path tracing computer graphics path tracing 2009 fabio pellacini 1

(1)

path tracing

(2)

path tracing

Monte Carlo algorithm for solving the rendering

equation

(3)

solving rendering equation

•  advantages

–  predictive simulation

•  can be used for architecture, engineering, …

–  photorealistic

•  if simulation if correct, images will look real

•  disadvantages

–  (really) slow

•  simulation of physics is computationally very expensive

–  need accurate geometry, materials and lights

(4)

rendering equation

[Bala]

x x

(5)

basic path tracing

•  need to evaluate radiance at x in direction Θ

•  determine visible point

•  look up emission

_la^]

(6)

basic path tracing

•  need to evaluate

•  use Monte Carlo estimation

(7)

basic path tracing

•  generate random direction Ψ

_i

with p(Ψ

_i

)

–  evaluate BRDF –  evaluate cosine –  evaluate

la]

(8)

basic path tracing

•  need to evaluate

•  determine visible point from x in direction Ψ

_i

•  in vacuum

[Bala]

(9)

basic path tracing

•  need to evaluate

•  recursively execute procedure

la]

(10)

russian roulette

•  when to stop recursion?

–  light bounce infinitely in the environment –  but every bounce has less energy

•  in many cases 3-4 bounces are enough

[Bala]

(11)

russian roulette

•  stop after k bounces

–  introduces bias (consistent error) in the solution –  need to make k large to capture all cases

•  Monte Carlo strategy (Russian Roulette) as stopping

criterion

–  pick a probability with which to stop the path –  at each intersection, test the path

(12)

russian roulette

[Dutré, Bekaert, Bala]

(13)

russian roulette

(14)

russian roulette

•  if f is a recursive integral, only continue with probability

•  but weight back the radiance

•  example: path tracing with

–  only 1 chance in 10 the ray will continue

–  estimate radiance of the ray multiplies by 10

–  intuition: instead of shooting 10 rays, shoot 1 but weight its contribution 10 times

(15)

russian roulette

•  how to choose the probability?

–  small fixed value: longer paths

•  slow but accurate

–  big fixed value: shorter paths

•  fast but inaccurate

–  proportional to integral of reflected light

•  adapts to material properties

•  darker patches will statistically shorten paths

•  exactly like in physics

(16)

basic path tracing pseudocode

computeImage()

foreach pixel (i,j)

estimatedRadiance[i,j] = 0 for s = 1 to #viewSamples generate Q in pixel (i,j) theta = (Q – E)/|Q-E|

x = trace(E,theta)

estimatedRadiance [i,j] +=

computeRadiance(x,-theta)

estimatedRadiance [i,j] /= #viewSamples computeRadiance(x, theta)

estimatedRadiance = basicPT(x, theta) return estimatedRadiance

(17)

basic path tracing pseudocode

basicPT(x, theta)

estimatedRadiance = Le(x, theta) if(not absorbed) // russian roulette for s = 1 to #radianceSamples

psi = generate random dir on hemisphere y = trace(x, psi)

estimatedRadiance +=

basicPT(y,-psi) * BRDF(x,psi,theta) * cos(Nx,psi) / pdf(psi)

estimatedRadiance /= #paths ^[Dutr

é, Bekaert, Bala]

(18)

basic path tracing intuition

(19)

basic path tracing intuition

(20)

basic path tracing performance

[Bala]

1 sample 16 samples 256 samples

(21)

monte carlo vs. deterministic integration

Monte Carlo Deterministic

[Bala]

(22)

next event estimation

•  in basic path tracing

–  if path does not hit a light, its radiance is 0

–  unlikely to hit a light by randomly picking dirs.

•  next event estimation

–  want to directly sample light sources

–  by splitting direct and indirect illumination estimation

•  two separate Monte Carlo processes

–  by using area formulation for direct illumination –  by using hemispherical formulation for indirect

(23)

direct and indirect illum. formulation

[Bala]

(24)

direct and indirect illum. formulation

[Bala]

(25)

direct and indirect illum. formulation

(26)

direct illum. – hemisphere sampling

(27)

direct illum. – area sampling

(28)

indirect illum. – hemisphere sampling

discard Le if ray hits light

(29)

indirect illum. – recursive evaluation

(30)

next event estimation performance

[Bala]

16 samples

without with

(31)

next event estimation performance

[Bala]

1 sample 4 samples

256 samples

(32)

direct illumination – one light

•  depends on

–  emitted radiance distribution Le –  how to pick points y on the light –  how many points to use

•  number of shadow rays

(33)

direct illumination – one light

•  each light type has its own physical models

–  angular distribution defines different light types –  flood, fill, spot, ect…

•  simplest model: emitted radiance is a constant

(34)

direct illumination – one light

•  uniform sampling of light area

–  simply sample that [0,1] square and rescale –  works fairly well in practice

–  slightly better techniques exists tough

(35)

direct illumination – many lights

(36)

direct illumination – many lights

•  how to allocate samples between different lights

–  various techniques, some quite advanced

(37)

direct illumination – many lights

•  split samples uniformly between lights

–  same as M light integrals with previous sampling

–  simple but inefficient

(38)

indirect illumination

•  depends on how to sample the hemisphere

–  uniform distribution

–  importance sampling: pick p to match integral

•  cosine distribution

•  BRDF distribution

•  BRDF*cosine distribution

(39)

indirect illumination – uniform dist.

(40)

indirect illumination – cosine dist.

[Bala]

(41)

indirect illumination – BRDF Dist.

(42)

**indirect illumination – BRDF*Cosine Dist.**

[Bala]

(43)

importance sampling performance

[Bala]

without with

(44)

pt pseudocode – pixel sampling

computeImage()

foreach pixel (i,j)

estimatedRadiance[i,j] = 0 for s = 1 to #viewSamples generate Q in pixel (i,j) theta = (Q – E)/|Q-E|

x = trace(E,theta)

estimatedRadiance [i,j] +=

computeRadiance(x,-theta)

estimatedRadiance [i,j] /= #viewSamples _[Dutr

é, Bekaert, Bala]

(45)

pt pseudocode – radiance estimation

computeRadiance(x,theta)

estimatedRadiance = Le(x,theta)

estimatedRadiance += directIllumination(x, theta) estimatedRadiance += indirectIllumination(x, theta) return estimatedRadiance

(46)

pt pseudocode – direct illumination

directIllumination(x,theta) estimatedRadiance = 0 for s = 1 to #shadowRays k = pick random light

y = generate random point on light k psi = (x-y) / |x-y|

estimatedRadiance += Le_k(y,-psi) *

BRDF(x,psi,tetha) * G(x,y) * V(x,y) /

(p(k)*p(y|k)) estimateRadiance /= #shadowRays

return estimatedRadiance

(47)

pt pseudocode – direct illumination UL

directIllumination(x,theta) estimatedRadiance = 0 for k = 1 to #lights

for s = 1 to #shadowRays / #lights

y = generate random point on light k psi = (x-y) / |x-y|

estimatedRadiance += Le_k(y,-psi) *

BRDF(x,psi,tetha) * G(x,y) * V(x,y) / p(y)

estimateRadiance /= #shadowRays _[Dutr

é, Bekaert, Bala]

(48)

pt pseudocode – indirect illumination

indirectIllumination(x,theta) estimatedRadiance = 0

if(not absorbed) // russian roulette for s = 1 to #indirectSamples

psi = generate random dir on hemisphere y = trace(x, psi)

estimatedRadiance +=

computeRadiance(y,-psi) * BRDF(x,psi,theta) *

cos(Nx,psi) / pdf(psi)

estimatedRadiance /= #indirectSamples return estimatedRadiance /(1-absorption)

(49)

beyond path tracing

bidirectional techniues

(50)

path tracing

•  perfectly accurate, but slow to converge

–  noise remains in the image for a long time

•  intuition: is there are bright reflections, we cannot

sample them directly

–  halogen lamps

•  idea: shoot paths from the eye and from the light

–  eye paths: work well for reflections –  light paths: pick up secondary sources –  in reality very complex

(51)

bidirectional path tracing

(52)

bidirectional path tracing

(53)

photon mapping

(54)

path tracing computer graphics path tracing 2009 fabio pellacini 1

path tracing

path tracing

Monte Carlo algorithm for solving the rendering

equation

solving rendering equation

• advantages

• disadvantages

rendering equation

x x

basic path tracing

• need to evaluate radiance at x in direction Θ

• determine visible point

• look up emission

basic path tracing

• need to evaluate

• use Monte Carlo estimation

basic path tracing

• generate random direction Ψ

with p(Ψ

)

basic path tracing

• need to evaluate

• determine visible point from x in direction Ψ

• in vacuum

basic path tracing

• need to evaluate

• recursively execute procedure

russian roulette

• when to stop recursion?

russian roulette

• stop after k bounces

• Monte Carlo strategy (Russian Roulette) as stopping

criterion

russian roulette

russian roulette

russian roulette

• if f is a recursive integral, only continue with probability

• but weight back the radiance

• example: path tracing with

russian roulette

• how to choose the probability?

basic path tracing pseudocode

basic path tracing pseudocode

basic path tracing intuition

basic path tracing intuition

basic path tracing performance

monte carlo vs. deterministic integration

next event estimation

• in basic path tracing

• next event estimation

direct and indirect illum. formulation

direct and indirect illum. formulation

direct and indirect illum. formulation

direct illum. – hemisphere sampling

direct illum. – area sampling

indirect illum. – hemisphere sampling

indirect illum. – recursive evaluation

next event estimation performance

next event estimation performance

direct illumination – one light

• depends on

direct illumination – one light

• each light type has its own physical models

• simplest model: emitted radiance is a constant

direct illumination – one light

• uniform sampling of light area

direct illumination – many lights

direct illumination – many lights

• how to allocate samples between different lights

direct illumination – many lights

• split samples uniformly between lights

indirect illumination

• depends on how to sample the hemisphere

indirect illumination – uniform dist.

indirect illumination – cosine dist.

indirect illumination – BRDF Dist.

indirect illumination – BRDF*Cosine Dist.

importance sampling performance

pt pseudocode – pixel sampling

•  advantages

•  disadvantages

•  need to evaluate radiance at x in direction Θ

•  determine visible point

•  look up emission

•  need to evaluate

•  use Monte Carlo estimation

•  generate random direction Ψ

•  need to evaluate

•  determine visible point from x in direction Ψ

•  in vacuum

•  need to evaluate

•  recursively execute procedure

•  when to stop recursion?

•  stop after k bounces

•  Monte Carlo strategy (Russian Roulette) as stopping

•  if f is a recursive integral, only continue with probability

•  but weight back the radiance

•  example: path tracing with

•  how to choose the probability?

•  in basic path tracing

•  next event estimation

•  depends on

•  each light type has its own physical models

•  simplest model: emitted radiance is a constant

•  uniform sampling of light area

•  how to allocate samples between different lights

•  split samples uniformly between lights

•  depends on how to sample the hemisphere

**indirect illumination – BRDF*Cosine Dist.**

•  perfectly accurate, but slow to converge

•  intuition: is there are bright reflections, we cannot

•  idea: shoot paths from the eye and from the light