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ύετϨʔγϯά
෦৺
!4IPDLFS@Y
ϨΠτϨ߹॓
https://sites.google.com/site/raytracingcamp3/
PATH TRACING
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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
) ɹΛαϯϓϧ͢Δ֬ີ 1%'
i
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ೖࣹํΛ֬తʹαϯϓϧɺޫݯʹͨΕد༩Λܭࢉ
ࢹ͔Βޫ༌ૹܦ࿏ΛτϨʔε
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ೖࣹํͷαϯϓϦϯά
%JGGVTF#4%' (MPTTZ#4%' 4QFDVMBS#4%'
ͲͷํࣅͨΑ͏ͳ ڸ໘ࣹํͷ
पғʹେ͖ͳ
ڸ໘ࣹํҎ֎ͷ
θϩ
গͳ͘ͱ#4%'͕େ͖ͳΛ࣋ͭํΛ
ଟ͘αϯϓϧ͢Δͷ͕·͍͠à#4%'ͷॏతαϯϓϦϯά
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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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ೖࣹํΛ֬తʹαϯϓϧɺޫݯʹͨΕد༩͕ͱΕΔ
ࢹ͔Βޫ༌ૹܦ࿏ΛτϨʔε
ͳ͔ͳ͔ͨΒͳ͍ʂ(ಛʹޫݯ͕খ͍͞ͱ͖)
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NEXT EVENT ESTIMATION
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NEXT EVENT ESTIMATION
ޫݯ্ͷΛ໌ࣔతʹαϯϓϧɺࢹઢܦ࿏ͱଓ͢Δ
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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;
}
γϟυϨΠΛඈͯ͠ःณΛνΣοΫ
ःณ͕ແ͚Εد༩Λੵ
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ʁʁʁ
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// 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;
}
͜ΕԿʁ
DPT߲ͭͱڑͷೋ
͔֬ʹͦΕͬΆ͍߲͕ͩ
ϨϯμϦϯάํఔࣜʹͦΜͳ߲͚͋ͬͨͬʁʁ
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ແ͍
DPT߲ͻͱͭͩ͠ڑೋແ͍
Lo
(x, o
) = Le
(x, o
) +
S2
Li
(x, i
)fs
(x, i, o
) |n · i
| d i
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୯Ґͷ
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֬ʹ୯Ґແ͍͕֬ີ 1%'
ʹ୯Ґ͕͋Δ
Lo
(x, o
) = Le
(x, o
) + Li
(x, i
)fs
(x, i, o
) |n · i
|
p (
i
)
#4%'ͷαϯϓϦϯάͰೖࣹํΛαϯϓϦϯά
ํ ཱମ֯
ʹؔ͢Δ1%'
[sr 1]
/FYU&WFOU&TUJNBUJPOͰޫݯ্ͷҐஔΛαϯϓϦϯά
ମද໘্ͷҐஔ ໘ੵ
ʹؔ͢Δ1%'
[m 2]
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Ґஔʹؔ͢ΔαϯϓϦϯάͰ
ѻ͏ϨϯμϦϯάํఔ͕ࣜएׯมΘΔ
Lo
(x x ) = Le
(x x ) +
M
Li
(x x)fs
(x x x )G(x x )dA
x
x
x
x
ੵ߲γʔϯதͷମද໘ɹ্ͷੵͱͯ͠มܗ͞ΕΔ
M
زԿ߲ɹੵͷมมͷ݁Ռ ϠίϏΞϯ
G
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#4%'ͷαϯϓϦϯά /FYU&WFOU&TUJNBUJPO
ͨͩ͠/FYU&WFOU&TUJNBUJPO͕ٯޮՌʹͳΔ͜ͱ
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MULTIPLE IMPORTANCE SAMPLING
ଟॏॏతαϯϓϦϯά
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Light
ޫBSDF
#4%'ͷد༩ʹԊͬͨॏతαϯϓϦϯά
#4%'د༩ʹԊͬͯೖࣹํαϯϓϧɿߴ͍֬Ͱߴ͍د༩
à͍ࢄ ޫݯ͕ྑ͍ॴʹ͋Ε
C =
1
N
N
i=1
f(¯
xi
)
pBSDF
(¯
xi
)
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Light
֦ࢄBSDF
#4%'ͷد༩ʹԊͬͨॏతαϯϓϦϯά
#4%'د༩ʹԊͬͯೖࣹํαϯϓϧɿ͍֬Ͱߴ͍د༩
àߴ͍ࢄ ͨ·ʹ͔ͨ͠Βͳ͍ͨΊ
C =
1
N
N
i=1
f(¯
xi
)
pBSDF
(¯
xi
)
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Light
֦ࢄBSDF
ޫݯ্ͷҐஔͷॏతαϯϓϦϯά
ޫݯ্ͷҐஔΛαϯϓϧͯ͠ଓɿߴ͍֬Ͱߴ͍د༩
à͍ࢄ #4%'ͷ͕ൺֱతҰ༷Ͱ͋Ε
C =
1
N
N
i=1
f(¯
xi
)
plight
(¯
xi
)
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Light
ޫBSDF
ޫݯ্ͷҐஔͷॏతαϯϓϦϯά
ޫݯ্ͷҐஔΛαϯϓϧͯ͠ଓɿ͍֬Ͱߴ͍د༩
àߴ͍ࢄ #4%'ͷ͕ඇҰ༷ͳͨΊ
C =
1
N
N
i=1
f(¯
xi
)
plight
(¯
xi
)
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ޫݯ໘ͷαϯϓϦϯά
#4%'ͷαϯϓϦϯά
n VEACH, E. 1997. Robust Monte Carlo methods for light transport simulation. PhD thesis, Stanford, CA, USA.
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ଟॏॏతαϯϓϦϯά
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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.*4ΣΠτ
Ұྫ: όϥϯεώϡʔϦεςΟοΫ
(ΣΠτ = ͦΕͧΕͷPDFͷॏΈ͖ฏۉ)
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)
(࣮ࡍʹPDFͷ୯ҐΛ߹ΘͤΔඞཁ͕͋Δ)
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όϥϯεώϡʔϦεςΟοΫɿཁ͢Δʹ
BSDF
ʮύε͖࣋ͬͯͨΑʯ
ʮਪఆ݁Ռʹࣗ৴ͷ΄Ͳʯ
BSDF
ʮʯ
ʮMJHIU͕Ծʹ͜ͷύεͭͬͨ͘ͳΒࣗ৴ʯ
lightʮʯ
ʮ͡Ό͋#4%'͕͖࣋ͬͯͨύεͷॏΈ
ɹ
ͬͯ͜ͱͰʯ
lightʮύε͖࣋ͬͯͨΑʯ
ʮਪఆ݁Ռʹࣗ৴ͷ΄Ͳʯ
lightʮʯ
ʮ#4%'͕Ծʹ͜ͷύεͭͬͨ͘ͳΒࣗ৴ʯ
BSDF
ʮʯ
ʮ͡Ό͋MJHIU͕͖࣋ͬͯͨύεͷॏΈ
ɹ
ͬͯ͜ͱͰʯ
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.*4ʹΑΔΣΠτ
.VMUJQMF*NQPSUBODF4BNQMJOH
n VEACH, E. 1997. Robust Monte Carlo methods for light transport simulation. PhD thesis, Stanford, CA, USA.
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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.*4ΣΠτ
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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.*4ΣΠτ
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DEPTH OF FIELD
ඃࣸքਂ
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ബϨϯζϞσϧΛ༻͍ͨ%P'දݱ
ཧɾखॱͷৄࡉ8ϖʔδࢀর
తʹड़Δͱʜ
xp
x0
x1
x1
x0
xp x1
γʔϯதͷҐஔ
ηϯαʔ໘
໘
Ϩϯζ্ͷɹΛαϯϓϧɺ
ϐΫηϧதͷҐஔɹʹରԠ͢Δ໘্ͷɹʹ͔ͬͯϨΠτϨʔε
ϨϯζҐஔͷαϯϓϧΛߦ͍ͬͯΔͷͰ1%'ɹɹɹͰد༩ΛׂΔ
pA
(x0
)
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࣮͢Δͱʜ
ϐϯϗʔϧ ബϨϯζϞσϧʹΑΔ%P'
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ͪΌΜͱϘέͯΔΈ͍͚ͨͩͲ
ϐϯϗʔϧͱ໌Δ͞ҧ͏͠ɺ
Ϩϯζͷେ͖͞ʹΑͬͯ໌Δ͞มΘΔʜ
ݱ࣮ͷΧϝϥͱҰॹ
Ϩϯζ͕େ͖͚ΕऔΓೖΕΔޫͷྔ͕૿͑ͯ໌Δ͘ͳΔ
àը૾ͷ໌Δ͞ΛҰఆʹอ͍ͪͨͷͳΒ
ηϯαʔͷײΛมߋ͢Δඞཁ͕͋Δ
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ηϯαʔͷײΛదʹઃఆ͢Δͱʜ
ϐϯϗʔϧ ബϨϯζϞσϧʹΑΔ%P'
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ͪΐͬͱͯ
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ϐϯϗʔϧΧϝϥബϨϯζϞσϧͰ
Ϩϯζ͕ۃݶʹখ͍͞߹ͱଊ͑Δ͜ͱ͕Ͱ͖Δ
à1%'ͷ͕ແݶͱͳΔͷͰɺ1%'ͰׂΔͱ݁Ռθϩ
Ϩϯζ͕େ͖͍ఔը૾͕໌Δ͍ͷͨΓલ
͡Ό͋ϐϯϗʔϧΧϝϥͷը૾ਅͬ҉ͳͷͰʜʁ
͋Δҙຯਖ਼ղ
ॳาతͳύετϨͷ࣮Ͱ
ͦͦϨϯζ໘ͷ1%'ͰׂͬͨΓ͍ͯ͠ͳ͍
à1%'ΛΩϟϯηϧ͢Δ͚ͩͷ
ແݶͷηϯαʔײ͕͋Δͷͱ͍͠
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͓͠·͍
*#-ͱ͔৭ʑ͔͚ͨͬͨ͠Ͳ
ͦͷ͏ͪ8ϖʔδʹॻ͖·͢
IUUQTSBZTQBDFYZ[$(