϯϓϦϯά
    

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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
    ∑
    

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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
    

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

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20. BIDIRECTIONAL PATH TRACING!
    VEACH-STYLE
    
    

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

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22. ࢹઢαϒύεͱޫݯαϒύεΛੜ੒ɺ֤௖఺Λ઀ଓ
    

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23. (2,2)
    (3,1) (1,3)
    ྫ௕͞ MIS
    ˎ
    
    ΋͋ΓಘΔ
    

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

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25. METROPOLIS LIGHT TRANSPORT
    

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

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

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

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

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

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

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

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

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

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35. Bidirectional Path Tracing
    

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36. Metropolis Light Transport
    

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37. PRIMARY SAMPLE SPACE MLT
    

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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
    
    

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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
    
    

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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
    
    

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41. PHOTON MAPPING
    

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43. ϑΥτϯτϨʔγϯά
    

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44. ϑΥτϯτϨʔγϯά
    

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45. ϑΥτϯτϨʔγϯά
    

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46. ϑΥτϯτϨʔγϯά ີ౓ਪఆ
    

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47. ϑΥτϯτϨʔγϯά ີ౓ਪఆ
    

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

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49. PROGRESSIVE PHOTON MAPPING
    

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50. ϑΥτϯϚοϐϯάͷ໰୊఺
    ਖ਼֬ͳً౓ਪఆʹ͸

    ແݶখͷ୳ࡧ൒ܘʹແݶݸͷϑΥτϯͱ͍͏৚͕݅ඞཁ
    ϝϞϦ΍ܭࢉίετ໘ͰෆՄೳʂ
    

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

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52. ແݶখͷ൒ܘʹແݶݸͷϑΥτϯͱ͍͏৚݅ʹ

    ϓϩάϨογϒʹۙͮ͘
    ϑΥτϯͷ୳ࡧ൒ܘΛ൓෮͝ͱʹॖݮ
    ൒ܘॖݮʗ౷ܭྔͷߋ৽
    

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53. STOCHASTIC PPM
    

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

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

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

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57. Bidirectional Path Tracing
    

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58. Progressive Photon Mapping
    

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59. Stochastic PPM
    

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60. PPM: PROBABILISTIC APPROACH
    

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

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62. ADAPTIVE !
    MARKOV CHAIN MONTE CARLO!
    PPM
    

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63. 11. 411.
    ͷ໰୊఺
    ՄࢹྖҬ
    ෆՄࢹͳϑΥτϯܦ࿏

    ʹແବͳܭࢉ
    ༗ޮͳϑΥτϯܦ࿏
    

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64. AMCMCPPM = PPM + PSSMLT + α
    ॳظͷՄࢹܦ࿏
    

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

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66. AMCMCPPM = PPM + PSSMLT + α
    ॳظͷՄࢹܦ࿏
    

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

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68. AMCMCPPM = PPM + PSSMLT + α
    

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

    தͷมҟΛ༻͍ͯ

    ܦ࿏Λੜ੒
    

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

    தͷมҟΛ༻͍ͯ

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

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71. ".$.$11.
    411.
    

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

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

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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
    

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

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76. BPT
    ܦ࿏ͷ࣍ݩ : A5
    ֦ுBPT
    ܦ࿏ͷ࣍ݩ : A6
    Vertex Perturbation
    ࢹઢύεͷ୺఺ΛͣΒͯ͠ޫઢύεͷ୺఺Λ௥Ճ

    Ծ૝తʹPPMͱ࣍ݩΛ߹ΘͤΔ
    

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77. ֦ு#15ͱ11.ͷ.*4
    

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78. BATHROOM
    Bidirectional Path Tracing
    

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79. BATHROOM
    Progressive Photon Mapping
    

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80. BATHROOM
    Unified Path Sampling
    

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81. PATH SPACE REGULARIZATION
    

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82. 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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83. Original MLT
    

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84. AMCMCPPM
    

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85. VCM(UPS)
    

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86. Regularized MLT
    

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87. Regularized MLT + ME
    

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88. MULTIPLEXED MLT
    

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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
    

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

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91. PSSMLT
    

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92. Original MLT
    

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93. Multiplexed MLT
    

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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
    

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95. 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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96. 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.!
    

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97. 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.!
    

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