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High-Order Similarity Relations in Radiative Transfer Shuang Zhao 1, Ravi Ramamoorthi 2, and Kavita Bala 1 1 Cornell University 2 University 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 Radiative transfer Scattering param. Appearance

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Rendering translucency Radiative transfer Scattering param. 2 Appearance 2 Radiative transfer Scattering param. 1 Appearance 1 Radiative transfer Scattering param. 1 Appearance 1 Radiative transfer Scattering param. 2 Appearance 2 ≈ ≠

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First-order methods Scattering param. 1 Scattering param. 2 Scattering param. 1 Scattering param. 2 First-order approx. Approx. identical appearance Cheaper to render Limited accuracy [Frisvad et al. 2007] [Arbree et al. 2011][Wang et al. 2009] [Jensen et al. 2001]

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Similarity theory Scattering param. 1 Scattering param. 2 First-order approx. First-order methods Scattering param. 1 Scattering param. 2 First-order approx. Similarity theory [Wyman et al. 1989] Scattering param. 1 Scattering param. 2 Similarity relations

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Similarity theory [Wyman et al. 1989] Scattering param. 1 Scattering param. 2 Similarity relations Provide fundamental insights into the structure of material parameter space

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Similarity theory [Wyman et al. 1989] Scattering param. 1 Scattering param. 2 Similarity relations 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-order similarity theory to computer graphics Novel algorithms benefiting forward & inverse rendering

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Our contribution: forward rendering Better accuracy Our approach User-specified (balancing performance and accuracy) Approx. identical appearance Cheaper to render Scattering param ~ 200 lines of MATLAB code Scattering param. 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 Absorbed Scattered Interaction

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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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n th Legendre moment Phase function moments Legendre polynomial 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 [Wyman et al. 1989] Similarity relations … identical appearance … Derivation in the paper Radiance low-frequency everywhere

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Order-N similarity relation Similarity relations … Higher order, Better accuracy Approximately identical appearance Radiance low-frequency everywhere

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Challenge Order-N similarity relation … … Original (given) Altered (unknown) ? ?

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Solving for Altered Parameters

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The problem Altered parameters ? … ?? Order-N similarity relation Constraints Forward Original parameters

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The problem Altered parameters ? ? … Order-N similarity relation … … Forward Original parameters

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The problem Altered parameters ? ? … Order-N similarity relation … Forward Original parameters

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Altered phase function … Altered parameters ? … ? Forward Original parameters Remaining unknown

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Altered phase function Altered parameters ? Forward Original parameters Remaining unknown Legendre moments of …

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Altered phase function Altered parameters ? ? Order-1 Order-2 Order-3 Order-4 … Finding highest satisfiable order N Normalization constraint

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Finding order N Given desired Legendre moments (Truncated Hausdorff moment problem) [Curto and Fialkow 1991] Phase function Hankel matrices built using are positive semi-definite exists 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 Problem: not uniquely specified InvalidValid

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Constructing altered phase function … 1 0 Need: has Legendre moments non-negative Represent as a tabulated function with pieces ? … 1 0

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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 0 Good 1 0 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 ValidInvalidValid 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 N Solve optimization Compute order N Solve optimization

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Application: Forward Rendering

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Our contribution: forward rendering Better accuracy Our approach Approx. identical appearance Cheaper to render Scattering param. 2 Scattering param. 1 Effort-free speedups! User-specified (balancing performance and accuracy)

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Application: forward rendering 0 1 No change in parameters large Better accuracy Lower speedup small Worse accuracy Greater 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 Original parameters Reduced scattering coefficient Satisfies order-1 similarity relation

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Power of high-order relations Altered parameters (Order-3) 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 Reference 473 min Ours 178 min (2.7X) Reference 23 min Ours 20 min Equal-timeEqual-sample

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Conclusion Order-N similarity relation … Introducing high-order similarity relations to graphics Proposing a practical algorithm to solve for altered parameters ? OriginalAltered

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Picking automatically and adaptively Alternative versions of similarity theory Future work

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Thank you! High-Order Similarity Relations in Radiative Transfer Shuang Zhao 1, Ravi Ramamoorthi 2, Kavita Bala 1 1 Cornell University, 2 University of California, San Diego Project website: (MATLAB code available!) Project website: (MATLAB code available!) Funding: NSF IIS grants , , Intel Science and Technology Center – Visual Computing Reference Ours (3.7X)

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Extra Slides

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Order-1 similarity relation Reduced scattering 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

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