Integral formulations of volumetric transmittancewjarosz/publications/georgiev...Integral...

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Derek Nowrouzezahrai Jaroslav Křivánek Wojciech Jarosz Iliyan Georgiev Zackary Misso Toshiya Hachisuka Integral formulations of volumetric transmittance

Transcript of Integral formulations of volumetric transmittancewjarosz/publications/georgiev...Integral...

Page 1: Integral formulations of volumetric transmittancewjarosz/publications/georgiev...Integral formulations of volumetric transmittance 7 T (a,b)

Derek Nowrouzezahrai Jaroslav Křivánek Wojciech Jarosz

Iliyan Georgiev Zackary Misso Toshiya Hachisuka

Integral formulations of volumetric transmittance

Page 2: Integral formulations of volumetric transmittancewjarosz/publications/georgiev...Integral formulations of volumetric transmittance 7 T (a,b)

Mahesh Kularatne

Sean Svadilfari

Page 3: Integral formulations of volumetric transmittancewjarosz/publications/georgiev...Integral formulations of volumetric transmittance 7 T (a,b)

Non-linear sphere tracing for rendering deformed signed distance fields

Rendering volumetrics

�3Integral formulations of volumetric transmittance

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Non-linear sphere tracing for rendering deformed signed distance fields

Rendering volumetrics

�4Integral formulations of volumetric transmittance

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Non-linear sphere tracing for rendering deformed signed distance fields

Rendering volumetrics

�5Integral formulations of volumetric transmittance

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Non-linear sphere tracing for rendering deformed signed distance fields

Rendering volumetrics

�6Integral formulations of volumetric transmittance

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Non-linear sphere tracing for rendering deformed signed distance fields

Rendering volumetrics

�7Integral formulations of volumetric transmittance

T (a, b)<latexit 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Non-linear sphere tracing for rendering deformed signed distance fields

Transmittance

�8Integral formulations of volumetric transmittance

a b

T (a, b) =L(a)

L(b)<latexit 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Page 9: Integral formulations of volumetric transmittancewjarosz/publications/georgiev...Integral formulations of volumetric transmittance 7 T (a,b)

Non-linear sphere tracing for rendering deformed signed distance fields

T (a, b) =L(a)

L(b)<latexit 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Transmittance

�9Integral formulations of volumetric transmittance

a b

⌧ =

Z b

aµ(x)dx

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T (a, b) = e�⌧<latexit 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Page 10: Integral formulations of volumetric transmittancewjarosz/publications/georgiev...Integral formulations of volumetric transmittance 7 T (a,b)

Non-linear sphere tracing for rendering deformed signed distance fields

Extinction

�10Integral formulations of volumetric transmittance

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a b

⌧ =

Z b

aµ(x)dx

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T (a, b) = e�⌧<latexit 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Non-linear sphere tracing for rendering deformed signed distance fields

Previous work

�11Integral formulations of volumetric transmittance

µ(x)<latexit 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bT (a, b) = ?<latexit 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Page 12: Integral formulations of volumetric transmittancewjarosz/publications/georgiev...Integral formulations of volumetric transmittance 7 T (a,b)

Non-linear sphere tracing for rendering deformed signed distance fields

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Previous work

�12Integral formulations of volumetric transmittance

Delta tracking*

bT (a, b) = ?<latexit 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*[Woodcock et al. 1965]

Page 13: Integral formulations of volumetric transmittancewjarosz/publications/georgiev...Integral formulations of volumetric transmittance 7 T (a,b)

Non-linear sphere tracing for rendering deformed signed distance fields

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Previous work

�13Integral formulations of volumetric transmittance

a b

Delta tracking*

bT (a, b) = ?<latexit 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*[Woodcock et al. 1965]

Page 14: Integral formulations of volumetric transmittancewjarosz/publications/georgiev...Integral formulations of volumetric transmittance 7 T (a,b)

Non-linear sphere tracing for rendering deformed signed distance fields

E[ bTDT (a, b)] = e�⌧<latexit 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µ(x)<latexit 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Previous work

�14Integral formulations of volumetric transmittance

a b

bTDT (a, b)<latexit 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sha1_base64="6c86WeDHw8ebvOAciOn5mgpdhvI=">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</latexit><latexit sha1_base64="6c86WeDHw8ebvOAciOn5mgpdhvI=">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</latexit>

Delta tracking*

bT (a, b) = ?<latexit 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*[Woodcock et al. 1965]

Page 15: Integral formulations of volumetric transmittancewjarosz/publications/georgiev...Integral formulations of volumetric transmittance 7 T (a,b)

Non-linear sphere tracing for rendering deformed signed distance fields

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Naive Monte Carlo

�15Integral formulations of volumetric transmittance

⌧ =

Z b

aµ(x)dx

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µ(x3)<latexit 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b⌧ =kX

j=0

µ(xj)

p(xj)<latexit 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Page 16: Integral formulations of volumetric transmittancewjarosz/publications/georgiev...Integral formulations of volumetric transmittance 7 T (a,b)

Non-linear sphere tracing for rendering deformed signed distance fields

Our contributions

�16Integral formulations of volumetric transmittance

T (a, b) = e�⌧<latexit 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• 3 integral formulations • Rendering analogies • MC design decisions

T (a, b) =

Z. . .

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Page 17: Integral formulations of volumetric transmittancewjarosz/publications/georgiev...Integral formulations of volumetric transmittance 7 T (a,b)

Non-linear sphere tracing for rendering deformed signed distance fields

Deriving an integral formulation

�17Integral formulations of volumetric transmittance

Z b

a�dL(x)

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Z b

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L(a)� L(b) =

Z b

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T (a, b) = 1 +

Z b

a�µ(x)T (x, b)dx

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�dL(x)

dx= �µ(x)L(x)

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Page 18: Integral formulations of volumetric transmittancewjarosz/publications/georgiev...Integral formulations of volumetric transmittance 7 T (a,b)

Non-linear sphere tracing for rendering deformed signed distance fields

First integral formulation

�18Integral formulations of volumetric transmittance

T (a, b) = 1 +

Z b

a�µ(x)T (x, b)dx

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T (a, b) = 1 +

Z b

a�µ(x)T (x, b)dx

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T (a, b) = e�R ba µ(x)dx

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Page 19: Integral formulations of volumetric transmittancewjarosz/publications/georgiev...Integral formulations of volumetric transmittance 7 T (a,b)

Non-linear sphere tracing for rendering deformed signed distance fields

Visualizing an estimator

�19Integral formulations of volumetric transmittance

a bT (a, b) = 1 +

Z b

a�µ(x)T (x, b)dx

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Page 20: Integral formulations of volumetric transmittancewjarosz/publications/georgiev...Integral formulations of volumetric transmittance 7 T (a,b)

Non-linear sphere tracing for rendering deformed signed distance fields

Visualizing an estimator

�20Integral formulations of volumetric transmittance

a bT (a, b) = 1 +

Z b

a�µ(x)T (x, b)dx

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sha1_base64="xHhmqj1dfl4A3elOrVzeTfuAOC4=">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</latexit><latexit sha1_base64="xHhmqj1dfl4A3elOrVzeTfuAOC4=">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</latexit><latexit sha1_base64="xHhmqj1dfl4A3elOrVzeTfuAOC4=">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</latexit>x

extinction lookup

ybT (x, b) = 1 +

�µ(y) bT (y, b)p(y)

<latexit 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bT (a, b) = 1 +�µ(x) bT (x, b)

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Page 21: Integral formulations of volumetric transmittancewjarosz/publications/georgiev...Integral formulations of volumetric transmittance 7 T (a,b)

Non-linear sphere tracing for rendering deformed signed distance fields

Analogy to previous work

�21Integral formulations of volumetric transmittance

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a bDelta trackingT (a, b) = 1 +

Z b

a�µ(x)T (x, b)dx

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Delta tracking Ratio tracking*

*[Novák et al. 2014]

Page 22: Integral formulations of volumetric transmittancewjarosz/publications/georgiev...Integral formulations of volumetric transmittance 7 T (a,b)

Non-linear sphere tracing for rendering deformed signed distance fields

Analogy to rendering

�22Integral formulations of volumetric transmittance

a b

T (a, b) = 1 +

Z b

a�µ(x)T (x, b)dx

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a

b

Path Tracing (Fredholm)Path Tracing*Volterra L(!) = Le(!) +

Z

⌦fr(!j)cos(✓)L(!j)d!j

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extinction lookup

BRDF evaluations

*[Kajiya 1986]

Page 23: Integral formulations of volumetric transmittancewjarosz/publications/georgiev...Integral formulations of volumetric transmittance 7 T (a,b)

Non-linear sphere tracing for rendering deformed signed distance fields

Better estimators?

�23Integral formulations of volumetric transmittance

a

b

Path Tracing (Fredholm)

a b

Volterra Path Tracing w/ NEE (Fredholm)

Next flight*

*[Cramer 1978]

Page 24: Integral formulations of volumetric transmittancewjarosz/publications/georgiev...Integral formulations of volumetric transmittance 7 T (a,b)

Non-linear sphere tracing for rendering deformed signed distance fields

So far

�24Integral formulations of volumetric transmittance

• Volterra integral formulation • Previous work • Unidirectional path tracing analogy

Page 25: Integral formulations of volumetric transmittancewjarosz/publications/georgiev...Integral formulations of volumetric transmittance 7 T (a,b)

Non-linear sphere tracing for rendering deformed signed distance fields

Other integral formulations

�25Integral formulations of volumetric transmittance

T (a, b) = 1 +

Z b

a�µ(x)T (x, b)dx

<latexit 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L(!) = Le(!) +

Z

⌦fr(!j)cos(✓)L(!j)d!j

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Volterra Rendering Equation

Page 26: Integral formulations of volumetric transmittancewjarosz/publications/georgiev...Integral formulations of volumetric transmittance 7 T (a,b)

Non-linear sphere tracing for rendering deformed signed distance fields

Other integral formulations

�26Integral formulations of volumetric transmittance

T (a, b) = 1 +

Z b

a�µ(x)T (x, b)dx

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L(!) = Le(!) +

Z

⌦fr(!j)cos(✓)L(!j)d!j

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Volterra Rendering Equation

Ij =1X

k=0

JkLe

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Neumann Series*

*[Veach 1997]

Page 27: Integral formulations of volumetric transmittancewjarosz/publications/georgiev...Integral formulations of volumetric transmittance 7 T (a,b)

Non-linear sphere tracing for rendering deformed signed distance fields

Other integral formulations

�27Integral formulations of volumetric transmittance

Neumann Series*

T (a, b) = 1 +

Z b

a�µ(x)T (x, b)dx

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L(!) = Le(!) +

Z

⌦fr(!j)cos(✓)L(!j)d!j

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Volterra Rendering Equation

Ij =1X

k=0

JkLe

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Power-seriesT (a, b) =

1X

k=0

(�⌧)k1

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*[Veach 1997]

Page 28: Integral formulations of volumetric transmittancewjarosz/publications/georgiev...Integral formulations of volumetric transmittance 7 T (a,b)

Non-linear sphere tracing for rendering deformed signed distance fields

Power series formulation

�28Integral formulations of volumetric transmittance

Single-termPrefix-sum

T (a, b) =1X

k=0

(�⌧)k1

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Page 29: Integral formulations of volumetric transmittancewjarosz/publications/georgiev...Integral formulations of volumetric transmittance 7 T (a,b)

Non-linear sphere tracing for rendering deformed signed distance fields

Other integral formulations

�29Integral formulations of volumetric transmittance

Neumann Series

T (a, b) = 1 +

Z b

a�µ(x)T (x, b)dx

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L(!) = Le(!) +

Z

⌦fr(!j)cos(✓)L(!j)d!j

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Volterra Rendering Equation

Ij =1X

k=0

JkLe

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Power-seriesT (a, b) =

1X

k=0

(�⌧)k1

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Path Integral*Ij =

Z

⌦f(x)dµ(x)

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Page 30: Integral formulations of volumetric transmittancewjarosz/publications/georgiev...Integral formulations of volumetric transmittance 7 T (a,b)

Non-linear sphere tracing for rendering deformed signed distance fields

Other integral formulations

�30Integral formulations of volumetric transmittance

Neumann Series

T (a, b) = 1 +

Z b

a�µ(x)T (x, b)dx

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L(!) = Le(!) +

Z

⌦fr(!j)cos(✓)L(!j)d!j

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Volterra Rendering Equation

Ij =1X

k=0

JkLe

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Power-seriesT (a, b) =

1X

k=0

(�⌧)k1

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Path Integral*Ij =

Z

⌦f(x)dµ(x)

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HypercubeT (a, b) =

Z

H

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1997]

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Non-linear sphere tracing for rendering deformed signed distance fields

Hypercube integral formulation

�31Integral formulations of volumetric transmittance

a

a

b

b

Hypercube (bidirectional MIS) Bidirectional path tracing*

*[Veach and Guibas 1994]

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Non-linear sphere tracing for rendering deformed signed distance fields

Estimator summary

�32Integral formulations of volumetric transmittance

Track-lengthRatio trackingNext-flight

Unidirectional-MISBidirectional-MIS

P-series ratioP-series next-flightP-series cumulativeP-series CMF

Volterra

Power series

Hypercube

Ours are in green

Z

H

f(x)dx<latexit 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Non-linear sphere tracing for rendering deformed signed distance fields

Results

�33Integral formulations of volumetric transmittance

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Non-linear sphere tracing for rendering deformed signed distance fields

Majorant / upper control

�34Integral formulations of volumetric transmittance

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Tight upper control Loose upper control

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Tight upper control

�35Integral formulations of volumetric transmittance

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Ratio tracking Next-flight

P-CMF

RMSE: 1.3e-2 RMSE: 3.6e-3

RMSE: 1.7e-6Reference

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Loose upper control

�37Integral formulations of volumetric transmittance

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Ratio tracking Next-flight

P-CMF

RMSE: 1.1e-2 RMSE: 1.0e-1

RMSE: 1.1e-2Reference

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Non-linear sphere tracing for rendering deformed signed distance fields

Non-bounding upper controls

�39Integral formulations of volumetric transmittance

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Ratio tracking (1x max) P-series Cumulative (1x max)

P-series Cumulative (0.5x max)

RMSE: 1.0e-2 RMSE: 8.3e-3

RMSE: 8.6e-3Ratio tracking (0.5x max)

RMSE: 2.7e-2

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Non-linear sphere tracing for rendering deformed signed distance fields

Stratification

�41Integral formulations of volumetric transmittance

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Ratio tracking (independent) P-series ratio (independent)

P-series ratio (stratified)

RMSE: 1.4e-2 RMSE: 1.4e-2

RMSE: 6.4e-3Ratio tracking (stratified)

RMSE: 7.9e-3

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Non-linear sphere tracing for rendering deformed signed distance fields

Best estimator?

�43Integral formulations of volumetric transmittance

• There is no “best” estimator • Loose controls - ratio • Non-bounding controls - power series cumulative • Other cases - power series CMF

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Conclusion

�44Integral formulations of volumetric transmittance

• Integral formulation for transmittance • Allows for flexible estimator design • Re-formulated previous work • 4 novel estimators

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Non-linear sphere tracing for rendering deformed signed distance fields

Future work

�45Integral formulations of volumetric transmittance

• Non-exponential media • Application of the different integral formulations to

sample scattering events • More MIS estimators

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Thank you!• Please visit: http://zackmisso.com • For the paper and supplemental

• Code repository (GitHub): • ZackMisso/TransmittanceEstimation

[email protected] [email protected] Misso Iliyan Georgiev