Monte Carlo Methods for Physically Based Volume Rendering - - PowerPoint PPT Presentation

Wojciech Jarosz

wjarosz@dartmouth.edu

Monte Carlo Methods for
 Physically Based Volume Rendering

Transmittance estimation

Transmittance estimation

y

x

T( x , y)

| {z }

T(x↔y)

Transmittance estimation

y

x

T( x , y)

| {z }

T(x↔y)

y

x

Transmittance estimation

T( x , y)

| {z }

T(x↔y)

y

x

T(x, y) = e−τ(x,y)

transmittance

Transmittance estimation

T( x , y)

| {z }

T(x↔y)

y

x

T(x, y) = e−τ(x,y)

transmittance

Transmittance estimation

T( x , y)

| {z }

T(x↔y)

y

x

T(x, y) = e−τ(x,y)

τ(x, y) =

Z t

0 µt(x − sω) ds

transmittance

Transmittance estimation

“optical thickness”

ω

T( x , y)

| {z }

T(x↔y)

y

x

T(x, y) = e−τ(x,y)

τ(x, y) =

Z t

0 µt(x − sω) ds

transmittance

Transmittance estimation

“optical thickness”

ω

T( x , y)

| {z }

T(x↔y)

y

x

T(x, y) = e−τ(x,y)

τ(x, y) =

Z t

0 µt(x − sω) ds

transmittance

Transmittance estimation

“optical thickness” | {z }

T(x↔y)

kx yk = t

| {z }

T(x↔y)

kx yk = t

T( x , y)

| {z }

T(x↔y)

Transmittance estimation

• 1. Estimators integrating optical thickness

y

x

T(x, y) = e−τ(x,y)

τ(x, y) =

Z t

0 µt(x − sω) ds

transmittance “optical thickness”

| {z }

T(x↔y)

kx yk = t

T( x , y)

| {z }

T(x↔y)

Transmittance estimation

• 1. Estimators integrating optical thickness
• 2. Estimators using free-flight sampling

y

x

T(x, y) = e−τ(x,y)

τ(x, y) =

Z t

0 µt(x − sω) ds

transmittance “optical thickness”

| {z }

T(x↔y)

kx yk = t

T( x , y)

| {z }

T(x↔y)

Transmittance estimation

• 1. Estimators integrating optical thickness
• 2. Estimators using free-flight sampling
• 3. Estimators using null collisions

y

x

T(x, y) = e−τ(x,y)

τ(x, y) =

Z t

0 µt(x − sω) ds

transmittance “optical thickness”

Homogeneous medium

| {z }

T(x↔y)

kx yk = t

T( x , y)

| {z }

T(x↔y)

µt

y

x

τ(x, y) =

Z t

0 µt(x − sω) ds

T(x, y) = e−τ(x,y)

Homogeneous medium

| {z }

T(x↔y)

kx yk = t

T( x , y)

| {z }

T(x↔y)

µt

y

x

τ(x, y) =

Z t

0 µt(x − sω) ds

T(x, y) = e−τ(x,y)

µt t

=

Homogeneous medium

| {z }

T(x↔y)

kx yk = t

T( x , y)

| {z }

T(x↔y)

µt

y

x

T(x, y) = e−τ

µt t

Homogeneous medium

| {z }

T(x↔y)

kx yk = t

T( x , y)

| {z }

T(x↔y)

µt

y

x

T(x, y) = e−τ

µt t

Homogeneous medium

| {z }

T(x↔y)

kx yk = t

T( x , y)

| {z }

T(x↔y)

µt

y

x

T(x, y) = e−τ

µt t

: “Expected Value” estimator

hT(t)iEV

Piecewise homogeneous medium

| { z }

∆1

| {z }

∆2

| {z }

T(x↔y)

kx yk = t

µt,1

µt,2

y

x

Piecewise homogeneous medium

| { z }

∆1

| {z }

∆2

| {z }

T(x↔y)

kx yk = t

h iEV h

1 iEV · h 2 iEV

• t = ∆1 + ∆2

µt,1

µt,2

y

x

Piecewise homogeneous medium

T(t) = e−τ(t) = e−(τ1+τ2) = e−(µt,1∆1+µt,2∆2)

| { z }

∆1

| {z }

∆2

| {z }

T(x↔y)

kx yk = t

h iEV h

1 iEV · h 2 iEV

• t = ∆1 + ∆2

µt,1

µt,2

y

x

Voxelize medium

T(t) = e−τ(t) = e−(τ1+τ2) = e−(µt,1∆1+µt,2∆2)

Voxelized medium (piecewise const.)

T(t) = e−τ(t) = e− ∑k

i τi = e− ∑k i µt,i ∆i

Voxelized medium (piecewise const.)

T(t) = e−τ(t) = e− ∑k

i τi = e− ∑k i µt,i ∆i

Regular tracking (piecewise constant)

T(t) = e−τ(t) = e− ∑k

i τi = e− ∑k i µt,i ∆i

τ1

Regular tracking (piecewise constant)

T(t) = e−τ(t) = e− ∑k

i τi = e− ∑k i µt,i ∆i

τ2

Regular tracking (piecewise constant)

T(t) = e−τ(t) = e− ∑k

i τi = e− ∑k i µt,i ∆i

τ3

Regular tracking (piecewise constant)

T(t) = e−τ(t) = e− ∑k

i τi = e− ∑k i µt,i ∆i

τ4

Regular tracking (piecewise constant)

T(t) = e−τ(t) = e− ∑k

i τi = e− ∑k i µt,i ∆i

τ5

Regular tracking (piecewise constant)

T(t) = e−τ(t) = e− ∑k

i τi = e− ∑k i µt,i ∆i

τ6

Regular tracking (piecewise constant)

T(t) = e−τ(t) = e− ∑k

i τi = e− ∑k i µt,i ∆i

τ7

Regular tracking (piecewise constant)

T(t) = e−τ(t) = e− ∑k

i τi = e− ∑k i µt,i ∆i

τ8

Regular tracking (piecewise constant)

T(t) = e−τ(t) = e− ∑k

i τi = e− ∑k i µt,i ∆i

τ9

Regular tracking (piecewise constant)

T(t) = e−τ(t) = e− ∑k

i τi = e− ∑k i µt,i ∆i

τ10

Regular tracking (piecewise constant)

T(t) = e−τ(t) = e− ∑k

i τi = e− ∑k i µt,i ∆i

τ11

Regular tracking (piecewise constant)

T(t) = e−τ(t) = e− ∑k

i τi = e− ∑k i µt,i ∆i

Regular tracking (piecewise constant)

T(t) = e−τ(t) = e− ∑k

i τi = e− ∑k i µt,i ∆i

Regular tracking (bi-linear/tri-linear)

T(t) = e−τ(t) = e− ∑k

i τi = e

Regular tracking (bi-linear/tri-linear)

T(t) = e−τ(t) = e− ∑k

i τi = e

analytic, but more complicated

Regular tracking (bi-linear/tri-linear)

T(t) = e−τ(t) = e− ∑k

i τi =

analytic, but more complicated

Regular tracking

✔ Integrates optical depth/transmittance in closed form

Regular tracking

✔ Integrates optical depth/transmittance in closed form ✔ Analytic/unbiased!

Regular tracking

✔ Integrates optical depth/transmittance in closed form ✔ Analytic/unbiased! ✘ Only for piecewise “simple” media

Regular tracking

✔ Integrates optical depth/transmittance in closed form ✔ Analytic/unbiased! ✘ Only for piecewise “simple” media ✘ No complex procedurals

Regular tracking

✔ Integrates optical depth/transmittance in closed form ✔ Analytic/unbiased! ✘ Only for piecewise “simple” media ✘ No complex procedurals ✘ Discretization artifacts

Regular tracking

✔ Integrates optical depth/transmittance in closed form ✔ Analytic/unbiased! ✘ Only for piecewise “simple” media ✘ No complex procedurals ✘ Discretization artifacts ✘ Expensive

Regular tracking

✔ Integrates optical depth/transmittance in closed form ✔ Analytic/unbiased! ✘ Only for piecewise “simple” media ✘ No complex procedurals ✘ Discretization artifacts ✘ Expensive ✘ No speed/quality knob

Regular tracking

✔ Integrates optical depth/transmittance in closed form ✔ Analytic/unbiased! ✘ Only for piecewise “simple” media ✘ No complex procedurals ✘ Discretization artifacts ✘ Expensive ✘ No speed/quality knob ✘ Not progressive

Estimate/approximate optical depth integral τ

Integrating

| {z }

T(x↔y)

kx yk = t

τ

τ(t) =

⇡

Z t

0 µt(t0) dt0

Estimate/approximate optical depth integral τ

Integrating

hT(t)iRM = ehτ(t)i

| {z }

T(x↔y)

kx yk = t

τ

τ(t) =

⇡

Z t

0 µt(t0) dt0

⇡

Z t

0 µt(t0) dt0

Ray marching (Quadrature)

hτ(t)iRS =

k

∑

i=1

µt(ti) ∆ti ⇡

hT(t)iRM = ehτ(t)i

⇡

Z t

0 µt(t0) dt0

Ray marching (Quadrature)

hτ(t)iRS =

k

∑

i=1

µt(ti) ∆ti ⇡

hT(t)iRM = ehτ(t)i

Riemann sum

Ray marching (Monte Carlo)

hT(t)iRM = ehτ(t)i

hτ(t)iMC =

k

∑

i=1

µt(ti) p(ti)k with ti ∝ p(ti)

Ray marching (Monte Carlo)

hT(t)iRM = ehτ(t)i

hτ(t)iMC =

k

∑

i=1

µt(ti) p(ti)k with ti ∝ p(ti)

Ray marching (Stratified Monte Carlo)

hT(t)iRM = ehτ(t)i

hτ(t)iSMC =

k

∑

i=1

µt(ti)∆t, with ti = (i ξ)∆t

Ray marching (Stratified Monte Carlo)

hT(t)iRM = ehτ(t)i

hτ(t)iSMC =

k

∑

i=1

µt(ti)∆t, with ti = (i ξ)∆t

✔ Medium not discretized, supports procedurals

Ray marching

✔ Medium not discretized, supports procedurals ✔ Useful speed/quality knob, and progressive, but…

Ray marching

✔ Medium not discretized, supports procedurals ✔ Useful speed/quality knob, and progressive, but… ✘ Has error: variance/banding/noise, and

Ray marching

✔ Medium not discretized, supports procedurals ✔ Useful speed/quality knob, and progressive, but… ✘ Has error: variance/banding/noise, and ✘ Bias (not even consistent when used progressively)!

Ray marching

✔ Medium not discretized, supports procedurals ✔ Useful speed/quality knob, and progressive, but… ✘ Has error: variance/banding/noise, and ✘ Bias (not even consistent when used progressively)! ✘ unbiased estimate of optical depth leads to biased estimate of transmittance since:

Ray marching

E h eXi

6= eE[X]

✔ Medium not discretized, supports procedurals ✔ Useful speed/quality knob, and progressive, but… ✘ Has error: variance/banding/noise, and ✘ Bias (not even consistent when used progressively)! ✘ unbiased estimate of optical depth leads to biased estimate of transmittance since: ✘ Overestimates transmittances (medium looks “thinner” when using large steps)

Ray marching

E h eXi

6= eE[X]

Transmittance from free-flight sampling

T(t)

Transmittance from free-flight sampling

T(t)

Transmittance from free-flight sampling

Collision estimator:

Counting free-flights

hT(t)iC

|) = K(ti t)

µt(ti)

T(t)

Counting free-flights

Collision estimator:

hT(t)iC

|) = K(ti t)

µt(ti)

K T(t)

Counting free-flights

Collision estimator:

hT(t)iC

|) = K(ti t)

µt(ti)

K T(t)

Express transmittance as an integral (convolution with delta)

Why does this work?

Express transmittance as an integral (convolution with delta)

Why does this work?

T(t) =

Z ∞

T(s) δ(s − t) ds

Express transmittance as an integral (convolution with delta)

Why does this work?

T(t) =

Z ∞

T(s) δ(s − t) ds

Delta function

Express transmittance as an integral (convolution with delta) Replace/blur into
 finite kernel

Why does this work?

T(t) =

Z ∞

T(s) δ(s − t) ds

Delta function

T(t) ≈

Z ∞

T(s) K(s − t) ds ≈

Express transmittance as an integral (convolution with delta) Replace/blur into
 finite kernel

Why does this work?

T(t) =

Z ∞

T(s) δ(s − t) ds

Delta function

T(t) ≈

Z ∞

T(s) K(s − t) ds ≈

s ≈ T(ti) K(ti − t) p(ti) .

Express transmittance as an integral (convolution with delta)

Why does this work?

T(t) =

Z ∞

T(s) δ(s − t) ds

T(t) ≈

Z ∞

T(s) K(s − t) ds ≈

s ≈ T(ti) K(ti − t) p(ti) .

Express transmittance as an integral (convolution with delta)

Why does this work?

T(t) =

Z ∞

T(s) δ(s − t) ds

T(t) ≈

Z ∞

T(s) K(s − t) ds ≈

s ≈ T(ti) K(ti − t) p(ti) .

hT(t)iC

= T(ti) K(|ti t|)

µt(ti)T(ti)

Express transmittance as an integral (convolution with delta)

Why does this work?

T(t) =

Z ∞

T(s) δ(s − t) ds

T(t) ≈

Z ∞

T(s) K(s − t) ds ≈

s ≈ T(ti) K(ti − t) p(ti) .

hT(t)iC

|) = K(ti t)

µt(ti)

= T(ti) K(|ti t|)

µt(ti)T(ti)

Track-length estimator (binary):

Counting free-flights

T(t)

t < ti

t

hT(t)iTL =

( 1 if ti > t

• therwise

hT(t)iTL =

( 1 if ti > t

• therwise

t < ti

Counting free-flights

Track-length estimator (binary):

t

T(t)

hT(t)iTL =

( 1 if ti > t

• therwise

t < ti

Counting free-flights

Track-length estimator (binary):

t

T(t)

Why does this work?

Start with Russian roulette (unbiased):

Why does this work?

hT(t)iRR =

(

T(t) P(accept)

if accept

• therwise.

Start with Russian roulette (unbiased): Using free-flight sampling for acceptance probability:

Why does this work?

hT(t)iRR =

(

T(t) P(accept)

if accept

• therwise.

P(ti > t) =

Z ∞

t p(s) ds

Start with Russian roulette (unbiased): Using free-flight sampling for acceptance probability:

Why does this work?

hT(t)iRR =

(

T(t) P(accept)

if accept

• therwise.

P(ti > t) =

Z ∞

t p(s) ds

s =

Z ∞

t µt(s) T(s) ds = T(t)

Start with Russian roulette (unbiased): Using free-flight sampling for acceptance probability: This gives an unbiased “track-length estimator":

Why does this work?

hT(t)iRR =

(

T(t) P(accept)

if accept

• therwise.

P(ti > t) =

Z ∞

t p(s) ds

hT(t)iTL =

(

T(t) P(ti>t) = 1

if ti > t

• therwise

s =

Z ∞

t µt(s) T(s) ds = T(t)

Transmittance from free-flight sampling

✔ Estimate transmittance given free-flight sampling

Transmittance from free-flight sampling

✔ Estimate transmittance given free-flight sampling Collision Estimator

• Count photons starting at x that land near y

Transmittance from free-flight sampling

✔ Estimate transmittance given free-flight sampling Collision Estimator

• Count photons starting at x that land near y

✘ Biased (transmittance is blurred due to kernel/bin size)

Transmittance from free-flight sampling

✔ Estimate transmittance given free-flight sampling Collision Estimator

• Count photons starting at x that land near y

✘ Biased (transmittance is blurred due to kernel/bin size) Track-length Estimator

• Count photons starting at x that travel past y

Transmittance from free-flight sampling

✔ Estimate transmittance given free-flight sampling Collision Estimator

• Count photons starting at x that land near y

✘ Biased (transmittance is blurred due to kernel/bin size) Track-length Estimator

• Count photons starting at x that travel past y

✔ Unbiased!

Transmittance from free-flight sampling

Transmittance estimation

• 1. Estimators integrating optical thickness
• 2. Estimators using free-flight sampling
• 3. Next: Estimators using null collisions

T(x, y) = e−τ(x,y)

τ(x, y) =

Z t

0 µt(x − sω) ds

transmittance

• ptical thickness

p(ti) ∝ T(ti)

free-flight sampling