パストレーシング / Path Tracing

レイトレ合宿3!!!のセミナーで使用した資料です。
スライドの趣旨：パストレーシングの基礎概念からおさらいしてNext Event EstimationとMultiple Importance Samplingまで。
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。

2017/12/8: パストレーシングの理論と実装について記事を書いています。
https://rayspace.xyz/CG/contents/path_tracing/
https://rayspace.xyz/CG/contents/path_tracing_implementation/

Twitter: @Shocker_0x15

Transcript

1. ύετϨʔγϯά
   ౉෦৺
   !4IPDLFS@Y
   ϨΠτϨ߹॓
   https://sites.google.com/site/raytracingcamp3/
   PATH TRACING
   

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2. ϨϯμϦϯάํఔࣜ
   Lo
   (x, o
   ) = Le
   (x, o
   ) +
   S2
   Li
   (x, i
   )fs
   (x, i, o
   ) |n · i
   | d i
   ϞϯςΧϧϩ .$
   ੵ෼
   Lo
   (x, o
   ) = Le
   (x, o
   ) + Li
   (x, i
   )fs
   (x, i, o
   ) |n · i
   |
   p (
   i
   )
   ϥϯμϜʹαϯϓϧ͞Εͨೖࣹํ޲
   i
   p (
   i
   ) ɹΛαϯϓϧ͢Δ֬཰ີ౓ 1%'
   
   i
   

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3. ೖࣹํ޲Λ֬཰తʹαϯϓϧɺޫݯʹ౰ͨΕ͹د༩Λܭࢉ
   ࢹ఺͔Βޫ༌ૹܦ࿏ΛτϨʔε
   

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4. ೖࣹํ޲ͷαϯϓϦϯά
   %JGGVTF#4%' (MPTTZ#4%' 4QFDVMBS#4%'
   Ͳͷํ޲΋ࣅͨΑ͏ͳ஋ ڸ໘൓ࣹํ޲ͷ
   पғʹେ͖ͳ஋
   ڸ໘൓ࣹํ޲Ҏ֎ͷ
   ஋͸θϩ
   গͳ͘ͱ΋#4%'͕େ͖ͳ஋Λ࣋ͭํ޲Λ
   ଟ͘αϯϓϧ͢Δͷ͕๬·͍͠à#4%'ͷॏ఺తαϯϓϦϯά
   

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5. for (int j = 0; j < ImageHeight; ++j) {
   
   
   for (int i = 0; i < ImageWidth; ++i) {
   
   
   
   px = i + rnd01();
   
   
   
   py = j + rnd01();
   
   
   
   ray, We, dirPDF = sampleCameraRay(px, py);
   
   
   
   alpha *= We / dirPDF;
   
   
   
   
   
   // シーンと交叉判定
   
   
   
   hit, surfPt = intersect(ray);
   
   
   
   if (!hit)
   
   
   
   
   break;
   
   
   
   
   
   while (true) {
   
   
   
   
   ...
   
   
   
   
   ray = Ray(surfPt.p, dir);
   
   
   
   
   
   
   // シーンと交叉判定
   
   
   
   
   hit, surfPt = intersect(ray);
   
   
   
   
   if (!hit)
   
   
   
   
   
   break;
   
   
   
   
   
   
   // Russian Roulette
   
   
   
   
   if (rnd01() >= ProbRR)
   
   
   
   
   
   break;
   
   
   
   
   alpha /= ProbRR;
   
   
   
   }
   
   
   }
   
   }
   
   // 光源にヒットした場合、寄与を蓄積
   
   if (surfPt.isEmitting())
   
   
   contribution += alpha * surfPt.Le(-ray.dir);
   
   
   
   // BSDFの重点的サンプリング
   
   fs, dir, dirPDF = surfPt.BSDF.sample(rnd01(), rnd01());
   
   alpha *= fs * absDot(surfPt.n, dir) / dirPDF;
   
   ϨϯμϦϯάํఔࣜ ͷ.$ੵ෼
   
   ͦͷ··
   Li
   (x, i
   )fs
   (x, i, o
   ) |n · i
   |
   p (
   i
   )
   

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6. ೖࣹํ޲Λ֬཰తʹαϯϓϧɺޫݯʹ౰ͨΕ͹د༩͕ͱΕΔ
   ࢹ఺͔Βޫ༌ૹܦ࿏ΛτϨʔε
   ͳ͔ͳ͔౰ͨΒͳ͍ʂ(ಛʹޫݯ͕খ͍͞ͱ͖)
   

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7. NEXT EVENT ESTIMATION
   

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

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9. for (int j = 0; j < ImageHeight; ++j) {
   
   
   for (int i = 0; i < ImageWidth; ++i) {
   
   
   
   px = i + rnd01();
   
   
   
   py = j + rnd01();
   
   
   
   ray, We, dirPDF = sampleCameraRay(px, py);
   
   
   
   alpha *= We / dirPDF;
   
   
   
   
   
   // シーンと交叉判定
   
   
   
   hit, surfPt = intersect(ray);
   
   
   
   if (!hit)
   
   
   
   
   break;
   
   
   
   
   
   if (surfPt.isEmitting())
   
   
   
   
   contribution += alpha * surfPt.Le(-ray.dir);
   
   
   
   
   
   while (true) {
   
   
   
   
   ...
   
   
   
   
   
   
   // BSDFの重点的サンプリング
   
   
   
   
   fs, dir, dirPDF = surfPt.BSDF.sample(rnd01(), rnd01());
   
   
   
   
   alpha *= fs * absDot(surfPt.n, dir) / dirPDF;
   
   
   
   
   
   
   ray = Ray(surfPt.p, dir);
   
   
   
   
   
   
   // シーンと交叉判定
   
   
   
   
   hit, surfPt = intersect(ray);
   
   
   
   
   if (!hit)
   
   
   
   
   
   break;
   
   
   
   
   
   
   // Russian Roulette
   
   
   
   
   if (rnd01() >= ProbRR)
   
   
   
   
   
   break;
   
   
   
   
   alpha /= ProbRR;
   
   
   
   }
   
   
   }
   
   }
   
   // 光源上の点を明示的にサンプルして現在の点と接続
   
   lightSurfPt, areaPDF = sampleLightPoint(rnd01(), rnd01());
   
   if (unoccluded(surfPt, lightSurfPt)) {
   
   
   shadowDir = normalize(lightSurfPt.p – surfPt.p);
   
   
   fs = surfPt.BSDF.eval(shadowDir, -ray.dir);
   
   
   G = absDot(surfPt.n, shadowDir) *
   
   absDot(lightSurfPt.n, shadowDir) /
   
   
   sqDistance(surfPt.p, lightSurfPt.p);
   
   
   contribution += alpha * fs * lightSurfPt.Le(-shadowDir) *
   
   G / areaPDF;
   
   }
   
   γϟυ΢ϨΠΛඈ͹ͯ͠ःณΛνΣοΫ
   ःณ͕ແ͚Ε͹د༩Λ஝ੵ
   

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10. ʁʁʁ
    

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11. // 光源上の点を明示的にサンプルして現在の点と接続
    
    lightSurfPt, areaPDF = sampleLightPoint(rnd01(), rnd01());
    
    if (unoccluded(surfPt, lightSurfPt)) {
    
    
    shadowDir = normalize(lightSurfPt.p – surfPt.p);
    
    
    fs = surfPt.BSDF.eval(shadowDir, -ray.dir);
    
    
    G = absDot(surfPt.n, shadowDir) *
    
    absDot(lightSurfPt.n, shadowDir) /
    
    
    sqDistance(surfPt.p, lightSurfPt.p);
    
    
    contribution += alpha * fs * lightSurfPt.Le(-shadowDir) *
    
    G / areaPDF;
    
    }
    
    ͜Ε͸Կʁ
    DPT߲ͭͱڑ཭ͷೋ৐
    ͔֬ʹͦΕͬΆ͍߲͕ͩ
    ϨϯμϦϯάํఔࣜʹͦΜͳ߲͚͋ͬͨͬʁʁ
    

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12. ແ͍
    DPT߲ͻͱͭͩ͠ڑ཭ೋ৐΋ແ͍
    Lo
    (x, o
    ) = Le
    (x, o
    ) +
    S2
    Li
    (x, i
    )fs
    (x, i, o
    ) |n · i
    | d i
    

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13. ୯Ґͷ࿩
    

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14. ֬཰ʹ୯Ґ͸ແ͍͕֬཰ີ౓ 1%'
    ʹ͸୯Ґ͕͋Δ
    Lo
    (x, o
    ) = Le
    (x, o
    ) + Li
    (x, i
    )fs
    (x, i, o
    ) |n · i
    |
    p (
    i
    )
    #4%'ͷαϯϓϦϯάͰ͸ೖࣹํ޲ΛαϯϓϦϯά
    ํ޲ ཱମ֯
    ʹؔ͢Δ1%'
    [sr 1]
    /FYU&WFOU&TUJNBUJPOͰ͸ޫݯ্ͷҐஔΛαϯϓϦϯά
    ෺ମද໘্ͷҐஔ ໘ੵ
    ʹؔ͢Δ1%'
    [m 2]
    

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15. Ґஔʹؔ͢ΔαϯϓϦϯάͰ͸
    ѻ͏ϨϯμϦϯάํఔ͕ࣜएׯมΘΔ
    Lo
    (x x ) = Le
    (x x ) +
    M
    Li
    (x x)fs
    (x x x )G(x x )dA
    x
    x
    x
    x
    ੵ෼߲͸γʔϯதͷ෺ମද໘ɹ্ͷੵ෼ͱͯ͠มܗ͞ΕΔ
    M
    زԿ߲ɹ͸ੵ෼ͷม਺ม׵ͷ݁Ռ ϠίϏΞϯ
    
    G
    

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16. #4%'ͷαϯϓϦϯά /FYU&WFOU&TUJNBUJPO
    ͨͩ͠/FYU&WFOU&TUJNBUJPO͕ٯޮՌʹͳΔ͜ͱ΋
    

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17. MULTIPLE IMPORTANCE SAMPLING
    ଟॏॏ఺తαϯϓϦϯά
    

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18. Light
    ޫ୔BSDF
    #4%'ͷد༩ʹԊͬͨॏ఺తαϯϓϦϯά
    #4%'د༩ʹԊͬͯೖࣹํ޲αϯϓϧɿߴ͍֬཰Ͱߴ͍د༩
    à௿͍෼ࢄ ޫݯ͕ྑ͍৔ॴʹ͋Ε͹
    
    C =
    1
    N
    N
    i=1
    f(¯
    xi
    )
    pBSDF
    (¯
    xi
    )
    

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19. Light
    ֦ࢄBSDF
    #4%'ͷد༩ʹԊͬͨॏ఺తαϯϓϦϯά
    #4%'د༩ʹԊͬͯೖࣹํ޲αϯϓϧɿ௿͍֬཰Ͱߴ͍د༩
    àߴ͍෼ࢄ ͨ·ʹ͔͠౰ͨΒͳ͍ͨΊ
    
    C =
    1
    N
    N
    i=1
    f(¯
    xi
    )
    pBSDF
    (¯
    xi
    )
    

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20. Light
    ֦ࢄBSDF
    ޫݯ্ͷҐஔͷॏ఺తαϯϓϦϯά
    ޫݯ্ͷҐஔΛαϯϓϧͯ͠઀ଓɿߴ͍֬཰Ͱߴ͍د༩
    à௿͍෼ࢄ #4%'ͷ஋͕ൺֱతҰ༷Ͱ͋Ε͹
    
    C =
    1
    N
    N
    i=1
    f(¯
    xi
    )
    plight
    (¯
    xi
    )
    

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21. Light
    ޫ୔BSDF
    ޫݯ্ͷҐஔͷॏ఺తαϯϓϦϯά
    ޫݯ্ͷҐஔΛαϯϓϧͯ͠઀ଓɿ௿͍֬཰Ͱߴ͍د༩
    àߴ͍෼ࢄ #4%'ͷ஋͕ඇҰ༷ͳͨΊ
    
    C =
    1
    N
    N
    i=1
    f(¯
    xi
    )
    plight
    (¯
    xi
    )
    

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22. ޫݯ໘ͷαϯϓϦϯά
    #4%'ͷαϯϓϦϯά
    n  VEACH, E. 1997. Robust Monte Carlo methods for light transport simulation. PhD thesis, Stanford, CA, USA.
    

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23. ଟॏॏ఺తαϯϓϦϯά
    C =
    1
    N
    N
    i=1
    f(¯
    xlight,i
    )
    plight
    (¯
    xlight,i
    )
    C =
    1
    N
    N
    i=1
    f(¯
    xBSDF,i
    )
    pBSDF
    (¯
    xBSDF,i
    )
    C =
    1
    N
    N
    i=1
    wBSDF
    (¯
    xBSDF,i
    ) f(¯
    xBSDF,i
    )
    pBSDF
    (¯
    xBSDF,i
    )
    + wlight
    (¯
    xlight,i
    ) f(¯
    xlight,i
    )
    plight
    (¯
    xlight,i
    )
    ϛοΫεʂ
    

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24. .*4΢ΣΠτ
    Ұྫ: όϥϯεώϡʔϦεςΟοΫ

    (΢ΣΠτ = ͦΕͧΕͷPDFͷॏΈ෇͖ฏۉ)
    C =
    1
    N
    N
    i=1
    wBSDF
    (¯
    xBSDF,i
    ) f(¯
    xBSDF,i
    )
    pBSDF
    (¯
    xBSDF,i
    )
    + wlight
    (¯
    xlight,i
    ) f(¯
    xlight,i
    )
    plight
    (¯
    xlight,i
    )
    wBSDF
    (¯
    x) = pBSDF
    (¯
    x)
    pBSDF
    (¯
    x) + plight
    (¯
    x) wlight
    (¯
    x) = plight
    (¯
    x)
    pBSDF
    (¯
    x) + plight
    (¯
    x)
    (࣮ࡍʹ͸PDFͷ୯ҐΛ߹ΘͤΔඞཁ͕͋Δ)
    

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25. όϥϯεώϡʔϦεςΟοΫɿཁ͢Δʹ
    BSDF
    ʮύε͖࣋ͬͯͨΑʯ
    ʮਪఆ݁Ռʹࣗ৴ͷ΄Ͳ͸ʯ
    BSDF
    ʮʯ
    ʮMJHIU͕Ծʹ͜ͷύεͭͬͨ͘ͳΒࣗ৴͸ʯ
    lightʮʯ
    ʮ͡Ό͋#4%'͕͖࣋ͬͯͨύεͷॏΈ͸
    ɹ
    ͬͯ͜ͱͰʯ
    lightʮύε͖࣋ͬͯͨΑʯ
    ʮਪఆ݁Ռʹࣗ৴ͷ΄Ͳ͸ʯ
    lightʮʯ
    ʮ#4%'͕Ծʹ͜ͷύεͭͬͨ͘ͳΒࣗ৴͸ʯ
    BSDF
    ʮʯ
    ʮ͡Ό͋MJHIU͕͖࣋ͬͯͨύεͷॏΈ͸
    ɹ
    ͬͯ͜ͱͰʯ
    

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26. .*4ʹΑΔ΢ΣΠτ഑෼
    .VMUJQMF*NQPSUBODF4BNQMJOH
    n  VEACH, E. 1997. Robust Monte Carlo methods for light transport simulation. PhD thesis, Stanford, CA, USA.
    

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27. for (int j = 0; j < ImageHeight; ++j) {
    
    
    for (int i = 0; i < ImageWidth; ++i) {
    
    
    
    px = i + rnd01(), py = j + rnd01();
    
    
    
    ray, We, dirPDF = sampleCameraRay(px, py);
    
    
    
    alpha *= We / dirPDF;
    
    
    
    hit, surfPt = intersect(ray);
    
    
    
    if (!hit)
    
    
    
    
    continue;
    
    
    
    
    
    if (surfPt.isEmitting())
    
    
    
    
    contribution += alpha * surfPt.Le(-ray.dir);
    
    
    
    
    
    while (true) {
    
    
    
    
    ...
    
    
    
    
    
    
    // BSDFの重点的サンプリング
    
    
    
    
    fs, dir, dirPDF = surfPt.BSDF.sample(rnd01(), rnd01());
    
    
    
    
    alpha *= fs * absDot(surfPt.n, dir) / dirPDF;
    
    
    
    
    
    
    ray = Ray(surfPt.p, dir);
    
    
    
    
    
    
    hit, surfPt = intersect(ray);
    
    
    
    
    if (!hit)
    
    
    
    
    
    break;
    
    
    
    
    
    
    
    
    
    ...
    
    
    
    
    
    
    // Russian Roulette
    
    
    
    
    if (rnd01() >= ProbRR)
    
    
    
    
    
    break;
    
    
    
    
    alpha /= ProbRR;
    
    
    
    }
    
    
    }
    
    }
    
    // 光源上の点を明示的にサンプリングして現在の点と接続
    
    lightSurfPt, lightPDF = sampleLightPoint(rnd01(), rnd01());
    
    if (unoccluded(surfPt, lightSurfPt)) {
    
    
    shadowDir = normalize(lightSurfPt.p – surfPt.p);
    
    
    cosShd = absDot(surfPt.n, shadowDir);
    
    
    cosLight = absDot(lightSurfPt.n, shadowDir);
    
    
    dist2 = sqDistance(surfPt.p, lightSurfPt.p);
    
    
    
    
    fs = surfPt.BSDF.eval(shadowDir, -ray.dir);
    
    
    G = cosShd * cosLight / dist2;
    
    
    
    
    // 単位をlightPDF[m^-2]に合わせる。
    
    
    bsdfPDF = surfPt.BSDF.evalDirectionalPDF(shadowDir, -ray.dir) *
    
    
    cosLight / dist2;
    
    
    MISWeight = lightPDF / (bsdfPDF + lightPDF);
    
    
    
    
    contribution += alpha * MISWeight *
    
    
    fs * lightSurfPt.Le(-shadowDir) *
    
    
    G / lightPDF;
    
    }
    
    /FYU&WFOU&TUJNBUJPO.*4΢ΣΠτ
    

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28. for (int j = 0; j < ImageHeight; ++j) {
    
    
    for (int i = 0; i < ImageWidth; ++i) {
    
    
    
    px = i + rnd01(), py = j + rnd01();
    
    
    
    ray, We, dirPDF = sampleCameraRay(px, py);
    
    
    
    alpha *= We / dirPDF;
    
    
    
    hit, surfPt = intersect(ray);
    
    
    
    if (!hit)
    
    
    
    
    continue;
    
    
    
    
    
    if (surfPt.isEmitting())
    
    
    
    
    contribution += alpha * surfPt.Le(-ray.dir);
    
    
    
    
    
    while (true) {
    
    
    
    
    ...
    
    
    
    
    
    
    // BSDFの重点的サンプリング
    
    
    
    
    fs, dir, dirPDF = surfPt.BSDF.sample(rnd01(), rnd01());
    
    
    
    
    alpha *= fs * absDot(surfPt.n, dir) / dirPDF;
    
    
    
    
    
    
    ray = Ray(surfPt.p, dir);
    
    
    
    
    
    
    hit, surfPt = intersect(ray);
    
    
    
    
    if (!hit)
    
    
    
    
    
    break;
    
    
    
    
    
    
    
    
    
    ...
    
    
    
    
    
    
    // Russian Roulette
    
    
    
    
    if (rnd01() >= ProbRR)
    
    
    
    
    
    break;
    
    
    
    
    alpha /= ProbRR;
    
    
    
    }
    
    
    }
    
    }
    
    // 光源にヒットした場合、寄与を蓄積
    
    if (surfPt.isEmitting()) {
    
    
    cosLight = absDot(ray.dir, surfPt.n);
    
    
    dist2 = sqDistance(ray.org, surfPt.p);
    
    
    
    
    bsdfPDF = dirPDF;
    
    
    // 単位をbsdfPDF[sr^-1]に合わせる。
    
    
    lightPDF = surfPt.evalAreaPDF() * dist2 / cosLight;
    
    
    MISWeight = bsdfPDF / (bsdfPDF + lightPDF);
    
    
    
    
    
    contribution += alpha * MISWeight * surfPt.Le(-ray.dir);
    
    }
    
    #4%'4BNQMJOH.*4΢ΣΠτ
    

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29. DEPTH OF FIELD
    ඃࣸքਂ౓
    

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30. ബϨϯζϞσϧΛ༻͍ͨ%P'දݱ
    ཧ࿦ɾखॱͷৄࡉ͸8ϖʔδࢀর
    ୺తʹड़΂Δͱʜ
    xp
    x0
    x1
    x1
    x0
    xp x1
    γʔϯதͷҐஔ
    ηϯαʔ໘
    ෺໘
    Ϩϯζ্ͷ఺ɹΛαϯϓϧɺ
    ϐΫηϧதͷҐஔɹʹରԠ͢Δ෺໘্ͷ఺ɹʹ޲͔ͬͯϨΠτϨʔε
    ϨϯζҐஔͷαϯϓϧΛߦ͍ͬͯΔͷͰ1%'ɹɹɹͰد༩ΛׂΔ
    pA
    (x0
    )
    

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31. ࣮૷͢Δͱʜ
    ϐϯϗʔϧ ബϨϯζϞσϧʹΑΔ%P'
    

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32. ͪΌΜͱϘέͯΔΈ͍͚ͨͩͲ
    ϐϯϗʔϧͱ໌Δ͞ҧ͏͠ɺ
    Ϩϯζͷେ͖͞ʹΑͬͯ໌Δ͞มΘΔʜ
    ݱ࣮ͷΧϝϥͱҰॹ
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34. ͪΐͬͱ଴ͯ
    

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35. ϐϯϗʔϧΧϝϥ͸ബϨϯζϞσϧͰ
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    à1%'ΛΩϟϯηϧ͢Δ͚ͩͷ
    ແݶͷηϯαʔײ౓͕͋Δͷͱ౳͍͠
    

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36. ͓͠·͍
    *#-ͱ͔৭ʑ࿩͔͚ͨͬͨ͠Ͳ
    ͦͷ͏ͪ8ϖʔδʹॻ͖·͢
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