Transcript Recent Advances in Light Transport

Edit this text to create a Heading




This subtitle is 20 points
BulletsRecent
are blue
Advances in Light
They
have 110% line
spacing, 2 points
before & after
Transport
Simulation:
Theory
&
Longer bullets in the form of a paragraph are harder to
Practice
read if there is insufficient line spacing. This is the
Introduction
to Markov
Chain
maximum
recommended
number of
linesand
per slide
(seven).
Sequential Monte Carlo
 Sub bullets look like this
Edit this text to create a Heading




This subtitle is 20 points
Bullets are blue
They have 110% line spacing, 2 points before & after
Longer bullets in the form of a paragraph are harder to
read if there is insufficient line spacing. This is the
maximum recommended number of lines per slide
Markov
Chains
(seven).
 Sub bullets look like this
Markov
Chain:
Introduction
Edit this
text to
create a Heading
•

•
•
This
subtitle
20 points
Imagine
a ismolecule
•
read if there is insufficient line spacing. This is the
Set
of all possible positions is a state space
maximum recommended number of lines per slide
(seven).
It takes a new position 𝑥
with some
•
moving randomly in space
Bullets are blue
Current molecule position 𝑥𝑛 - current state
They have 110% line spacing, 2 points before & after
Time
discrete
=of0,1,2,
. .), 𝑥0 are
- initial
state
Longeris
bullets
in the (𝑛
form
a paragraph
harder
to
𝑛+1
 Sub bullets look
this
probability
𝑇(𝑥like
𝑛+1 𝑥𝑛 based on position 𝑥𝑛
3
Markov
Chain
Edit this
text to create a Heading
•


•
This
subtitlewalk
is 20 implies
points
Random
a transition probability
Bullets
𝑇(𝑥𝑛+1are=blue
𝑗 𝑥𝑛 = 𝑖 ≡ 𝑇𝑖→𝑗 for each move
They have 110% line spacing, 2 points before & after
At
each
move
theform
chain
a posterior
Longer
bullets
in the
of a forms
paragraph
are harder to
distribution
stateline
space
read
if there is over
insufficient
spacing. This is the
maximum recommended number of lines per slide
– A histogram of all visited states up to move 𝑛
(seven).
• Detailed
balance
defined as 𝑇𝑖→𝑗 = 𝑇𝑗→𝑖
 Sub bullets
look like this
4
Markov
Chain
Edit this
text to create a Heading
This subtitle
is 20 pointsto
• Posterior
converges



Burn-in
area
𝒙𝟎
the
target
distribution if
Bullets
are blue
the
balance
Theydetailed
have 110%
line spacing, 2 points before & after
obeyed
and all
states
Longer bullets
in the
formare
of a paragraph are harder to
reachable
read if there(ergodicity)
is insufficient line spacing. This is the
maximum
of lines per slide
• With
“bad”recommended
initial state 𝑥number
0
(seven).
the
start-up bias (burn-in
 Sub bullets
like this
phase)
can look
be significant
Equilibrium
𝒙𝟎
𝒙𝟎
5
Edit this text to create a Heading




This subtitle is 20 points
Bullets are blue
They have 110% line spacing, 2 points before & after
Longer bullets in the form of a paragraph are harder to
read if there is insufficient line spacing. This is the
maximum recommended number of lines per slide
Metropolis-Hastings
Algorithm
(seven).
 Sub bullets look like this
Metropolis-Hastings
(MH)
Algorithm
Edit this text to create
a Heading
•
•


Goal:
Random
walk
according to a desired function 𝑓
This subtitle
is 20
points
Define conditional
Bullets
are blue rejection sampling probability
𝑓 𝑥𝑗2 points
𝑓𝑗 before & after
They have 110% line spacing,
𝑎𝑖→𝑗 =
=
𝑓
𝑥
𝑓𝑖
Longer bullets in the form of a paragraph
are harder to
𝑖
– 𝑎𝑖→𝑗
is acceptance
probability
state 𝑖 for
proposal
read
if there
is insufficient
line at
spacing.
This
is the state 𝑗
recommended
number
of𝑇lines
• maximum
Detailed balance
is affected
as 𝑎𝑖→𝑗
𝑎𝑗→𝑖slide
𝑇𝑗→𝑖
𝑖→𝑗 =per
• (seven).
Posterior distribution is then proportional to 𝑓
– Sub
bulletstolook
like this
Accurate
a scaling
factor = normalization constant
7
Metropolis-Hastings:
Example
Edit this text to create
a Heading




This subtitle is 20 points
Bullets are blue
𝒙𝟎
They have 110% line spacing, 2 points before & after
Longer bullets in the form of a paragraph are harder to
read if there is insufficient line spacing. This is the
maximum recommended number of lines per slide
(seven).
 Sub bullets look like this
8
Metropolis-Hastings:
Example
Edit this text to create
a Heading




This subtitle is 20 points
𝒙𝟏
Bullets are blue
𝒙𝟎
𝑎𝑥0→𝑥1 =
ℕ(𝑥1 )
>1
ℕ 𝑥0
They have 110% line spacing, 2 points before & after
Longer bullets in the form of a paragraph are harder to
read if there is insufficient line spacing. This is the
maximum recommended number of lines per slide
ℕ(𝒙𝟏 )
(seven).
 Sub bullets look like this
ℕ(𝒙𝟎 )
9
Metropolis-Hastings:
Example
Edit this text to create
a Heading




This subtitle is 20 points
𝒙𝟏
𝒙𝟐 ′
Bullets are blue
𝑎𝑥1→𝑥2 =
ℕ(𝑥2 )
≪1
ℕ 𝑥1
They have 110% line spacing, 2 points before & after
Longer bullets in the form of a paragraph are harder to
read if there is insufficient line spacing. This is the
maximum recommended number of lines per slide
(seven).
 Sub bullets look like this
10
Metropolis-Hastings:
Example
Edit this text to create
a Heading




This subtitle is 20 points
𝒙𝟐 𝒙𝟑
Bullets are blue
They have 110% line spacing, 2 points before & after
Longer bullets in the form of a paragraph are harder to
read if there is insufficient line spacing. This is the
maximum recommended number of lines per slide
(seven).
 Sub bullets look like this
11
Metropolis-Hastings:
Example
Edit this text to create
a Heading




This subtitle is 20 points
𝒙𝟒 ′
Bullets are blue
𝒙𝟑
They have 110% line spacing, 2 points before & after
Longer bullets in the form of a paragraph are harder to
read if there is insufficient line spacing. This is the
maximum recommended number of lines per slide
(seven).
 Sub bullets look like this
12
Metropolis-Hastings:
Example
Edit this text to create
a Heading




This subtitle is 20 points
Bullets are blue
𝒙𝟒
𝒙𝟓 ′
They have 110% line spacing, 2 points before & after
Longer bullets in the form of a paragraph are harder to
read if there is insufficient line spacing. This is the
maximum recommended number of lines per slide
(seven).
 Sub bullets look like this
13
Metropolis-Hastings:
Example
Edit this text to create
a Heading




This subtitle is 20 points𝒏 = 𝟐𝟎
Bullets are blue
They have 110% line spacing, 2 points before & after
Longer bullets in the form of a paragraph are harder to
read if there is insufficient line spacing. This is the
maximum recommended number of lines per slide
(seven).
 Sub bullets look like this
14
Metropolis-Hastings:
Example
Edit this text to create
a Heading




𝒏 = 𝟐𝟎𝟎
This subtitle is 20 points
Bullets are blue
They have 110% line spacing, 2 points before & after
Longer bullets in the form of a paragraph are harder to
read if there is insufficient line spacing. This is the
maximum recommended number of lines per slide
(seven).
 Sub bullets look like this
15
Metropolis-Hastings:
Example
Edit this text to create
a Heading




𝒏 = 𝟐𝟎𝟎𝟎
This subtitle is 20 points
Bullets are blue
They have 110% line spacing, 2 points before & after
Longer bullets in the form of a paragraph are harder to
read if there is insufficient line spacing. This is the
maximum recommended number of lines per slide
(seven).
 Sub bullets look like this
16
Importance
Sampling
Edit this text
to create a Heading
•

•


•
This
subtitle
is 20proposals
points
Cannot
fetch
directly from 𝑓
Bullets are blue
Generate a proposal 𝑗 from some pdf 𝑇𝑗
They have 110% line spacing, 2 points before & after
– Similar to importance sampling in Monte Carlo
Longer bullets in the form of a paragraph are harder to
– 𝑇 ifcan
depend
on the current
state 𝑖This
: 𝑇𝑖→𝑗
read
there
is insufficient
line spacing.
is the
maximum
recommended
number
of lines per slide
Acceptance
probability
is then
(seven).
𝑓𝑗
𝑓𝑖
 Sub bullets look
like=this
𝑎𝑖→𝑗
/
𝑇𝑖→𝑗
𝑇𝑗→𝑖
17
Correspondence
Edit this text to Table
create a Heading
 ThisOrdinary
subtitle
is 20
points
Monte
Carlo
Markov chain Monte Carlo
 Bullets are blue
1
Mixing rate, depends on multiple factors,
Convergence rate, usually 𝑂( )
𝑁 spacing,
can be
𝑂 𝛾 𝑁 , 𝛾 ∈&(0;
1)
 They have 110% line
2 geometric
points before
after
Convergence
Longer bullets in the form ofConvergence
a paragraph
are harder
of the posterior
to the to
to an expected value
target distribution (e.g., in total variation)
read if there is insufficient line
spacing. This is the
Transitionof
probability
𝑇𝑖→𝑗 slide
Importance
sampling
distribution 𝑝(𝑥) number
maximum
recommended
lines per
(seven).
Variance
of the estimate
Acceptance rate, correlation of samples
 Sub
bullets
Number
of samples
look like this
Number of moves (mutations)
18
Edit this text to create a Heading




This subtitle is 20 points
Bullets are blue
They have 110% line spacing, 2 points before & after
Longer bullets in the form of a paragraph are harder to
read if there is insufficient line spacing. This is the
maximum recommended number of lines per slide
Metropolis
Light Transport
(seven).
 Sub bullets look like this
Image
Generation
Edit this
text to create a Heading
•


•
•
This
subtitle
is 20 points
Reduce
per-pixel
integrals
to a single integral
Bullets are blue
– Each pixel has an individual filter function then
They have 110% line spacing, 2 points before & after
Compute
thein distribution
over theare
image
Longer bullets
the form of a paragraph
harderplane
to
read
is insufficient
spacing. Thispixels
is the
– Binif there
this distribution
intoline
corresponding
maximum recommended number of lines per slide
Walk
(seven).over the image plane
 Sub bullets look like this
20
Metropolis
Light
Edit this text
to Transport
create a Heading
•
•

•

This
subtitle
is 20
State
space
= points
space
of full paths, path space
Bullets
arethe
bluefunction 𝑓 for light transport?
What is
They have 110% line spacing, 2 points before & after
Interested in flux arriving at image plane
Longer bullets in the form of a paragraph are harder to
read if there is insufficient line spacing. This is the
maximum recommended number of lines per slide
(seven).
 Sub bullets look like this
21
Measurement
Edit this textContribution
to create a Heading
•



This
subtitle is 20 contribution
points
Measurement
𝑓 for 𝑘-length path
Bullets are blue
𝑓(𝑥𝑘 ) = 𝐿e 𝐺
𝜌𝑘 𝐺𝑘 𝑊e
They have 110% line spacing, 𝑘−1
2 points before & after
Longer bullets in the form of a paragraph are harder to
read
𝑊e if there is insufficient line spacing.𝜌This is the 𝐿e
𝜌
𝐺 recommended
maximum
number
of
𝐺
𝐺 lines per
𝐺 slide
𝐺
𝐺
(seven).
𝜌
 Sub bullets look like this
𝜌
𝜌
22
Measurement
Edit this textContribution
to create a Heading




𝑑𝑄
This subtitle is 20 points 𝑑𝑄
𝑓 𝑥𝑘 =
=
𝑊/(𝑚2 )𝑘
𝑑𝜇𝑘
𝑘 𝑑𝐴𝑘
Bullets are blue
– Flux
through
differential
a path& after
They
have
110%all
line
spacing, 2areas
pointsofbefore
𝑑𝐴
𝑑𝐴 form of a paragraph
Longer bullets in the
are harder to
𝑑𝐴read if there is insufficient line spacing. This is the
𝑑𝐴
maximum recommended number of lines per slide
(seven).
 Sub bullets look like this
𝑑𝐴
𝑑𝐴
𝑑𝐴
23
Comparing
Paths
Edit this text
to create a Heading
•


•
•
This
points
MH subtitle
needsisto20compare
two states (paths)
Bullets are blue
• Use flux through the infinitesimal path beam
They have 110% line spacing, 2 points before & after
Directly
comparable
pathsto
Longer bullets
in the form for
of aequal-length
paragraph are harder
read
if there isflows
insufficient
line through
spacing.each
This is
the
– Compare
of energy
path
maximum recommended number of lines per slide
For
different lengths the measure is different
(seven).
– Sub
bullets
look like this
Always
compare
fluxes going through each path
24
Path
EditIntegral
this text to create a Heading
•

•

This
20 points
For subtitle
path ofis length
𝑘 : 𝐼𝑘 = Ω𝑘 𝑓 𝑥 𝑑𝜇𝑘 (𝑥)
Bullets are blue
Combine
all path
lengths2into
a before
single&integral
They have 110%
line spacing,
points
after
Longer
the formfor
of a
are harder to
– Use bullets
unified in
measure
allparagraph
paths
∞
read if there is insufficient
line
spacing. This is the
𝑑𝜇 𝐷 =
𝑑𝜇𝑘 (𝐷 ∩ Ω𝑘 )
maximum recommended𝑘=1number of lines per slide
– Compare paths of different length
(seven).
– Sub
bullets look
like this
Compare
groups
of paths
25
Metropolis
Light
Edit this text
to Transport
create a Heading
1.This
subtitle isinitial
20 points
Generate
path 𝑥0 using PT/BDPT
 Bullets are blue
2.
Mutate with some transition probability 𝑇𝑥𝑖→𝑥𝑗
 They have 110% line spacing, 2 points before & after
3.Longer
bullets
the form
of a paragraph
are𝑎harder
to
Accept
newinpath
𝑥𝑗 with
probability
𝑥 →𝑥
𝑖
𝑗
read if there is insufficient line spacing. This is the
4.maximum
Accumulate
contribution
to ofthe
image
plane
recommended
number
lines
per slide
(seven).
5. Go to step 2
 Sub bullets look like this
𝑥1
𝑥0
𝑥2
26
Advantages
Edit this text to create a Heading
•

•

•
This
subtitle
is 20
More
robust
topoints
complex
light paths
Bullets
are blue successful paths
– Remembers
They
havecoherence
110% line spacing,
2 points
before & after
Utilizes
of image
pixels
Longer
bullets
in the form
of a paragraph are harder to
– Explores
features
faster
read if there is insufficient line spacing. This is the
Cheaper
samples number of lines per slide
maximum recommended
– Correlated
(seven).
• Flexible
path
(mutations)
 Sub bullets
look generators
like this
27
Variations
and to
Improvements
Edit this text
create a Heading
•



This
subtitle
is 20 points
Energy
redistribution
path tracing [Cline05]
Bullets are blue
– Run many short Markov chains for each seed
They have 110% line spacing, 2 points before & after
– Adaptive number of chains according to path energy
Longer bullets in the form of a paragraph are harder to
– In ifspirit
lensline
mutation
read
thereofisVeach’s
insufficient
spacing. This is the
maximum recommended number of lines per slide
(seven).
 Sub bullets look like this
28
Variations
and to
Improvements
Edit this text
create a Heading
•



This
subtitle
is 20 points[Kitaoka09]
Replica
exchange
Bullets are blue
– Run separate Markov chain(s) for specific features
They have 110% line spacing, 2 points before & after
– Exchange
discovered
between
chainsto
Longer
bulletsthe
in the
form of a paths
paragraph
are harder
Chain
read
if 1there is insufficient line spacing. This is the
maximum recommended number of lines per slide
Chain 2
(seven).
 Sub bullets look like this
Chain 3
29
Edit this text to create a Heading




This subtitle is 20 points
Bullets are blue
They have 110% line spacing, 2 points before & after
Longer bullets in the form of a paragraph are harder to
read if there is insufficient line spacing. This is the
maximum recommended number of lines per slide
Normalization
(seven).
and
Bias in MLT
 Sub Start-up
bullets look like this
Differences
to MCMC
Edit this text
to create a Heading
•



This
is 20
We subtitle
do have
a points
good
alternative sampler
Bullets are blue
– Path tracer / bidirectional path tracer
They have 110% line spacing, 2 points before & after
– Easybullets
to compute
normalization
constant
Longer
in the form
of a paragraph
are harder to
if there is bias,
insufficient
spacing.
is the
• read
No start-up
startline
within
theThis
equilibrium
maximum recommended number of lines per slide
– Start many chains stratified over path space
(seven).
–
well
with
parallel MLT
 Scales
Sub bullets
look
like massively
this
31
Edit this text to create a Heading




This subtitle is 20 points
Bullets are blue
They have 110% line spacing, 2 points before & after
Longer bullets in the form of a paragraph are harder to
read if there is insufficient line spacing. This is the
maximum recommended number of lines per slide
Mutation
Strategies and
(seven).
Their
Properties
 Sub bullets
look like this
Good
Criteria
Edit Mutation
this text to
create a Heading
•

•

This
subtitle is 20
points
Lightweight
mutation:
change a few vertices
Bullets are blue
Low correlation of samples
They have 110% line spacing, 2 points before & after
– Largebullets
stepsininthe
path
space
Longer
form
of a paragraph are harder to
if there is insufficient line spacing. This is the
• read
Good
stratification over the image plane
maximum recommended number of lines per slide
– Hard to control, usually done by re-seeding
(seven).
 Sub
bullets look like this
• It’s
OK
to have many specialized mutations
33
Edit this text to create a Heading




This subtitle is 20 points
Bullets are blue
They have 110% line spacing, 2 points before & after
Longer bullets in the form of a paragraph are harder to
read if there is insufficient line spacing. This is the
maximum recommended number of lines per slide
Existing
Mutation Strategies
(seven).
 Sub bullets look like this
Veach
Mutations
Edit this
text to create a Heading
•


•
This
subtitle
is 20 points
Minimal
changes
to
the path
Lens
perturbation
Bullets are blue
– Lens, caustics, multi-chain perturbations
They have 110% line spacing, 2 points before & after
Large
changes
theofpath
Longer bullets
in thetoform
a paragraph are harder to
Caustics
perturbation
read
if there is insufficient
– Bidirectional
mutation line spacing. This is the
maximum recommended number of lines per slide
• BDPT-like large step
(seven).
Bidirectional
mutation
–
Lens
mutation
 Sub bullets look like this
• stratified seeding on the image plane
35
Kelemen
Edit thisMutation
text to create a Heading
•

•
•
This
subtitle
is 20 pointsvector
Mutate
a “random”
that maps to a path
Bullets are blue
Symmetric perturbation of “random” numbers
They have 110% line spacing, 2 points before & after
Use
the
“random”
vector
for importance
pdfs
Longer
bullets
in the form
of a paragraph
are harder
to
read if there is insufficient line spacing. This is the
– Primary space: importance function domain
maximum recommended number of lines per slide
– Assume the importance sampling is good
(seven).
 Sub bullets look like this
36
Kelemen
Part IIa Heading
Edit thisMutation,
text to create
•


•

This
subtitle is 20
points
Acceptance
probability
𝑎𝑖→𝑗 = 𝑓𝑗 /𝑝𝑗 / 𝑓𝑖 /𝑝𝑖
Bullets
– Easyare
to blue
compute: just take values from PT/BDPT
They have 110% line spacing, 2 points before & after
Large step: pure PT / BDPT step
Longer bullets in the form of a paragraph are harder to
– Generate primary sample (random vector) anew
read if there is insufficient line spacing. This is the
maximum recommended number of lines per slide
(seven).
 Sub bullets look like this
37
Manifold
Mutation
Edit thisExploration
text to create
a Heading
•



This
subtitle is 20 points
Mutate/connect
while
keeping path structure
Bullets
– Workare
in blue
the local parameterization of current path
They
have
110%through
line spacing,
2 points
before & after
– Can
connect
a specular
chain
– Eliminates/fixes
dimensions
Longer
bullets in theintegration
form of a paragraph
are harder to
to keep
𝑓 constantline
by obeying
constraints
read• ifTries
there
is insufficient
spacing.
This is the
maximum recommended number of lines per slide
(seven).
 Sub bullets look like this
S
38
Combinations
Edit this text to create a Heading
•



This
subtitleexploration
is 20 points
Manifold
can be combined
Bullets are blue
– With Veach mutation strategies in MLT
They have 110% line spacing, 2 points before & after
– Withbullets
energyinredistribution
tracing
Longer
the form of a path
paragraph
are harder to
if thereKelemen’s
is insufficient and
line spacing.
This
is the
• read
Combine
Veach’s
mutations?
maximum recommended number of lines per slide
– Possible, yet unexplored option
(seven).
 Sub bullets look like this
39
Edit this text to create a Heading




This subtitle is 20 points
Bullets are blue
They have 110% line spacing, 2 points before & after
Longer bullets in the form of a paragraph are harder to
read if there is insufficient line spacing. This is the
maximum recommended number of lines per slide
Population
Monte Carlo
(seven).
Light
Transport
 Sub bullets
look like this
Population
Monte
Carlo a
Framework
Edit this text
to create
Heading
•


•

•
This
is 20 points
Usesubtitle
a population
of
Markov chains
Bullets
blue on top of Metropolis-Hastings
– Can are
operate
They have 110% line spacing, 2 points before & after
Rebalance the workload
Longer bullets in the form of a paragraph are harder to
– Weakest
chains
are eliminated
read
if there is
insufficient
line spacing. This is the
– Strongest
chains are forked
multiple
maximum
recommended
numberinto
of lines
per slide
(seven).
Use mixture of mutations, adapt to the data
 Sub bullets look like this
– Select optimal mutation on the fly
41
Population
Monte
Carlo a
ERPT
[Lai07]
Edit this text
to create
Heading
•


•
This
subtitle
is 20 points
Spawn
a population
of chains with paths
Bullets are blue
– Do elimination and reseeding based on path energy
They have 110% line spacing, 2 points before & after
Use
many
with
differentareparameters
Longer
bulletsmutations
in the form of
a paragraph
harder to
read
if there isthem
insufficient
line based
spacing.
is the
– Reweight
on-the-fly
onThis
the efficiency
maximum recommended number of lines per slide
– Lens and caustics perturbations in the original paper
(seven).
 Sub
bullets
look like
this with manifold exploration
• We
will
show
PMC
42
Edit this text to create a Heading




This subtitle is 20 points
Bullets are blue
They have 110% line spacing, 2 points before & after
Longer bullets in the form of a paragraph are harder to
read if You
therefor
is insufficient
line spacing. This is the
Thank
Your attention.
maximum recommended number of lines per slide
Part
one questions?
(seven).
 Sub bullets look like this