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High-Order Similarity Relations in Radiative Transfer
Shuang Zhao1, Ravi Ramamoorthi2, and Kavita Bala1
1Cornell University2University of California, San Diego
Translucency is everywhere
Food
Skin
Jewelry
Architecture
Slide courtesy of Ioannis Gkioulekas
Rendering translucency
Radiativetransfer
Scatteringparam.
Appearance
Rendering translucency
Radiativetransfer
Scatteringparam. 2
Appearance 2
Radiativetransfer
Scatteringparam. 1
Appearance 1
Radiativetransfer
Scatteringparam. 1
Appearance 1
Radiativetransfer
Scatteringparam. 2
Appearance 2
≈≠
First-order methods
Scatteringparam. 1
Scatteringparam. 2
Scatteringparam. 1
Scatteringparam. 2
First-order approx.
Approx. identical appearance
Cheaper to render
Limitedaccuracy
[Frisvad et al. 2007] [Arbree et al. 2011][Wang et al. 2009][Jensen et al. 2001]
Similarity theory
Scatteringparam. 1
Scatteringparam. 2
First-order approx.First-ordermethods
Scatteringparam. 1
Scatteringparam. 2
First-order approx.
Similaritytheory
[Wyman et al. 1989]
Scatteringparam. 1
Scatteringparam. 2
Similarityrelations
Similarity theory
Similaritytheory
[Wyman et al. 1989]
Scatteringparam. 1
Scatteringparam. 2
Similarityrelations
Provide fundamental insights into thestructure of material parameter space
Similarity theory
Similaritytheory
[Wyman et al. 1989]
Scatteringparam. 1
Scatteringparam. 2
Similarityrelations
Originates in applied optics[Wyman et al. 1989]
Similar ideas explored in neutron transfer(Condensed History Monte Carlo)
[Prinja & Franke 2005], [Bhan & Spanier 2007], …
Our contribution
Introducing high-ordersimilarity theory tocomputer graphics
Novel algorithmsbenefiting forward &
inverse rendering
Our contribution: forward rendering
BetteraccuracyOur
approach
User-specified(balancing performance and accuracy)
Approx. identical appearance
Cheaper to render
Scatteringparam. 2
100 ~ 200 lines of MATLAB code
Scatteringparam. 1
Up to 10X speedup
Our contribution: inverse rendering
Parameter space 1
Reparameterize
Parameter space 2
Gradient descent methods perform
much better
Background
Material scattering parameters
Extinction coefficient
Scattering coefficient
Phase function
Light particle
Absorption coefficient
AbsorbedScatteredInteraction
Phase function
Scattered
Probability density for , parameterized as
Isotropic scattering
Forward
Forward scattering
Forward
Similarity Theory
nth Legendremoment
Phase function moments
Legendrepolynomial
For a phase function
“Average cosine”
Similarity relations
Low-frequency radiance
Band-limited up to order-N in spherical harmonics domain
…
[Wyman et al. 1989]
Order-N similarity relation[W
yman et al. 1989]
Similarity relations
…
identical appearance
…
Derivationin the paper
Radiancelow-frequency
everywhere
Order-N similarity relation
Similarity relations
…
Higher order,Better accuracy
Approximatelyidentical appearanceRadiance
low-frequencyeverywhere
Challenge
Order-N similarity relation
…
Order-N similarity relation
…
Original(given)
Altered(unknown)
??
Solving forAltered Parameters
The problem
Altered parameters ?
…
??O
rder
-N
sim
ilarit
y re
latio
nC
onstraints
Forward
Original parameters
The problem
Altered parameters ??
…
Ord
er-N
si
mila
rity
rela
tion
…
Ord
er-N
si
mila
rity
rela
tion
…
Forward
Original parameters
The problem
Altered parameters ??
…
Ord
er-N
si
mila
rity
rela
tion
…
Forward
Original parameters
Altered phase function
…
Altered parameters ?
…
Altered parameters ?
Forward
Original parameters
Remainingunknown
Altered phase function
Altered parameters ?
Forward
Original parameters
Remainingunknown
Legendre moments of
…
Legendre moments of
Altered phase function
Altered parameters ?
Altered parameters ?
Order-1
Order-2
Order-3
Order-4…
Finding highest satisfiable order N
Normalizationconstraint
Finding order N
Given desired Legendre moments
(Truncated Hausdorff moment problem)[Curto and Fialkow 1991]
Phase functionHankel matrices builtusing are
positive semi-definiteexists
Existence condition
Does phase function exist?
Finding order N
Altered parameters ?
Order-1
Order-2
Order-3
Order-4…
Finding highest satisfiable order N
Altered phase function
Altered parameters ?
Order-3
Order-3
Problem: not uniquely specified
Invalid Valid Valid
Constructing altered phase function
…
-1 10
Need:has Legendre moments
non-negative
Represent as a tabulated function with pieces
?…
-1 10
Constructing altered phase function
Need:
Represent as a tabulated function with pieces
?Const.
Constructing altered phase function
Solve subject to
Smoothness term(favoring “uniform” solutions)
-1 10
Good
-1 10
Bad
Constructing altered phase function
Solve subject to
Quadratic programming
• Standard problem
• Solvable with many tools/libraries• MATLAB, Gurobi, CVXOPT, …
• Our MATLAB code is available online
Constructing altered phase function
Altered parameters ?
Order-3
ValidInvalid Valid
Our approach
Forward
Altered parameters
Constructing altered phase function
Summary
Forward
Original parameters
Forward
Altered parameters
Forward
Altered parameters
Compute order NSolve optimization
Application:Forward Rendering
Our contribution: forward rendering
BetteraccuracyOur
approach
Approx. identical appearance
Cheaper to render
Scatteringparam. 2
Scatteringparam. 1
Effort-free speedups!
User-specified(balancing performance and accuracy)
Application: forward rendering
0 1
No changein parameters
large
Better accuracyLower speedup
small
Worse accuracyGreater speedup
Perform test renderings to find optimal
Reuse for high-resolution renderings or videos
is a good start
Experimental Results
Performance vs. accuracy
α = 0.05 (44 min, 8.0X)
Relative error 0%
30%
Reference (350 min)
Performance vs. accuracy
Reference (350 min) α = 0.05 (44 min, 8.0X)
Relative error
α = 0.10 (63 min, 5.6X)
Relative error 0%
30%
0%
30%
Performance vs. accuracy
α = 0.20 (103 min, 3.4X)
Relative error
α = 0.30 (126 min, 2.8X)
Relative error 0%
30%
α = 0.10 (63 min, 5.6X)
Relative error
α = 0.10 (63 min, 5.6X)
Relative error
Visually identical
Power of high-order relations
Used by first-order methods:
Altered parameters(Order-1)
Forward
Forward
Original parameters
Reduced scatteringcoefficient
Satisfies order-1similarity relation
Power of high-order relations
Altered parameters(Order-3)
Forward
Forward
Original parameters
Altered parameters(Order-1)
Forward
Power of high-order relations
Altered parameters(Order-3)
Original parameters
Altered parameters(Order-1)
Original parameters
Altered parameters(Order-1)
Altered parameters(Order-3)
426 min (reference) 119 min (3.6X) 115 min (3.7X)
More renderings
Reference473 min
Ours178 min (2.7X)
Reference23 min
Ours20 min
Equal-timeEqual-sample
Conclusion
Order-N similarity relation
…
Introducing high-ordersimilarity relations to graphics
Proposing a practical algorithmto solve for altered parameters
?Original Altered
• Picking automatically and adaptively
• Alternative versions of similarity theory
Future work
Thank you!
High-Order Similarity Relationsin Radiative Transfer
Shuang Zhao1, Ravi Ramamoorthi2, Kavita Bala1
1Cornell University, 2University of California, San Diego
Project website: (MATLAB code available!)
www.cs.cornell.edu/projects/translucency
Funding:NSF IIS grants 1011832, 1011919, 1161645Intel Science and Technology Center – Visual Computing
Reference
Ours (3.7X
)
Extra Slides
Order-1 similarity relation
Order-1 similarity relation
Reducedscattering coefficient
Special case (used by diffusion methods):
Order-N similarity relation
…
Prior work: solving for altered parameters
[Wyman et al. 1989]
fixed such that
given by the user
Discrete scattering angle [Prinja & Franke 2005]
Represent as the sum of delta functions
“Spiky” phase functions do not perform as well as“uniform” ones for rendering applications
Constructing altered phase function
Represent as a tabulated function with pieces
Quadratic programming
Solve subject to
Hankel matrices built using being positive semi-definite
Existence condition
Performance vs. accuracy
Reference (350 min)
Discarded Slides
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