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Monte Carlo Path Tracing

COS 526, Fall 2016

Tom Funkhouser

Slides from Rusinkiewicz, Shirley

Outline

• Motivation

• Monte Carlo integration

• Variance reduction techniques

• Monte Carlo path tracing

• Sampling techniques

• Conclusion

Motivation

• Rendering = integration

– Antialiasing

– Soft shadows

– Indirect illumination

– Caustics

Motivation


– Antialiasing

– Soft shadows


– Caustics

Surface

Eye

Pixel

x

dAe)L(xLS

P

Motivation


– Antialiasing

– Soft shadows


– Caustics

Surface

Eye

Light

x

x’

dAxxGxxx)VxL(exxxxfe)(x,xL)wL(x,S

re ),(),(),,(

Motivation


– Antialiasing

– Soft shadows


– Caustics

Herf


re ),(),(),,(

Motivation


– Antialiasing

– Soft shadows


– Caustics

wdnw)w(x,Lwwxf)w(x,L)w(x,L ireo

)(),,(

Surface

Eye

Light

x

w

Surface

w’

Motivation


– Antialiasing

– Soft shadows


– Caustics

Debevec


)(),,(

Motivation


– Antialiasing

– Soft shadows


– Caustics

Diffuse Surface

Eye

Light

x

w

Specular

Surface

w’


)(),,(

Motivation


– Antialiasing

– Soft shadows


– Caustics

Jensen


)(),,(

Challenge

• Rendering integrals are difficult to evaluate

– Multiple dimensions

– Discontinuities

• Partial occluders

• Highlights

• Caustics

Drettakis


re ),(),(),,(

Challenge

• Rendering integrals are difficult to evaluate

– Multiple dimensions

– Discontinuities

• Partial occluders

• Highlights

• Caustics

Jensen


re ),(),(),,(

Outline

• Motivation





• Conclusion

Integration in 1D

x=1

f(x)

?)(

1

0

dxxf

Slide courtesy of

Peter Shirley

We can approximate

x=1

f(x)g(x)

1

0

1

0

)()( dxxgdxxf

Slide courtesy of

Peter Shirley

Or we can average

x=1

f(x)

E(f(x))

))(()(

1

0

xfEdxxf

Slide courtesy of

Peter Shirley

Estimating the average

x1

f(x)

xN

N

i

ixfN

dxxf1

1

0

)(1

)(

E(f(x))

Slide courtesy of

Peter Shirley

Other Domains

x=b

f(x)< f >ab

x=a

N

i

i

b

a

xfN

abdxxf

1

)()(

Slide courtesy of

Peter Shirley

Multidimensional Domains

• Same ideas apply for integration over …

– Pixel areas

– Surfaces

– Projected areas

– Directions

– Camera apertures

– Time

– Paths

N

i

i

UGLY

xfN

dxxf1

)(1

)(

Surface

Eye

Pixel

x

Efficiency?

x1 xN

E(f(x))

2

1

))](()([1

)( xfExfN

xfVarN

i

i

Efficiency?

x1 xN

E(f(x))

)(1

))(( xfVarN

xfEVar

Variance decreases as 1/N

Error decreases as 1/sqrt(N)

Outline

• Motivation





• Conclusion

Variance Reduction Techniques

• Stratified sampling

• Importance sampling

• Metropolis sampling

• Quasi-random

N

i

ixfN

dxxf1

1

0

)(1

)(

Stratified Sampling

• Estimate subdomains separately

x1 xN

Ek(f(x))

Arvo

Stratified Sampling

• This is still unbiased

M

k

ii

N

i

iN

FNN

xfN

F

1

1

1

)(1

x1 xN

Ek(f(x))

Stratified Sampling

• Less overall variance if

less variance in subdomains

M

k

iiN FVarNN

FVar1

2

1

x1 xN

Ek(f(x))

Variance Reduction Techniques

• Stratified sampling

• Importance sampling

• Metropolis sampling

• Quasi-random

N

i

ixfN

dxxf1

1

0

)(1

)(

Importance Sampling

• Put more samples where f(x) is bigger

)(

)(

1)(

1

i

ii

N

i

i

xp

xfY

YN

dxxf

x1 xN

E(f(x))

Importance Sampling

• This is still unbiased

x1 xN

E(f(x))

dxxf

dxxpxp

xf

dxxpxYYE i

)(

)()(

)(

)()(

for all N

Importance Sampling

• How do we draw samples with probability

proportional to function value?

Pro

babili

ty

0

1

Importance Sampling

• Sampling uniform distribution:

– Use random number generator

Pro

babili

ty

0

1

Importance Sampling

• Sampling specific probability distribution:

– Function inversion

– Rejection

Pro

babili

ty

0

1

Importance Sampling



– RejectionC

um

ula

tive

Pro

babili

ty

0

1

Importance Sampling



– RejectionC

um

ula

tive

Pro

babili

ty

0

1

y

Importance Sampling



– RejectionC

um

ula

tive

Pro

babili

ty

0

1

y

x

Importance Sampling



– RejectionC

um

ula

tive

Pro

babili

ty

0

1

Importance Sampling



– Rejection

Pro

babili

ty

0

1

Importance Sampling



– Rejection

Pro

babili

ty

0

1

x

x

x

x x

xx

x

x x

Combining Multiple PDFs

• Balance heuristic

– Use combination of samples generated for each PDF

– Number of samples for each PDF chosen by weights

– Near optimal

Outline

• Motivation





• Conclusion


• Integrate radiance

for each pixel

by sampling paths

randomly

Diffuse Surface

Eye

Light

x

Specular

Surface

Pixel


)(),,(

Monte Carlo Path Tracer

• For each pixel, repeat n times:

– Choose a ray with p=camera, d=(q,f) within pixel

– Pixel color += (1/n) * TracePath(p, d)

• Use stratified sampling

to select rays within

each pixel

N

i

i

UGLY

xfN

dxxf1

)(1

)(

Surface

Eye

Pixel

x

TracePath

• TracePath(p, d) returns (r,g,b):

– Trace ray (p, d) to find nearest intersection p’

– Sample radiance leaving p’ towards p

Surface

x p’

pd

TracePath

• Can sample radiance however we want, but

contribution weighted by 1/probability

)(

)(

1)(

1

i

ii

N

i

i

xp

xfY

YN

dxxf

x1 xN

E(f(x))



– If random() < pemit then

• Emitted:

return (1/ pemit) * (Lered, Legreen, Leblue)

• Reflected:

generate ray in random direction d’

return (1/ (1pemit)) * fr(d d’) * (nd’) * TracePath(p’, d’)

TracePath



– If Le = (0,0,0) then pemit = 0

else if fr = (0,0,0) then pemit = 1

else pemit = .9


• Emitted:


• Reflected:

generate ray in random direction d’

return (1/ (1pemit)) * fr(d d’) * (nd’) * TracePath(p’, d’)

TracePath

• Reflected case:

– Pick a light source

– Trace a ray towards that light

– Trace a ray anywhere except for that light

• Rejection sampling

– Divide by probabilities

• plight = 1/(solid angle of light) for ray to light source

• (1 – the above) for non-light ray

TracePath

• TracePath(p, d) returns (r,g,b):– Trace ray (p, d) to find nearest intersection p’



else pemit = .9


• Emitted:


• Reflected:

generate ray in random direction d’ towards a light

Lr = (1/2 *plight) * fr(d d’) * (nd’) * TracePath(p’, d’)

generate ray in random direction d’ not towards the light

Lr += (1/2*(1-plight)) * fr(d d’) * (nd’) * TracePath(p’, d’)

return (1/ (1pemit)) * Lr

TracePath

Reflected Ray Sampling

• Uniform directional sampling:

how to generate random ray on hemisphere?


• Option #1: rejection sampling

– Generate random numbers (x,y,z),

with x,y,z in –1..1

– If x2+y2+z2 > 1, reject

– Normalize (x,y,z)

– If pointing into surface (ray dot n < 0), flip


• Option #2: inversion method

– In polar coords, density must be proportional to sin q

(remember d(solid angle) = sin q dq df)

– Integrate, invert cos-1

• So, recipe is

– Generate f in 0..2p

– Generate z in 0..1

– Let q = cos-1 z

– (x,y,z) = (sin q cos f, sin q sin f, cos q)

BRDF Importance Sampling

• Better than uniform sampling:

importance sampling

• Because you divide by probability, ideally:

probability fr * cos qi

• [Lafortune, 1994]:

pp

ww n

sdoir

nkkxf cos

2

21),,(


• For cosine-weighted Lambertian:

– Density = cos q sin q

– Integrate, invert cos-1(sqrt)

• So, recipe is:



– Let q = cos-1 (sqrt(z))


• Phong BRDF: fr cosn where is angle

between outgoing ray and ideal mirror direction

• Constant scale = ks(n+2)/(2p)

• Ideally we would sample this times cos qi

– Difficult!

– Easier to sample BRDF itself, then multiply by cos qi

– That’s OK – still better than random sampling


• Recipe for sampling specular term:


– Let = cos-1 (z1/(n+1))


• This gives direction w.r.t. ideal mirror

direction


• Recipe for combining terms:

– r = random()

– If (r < kd) then

• d’ = sample diffuse direction

• weight = 1/kd

– else if (r < kd + ks) then

• d’ = sample specular direction

• weight = 1/ks

– else

• terminate ray

• TracePath(p, d) returns (r,g,b):– Trace ray (p, d) to find nearest intersection p’



else pemit = .9


• Emitted:


• Reflected:

generate ray in random direction d’ towards a light

Lr = (1/2 *plight) * fr(d d’) * (nd’) * TracePath(p’, d’)

generate ray in random direction d’ not towards the light

Lr += (1/2*(1-plight)) * fr(d d’) * (nd’) * TracePath(p’, d’)

return (1/ (1pemit)) * Lr

Recap


• Advantages

– Any type of geometry (procedural, curved, ...)

– Any type of BRDF (specular, glossy, diffuse, ...)

– Samples all types of paths (L(SD)*E)

– Accuracy controlled at pixel level

– Low memory consumption

– Unbiased - error appears as noise in final image

• Disadvantages

– Slow convergence

– Noise in final image


Big diffuse light source, 20 minutes

Jensen


1000 paths/pixel

Jensen

Summary

• Monte Carlo Integration Methods

– Very general

– Good for complex functions with high dimensionality

– Converge slowly (but error appears as noise)

• Conclusion

– Preferred method for difficult scenes

– Noise removal (filtering) and

irradiance caching (photon maps)

used in practice

More Information

• Books– Realistic Ray Tracing, Peter Shirley

– Realistic Image Synthesis Using Photon Mapping, Henrik Wann Jensen

• Theses– Robust Monte Carlo Methods for Light Transport Simulation, Eric Veach

– Mathematical Models and Monte Carlo Methods for Physically Based

Rendering, Eric La Fortune

• Course Notes– Mathematical Models for Computer Graphics, Stanford, Fall 1997

– State of the Art in Monte Carlo Methods for Realistic Image Synthesis,

Course 29, SIGGRAPH 2001

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