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![Page 1: High-Order Similarity Relations in Radiative Transfer Shuang Zhao 1, Ravi Ramamoorthi 2, and Kavita Bala 1 1 Cornell University 2 University of California,](https://reader037.fdocuments.us/reader037/viewer/2022110320/56649cb65503460f9497b803/html5/thumbnails/1.jpg)
High-Order Similarity Relations in Radiative Transfer
Shuang Zhao1, Ravi Ramamoorthi2, and Kavita Bala1
1Cornell University2University of California, San Diego
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Translucency is everywhere
Food
Skin
Jewelry
Architecture
Slide courtesy of Ioannis Gkioulekas
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Rendering translucency
Radiativetransfer
Scatteringparam.
Appearance
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Rendering translucency
Radiativetransfer
Scatteringparam. 2
Appearance 2
Radiativetransfer
Scatteringparam. 1
Appearance 1
Radiativetransfer
Scatteringparam. 1
Appearance 1
Radiativetransfer
Scatteringparam. 2
Appearance 2
≈≠
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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]
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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
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Similarity theory
Similaritytheory
[Wyman et al. 1989]
Scatteringparam. 1
Scatteringparam. 2
Similarityrelations
Provide fundamental insights into thestructure of material parameter space
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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], …
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Our contribution
Introducing high-ordersimilarity theory tocomputer graphics
Novel algorithmsbenefiting forward &
inverse rendering
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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
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Our contribution: inverse rendering
Parameter space 1
Reparameterize
Parameter space 2
Gradient descent methods perform
much better
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Background
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Material scattering parameters
Extinction coefficient
Scattering coefficient
Phase function
Light particle
Absorption coefficient
AbsorbedScatteredInteraction
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Phase function
Scattered
Probability density for , parameterized as
Isotropic scattering
Forward
Forward scattering
Forward
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Similarity Theory
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nth Legendremoment
Phase function moments
Legendrepolynomial
For a phase function
“Average cosine”
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Similarity relations
Low-frequency radiance
Band-limited up to order-N in spherical harmonics domain
…
[Wyman et al. 1989]
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Order-N similarity relation[W
yman et al. 1989]
Similarity relations
…
identical appearance
…
Derivationin the paper
Radiancelow-frequency
everywhere
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Order-N similarity relation
Similarity relations
…
Higher order,Better accuracy
Approximatelyidentical appearanceRadiance
low-frequencyeverywhere
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Challenge
Order-N similarity relation
…
Order-N similarity relation
…
Original(given)
Altered(unknown)
??
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Solving forAltered Parameters
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The problem
Altered parameters ?
…
??O
rder
-N
sim
ilarit
y re
latio
nC
onstraints
Forward
Original parameters
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The problem
Altered parameters ??
…
Ord
er-N
si
mila
rity
rela
tion
…
Ord
er-N
si
mila
rity
rela
tion
…
Forward
Original parameters
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The problem
Altered parameters ??
…
Ord
er-N
si
mila
rity
rela
tion
…
Forward
Original parameters
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Altered phase function
…
Altered parameters ?
…
Altered parameters ?
Forward
Original parameters
Remainingunknown
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Altered phase function
Altered parameters ?
Forward
Original parameters
Remainingunknown
Legendre moments of
…
Legendre moments of
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Altered phase function
Altered parameters ?
Altered parameters ?
Order-1
Order-2
Order-3
Order-4…
Finding highest satisfiable order N
Normalizationconstraint
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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?
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Finding order N
Altered parameters ?
Order-1
Order-2
Order-3
Order-4…
Finding highest satisfiable order N
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Altered phase function
Altered parameters ?
Order-3
Order-3
Problem: not uniquely specified
Invalid Valid Valid
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Constructing altered phase function
…
-1 10
Need:has Legendre moments
non-negative
Represent as a tabulated function with pieces
?…
-1 10
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Constructing altered phase function
Need:
Represent as a tabulated function with pieces
?Const.
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Constructing altered phase function
Solve subject to
Smoothness term(favoring “uniform” solutions)
-1 10
Good
-1 10
Bad
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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
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Constructing altered phase function
Altered parameters ?
Order-3
ValidInvalid Valid
Our approach
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Forward
Altered parameters
Constructing altered phase function
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Summary
Forward
Original parameters
Forward
Altered parameters
Forward
Altered parameters
Compute order NSolve optimization
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Application:Forward Rendering
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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)
![Page 40: High-Order Similarity Relations in Radiative Transfer Shuang Zhao 1, Ravi Ramamoorthi 2, and Kavita Bala 1 1 Cornell University 2 University of California,](https://reader037.fdocuments.us/reader037/viewer/2022110320/56649cb65503460f9497b803/html5/thumbnails/40.jpg)
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
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Experimental Results
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Performance vs. accuracy
α = 0.05 (44 min, 8.0X)
Relative error 0%
30%
Reference (350 min)
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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%
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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
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Power of high-order relations
Used by first-order methods:
Altered parameters(Order-1)
Forward
Forward
Original parameters
Reduced scatteringcoefficient
Satisfies order-1similarity relation
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Power of high-order relations
Altered parameters(Order-3)
Forward
Forward
Original parameters
Altered parameters(Order-1)
Forward
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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)
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More renderings
Reference473 min
Ours178 min (2.7X)
Reference23 min
Ours20 min
Equal-timeEqual-sample
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Conclusion
Order-N similarity relation
…
Introducing high-ordersimilarity relations to graphics
Proposing a practical algorithmto solve for altered parameters
?Original Altered
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• Picking automatically and adaptively
• Alternative versions of similarity theory
Future work
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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
)
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Extra Slides
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Order-1 similarity relation
Order-1 similarity relation
Reducedscattering coefficient
Special case (used by diffusion methods):
Order-N similarity relation
…
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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
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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
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Performance vs. accuracy
Reference (350 min)
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Discarded Slides