at the boundary between light transport simulation …hachisuka/mcqmc2014.pdflight transport...

At the Boundary Between Light Transport Simulation and

Computational Statistics

Toshiya Hachisuka

Aarhus University

MCQMC2014

At the boundary

2

Ij = E

fj(X)

p(X)

Me

What’s often considered

• Graphics communities “just use” statistics

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StatisticsGraphics

What I hope to see

• Statistics communities take something from graphics

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StatisticsGraphics

Aim of this talk

• Motivate you to facilitate cross-pollination

• Both ways (graphics statistics)

• Using two examples

• Progressive density estimation

• Primary space serial tempering

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“Progressive Photon Mapping” Hachisuka et al. ACM SIGGRAPH Asia 2008

“Multiplexed Metropolis Light Transport” Hachisuka et al. ACM SIGGRAPH 2014 (conditionally accepted)

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Progressive Density Estimation

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Manifold of non-zero valuesΩ∞

L =

Ω∞

fjXdµ(X)

L =

Ω∞

fjXdµ(X)

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Ω∞

xi ∼ p(xi)≈ 1

N

i

fj (xi)

p(xi)

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Eye

Transparent Layer

Light source

Matte Surface

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11

θ1

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θ1

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θ2

θ1

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θ2

θ1

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Probability of sampling this path is nearly zero

θ2

θ1

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Probability of sampling this path is nearly zero

θ2

θ1

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Probability of hitting at the same point is zero

θ2 θ1

Manifold of non-samplable paths

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θ1

θ2

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Light source

Eye

Glass

Matte surface

Matte surface

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How can we capture non-samplable paths?

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Progressive Density Estimation

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θ2 θ1

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θ2 θ1

Allow approximate connection within distance

r

r

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θ2 θ1

Allow approximate connection within distance r... and progressively reduce r

Progressive density estimation

• Algorithmically convergent density estimation

• Add gradually reducing bias to relax the problem

• First to solve the issue of non-samplable paths

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[Hachisuka et al. 2008]

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Initial pass

Image

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Initial pass

j-th pixel

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Initial pass

xj Visible point

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Initial pass

Kernelr0

r0Initial kernel bandwidth:

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1st passMonte Carlo photon tracing

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1st pass

Photons

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1st pass

Range query with r0

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1st pass

Reduce radius into r1

Radius reduction

• Reduce radius based on sample statistics

• Number of photons found so far

• Number of newly found photons

• Reduction rate (user-defined)

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rn+1 =Nn + αMn

Nn +Mnrn

Nn

Mn

α ∈]0, 1[

Nn+1 = Nn + αMn

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1st pass

r1 =N0 + αM0

N0 +M0r0 =

0 + α4

0 + 4r0

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1st pass

N1 = N0 + αM0 = 0 + α4

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2nd pass

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2nd pass

New set of photons

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2nd pass

Range query with r1

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2nd pass

r2 =N1 + αM1

N1 +M1r1 N2 = N1 + αM1

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3rd and n-th passes

rn+1 =Nn + αMn

Nn +Mnrn Nn+1 = Nn + αMn

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MC integration Progressive density estimation

Equal time

Convergence

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• Converges to the correct solution

• Infinite number of neighboring samples

• Infinitely small radius

L = limn→∞

Nn

πr2n

• Converges to the correct solution

•

•

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Convergence

limi→∞

Ni(Ri) = ∞limi→∞

Ri(Ni) = 0

L = limn→∞

Nn

πr2n

• Converges to the correct solution

•

•

• Convergence rate

• Variance

• Bias

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Convergence

limi→∞

Ni(Ri) = ∞limi→∞

Ri(Ni) = 0

L = limn→∞

Nn

πr2n

O(n−α)

O(nα−1)α =

2

3Balanced with

Relationship to recursive density estimation

• Recursive density estimation

• Predefined sequence of radii

• Assume stationary density distribution

• Progressive density estimation

• Adaptive sequence from sample statistics

• No assumption on stationarity

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[Wolverton and Wagner 1969, Yamato 1971, Knaus and Zwicker 2011]

[Hachisuka et al. 2008, Hachisuka and Jensen 2009]

Progressive vs Ordinary

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Progressive Ordinary

Computation Unbounded Unbounded

Storage Bounded Unbounded

Convergence Single run Multiple runs

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Primary Space Serial Tempering

Multiple sampling methods

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Multiple sampling methods

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Multiple sampling methods

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Multiple sampling methods

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Multiple sampling methods

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p0(X)

p1(X) p2(X)

p3(X)

Given the same integrand, there are many known PDFs

Multiple importance sampling

• Utilizes samples from multiple PDFs by weighting

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[Veach and Guibas 1995]

f(X)

Multiple importance sampling

• Utilizes samples from multiple PDFs by weighting

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[Veach and Guibas 1995]

p0(X) p1(X)

f(X)

Multiple importance sampling

• Importance sampling

• Multiple importance sampling

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[Veach and Guibas 1995]

f(X)dµ(X) ≈ 1

N

i

f(xi)

p0(xi)p0(xi)

f(X)dµ(X) ≈ 1

N

w0(xi)

f(xi)

p0(xi)+ w1(xi)

f(xi)

p1(xi)

p0(xi) p1(xi)

1

N

i

f(xi)

p1(xi)p1(xi)or

Uses both

Markov chain Monte Carlo sampling

• Requires no knowledge of PDFs

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f(X)

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How can we combine MIS and MCMC?

Combination

• Multiple importance sampling

• Ordinary Monte Carlo method

• Utilizes the knowledge of all the PDFs

• Markov chain Monte Carlo sampling

• Markov chain Monte Carlo method

• No usage (or requirement) of PDFs

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Primary Space Serial Tempering

Primary sample space

• Hypercube of random numbers

• Mapping from a point to a sample

• Inverse CDF = mapping function

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[Kelemen et al. 2002]

U ∈ ]0, 1[N

C(U) =f(P−1(U))

p(P−1(U))

f(X)dµ(X) ≈ 1

N

i

f(xi)

p(xi)=

1

N

i

C(ui)

Primary sample space

• Hypercube of random numbers

• Mapping from a point to a sample

• Inverse CDF = mapping function

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[Kelemen et al. 2002]

Point in the hypercubePath in the sample space

U ∈ ]0, 1[N

X = P−1(U)X U

Serial tempering

• MCMC sampling of a sum of multiple distributions

• Extended states of the Markov chain

• Index j is updated via MCMC

• Extended state is originally temperature

• Still does not use any PDFs

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x ∼

j

fj(x)

(x, j)

[Marinari and Parisi 1992, Geyer and Thompson 1995]

Primary space serial tempering

• Use a weighted sum of primary space distributions

• Extended states is now

• Index j is the index to the j-th PDF/CDF

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u ∼

j

wj(u)Cj(u)

wj(U) = w(P−1j (U))

(u, j)

Cj(U) =f(P−1

j (U))

pj(P−1j (U))

[Hachisuka et al. 2014] (TBA)

MIS weightPrimary space distribution

Original primary space

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C(U)

x = P−1(U)

Multiplexed primary space

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w3(U)C3(U)w2(U)C2(U)w1(U)C1(U)w0(U)C0(U)

x = P−1j (U)

Primary space serial tempering

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w3(U)C3(U)w2(U)C2(U)w1(U)C1(U)w0(U)C0(U)

x = P−10 (U)

Primary space serial tempering

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w3(U)C3(U)w2(U)C2(U)w1(U)C1(U)w0(U)C0(U)

x = P−10 (U)

Primary space serial tempering

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w3(U)C3(U)w2(U)C2(U)w1(U)C1(U)w0(U)C0(U)

x = P−11 (U)

Primary space serial tempering

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w3(U)C3(U)w2(U)C2(U)w1(U)C1(U)w0(U)C0(U)

x = P−13 (U)

Primary space serial tempering

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w3(U)C3(U)w2(U)C2(U)w1(U)C1(U)w0(U)C0(U)

x = P−13 (U)

[Veach and Guibas 1997]Metropolis-Hastings

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Equal time

Primary space only

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[Kelemen et al. 2002]

Equal time

Primary space serial tempering

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Equal time

[Veach and Guibas 1997]Metropolis-Hastings

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Equal time

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Summary

Statistics / Graphics

• Progressive density estimation

• Statistics: new density estimator

• Graphics: simulation of complex light paths

• Primary space serial tempering

• Statistics: MCMC driven MIS

• Graphics: robust and efficient path sampling

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What I hope to see

• Statistics communities take something from graphics

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StatisticsGraphics

StatisticsGraphics

What I really hope to see

• We collaborate to develop something interesting!

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